CoCalc -- twistedlseries-alex-1.sagews

Project: Harvard Workshop: Distribution of modular symbols and L-values

Path: code / alex / psage / twistedlseries-alex-1.sagews

Views: ¹⁸⁴⁶⁰⁷

attach("twistedlseries.sage")

psage/modform/hilbert/sqrt5/sqrt5.py:852: DeprecationWarning: python.pxi is deprecated, use "from cpython cimport *" instead
See http://trac.sagemath.org/20158 for details.
  from tables import primes_of_bounded_norm
psage/number_fields/__init__.py:1: DeprecationWarning: import "cysignals/signals.pxi" instead of "sage/ext/interrupt.pxi"
See http://trac.sagemath.org/20002 for details.
  import sqrt5

E = EllipticCurve(F, [1, a + 1, a, a, 0])
E
L = LSeriesEllipticCurveSqrt5(E)
#L = LSeriesEllipticCurve(E)

Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1

value_search(E,2,3000)

Fractional ideal (a - 6) 29 Fractional ideal (a - 6) [1]
raw value 4.27757631063698
normalisd -2.00000000000000 - 1.11328149605248e-15*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (a + 5) 29 Fractional ideal (a + 5) [1]
raw value -8.64054724157950e-19
normalisd 4.03992663793896e-19 + 2.24878778571347e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a - 10) 89 Fractional ideal (a - 10) [1]
raw value -8.44637764064513e-19
normalisd 6.91828946321773e-19 + 4.15225609871438e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a + 9) 89 Fractional ideal (a + 9) [1]
raw value 4.68174963700270e-20
normalisd -3.83474437932251e-20 - 2.30155746166868e-35*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (9*a - 5) 101 Fractional ideal (9*a - 5) [1]
raw value 1.61261698518588e-18
normalisd 1.40710059738480e-18 + 7.92765727097779e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (9*a - 4) 101 Fractional ideal (9*a - 4) [1]
raw value -1.24047460398914e-18
normalisd -1.08238507491139e-18 - 5.50033536146272e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (13) 169 Fractional ideal (13) [1]
raw value -3.85501338470470e-18
normalisd 4.35113415080682e-18 - 1.85797281440714e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a - 14) 181 Fractional ideal (a - 14) [1]
raw value -5.53758828742112e-18
normalisd -6.46834508766171e-18 + 3.68309749065901e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a + 13) 181 Fractional ideal (a + 13) [1]
raw value 1.71221176742225
normalisd 2.00000000000000 - 1.13880673982113e-15*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (-13*a + 8) 209 (Fractional ideal (-3*a + 1)) * (Fractional ideal (-4*a + 1)) [1, 1]
raw value -7.73308306955716e-18
normalisd 9.70641932492945e-18 + 3.27981095764292e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (13*a - 5) 209 (Fractional ideal (-3*a + 2)) * (Fractional ideal (4*a - 3)) [1, 1]
raw value 1.59339563039399
normalisd -2.00000000000000 - 9.67626228555456e-16*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (3*a - 17) 229 Fractional ideal (3*a - 17) [1]
raw value -3.72034042803380e-18
normalisd 4.88802764587848e-18 - 2.76132081945557e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-3*a - 14) 229 Fractional ideal (-3*a - 14) [1]
raw value 6.08890243273603
normalisd -8.00000000000002 + 4.51932111600701e-15*I
algdeprts [(-8.00000000000000, 1)]
Fractional ideal (17) 289 Fractional ideal (17) [1]
raw value -1.73408906578412e-19
normalisd 2.55949067694519e-19 + 1.13664219234319e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-4*a + 21) 341 (Fractional ideal (-3*a + 2)) * (Fractional ideal (5*a - 3)) [1, 1]
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.469818993461713, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.731744473415701, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.831380074346103, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.955726826095017, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.04287727895700, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.12076264926057, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.22934955958704, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.41660663949679, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=2.71238258310115, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.246018276556605, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.642728870707071, which is not sufficiently close to 1
raw value 1.24743974014669
normalisd 2.00000000000000 + 1.27458528784333e-15*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (4*a + 17) 341 (Fractional ideal (-3*a + 1)) * (Fractional ideal (5*a - 2)) [1, 1]
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.469806140534232, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.728874717785074, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.820888845633422, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.925974249529631, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.996648769992518, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.05848592218199, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.11618865135178, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.12632815115418, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.09329047688828, which is not sufficiently close to 1
raw value 2.49487948029338
normalisd 3.99999999999999 + 2.54917057568667e-15*I
algdeprts [(4.00000000000000, 1)]
Fractional ideal (-17*a + 5) 349 Fractional ideal (-17*a + 5) [1]
raw value 11.0975359574666
normalisd -18.0000000000003 + 8.55775814327889e-14*I
algdeprts [(-18.0000000000000, 1)]
Fractional ideal (17*a - 12) 349 Fractional ideal (17*a - 12) [1]
raw value 4.93223820331848
normalisd -8.00000000000012 + 3.71311617216712e-14*I
algdeprts [(-8.00000000000000, 1)]
Fractional ideal (3*a - 21) 369 (Fractional ideal (3)) * (Fractional ideal (a - 7)) [1, 1]
raw value 4.85533189848648e-18
normalisd -8.09776811669406e-18 + 2.97191311430776e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (3*a + 18) 369 (Fractional ideal (3)) * (Fractional ideal (a + 6)) [1, 1]
raw value 2.54650274396144e-18
normalisd -4.24708118008429e-18 + 1.46051013408935e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-18*a + 11) 401 Fractional ideal (-18*a + 11) [1]
raw value 8.12340725704607e-19
normalisd 1.41235425821411e-18 - 1.41729502294015e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (18*a - 7) 401 Fractional ideal (18*a - 7) [1]
raw value 1.15033564841131
normalisd 1.99999999999999 - 8.09451625039899e-16*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (21) 441 (Fractional ideal (3)) * (Fractional ideal (7)) [1, 1]
raw value -7.07408068383737e-18
normalisd -1.28980048041774e-17 + 3.40944331079133e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a - 22) 461 Fractional ideal (a - 22) [1]
raw value -7.28355023444207e-18
normalisd -1.35777189037468e-17 - 7.16823645161362e-32*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a + 21) 461 Fractional ideal (a + 21) [1]
raw value 1.07286802533996
normalisd 1.99999999999997 + 1.05588228809707e-14*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (4*a + 21) 509 Fractional ideal (4*a + 21) [1]
raw value 1.02102852118508
normalisd -2.00000000000002 + 8.63139372631161e-15*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (-4*a + 25) 509 Fractional ideal (-4*a + 25) [1]
raw value -1.98885366370030e-17
normalisd 3.89578473555648e-17 - 1.68130259627712e-31*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (5*a + 21) 521 Fractional ideal (5*a + 21) [1]
raw value 1.94785465167661e-18
normalisd 3.86018962576967e-18 - 1.20165550775937e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-5*a + 26) 521 Fractional ideal (-5*a + 26) [1]
raw value -1.27752812802864e-17
normalisd -2.53176017122239e-17 + 7.88122825305140e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (3*a + 22) 541 Fractional ideal (3*a + 22) [1]
raw value -7.23705724369818e-20
normalisd -1.46148334704980e-19 - 9.24317864145188e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-3*a + 25) 541 Fractional ideal (-3*a + 25) [1]
raw value -2.16764338563248e-17
normalisd -4.37743492108356e-17 - 2.76851686668767e-31*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (21*a - 12) 549 (Fractional ideal (3)) * (Fractional ideal (7*a - 4)) [1, 1]
raw value -3.14018576619297e-17
normalisd 6.38814468161796e-17 + 4.08632184886659e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (21*a - 9) 549 (Fractional ideal (3)) * (Fractional ideal (7*a - 3)) [1, 1]
raw value 0.983129194061283
normalisd -2.00000000000000 + 9.47663280684452e-16*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (22*a - 7) 589 (Fractional ideal (4*a - 3)) * (Fractional ideal (5*a - 3)) [1, 1]
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.468208638001472, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.745267579437347, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.780039970458374, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.953559587449065, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.991198530766234, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.03192645211244, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.49235842962703, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=1.87960865562082, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=3.63638023803632, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.688325860737610, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.760919185853027, which is not sufficiently close to 1
raw value 0.949159322166882
normalisd -2.00000000000000 + 9.88112364047340e-16*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (22*a - 15) 589 (Fractional ideal (-4*a + 1)) * (Fractional ideal (5*a - 2)) [1, 1]
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.468182866668352, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.733131527186299, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.758944919677417, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.845536747603196, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.859737334634212, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.873732371279926, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.936813696698266, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.946906927956033, which is not sufficiently close to 1
unable to determine epsilon from functional equation working to precision 53, since we get epsilon=0.956184939373125, which is not sufficiently close to 1
raw value 7.59327457733506
normalisd -16.0000000000000 + 3.07412735481394e-15*I
algdeprts [(-16.0000000000000, 1)]
Fractional ideal (a - 26) 649 (Fractional ideal (-3*a + 1)) * (Fractional ideal (7*a - 5)) [1, 1]
raw value -1.10345766872183e-19
normalisd 2.44068232679393e-19 - 7.73274250412063e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a + 25) 649 (Fractional ideal (-3*a + 2)) * (Fractional ideal (7*a - 2)) [1, 1]
raw value -2.36290342748156e-17
normalisd 5.22638683734487e-17 - 1.65586087121798e-31*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (4*a + 25) 709 Fractional ideal (4*a + 25) [1]
raw value 3.46045844719186
normalisd -7.99999999999978 - 5.45541316832748e-14*I
algdeprts [(-8.00000000000000, 1)]
Fractional ideal (-4*a + 29) 709 Fractional ideal (-4*a + 29) [1]
raw value -8.59521178919773e-17
normalisd 1.98706892057553e-16 + 1.35503524446029e-30*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (25*a - 17) 761 Fractional ideal (25*a - 17) [1]
raw value -9.21961151138312e-19
normalisd -2.20819885637982e-18 - 2.14621495694184e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (25*a - 8) 761 Fractional ideal (25*a - 8) [1]
raw value -3.91664436722600e-19
normalisd -9.38079614512676e-19 - 4.71919162906110e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (25*a - 16) 769 Fractional ideal (25*a - 16) [1]
raw value 3.32271878182960
normalisd -8.00000000000003 - 2.97865356716007e-15*I
algdeprts [(-8.00000000000000, 1)]
Fractional ideal (-25*a + 9) 769 Fractional ideal (-25*a + 9) [1]
raw value 0.830679695457400
normalisd -2.00000000000001 - 1.79359784689209e-15*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (25*a - 12) 781 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a - 9)) [1, 1]
raw value -4.68426082213158e-18
normalisd -1.13657949455130e-17 + 1.47649602480222e-32*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (25*a - 13) 781 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a + 8)) [1, 1]
raw value 7.41845996717151
normalisd 18.0000000000001 - 2.14525277631348e-14*I
algdeprts [(18.0000000000000, 1)]
Fractional ideal (26*a - 19) 809 Fractional ideal (26*a - 19) [1]
raw value 0.809883412466161
normalisd -2.00000000000001 + 1.03828745238516e-15*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (-26*a + 7) 809 Fractional ideal (-26*a + 7) [1]
raw value 8.36285351070056e-19
normalisd -2.06519935634564e-18 + 1.73315103138733e-33*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (26*a - 11) 841 (Fractional ideal (a - 6))^2 [1]
raw value 4.27757631063698
normalisd -2.00000000000000 - 1.11328149605248e-15*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (29) 841 (Fractional ideal (a + 5)) * (Fractional ideal (a - 6)) [1, 1]
raw value -1.48974952441026e-18
normalisd -3.75097771621277e-18 + 5.02602634099434e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-26*a + 15) 841 (Fractional ideal (a + 5))^2 [1]
raw value -8.64054724157950e-19
normalisd 4.03992663793896e-19 + 2.24878778571347e-34*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a - 30) 869 (Fractional ideal (-3*a + 2)) * (Fractional ideal (-8*a + 3)) [1, 1]
raw value -1.26873858230788e-18
normalisd 3.24724656396224e-18 - 2.54744535905473e-32*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (a + 29) 869 (Fractional ideal (-3*a + 1)) * (Fractional ideal (-8*a + 5)) [1, 1]
raw value 3.84081770825930e-17
normalisd -9.83030096181298e-17 + 7.71181185968387e-31*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (4*a + 29) 941 Fractional ideal (4*a + 29) [1]
raw value 1.65630383787047e-17
normalisd 4.41131479656183e-17 - 6.39738976476361e-31*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-4*a + 33) 941 Fractional ideal (-4*a + 33) [1]
raw value 4.96169590084308e-19
normalisd 1.32147266962627e-18 - 1.91642993550833e-32*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (3*a - 33) 981 (Fractional ideal (3)) * (Fractional ideal (a - 11)) [1, 1]
raw value 6.61918882981481
normalisd 18.0000000000000 - 2.15498171196667e-14*I
algdeprts [(18.0000000000000, 1)]
Fractional ideal (3*a + 30) 981 (Fractional ideal (3)) * (Fractional ideal (a + 10)) [1, 1]
raw value 0.735465425534979
normalisd 2.00000000000001 - 2.27390536759046e-15*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (29*a - 8) 1009 Fractional ideal (29*a - 8) [1]
raw value 1.93131198180754e-19
normalisd

Too many output messages: 257 (at most 256 per cell -- type 'smc?' to learn how to raise this limit): attempting to terminate...

value_search(E,3,1000)

Fractional ideal (7) 49 Fractional ideal (7) [1]
raw value 5.01151740011611e-16 + 5.55709521912110e-16*I
normalisd 8.43838963899053e-17 - 4.46894271673792e-16*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (2*a + 11) 139 Fractional ideal (2*a + 11) [1]
raw value 1.94848534487254 - 0.144575522568630*I
normalisd 0.999999999999992 + 1.73205080756887*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-2*a + 13) 139 Fractional ideal (-2*a + 13) [1]
raw value -1.09944874774612 - 1.61515004627698*I
normalisd 0.999999999999997 - 1.73205080756887*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (11*a - 6) 151 Fractional ideal (11*a - 6) [1]
raw value 1.28139749191622 - 1.36826182656356*I
normalisd -1.99999999999999 + 7.71140384322982e-16*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (11*a - 5) 151 Fractional ideal (11*a - 5) [1]
raw value 3.84419247574867 - 4.10478547969069*I
normalisd -5.99999999999998 - 4.31838615220870e-15*I
algdeprts [(-6.00000000000000, 1)]
Fractional ideal (13) 169 Fractional ideal (13) [1]
raw value 1.54579562381928 + 0.866227961755149*I
normalisd 2.00000000000000 - 1.69650884551056e-15*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (3*a + 13) 199 Fractional ideal (3*a + 13) [1]
raw value 1.29290702755217 + 0.997438946405499*I
normalisd 2.00000000000001 - 7.09449153577143e-15*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (-3*a + 16) 199 Fractional ideal (-3*a + 16) [1]
raw value 1.51026098008723 - 0.620970857388854*I
normalisd 0.999999999999999 - 1.73205080756889*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (3*a - 20) 331 Fractional ideal (3*a - 20) [1]
raw value 5.51133057385612 - 3.11499430386439*I
normalisd 9.99999999999997 - 3.08456153729193e-15*I
algdeprts [(10.0000000000000, 1)]
Fractional ideal (-3*a - 17) 331 Fractional ideal (-3*a - 17) [1]
raw value 0.0116002174275345 - 1.26608988750909*I
normalisd 0.999999999999997 - 1.73205080756887*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (19) 361 (Fractional ideal (4*a - 3)) * (Fractional ideal (-4*a + 1)) [1, 1]
raw value 3.31254869309870 - 1.50202417497642*I
normalisd -2.99999999999999 + 5.19615242270663*I
algdeprts [(-3.00000000000000 - 5.19615242270663*I, 1), (-3.00000000000000 + 5.19615242270663*I, 1)]
Fractional ideal (3*a + 22) 541 Fractional ideal (3*a + 22) [1]
raw value 0.220342943690387 - 0.965548904601734*I
normalisd -0.999999999999960 - 1.73205080756884*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-3*a + 25) 541 Fractional ideal (-3*a + 25) [1]
raw value 5.67816811095919 - 1.75171119312209*I
normalisd 5.99999999999994 - 10.3923048454130*I
algdeprts [(6.00000000000000 - 10.3923048454133*I, 1), (6.00000000000000 + 10.3923048454133*I, 1)]
Fractional ideal (-3*a - 23) 589 (Fractional ideal (-4*a + 1)) * (Fractional ideal (5*a - 3)) [1, 1]
raw value 1.86011792285857 - 2.15594343218160*I
normalisd 3.00000000000000 - 5.19615242270663*I
algdeprts [(3.00000000000000 - 5.19615242270663*I, 1), (3.00000000000000 + 5.19615242270663*I, 1)]
Fractional ideal (22*a - 7) 589 (Fractional ideal (4*a - 3)) * (Fractional ideal (5*a - 3)) [1, 2]
raw value 4.65484426813973 - 3.28099800642935*I
normalisd 12.0000000000000 - 9.87059691933417e-15*I
algdeprts [(12.0000000000000, 1)]
Fractional ideal (22*a - 15) 589 (Fractional ideal (-4*a + 1)) * (Fractional ideal (5*a - 2)) [1, 2]
raw value -0.0856675815439997 - 0.945285398347338*I
normalisd 1.00000000000001 - 1.73205080756886*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (3*a - 26) 589 (Fractional ideal (4*a - 3)) * (Fractional ideal (5*a - 2)) [1, 1]
raw value 2.99370444680520e-16 + 6.82769503939551e-17*I
normalisd 6.46591406839553e-16 + 2.32624427287458e-17*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-5*a - 23) 619 Fractional ideal (-5*a - 23) [1]
raw value 1.16293111365538e-15 - 3.77673294732337e-15*I
normalisd -7.81187150752790e-15 - 3.44115654976350e-15*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-5*a + 28) 619 Fractional ideal (-5*a + 28) [1]
raw value 0.850102587652340 + 0.366833047835574*I
normalisd 0.999999999999994 - 1.73205080756885*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (23*a - 12) 661 Fractional ideal (23*a - 12) [1]
raw value -1.05030706605215 - 1.45187533489213*I
normalisd -1.99999999999996 - 3.46410161513771*I
algdeprts [(-2.00000000000000 - 3.46410161513775*I, 1), (-2.00000000000000 + 3.46410161513775*I, 1)]
Fractional ideal (23*a - 11) 661 Fractional ideal (23*a - 11) [1]
raw value 2.67377168425605 - 0.275482400294099*I
normalisd 5.99999999999991 + 2.64501151822783e-14*I
algdeprts [(6.00000000000000, 1)]
Fractional ideal (4*a + 25) 709 Fractional ideal (4*a + 25) [1]
raw value 1.16293111365538e-15 - 3.77673294732337e-15*I
normalisd -7.81187150752790e-15 - 3.44115654976350e-15*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-5*a + 28) 619 Fractional ideal (-5*a + 28) [1]
raw value 0.850102587652340 + 0.366833047835574*I
normalisd 0.999999999999994 - 1.73205080756885*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (23*a - 12) 661 Fractional ideal (23*a - 12) [1]
raw value -1.05030706605215 - 1.45187533489213*I
normalisd -1.99999999999996 - 3.46410161513771*I
algdeprts [(-2.00000000000000 - 3.46410161513775*I, 1), (-2.00000000000000 + 3.46410161513775*I, 1)]
Fractional ideal (23*a - 11) 661 Fractional ideal (23*a - 11) [1]
raw value 2.67377168425605 - 0.275482400294099*I
normalisd 5.99999999999991 + 2.64501151822783e-14*I
algdeprts [(6.00000000000000, 1)]
Fractional ideal (4*a + 25) 709 Fractional ideal (4*a + 25) [1]
raw value 0.407925749775956 + 0.762902270422672*I
normalisd -0.999999999999960 - 1.73205080756885*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-4*a + 29) 709 Fractional ideal (-4*a + 29) [1]
raw value 3.65383897522305 - 5.87780157900103*I
normalisd -7.99999999999996 + 13.8564064605506*I
algdeprts [(-8.00000000000000 - 13.8564064605510*I, 1), (-8.00000000000000 + 13.8564064605510*I, 1)]
Fractional ideal (a - 29) 811 Fractional ideal (a - 29) [1]
raw value 2.42211496159430 - 0.148329222983796*I
normalisd 3.00000000000004 + 5.19615242270670*I
algdeprts [(3.00000000000000 - 5.19615242270663*I, 1), (3.00000000000000 + 5.19615242270663*I, 1)]
Fractional ideal (a + 28) 811 Fractional ideal (a + 28) [1]
raw value 0.807371653864767 - 0.0494430743279323*I
normalisd 1.00000000000001 + 1.73205080756890*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-26*a + 9) 829 Fractional ideal (-26*a + 9) [1]
raw value -0.708540684817226 - 0.371560340655551*I
normalisd -2.00000000000000 + 2.00496499923975e-15*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (-26*a + 17) 829 Fractional ideal (-26*a + 17) [1]
raw value -0.0649792967242121 - 1.59878880598863*I
normalisd -2.00000000000001 - 3.46410161513776*I
algdeprts [(-2.00000000000000 - 3.46410161513775*I, 1), (-2.00000000000000 + 3.46410161513775*I, 1)]
Fractional ideal (3*a - 32) 919 Fractional ideal (3*a - 32) [1]
raw value 1.47488307737899 - 1.73819870330501*I
normalisd 2.99999999999996 + 5.19615242270652*I
algdeprts [(3.00000000000000 - 5.19615242270663*I, 1), (3.00000000000000 + 5.19615242270663*I, 1)]
Fractional ideal (-3*a - 29) 919 Fractional ideal (-3*a - 29) [1]
raw value 0.747588590858935 + 0.136062286989825*I
normalisd -0.999999999999974 + 1.73205080756885*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (31) 961 (Fractional ideal (5*a - 2)) * (Fractional ideal (5*a - 3)) [1, 1]
raw value 4.32662646875764e-16 - 3.69338416262855e-16*I
normalisd 9.47802756605315e-17 - 1.52816668209505e-15*I
algdeprts [(0.000000000000000, 1)]
[5.01151740011611e-16 + 5.55709521912110e-16*I, 1.94848534487254 - 0.144575522568630*I, -1.09944874774612 - 1.61515004627698*I, 1.28139749191622 - 1.36826182656356*I, 3.84419247574867 - 4.10478547969069*I, 1.54579562381928 + 0.866227961755149*I, 1.29290702755217 + 0.997438946405499*I, 1.51026098008723 - 0.620970857388854*I, 5.51133057385612 - 3.11499430386439*I, 0.0116002174275345 - 1.26608988750909*I, 3.31254869309870 - 1.50202417497642*I, 0.220342943690387 - 0.965548904601734*I, 5.67816811095919 - 1.75171119312209*I, 1.86011792285857 - 2.15594343218160*I, 4.65484426813973 - 3.28099800642935*I, -0.0856675815439997 - 0.945285398347338*I, 2.99370444680520e-16 + 6.82769503939551e-17*I, 1.16293111365538e-15 - 3.77673294732337e-15*I, 0.850102587652340 + 0.366833047835574*I, -1.05030706605215 - 1.45187533489213*I, 2.67377168425605 - 0.275482400294099*I, 0.407925749775956 + 0.762902270422672*I, 3.65383897522305 - 5.87780157900103*I, 2.42211496159430 - 0.148329222983796*I, 0.807371653864767 - 0.0494430743279323*I, -0.708540684817226 - 0.371560340655551*I, -0.0649792967242121 - 1.59878880598863*I, 1.47488307737899 - 1.73819870330501*I, 0.747588590858935 + 0.136062286989825*I, 4.32662646875764e-16 - 3.69338416262855e-16*I]

value_search(E,5,1000,checkfe=True)

Fractional ideal (11) 121 (Fractional ideal (-3*a + 2)) * (Fractional ideal (-3*a + 1)) [1, 1]
2.64349400471371 + 2.11967871863359*I
Fractional ideal (11) 121 (Fractional ideal (-3*a + 2)) * (Fractional ideal (-3*a + 1)) [2, 2]
1.11220914931987 + 0.661861385341200*I
Fractional ideal (13*a - 7) 211 Fractional ideal (13*a - 7) [1]
0.270190673698379 - 0.542132654592904*I
Fractional ideal (13*a - 7) 211 Fractional ideal (13*a - 7) [2]
3.11862573530779 + 2.74065169134376*I
Fractional ideal (13*a - 6) 211 Fractional ideal (13*a - 6) [1]
0.537246425899483 - 0.279780437464388*I
Fractional ideal (13*a - 6) 211 Fractional ideal (13*a - 6) [2]
1.64280629864264 - 3.81289727573790*I
Fractional ideal (15*a - 8) 281 Fractional ideal (15*a - 8) [1]
0.799630931041636 + 1.11756918601580*I
Fractional ideal (15*a - 8) 281 Fractional ideal (15*a - 8) [2]
0.668896193321883 + 1.20039509911113*I
Fractional ideal (15*a - 7) 281 Fractional ideal (15*a - 7) [1]
-0.0199513470320928 + 2.74828746483946*I
Fractional ideal (15*a - 7) 281 Fractional ideal (15*a - 7) [2]
1.86988657985847 - 2.01420113794708*I
Fractional ideal (a + 18) 341 (Fractional ideal (-3*a + 1)) * (Fractional ideal (5*a - 3)) [2, 1]
1.45101120578398 - 0.521617430312478*I
Fractional ideal (a + 18) 341 (Fractional ideal (-3*a + 1)) * (Fractional ideal (5*a - 3)) [1, 3]
0.0563577153762502 + 4.03640637307097*I
Fractional ideal (-4*a + 21) 341 (Fractional ideal (-3*a + 2)) * (Fractional ideal (5*a - 3)) [1, 2]
-0.187408727583072 + 0.227149243377430*I
Fractional ideal (-4*a + 21) 341 (Fractional ideal (-3*a + 2)) * (Fractional ideal (5*a - 3)) [2, 4]
3.35911784120422 + 4.07915613997706*I
Fractional ideal (4*a + 17) 341 (Fractional ideal (-3*a + 1)) * (Fractional ideal (5*a - 2)) [1, 2]
1.95432257382921 - 0.504540808641061*I
Fractional ideal (4*a + 17) 341 (Fractional ideal (-3*a + 1)) * (Fractional ideal (5*a - 2)) [2, 4]
-0.414566859796096 + 0.650010680287581*I
Fractional ideal (a - 19) 341 (Fractional ideal (-3*a + 2)) * (Fractional ideal (5*a - 2)) [2, 1]
0.457497113063718 + 0.133150979825481*I
Fractional ideal (a - 19) 341 (Fractional ideal (-3*a + 2)) * (Fractional ideal (5*a - 2)) [1, 3]
3.11978478978151 + 0.965738790066169*I
Fractional ideal (4*a - 23) 421 Fractional ideal (4*a - 23) [1]
-0.173473179001174 - 0.392171167853184*I
Fractional ideal (4*a - 23) 421 Fractional ideal (4*a - 23) [2]
-1.05352097604217 - 2.74391401694691*I
Fractional ideal (-4*a - 19) 421 Fractional ideal (-4*a - 19) [1]
2.50837070501377 - 2.24761113109585*I
Fractional ideal (-4*a - 19) 421 Fractional ideal (-4*a - 19) [2]
2.82480889498959 + 1.83415759878612*I
Fractional ideal (19*a - 10) 451 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a - 7)) [1, 2]
1.74195490107801 + 0.214214428817783*I
Fractional ideal (19*a - 10) 451 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a - 7)) [2, 4]
-0.0519138603730805 - 0.668366584325319*I
Fractional ideal (-3*a - 20) 451 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a - 7)) [2, 1]
2.71280780767583 + 0.839636835607123*I
Fractional ideal (-3*a - 20) 451 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a - 7)) [1, 3]
0.157124164344829 + 0.383367887529125*I
Fractional ideal (3*a - 23) 451 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a + 6)) [2, 1]
1.01162633804443 - 1.43419205515797*I
Fractional ideal (3*a - 23) 451 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a + 6)) [1, 3]
0.570282728927224 + 0.352401119505667*I
Fractional ideal (19*a - 9) 451 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a + 6)) [1, 2]
3.66283754896376 + 3.41692663384365*I
Fractional ideal (19*a - 9) 451 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a + 6)) [2, 4]
2.38804832930688 - 0.986301878397431*I
Fractional ideal (a - 22) 461 Fractional ideal (a - 22) [1]
0.592624522722177 + 0.297416455467599*I
Fractional ideal (a - 22) 461 Fractional ideal (a - 22) [2]
0.355187803254333 - 1.69921118490180*I
Fractional ideal (a + 21) 461 Fractional ideal (a + 21) [1]
1.04198734456946 - 1.05478160693580*I
Fractional ideal (a + 21) 461 Fractional ideal (a + 21) [2]
2.87586123719103 - 2.60706871347921*I
Fractional ideal (5*a + 21) 521 Fractional ideal (5*a + 21) [1]
1.44989533175720 - 1.72923169012777*I
Fractional ideal (5*a + 21) 521 Fractional ideal (5*a + 21) [2]
-1.05605483689566 - 1.99428856805196*I
Fractional ideal (-5*a + 26) 521 Fractional ideal (-5*a + 26) [1]
2.26678589086563 + 3.62459690292654*I
Fractional ideal (-5*a + 26) 521 Fractional ideal (-5*a + 26) [2]
0.0335560110819777 - 0.235865153603701*I
Fractional ideal (2*a + 25) 671 (Fractional ideal (-3*a + 1)) * (Fractional ideal (7*a - 4)) [1, 1]
3.16173309874920 - 2.62076356138794*I
Fractional ideal (2*a + 25) 671 (Fractional ideal (-3*a + 1)) * (Fractional ideal (7*a - 4)) [2, 2]
1.72223827055726 - 1.23318746438282*I
Fractional ideal (5*a - 29) 671 (Fractional ideal (-3*a + 2)) * (Fractional ideal (7*a - 4)) [2, 3]
0.576962938871172 + 1.31813344465100*I
Fractional ideal (5*a - 29) 671 (Fractional ideal (-3*a + 2)) * (Fractional ideal (7*a - 4)) [1, 4]
0.110703846598091 + 0.538336702135814*I
Fractional ideal (-5*a - 24) 671 (Fractional ideal (-3*a + 1)) * (Fractional ideal (7*a - 3)) [2, 3]
-0.356582706471412 - 0.814651270502296*I
Fractional ideal (-5*a - 24) 671 (Fractional ideal (-3*a + 1)) * (Fractional ideal (7*a - 3)) [1, 4]
0.179122586481065 + 0.871047081447274*I
Fractional ideal (-2*a + 27) 671 (Fractional ideal (-3*a + 2)) * (Fractional ideal (7*a - 3)) [1, 1]
5.37730432650301 - 4.45725265139172*I
Fractional ideal (-2*a + 27) 671 (Fractional ideal (-3*a + 2)) * (Fractional ideal (7*a - 3)) [2, 2]
0.828521246130364 - 0.593252415864733*I
Fractional ideal (3*a + 25) 691 Fractional ideal (3*a + 25) [1]
1.06157700434569 + 0.582794376197248*I
Fractional ideal (3*a + 25) 691 Fractional ideal (3*a + 25) [2]
2.60691982850923 + 1.80448353426136*I
Fractional ideal (-3*a + 28) 691 Fractional ideal (-3*a + 28) [1]
0.285390297333253 + 0.605560846196166*I
Fractional ideal (-3*a + 28) 691 Fractional ideal (-3*a + 28) [2]
1.31780852265751 - 4.39510688945541*I
Fractional ideal (25*a - 12) 781 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a - 9)) [2, 3]
3.83644169730400e-17 - 7.04724581904932e-17*I
Fractional ideal (25*a - 12) 781 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a - 9)) [1, 4]
3.42977727319086e-17 + 1.84234224012911e-16*I
Fractional ideal (5*a + 26) 781 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a - 9)) [1, 1]
1.60607128200680 - 4.55025585434421*I
Fractional ideal (5*a + 26) 781 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a - 9)) [2, 2]
-0.0260551329148557 + 0.703531163673378*I
Fractional ideal (-5*a + 31) 781 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a + 8)) [1, 1]
-0.404475524735724 - 0.309705325184351*I
Fractional ideal (-5*a + 31) 781 (Fractional ideal (-3*a + 1)) * (Fractional ideal (a + 8)) [2, 2]
0.823326180236132 - 1.04923578306954*I
Fractional ideal (25*a - 13) 781 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a + 8)) [2, 3]
-0.105658214514747 - 0.817473463628480*I
Fractional ideal (25*a - 13) 781 (Fractional ideal (-3*a + 2)) * (Fractional ideal (a + 8)) [1, 4]
0.824066632887496 + 0.0184582391536358*I
Fractional ideal (-27*a + 19) 881 Fractional ideal (-27*a + 19) [1]
-1.24642063263489 - 1.45856298909875*I
Fractional ideal (-27*a + 19) 881 Fractional ideal (-27*a + 19) [2]
1.63756401143936 - 4.74848717731481*I
Fractional ideal (27*a - 8) 881 Fractional ideal (27*a - 8) [1]
0.504187736970598 + 0.590001122773605*I
Fractional ideal (27*a - 8) 881 Fractional ideal (27*a - 8) [2]
0.253017554455786 - 0.733681617680901*I
Fractional ideal (31) 961 (Fractional ideal (5*a - 2)) * (Fractional ideal (5*a - 3)) [1, 1]
-0.405261681029562 + 1.13196904833940*I
Fractional ideal (31) 961 (Fractional ideal (5*a - 2)) * (Fractional ideal (5*a - 3)) [2, 2]
0.350128779134148 + 0.297184649980406*I
Fractional ideal (a - 32) 991 Fractional ideal (a - 32) [1]
5.69044556466475 - 0.805705773491573*I
Fractional ideal (a - 32) 991 Fractional ideal (a - 32) [2]
0.639827141127917 - 0.541952664851865*I
Fractional ideal (a + 31) 991 Fractional ideal (a + 31) [1]
2.75339013068939 - 2.66439309673583*I
Fractional ideal (a + 31) 991 Fractional ideal (a + 31) [2]
0.211806768906683 + 0.517322903638813*I
0.211806768906683 + 0.517322903638813*I

value_search(E,7,1000)

chi = NFChar([F.prime_above(139)],[1],3)
LEchi = L.twist(chi, epsilon='solve')
omega = prod([E.period_lattice(phi).basis(prec=1000)[0] for phi in F.embeddings(RR)])
val = LEchi(1)
val

-1.09944874774612 - 1.61515004627698*I

normalised = val*chi.conjugate().gauss_sum()*CC(sqrt(5))/omega
normalised
algdep(normalised, 2, known_bits=53).roots(CC)

-2*a + 1
0.999999999999997 - 1.73205080756887*I
[(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]