Properties

Label 3969.2.a.bi
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} - 736 x^{3} + 96 x^{2} + 60 x - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 441)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{8} ) q^{4} -\beta_{10} q^{5} + ( 1 + \beta_{8} - \beta_{9} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{8} ) q^{4} -\beta_{10} q^{5} + ( 1 + \beta_{8} - \beta_{9} ) q^{8} + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{10} + ( 2 - \beta_{2} + \beta_{8} + \beta_{11} ) q^{11} + ( -\beta_{1} + \beta_{3} + \beta_{6} - \beta_{10} ) q^{13} + ( 1 + \beta_{2} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{16} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} ) q^{19} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{20} + ( -1 + 2 \beta_{2} - \beta_{5} - \beta_{9} + \beta_{11} ) q^{22} + ( 3 - \beta_{2} ) q^{23} + ( 1 - \beta_{2} - \beta_{5} - \beta_{9} + \beta_{11} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} ) q^{26} + ( 1 + \beta_{2} + \beta_{9} - 2 \beta_{11} ) q^{29} + ( -\beta_{1} + \beta_{4} + 2 \beta_{7} - \beta_{10} ) q^{31} + ( 4 + \beta_{2} + \beta_{5} + 2 \beta_{8} - \beta_{11} ) q^{32} + ( \beta_{3} + \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{34} + ( 1 + \beta_{2} + \beta_{5} + \beta_{8} + \beta_{11} ) q^{37} + ( 2 \beta_{1} + 3 \beta_{3} + \beta_{6} - \beta_{7} ) q^{38} + ( \beta_{4} - 3 \beta_{6} - \beta_{10} ) q^{40} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{41} + ( -1 + 3 \beta_{2} + \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{43} + ( 5 - \beta_{2} + 2 \beta_{8} + \beta_{9} ) q^{44} + ( -3 + 3 \beta_{2} - \beta_{8} ) q^{46} + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{47} + ( -\beta_{2} + \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{50} + ( \beta_{1} + 2 \beta_{6} - 3 \beta_{7} + \beta_{10} ) q^{52} + ( 2 + \beta_{2} - \beta_{5} + 2 \beta_{9} - \beta_{11} ) q^{53} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{6} + \beta_{7} - 2 \beta_{10} ) q^{55} + ( -1 + 4 \beta_{2} + \beta_{5} - \beta_{8} - 2 \beta_{11} ) q^{58} + ( -3 \beta_{1} + 4 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{10} ) q^{59} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{61} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} ) q^{62} + ( 2 + 5 \beta_{2} - \beta_{5} + \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{64} + ( 6 - 3 \beta_{2} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{65} + ( 1 - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{67} + ( -5 \beta_{3} - \beta_{4} - \beta_{7} + 2 \beta_{10} ) q^{68} + ( 5 - \beta_{2} - \beta_{9} + 2 \beta_{11} ) q^{71} + ( \beta_{1} + 2 \beta_{3} + \beta_{6} - \beta_{7} - \beta_{10} ) q^{73} + ( 5 + \beta_{2} - \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{74} + ( -4 \beta_{3} - \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{76} + ( -2 + 2 \beta_{2} - \beta_{5} - 3 \beta_{8} + \beta_{9} ) q^{79} + ( -\beta_{1} + 5 \beta_{3} + \beta_{4} - 2 \beta_{6} + 3 \beta_{7} + \beta_{10} ) q^{80} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + 4 \beta_{7} + \beta_{10} ) q^{82} + ( \beta_{6} - 3 \beta_{7} ) q^{83} + ( -1 + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{11} ) q^{85} + ( 5 + 3 \beta_{2} - \beta_{5} - \beta_{9} - \beta_{11} ) q^{86} + ( -1 + 4 \beta_{2} + \beta_{5} - \beta_{8} - 2 \beta_{11} ) q^{88} + ( \beta_{1} - \beta_{6} - 2 \beta_{7} - 2 \beta_{10} ) q^{89} + ( 2 - 2 \beta_{2} + 2 \beta_{8} + \beta_{9} ) q^{92} + ( -4 \beta_{1} - 2 \beta_{4} + 3 \beta_{6} - \beta_{7} + 2 \beta_{10} ) q^{94} + ( 5 - \beta_{2} - 2 \beta_{5} - \beta_{8} - 2 \beta_{9} ) q^{95} + ( \beta_{1} - 4 \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{7} + \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{2} + 12q^{4} + 12q^{8} + O(q^{10}) \) \( 12q + 4q^{2} + 12q^{4} + 12q^{8} + 20q^{11} + 12q^{16} + 32q^{23} + 12q^{25} + 16q^{29} + 48q^{32} + 12q^{37} + 56q^{44} - 24q^{46} - 4q^{50} + 32q^{53} + 48q^{64} + 60q^{65} + 12q^{67} + 56q^{71} + 68q^{74} - 12q^{79} - 12q^{85} + 76q^{86} + 16q^{92} + 64q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} - 736 x^{3} + 96 x^{2} + 60 x - 7\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-29600 \nu^{11} + 377504 \nu^{10} - 178989 \nu^{9} - 9010433 \nu^{8} + 14881733 \nu^{7} + 63630698 \nu^{6} - 115990022 \nu^{5} - 141596603 \nu^{4} + 217572487 \nu^{3} + 88708281 \nu^{2} - 75709700 \nu - 10235919\)\()/5915292\)
\(\beta_{2}\)\(=\)\((\)\(-153653 \nu^{11} + 1106428 \nu^{10} + 2049118 \nu^{9} - 26580896 \nu^{8} + 9534853 \nu^{7} + 191938557 \nu^{6} - 156257935 \nu^{5} - 454973007 \nu^{4} + 273870783 \nu^{3} + 316226486 \nu^{2} - 16532156 \nu - 14106727\)\()/8872938\)
\(\beta_{3}\)\(=\)\((\)\(-153653 \nu^{11} + 1106428 \nu^{10} + 2049118 \nu^{9} - 26580896 \nu^{8} + 9534853 \nu^{7} + 191938557 \nu^{6} - 156257935 \nu^{5} - 454973007 \nu^{4} + 273870783 \nu^{3} + 316226486 \nu^{2} - 25405094 \nu - 14106727\)\()/8872938\)
\(\beta_{4}\)\(=\)\((\)\(377200 \nu^{11} - 1719752 \nu^{10} - 7584581 \nu^{9} + 39941737 \nu^{8} + 30456613 \nu^{7} - 273622386 \nu^{6} + 81347402 \nu^{5} + 585403491 \nu^{4} - 356348877 \nu^{3} - 349019797 \nu^{2} + 246718636 \nu + 4252571\)\()/17745876\)
\(\beta_{5}\)\(=\)\((\)\(-352181 \nu^{11} + 1253056 \nu^{10} + 8078080 \nu^{9} - 28246487 \nu^{8} - 50779916 \nu^{7} + 183651234 \nu^{6} + 68245598 \nu^{5} - 351832803 \nu^{4} + 62503395 \nu^{3} + 201819119 \nu^{2} - 79593119 \nu - 37815883\)\()/8872938\)
\(\beta_{6}\)\(=\)\((\)\(-491816 \nu^{11} + 1331248 \nu^{10} + 12444820 \nu^{9} - 28741478 \nu^{8} - 96673697 \nu^{7} + 170701317 \nu^{6} + 258297167 \nu^{5} - 237915981 \nu^{4} - 203137878 \nu^{3} + 10654406 \nu^{2} + 4887547 \nu - 16670305\)\()/8872938\)
\(\beta_{7}\)\(=\)\((\)\(371053 \nu^{11} - 814814 \nu^{10} - 9883901 \nu^{9} + 16809322 \nu^{8} + 83640367 \nu^{7} - 89740203 \nu^{6} - 259040476 \nu^{5} + 67595349 \nu^{4} + 248276754 \nu^{3} + 102871706 \nu^{2} - 24935219 \nu - 16913407\)\()/5915292\)
\(\beta_{8}\)\(=\)\((\)\(491816 \nu^{11} - 1331248 \nu^{10} - 12444820 \nu^{9} + 28741478 \nu^{8} + 96673697 \nu^{7} - 170701317 \nu^{6} - 258297167 \nu^{5} + 237915981 \nu^{4} + 203137878 \nu^{3} - 6217937 \nu^{2} - 4887547 \nu - 5512040\)\()/4436469\)
\(\beta_{9}\)\(=\)\((\)\(-410574 \nu^{11} + 1474988 \nu^{10} + 9575126 \nu^{9} - 33756309 \nu^{8} - 63418903 \nu^{7} + 226224623 \nu^{6} + 115238431 \nu^{5} - 462927506 \nu^{4} - 36274042 \nu^{3} + 258774249 \nu^{2} + 4474145 \nu - 13564244\)\()/2957646\)
\(\beta_{10}\)\(=\)\((\)\(-3199606 \nu^{11} + 10762397 \nu^{10} + 76422230 \nu^{9} - 243920686 \nu^{8} - 533002183 \nu^{7} + 1605699294 \nu^{6} + 1130271061 \nu^{5} - 3146428860 \nu^{4} - 642900054 \nu^{3} + 1630110649 \nu^{2} + 154189073 \nu - 100038029\)\()/17745876\)
\(\beta_{11}\)\(=\)\((\)\(-604072 \nu^{11} + 2002445 \nu^{10} + 14514229 \nu^{9} - 45298156 \nu^{8} - 102681496 \nu^{7} + 297294947 \nu^{6} + 228532297 \nu^{5} - 579246461 \nu^{4} - 158897277 \nu^{3} + 299857215 \nu^{2} + 52483887 \nu - 18960010\)\()/2957646\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(-\beta_{3} + \beta_{2}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + 2 \beta_{6} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{9} + \beta_{8} + 3 \beta_{6} - 3 \beta_{4} - 8 \beta_{3} + 10 \beta_{2} - 3 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + 19 \beta_{8} - 8 \beta_{7} + 28 \beta_{6} + \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + \beta_{2} - 4 \beta_{1} + 55\)
\(\nu^{5}\)\(=\)\(-\beta_{11} + 10 \beta_{10} - 28 \beta_{9} + 30 \beta_{8} - 10 \beta_{7} + 60 \beta_{6} + \beta_{5} - 55 \beta_{4} - 79 \beta_{3} + 121 \beta_{2} - 55 \beta_{1} + 32\)
\(\nu^{6}\)\(=\)\(-2 \beta_{11} + 12 \beta_{10} - 41 \beta_{9} + 317 \beta_{8} - 176 \beta_{7} + 410 \beta_{6} + 39 \beta_{5} - 100 \beta_{4} + 82 \beta_{3} + 45 \beta_{2} - 112 \beta_{1} + 726\)
\(\nu^{7}\)\(=\)\(-53 \beta_{11} + 266 \beta_{10} - 532 \beta_{9} + 644 \beta_{8} - 294 \beta_{7} + 1078 \beta_{6} + 55 \beta_{5} - 903 \beta_{4} - 862 \beta_{3} + 1624 \beta_{2} - 931 \beta_{1} + 756\)
\(\nu^{8}\)\(=\)\(-136 \beta_{11} + 432 \beta_{10} - 993 \beta_{9} + 5138 \beta_{8} - 3136 \beta_{7} + 6336 \beta_{6} + 851 \beta_{5} - 2016 \beta_{4} + 1320 \beta_{3} + 1271 \beta_{2} - 2416 \beta_{1} + 10379\)
\(\nu^{9}\)\(=\)\(-1387 \beta_{11} + 5238 \beta_{10} - 9125 \beta_{9} + 12427 \beta_{8} - 6450 \beta_{7} + 18993 \beta_{6} + 1561 \beta_{5} - 14610 \beta_{4} - 9904 \beta_{3} + 23300 \beta_{2} - 15714 \beta_{1} + 15729\)
\(\nu^{10}\)\(=\)\(-4052 \beta_{11} + 10676 \beta_{10} - 20438 \beta_{9} + 83197 \beta_{8} - 52708 \beta_{7} + 101398 \beta_{6} + 15750 \beta_{5} - 37870 \beta_{4} + 19396 \beta_{3} + 29184 \beta_{2} - 46986 \beta_{1} + 155772\)
\(\nu^{11}\)\(=\)\(-28918 \beta_{11} + 93324 \beta_{10} - 151655 \beta_{9} + 228997 \beta_{8} - 127292 \beta_{7} + 331947 \beta_{6} + 35166 \beta_{5} - 237303 \beta_{4} - 117479 \beta_{3} + 349639 \beta_{2} - 265199 \beta_{1} + 306345\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.762097
−3.59052
−2.71409
0.114341
1.34584
−1.48259
2.51703
−0.311400
0.312397
3.14082
4.12935
1.30092
−2.17631 0 2.73633 −1.26829 0 0 −1.60248 0 2.76019
1.2 −2.17631 0 2.73633 1.26829 0 0 −1.60248 0 −2.76019
1.3 −1.29987 0 −0.310333 −3.52584 0 0 3.00314 0 4.58314
1.4 −1.29987 0 −0.310333 3.52584 0 0 3.00314 0 −4.58314
1.5 −0.0683740 0 −1.99532 −2.66379 0 0 0.273176 0 0.182134
1.6 −0.0683740 0 −1.99532 2.66379 0 0 0.273176 0 −0.182134
1.7 1.10281 0 −0.783802 −0.105466 0 0 −3.07001 0 −0.116309
1.8 1.10281 0 −0.783802 0.105466 0 0 −3.07001 0 0.116309
1.9 1.72661 0 0.981184 −3.51231 0 0 −1.75910 0 −6.06439
1.10 1.72661 0 0.981184 3.51231 0 0 −1.75910 0 6.06439
1.11 2.71513 0 5.37195 −1.58639 0 0 9.15528 0 −4.30727
1.12 2.71513 0 5.37195 1.58639 0 0 9.15528 0 4.30727
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.2.a.bi 12
3.b odd 2 1 3969.2.a.bh 12
7.b odd 2 1 inner 3969.2.a.bi 12
9.c even 3 2 1323.2.f.h 24
9.d odd 6 2 441.2.f.h 24
21.c even 2 1 3969.2.a.bh 12
63.g even 3 2 1323.2.g.h 24
63.h even 3 2 1323.2.h.h 24
63.i even 6 2 441.2.h.h 24
63.j odd 6 2 441.2.h.h 24
63.k odd 6 2 1323.2.g.h 24
63.l odd 6 2 1323.2.f.h 24
63.n odd 6 2 441.2.g.h 24
63.o even 6 2 441.2.f.h 24
63.s even 6 2 441.2.g.h 24
63.t odd 6 2 1323.2.h.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.h 24 9.d odd 6 2
441.2.f.h 24 63.o even 6 2
441.2.g.h 24 63.n odd 6 2
441.2.g.h 24 63.s even 6 2
441.2.h.h 24 63.i even 6 2
441.2.h.h 24 63.j odd 6 2
1323.2.f.h 24 9.c even 3 2
1323.2.f.h 24 63.l odd 6 2
1323.2.g.h 24 63.g even 3 2
1323.2.g.h 24 63.k odd 6 2
1323.2.h.h 24 63.h even 3 2
1323.2.h.h 24 63.t odd 6 2
3969.2.a.bh 12 3.b odd 2 1
3969.2.a.bh 12 21.c even 2 1
3969.2.a.bi 12 1.a even 1 1 trivial
3969.2.a.bi 12 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{6} - 2 T_{2}^{5} - 7 T_{2}^{4} + 12 T_{2}^{3} + 10 T_{2}^{2} - 14 T_{2} - 1 \)
\( T_{5}^{12} - 36 T_{5}^{10} + 465 T_{5}^{8} - 2580 T_{5}^{6} + 5850 T_{5}^{4} - 4470 T_{5}^{2} + 49 \)
\( T_{11}^{6} - 10 T_{11}^{5} + 8 T_{11}^{4} + 134 T_{11}^{3} - 211 T_{11}^{2} - 160 T_{11} + 283 \)
\( T_{13}^{12} - 78 T_{13}^{10} + 2283 T_{13}^{8} - 31500 T_{13}^{6} + 211779 T_{13}^{4} - 621030 T_{13}^{2} + 502681 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 - 14 T + 10 T^{2} + 12 T^{3} - 7 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$3$ \( T^{12} \)
$5$ \( 49 - 4470 T^{2} + 5850 T^{4} - 2580 T^{6} + 465 T^{8} - 36 T^{10} + T^{12} \)
$7$ \( T^{12} \)
$11$ \( ( 283 - 160 T - 211 T^{2} + 134 T^{3} + 8 T^{4} - 10 T^{5} + T^{6} )^{2} \)
$13$ \( 502681 - 621030 T^{2} + 211779 T^{4} - 31500 T^{6} + 2283 T^{8} - 78 T^{10} + T^{12} \)
$17$ \( 707281 - 926316 T^{2} + 378849 T^{4} - 57860 T^{6} + 3690 T^{8} - 102 T^{10} + T^{12} \)
$19$ \( 555025 - 1725612 T^{2} + 838386 T^{4} - 135382 T^{6} + 7077 T^{8} - 144 T^{10} + T^{12} \)
$23$ \( ( 47 - 262 T + 415 T^{2} - 288 T^{3} + 98 T^{4} - 16 T^{5} + T^{6} )^{2} \)
$29$ \( ( 18395 - 11498 T + 772 T^{2} + 594 T^{3} - 73 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$31$ \( 60025 - 1738422 T^{2} + 1344870 T^{4} - 280452 T^{6} + 12201 T^{8} - 192 T^{10} + T^{12} \)
$37$ \( ( 4507 - 2502 T - 852 T^{2} + 656 T^{3} - 81 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$41$ \( 2344593241 - 522367698 T^{2} + 42495537 T^{4} - 1604108 T^{6} + 30486 T^{8} - 282 T^{10} + T^{12} \)
$43$ \( ( 65005 - 43164 T + 7134 T^{2} + 522 T^{3} - 171 T^{4} + T^{6} )^{2} \)
$47$ \( 2393209 - 6701520 T^{2} + 1909590 T^{4} - 207826 T^{6} + 9969 T^{8} - 192 T^{10} + T^{12} \)
$53$ \( ( 32827 - 11092 T - 4477 T^{2} + 1406 T^{3} - 28 T^{4} - 16 T^{5} + T^{6} )^{2} \)
$59$ \( 40309801 - 428252034 T^{2} + 54884202 T^{4} - 2442192 T^{6} + 44865 T^{8} - 354 T^{10} + T^{12} \)
$61$ \( 50481025 - 350332752 T^{2} + 66477273 T^{4} - 3061898 T^{6} + 52734 T^{8} - 384 T^{10} + T^{12} \)
$67$ \( ( -15893 - 4920 T + 2073 T^{2} + 420 T^{3} - 96 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$71$ \( ( -4657 + 6338 T - 2144 T^{2} - 282 T^{3} + 227 T^{4} - 28 T^{5} + T^{6} )^{2} \)
$73$ \( 80089 - 463488 T^{2} + 471006 T^{4} - 109578 T^{6} + 7257 T^{8} - 168 T^{10} + T^{12} \)
$79$ \( ( 8207 - 18786 T + 11046 T^{2} - 262 T^{3} - 219 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$83$ \( 107723641 - 153715560 T^{2} + 58696773 T^{4} - 2925660 T^{6} + 53814 T^{8} - 402 T^{10} + T^{12} \)
$89$ \( 6477025 - 30891318 T^{2} + 7537917 T^{4} - 569480 T^{6} + 18222 T^{8} - 246 T^{10} + T^{12} \)
$97$ \( 64362167809 - 10564360530 T^{2} + 611276478 T^{4} - 15142740 T^{6} + 155097 T^{8} - 666 T^{10} + T^{12} \)
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