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	__sig__11��]<p�����^ComposeQuotients,Dx	__sig__17cess0HN6�Checks whether two soluble quotients have the same kernel. If not and Construct is true, then a bigger quotient will be constructed, where the kernel is the intersection of both kernels.OOOO������D<8Q<@����REquivalentQuotientsS��������<T6�Construct a bigger soluble quotient by intersecting the kernels of the given quotient. The return values are a new soluble quotient process and maps from the new to the given soluble groups.UOOO����\<PV
	__sig__10W<X����dSolubleQuotientProcess e���������f$rStart the soluble quotient algorithm for a finitely presented group F with expected order of the quotient being n.��g"O��������<�@h
	__sig__13�i<�����Tj,�Start the soluble quotient algorithm for a finitely presented group F with expected order of the quotient being the factorized integer f.0H`x_��������l`&xStart the soluble quotient algorithm for a finitely presented group F without any information about the relevant primes.a"O��������<�b
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	__sig__28���<������0h�
SplitCollector ���������$�,�Setup the collector for a standard ssplit extension of SQP with algebra RG, R the prime ring in characteristic p. ws defines the weights of a series.`
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	__sig__22���<@�����O����������T<H��	__sig__23`x��r$sStart the soluble quotient algorithm for a finitely presented group F with relevant primes restricted by the set s.s"O�������<�u<�����v"kSet up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p.wO������������<�y<�����zNonsplitCollector{���������|,�Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails.}O�����������<�<������6�Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails. ws defines the weights of a series.�O����������<�
	__sig__19�<�����B�Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails. ws defines the weights of a series. epi is an epimorphism onto another soluble group.�O����������<�
	__sig__20�<�����$qSet up the collector for a standard split extension of SQP with algebra RG, R the prime ring in characteristic p.�O����������<,< �
	__sig__21�<(�����4� Calculates the split extension lift for the module M. The return is -1 iff the solution space does not exist in general,otherwise it is the Fq-dimension of the space (could be 0).d�SplitExtensionSpace�������������:� Calculates the split extension lift for the modules in the list. Returned is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).0�
	__sig__29��<�����l�8�Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. ws defines the weights of a series. epi is an epimorphism onto another soluble group.�<P�����6�Calculates the G-modules in characteristic p with respect to the given options. The return value is 0 iff no such module exists, otherwise it is the index of the list of modules in SQP.�O��������td<X�
	__sig__24�<`�������������\�7Calculates the modules for all (known) relevant primes.�O��������|<p�
	__sig__25�<x�����image��T4Append/Find the module M in the list of modules in SQP. The first return value is the index of the module in the char p list in SQP. The second return value is false iff the index belongs to an isomophic module. (It might happen that the isomorphism is the identity; relevant is the internal data structure.)�O�������<��
	__sig__26�<������DeleteSplitSolutionspaceH`���������d�XDelete the k-th split solution space of the i-th p-module as the actual solution space. (@�O�����������<x��
	__sig__39���<�������DeleteNonsplitSolutionspace,t���������|�OO����������<���O����������<���>� Calculates the split extension lift for the modules in the l-th list in SQP. The return is a sequence of: -1 iff the solution space does not exist in general,otherwise it is the Fq-dimension of the space (could be 0).(@�O����������<�t�
	__sig__30��<�����,�
	__sig__31���<�����h���&zBuild the split extension of the i-th p-module for the k-th solution space. A set of linear combinations can be specified.�	�	�<�����,�LiftSplitExtension\t�����������XBuild the split extension of the l-th list of p-modules for their actual solution space.���OO��������<���
	__sig__41`��<�����|�IsNodePl�<� Calculates the split extension lift for the p-modules stored in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O����������<��@� Calculates the split extension lift for all modules stored in SQP. The return are sequences for each prime of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������<��
	__sig__32�<�����
	__sig__40tionspace�6� Calculates the nonsplit extension lift for the module M. The return is -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������$<�
	__sig__33�<�����NonsplitExtensionSpace����������<�Calculates the nonsplit extension lift for the modules in the list. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).��O��������4,< �
	__sig__34�<(�����>�Calculates the nonsplit extension lift for the modules in the l-th list in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������D<<0�
	__sig__35�<8�����<� Calculates the split extension lift for the p-modules stored in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������TL<@�
	__sig__36�<H�����@� Calculates the nonsplit extension lift for all modules stored in SQP. The return are sequences for each prime of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������\<P�
	__sig__37�<X�����XDelete the k-th split solution space of the i-th p-module as the actual solution space. �O����������l<`�
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	__sig__65�c<�!����Td7Returns the soluble group and the epimorphism from SQP.<l�eO�������!<�!hf
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	__sig__75���<h"����x�BDelete the collector for non-split extensions in characteristic p.x��O�����������"|"<p"|�
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	__sig__67(@m<�!�����n
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	__sig__74��<P"����8!�KeepSplitAbelian� ����������#�(�Return the sequence useful for the Collector setup. It determines that the presentation of SQP and of SQR SQP are kept unchanged.�
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	__sig__83ete��>Delete all soluble quotient processes which are related to SQPx�O�����������"<�"�!�!Delete all the collectors of SQP.�O�����������"<�"�
	__sig__79�<�"�����-Delete the soluble quotient process variable.�O�����������"<�"�
	__sig__80�<�"�����DeleteProcessComplete$����������"�(Return the sequence useful for the Collector setup. It determines that the order of the pc generators in SQP is kept unchanged.��
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	__sig__82�<�"�����KeepGeneratorAction�$���������$#�(�Return the sequence useful for the Collector setup. It determines that the order of the pc generators in SQRSQP is kept unchanged.�
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__sig__105�$�$)	<�$����*	SplitAbelianSection+	���������$,	>�Determine the maximal abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.-	OO�������$<�$.	

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	__sig__11<,\,�\q0r���� .� cThe totally ramified local ring with prime p and precision n defined by the Eisenstein polynomial g�2�2����������LrDr\q8rl6�
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	__sig__13DC|C�\qPr����pE� dThe local ring with prime p, inertia degree f and precision n defined by the Eisenstein polynomial gLPL����������dr\qXr P�
	__sig__14Q(Q�\q`r�����R�JThe uniformizing element (an element with valuation 1) of the local ring LWlW�����������tr\qhr,[�
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	__sig__16`lxl�\q�r����To�eps3�,HackobjCoerceJacCurveElt@V-���������p.Print j at level l�W/a�����������p�p�p0
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__sig__5isor�78A divisor in the class of j��79a^���������p�p�p�@:
__sig__4��7;�p�p������7<RepresentativeDivisor=���������p�JThe underlying point in the projective plane of the single place divisor d���^T��������<om0o`�m8o������1The sequence of points underlying the places of d���^��������Lom@o$�
	__sig__29��mHo����<�UnderlyingPoints\|���������Do�EThe sequence of points forming the support of the effective divisor d"�^��������domXoH%�
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SupportPoints*���������\o�$vA divisor linearly equivalent to d but in a normal form (only in the case that the curve is a nonsingular plane cubic)l2�^^��������|ompo�5�
	__sig__31�7t8�mxo����4:�#Degree 1 divisor corresponding to d�;�;�^^���������om�o$?�
	__sig__32�@�@�m�o�����B�NormalFormDivisor�D�D����������o�
	__sig__34Function�[A rational function on projective plane whose divisor plus the normal form of d is d itself$OdO�^����������om�o@R�
	__sig__33 TtT�m�o����V�NormalFormRationalFunctionLXhX����������o�ZTrue iff d is a principal divisor in which case also return a rational function defining d�_,`�^���������om�oHd�m�o����pf�
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	__sig__20�$�m�n����t�EThe sequence of places forming the support of the effective divisor d��^���������nm�n� �
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	__sig__228,X,�m�n����.�
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	__sig__23�:�:�m�n����,<�UnderlyingPlaces=\=t=����������n�_^^��������o�nm�D�
	__sig__24�F�F�m�n����H�"The component of d associated to p4LLL�T^^��������om�nP�
	__sig__25Q$Q�mo�����R�_^^��������$oomhW�
	__sig__26�X�X�mo�����Z�%��������,��%|��������L�Ȗ@�<
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__sig__81�1�1�%Ȗ�������2�%EquationEvaluation�4�4�%��������|��%@Returns the value of the defining equation of E, evaluated at P.:�:�:�%|����������Ȗ���=�%
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EllipticIsogeny�k�%��������̗���������o�EThe divisor d with all occurances of p removed from its factorisation�a�T^^��������,om o8g�
	__sig__27�hi�m(o�����j#�	__sig__28Dn\n�n�n�ohp����$True iff J is equal to I��``��������|p�oppd
__sig__8\t�oxp�����
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jacobianlt���%����������̔����%̔��������%)/home/was/magma/Kohel/elliptic_schemes.mg�Y��a��%V)/home/was/magma/Kohel/elliptic_schemes.mg0Z��a��%"Elliptic scheme construction of E.�%�%�%||���������ܖȖЖ�)�%
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	__sig__35�	�	�m�o����d�AreLinearlyEquivalent�����������o�3/usr/local/magma-2.5/package/AlgGeom/divisors/jac.mc�8l��3/usr/local/magma-2.5/package/AlgGeom/divisors/jac.mXc�8l��W`���������o�o�o��
__sig__0d|��o�o������The jacobian of C�!�!�X`���������p�o�o�o�$�
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__sig__22 383�op�����4�&The degree f unramified extension of L�������������r�r\q�r8
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	__sig__25ho�oHackobjCoerceJacCurve`7��������pPrint J at level ll;�;`����������$p�op�>
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	__sig__13�8�8r%̔ĕ����h:s%"The (rational) factorization of x.�;<t%��������ؕ̔̕|?u%
	__sig__14�@�@v%�̔ԕX��Bw%Returns the first n primes.XEpEx%���������̔ܕXKy%
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	__sig__174dLd�%̔�����tf�%

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	__sig__18���\q�r������JThe totally ramified extension of L defined by the Eisenstein polynomial g,t������������r�r\q�r��
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	__sig__21�384�\q�r����x5�@The unramified local ring with prime p, degree f and precision n9�94:�����������r�r\q�r�<�
	__sig__22�=�=�\q�r����,@�SThe totally ramified local ring with prime p defined by the Eisenstein polynomial g�D�D����������s�r\q�r�I�
	__sig__234LLL�\q�r�����M� cThe totally ramified local ring with prime p and precision n defined by the Eisenstein polynomial g�Q�Q����������ss\qsXV�
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__sig__5&�&�&28s�t����$)3^Returns the the space of modular symbols of weight k for Gamma_0(N), over the finite field Fp./4t���������t�t8s�t<25
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__sig__32148p#8s������L8q#SupersingularPolynomial�Cr#����������s#,�Returns a list of the supersingular j-invariants in characteristic p. However, it is *much* faster to use the command SupersingularBasis(Mestre(p)).�Nt#��������؍8s̍PTv#8sԍ����Ww#6�Given a new factor A=A_f and a prime ell, returns Norm((ell+1-a_ell(f))*(ell+1+a_ell(f))). This is an integer divisible by the congruence primes relating A to a form at level N(A)*ell.f�fx#t���������8s܍ky#

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__sig__8;<;T;+%���������<��� >�����=,%CremonaReferences@,@-%��������>.%HThe Cremona database reference for linear Hecke eigenspace V of level N.GDG\G/%t��������Ĕ�����L0%
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%HeckeLinearDecomposition;p;�;%��������L�%(�The sequence of one dimensional vector subspaces stabilized by the Hecke operators, priority given to the divisors of the level N.HD`D%t��������P�H���<�H%
__sig__3J�JXK%��D�����M%'The sequence of linear Hecke subspaces.�O�O�O%t��������<�X���L�(S%
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$�InitialManinSymbolGenListToM%(&��������� v�t��������<v8s`-�8s8v�����/����������v�QReturns a bound b such that T1, ..., Tb generate the Hecke algebra as a Z-module.`
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	__sig__31� � �8s�v����"�HReturn vector space with basis the lattice defined by the elements of B.$ %H%�t���������v8s�v$)�
	__sig__32@+�+�8s�v����\-�VectorSpaceZBasisd/|/����������v�$sThis function computes the Hecke module generated by a vector v. The result is returned as a subspace with Z-basis. 6h6�utt��������w�v8s�v�:�
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	HeckeSpan$>�>����������v�$uThis function computes the Hecke module generated by a basis of V. The result is returned as a subspace with Z-basis.\G�ttt��������w8sw�L�
	__sig__34�M�M�8sw����P�Compute data needed by FastTn.�Q�t���������(ww8sw�U�
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HeckeOperator�;���������Hv�ECompute the n-th Hecke operator Tn on the space M of modular symbols.�@�t��������hv8s\vlE�
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	__sig__27�OP�48spvX 
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TnSparseR�R�R���������lv�"gCompute the action of the Hecke operator defined by the Heilbronn matrices Heil on the sparse vector v.�Z�Z�Z�t���������v8s�vd_�
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	__sig__30ond�d�)Return sparse representation of vector v.hTh�u���������v8s�vtl�
	__sig__29�n�n�48s�vX,
pq�SparseRepresentationu�u�,�Compute the characteristic polynomial of the p-th Hecke operator Tn on the space M of modular symbols. Uses the modular algorithm, without proof.`
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	__sig__46x��8s�w���������������w�,� Suppose A is a linear transformation of V which leaves the lattice L invariant. Returns A restricted to W, with respect to the basis Basis(W).������������x�w8s�w�
	__sig__47���8s�w����� �$�������� �����#�$
NewtonSlopes%H%�$����������$	__sig__15ctorIndexesT)�$8�Make a list of forms in S_k(N,eps), with eps^2=1, and the correspoding index in the maximal order. The format of the list is [[degrees of eps], isogeny class, disc(O_f), index in max. order].<2T2l2�$����������8���,��5�$
	__sig__14�7t8�$��4�����4:�$QuadraticModularFactorIndexes�;�$��������0��$����������L���$A�$��H�����xC�$SlopesOfUpGamma0ETElE�$��������D��$��������`���|M�$
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factorpadic�Q�Q�$��������X��$JReturn the p-adic slope alpha part of the polynomial f, to precision prec.�X�X�$��������x���l�]�$
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	__sig__19a0f4g�$,�Returns the slope alpha part of the characteristic polynomial of T_p acting on S_k(Gamma_0(N)), with p-adic computations done to precision prec.noPo�$��������������,v�$
	__sig__18@w�w�?Compute action of Transpose(Tn) on the Hecke-stable subspace V.P����t��������XwTw8sHw`�48sPwX`
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	__sig__39��8s\w����T�?Compute action of Transpose(Tn) on the Hecke-stable subspace V.����tt�������xwpw8sdw��
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	__sig__41�*�*�8s|w�����,�LCompute action of Transpose(Tn), n in plist, on the Hecke-stable subspace V.01�tt��������w�w8s�wh4�
	__sig__42`5x5�8s�w����t8�LCompute action of Transpose(Tn), n in nlist, on the Hecke-stable subspace V.;�;�tt��������w8s�w�>�
	__sig__43[email protected][email protected]�8s�w����B�@�Compute the action of the Atkin-Lehner involution Wq on M, when this makes sense (i.e., trivial character, even weight). The Atkin-Lehner map Wq is rescaled so that it is an involution, except when k>2 and char(F) divides q.M�M�M�t���������}�w8s�w�Q�
	__sig__44�R$S�8s�w����0U�Wq(VXV����������w4SkbolsZ�,�Compute the conjugation involution * on V. This involution is defined by the 2x2 matrix [-1,0,0,1]; it sends X^i*Y^j{u,v} to (-1)^j*X^i*Y^j {-u,-v}.apa�t���������w8s�w@f�
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StarInvolution�k����������w�t���������w8s�w,v�Tnpoly w"DCompute the action of T_p on a subspace of integral modular symbols.P#t���������x8s�x(b$CoefficientValuationpc$��������|�l$__sig__8ionPrimeToP@d$ dReturns the valuation of the ideal generated by the coefficients a_n of f with n not divisible by p.te$#���������������g$��������hh$CoefficientValuationPrimeToP � i$����������j$/Cast g, which should be in Z[[q]], to lie in R.�$ %H%k$#$#��������������$)m$���������+n$
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__sig__99<9�9s$��ȑ�����:t$TruncatePowerSeries�;�;u$��������đv$"jReturns a sequence of tuples <[N,iso,dim],[cp1,cp2,...],[wp1,wp2,...]>, one for each new factor of J_0(N).@CxCw$������������ؑ�Gx$
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ComponentGroups�M{$��������ܑ|$ cReturns a sequence of triples <dim,[cp1,cp2,...],[wp1,wp2,...]>, one for each new factor of J_0(N).U0U}$���������������tY~$
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	__sig__12�d�d�$�������4g�$ComponentGroupsPrintiHi�$����������$*�Returns the sequences of slopes of the Newton polygon of f with at the prime p. The slopes are the valuations of the p-adic roots of f.vv,v�$��������w�wDx�
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	__sig__48��8sx����] Suppose A leaves W invariant. Returns A restricted to W, with respect to the basis Basis(W).�8��������(x x8sx8!
	__sig__490"X"8sx�����#	2� Suppose A is a linear transformation of V which leaves the subspace W spanned by the vectors listed in B invariant. Returns A restricted to W, with respect to the basis B.+�+
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>�Suppose B is a basis for an n-dimensional subspace of some ambient space and A is an nxn matrix. Then A defines a linear transformation of the space spanned by B. This function returns the kernel of that transformation.�;�;t��������Lx@x8s4x$?
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	__sig__52�M�M48sPxX�
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HeckeOperatorOn�]��������\x9Compute the action of Tn on the integral modular symbols.�cdt��������|x8spxi
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	__sig__55v,v%8s�x�����w9$��������Ԑ:$PGiven a space M of modular symbols and a field F, attempt to compute M tensor F.
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	OrderList��D$���������E$ZTwists the q-expansion f by the Dirichlet character eps, thus returning sum eps(n)*a_n(f).�%�%F$##�������� �����)G$
__sig__1+, ,H$��������-J$����������$fcpableesK$0�Compute the Zp-rational newforms associated to f. The precision of Zp is of course not infinite; it is actually O(p^20), which could cause problems. Be careful!8�8$9L$��������D�8���,��;M$
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__sig__3I0JTJS$��H������LT$4Returns a list of the the Zp-rational newforms in M.O�OU$t��������\���P�$SV$
__sig__4T�T�TW$��X������VX$?Returns the power series sum_{n not divisible by p} a_n(f) q^n.`ZtZ�ZY$##��������l���`�_Z$
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	__sig__678,X,_48sxyX.`VReturns the space Mk(NN,eps') associated to Mk(N,eps). Here NN must be a divisor of N.<2att���������y8s�yx5b
	__sig__687`7c48s�yX�9dOldModularSymbols;$;e���������yf Returns the p-new subspace of M.>?$?gtt���������y8s�yxCh
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	__sig__70 TtTo48s�yXVppNewDualSubspaceXX4Xq���������yrtt���������y8s_s
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	__sig__76���48s4zX4��#@Create a table of odd parts of BSD data for N between N1 and N2.!�!�!�#����������8���8s���$�#TableCongruenceR&l&�&�#����������#>Append the level N of the congruence table to the file "name".\-�#����������Џ8sďL1�#

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TableConnected\<�#��������ԏ�#*�Create a table of connectedness of abelian intersection graph in the "box" T_k(Gamma_0(N)), where N1 <= N <= N2 (N prime) and k1 <= k <= k2.ElE�#�����������8s�TK�#

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__sig__350�S�S$8s�������U$Returns the odd part of n.�WX$�����������8s��[$

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$DotProdDl$���������$*�Compute the absolute value of the determinant of the matrix got by finding a Z-lattice basis for the Z-span of the rows in the sequence L.�x�x$

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	__sig__74�t�48szX,\�#(Create a table of factored characteristic polynomials of Tp on S_k(Gamma_0(N)) for N between N1 and N2 and p between p1 and p2.�	�	�	�#�����������8s���#8s������
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__sig__341�FG�#8sd������H�#BCreate a table of rational parts of special values of L-functions.hM�M�#����������x�8sl�(Q�#

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__sig__344�v�v�#8s�������x�#8s�������y�CuspidalNewDualSubspace����������0z�#Returns the new subspace of M dual.�!�!�tt��������Pz8sDz�$�48sLzX8�&�
NewDualSubspace�'���������Hz�8�Computes the image in M2 of x in M1 under the theta operator q*d/dq : M1 --> M2. Thus M1 and M2 should be spaces of modular symbols mod ell of the same level, and Weight(M2) = Weight(M1)+ell+1.T3l3�uttu��������hz8s\zd7�
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�#��������t��#�� Given an integer N that is coprime to 5; a sequence [d1,...,dr] of degrees which define a mod N character eps; and a sequence [p1,p2,...,pn] of primes p such that a_p=0; this function computes the subspace ker(Tp1') meet ker(Tp2') ... meet ker(Tpn') of the space M_2(Gamma_1(5*N),eps_5*eps;F5bar)^+. (Primes denote transpose.) The first argument returned is the subspace W, and the second is the space of modular symbols. The subspace W is a subspace of the dual and can be computed efficiently using FastTp.�#�#tt��������8s���&�#

__sig__331�(�(�#8s�������+�#����������$OddPart/�#"hReturns the modular symbols factor corresponding to the nth curve from the paper "Empirical evidence..."3�3�3�#t������������8s��t8�#

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EvidenceCurve,<�#�����������#4�Returns the modular symbols factor corresponding to the curve from the paper "Empirical evidence..." labeled s. Here s is given by the notation in that paper. E.g., "29", or "65,A".,G�#t��������Ď8s��|L�#

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	__sig__80�UV�8s�z����4X�WindingSubmoduleYZ0Z����������z�9Returns the winding element X^(i-1)*Y^(k-2-(i-1))*{0,oo}.D`\`�tu���������z8s�z�d�
	__sig__81Xfpf�8s�z����i�<Returns the submodule Te spanned by the ith winding element.ltl�tt���������z8s�zdt�
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__sig__301�=�=#8s������,@#OCompute the dimension of the space of cusp forms of character eps and weight k.�D�D�D#��������� ��8s��I#

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DimensionS2�x�x#��������D�1#TpSSularJ�&xGiven a Manin symbol [P(X,Y),[u,v]], this function computes the corresponding element of the space M of Modular symbols.	�	8
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	__sig__89H`�48s@{Xx��
	__sig__93bol�
�-Returns the modular symbol {alpha,beta} in M.��tu��������d{T{8sH{��
	__sig__90<\�8sP{�����ConvFromModularSymbol���������L{�HGiven a modular symbol P(X,Y)*{alp,beta}, FromModularSymbol returns the corresponding element of M. The input is a pair <P,x>, where P is a polynomial in X and Y, and x=[[a,b],[c,d]] is a pair of elements of P^1(Q), where [a,b] <--> a/b and [c,d] <--> c/d.)�)�)�tu��������t{l{8s`{/�
	__sig__9140l0�48sh{X��1�NGiven a sequence of modular symbols P(X,Y)*{alp,beta}, FromModularSymbol returns the corresponding element of M. The input is a sequence of pairs <P,x>, where P is a polynomial in X and Y, and x=[[a,b],[c,d]] is a pair of elements of P^1(Q), where [a,b] <--> a/b and [c,d] <--> c/d.t<�<�tu��������|{8sp{[email protected]�
	__sig__92�A�A�48sx{X��D�RTests the modular symbols conversion routines. Must always return the basis for M.�HI�t�����������{8s�{�M�8s�{����P�
TestConvPQ$Q����������{�
	__sig__95bols0U�=Return the sub Z-module M_k(N,Z) of integral modular symbols.�X�tt���������{8s�{]�
	__sig__94`^�^�8s�{�����`�IntegralModularSymbolsXb����������{�tt���������{8s j�8s�{����Dl�IsogenyCodeToInteger$����������{���������|�{8s��
	__sig__98���48s�{X��
SmallestPrimeNondivisor���������{KReturn the smallest prime number ell not dividing N and such that ell >= p.x���������|8s||
	__sig__99t�48s|X�|!Returns the nth prime.�"��������$|8s|(&	48s |X��'

NthPrime)@)X)��������|1Returns the integer n so that p is the nth prime.�/0
��������L|<|8s0|83

__sig__101�4�448s8|X�l6
PrimePos8�8�8��������4|HReturns the position of the divisor p of N, or 0 if p does not divide N.=`=x=��������T|8sH|�A

__sig__102DC|C8sP|����pEFast, non-proved, charpoly.�G�G��������d|8sX|0M

__sig__103�NO48s`|X��P
ModularCharpoly�Q��������\|��������x|8slX

__sig__104Z4Z48st|X��[
FactorCharpoly�] ��������p|!>Compute integer kernel of the not-necessarily-square matrix A.d"w8���������|�|8s�|i#

__sig__105<jTj$8s�|����xl%

IntegerKernel�n&���������|'>Compute integer kernel of the not-necessarily-square matrix A.�w(D8���������|�|8s�|Xz)
__sig__106�{�{�{�{����������{YIsNewnns�FReturns the n-th isogeny coding. The coding goes A,B,C,...,Z,AA,BB,...�x����������{8s�{{�"����������"H�Let eps_p denote the character obtained by restricting eps to the p-factor. This function returns eps_p(n). When gcd(p,n)>1 this is 0. When gcd(p,n)=1, let m be an element of (Z/NZ)^* which is 1 mod q for all q neq p and is n at p. Then eps_p(n) = eps(m).���"����������8�8s,���"8s4������"�����������D�8s�"

__sig__292���"48s@�X|� �"NReturns true iff M>=1 divides Modulus(eps) and is divisible by Conductor(eps).�#�"���������T�8sH� '�"

__sig__293)()�"48sP�X��+�"

CanReduceModM`-�"��������L��"

__sig__296ersP1�"*�Given a Dirichlet character eps, such that M divides Level(eps) and Conductor(eps) divides M, return the associated mod M Dirichlet character.7�"����������l�8s`�(;�"

__sig__294<0<�"48sh�X�x=�"AssociatedModMCharacter|?�"��������d��"GReturns a sequence consisting of all of the mod N Dirichlet characters.�F�FG�"����������8sx�PL�"

__sig__295hM�M�"8s�������O�"AllDirichletCharacters�P�"��������|��"
__sig__297erClassesU4U�" aReturns a sequence consisting of the Gal(Qbar/Q)-conjugacy classes of mod N Dirichlet characters.LZdZ�"����������8s���^�"8s�������`�"AllDirichletCharacterClassesb�b�"�����������" fPrint a Dirichlet character. This is a temporary function, while I wait for classes or "hack objects".lk�"�������������8s���p�"8s�������u�".�Let P be a sequence of power series (or polynomials) over the integers. This function returns a basis for the Z-linear relations satisfied by these series.�z�z�"

__sig__298�{|*8s�|�����+'Compute integer kernel of the matrix A.	(	@	,8���������|8s�|D-

__sig__107
4
.8s�|����l/ECompute integer kernel of the matrix A, and return as integral space.x08���������|8s�|@1

__sig__1080H28s�|����L3
IntegerKernelZ4���������|5UCompute the linear combinations of the elements of B defined by the elements of Comb.�$6���������|8s�|�(7

__sig__109�*�*848s�|X�,�"Create a Dirichlet character modulo N. The input zeta is a root of unity in some field. The second input "degs" is a list of nonnegative integers. We explain the notation we use for degs. There is an isomorphism Z/NZ ---> (Z/p1^(i1)Z) x (Z/(p2)^(i2)Z) x ... with p1 < p2 < ... the primes dividing N. Passing to units we find that each factor in the right hand side is a cyclic group, except possibly when p1=2 in which case the cyclic group is trivial, or a group of order 2 times a cyclic group (possibly trivial). The degs specify the order that the image of a primitive root of each Z/p^iZ should map to. The primitive root used is the one computed using the function PrimitiveRoot(p^i). In the special case p=2 and i>=3, one should supply supply two entries corresponding to 2 in the degs sequence, because the group (Z/2^nZ)^* is generated by -1 and 5. The entries in the gens sequence correspond bijectively, and in order, to the prime divisors of N, listed in increasing order. Note: Not every character can be described in this way. $/L/�"���������Ȋ��8s��l2�"48s��XX84�"'Create the trivial Dirichlet character.h6�6�6�"���������؊Њ8sĊ�:�"

__sig__285�;�;�"8s̊����,=�"]Create the trivial Dirichlet character with values in the prime field of characteristic p>=0.�A�"����������8sԊ�F�"

__sig__286�GH�"8s܊�����K�"^Evaluate the character eps on the matrix gamma=[a,b, c,d] in Gamma_0(N). The result is eps(a).P�"�������������8s��S�"

__sig__287U0U�"48s�XhhW�",Evaluate the character eps at the integer n.Z0Z�"������������8s�^�"

__sig__288�_,`�"48s��Xl�a�"4Evaluate the character eps at the integer (n mod N).eLe�"L�����������8s� j�"

__sig__289�k�k�"48s�XpXn�",Evaluate the character eps at the integer n.rdt�"L����������0� �8s�Dx�"

__sig__290(y<y�"8s�����Tz�"
EvaluateAtp�{�{�"
__sig__291�|�|�|�|9LinearCombinations�/�/:���������|;tt��������}�|8s5<

__sig__110�6�6=8s�|�����9>
IntegerKernelOn�:?���������|@F�Suppose B is a basis for an n-dimensional subspace of some ambient space and A is an nxn matrix. Then A defines a linear transformation of the space spanned by B. This function returns the integer kernel of that transformation, as a subspace with basis.\GAt��������}8s}�LB

__sig__111�M�MC8s}����PD Compute the lattice index [V:W].Q�Q�QE����������}8s}VF

__sig__112�WXG8s}�����YH
LatticeIndex[([I��������}J"gReturns true if and only if A is a root in the ModSymParent tree, i.e., if and only if A has no parent.8c�c�cKt��������4}8s(}�hL

__sig__113�i jM48s0}X DlN
ModSymIsRootnXnO��������,}P3Returns the parent of the modular symbols factor A.�v�vQtt��������L}8s@}zR

__sig__114�z{S48sH}X$||�|�|^
	OldFactor��_��������l}`(List the dimensions of the factors in D.
`
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a���������}8s�}0
b

__sig__1170c8s�}�����dList the Shas of factors in D.te��������؄�}8s�}<f

__sig__118,Dg8s�}����HhSha��i���������}~J0forms"j6Returns the number of newforms in the decomposition D.�$k���������}8s�}t(l

__sig__119�)*m8s�}�����,�"(�Using the standard vector std, this function creates a Dirichlet character modulo N that takes values in a field of characteristic p.l2�"��������������8s|��5�"8s�������8�"DirichletCharacterL:d:�"�����������"(Create any Dirichlet character modulo N in characteristic p. The input "pows" is a list of nonnegative integers. There is an isomorphism Z/NZ ---> (Z/p1^(i1)Z) x (Z/(p2)^(i2)Z) x ... with p1 < p2 < ... the primes dividing N. Passing to units we find that the right hand side is a cyclic groups, except possibly when p1=2 in which case the cyclic group is trivial, or a group of order 2 times a cyclic group (possibly trivial). Let zeta be a primitive rth root of unity in the algebraic closure of the prime field of characteristic p. The pows specify the power of zeta that a primitive root (computed using PrimitiveRoot(p^i)) of each Z/p^iZ maps to. The primitive root used is the one computed using the function PrimitiveRoot(p^i). In the special case p1=2 and i1>=3, the user must supply two entries in the pows sequence. (But if i1=1 or 2, then the user must supply exactly one entry.) The group (Z/2^nZ)^* is generated by -1 and 5. The entries in the pows sequence correspond bijectively, and in order, to the prime divisors of N, listed in increasing order. i0iHi�"�������������8s���m�"

__sig__282�o�o�"48s��XL�t�"�������������8s���x�"

__sig__283Pyhy�"48s��XT�z�"Create Dirichlet character.(|@|�"

__sig__284P}d}T
ModSymParent U��������D}Vt��������`}8s�W

__sig__115��X48s\}X(�
Z��������X}[tt��������t}8s�\

__sig__116<\]8sp}�����-Returns the sum of all cuspidal factors of M. �tt��������T~8sH~�
�8sP~�����F"

__sig__263��G"8s������
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ClassicalPeriodI"���������J"@Returns the classical period r_j(f) = int_{0}^{ioo} f(z) z^j dz.@`K"tV��������4�8s(�LL"

__sig__264�M"8s0�����<!N"NCompute the G_1 function defined in section 2.13 of Cremona's Algorithms book.$O"VV��������D�8s8��'P"

__sig__265@)X)Q"8s@�����$,"����������8s��p0�"48s�X0�1�"
CharacterLevel�2�"�����������"0 Return true iff the Dirichlet character is odd.89(9�"����������8s�<�"

__sig__276�<=�"8s������>�"F Returns the field in which the Dirichlet character eps takes values. �B�"���������,�8s �`G�"

__sig__277�HI�"48s(�X8PL�"

FieldOfValues�M�"��������$��"

__sig__279magesQ�"���������@�8s4U�"

__sig__278pV�V�"48s<�X<�X�"DirichletCharacterImagesZ�Z�Z�"��������8��"-Returns the standard vector that defines eps.�`�"���������X�8sL� e�"8sT������g�"
StandardVectorLi�"��������P��";Returns the orders of the images of the factors (Z/p^iZ)^*.p,p�"���������p�8sd��v�"

__sig__280lx�x�"8sl������y�"
ImageOrderspz�z�"��������h��"
__sig__281|}�}�}�}nNumberOfNewforms��o���������}x

__sig__121tors"p>Returns the number of cuspidal factors in the decomposition D.�$q���������}8s�}x(r

__sig__120�)*s8s�}�����,tNumberOfCuspidalFactors/u���������}v0List the Atkin-Lehner signs of the factors in D.3T3l3w���������}8s�}d7y8s�}�����9zBCompute J_0(N). (Really this is a decomposition of H1(X_0(N),Z)^+.H<`<{����������}8s�}[email protected]|

__sig__122tA�A}8s�}����`D�FullCuspidalFactor� ���������L~�9Compute the Hecke decomposition of the old subspace of M.(�t��������l~8s`~�
�

__sig__128���8sh~������OldDecomposition�$���������d~�;Compute the Hecke decomposition of the *NEW* subspace of M.���t��������x��~8sx~� �

__sig__129""�48s�~Xtx#�;Compute the Hecke decomposition of the *NEW* subspace of M.<&T&�t����������~8s�~�*�

__sig__130x,�,�48s�~Xx/�NewDecomposition0�0�0����������~�%Compute the Hecke decomposition of M.�4�t����������~8s�~�9�

__sig__131|:�:�8s�~�����;�FullDecomposition�<�<����������~�tt���������~8s\D�

__sig__132TElE�8s�~�����G����������~�t���������~8s�O�

__sig__133�P�P�48s�~X��Q�

IsCuspidal�ST����������~�WqOnWX�t���������~8s�[�

__sig__134t]�]�48s�~X��_�
IsEisenstein`8a����������~�:Compute the action of the Atkin-Lehner involution Wq on A.hTh�t��������8s�~tl�

__sig__135�n�n�8s�~����pq����������~�5Returns the sign of the Atkin-Lehner involution on A.�x�t��������8s�{�

__sig__136�|�|�8s����~���������}�GetVtion�#Compute the SemiDecomposition of M.dL|L�t���������~~8s~LP�

__sig__123<QTQ�8s~����$S�SemiDecompositionU0U���������~�#Compute the SemiDecomposition of M.HZ`Z�t���������,~$~8s~�^�

__sig__124D`\`�8s ~�����a�ACompute the SemiDecomposition of the subspace Vdual inside Mdual.Xfpf�tt���������<~4~8s(~,k�

__sig__125�l�l�8s0~�����o�SCompute the SemiDecomposition of the subspace Vdual inside Mdual, starting with Tp.xv�v�tt���������D~8s8~�y�

__sig__126�z�z�8s@~����@|�

__sig__127P}d}�

DecompFast�����������8�����������T8s`�

__sig__139Xp�48sPX��
���������`8s�

__sig__140�$�48s\X�t�
LabelHelper�<���������X�Printx�OLabel the factors in the decomposition D, then return the sorted decomposition.�!""���������x8slH%�

__sig__141l&�&�48stX�t()"t������������8s\-+"8s������|/,"EllipticInvariants�01-"����������.".Computes w1, w2, c4, c6, and j to precision n.H5/"t��������Ĉ8s��4:0"

__sig__258;$;1"8s������\<2",Returns the abelian variety associated to A.>�>3"t��������Ԉ8sȈ�B4"

__sig__259�D�D5"8sЈ����,G6"
AbelianVariety�H7"��������̈8"HComputes an elliptic curve associated to the weight-two rational modular symbols factor A (this curve is well-defined up to isogeny). By the Shimura-Taniyama theorem the computed curve is correct (assuming correctness of our implementation of the algorithms).�VhW9"t���������8s�([:"

__sig__260]];"8s�����_<"tV�����������8s8c="

__sig__261�de>"8s������g?"
	LAnalytic0iHi@"���������A"&yCompute L(A,j) for j a "critical integer", i.e., one of the integers 1,...,k-1. Use at least n terms of the q-expansions.�u�uB"tV���������8s�yC"

__sig__262�yzD"8s�����X{E"tV��������,��8sp~X"DerivG20�SignWN ��������������������,8s��

__sig__137���8s(�����
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DecompWq�����������$�t����������L@8s\�

__sig__138,�48s<X��/ *�Give the subspace A of the DUAL of M, attach necessary data to make it a factor of M. For this to make sense, A should be Hecke stable.�	�	<
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2 8sĀ����4@ DecompZ3 t��������Ԁ8s�4 

__sig__160��5 48sЀX�6 
DecompFastTp�7 ��������̀8 t���������8s�"9 

__sig__161�#�#: 8s�����L%; DecompCharpolyTp&�&�&< ���������= tt����������8s .> 

__sig__162�/0? 8s�������1
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__sig__2519(9"8s<������:"

PeriodLattice�;"��������8�"VComputes the complex period lattice associated to A using n terms of the q-expansions.�A"t��������X�8sL��F"

__sig__252�G�G"8sT�����XK"tV��������t�d�8shO"

__sig__253hP�P"8s`������Q"

RealVolume�S�S"��������\�".�Computes the volume of A(R), R the RealField. This function returns the volume of the identity component times the number RealTamagawa(A) of real components.�]"tV��������|�8sp��a"

__sig__254�c�c"8sx������e "tV������������8sTj!"

__sig__255�kHl""8s�������n#"
ImaginaryVolume�p$"����������%"8Computes the volume of A(iR), the pure imaginary points.x�x�x&"tV����������8s���{'"

__sig__256�|�|("8s������~*"

__sig__257 4�
LabelFactors����������p�t������������8sL%�

__sig__142p&�&�48s�X�x(�����������Print the factors in D.�-�-�-��������������8s��1�

__sig__143�2�2�48s�X��4 newforms�8Return the number of new factors of the decomposition D.9�9�9����������8s��<�

__sig__144�=�=�8s������? NumNewformFactors�  ��������� t��������(�8s�	 

__sig__149��
 8s$������
 �������� �# Factor
 D�Returns the modular factor "Nk[Weight][IsogenyClass]", where N is the level, k is the weight, and [IsogenyClass] is a letter such as "A", "B", etc. An alternative input format is "N[IsogenyClass]" in which case the weight is assumed to be two.�� t��������L�@�8s4�H# 

__sig__150 $X$ 8s<�����$& 

ModularFactor' ��������8� t��������X�T�8sL/ 

__sig__151�0�0 8sP�����2 t��������d�`�8sH5 

__sig__152�67 8s\������9 tt��������t�l�8s\< 

__sig__153\=t= 8sh�����x? >�Returns the index into the D list of the new factor with the indicated isogeny class. If the iso classes have not yet been computed yet, they are computed. If there is no factor of class iso this function returns 0.TJ�JTK t��������|�8sp�dO 

__sig__154dP|P 8sx������Q  tt������������8sV! 

__sig__155�WX" 8s�������Y$ ����������% tt������������8s�`& 

__sig__156�a�a' 8s������Hd( t������������8sHi) 

__sig__157hj�j* 8s�������l+ >�Returns the index into the D list of the new factor with the indicated isogeny class. If the iso classes have not yet been computed yet, they are computed. If there is no factor of class iso this function returns 0.<yPyhy, t������������8s��@|- 

__sig__158P}d}. 8s�������~1 

__sig__159���

NumOldFactors�A�����������.Return the indexes of new factors in the list.�G�����������8s�,M�

__sig__145�NO�8s�����|P�

NewFactors�Q�Q�����������.Return the indexes of new factors in the list.hW�����������8s�([�

__sig__146]]�8s�����_�
NewformFactors�`�����������8Return the number of new factors of the decomposition D.g�g�g����������8s��k�

__sig__147�m�m�8s�����(p 

NumNewFactors�t ��������� 8Return the number of new factors of the decomposition D.y�y�y ���������8s�|| 

__sig__148x}�} 8s�����K 
	CuspOrder��L �������� �M  eCompute the order in A of the difference (x)-(y) of the cusps defined by the 2-tuples alpha and beta.N t����������@�8s4��O 

__sig__165��P 8s<�����t� LRatiotionQ $qComputes the following upper bound on the torsion subgroup of A: gcd { #A(F_p) : 3 <= p <= 19, p not dividing N }��R t������`�P�8sD�8!S 

__sig__1660"X"T 8sL������#U 
TorsionBound$�$V ��������H�W $sComputes the following upper bound on the torsion subgroup of A: gcd { #A(F_p) : 3 <= p <= maxp, p not dividing N }�-�-X t��������h�8s\��1Y 

__sig__167�2�2Z 8sd�����h4c 

__sig__169dMapx5[ bhCompute the rational period mapping. The period mapping is scaled so that the integral modular symbols SkZ are taken surjectively onto the lattice Z^d. Note: the choice of rational period mapping is well-defined only once A is created, and can be different if A is created again; i.e., it is with respect to the non-canonical basis used internally to define A.D�D�D\ tw��������x�8sl�I] 

__sig__168�KL^ 8st�����|M_ ScaledRationalPeriodMap�O` ��������p�a ICompute the order in A of the cusp defined by X^i*Y^{k-2-i}*{alpha,beta}.XU�Ub t����������8s���Yd 8s������|[e t8����������8s,`f 

__sig__170Papag 8s�������ch 

ModularKernelLei ����������j B�Compute the modular degree of A. This is the degree of the canonical map from the dual of A to A. When A corresponds to an elliptic curve, this is the SQUARE of what is frequently referred to as the "modular degree" in the literature.xv�vk t8����������8s���yl 

__sig__171�z�zm 8s������@|n 

ModularDegreed}o ����������r 

__sig__172�,�A ���������B tt���������8s�
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__sig__163x�D 8s������E 
DecompZdual�
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F ���������G 2Compute the order in A of the difference (0)-(oo).<TH t��������8�(�8s�,I 

__sig__164��J 8s$�����h� 9Return the level M of the character group of X_0(pM)/F_p.��� ����������8s���
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__sig__186�� 8s�����4
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XGroup_M
4� ����������� >�Compute the n-th Hecke operator on the toric part (+Eisenstein series) of the closed fiber of the Neron model of J_0(pM) over F_p. The second argument returned is a diagonal matrix which is double the weight vector.|��� ����������8s�"� 

__sig__1874#L#� 8s������$i!uu������������8s��x(k!8s�������*m!����������|!

__sig__223iety�/n!;Returns the dimension of the abelian variety attached to A.(2@2o!t��������ą8s��|5p!

__sig__2217d7q!48s��X �9r!DimensionAbelianVariety(;s!����������t!fxAttempt to compute the q-expansion of one of the Galois conjugate modular forms associated to A. This function uses theta series, so if successful it can be used to compute a very large number of Fourier coefficients quickly. If there is no prime which exactly divides Level(A), Weight(A) gt 2, the character of A is nontrivial, then the function qEigenform(A,prec) is called.MN�Nu!t#��������܅8sЅ�Qv!

__sig__222�S�Sw!8s؅�����Ux!
qEigenformThetalWy!��������ԅz!5Returns the at least 7 terms of the q-expansion of A. ]{!t#����������8s�<a}!8s�����<c~!

qEigenform�d e!����������! fReturns the q-expansion of one of the eigenforms associated to A, computed to precision at least prec.�m�!t#���������8s��u�!

__sig__224|v�v�!48s�X4�xR"DerivG1ts�!*�Returns generators for the saturation of the Z-module generated by the sequence of q-expansions. Uses the smallest degree as precision.�}�}�}�!���������8s��^"DerivG3ԁp .�Returns the congruence number r = [S_k(N,Z) : (This) + (Complement)], along with the structure of the quotient; q-expansions are computed to precision prec.%�%q t8������́8s���)s 8sȁ����\,t 
CongruenceR�-�-u ��������āv 8�Computes the invariants of the group-theoretic intersection of the abelian varieties corresponding to the modular factors A and B. If the factors are not distinct, then the 0 group is returned.9(9w tt8����������8s؁<x 

__sig__173�<=y 8s������>� B�For an integer s in the critical strip (1<=s<=k-1), return the quotient L_A(s)*(s-1)! / (2pi)^(s-1)*Omega, which is a rational number. Here Omega is the volume of A(R), if s is odd, and the volume of the -1 eigenspace for conjugation.��� t��������P�8sD��� 

__sig__178(�� 8sL������� BdEDing� TReturns the odd part of the rational part of L(A,s); potentially faster than LRatio.�� t��������l�`�8sT�8!� 

__sig__1790"X"� 8s\������#� 

LRatioOddPart�$� ��������X�� t��������t�8s ,� 

__sig__180�-�-� 8sp������/�  Returns true iff L(A,eps,1) = 0.1�1�1� �t����������8sx�5� 

__sig__181�6�6� 8s�������9� LEpsilonVanishes:�:�:� ��������|�� <Returns the degrees d dividing p-1 such that L(A,eps,1) = 0.?x?� t����������8s��\D� 

__sig__182TElE� 8s�������G� LEpsilonVanishing0JTJ� ����������� TM�M�N� 
Z[25\� precS��X�� "iReturn the character group X of the toric part of the closed fiber of the Neron model of J_0(p) over F_p.�UV� ���������ԂĂ8s��`Z� 

__sig__183�[�[� 8s�������]� XGroupd_� ����������� .�Return the character group X of the toric part of the closed fiber of the Neron model of J_0(pM) over F_p. [RIGHT NOW RETURNS THE EISENSTEIN PART AS WELL.]�i j� ���������t�܂8sЂPo� 

__sig__184�ppq� 8s؂�����u� BReturn the characteristic p of the character group of X_0(pM)/F_p.�x�x� ����������8s��{� 

__sig__185�|�|� 8s�����~� 
XGroup_p0� ���������� TXn��z AbelianIntersectionpA�A{ ��������܁| LReturns the group-theoretic intersection of a sequence S of modular factors.HI} 8����������8s��M~ 

__sig__174�O�O 8s������$Q� 8�Compute "[Phi(SkZ^+):Phi(W)]". If Plus is false, compute instead "[Phi(SkZ^-):Phi(W)]". It should be the case that W tensor Q is contained in SkZ^+ tensor Q (or SkZ^- tensor Q, when Plus is false).0Z� tt���������8s�^� 

__sig__175�_,`� 8s������a� 
PeriodIndex�c�c� ���������� PReturns the lattice spanned by the basis of W under the period map defined by A.kPkhk� tt����������$�8s��p� 

__sig__176,udu� 8s ������v� 
PeriodImage�wDx� ���������� t��������H�8�8s�|� 

__sig__177�}�}� 8s4�����0� ��������0�� 
PhiX_and_mX��� ��������D�� /Returns the dimension of the character group X.�
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__sig__190Ph� 8s`������� 

XDimension�(� ��������\�� 6�Returns the character group of the toric part of the closed fiber at p of the space M of modular symbols. This only makes sense when p exactly divides the level of M and M has weight two.� � � t�����������|�8sp��#� 

__sig__191 %H%� 8sx������&� *�Returns the character group corresponding to the largest prime p which exactly divides the level. If no such p exists, an error results.-..� t�����������8s��2� 

__sig__192343� 8s�������4� =Compute XGroup for xmallest prime exactly dividing the level.$9� tt������������8s���;� 

__sig__193�<�<� 8s�������>� ����������� (�Return the factor of the character group corresponding to the p-adic rigid analytic optimal and co-optimal quotient associated to A.GH� t8����������8s��|M� 

__sig__194$OdO� 8s�������P� ]Compute the p-adic modular degree of A, at the largest prime which exactly divides the level.0U� t��������ԃă8s��tY� 

__sig__195�Z�Z� 8s������]� PhiX]^� �����������  fCompute the order of the image of the component group of J_0(N) in the component group of A, all at p.pf� t��������܃8sЃ,k� 

__sig__196�l�l� 8s؃�����ol!Dotceor� YCompute the p-adic modular degree of A, at largest prime which exactly divides the level.�wDx� t�����������8s��z� 

__sig__197�{|� 8s�����d}� 
XModularDegree�~� ���������� 'Compute the p-adic modular degree of A.<�d���� ���������� XGroupV� Returns the monodromy weights.�	� ���������P�4�8s(��� 

__sig__188�
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� 8s0������ MonodromyWeights�� ��������,�� 4Computes the quantities PhiX and mX associated to V.�� 8��������L�8s@��� 

__sig__189� � � 8sH�����">!.�Let T be an nxn matrix over K with irreducible characteristic polynomial f. This function returns an eigenvector for T over the extension field K[x]/(f(x)).
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__sig__212��A!48s�X�
B!8�The elements of gens should all lie in a field Q[x]/(f(x)). It is assumed that they generate a subring O of Q[x]/(f(x)), as a *Z-module*! This function returns the discriminant of this Z-module.��C!���������8s��D!

__sig__2134lE!8s������ F!
ZAlgDisc!�!�!G!���������H!*�If A is newform modular factor, compute the discriminant of the ring generated by the Fourier coefficients of one of the q-expansions of A.�)�)I!t��������,�8s �/J!

__sig__21480p0K!8s(������1L!
DiscriminantOf�2M!��������$�N!*�If A is newform modular factor, compute the discriminant of the ring generated by the Fourier coefficients of one of the q-expansions of A.p;�;O!t��������D�8s8��>P!

__sig__215[email protected][email protected]Q!8s@�����BR!
DiscriminantKf�DS!��������<�T!KCompute the motive B made from the *=sign subspace of the modular motive A.hL�LU!tt��������\�8sP�PPV!

__sig__216@QXQW!8sX�����(SX!
SignedSubspace�TY!��������T�Z!+Compute eigenvector for sign subspace of A.LZdZ[!tu��������|�t�8sh��^\!

__sig__217H```]!48sp�X�a^!$pReturns an eigenvector of the Hecke algebra on A over a polynomial extension of the ground field. A must be new.h�hi_!tu����������8sx�m`!

__sig__218lo�oa!48s��Xhtb!ZReturns the images of the ith standard basis vector under the Hecke operators Tp for p<=n.�x�xc!t����������8s��\{d!

__sig__219l|�|e!48s��X�}f!
HeckeImages�~g!����������h!#Compute the dot product of v and w.��̂j!

__sig__220�,�� t���������8s��X&!

__sig__198@'�'!8s������)!+Computes the quantity PhiX associated to V.-`-!88�������8s�P1!

__sig__199X2p2!8s�����<4!SReturns the Mestre module of M. The level must be prime and the weight must be two.9(9!t8��������$�8s�<!

__sig__200�<=	!8s ������>
!
MestreGroup[email protected][email protected]!���������!4�Return the factor of the character group corresponding (maybe) to the p-adic rigid analytic optimal quotient associated to A. The representation is via the Mestre construction.L�LM
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__sig__201�Q�Q!8s8�����T !LLet v be a choice of eigenvector, corresponding to A, on the module of supersingular points. This function returns sum w_i (v_i)^n, a quantity which, by Gross-Zagier, is relevant to the computation of the order of vanishing of L(A,s) at 1. Currently Conductor(A) must be prime.�
!!t������������8s���#!8s�������$!(�Compute the order of the group of *geometric* points of the component group at the largest prime that exactly divides the level.0H%!t������������8s���!&!

__sig__206�"�"'!8s������$(!
ComponentGroup�%)!����������*!:�Compute the order of the group of *geometric* points of the component group at p, assuming that p exactly divides the level. If working in the +1 quotient, then only the odd part of the order is returned.1�1+!t����������8s���4,!

__sig__207 6h6-!8s������$9.!:�Compute the order of the group of Fp rational points of the component group of A at the largest prime which exactly divides the level of A. WARNING: Stein has not yet nailed down the power of 2 when Wp=+1![email protected]/!t��������ȄĄ8s���D0!

__sig__208�F�F1!8s�������H2!t��������Є8s�M3!

__sig__209�O�O4!8s̄�����P5!��Try to compute the odd part of the order of Sha. Assuming BSD we obtain a number which is either zero or divides the true order of Sha, and only misses primes where the representations are reducible. The following caveats apply: (a) We assume the BSD formula. (b) We use the upper bound on torsion coming from Hecke operators, so the result of this function might be too small. (C) At primes whose square divide the level we do not know how to compute cp; however such cp can only be divisible by p and primes dividing the order of the torsin group. We simply make our conjectural value of Sha coprime to such primes when there is a p whose square divides the level.@nXn6!t���������8sԄ�u7!

__sig__210�v�v8!8s܄�����x9!u�����������8sX{:!

__sig__211h|||;!8s������}<!
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MestreGroupVVXV!��������4�!t����������\�P�8s\]!

__sig__202�^_!8sL������`!MestreEigenvectorpb�b!��������H�!88����������d�8sPj!

__sig__203�kDl!8s`������n!6�This function returns sum w_i (e_i)^n, a quantity which, by Gross-Zagier, is relevant to the computation of the order of vanishing of L(A,s) at 1. Currently Conductor(A) must be prime.yPyhy!u8������������t�8sh�@|!

__sig__204P}d}!8sp������~!
MestrePowerSum�!��������l�"!

__sig__205���!
SaturatePolySeq��!����������!,�Returns the index and structure of the Z-module generated by the sequence of q-expansions in its saturation. Uses the smallest degree as precision.
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__sig__226Ht�!8s0�������!IndexInSaturationt��!��������,��!
__sig__228ntegralBasis��!MReturns the linear combination of qIntegralBasis(A) which gives Eigenform(A)."�!tt��������L�8s@�H%�!

__sig__227l&�&�!8sH�����t(�!#tV��������ć��8s��\-�!8s������|/�!FastPeriodIntegral�01�!�����������!#tV��������̇8s�6�!

__sig__245�8$9�!8sȇ�����:�!<�Given a homogeneous polynomial P(X,Y) of degree k-2, a 2x2 matrix g=[a,b,c,d], and a q-expansion f of a weight k modular form, this function returns the period <f, P {oo,g(oo)}> = Int_{oo,g(oo)} P(z,1) f(z) dz.HB�B�!#tV��������܇8sЇ\G�!

__sig__246�HI�!8s؇����LL�!SlowPeriodIntegral�M�M�!��������ԇ�!t�������8sT�!

__sig__247XU�U�!8s������W�!PeriodGeneratorsXHYtY�!����������!t��������,��8s\`"

__sig__248�a�a"8s�����d"

PeriodMapping�e"����������"(Apply the period mapping Phi to v in Mk.k�k�k"uu���������8s�pq"

__sig__249|u�u"8s������v"ApplyPeriodMapping�x�x	"���������
":�Computes the complex period mapping associated to A using n terms of the q-expansions. The period map is a homomorphism Phi:Mk(N,eps;Q)--> C^d, where d=dim A. Furthermore, A(C) := C^d/Phi(Sk(N,eps;Z)).؀�"t��������4�8s(�(�"

__sig__250`����!

__sig__225� �!8s�����P�!

__sig__235���!8s�������!

__sig__240gationg�!3Returns the map r_A : M_k(Q) ----> M_k(Q)/Ker(Phi).Ld�!t���������8s��
�!

__sig__236��!8s�������!RationalPeriodMappingX�!�����������!EReturns the map r_A : M_k(Q) ----> M_k(Q)/(Ker(Phi)+(sign quotient)).�!t���������8s�|!�!

__sig__237�"�"�!8s������#�!SignedRationalPeriodMapping�%�%�!����������!8Returns a basis for the image of S_k(Z) in M_k/Ker(Phi).,<,\,�!t��������0�8s$��0�!

__sig__238�1�1�!8s,�����83�!RationalPeriodLattice�4�!��������(��!XReturns matrix of "conjugation" with respect to the basis RationalPeriodLatticeBasis(A).;�;�;�!t��������H�8s<�(?�!

__sig__239�@�@�!8sD������B�!RationalPeriodConjugationE@E�!��������@��!NComputes the number of real components of the abelian variety associated to A.0M�!t��������`�8sT��P�!8s\�����DR�!ImaginaryTamagawa�T�T�!��������X��!NComputes the number of real components of the abelian variety associated to A.,[�!t��������x�8sl��_�!

__sig__241�`<a�!8st�����<c�!
RealTamagawad e�!��������p��!EComputes <f, {alpha,oo}> for alpha any point in the upper half plane.Hl�!V#V������������8s���q�!

__sig__242�u�u�!8s������$w�!
PeriodIntegral�x�!�����������!FComputes <f, P{alpha,oo}> for alpha any point in the upper half plane.�|�!V#V����������8s����!

__sig__243���!8s���������!Compute <f, {oo, g(oo)}>.D�X��!

__sig__244��̅�!EigenformInTermsOfIntegralBasis�
�!��������D��!JReturn an integral basis for the sum of the spaces in the decomposition D.���!��������t�d�8sX�X�!8s`�������!
qIntegralBasis��!��������\��!SReturn an integral basis for the sum of the cuspidal spaces in the decomposition D.T!|!�!����������|�8sp�\$�!

__sig__229�%�%�!8sx����� '�!t������������8s\,�!

__sig__230�-�-�!8s������0�!B�Compute an integral basis for the space spanned by the Galois conjugates associated to A. If A is a space of modular symbols, returns an integral basis for the corresponding space of cusp forms. The base field must be the rationals.9�9�9�!t����������8s���<�!

__sig__231�=�=�!8s�������?�!/Compute the sum of two modular symbols factors.�B�B�B�!ttt����������8s���G�!

__sig__232�I�I�!8s�������L�!

AddFactors�M�N�!�����������!<�Compute the intersection of two modular symbols factors. If the intersection is zero-dimensional use the function "Intersection" to compute the structure of the corresponding intersection of abelian varieties.Z0Z�!ttt����������8s��^�!

__sig__233�_,`�!8s�������a�!IntersectFactorsc�cd�!�����������!&~Given a list polyprimes of pairs <p, f(x)> computes the following subspace of the dual of M: intersect ker(f(Transpose(Tp))). �m�!tt���������؆8s̆�u�!

__sig__234xv�v�!8sԆ�����x�!
DualSubspaceyhy�!��������І�!<� Given a sequence polyprimes [<p1, f1(x)>,...<pr,fr(x)>] of pairs a prime and a polynomial, this function computes the following subspace of the subspace W of the dual of M: intersect ker(f(Transpose(Tp))). Ё���!ttt���������8s���S"��������<�T"NCompute the G_2 function defined in section 2.13 of Cremona's Algorithms book.�U"VV��������\�8sP�0V"

__sig__266(@W"8sX������Y"��������T�Z"NCompute the G_3 function defined in section 2.13 of Cremona's Algorithms book.|["VV��������t�8sh�h \"

__sig__267�!�!]"8sp�����#_"��������l�`"6�Compute (d!)*Prod L^(dr)(fi,1) over the d Galois conjugates of f. TODO: When both r > 1 and d>1 the normalization is wrong! It's trivial combinatoric to work out but I have not done it.�-�-a"#V����������8s���1b"

__sig__268�2�2c"8s�������4d"
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__sig__269$>�>h"8s������[email protected]i"tV������������8s�Dj"

__sig__270�F�Fk"8s�������Hl"8�Return the leading coefficient of Taylor expansion about the critical integer j and the analytic rank of the the L-series associated to A. The precision used is at least n terms of the q-expansion.�Qm"tV��������8s��Vn"

__sig__271�WXo"8s�������Yx"

__sig__273ield([p"4�Returns two arguments, a v-adic field K and an element zeta in K that is a primitive pth root of unity. Here v should be a prime number, or 0; if v=0 then K=the complex numbers.�cdq"����������̉8s��ir"

__sig__2728jPjs"8sȉ����tlt"CompleteCyclotomicField�tu"��������ĉv"> Returns true iff the character eps is the trivial character. �yw"����������8s؉�|y"48s�X(�}z") Returns the level of the character eps. ��{"����������8s��|"

__sig__274D�X�}"48s�X,��~") Returns the level of the character eps. ����"
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__sig__312Hh@#8s�������A#KComputes Tp on the Mestre graphs module. p must be a prime between 2 and 7.�,�B#8���������8s�h�C#

__sig__313l���D#8s�����H��$SlopeAlphaPartGamma0��$�����������$4Return the smallest valuation of a coefficient of f.�$���������������$��������0�$
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__sig__1�==%̔�����>%.Returns the Chinese remainder lift of a and b.?%rrL����������̔��<?@%
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