CoCalc Public Fileswww / tables / serremodpq / serremodpq.tex
Author: William A. Stein
1% serremodpq.tex
2
3\documentclass{article}
4\title{Serre's Conjecture Mod $pq$?}
5\author{B. Mazur\and W.A. Stein}
6\date{January 11, 1999}
7\newcommand{\comment}[1]{}
8\begin{document}
9\maketitle
10\tableofcontents
11
12\section{Introduction}
13
14I haven't yet written an introduction, but here are some helpful remarks
15as to how to read the table.  Please see Barry Mazur's letter below
16which motivated the computation.  Basically, when both an $f$ and $g$
17occur together in a given row, then Serre's Conjecture mod $pq$''
18for a certain representation has an easy positive answer.
19
20The rows of the congruence table contain an $f$ whenever the
21corresponding eigenform
22at level $NM$ is congruent to the reduction of $f$ modulo the prime $P$ listed
23in the first column, at least to the precision computed.
24Similarly for $g$.  In some cases the program did not perform
25a complete check that the forms were congruent to the precision
26computed, but only checked, with {\em high} probability,
27that there is a congruence to that precision.
28In such cases this is indicated
29by a question mark subscript.
30In many cases Ribet's theorem implies
31that there must be a congruence.
32
33I do not know if Sturm's result can be applied in
34this context because the congruence is only checked
35for those $a_p$ for which $p$ is prime to $NMpq$.
36
37In a few cases it was too much extra work to reduce the prime level $\leq 97$
38eigenforms modulo $2$, and so this is not done, even though $2$ might
39be among the canidate pairs $(P, Q)$.  This is indicated above the
40relevant table.
41
42The eigenforms refered to in the congruence table are uniquely
43determined, up to conjugacy, by a factor of the characteristic
44polynomial of some Hecke operator.  These factors are given, so that
45further investigation of particular eigenforms is possible.
46
47To understand why this table was computed, see the letter from Barry
48Mazur in the last section.  Note that the congruence table does {\em not}
49omit primes for which the corresponding representations are reducible.
50
51\section{Letters}
52\newcommand{\Q}{\mathbf{Q}}
53\newcommand{\GL}{{\rm GL}}
54\newcommand{\Z}{\mathbf{Z}}
55\newcommand{\mod}{{\rm\,\, mod\,\,}}
56\begin{verbatim}
57From: Barry Mazur
58Subject: SERRE'S CONJECTURE MOD PQ
59To: William Stein
60\end{verbatim}
61
62Dear William,
63
64   To put perspective on the subject of this e-mail, let me begin with
65an utterly untestable version of what I was suggesting to you.  Let $p$
66and $q$ be distinct prime numbers and $G$ the Galois group of the maximal
67algebraic extension of $\Q$ unramified outside a finite set $S$ of primes.
68Suppose that we have an odd continuous representation
69$r: G \rightarrow \GL_2(\Z/pq\Z)$
70which is irreducible when reduced mod $p$ and mod $q$.  Is $r$ "modular"?
71
72   To begin to give {\em some} shape to the above question, we might as well
73assume the
74classical'' Serre conjecture for single primes, so we can start out by
75giving
76ourselves two modular newforms $f$ and $g$ and then look for a modular newform
77$h$ that
78is congruent mod $p$ to $f$ and mod $q$ to $g$ -- under the assumption that the
79associated
80Galois representations mod $p$ and mod $q$ respectively are irreducible.  I
81should
82be more careful about what I mean mod $p$'' and mod $q$'':  I really mean to
83take $p$ and $q$ prime ideals in the ring generated by the Fourier coefficients
84of $f$ and $g$, and ditto with regard to $h$.  To simplify one's life, one
85can "start out" by working only with pairs $f$ and $g$ of newforms whose ring
86of Fourier coefficients is $\Z$  (i.e., the $f$ and $g$ in question correspond to
87elliptic curves).  But even if one does this, one has to face the
88possibility
89(the probability'', in fact) that $h$, if it exists at all, no longer has its
90ring of
91Fourier coefficients equal to $\Z$.
92
93 Of course, phrased this way, our qeustion is still relatively
94unfalsifiable''
95so we have better be more specific.
96
97   Ken and I thought a bit about how to shape a more specific testable
98question,
99and here is what we came up with-- very tentatively still.  Suppose that $N$
100and $M$ are
101distinct prime numbers, and $f$ and $g$ are newforms of weight two on
102$X_0(N)$
103and $X_0(M)$
104respectively.  Now let $p$ be a prime number (or ideal) dividing either
105$1+M- a_M(f)$   or
106$1+M + a_M(f)$,  and let $q$ be a prime number (or ideal)  dividing either
107$1+N - a_N(g)$   or  $1+N + a_N(g)$.  Let us suppose that the residual
108characteristics
109of the ideals $p$ and $q$ are distinct  (i.e., if we are in a case where $p$ and
110$q$ are
111prime numbers, we just want them to be distinct).  Suppose also that the
112corresponding Galois representations associated to $f \mod p$ and $g \mod q$
113are irreducible.  Let us say that we have maximum success''
114if for any such choice of $f,g, p, q$  there is a modular newform $h$ of weight
115two on $X_0(NM)$
116which is congruent to $f \mod p$ and to $g \mod q$.
117Here $a_N(f)$ means the $N$th Fourier coefficient of $f$, etc.
118
119   The reason why we call the above maximum success'' is that we are
120putting down
121the evident'' conditions necessary for there to be such a newform $h$ (and
122looking
123for such an $h$ at the minimal conceivable level, and weight).  It is
124probably
125unreasonable to expect maximum success'', but initial experiments might
126give us a sense of
127how far from the mark maximal success'' actually  is, in which case we will
128weaken our
129expectations accordingly...  I should also say, that the first case is
130$N=11$ and $M=17$,
131where the level of the sought for $h$ is $187$, i.e., not too large yet...
132
133   Does this sound as if it is a conceivable, fun, etc. computation to
134make?
135
136     Barry
137
138\begin{verbatim}
139Date: Mon, 26 Oct 1998 12:46:09 -0500
140\end{verbatim}
141
142Dear William,
143
144    That's terrific!  Actually I had debated about whether to exclude the
145prime $2$ from the search-- it being slightly problematic for level-raising,
146level-lowering issues. I also excluded reducible representations as being
147significantly more problematic on a number of other grounds. But, while you
148are systemtically doing the computation, it might make sense not to
149exclude reducible representations  from the search, but just indicate
150in the data when they are reducible.  After all, we may also be getting
151some
152maximal succeses'' there too  (but if we get a "failure" for a
153reducible case, I wouldn't worry).  Keep me posted!
154
155Barry
156
157\begin{verbatim}
158Date: Tue, 3 Nov 1998 20:32:35 +0000 (GMT)
159From: Kevin Buzzard <buzzard@ic.ac.uk>
160To: was@bmw.autobahn.org
161Subject: Re: congruence table
162
163I looked through your tables a fair bit yesterday and can't find anything
164which smacks of contradiction, which is very comforting. It all seems
165to fit together very well, which means that I am getting more and more
166confident in your programs!
167
168[NB my experience with these programs is that if they "usually" work
169then they still might not work 100% because if numbers get too big
170then there's weird overflows and so maybe things don't work then---
171but I don't have any evidence at all that your programs don't
172work 100%]
173
174Anyway, here's some random comments about them. When you say
175in the intro that "in _some_ cases the program didn't perform
176a complete check of congruence"---really, you never performed
177a _complete_ check. If you mean "complete check up to p=***"
178whatever *** is, maybe you should say this. Also it's a bit
179strange that you then go on to talk about the ?? cases, and
180say that they're congruences with a "high probability", whereas
181in reality we're *always* only dealing with probabilities (except,
182I guess, in cases where Ken's theorem predicts that a form
183exists and only 1 form can possibly work at level NM). In practice
184it's very difficult to verify that forms are actually congruent
185because the effective Chebatorev bounds aren't good enough.
186\end{verbatim}
187
188\newpage
189\textheight=1.28\textheight
190\textwidth=1.35\textwidth
191\voffset=-1.3in
192\hoffset=-.9in
193\section{Table of Congruences}
194\noindent\begin{minipage}{\textwidth}
195\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=17A}$}\\
196\begin{tabular}{|cc|cccccc|}\hline
197$P$&$Q$&187A&187B&187C&187D&187E&187F\\\hline
198$2$&$3$&$f$&$f$&$fg$&$f$&$f$&$g$\\\hline
199$5$&$2$&&&&&$g$&\\\hline
200$5$&$3$&&&$g$&&&$g$\\\hline
201\end{tabular}\end{minipage}
202
203
204
205\noindent\begin{minipage}{\textwidth}
206\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=19A}$}\\
207\begin{tabular}{|cc|cccc|}\hline
208$P$&$Q$&209A&209B&209C&209D\\\hline
209$2$&$3$&$f$&$f_?$&$f_?g$&$fg$\\\hline
210$2$&$5$&$f$&$f_?$&$f_?g$&$f$\\\hline
211$5$&$3$&&$f$&$g$&$g$\\\hline
212\end{tabular}\end{minipage}
213
214
215
216\noindent\begin{minipage}{\textwidth}
217\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=23A}$}\\
218\begin{tabular}{|cc|cccc|}\hline
219$P$&$Q$&253A&253B&253C&253D\\\hline
220$23$&$({x + {8}},11) 221$&&&&$f$\\\hline
222$23$&$({x + {15}},19) 223$&$g$&&&$f$\\\hline
224$23$&$({x^{2} + x + 1},2) 225$&&&$g$&$fg$\\\hline
226$23$&$({x + {3}},5) 227$&&$g$&&$f$\\\hline
228$5$&$({x + {8}},11) 229$&&$f$&&\\\hline
230$5$&$({x + {15}},19) 231$&$g$&$f$&&\\\hline
232$5$&$({x^{2} + x + 1},2) 233$&&$f$&$g$&$g$\\\hline
234\end{tabular}\end{minipage}
235
236
237
238\noindent\begin{minipage}{\textwidth}
239\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=29A}$}\\
240\begin{tabular}{|cc|ccccc|}\hline
241$P$&$Q$&319A&319B&319C&319D&319E\\\hline
242$2$&$({x + {14}},167) 243$&$f$&&$f$&&$g$\\\hline
244$2$&$({x + {7}},17) 245$&$f$&$g$&$f$&&\\\hline
246$2$&$({x + {4}},7) 247$&$f$&&$f$&&\\\hline
248$3$&$({x + {14}},167) 249$&&$f$&&$f$&$g$\\\hline
250$3$&$({x + {7}},17) 251$&&$fg$&&$f$&\\\hline
252$3$&$({x + {4}},7) 253$&&$f$&&$f$&\\\hline
254$5$&$({x + {14}},167) 255$&&&&$f$&$g$\\\hline
256$5$&$({x + {7}},17) 257$&&$g$&&$f$&\\\hline
258$5$&$({x + {4}},7) 259$&&&&$f$&\\\hline
260\end{tabular}\end{minipage}
261
262
263
264\noindent\begin{minipage}{\textwidth}
265\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=19A}$}\\
266\begin{tabular}{|cc|cccccc|}\hline
267$P$&$Q$&323A&323B&323C&323D&323E&323F\\\hline
268$2$&$3$&$f$&&$f$&$f_?g$&$f_?$&\\\hline
269$2$&$5$&$f$&&$f$&$f_?$&$f_?$&$g$\\\hline
270$2$&$7$&$f$&$g$&$f$&$f_?$&$f_?$&\\\hline
271$3$&$5$&&$f$&&&&$g$\\\hline
272$3$&$7$&&$fg$&&&&\\\hline
273\end{tabular}\end{minipage}
274
275
276\noindent\begin{minipage}{\textwidth}
277\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=31A}$}\\
278\begin{tabular}{|cc|cccc|}\hline
279$P$&$Q$&341A&341B&341C&341D\\\hline
280$13$&$({x^{2} + x + 1},2) 281$&&$g_?$&&$fg_?$\\\hline
282$13$&$({x + {2}},5) 283$&$g$&&&$f$\\\hline
284$13$&$({x^{2} + {6}x + {6}},7) 285$&&&$g_?$&$f$\\\hline
286$3$&$({x^{2} + x + 1},2) 287$&&$g_?$&&$f_?g_?$\\\hline
288$3$&$({x + {2}},5) 289$&$g$&&&$f_?$\\\hline
290$3$&$({x^{2} + {6}x + {6}},7) 291$&&&$g_?$&$f_?$\\\hline
292$5$&$({x^{2} + x + 1},2) 293$&$f$&$g_?$&&$g_?$\\\hline
294$5$&$({x^{2} + {6}x + {6}},7) 295$&$f$&&$g_?$&\\\hline
296\end{tabular}\end{minipage}
297
298
299
300\noindent\begin{minipage}{\textwidth}
301\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=23A}$}\\
302\begin{tabular}{|cc|ccccc|}\hline
303$P$&$Q$&391A&391B&391C&391D&391E\\\hline
304$2$&$({x + {99}},109) 305$&&&$g$&&$f_?$\\\hline
306$2$&$({x + {8}},11) 307$&&&&&$f_?$\\\hline
308$2$&$({x + {3}},5) 309$&&$g$&&&$f_?$\\\hline
310$5$&$({x + {99}},109) 311$&&$f$&$g$&&\\\hline
312$5$&$({x + {8}},11) 313$&&$f$&&&\\\hline
314$5$&$({x^{2} + x + 1},2) 315$&&$f$&&$g$&$g_?$\\\hline
316$7$&$({x + {99}},109) 317$&&&$g$&&$f_?$\\\hline
318$7$&$({x + {8}},11) 319$&&&&&$f_?$\\\hline
320$7$&$({x^{2} + x + 1},2) 321$&&&&$g$&$f_?g_?$\\\hline
322$7$&$({x + {3}},5) 323$&&$g$&&&$f_?$\\\hline
324\end{tabular}\end{minipage}
325
326
327
328\noindent\begin{minipage}{\textwidth}
329\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=37A}$}\\
330\begin{tabular}{|cc|cccc|}\hline
331$P$&$Q$&407A&407B&407C&407D\\\hline
332$41$&$17$&&&$fg$&\\\hline
333$41$&$7$&$g$&&$f$&\\\hline
334$5$&$17$&&&$g$&\\\hline
335$5$&$7$&$g$&&&\\\hline
336$7$&$17$&&$f$&$g$&\\\hline
337\end{tabular}\end{minipage}
338
339\noindent\begin{minipage}{\textwidth}
340\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=37B}$}\\
341\begin{tabular}{|cc|cccc|}\hline
342$P$&$Q$&407A&407B&407C&407D\\\hline
343$41$&$3$&&$g$&$f$&$g$\\\hline
344$41$&$5$&&&$f$&$g$\\\hline
345$5$&$3$&&$g$&&$g$\\\hline
346$7$&$3$&&$fg$&&$g$\\\hline
347$7$&$5$&&$f$&&$g$\\\hline
348\end{tabular}\end{minipage}
349
350
351
352\noindent\begin{minipage}{\textwidth}
353\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=23A}$}\\
354\begin{tabular}{|cc|cccccccc|}\hline
355$P$&$Q$&437A&437B&437C&437D&437E&437F&437G&437H\\\hline
356$2$&$({x + {4}},11) 357$&$f$&$f$&$g$&$f$&&$f_?$&$f_?$&$f_?$\\\hline
358$2$&$({x + {8}},11) 359$&$f$&$f$&&$f$&&$f_?$&$f_?$&$f_?$\\\hline
360$2$&$({x^{2} + x + {2}},3) 361$&$f$&$f$&&$f$&&$f_?$&$f_?g$&$f_?$\\\hline
362$3$&$({x + {4}},11) 363$&&&$g$&&&&&$f$\\\hline
364$3$&$({x + {8}},11) 365$&&&&&&&&$f$\\\hline
366$3$&$({x^{2} + x + 1},2) 367$&&&&&&$g_?$&&$fg_?$\\\hline
368\end{tabular}\end{minipage}
369
370
371
372\noindent\begin{minipage}{\textwidth}
373\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=41A}$}  {\small (char 2 primes for $g$ omitted)}\\
374\begin{tabular}{|cc|ccccc|}\hline
375$P$&$Q$&451A&451B&451C&451D&451E\\\hline
376$17$&$({x + {2}},5) 377$&&$g$&$f$&&\\\hline
378$17$&$({x + {413}},863) 379$&&&$f$&$g$&\\\hline
380$2$&$({x + {2}},5) 381$&&$g$&&$f_?$&$f$\\\hline
382$2$&$({x + {413}},863) 383$&&&&$f_?g$&$f$\\\hline
384$5$&$({x + {413}},863) 385$&&$f$&&$g$&\\\hline
386\end{tabular}\end{minipage}
387
388
389
390\noindent\begin{minipage}{\textwidth}
391\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=43A}$}\\
392\begin{tabular}{|cc|ccccccc|}\hline
393$P$&$Q$&473A&473B&473C&473D&473E&473F&473G\\\hline
394$19$&$3$&&&&&&$fg$&$g$\\\hline
395$19$&$5$&&$g$&&&&$f$&\\\hline
396$2$&$3$&&&&&$f_?$&$g$&$fg$\\\hline
397$2$&$5$&&$g$&&&$f_?$&&$f$\\\hline
398$5$&$3$&$f$&&&&&$g$&$g$\\\hline
399\end{tabular}\end{minipage}
400
401\noindent\begin{minipage}{\textwidth}
402\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=43B}$}\\
403\begin{tabular}{|cc|ccccccc|}\hline
404$P$&$Q$&473A&473B&473C&473D&473E&473F&473G\\\hline
405$19$&$({x + {62}},113) 406$&&&&&$g$&$f$&\\\hline
407$19$&$({x + {5}},23) 408$&&&$g$&&&$f$&\\\hline
409$19$&$({x + {4}},7) 410$&&&&&&$f$&\\\hline
411$2$&$({x + {62}},113) 412$&&&&&$f_?g$&&$f$\\\hline
413$2$&$({x + {5}},23) 414$&&&$g$&&$f_?$&&$f$\\\hline
415$2$&$({x + {4}},7) 416$&&&&&$f_?$&&$f$\\\hline
417$5$&$({x + {62}},113) 418$&$f$&&&&$g$&&\\\hline
419$5$&$({x + {5}},23) 420$&$f$&&$g$&&&&\\\hline
421$5$&$({x + {4}},7) 422$&$f$&&&&&&\\\hline
423\end{tabular}\end{minipage}
424
425
426
427\noindent\begin{minipage}{\textwidth}
428\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=29A}$}\\
429\begin{tabular}{|cc|cccccccc|}\hline
430$P$&$Q$&493A&493B&493C&493D&493E&493F&493G&493H\\\hline
431$2$&$({x + {24}},31) 432$&&&&&$f$&$f$&$f_?g$&\\\hline
433$2$&$({x + {4}},7) 434$&&&&$g$&$f$&$f$&$f_?$&\\\hline
435$3$&$({x + 1},2) 436$&$fg$&$g$&$g$&&&$g$&&$f$\\\hline
437$3$&$({x + {24}},31) 438$&$f$&&&&&&$g$&$f$\\\hline
439$3$&$({x + {4}},7) 440$&$f$&&&$g$&&&&$f$\\\hline
441\end{tabular}\end{minipage}
442
443
444
445\noindent\begin{minipage}{\textwidth}
446\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=47A}$}\\
447\begin{tabular}{|cc|ccccccccccc|}\hline
448$P$&$Q$&517A&517B&517C&517D&517E&517F&517G&517H&517I&517J&517K\\\hline
449$2$&$({x + {20}},23) 450$&$f$&$f$&$f_?$&&&$f_?$&&&$f_?$&$f_?$&$f$\\\hline
451$2$&$({x + 1},3) 452$&$f$&$fg$&$f_?$&&&$f_?g$&&$g_?$&$f_?$&$f_?$&$fg$\\\hline
453$2$&$({x + {8}},31) 454$&$f$&$f$&$f_?g$&&&$f_?$&&&$f_?$&$f_?$&$f$\\\hline
455$2$&$({x + {2}},7) 456$&$f$&$f$&$f_?$&$g$&&$f_?$&&&$f_?$&$f_?$&$f$\\\hline
457$5$&$({x^{4} + x^{3} + x^{2} + x + 1},2) 458$&&&&&&&&&$g_?$&$g_?$&\\\hline
459$5$&$({x + {20}},23) 460$&&&&&&&&&&&\\\hline
461$5$&$({x + 1},3) 462$&&$g$&&&&$g$&&$g_?$&&&$g$\\\hline
463$5$&$({x + {8}},31) 464$&&&$g$&&&&&&&&\\\hline
465$5$&$({x + {2}},7) 466$&&&&$g$&&&&&&&\\\hline
467$7$&$({x^{4} + x^{3} + x^{2} + x + 1},2) 468$&&&&&$f$&&&&$g_?$&$g_?$&\\\hline
469$7$&$({x + {20}},23) 470$&&&&&$f$&&&&&&\\\hline
471$7$&$({x + 1},3) 472$&&$g$&&&$f$&$g$&&$g_?$&&&$g$\\\hline
473$7$&$({x + {8}},31) 474$&&&$g$&&$f$&&&&&&\\\hline
475\end{tabular}\end{minipage}
476
477
478
479\noindent\begin{minipage}{\textwidth}
480\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=31A}$}\\
481\begin{tabular}{|cc|ccccc|}\hline
482$P$&$Q$&527A&527B&527C&527D&527E\\\hline
483$2$&$({x + {98}},109) 484$&&&&$g$&$f_?$\\\hline
485$2$&$({x + {7}},11) 486$&$g$&&&&$f_?$\\\hline
487$2$&$({x + {2}},5) 488$&&&&&$f_?$\\\hline
489$3$&$({x + {98}},109) 490$&&&&$g$&$f_?$\\\hline
491$3$&$({x + {7}},11) 492$&$g$&&&&$f_?$\\\hline
493$3$&$({x^{2} + x + 1},2) 494$&&&$g$&&$f_?g_?$\\\hline
495$3$&$({x + {2}},5) 496$&&&&&$f_?$\\\hline
497$7$&$({x + {98}},109) 498$&$f$&&&$g$&\\\hline
499$7$&$({x + {7}},11) 500$&$fg$&&&&\\\hline
501$7$&$({x^{2} + x + 1},2) 502$&$f$&&$g$&&$g_?$\\\hline
503$7$&$({x + {2}},5) 504$&$f$&&&&\\\hline
505\end{tabular}\end{minipage}
506
507
508
509
510\noindent\begin{minipage}{\textwidth}
511\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=29A}$}\\
512\begin{tabular}{|cc|cccccccc|}\hline
513$P$&$Q$&551A&551B&551C&551D&551E&551F&551G&551H\\\hline
514$2$&$({x^{2} + {2}x + {12}},13) 515$&&$f$&$f$&&$f$&$f_?$&$f_?g$&\\\hline
516$2$&$({x + {4}},7) 517$&&$f$&$f$&&$f$&$f_?$&$f_?$&\\\hline
518$2$&$({x + {5}},7) 519$&&$f$&$fg$&&$f$&$f_?$&$f_?$&\\\hline
520$3$&$({x^{2} + {2}x + {12}},13) 521$&&&&&&$f_?$&$g$&\\\hline
522$3$&$({x + 1},2) 523$&$g$&&&$g$&&$f_?g_?$&$g_?$&$g$\\\hline
524$3$&$({x + {4}},7) 525$&&&&&&$f_?$&&\\\hline
526$3$&$({x + {5}},7) 527$&&&$g$&&&$f_?$&&\\\hline
528\end{tabular}\end{minipage}
529
530
531
532\noindent\begin{minipage}{\textwidth}
533\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=53A}$}\\
534\begin{tabular}{|cc|ccccccccc|}\hline
535$P$&$Q$&583A&583B&583C&583D&583E&583F&583G&583H&583I\\\hline
536$2$&$3$&&$f$&$fg$&$f$&$fg$&&$f_?$&$f_?$&$f_?$\\\hline
537$3$&$2$&$fg$&&&&&$g_?$&$f$&$g_?$&\\\hline
538$5$&$2$&$g$&&&&&$g_?$&&$g_?$&\\\hline
539$5$&$3$&&&$g$&&$g$&&&&\\\hline
540\end{tabular}\end{minipage}
541
542\noindent\begin{minipage}{\textwidth}
543\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=53B}$}\\
544\begin{tabular}{|cc|ccccccccc|}\hline
545$P$&$Q$&583A&583B&583C&583D&583E&583F&583G&583H&583I\\\hline
546$2$&$({x + {10}},13) 547$&&$f$&$f$&$f$&$f$&&$f_?$&$f_?$&$f_?$\\\hline
548$2$&$({x + {224}},281) 549$&&$f$&$f$&$f$&$f$&&$f_?$&$f_?$&$f_?g$\\\hline
550$2$&$({x + {5}},43) 551$&&$f$&$f$&$f$&$f$&&$f_?g$&$f_?$&$f_?$\\\hline
552$3$&$({x + {10}},13) 553$&$f$&&&&&&$f$&&\\\hline
554$3$&$({x + 1},2) 555$&$fg$&&&&&$g_?$&$f$&$g_?$&\\\hline
556$3$&$({x + {224}},281) 557$&$f$&&&&&&$f$&&$g$\\\hline
558$3$&$({x + {5}},43) 559$&$f$&&&&&&$fg$&&\\\hline
560$5$&$({x + {10}},13) 561$&&&&&&&&&\\\hline
562$5$&$({x + 1},2) 563$&$g$&&&&&$g_?$&&$g_?$&\\\hline
564$5$&$({x + {224}},281) 565$&&&&&&&&&$g$\\\hline
566$5$&$({x + {5}},43) 567$&&&&&&&$g$&&\\\hline
568\end{tabular}\end{minipage}
569
570
571
572\noindent\begin{minipage}{\textwidth}
573\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=31A}$}\\
574\begin{tabular}{|cc|ccccc|}\hline
575$P$&$Q$&589A&589B&589C&589D&589E\\\hline
576$2$&$({x + {2}},5) 577$&&$f$&$f$&$f_?g$&$f$\\\hline
578$2$&$({x + {29}},79) 579$&&$f$&$f$&$f_?g$&$f$\\\hline
580$2$&$({x + {49}},79) 581$&&$f$&$f$&$f_?$&$fg$\\\hline
582$3$&$({x + {2}},5) 583$&&&&$g$&$f$\\\hline
584$3$&$({x + {29}},79) 585$&&&&$g$&$f$\\\hline
586$3$&$({x + {49}},79) 587$&&&&&$fg$\\\hline
588$7$&$({x + {2}},5) 589$&$f$&&&$g$&\\\hline
590$7$&$({x + {29}},79) 591$&$f$&&&$g$&\\\hline
592$7$&$({x + {49}},79) 593$&$f$&&&&$g$\\\hline
594\end{tabular}\end{minipage}
595
596
597
598\noindent\begin{minipage}{\textwidth}
599\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=37A}$}\\
600\begin{tabular}{|cc|cccccccc|}\hline
601$P$&$Q$&629A&629B&629C&629D&629E&629F&629G&629H\\\hline
602$2$&$3$&$g$&&&&$f_?$&$f_?g$&$fg$&\\\hline
603$3$&$2$&&&$fg$&&$g_?$&$f$&$g$&$g$\\\hline
604$5$&$2$&&$f$&$g$&&$g_?$&&$g$&$g$\\\hline
605$5$&$3$&$g$&$f$&&&&$g$&$g$&\\\hline
606\end{tabular}\end{minipage}
607
608\noindent\begin{minipage}{\textwidth}
609\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=37B}$}\\
610\begin{tabular}{|cc|cccccccc|}\hline
611$P$&$Q$&629A&629B&629C&629D&629E&629F&629G&629H\\\hline
612$2$&$3$&&&&&$f_?g$&$f_?$&$f$&\\\hline
613$3$&$2$&&&$fg$&&$g_?$&$f$&$g$&$g$\\\hline
614$5$&$2$&&$f$&$g$&&$g_?$&&$g$&$g$\\\hline
615$5$&$3$&&$f$&&&$g$&&&\\\hline
616\end{tabular}\end{minipage}
617
618
619
620\noindent\begin{minipage}{\textwidth}
621\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=59A}$}  {\small (char 2 primes for $g$ omitted)}\\
622\begin{tabular}{|cc|cccccc|}\hline
623$P$&$Q$&649A&649B&649C&649D&649E&649F\\\hline
624$11$&$({x + {9}},19) 625$&&&$fg$&&&\\\hline
626$11$&$({x + {1940}},2671) 627$&&&$f$&$g$&&\\\hline
628$11$&$({x + {26}},29) 629$&&&$f$&&&\\\hline
630$11$&$({x + 1},7) 631$&&$g$&$f$&&&\\\hline
632$13$&$({x + {9}},19) 633$&&&$g$&&$f$&\\\hline
634$13$&$({x + {1940}},2671) 635$&&&&$g$&$f$&\\\hline
636$13$&$({x + {26}},29) 637$&&&&&$f$&\\\hline
638$13$&$({x + 1},7) 639$&&$g$&&&$f$&\\\hline
640$5$&$({x + {9}},19) 641$&&&$g$&&$f$&\\\hline
642$5$&$({x + {1940}},2671) 643$&&&&$g$&$f$&\\\hline
644$5$&$({x + {26}},29) 645$&&&&&$f$&\\\hline
646$5$&$({x + 1},7) 647$&&$g$&&&$f$&\\\hline
648\end{tabular}\end{minipage}
649
650
651
652\noindent\begin{minipage}{\textwidth}
653\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=29A}$}\\
654\begin{tabular}{|cc|cccc|}\hline
655$P$&$Q$&667A&667B&667C&667D\\\hline
656$({x + {4}},11) 657$&$({x + {52}},113) 658$&$f$&&&$g$\\\hline
659$({x + {4}},11) 660$&$({x + 1},2) 661$&$f$&$g$&$g_?$&\\\hline
662$({x + {4}},11) 663$&$({x + {19}},23) 664$&$fg$&&&\\\hline
665$({x + {4}},11) 666$&$({x + {4}},7) 667$&$f$&&&\\\hline
668$({x + {8}},11) 669$&$({x + {52}},113) 670$&&&&$g$\\\hline
671$({x + {8}},11) 672$&$({x + 1},2) 673$&&$g$&$g_?$&\\\hline
674$({x + {8}},11) 675$&$({x + {19}},23) 676$&$g$&&&\\\hline
677$({x + {8}},11) 678$&$({x + {4}},7) 679$&&&&\\\hline
680$({x^{2} + x + {2}},3) 681$&$({x + {52}},113) 682$&$f$&&$f$&$g$\\\hline
683$({x^{2} + x + {2}},3) 684$&$({x + 1},2) 685$&$f$&$g$&$fg_?$&\\\hline
686$({x^{2} + x + {2}},3) 687$&$({x + {19}},23) 688$&$fg$&&$f$&\\\hline
689$({x^{2} + x + {2}},3) 690$&$({x + {4}},7) 691$&$f$&&$f$&\\\hline
692\end{tabular}\end{minipage}
693
694
695
696\noindent\begin{minipage}{\textwidth}
697\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=61A}$}\\
698\begin{tabular}{|cc|cccc|}\hline
699$P$&$Q$&671A&671B&671C&671D\\\hline
700$2$&$17$&&$f$&$f$&$g$\\\hline
701$2$&$7$&&$fg$&$f$&\\\hline
702$37$&$17$&&&&$fg$\\\hline
703$37$&$7$&&$g$&&$f$\\\hline
704$5$&$17$&$f$&&&$g$\\\hline
705$5$&$7$&$f$&$g$&&\\\hline
706\end{tabular}\end{minipage}
707
708\noindent\begin{minipage}{\textwidth}
709\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=61B}$}\\
710\begin{tabular}{|cc|cccc|}\hline
711$P$&$Q$&671A&671B&671C&671D\\\hline
712$2$&$({x + {3}},13) 713$&&$f$&$fg$&\\\hline
714$2$&$({x + {9}},17) 715$&$g$&$f$&$f$&\\\hline
716$2$&$({x + {16}},331) 717$&&$f$&$fg$&\\\hline
718$2$&$({x + {2}},5) 719$&$g$&$f$&$f$&\\\hline
720$37$&$({x + {3}},13) 721$&&&$g$&$f$\\\hline
722$37$&$({x + {9}},17) 723$&$g$&&&$f$\\\hline
724$37$&$({x + {16}},331) 725$&&&$g$&$f$\\\hline
726$37$&$({x + {2}},5) 727$&$g$&&&$f$\\\hline
728$5$&$({x + {3}},13) 729$&$f$&&$g$&\\\hline
730$5$&$({x + {9}},17) 731$&$fg$&&&\\\hline
732$5$&$({x + {16}},331) 733$&$f$&&$g$&\\\hline
734\end{tabular}\end{minipage}
735
736
737
738\noindent\begin{minipage}{\textwidth}
739\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=41A}$}  {\small (char 2 primes for $g$ omitted)}\\
740\begin{tabular}{|cc|cccccc|}\hline
741$P$&$Q$&697A&697B&697C&697D&697E&697F\\\hline
742$2$&$({x + {2}},5) 743$&$f$&$f_?$&$f$&&$f_?$&$f_?$\\\hline
744$2$&$({x^{2} + {4}x + {2}},5) 745$&$f$&$f_?$&$f$&$g$&$f_?$&$f_?$\\\hline
746$3$&$({x + {2}},5) 747$&$f$&&&&$f$&\\\hline
748$3$&$({x^{2} + {4}x + {2}},5) 749$&$f$&&&$g$&$f$&\\\hline
750\end{tabular}\end{minipage}
751
752
753
754\noindent\begin{minipage}{\textwidth}
755\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=37A}$}\\
756\begin{tabular}{|cc|ccccccccc|}\hline
757$P$&$Q$&703A&703B&703C&703D&703E&703F&703G&703H&703I\\\hline
758$2$&$5$&$f$&&&&&$f_?$&$fg$&$f_?g$&\\\hline
759$3$&$2$&&$g$&$g$&$f$&$g$&$g_?$&&$g_?$&$g$\\\hline
760$3$&$5$&&&&$f$&&&$g$&$g$&\\\hline
761$5$&$2$&&$g$&$g$&&$g$&$g_?$&&$g_?$&$fg$\\\hline
762\end{tabular}\end{minipage}
763
764\noindent\begin{minipage}{\textwidth}
765\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=37B}$}\\
766\begin{tabular}{|cc|ccccccccc|}\hline
767$P$&$Q$&703A&703B&703C&703D&703E&703F&703G&703H&703I\\\hline
768$2$&$11$&$f$&&&&&$f_?g$&$f$&$f_?$&\\\hline
769$2$&$3$&$f$&&&$g$&&$f_?$&$f$&$f_?$&\\\hline
770$3$&$11$&&&&$f$&&$g$&&&\\\hline
771$3$&$2$&&$g$&$g$&$f$&$g$&$g_?$&&$g_?$&$g$\\\hline
772$5$&$11$&&&&&&$g$&&&$f$\\\hline
773$5$&$2$&&$g$&$g$&&$g$&$g_?$&&$g_?$&$fg$\\\hline
774$5$&$3$&&&&$g$&&&&&$f$\\\hline
775\end{tabular}\end{minipage}
776
777
778
779\noindent\begin{minipage}{\textwidth}
780\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=31A}$}\\
781\begin{tabular}{|cc|ccccccc|}\hline
782$P$&$Q$&713A&713B&713C&713D&713E&713F&713G\\\hline
783$({x + {4}},11) 784$&$({x^{2} + x + 1},2) 785$&&$g$&&&$fg_?$&&\\\hline
786$({x + {4}},11) 787$&$({x + {5}},29) 788$&&&&&$f$&&$g$\\\hline
789$({x + {4}},11) 790$&$({x + {2}},5) 791$&&&&&$f$&&\\\hline
792$({x + {8}},11) 793$&$({x^{2} + x + 1},2) 794$&&$g$&&&$g_?$&&\\\hline
795$({x + {8}},11) 796$&$({x + {5}},29) 797$&&&&&&&$g$\\\hline
798$({x + {8}},11) 799$&$({x + {2}},5) 800$&&&&&&&\\\hline
801$({x + {10}},89) 802$&$({x + {7}},11) 803$&&&&$g$&&&$f$\\\hline
804$({x + {10}},89) 805$&$({x^{2} + x + 1},2) 806$&&$g$&&&$g_?$&&$f$\\\hline
807$({x + {10}},89) 808$&$({x + {5}},29) 809$&&&&&&&$fg$\\\hline
810$({x + {10}},89) 811$&$({x + {2}},5) 812$&&&&&&&$f$\\\hline
813$({x + {80}},89) 814$&$({x + {7}},11) 815$&&&&$g$&$f_?$&&\\\hline
816$({x + {80}},89) 817$&$({x^{2} + x + 1},2) 818$&&$g$&&&$f_?g_?$&&\\\hline
819$({x + {80}},89) 820$&$({x + {5}},29) 821$&&&&&$f_?$&&$g$\\\hline
822$({x + {80}},89) 823$&$({x + {2}},5) 824$&&&&&$f_?$&&\\\hline
825\end{tabular}\end{minipage}
826
827
828
829
830
831\noindent\begin{minipage}{\textwidth}
832\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=43A}$}\\
833\begin{tabular}{|cc|cccccc|}\hline
834$P$&$Q$&731A&731B&731C&731D&731E&731F\\\hline
835$2$&$3$&$g$&$f$&&&$f_?$&$f_?g_?$\\\hline
836$2$&$5$&&$f$&$g$&&$f_?$&$f_?$\\\hline
837$2$&$7$&&$f$&&&$f_?$&$f_?g$\\\hline
838$3$&$5$&&&$g$&&&$f_?$\\\hline
839$3$&$7$&&&&&&$f_?g$\\\hline
840$5$&$3$&$g$&&&$f$&&$g_?$\\\hline
841$5$&$7$&&&&$f$&&$g$\\\hline
842\end{tabular}\end{minipage}
843
844\noindent\begin{minipage}{\textwidth}
845\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=43B}$}\\
846\begin{tabular}{|cc|cccccc|}\hline
847$P$&$Q$&731A&731B&731C&731D&731E&731F\\\hline
848$2$&$({x + {5}},23) 849$&&$f$&&$g$&$f_?$&$f_?$\\\hline
850$2$&$({x + {272}},521) 851$&&$f$&&&$f_?g$&$f_?$\\\hline
852$2$&$({x + {4}},7) 853$&&$f$&&&$f_?$&$f_?$\\\hline
854$3$&$({x + {5}},23) 855$&&&&$g$&&$f_?$\\\hline
856$3$&$({x + {272}},521) 857$&&&&&$g$&$f_?$\\\hline
858$3$&$({x + {4}},7) 859$&&&&&&$f_?$\\\hline
860$5$&$({x + {5}},23) 861$&&&&$fg$&&\\\hline
862$5$&$({x + {272}},521) 863$&&&&$f$&$g$&\\\hline
864$5$&$({x + {4}},7) 865$&&&&$f$&&\\\hline
866\end{tabular}\end{minipage}
867
868
869
870\noindent\begin{minipage}{\textwidth}
871\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=67A}$}\\
872\begin{tabular}{|cc|cccccc|}\hline
873$P$&$Q$&737A&737B&737C&737D&737E&737F\\\hline
874$3$&$2$&$fg$&$g$&$g_?$&$g$&$g_?$&\\\hline
875$5$&$2$&$g$&$g$&$g_?$&$g$&$g_?$&$f$\\\hline
876$61$&$2$&$g$&$g$&$fg_?$&$g$&$g_?$&\\\hline
877\end{tabular}\end{minipage}
878
879\noindent\begin{minipage}{\textwidth}
880\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=67B}$}\\
881\begin{tabular}{|cc|cccccc|}\hline
882$P$&$Q$&737A&737B&737C&737D&737E&737F\\\hline
883$3$&$({x + {4}},11) 884$&$f$&&&&&$g$\\\hline
885$3$&$({x + {8}},11) 886$&$f$&&&&&\\\hline
887$3$&$({x^{2} + x + {12}},13) 888$&$f$&&&&$g$&\\\hline
889$5$&$({x + {4}},11) 890$&&&&&&$fg$\\\hline
891$5$&$({x + {8}},11) 892$&&&&&&$f$\\\hline
893$5$&$({x^{2} + x + {12}},13) 894$&&&&&$g$&$f$\\\hline
895$61$&$({x + {4}},11) 896$&&&$f$&&&$g$\\\hline
897$61$&$({x + {8}},11) 898$&&&$f$&&&\\\hline
899$61$&$({x^{2} + x + {12}},13) 900$&&&$f$&&$g$&\\\hline
901\end{tabular}\end{minipage}
902
903\noindent\begin{minipage}{\textwidth}
904\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=67C}$}\\
905\begin{tabular}{|cc|cccccc|}\hline
906$P$&$Q$&737A&737B&737C&737D&737E&737F\\\hline
907$3$&$({x + {65}},139) 908$&$f$&&&$g$&&\\\hline
909$3$&$({x + {77}},139) 910$&$f$&&$g$&&&\\\hline
911$5$&$({x + {65}},139) 912$&&&&$g$&&$f$\\\hline
913$5$&$({x + {77}},139) 914$&&&$g$&&&$f$\\\hline
915$61$&$({x + {65}},139) 916$&&&$f$&$g$&&\\\hline
917$61$&$({x + {77}},139) 918$&&&$fg$&&&\\\hline
919\end{tabular}\end{minipage}
920
921
922
923\noindent\begin{minipage}{\textwidth}
924\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=41A}$}  {\small (char 2 primes for $g$ omitted)}\\
925\begin{tabular}{|cc|ccccc|}\hline
926$P$&$Q$&779A&779B&779C&779D&779E\\\hline
927$2$&$({x + {2}},5) 928$&$f$&$f$&$f$&$f_?$&$f_?g_?$\\\hline
929$2$&$({x + {171}},607) 930$&$f$&$fg$&$f$&$f_?$&$f_?$\\\hline
931$2$&$({x + {299}},929) 932$&$f$&$f$&$f$&$f_?$&$f_?g$\\\hline
933$3$&$({x + {2}},5) 934$&&&&$f$&$g_?$\\\hline
935$3$&$({x + {171}},607) 936$&&$g$&&$f$&\\\hline
937$3$&$({x + {299}},929) 938$&&&&$f$&$g$\\\hline
939\end{tabular}\end{minipage}
940
941
942
943\noindent\begin{minipage}{\textwidth}
944\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=71A}$}\\
945\begin{tabular}{|cc|ccccccc|}\hline
946$P$&$Q$&781A&781B&781C&781D&781E&781F&781G\\\hline
947$23$&$({x^{3} + x + 1},2) 948$&&&&&$g_?$&&$fg_?$\\\hline
949$23$&$({x},3) 950$&$g$&$g$&$g$&$g$&&&$f$\\\hline
951$23$&$({x + {2}},5) 952$&&&&$g$&&&$f$\\\hline
953$23$&$({x + {20}},53) 954$&&&&&&$g$&$f$\\\hline
955$3$&$({x^{3} + x + 1},2) 956$&&&$f$&&$g_?$&&$f_?g_?$\\\hline
957$3$&$({x + {2}},5) 958$&&&$f$&$g$&&&$f_?$\\\hline
959$3$&$({x + {20}},53) 960$&&&$f$&&&$g$&$f_?$\\\hline
961$5$&$({x^{3} + x + 1},2) 962$&&&&$f$&$g_?$&&$g_?$\\\hline
963$5$&$({x},3) 964$&$g$&$g$&$g$&$fg$&&&\\\hline
965$5$&$({x + {20}},53) 966$&&&&$f$&&$g$&\\\hline
967\end{tabular}\end{minipage}
968
969\noindent\begin{minipage}{\textwidth}
970\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=71B}$}\\
971\begin{tabular}{|cc|ccccccc|}\hline
972$P$&$Q$&781A&781B&781C&781D&781E&781F&781G\\\hline
973$23$&$({x^{3} + x^{2} + 1},2) 974$&&&&&$g_?$&&$fg_?$\\\hline
975$23$&$({x},3) 976$&$g$&$g$&$g$&$g$&&&$f$\\\hline
977$23$&$({x + {27}},61) 978$&&&&&&$g_?$&$f$\\\hline
979$23$&$({x + {4}},7) 980$&&&&&&&$f$\\\hline
981$3$&$({x^{3} + x^{2} + 1},2) 982$&&&$f$&&$g_?$&&$f_?g_?$\\\hline
983$3$&$({x + {27}},61) 984$&&&$f$&&&$g_?$&$f_?$\\\hline
985$3$&$({x + {4}},7) 986$&&&$f$&&&&$f_?$\\\hline
987$5$&$({x^{3} + x^{2} + 1},2) 988$&&&&$f$&$g_?$&&$g_?$\\\hline
989$5$&$({x},3) 990$&$g$&$g$&$g$&$fg$&&&\\\hline
991$5$&$({x + {27}},61) 992$&&&&$f$&&$g_?$&\\\hline
993$5$&$({x + {4}},7) 994$&&&&$f$&&&\\\hline
995\end{tabular}\end{minipage}
996
997
998\noindent\begin{minipage}{\textwidth}
999\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=47A}$}\\
1000\begin{tabular}{|cc|ccccccc|}\hline
1001$P$&$Q$&799A&799B&799C&799D&799E&799F&799G\\\hline
1002$2$&$({x + {20}},23) 1003$&$f$&&$f$&$f_?$&&$f_?$&$f$\\\hline
1004$2$&$({x + 1},3) 1005$&$f$&&$f$&$f_?g_?$&&$f_?$&$fg$\\\hline
1006$2$&$({x + {3495}},43991) 1007$&$f$&&$f$&$f_?g$&&$f_?$&$f$\\\hline
1008$2$&$({x + {55}},937) 1009$&$f$&&$f$&$f_?$&$g$&$f_?$&$f$\\\hline
1010$3$&$({x + {20}},23) 1011$&&&&&$f$&$f$&\\\hline
1012$3$&$({x + {3495}},43991) 1013$&&&&$g$&$f$&$f$&\\\hline
1014$3$&$({x + {55}},937) 1015$&&&&&$fg$&$f$&\\\hline
1016\end{tabular}\end{minipage}
1017
1018
1019
1020\noindent\begin{minipage}{\textwidth}
1021\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=73A}$}\\
1022\begin{tabular}{|cc|cccccc|}\hline
1023$P$&$Q$&803A&803B&803C&803D&803E&803F\\\hline
1024$13$&$2$&&&$f$&$g_?$&&\\\hline
1025$13$&$5$&&&$f$&$g$&&\\\hline
1026$13$&$7$&&&$f$&&&$g$\\\hline
1027$2$&$5$&&$f$&&$f_?g$&$f$&\\\hline
1028$2$&$7$&&$f$&&&$f$&$g$\\\hline
1029$3$&$2$&&&$f_?$&$g_?$&&\\\hline
1030$3$&$5$&&&$f_?$&$g$&&\\\hline
1031$3$&$7$&&&$f_?$&&&$g$\\\hline
1032$5$&$2$&&&&$g_?$&&\\\hline
1033$5$&$7$&&&&&&$g$\\\hline
1034$7$&$2$&&&&$g_?$&&$f$\\\hline
1035$7$&$5$&&&&$g$&&$f$\\\hline
1036\end{tabular}\end{minipage}
1037
1038\noindent\begin{minipage}{\textwidth}
1039\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=73B}$}\\
1040\begin{tabular}{|cc|cccccc|}\hline
1041$P$&$Q$&803A&803B&803C&803D&803E&803F\\\hline
1042$13$&$({x + {14}},23) 1043$&&&$f$&&&$g$\\\hline
1044$13$&$({x},3) 1045$&&$g$&$f$&&&\\\hline
1046$13$&$({x + {15}},79) 1047$&&&$f$&$g$&&\\\hline
1048$2$&$({x + {14}},23) 1049$&&$f$&&&$f$&$g$\\\hline
1050$2$&$({x},3) 1051$&&$fg$&&&$f$&\\\hline
1052$2$&$({x + {15}},79) 1053$&&$f$&&$g$&$f$&\\\hline
1054$3$&$({x + {14}},23) 1055$&&&$f_?$&&&$g$\\\hline
1056$3$&$({x + {15}},79) 1057$&&&$f_?$&$g$&&\\\hline
1058$5$&$({x + {14}},23) 1059$&&&&&&$g$\\\hline
1060$5$&$({x},3) 1061$&&$g$&&&&\\\hline
1062$5$&$({x + {15}},79) 1063$&&&&$g$&&\\\hline
1064$7$&$({x + {14}},23) 1065$&&&&&&$fg$\\\hline
1066$7$&$({x},3) 1067$&&$g$&&&&$f$\\\hline
1068$7$&$({x + {15}},79) 1069$&&&&$g$&&$f$\\\hline
1070\end{tabular}\end{minipage}
1071
1072\noindent\begin{minipage}{\textwidth}
1073\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=73C}$}\\
1074\begin{tabular}{|cc|cccccc|}\hline
1075$P$&$Q$&803A&803B&803C&803D&803E&803F\\\hline
1076$13$&$({x + {100}},109) 1077$&&&$f$&&$g$&\\\hline
1078$13$&$({x + {15}},181) 1079$&&&$fg$&&&\\\hline
1080$2$&$({x + {100}},109) 1081$&&$f$&&&$fg$&\\\hline
1082$2$&$({x + {15}},181) 1083$&&$f$&$g$&&$f$&\\\hline
1084$3$&$({x + {100}},109) 1085$&&&$f_?$&&$g$&\\\hline
1086$3$&$({x + {15}},181) 1087$&&&$f_?g$&&&\\\hline
1088$5$&$({x + {100}},109) 1089$&&&&&$g$&\\\hline
1090$5$&$({x + {15}},181) 1091$&&&$g$&&&\\\hline
1092$7$&$({x + {100}},109) 1093$&&&&&$g$&$f$\\\hline
1094$7$&$({x + {15}},181) 1095$&&&$g$&&&$f$\\\hline
1096\end{tabular}\end{minipage}
1097
1098
1099
1100\noindent\begin{minipage}{\textwidth}
1101\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=43A}$}\\
1102\begin{tabular}{|cc|cccccc|}\hline
1103$P$&$Q$&817A&817B&817C&817D&817E&817F\\\hline
1104$3$&$11$&&&$f_?$&&&$g_?$\\\hline
1105$3$&$2$&$g$&$g$&$f_?$&$g_?$&&$g_?$\\\hline
1106$43$&$11$&&&&&&$fg_?$\\\hline
1107$43$&$2$&$g$&$g$&&$g_?$&&$fg_?$\\\hline
1108$43$&$3$&&&&$g$&&$f$\\\hline
1109$5$&$11$&$f$&&&&&$g_?$\\\hline
1110$5$&$2$&$fg$&$g$&&$g_?$&&$g_?$\\\hline
1111$5$&$3$&$f$&&&$g$&&\\\hline
1112\end{tabular}\end{minipage}
1113
1114\noindent\begin{minipage}{\textwidth}
1115\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=43B}$}\\
1116\begin{tabular}{|cc|cccccc|}\hline
1117$P$&$Q$&817A&817B&817C&817D&817E&817F\\\hline
1118$3$&$({x + {11}},17) 1119$&&&$f_?g_?$&&&\\\hline
1120$3$&$({x},2) 1121$&$g$&$g$&$f_?$&$g_?$&&$g_?$\\\hline
1122$3$&$({x + {4}},7) 1123$&&&$f_?$&&&\\\hline
1124$3$&$({x + {70}},79) 1125$&&&$f_?$&&$g$&\\\hline
1126$43$&$({x + {11}},17) 1127$&&&$g_?$&&&$f$\\\hline
1128$43$&$({x},2) 1129$&$g$&$g$&&$g_?$&&$fg_?$\\\hline
1130$43$&$({x + {4}},7) 1131$&&&&&&$f$\\\hline
1132$43$&$({x + {70}},79) 1133$&&&&&$g$&$f$\\\hline
1134$5$&$({x + {11}},17) 1135$&$f$&&$g_?$&&&\\\hline
1136$5$&$({x},2) 1137$&$fg$&$g$&&$g_?$&&$g_?$\\\hline
1138$5$&$({x + {4}},7) 1139$&$f$&&&&&\\\hline
1140$5$&$({x + {70}},79) 1141$&$f$&&&&$g$&\\\hline
1142\end{tabular}\end{minipage}
1143
1144
1145
1146\noindent\begin{minipage}{\textwidth}
1147\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=37A}$}\\
1148\begin{tabular}{|cc|cccccccc|}\hline
1149$P$&$Q$&851A&851B&851C&851D&851E&851F&851G&851H\\\hline
1150$({x + {8}},11) 1151$&$13$&&&&&&$g$&&\\\hline
1152$({x + {8}},11) 1153$&$2$&$g$&&&$g$&$g_?$&&&$g$\\\hline
1154$({x^{2} + x + 1},2) 1155$&$11$&&&&&$g$&$f_?$&$f_?$&\\\hline
1156$({x^{2} + x + 1},2) 1157$&$13$&&&&&&$f_?g$&$f_?$&\\\hline
1158$({x + {19}},31) 1159$&$11$&&&&$f$&$g$&&&\\\hline
1160$({x + {19}},31) 1161$&$13$&&&&$f$&&$g$&&\\\hline
1162$({x + {19}},31) 1163$&$2$&$g$&&&$fg$&$g_?$&&&$g$\\\hline
1164$({x + {360}},379) 1165$&$11$&&&&&$fg$&&&\\\hline
1166$({x + {360}},379) 1167$&$13$&&&&&$f$&$g$&&\\\hline
1168$({x + {360}},379) 1169$&$2$&$g$&&&$g$&$fg_?$&&&$g$\\\hline
1170\end{tabular}\end{minipage}
1171
1172\noindent\begin{minipage}{\textwidth}
1173\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=37B}$}\\
1174\begin{tabular}{|cc|cccccccc|}\hline
1175$P$&$Q$&851A&851B&851C&851D&851E&851F&851G&851H\\\hline
1176$({x + {8}},11) 1177$&$2$&$g$&&&$g$&$g_?$&&&$g$\\\hline
1178$({x + {8}},11) 1179$&$3$&&$g_?$&&&&&$g$&\\\hline
1180$({x + {8}},11) 1181$&$5$&$g$&&&&&&&\\\hline
1182$({x^{2} + x + 1},2) 1183$&$3$&&$g_?$&&&&$f_?$&$f_?g$&\\\hline
1184$({x^{2} + x + 1},2) 1185$&$5$&$g$&&&&&$f_?$&$f_?$&\\\hline
1186$({x + {19}},31) 1187$&$2$&$g$&&&$fg$&$g_?$&&&$g$\\\hline
1188$({x + {19}},31) 1189$&$3$&&$g_?$&&$f$&&&$g$&\\\hline
1190$({x + {19}},31) 1191$&$5$&$g$&&&$f$&&&&\\\hline
1192$({x + {360}},379) 1193$&$2$&$g$&&&$g$&$fg_?$&&&$g$\\\hline
1194$({x + {360}},379) 1195$&$3$&&$g_?$&&&$f$&&$g$&\\\hline
1196$({x + {360}},379) 1197$&$5$&$g$&&&&$f$&&&\\\hline
1198\end{tabular}\end{minipage}
1199
1200
1201
1202\noindent\begin{minipage}{\textwidth}
1203\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=79A}$}\\
1204\begin{tabular}{|cc|cccccccc|}\hline
1205$P$&$Q$&869A&869B&869C&869D&869E&869F&869G&869H\\\hline
1206$2$&$5$&&&$f$&&$g_?$&&&$f_?$\\\hline
1207$2$&$7$&&&$f$&&&&$g$&$f_?$\\\hline
1208$3$&$2$&$fg$&&&$g$&&$f$&$g_?$&$g_?$\\\hline
1209$3$&$5$&$f$&&&&$g_?$&$f$&&\\\hline
1210$3$&$7$&$f$&&&&&$f$&$g$&\\\hline
1211$5$&$2$&$g$&&&$g$&&&$fg_?$&$g_?$\\\hline
1212$5$&$7$&&&&&&&$fg$&\\\hline
1213$7$&$2$&$g$&&&$g$&&&$fg_?$&$g_?$\\\hline
1214$7$&$5$&&&&&$g_?$&&$f$&\\\hline
1215\end{tabular}\end{minipage}
1216
1217\noindent\begin{minipage}{\textwidth}
1218\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=79B}$}\\
1219\begin{tabular}{|cc|cccccccc|}\hline
1220$P$&$Q$&869A&869B&869C&869D&869E&869F&869G&869H\\\hline
1221$2$&$({x + {10}},13) 1222$&&&$f$&&&&&$f_?$\\\hline
1223$2$&$({x + {1637}},1861) 1224$&&&$f$&&&&&$f_?g$\\\hline
1225$2$&$({x + {3680}},5927) 1226$&&&$f$&&&$g$&&$f_?$\\\hline
1227$2$&$({x + {55}},61) 1228$&&&$f$&&&&&$f_?g$\\\hline
1229$3$&$({x + {10}},13) 1230$&$f$&&&&&$f$&&\\\hline
1231$3$&$({x + {1637}},1861) 1232$&$f$&&&&&$f$&&$g$\\\hline
1233$3$&$({x + 1},2) 1234$&$fg$&&&$g$&&$f$&$g_?$&$g_?$\\\hline
1235$3$&$({x + {3680}},5927) 1236$&$f$&&&&&$fg$&&\\\hline
1237$3$&$({x + {55}},61) 1238$&$f$&&&&&$f$&&$g$\\\hline
1239$5$&$({x + {10}},13) 1240$&&&&&&&$f$&\\\hline
1241$5$&$({x + {1637}},1861) 1242$&&&&&&&$f$&$g$\\\hline
1243$5$&$({x + 1},2) 1244$&$g$&&&$g$&&&$fg_?$&$g_?$\\\hline
1245$5$&$({x + {3680}},5927) 1246$&&&&&&$g$&$f$&\\\hline
1247$5$&$({x + {55}},61) 1248$&&&&&&&$f$&$g$\\\hline
1249$7$&$({x + {10}},13) 1250$&&&&&&&$f$&\\\hline
1251$7$&$({x + {1637}},1861) 1252$&&&&&&&$f$&$g$\\\hline
1253$7$&$({x + 1},2) 1254$&$g$&&&$g$&&&$fg_?$&$g_?$\\\hline
1255$7$&$({x + {3680}},5927) 1256$&&&&&&$g$&$f$&\\\hline
1257$7$&$({x + {55}},61) 1258$&&&&&&&$f$&$g$\\\hline
1259\end{tabular}\end{minipage}
1260
1261
1262
1263\noindent\begin{minipage}{\textwidth}
1264\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=47A}$}\\
1265\begin{tabular}{|cc|cccc|}\hline
1266$P$&$Q$&893A&893B&893C&893D\\\hline
1267$17$&$({x + {304}},1373) 1268$&$g$&&&$f$\\\hline
1269$17$&$({x + {37}},139) 1270$&&&&$fg$\\\hline
1271$17$&$({x^{4} + x^{3} + x^{2} + x + 1},2) 1272$&&$g_?$&$g$&$f$\\\hline
1273$17$&$({x + {20}},23) 1274$&&&&$f$\\\hline
1275$17$&$({x + 1},3) 1276$&&&&$fg$\\\hline
1277$17$&$({x + {2}},7) 1278$&$g_?$&&&$f$\\\hline
1279$3$&$({x + {304}},1373) 1280$&$g$&$f$&&\\\hline
1281$3$&$({x + {37}},139) 1282$&&$f$&&$g$\\\hline
1283$3$&$({x^{4} + x^{3} + x^{2} + x + 1},2) 1284$&&$fg_?$&$g$&\\\hline
1285$3$&$({x + {20}},23) 1286$&&$f$&&\\\hline
1287$3$&$({x + {2}},7) 1288$&$g_?$&$f$&&\\\hline
1289$5$&$({x + {304}},1373) 1290$&$g$&$f$&&\\\hline
1291$5$&$({x + {37}},139) 1292$&&$f$&&$g$\\\hline
1293$5$&$({x^{4} + x^{3} + x^{2} + x + 1},2) 1294$&&$fg_?$&$g$&\\\hline
1295$5$&$({x + {20}},23) 1296$&&$f$&&\\\hline
1297$5$&$({x + 1},3) 1298$&&$f$&&$g$\\\hline
1299$5$&$({x + {2}},7) 1300$&$g_?$&$f$&&\\\hline
1301\end{tabular}\end{minipage}
1302
1303
1304
1305\noindent\begin{minipage}{\textwidth}
1306\vspace{.2in}\noindent{\large $\mathbf{f=29A, \, g=31A}$}\\
1307\begin{tabular}{|cc|cccccc|}\hline
1308$P$&$Q$&899A&899B&899C&899D&899E&899F\\\hline
1309$({x + {52}},113) 1310$&$({x^{2} + x + 1},2) 1311$&&&$g$&&$g_?$&$f$\\\hline
1312$({x + {52}},113) 1313$&$({x + {12}},31) 1314$&&&&&&$fg$\\\hline
1315$({x + {52}},113) 1316$&$({x + {2}},5) 1317$&&&&$g$&&$f$\\\hline
1318$({x + {52}},113) 1319$&$({x + {43}},61) 1320$&&&&$g$&&$f$\\\hline
1321$({x + {41}},47) 1322$&$({x^{2} + x + 1},2) 1323$&&&$g$&&$fg_?$&\\\hline
1324$({x + {41}},47) 1325$&$({x + {12}},31) 1326$&&&&&$f$&$g$\\\hline
1327$({x + {41}},47) 1328$&$({x + {2}},5) 1329$&&&&$g$&$f$&\\\hline
1330$({x + {41}},47) 1331$&$({x + {43}},61) 1332$&&&&$g$&$f$&\\\hline
1333$({x^{2} + {2}x + {4}},5) 1334$&$({x^{2} + x + 1},2) 1335$&&&$g$&&$fg_?$&\\\hline
1336$({x^{2} + {2}x + {4}},5) 1337$&$({x + {12}},31) 1338$&&&&&$f$&$g$\\\hline
1339$({x^{2} + {2}x + {4}},5) 1340$&$({x + {43}},61) 1341$&&&&$g$&$f$&\\\hline
1342$({x + {4}},7) 1343$&$({x^{2} + x + 1},2) 1344$&&&$g$&&$g_?$&\\\hline
1345$({x + {4}},7) 1346$&$({x + {12}},31) 1347$&&&&&&$g$\\\hline
1348$({x + {4}},7) 1349$&$({x + {2}},5) 1350$&&&&$g$&&\\\hline
1351$({x + {4}},7) 1352$&$({x + {43}},61) 1353$&&&&$g$&&\\\hline
1354\end{tabular}\end{minipage}
1355
1356
1357
1358\noindent\begin{minipage}{\textwidth}
1359\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=53A}$}\\
1360\begin{tabular}{|cc|cccccccccc|}\hline
1361$P$&$Q$&901A&901B&901C&901D&901E&901F&901G&901H&901I&901J\\\hline
1362$2$&$3$&$f$&&$fg$&&$g$&&$f_?$&&$f_?$&$f_?$\\\hline
1363$2$&$5$&$fg$&&$f$&&&&$f_?$&&$f_?$&$f_?$\\\hline
1364$2$&$7$&$f$&&$f$&&&&$f_?$&&$f_?$&$f_?g$\\\hline
1365$3$&$5$&$g$&&&&&$f$&&&&$f_?$\\\hline
1366$3$&$7$&&&&&&$f$&&&&$f_?g$\\\hline
1367$5$&$3$&&&$g$&&$fg$&&&&&\\\hline
1368$5$&$7$&&&&&$f$&&&&&$g$\\\hline
1369\end{tabular}\end{minipage}
1370
1371\noindent\begin{minipage}{\textwidth}
1372\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=53B}$}\\
1373\begin{tabular}{|cc|cccccccccc|}\hline
1374$P$&$Q$&901A&901B&901C&901D&901E&901F&901G&901H&901I&901J\\\hline
1375$2$&$({x + {47}},113) 1376$&$f$&&$f$&&&&$f_?g$&&$f_?$&$f_?$\\\hline
1377$2$&$({x + {10}},13) 1378$&$f$&&$f$&&&&$f_?$&&$f_?$&$f_?$\\\hline
1379$2$&$({x + {2078}},4139) 1380$&$f$&&$f$&&&&$f_?$&&$f_?g$&$f_?$\\\hline
1381$2$&$({x + {3}},5) 1382$&$f$&&$f$&&&$g$&$f_?$&&$f_?$&$f_?$\\\hline
1383$3$&$({x + {47}},113) 1384$&&&&&&$f$&$g$&&&$f_?$\\\hline
1385$3$&$({x + {10}},13) 1386$&&&&&&$f$&&&&$f_?$\\\hline
1387$3$&$({x + {2078}},4139) 1388$&&&&&&$f$&&&$g$&$f_?$\\\hline
1389$3$&$({x + {3}},5) 1390$&&&&&&$fg$&&&&$f_?$\\\hline
1391$5$&$({x + {47}},113) 1392$&&&&&$f$&&$g$&&&\\\hline
1393$5$&$({x + {10}},13) 1394$&&&&&$f$&&&&&\\\hline
1395$5$&$({x + {2078}},4139) 1396$&&&&&$f$&&&&$g$&\\\hline
1397\end{tabular}\end{minipage}
1398
1399
1400
1401\noindent\begin{minipage}{\textwidth}
1402\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=83A}$}\\
1403\begin{tabular}{|cc|ccccccc|}\hline
1404$P$&$Q$&913A&913B&913C&913D&913E&913F&913G\\\hline
1405$13$&$3$&&&&$g$&$fg$&&\\\hline
1406$13$&$5$&&&&$g$&$f$&&\\\hline
1407$2$&$3$&&$f$&&$fg$&$g$&&\\\hline
1408$2$&$5$&&$f$&&$fg$&&&\\\hline
1409$3$&$5$&&&&$g$&$f$&&$f_?$\\\hline
1410$5$&$3$&&&&$g$&$g$&&\\\hline
1411\end{tabular}\end{minipage}
1412
1413\noindent\begin{minipage}{\textwidth}
1414\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=83B}$}\\
1415\begin{tabular}{|cc|ccccccc|}\hline
1416$P$&$Q$&913A&913B&913C&913D&913E&913F&913G\\\hline
1417$13$&$({x + {5727}},10007) 1418$&&&$g$&&$f$&&\\\hline
1419$13$&$({x + {31}},197) 1420$&&&&&$f$&&$g$\\\hline
1421$13$&$({x + {5}},37) 1422$&&&&&$f$&$g$&\\\hline
1423$13$&$({x + {69}},379) 1424$&&&&&$f$&&$g$\\\hline
1425$13$&$({x + {38}},41) 1426$&&&&&$f$&&\\\hline
1427$13$&$({x + 1},5) 1428$&$g$&&&&$f$&&\\\hline
1429$2$&$({x + {5727}},10007) 1430$&&$f$&$g$&$f$&&&\\\hline
1431$2$&$({x + {31}},197) 1432$&&$f$&&$f$&&&$g$\\\hline
1433$2$&$({x + {5}},37) 1434$&&$f$&&$f$&&$g$&\\\hline
1435$2$&$({x + {69}},379) 1436$&&$f$&&$f$&&&$g$\\\hline
1437$2$&$({x + {38}},41) 1438$&&$f$&&$f$&&&\\\hline
1439$2$&$({x + 1},5) 1440$&$g$&$f$&&$f$&&&\\\hline
1441$3$&$({x + {5727}},10007) 1442$&&&$g$&&$f$&&$f_?$\\\hline
1443$3$&$({x + {31}},197) 1444$&&&&&$f$&&$f_?g$\\\hline
1445$3$&$({x + {5}},37) 1446$&&&&&$f$&$g$&$f_?$\\\hline
1447$3$&$({x + {69}},379) 1448$&&&&&$f$&&$f_?g$\\\hline
1449$3$&$({x + {38}},41) 1450$&&&&&$f$&&$f_?$\\\hline
1451$3$&$({x + 1},5) 1452$&$g$&&&&$f$&&$f_?$\\\hline
1453$5$&$({x + {5727}},10007) 1454$&&&$g$&&&&\\\hline
1455$5$&$({x + {31}},197) 1456$&&&&&&&$g$\\\hline
1457$5$&$({x + {5}},37) 1458$&&&&&&$g$&\\\hline
1459$5$&$({x + {69}},379) 1460$&&&&&&&$g$\\\hline
1461$5$&$({x + {38}},41) 1462$&&&&&&&\\\hline
1463\end{tabular}\end{minipage}
1464
1465
1466
1467\noindent\begin{minipage}{\textwidth}
1468\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=41A}$}  {\small (char 2 primes for $g$ omitted}\\
1469\begin{tabular}{|cc|ccccc|}\hline
1470$P$&$Q$&943A&943B&943C&943D&943E\\\hline
1471$({x + {8}},11) 1472$&$({x + {7}},13) 1473$&&&$g$&&\\\hline
1474$({x + {8}},11) 1475$&$({x + {10}},37) 1476$&&&$g$&&\\\hline
1477$({x + {8}},11) 1478$&$({x + {2}},5) 1479$&&&&&\\\hline
1480$({x + {8}},11) 1481$&$({x + {31}},67) 1482$&&&&&$g$\\\hline
1483$({x + {124}},151) 1484$&$({x + {7}},13) 1485$&&&$g$&&$f$\\\hline
1486$({x + {124}},151) 1487$&$({x + {10}},37) 1488$&&&$g$&&$f$\\\hline
1489$({x + {124}},151) 1490$&$({x + {2}},5) 1491$&&&&&$f$\\\hline
1492$({x + {124}},151) 1493$&$({x + {31}},67) 1494$&&&&&$fg$\\\hline
1495$({x + {13}},31) 1496$&$({x + {7}},13) 1497$&&&$g$&$f_?$&\\\hline
1498$({x + {13}},31) 1499$&$({x + {10}},37) 1500$&&&$g$&$f_?$&\\\hline
1501$({x + {13}},31) 1502$&$({x + {2}},5) 1503$&&&&$f_?$&\\\hline
1504$({x + {13}},31) 1505$&$({x + {31}},67) 1506$&&&&$f_?$&$g$\\\hline
1507$({x + {34}},59) 1508$&$({x + {7}},13) 1509$&&&$g$&$f$&\\\hline
1510$({x + {34}},59) 1511$&$({x + {10}},37) 1512$&&&$g$&$f$&\\\hline
1513$({x + {34}},59) 1514$&$({x + {2}},5) 1515$&&&&$f$&\\\hline
1516\$({x + {34}},