Properties

Label 160.6.c.c
Level 160
Weight 6
Character orbit 160.c
Analytic conductor 25.661
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 165 x^{10} + 9528 x^{8} + 254984 x^{6} + 3245664 x^{4} + 15975501 x^{2} + 588289\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{44}\cdot 5^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( -5 + \beta_{5} + \beta_{8} ) q^{5} -\beta_{9} q^{7} + ( -142 + 7 \beta_{1} + 2 \beta_{7} - 2 \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + ( -5 + \beta_{5} + \beta_{8} ) q^{5} -\beta_{9} q^{7} + ( -142 + 7 \beta_{1} + 2 \beta_{7} - 2 \beta_{8} ) q^{9} + \beta_{3} q^{11} + ( -3 \beta_{5} + 11 \beta_{6} ) q^{13} + ( \beta_{3} + 13 \beta_{4} + \beta_{10} ) q^{15} + ( 2 \beta_{5} - \beta_{6} - 7 \beta_{7} - 7 \beta_{8} ) q^{17} + ( -\beta_{2} + 3 \beta_{3} + \beta_{10} + \beta_{11} ) q^{19} + ( 80 - 68 \beta_{1} + 3 \beta_{7} - 3 \beta_{8} ) q^{21} + ( 36 \beta_{4} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{23} + ( 150 - 75 \beta_{1} + 10 \beta_{5} - 25 \beta_{6} - 25 \beta_{7} - 15 \beta_{8} ) q^{25} + ( -48 \beta_{4} - 14 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{27} + ( -1740 - 106 \beta_{1} + 42 \beta_{7} - 42 \beta_{8} ) q^{29} + ( 4 \beta_{2} - 4 \beta_{3} - 7 \beta_{10} - 7 \beta_{11} ) q^{31} + ( -88 \beta_{5} + 191 \beta_{6} - 69 \beta_{7} - 69 \beta_{8} ) q^{33} + ( 5 \beta_{2} - 7 \beta_{3} + 9 \beta_{4} + 20 \beta_{9} + 3 \beta_{10} + 5 \beta_{11} ) q^{35} + ( 98 \beta_{5} + 118 \beta_{6} - 53 \beta_{7} - 53 \beta_{8} ) q^{37} + ( -14 \beta_{2} - 22 \beta_{3} ) q^{39} + ( -9105 - 103 \beta_{1} + 80 \beta_{7} - 80 \beta_{8} ) q^{41} + ( -261 \beta_{4} + 26 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} ) q^{43} + ( -6265 + 10 \beta_{1} + 28 \beta_{5} + 155 \beta_{6} + 120 \beta_{7} - 167 \beta_{8} ) q^{45} + ( 42 \beta_{4} + 101 \beta_{9} + 5 \beta_{10} - 5 \beta_{11} ) q^{47} + ( -2418 - 833 \beta_{1} - 74 \beta_{7} + 74 \beta_{8} ) q^{49} + ( 17 \beta_{2} - 12 \beta_{3} - 7 \beta_{10} - 7 \beta_{11} ) q^{51} + ( 317 \beta_{5} + 65 \beta_{6} - 68 \beta_{7} - 68 \beta_{8} ) q^{53} + ( 30 \beta_{2} + 13 \beta_{3} - 381 \beta_{4} - 105 \beta_{9} - 2 \beta_{10} + 5 \beta_{11} ) q^{55} + ( -194 \beta_{5} + 365 \beta_{6} - 367 \beta_{7} - 367 \beta_{8} ) q^{57} + ( -63 \beta_{2} + 41 \beta_{3} + 3 \beta_{10} + 3 \beta_{11} ) q^{59} + ( 5820 - 1322 \beta_{1} + 221 \beta_{7} - 221 \beta_{8} ) q^{61} + ( 720 \beta_{4} - 107 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{63} + ( 185 - 595 \beta_{1} - 602 \beta_{5} + 390 \beta_{6} - 390 \beta_{7} - 12 \beta_{8} ) q^{65} + ( 867 \beta_{4} - 210 \beta_{9} + 16 \beta_{10} - 16 \beta_{11} ) q^{67} + ( -13870 + 22 \beta_{1} + 553 \beta_{7} - 553 \beta_{8} ) q^{69} + ( 26 \beta_{2} + 74 \beta_{3} + 43 \beta_{10} + 43 \beta_{11} ) q^{71} + ( -672 \beta_{5} - 453 \beta_{6} - 565 \beta_{7} - 565 \beta_{8} ) q^{73} + ( 75 \beta_{2} + 10 \beta_{3} + 1155 \beta_{4} + 150 \beta_{9} - 15 \beta_{10} - 25 \beta_{11} ) q^{75} + ( 1541 \beta_{5} + 311 \beta_{6} + 603 \beta_{7} + 603 \beta_{8} ) q^{77} + ( -130 \beta_{2} + 64 \beta_{3} + \beta_{10} + \beta_{11} ) q^{79} + ( -14726 + 737 \beta_{1} - 524 \beta_{7} + 524 \beta_{8} ) q^{81} + ( -1509 \beta_{4} + 78 \beta_{9} + 32 \beta_{10} - 32 \beta_{11} ) q^{83} + ( 12630 - 50 \beta_{1} - 31 \beta_{5} - 175 \beta_{6} + 525 \beta_{7} + 169 \beta_{8} ) q^{85} + ( -2086 \beta_{4} + 212 \beta_{9} - 42 \beta_{10} + 42 \beta_{11} ) q^{87} + ( -1040 + 2190 \beta_{1} - 622 \beta_{7} + 622 \beta_{8} ) q^{89} + ( 47 \beta_{2} - 22 \beta_{3} + 43 \beta_{10} + 43 \beta_{11} ) q^{91} + ( 834 \beta_{5} + 284 \beta_{6} + 1372 \beta_{7} + 1372 \beta_{8} ) q^{93} + ( 110 \beta_{2} - 41 \beta_{3} - 1983 \beta_{4} - 35 \beta_{9} - 21 \beta_{10} - 40 \beta_{11} ) q^{95} + ( -100 \beta_{5} - 965 \beta_{6} - 487 \beta_{7} - 487 \beta_{8} ) q^{97} + ( -141 \beta_{2} - 277 \beta_{3} - 69 \beta_{10} - 69 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 60q^{5} - 1676q^{9} + O(q^{10}) \) \( 12q - 60q^{5} - 1676q^{9} + 688q^{21} + 1500q^{25} - 21304q^{29} - 109672q^{41} - 75140q^{45} - 32348q^{49} + 64552q^{61} - 160q^{65} - 166352q^{69} - 173764q^{81} + 151360q^{85} - 3720q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 165 x^{10} + 9528 x^{8} + 254984 x^{6} + 3245664 x^{4} + 15975501 x^{2} + 588289\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -170020 \nu^{10} - 24454244 \nu^{8} - 1102611568 \nu^{6} - 20045402992 \nu^{4} - 128060923628 \nu^{2} + 2177766131 \)\()/ 200642913 \)
\(\beta_{2}\)\(=\)\((\)\( -411520 \nu^{10} - 60010112 \nu^{8} - 2780210368 \nu^{6} - 52857554368 \nu^{4} - 362353494656 \nu^{2} - 34380714880 \)\()/66880971\)
\(\beta_{3}\)\(=\)\((\)\( -5323964 \nu^{10} - 786315016 \nu^{8} - 37088343272 \nu^{6} - 711962887784 \nu^{4} - 4837404881512 \nu^{2} - 96708608876 \)\()/ 200642913 \)
\(\beta_{4}\)\(=\)\((\)\( -1310906 \nu^{11} - 195210058 \nu^{9} - 9277834112 \nu^{7} - 175028468600 \nu^{5} - 1019260920568 \nu^{3} + 2576633734216 \nu \)\()/ 51297704757 \)
\(\beta_{5}\)\(=\)\((\)\( 1957360 \nu^{11} + 286234304 \nu^{9} + 13567743616 \nu^{7} + 284453317984 \nu^{5} + 2688495819008 \nu^{3} + 8621360914288 \nu \)\()/ 51297704757 \)
\(\beta_{6}\)\(=\)\((\)\( -67875062 \nu^{11} - 10019691676 \nu^{9} - 471366569900 \nu^{7} - 8947051389500 \nu^{5} - 57637006550236 \nu^{3} + 38667851889478 \nu \)\()/ 153893114271 \)
\(\beta_{7}\)\(=\)\((\)\(85161215 \nu^{11} + 484184090 \nu^{10} + 12477922798 \nu^{9} + 70984599790 \nu^{8} + 581043147182 \nu^{7} + 3308494583420 \nu^{6} + 11012560403798 \nu^{5} + 62703306419780 \nu^{4} + 74871373684486 \nu^{3} + 421179959346250 \nu^{2} + 19752862444721 \nu + 17382114505340\)\()/ 153893114271 \)
\(\beta_{8}\)\(=\)\((\)\(85161215 \nu^{11} - 484184090 \nu^{10} + 12477922798 \nu^{9} - 70984599790 \nu^{8} + 581043147182 \nu^{7} - 3308494583420 \nu^{6} + 11012560403798 \nu^{5} - 62703306419780 \nu^{4} + 74871373684486 \nu^{3} - 421179959346250 \nu^{2} + 19752862444721 \nu - 17382114505340\)\()/ 153893114271 \)
\(\beta_{9}\)\(=\)\((\)\( 175955038 \nu^{11} + 25169463050 \nu^{9} + 1119493454860 \nu^{7} + 19636262498968 \nu^{5} + 109318479348956 \nu^{3} - 173990789836964 \nu \)\()/ 153893114271 \)
\(\beta_{10}\)\(=\)\((\)\(-2541138896 \nu^{11} - 3284674432 \nu^{10} - 372595783468 \nu^{9} - 475746157328 \nu^{8} - 17374777187732 \nu^{7} - 21714641655616 \nu^{6} - 329909822608880 \nu^{5} - 401095958765632 \nu^{4} - 2234157186356428 \nu^{3} - 2626945209511376 \nu^{2} - 317433643570484 \nu - 240122345851648\)\()/ 153893114271 \)
\(\beta_{11}\)\(=\)\((\)\(2541138896 \nu^{11} - 3284674432 \nu^{10} + 372595783468 \nu^{9} - 475746157328 \nu^{8} + 17374777187732 \nu^{7} - 21714641655616 \nu^{6} + 329909822608880 \nu^{5} - 401095958765632 \nu^{4} + 2234157186356428 \nu^{3} - 2626945209511376 \nu^{2} + 317433643570484 \nu - 240122345851648\)\()/ 153893114271 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{8} - \beta_{7} - 5 \beta_{6} - 2 \beta_{5} + 40 \beta_{4}\)\()/160\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{11} - \beta_{10} - 48 \beta_{8} + 48 \beta_{7} + 8 \beta_{3} + 5 \beta_{2} + 120 \beta_{1} - 8840\)\()/320\)
\(\nu^{3}\)\(=\)\((\)\(16 \beta_{11} - 16 \beta_{10} - 40 \beta_{9} - 130 \beta_{8} - 130 \beta_{7} + 910 \beta_{6} - 755 \beta_{5} - 3576 \beta_{4}\)\()/320\)
\(\nu^{4}\)\(=\)\((\)\(-80 \beta_{11} - 80 \beta_{10} + 9816 \beta_{8} - 9816 \beta_{7} - 1520 \beta_{3} - 995 \beta_{2} - 14040 \beta_{1} + 876040\)\()/640\)
\(\nu^{5}\)\(=\)\((\)\(-835 \beta_{11} + 835 \beta_{10} + 1930 \beta_{9} + 9416 \beta_{8} + 9416 \beta_{7} - 36920 \beta_{6} + 40567 \beta_{5} + 112850 \beta_{4}\)\()/160\)
\(\nu^{6}\)\(=\)\((\)\(12696 \beta_{11} + 12696 \beta_{10} - 839160 \beta_{8} + 839160 \beta_{7} + 128112 \beta_{3} + 79845 \beta_{2} + 1000440 \beta_{1} - 58009960\)\()/640\)
\(\nu^{7}\)\(=\)\((\)\(142620 \beta_{11} - 142620 \beta_{10} - 319200 \beta_{9} - 1729042 \beta_{8} - 1729042 \beta_{7} + 5885470 \beta_{6} - 6833759 \beta_{5} - 16637360 \beta_{4}\)\()/320\)
\(\nu^{8}\)\(=\)\((\)\(-586649 \beta_{11} - 586649 \beta_{10} + 34087152 \beta_{8} - 34087152 \beta_{7} - 5190248 \beta_{3} - 3135725 \beta_{2} - 38276760 \beta_{1} + 2171360360\)\()/320\)
\(\nu^{9}\)\(=\)\((\)\(-5783906 \beta_{11} + 5783906 \beta_{10} + 12748460 \beta_{9} + 71743835 \beta_{8} + 71743835 \beta_{7} - 233642345 \beta_{6} + 274570960 \beta_{5} + 645384276 \beta_{4}\)\()/160\)
\(\nu^{10}\)\(=\)\((\)\(24339950 \beta_{11} + 24339950 \beta_{10} - 1362127056 \beta_{8} + 1362127056 \beta_{7} + 207342440 \beta_{3} + 123500345 \beta_{2} + 1499141520 \beta_{1} - 84593817040\)\()/160\)
\(\nu^{11}\)\(=\)\((\)\(461870614 \beta_{11} - 461870614 \beta_{10} - 1010984020 \beta_{9} - 5773585979 \beta_{8} - 5773585979 \beta_{7} + 18531101945 \beta_{6} - 21831110698 \beta_{5} - 50835708524 \beta_{4}\)\()/160\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
8.90339i
4.23250i
4.32684i
5.17887i
0.192622i
4.71554i
4.71554i
0.192622i
5.17887i
4.32684i
4.23250i
8.90339i
0 26.2718i 0 48.8634 27.1545i 0 23.9306i 0 −447.206 0
129.2 0 26.2718i 0 48.8634 + 27.1545i 0 23.9306i 0 −447.206 0
129.3 0 19.0114i 0 −46.9000 30.4203i 0 91.0286i 0 −118.434 0
129.4 0 19.0114i 0 −46.9000 + 30.4203i 0 91.0286i 0 −118.434 0
129.5 0 9.81633i 0 −16.9634 53.2658i 0 222.821i 0 146.640 0
129.6 0 9.81633i 0 −16.9634 + 53.2658i 0 222.821i 0 146.640 0
129.7 0 9.81633i 0 −16.9634 53.2658i 0 222.821i 0 146.640 0
129.8 0 9.81633i 0 −16.9634 + 53.2658i 0 222.821i 0 146.640 0
129.9 0 19.0114i 0 −46.9000 30.4203i 0 91.0286i 0 −118.434 0
129.10 0 19.0114i 0 −46.9000 + 30.4203i 0 91.0286i 0 −118.434 0
129.11 0 26.2718i 0 48.8634 27.1545i 0 23.9306i 0 −447.206 0
129.12 0 26.2718i 0 48.8634 + 27.1545i 0 23.9306i 0 −447.206 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.c.c 12
4.b odd 2 1 inner 160.6.c.c 12
5.b even 2 1 inner 160.6.c.c 12
5.c odd 4 1 800.6.a.z 6
5.c odd 4 1 800.6.a.ba 6
8.b even 2 1 320.6.c.k 12
8.d odd 2 1 320.6.c.k 12
20.d odd 2 1 inner 160.6.c.c 12
20.e even 4 1 800.6.a.z 6
20.e even 4 1 800.6.a.ba 6
40.e odd 2 1 320.6.c.k 12
40.f even 2 1 320.6.c.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.c 12 1.a even 1 1 trivial
160.6.c.c 12 4.b odd 2 1 inner
160.6.c.c 12 5.b even 2 1 inner
160.6.c.c 12 20.d odd 2 1 inner
320.6.c.k 12 8.b even 2 1
320.6.c.k 12 8.d odd 2 1
320.6.c.k 12 40.e odd 2 1
320.6.c.k 12 40.f even 2 1
800.6.a.z 6 5.c odd 4 1
800.6.a.z 6 20.e even 4 1
800.6.a.ba 6 5.c odd 4 1
800.6.a.ba 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 1148 T_{3}^{4} + 350800 T_{3}^{2} + 24038400 \) acting on \(S_{6}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 310 T^{2} + 120679 T^{4} - 26699028 T^{6} + 7125974271 T^{8} - 1080903164310 T^{10} + 205891132094649 T^{12} )^{2} \)
$5$ \( ( 1 + 30 T + 75 T^{2} - 123500 T^{3} + 234375 T^{4} + 292968750 T^{5} + 30517578125 T^{6} )^{2} \)
$7$ \( ( 1 - 42334 T^{2} + 748334111 T^{4} - 10497411086308 T^{6} + 211385864339918639 T^{8} - \)\(33\!\cdots\!34\)\( T^{10} + \)\(22\!\cdots\!49\)\( T^{12} )^{2} \)
$11$ \( ( 1 + 220194 T^{2} + 46554422967 T^{4} + 4986677846505148 T^{6} + \)\(12\!\cdots\!67\)\( T^{8} + \)\(14\!\cdots\!94\)\( T^{10} + \)\(17\!\cdots\!01\)\( T^{12} )^{2} \)
$13$ \( ( 1 - 921822 T^{2} + 378030377607 T^{4} - 125995996576493956 T^{6} + \)\(52\!\cdots\!43\)\( T^{8} - \)\(17\!\cdots\!22\)\( T^{10} + \)\(26\!\cdots\!49\)\( T^{12} )^{2} \)
$17$ \( ( 1 - 7508390 T^{2} + 24616526523247 T^{4} - 45351621021996027220 T^{6} + \)\(49\!\cdots\!03\)\( T^{8} - \)\(30\!\cdots\!90\)\( T^{10} + \)\(81\!\cdots\!49\)\( T^{12} )^{2} \)
$19$ \( ( 1 + 8862994 T^{2} + 40251921150215 T^{4} + \)\(12\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!15\)\( T^{8} + \)\(33\!\cdots\!94\)\( T^{10} + \)\(23\!\cdots\!01\)\( T^{12} )^{2} \)
$23$ \( ( 1 - 27906430 T^{2} + 367961637958719 T^{4} - \)\(29\!\cdots\!08\)\( T^{6} + \)\(15\!\cdots\!31\)\( T^{8} - \)\(47\!\cdots\!30\)\( T^{10} + \)\(71\!\cdots\!49\)\( T^{12} )^{2} \)
$29$ \( ( 1 + 5326 T + 46243827 T^{2} + 133362268948 T^{3} + 948514025927223 T^{4} + 2240686724556870526 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} )^{4} \)
$31$ \( ( 1 + 10594234 T^{2} - 153174449706673 T^{4} + \)\(19\!\cdots\!88\)\( T^{6} - \)\(12\!\cdots\!73\)\( T^{8} + \)\(71\!\cdots\!34\)\( T^{10} + \)\(55\!\cdots\!01\)\( T^{12} )^{2} \)
$37$ \( ( 1 - 234974254 T^{2} + 32080895329334807 T^{4} - \)\(26\!\cdots\!92\)\( T^{6} + \)\(15\!\cdots\!43\)\( T^{8} - \)\(54\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!49\)\( T^{12} )^{2} \)
$41$ \( ( 1 + 27418 T + 531273543 T^{2} + 6402679071436 T^{3} + 61551334383790143 T^{4} + \)\(36\!\cdots\!18\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} )^{4} \)
$43$ \( ( 1 - 613900486 T^{2} + 182904754783784951 T^{4} - \)\(33\!\cdots\!72\)\( T^{6} + \)\(39\!\cdots\!99\)\( T^{8} - \)\(28\!\cdots\!86\)\( T^{10} + \)\(10\!\cdots\!49\)\( T^{12} )^{2} \)
$47$ \( ( 1 - 579920494 T^{2} + 165136673035562991 T^{4} - \)\(38\!\cdots\!48\)\( T^{6} + \)\(86\!\cdots\!59\)\( T^{8} - \)\(16\!\cdots\!94\)\( T^{10} + \)\(14\!\cdots\!49\)\( T^{12} )^{2} \)
$53$ \( ( 1 - 2026325390 T^{2} + 1866514545072595447 T^{4} - \)\(99\!\cdots\!20\)\( T^{6} + \)\(32\!\cdots\!03\)\( T^{8} - \)\(61\!\cdots\!90\)\( T^{10} + \)\(53\!\cdots\!49\)\( T^{12} )^{2} \)
$59$ \( ( 1 + 55950786 T^{2} + 253972763776609047 T^{4} - \)\(45\!\cdots\!68\)\( T^{6} + \)\(12\!\cdots\!47\)\( T^{8} + \)\(14\!\cdots\!86\)\( T^{10} + \)\(13\!\cdots\!01\)\( T^{12} )^{2} \)
$61$ \( ( 1 - 16138 T + 744681843 T^{2} - 29097452426876 T^{3} + 628955530019662743 T^{4} - \)\(11\!\cdots\!38\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} )^{4} \)
$67$ \( ( 1 - 2173336374 T^{2} + 6882868324909910631 T^{4} - \)\(82\!\cdots\!68\)\( T^{6} + \)\(12\!\cdots\!19\)\( T^{8} - \)\(72\!\cdots\!74\)\( T^{10} + \)\(60\!\cdots\!49\)\( T^{12} )^{2} \)
$71$ \( ( 1 + 942628714 T^{2} + 4622069102042239647 T^{4} + \)\(48\!\cdots\!68\)\( T^{6} + \)\(15\!\cdots\!47\)\( T^{8} + \)\(99\!\cdots\!14\)\( T^{10} + \)\(34\!\cdots\!01\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 4736672630 T^{2} + 15959781190221205247 T^{4} - \)\(37\!\cdots\!40\)\( T^{6} + \)\(68\!\cdots\!03\)\( T^{8} - \)\(87\!\cdots\!30\)\( T^{10} + \)\(79\!\cdots\!49\)\( T^{12} )^{2} \)
$79$ \( ( 1 + 454164826 T^{2} + 18487132928554429487 T^{4} - \)\(50\!\cdots\!28\)\( T^{6} + \)\(17\!\cdots\!87\)\( T^{8} + \)\(40\!\cdots\!26\)\( T^{10} + \)\(84\!\cdots\!01\)\( T^{12} )^{2} \)
$83$ \( ( 1 - 11599602390 T^{2} + 72197091204656028039 T^{4} - \)\(31\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!11\)\( T^{8} - \)\(27\!\cdots\!90\)\( T^{10} + \)\(37\!\cdots\!49\)\( T^{12} )^{2} \)
$89$ \( ( 1 + 930 T + 9446079447 T^{2} + 231471892494140 T^{3} + 52747469192025044703 T^{4} + \)\(28\!\cdots\!30\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} )^{4} \)
$97$ \( ( 1 - 35609466630 T^{2} + \)\(64\!\cdots\!47\)\( T^{4} - \)\(68\!\cdots\!40\)\( T^{6} + \)\(47\!\cdots\!03\)\( T^{8} - \)\(19\!\cdots\!30\)\( T^{10} + \)\(40\!\cdots\!49\)\( T^{12} )^{2} \)
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