Properties

Label 240.3.c.e
Level $240$
Weight $3$
Character orbit 240.c
Analytic conductor $6.540$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 34 x^{10} + 305 x^{8} + 616 x^{6} + 305 x^{4} + 34 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{7} q^{5} -\beta_{9} q^{7} + ( 1 - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{7} q^{5} -\beta_{9} q^{7} + ( 1 - \beta_{4} ) q^{9} + ( -\beta_{3} + \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{13} + ( -1 - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{10} ) q^{15} + ( \beta_{5} + \beta_{7} + \beta_{8} ) q^{17} + ( -1 + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{10} - \beta_{11} ) q^{19} + ( -\beta_{3} + 2 \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{21} + ( 3 \beta_{1} + \beta_{2} + \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{23} + ( 2 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{9} ) q^{25} + ( \beta_{1} + 3 \beta_{5} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{27} + ( 1 - 3 \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - \beta_{10} + \beta_{11} ) q^{29} + ( 5 - \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{31} + ( -\beta_{1} + \beta_{2} - 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{33} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - \beta_{10} + \beta_{11} ) q^{35} + ( 5 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{37} + ( 9 + 3 \beta_{3} + \beta_{4} - \beta_{6} + 5 \beta_{7} - 5 \beta_{8} ) q^{39} + ( -1 + \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{41} + ( -9 \beta_{1} + 3 \beta_{2} ) q^{43} + ( -7 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} - 4 \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{45} + ( -3 \beta_{1} - \beta_{2} - 5 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} ) q^{47} + ( -18 - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} - 3 \beta_{10} + 3 \beta_{11} ) q^{49} + ( -3 - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{51} + ( 12 \beta_{1} + 4 \beta_{2} - 2 \beta_{5} + 5 \beta_{7} + 5 \beta_{8} ) q^{53} + ( 6 + 7 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} + 2 \beta_{11} ) q^{55} + ( -3 \beta_{1} - \beta_{2} - 9 \beta_{5} - 5 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{57} + ( -2 + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{59} + ( 11 + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{10} - \beta_{11} ) q^{61} + ( -3 \beta_{1} + 5 \beta_{2} - 3 \beta_{5} - 7 \beta_{7} - 7 \beta_{8} - \beta_{10} - \beta_{11} ) q^{63} + ( -1 - 12 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 7 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{65} + ( 5 \beta_{1} - 3 \beta_{2} - 2 \beta_{5} - 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{67} + ( 25 - 4 \beta_{3} + \beta_{4} + \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{69} + ( 2 \beta_{3} - 2 \beta_{4} - 12 \beta_{7} + 12 \beta_{8} ) q^{71} + ( -14 \beta_{1} + 2 \beta_{2} - 4 \beta_{5} + 2 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{73} + ( -24 + 2 \beta_{1} - 5 \beta_{2} + \beta_{3} - 3 \beta_{4} - 9 \beta_{5} - 3 \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{75} + ( -12 \beta_{1} - 4 \beta_{2} + 12 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} ) q^{77} + ( -39 + 3 \beta_{3} + 5 \beta_{4} - 3 \beta_{6} - \beta_{10} + \beta_{11} ) q^{79} + ( -16 + 7 \beta_{3} - 3 \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{81} + ( 15 \beta_{1} + 5 \beta_{2} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{83} + ( 27 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{9} - 2 \beta_{11} ) q^{85} + ( -\beta_{1} - 5 \beta_{2} + 9 \beta_{5} - 8 \beta_{7} - 8 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{87} + ( 6 \beta_{3} - 6 \beta_{4} - 12 \beta_{7} + 12 \beta_{8} ) q^{89} + ( 8 \beta_{4} - 4 \beta_{10} + 4 \beta_{11} ) q^{91} + ( 3 \beta_{1} + 9 \beta_{2} + 9 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} + 9 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{93} + ( -1 - 12 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + 7 \beta_{4} + 8 \beta_{5} + \beta_{6} + 6 \beta_{8} + \beta_{10} - \beta_{11} ) q^{95} + ( -8 \beta_{1} - 4 \beta_{5} - 4 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{97} + ( 46 - \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{9} + O(q^{10}) \) \( 12 q + 8 q^{9} - 16 q^{15} + 4 q^{21} + 36 q^{25} + 48 q^{31} + 128 q^{39} - 68 q^{45} - 252 q^{49} - 48 q^{51} + 48 q^{55} + 144 q^{61} + 268 q^{69} - 304 q^{75} - 432 q^{79} - 188 q^{81} + 336 q^{85} + 560 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 34 x^{10} + 305 x^{8} + 616 x^{6} + 305 x^{4} + 34 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} - \nu^{10} + 35 \nu^{9} - 35 \nu^{8} + 340 \nu^{7} - 340 \nu^{6} + 956 \nu^{5} - 956 \nu^{4} + 1261 \nu^{3} - 1261 \nu^{2} + 1295 \nu - 647 \)\()/216\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} - \nu^{10} - 35 \nu^{9} - 35 \nu^{8} - 340 \nu^{7} - 340 \nu^{6} - 956 \nu^{5} - 956 \nu^{4} - 1261 \nu^{3} - 1261 \nu^{2} - 1295 \nu - 647 \)\()/72\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - 4 \nu^{10} + 25 \nu^{9} - 138 \nu^{8} + 4 \nu^{7} - 1288 \nu^{6} - 1964 \nu^{5} - 3072 \nu^{4} - 3875 \nu^{3} - 2380 \nu^{2} - 851 \nu - 282 \)\()/36\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{11} - 4 \nu^{10} + 173 \nu^{9} - 138 \nu^{8} + 1628 \nu^{7} - 1288 \nu^{6} + 4028 \nu^{5} - 3072 \nu^{4} + 3641 \nu^{3} - 2380 \nu^{2} + 1289 \nu - 282 \)\()/36\)
\(\beta_{5}\)\(=\)\((\)\( -35 \nu^{10} - 1189 \nu^{8} - 10640 \nu^{6} - 21220 \nu^{4} - 9755 \nu^{2} - 577 \)\()/27\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{11} + 64 \nu^{10} + 99 \nu^{9} + 2172 \nu^{8} + 816 \nu^{7} + 19384 \nu^{6} + 1032 \nu^{5} + 38208 \nu^{4} - 117 \nu^{3} + 17128 \nu^{2} + 219 \nu + 1038 \)\()/18\)
\(\beta_{7}\)\(=\)\((\)\( -137 \nu^{11} - 26 \nu^{10} - 4649 \nu^{9} - 880 \nu^{8} - 41480 \nu^{7} - 7796 \nu^{6} - 81680 \nu^{5} - 14860 \nu^{4} - 36521 \nu^{3} - 5966 \nu^{2} - 2393 \nu - 280 \)\()/36\)
\(\beta_{8}\)\(=\)\((\)\( 137 \nu^{11} - 26 \nu^{10} + 4649 \nu^{9} - 880 \nu^{8} + 41480 \nu^{7} - 7796 \nu^{6} + 81680 \nu^{5} - 14860 \nu^{4} + 36521 \nu^{3} - 5966 \nu^{2} + 2393 \nu - 280 \)\()/36\)
\(\beta_{9}\)\(=\)\((\)\( -655 \nu^{11} - 22205 \nu^{9} - 197572 \nu^{7} - 383900 \nu^{5} - 162043 \nu^{3} - 7337 \nu \)\()/108\)
\(\beta_{10}\)\(=\)\((\)\( -713 \nu^{11} - 167 \nu^{10} - 24191 \nu^{9} - 5665 \nu^{8} - 215732 \nu^{7} - 50492 \nu^{6} - 423692 \nu^{5} - 98884 \nu^{4} - 186485 \nu^{3} - 42923 \nu^{2} - 10235 \nu - 2461 \)\()/72\)
\(\beta_{11}\)\(=\)\((\)\( -733 \nu^{11} + 73 \nu^{10} - 24883 \nu^{9} + 2471 \nu^{8} - 222244 \nu^{7} + 21892 \nu^{6} - 439804 \nu^{5} + 41660 \nu^{4} - 201049 \nu^{3} + 16069 \nu^{2} - 15391 \nu + 491 \)\()/72\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{11} - 4 \beta_{10} + 6 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 18 \beta_{1} + 1\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} - \beta_{10} - 4 \beta_{6} - 9 \beta_{5} + 6 \beta_{4} + 4 \beta_{3} - 12 \beta_{2} - 36 \beta_{1} - 140\)\()/24\)
\(\nu^{3}\)\(=\)\((\)\(64 \beta_{11} + 86 \beta_{10} - 114 \beta_{9} + 105 \beta_{8} - 105 \beta_{7} + 11 \beta_{6} - 75 \beta_{5} + 102 \beta_{4} - 80 \beta_{3} + 16 \beta_{2} - 198 \beta_{1} - 11\)\()/48\)
\(\nu^{4}\)\(=\)\((\)\(6 \beta_{8} + 6 \beta_{7} + 18 \beta_{6} + 45 \beta_{5} - 18 \beta_{4} - 18 \beta_{3} + 36 \beta_{2} + 108 \beta_{1} + 382\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-1466 \beta_{11} - 1744 \beta_{10} + 1950 \beta_{9} - 2649 \beta_{8} + 2649 \beta_{7} - 139 \beta_{6} + 1605 \beta_{5} - 1734 \beta_{4} + 1456 \beta_{3} - 64 \beta_{2} + 3402 \beta_{1} + 139\)\()/48\)
\(\nu^{6}\)\(=\)\((\)\(-305 \beta_{11} + 305 \beta_{10} - 1296 \beta_{8} - 1296 \beta_{7} - 2452 \beta_{6} - 6345 \beta_{5} + 1842 \beta_{4} + 2452 \beta_{3} - 4140 \beta_{2} - 12420 \beta_{1} - 42620\)\()/24\)
\(\nu^{7}\)\(=\)\((\)\(31516 \beta_{11} + 35630 \beta_{10} - 35322 \beta_{9} + 59883 \beta_{8} - 59883 \beta_{7} + 2057 \beta_{6} - 33573 \beta_{5} + 31638 \beta_{4} - 27524 \beta_{3} - 980 \beta_{2} - 64206 \beta_{1} - 2057\)\()/48\)
\(\nu^{8}\)\(=\)\(405 \beta_{11} - 405 \beta_{10} + 1380 \beta_{8} + 1380 \beta_{7} + 2205 \beta_{6} + 5787 \beta_{5} - 1395 \beta_{4} - 2205 \beta_{3} + 3420 \beta_{2} + 10260 \beta_{1} + 34642\)
\(\nu^{9}\)\(=\)\((\)\(-663278 \beta_{11} - 731260 \beta_{10} + 674682 \beta_{9} - 1291845 \beta_{8} + 1291845 \beta_{7} - 33991 \beta_{6} + 697269 \beta_{5} - 607722 \beta_{4} + 539740 \beta_{3} + 44084 \beta_{2} + 1262286 \beta_{1} + 33991\)\()/48\)
\(\nu^{10}\)\(=\)\((\)\(-237761 \beta_{11} + 237761 \beta_{10} - 752976 \beta_{8} - 752976 \beta_{7} - 1116724 \beta_{6} - 2950551 \beta_{5} + 641202 \beta_{4} + 1116724 \beta_{3} - 1657428 \beta_{2} - 4972284 \beta_{1} - 16638620\)\()/24\)
\(\nu^{11}\)\(=\)\((\)\(13822672 \beta_{11} + 15043898 \beta_{10} - 13332606 \beta_{9} + 27258279 \beta_{8} - 27258279 \beta_{7} + 610613 \beta_{6} - 14433285 \beta_{5} + 12050202 \beta_{4} - 10828976 \beta_{3} - 1165280 \beta_{2} - 25370730 \beta_{1} - 610613\)\()/48\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.220185i
0.220185i
0.304307i
0.304307i
0.723561i
0.723561i
1.38205i
1.38205i
3.28615i
3.28615i
4.54164i
4.54164i
0 −2.72256 1.26002i 0 −0.689011 + 4.95230i 0 0.735748i 0 5.82469 + 6.86097i 0
209.2 0 −2.72256 + 1.26002i 0 −0.689011 4.95230i 0 0.735748i 0 5.82469 6.86097i 0
209.3 0 −2.49147 1.67109i 0 4.19906 2.71439i 0 12.7692i 0 3.41489 + 8.32698i 0
209.4 0 −2.49147 + 1.67109i 0 4.19906 + 2.71439i 0 12.7692i 0 3.41489 8.32698i 0
209.5 0 −0.938195 2.84952i 0 −4.88807 + 1.05205i 0 6.81219i 0 −7.23958 + 5.34682i 0
209.6 0 −0.938195 + 2.84952i 0 −4.88807 1.05205i 0 6.81219i 0 −7.23958 5.34682i 0
209.7 0 0.938195 2.84952i 0 4.88807 1.05205i 0 6.81219i 0 −7.23958 5.34682i 0
209.8 0 0.938195 + 2.84952i 0 4.88807 + 1.05205i 0 6.81219i 0 −7.23958 + 5.34682i 0
209.9 0 2.49147 1.67109i 0 −4.19906 + 2.71439i 0 12.7692i 0 3.41489 8.32698i 0
209.10 0 2.49147 + 1.67109i 0 −4.19906 2.71439i 0 12.7692i 0 3.41489 + 8.32698i 0
209.11 0 2.72256 1.26002i 0 0.689011 4.95230i 0 0.735748i 0 5.82469 6.86097i 0
209.12 0 2.72256 + 1.26002i 0 0.689011 + 4.95230i 0 0.735748i 0 5.82469 + 6.86097i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.c.e 12
3.b odd 2 1 inner 240.3.c.e 12
4.b odd 2 1 120.3.c.a 12
5.b even 2 1 inner 240.3.c.e 12
5.c odd 4 2 1200.3.l.y 12
8.b even 2 1 960.3.c.j 12
8.d odd 2 1 960.3.c.k 12
12.b even 2 1 120.3.c.a 12
15.d odd 2 1 inner 240.3.c.e 12
15.e even 4 2 1200.3.l.y 12
20.d odd 2 1 120.3.c.a 12
20.e even 4 2 600.3.l.g 12
24.f even 2 1 960.3.c.k 12
24.h odd 2 1 960.3.c.j 12
40.e odd 2 1 960.3.c.k 12
40.f even 2 1 960.3.c.j 12
60.h even 2 1 120.3.c.a 12
60.l odd 4 2 600.3.l.g 12
120.i odd 2 1 960.3.c.j 12
120.m even 2 1 960.3.c.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.c.a 12 4.b odd 2 1
120.3.c.a 12 12.b even 2 1
120.3.c.a 12 20.d odd 2 1
120.3.c.a 12 60.h even 2 1
240.3.c.e 12 1.a even 1 1 trivial
240.3.c.e 12 3.b odd 2 1 inner
240.3.c.e 12 5.b even 2 1 inner
240.3.c.e 12 15.d odd 2 1 inner
600.3.l.g 12 20.e even 4 2
600.3.l.g 12 60.l odd 4 2
960.3.c.j 12 8.b even 2 1
960.3.c.j 12 24.h odd 2 1
960.3.c.j 12 40.f even 2 1
960.3.c.j 12 120.i odd 2 1
960.3.c.k 12 8.d odd 2 1
960.3.c.k 12 24.f even 2 1
960.3.c.k 12 40.e odd 2 1
960.3.c.k 12 120.m even 2 1
1200.3.l.y 12 5.c odd 4 2
1200.3.l.y 12 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{6} + 210 T_{7}^{4} + 7680 T_{7}^{2} + 4096 \)
\( T_{17}^{6} - 264 T_{17}^{4} + 13968 T_{17}^{2} - 165888 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 531441 - 26244 T^{2} + 4455 T^{4} + 504 T^{6} + 55 T^{8} - 4 T^{10} + T^{12} \)
$5$ \( 244140625 - 7031250 T^{2} - 230625 T^{4} + 22500 T^{6} - 369 T^{8} - 18 T^{10} + T^{12} \)
$7$ \( ( 4096 + 7680 T^{2} + 210 T^{4} + T^{6} )^{2} \)
$11$ \( ( 1083392 + 34944 T^{2} + 336 T^{4} + T^{6} )^{2} \)
$13$ \( ( 6553600 + 157824 T^{2} + 768 T^{4} + T^{6} )^{2} \)
$17$ \( ( -165888 + 13968 T^{2} - 264 T^{4} + T^{6} )^{2} \)
$19$ \( ( -5648 - 780 T + T^{3} )^{4} \)
$23$ \( ( -13148192 + 289440 T^{2} - 1050 T^{4} + T^{6} )^{2} \)
$29$ \( ( 807698432 + 4295808 T^{2} + 4416 T^{4} + T^{6} )^{2} \)
$31$ \( ( 31104 - 1692 T - 12 T^{2} + T^{3} )^{4} \)
$37$ \( ( 237899776 + 1828992 T^{2} + 2592 T^{4} + T^{6} )^{2} \)
$41$ \( ( 772087808 + 4826112 T^{2} + 4356 T^{4} + T^{6} )^{2} \)
$43$ \( ( 1224440064 + 4199040 T^{2} + 4050 T^{4} + T^{6} )^{2} \)
$47$ \( ( -2393766432 + 8026272 T^{2} - 5754 T^{4} + T^{6} )^{2} \)
$53$ \( ( -21525635072 + 34334352 T^{2} - 11400 T^{4} + T^{6} )^{2} \)
$59$ \( ( 1313998848 + 22331520 T^{2} + 11856 T^{4} + T^{6} )^{2} \)
$61$ \( ( 1984 - 348 T - 36 T^{2} + T^{3} )^{4} \)
$67$ \( ( 1763584 + 4443264 T^{2} + 5106 T^{4} + T^{6} )^{2} \)
$71$ \( ( 124856041472 + 105400320 T^{2} + 19200 T^{4} + T^{6} )^{2} \)
$73$ \( ( 238331428864 + 163399680 T^{2} + 26400 T^{4} + T^{6} )^{2} \)
$79$ \( ( -234400 - 3516 T + 108 T^{2} + T^{3} )^{4} \)
$83$ \( ( -387755552 + 52514208 T^{2} - 14586 T^{4} + T^{6} )^{2} \)
$89$ \( ( 278628139008 + 168708096 T^{2} + 27648 T^{4} + T^{6} )^{2} \)
$97$ \( ( 16777216 + 1572864 T^{2} + 13344 T^{4} + T^{6} )^{2} \)
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