Properties

Label 8004.2.a.g
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{3} -\beta_{1} q^{5} + \beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -\beta_{1} q^{5} + \beta_{3} q^{7} + q^{9} -\beta_{2} q^{11} + \beta_{8} q^{13} + \beta_{1} q^{15} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{17} -\beta_{4} q^{19} -\beta_{3} q^{21} + q^{23} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{25} - q^{27} - q^{29} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{31} + \beta_{2} q^{33} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{35} + ( -3 + \beta_{2} - \beta_{5} - \beta_{8} + \beta_{11} ) q^{37} -\beta_{8} q^{39} + ( -1 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{41} + ( \beta_{2} - \beta_{3} - \beta_{9} + \beta_{11} ) q^{43} -\beta_{1} q^{45} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{4} + \beta_{6} + \beta_{11} ) q^{49} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{51} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{53} + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{55} + \beta_{4} q^{57} + ( -1 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{59} + ( -2 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} ) q^{61} + \beta_{3} q^{63} + ( -1 + 2 \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{65} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{67} - q^{69} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{7} + \beta_{11} ) q^{71} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{73} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{75} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{11} ) q^{77} + ( 2 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{79} + q^{81} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{83} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{85} + q^{87} + ( -3 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{10} + \beta_{11} ) q^{89} + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{91} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{93} + ( -3 + \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{95} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{97} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-8071 \nu^{11} + 19357 \nu^{10} + 231201 \nu^{9} - 479516 \nu^{8} - 2165385 \nu^{7} + 3873043 \nu^{6} + 7143630 \nu^{5} - 11289370 \nu^{4} - 2416332 \nu^{3} + 6816868 \nu^{2} - 11305176 \nu + 770552\)\()/1181592\)
\(\beta_{3}\)\(=\)\((\)\(-58412 \nu^{11} + 150230 \nu^{10} + 1862577 \nu^{9} - 3812431 \nu^{8} - 21650127 \nu^{7} + 29877380 \nu^{6} + 111631233 \nu^{5} - 64812155 \nu^{4} - 218775012 \nu^{3} - 63772828 \nu^{2} + 2829120 \nu + 11033668\)\()/4135572\)
\(\beta_{4}\)\(=\)\((\)\(-237136 \nu^{11} + 958345 \nu^{10} + 6749571 \nu^{9} - 26540003 \nu^{8} - 67159668 \nu^{7} + 247389829 \nu^{6} + 284027205 \nu^{5} - 890169226 \nu^{4} - 447721968 \nu^{3} + 927980284 \nu^{2} + 65617524 \nu - 69925000\)\()/16542288\)
\(\beta_{5}\)\(=\)\((\)\(643238 \nu^{11} - 1717169 \nu^{10} - 20607159 \nu^{9} + 44452597 \nu^{8} + 241331934 \nu^{7} - 362660321 \nu^{6} - 1259064135 \nu^{5} + 913186418 \nu^{4} + 2549822760 \nu^{3} + 226067380 \nu^{2} - 342835836 \nu - 32135128\)\()/16542288\)
\(\beta_{6}\)\(=\)\((\)\(227217 \nu^{11} - 656309 \nu^{10} - 7166323 \nu^{9} + 17182508 \nu^{8} + 82610957 \nu^{7} - 143477867 \nu^{6} - 426165362 \nu^{5} + 389223760 \nu^{4} + 866406388 \nu^{3} - 18498844 \nu^{2} - 162478008 \nu - 13046896\)\()/5514096\)
\(\beta_{7}\)\(=\)\((\)\(756551 \nu^{11} - 1925114 \nu^{10} - 24168090 \nu^{9} + 49854049 \nu^{8} + 280180599 \nu^{7} - 411626960 \nu^{6} - 1435974561 \nu^{5} + 1092840230 \nu^{4} + 2853762180 \nu^{3} + 39670792 \nu^{2} - 445104660 \nu - 37571512\)\()/16542288\)
\(\beta_{8}\)\(=\)\((\)\(324494 \nu^{11} - 886849 \nu^{10} - 10152007 \nu^{9} + 23105733 \nu^{8} + 114673742 \nu^{7} - 193749385 \nu^{6} - 570735599 \nu^{5} + 547270074 \nu^{4} + 1101125008 \nu^{3} - 145530924 \nu^{2} - 173254620 \nu + 21253000\)\()/5514096\)
\(\beta_{9}\)\(=\)\((\)\(1038679 \nu^{11} - 2548933 \nu^{10} - 33488751 \nu^{9} + 66191846 \nu^{8} + 390702759 \nu^{7} - 547193839 \nu^{6} - 2000252604 \nu^{5} + 1436390248 \nu^{4} + 3942455436 \nu^{3} + 142017764 \nu^{2} - 695410800 \nu - 33296528\)\()/16542288\)
\(\beta_{10}\)\(=\)\((\)\(-545408 \nu^{11} + 1592693 \nu^{10} + 17173041 \nu^{9} - 42407701 \nu^{8} - 196470678 \nu^{7} + 366059189 \nu^{6} + 997238067 \nu^{5} - 1082368964 \nu^{4} - 1977984000 \nu^{3} + 385348124 \nu^{2} + 341574492 \nu - 35157824\)\()/8271144\)
\(\beta_{11}\)\(=\)\((\)\(1186027 \nu^{11} - 3327328 \nu^{10} - 37462464 \nu^{9} + 87445547 \nu^{8} + 429697791 \nu^{7} - 737802358 \nu^{6} - 2179409127 \nu^{5} + 2066374630 \nu^{4} + 4256483244 \nu^{3} - 366651712 \nu^{2} - 490498764 \nu + 36924904\)\()/16542288\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} - \beta_{8} + \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{10} + \beta_{9} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 10 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-16 \beta_{11} + \beta_{9} - 11 \beta_{8} + 11 \beta_{7} - 4 \beta_{6} + 32 \beta_{5} - 16 \beta_{4} + 11 \beta_{3} - 11 \beta_{2} + 14 \beta_{1} + 68\)
\(\nu^{5}\)\(=\)\(-23 \beta_{11} + 22 \beta_{10} + 20 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} - 9 \beta_{6} + 32 \beta_{5} - 49 \beta_{4} - 25 \beta_{3} + 22 \beta_{2} + 115 \beta_{1} + 88\)
\(\nu^{6}\)\(=\)\(-264 \beta_{11} + \beta_{10} + 32 \beta_{9} - 133 \beta_{8} + 131 \beta_{7} - 97 \beta_{6} + 504 \beta_{5} - 266 \beta_{4} + 102 \beta_{3} - 124 \beta_{2} + 208 \beta_{1} + 902\)
\(\nu^{7}\)\(=\)\(-502 \beta_{11} + 362 \beta_{10} + 340 \beta_{9} - 58 \beta_{8} + 120 \beta_{7} - 264 \beta_{6} + 760 \beta_{5} - 952 \beta_{4} - 471 \beta_{3} + 371 \beta_{2} + 1485 \beta_{1} + 1682\)
\(\nu^{8}\)\(=\)\(-4357 \beta_{11} + 94 \beta_{10} + 724 \beta_{9} - 1744 \beta_{8} + 1742 \beta_{7} - 1889 \beta_{6} + 7975 \beta_{5} - 4518 \beta_{4} + 733 \beta_{3} - 1377 \beta_{2} + 3303 \beta_{1} + 13128\)
\(\nu^{9}\)\(=\)\(-10242 \beta_{11} + 5512 \beta_{10} + 5601 \beta_{9} - 1365 \beta_{8} + 2655 \beta_{7} - 5768 \beta_{6} + 15925 \beta_{5} - 17273 \beta_{4} - 8108 \beta_{3} + 5663 \beta_{2} + 21017 \beta_{1} + 31070\)
\(\nu^{10}\)\(=\)\(-72074 \beta_{11} + 3343 \beta_{10} + 14471 \beta_{9} - 24328 \beta_{8} + 25276 \beta_{7} - 34169 \beta_{6} + 127734 \beta_{5} - 77380 \beta_{4} + 960 \beta_{3} - 14696 \beta_{2} + 55027 \beta_{1} + 202616\)
\(\nu^{11}\)\(=\)\(-197916 \beta_{11} + 82860 \beta_{10} + 92278 \beta_{9} - 29324 \beta_{8} + 52846 \beta_{7} - 113038 \beta_{6} + 311629 \beta_{5} - 305427 \beta_{4} - 134565 \beta_{3} + 83314 \beta_{2} + 317839 \beta_{1} + 564322\)

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
4.15588
3.12492
3.06377
2.52462
0.425546
0.305886
−0.0891245
−0.413592
−1.42583
−2.48147
−2.64977
−3.54084
0 −1.00000 0 −4.15588 0 −1.11441 0 1.00000 0
1.2 0 −1.00000 0 −3.12492 0 4.33538 0 1.00000 0
1.3 0 −1.00000 0 −3.06377 0 3.59303 0 1.00000 0
1.4 0 −1.00000 0 −2.52462 0 −2.76852 0 1.00000 0
1.5 0 −1.00000 0 −0.425546 0 −4.01837 0 1.00000 0
1.6 0 −1.00000 0 −0.305886 0 −0.140026 0 1.00000 0
1.7 0 −1.00000 0 0.0891245 0 2.52085 0 1.00000 0
1.8 0 −1.00000 0 0.413592 0 2.75070 0 1.00000 0
1.9 0 −1.00000 0 1.42583 0 −1.49015 0 1.00000 0
1.10 0 −1.00000 0 2.48147 0 −0.0448560 0 1.00000 0
1.11 0 −1.00000 0 2.64977 0 −1.93393 0 1.00000 0
1.12 0 −1.00000 0 3.54084 0 2.31030 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8004.2.a.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.g 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{12} + \cdots\)
\(T_{7}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( 1 + T )^{12} \)
$5$ \( 16 - 136 T - 616 T^{2} + 1156 T^{3} + 3304 T^{4} - 2146 T^{5} - 1714 T^{6} + 697 T^{7} + 347 T^{8} - 80 T^{9} - 31 T^{10} + 3 T^{11} + T^{12} \)
$7$ \( -56 - 1704 T - 10516 T^{2} - 7720 T^{3} + 7026 T^{4} + 5494 T^{5} - 2177 T^{6} - 1320 T^{7} + 382 T^{8} + 127 T^{9} - 33 T^{10} - 4 T^{11} + T^{12} \)
$11$ \( -448 - 6400 T + 176 T^{2} + 24304 T^{3} + 11244 T^{4} - 13640 T^{5} - 5739 T^{6} + 2827 T^{7} + 823 T^{8} - 211 T^{9} - 48 T^{10} + 5 T^{11} + T^{12} \)
$13$ \( -280 - 1364 T + 4295 T^{2} + 11100 T^{3} - 3088 T^{4} - 11389 T^{5} - 1004 T^{6} + 3272 T^{7} + 545 T^{8} - 313 T^{9} - 53 T^{10} + 6 T^{11} + T^{12} \)
$17$ \( -113154 - 127110 T + 237933 T^{2} + 294484 T^{3} - 55532 T^{4} - 119254 T^{5} - 5262 T^{6} + 13235 T^{7} + 1395 T^{8} - 530 T^{9} - 69 T^{10} + 7 T^{11} + T^{12} \)
$19$ \( 57724 - 21494 T - 101934 T^{2} + 34571 T^{3} + 60097 T^{4} - 15948 T^{5} - 15116 T^{6} + 2642 T^{7} + 1602 T^{8} - 158 T^{9} - 70 T^{10} + 3 T^{11} + T^{12} \)
$23$ \( ( -1 + T )^{12} \)
$29$ \( ( 1 + T )^{12} \)
$31$ \( -151516160 + 146248512 T + 8056064 T^{2} - 35989228 T^{3} + 5733608 T^{4} + 2140788 T^{5} - 485448 T^{6} - 51919 T^{7} + 14964 T^{8} + 545 T^{9} - 202 T^{10} - 2 T^{11} + T^{12} \)
$37$ \( 5033728 - 3067704 T - 9890620 T^{2} - 280061 T^{3} + 6095318 T^{4} + 3664389 T^{5} + 818856 T^{6} + 29077 T^{7} - 17745 T^{8} - 2537 T^{9} + 2 T^{10} + 20 T^{11} + T^{12} \)
$41$ \( 31140960 - 44366832 T - 43587384 T^{2} + 24242548 T^{3} + 8334652 T^{4} - 2823212 T^{5} - 682530 T^{6} + 108949 T^{7} + 24899 T^{8} - 1046 T^{9} - 287 T^{10} + 3 T^{11} + T^{12} \)
$43$ \( 139228272 - 304355538 T + 215658776 T^{2} - 39098509 T^{3} - 17321935 T^{4} + 7339692 T^{5} - 246921 T^{6} - 225958 T^{7} + 23652 T^{8} + 1974 T^{9} - 289 T^{10} - 5 T^{11} + T^{12} \)
$47$ \( -238331376 + 75094128 T + 117066420 T^{2} - 63562060 T^{3} + 505088 T^{4} + 4954540 T^{5} - 643403 T^{6} - 113629 T^{7} + 22829 T^{8} + 722 T^{9} - 262 T^{10} + T^{12} \)
$53$ \( 12000192 + 78160608 T + 144729024 T^{2} + 84122176 T^{3} + 9114076 T^{4} - 4835498 T^{5} - 943018 T^{6} + 98893 T^{7} + 24931 T^{8} - 880 T^{9} - 267 T^{10} + 3 T^{11} + T^{12} \)
$59$ \( 491629824 + 105110784 T - 960529536 T^{2} - 800075424 T^{3} - 208627056 T^{4} - 6618408 T^{5} + 5311516 T^{6} + 689424 T^{7} - 16243 T^{8} - 7068 T^{9} - 214 T^{10} + 20 T^{11} + T^{12} \)
$61$ \( -509206528 - 55350656 T + 302721856 T^{2} + 69452272 T^{3} - 41184304 T^{4} - 12995360 T^{5} + 570372 T^{6} + 413311 T^{7} + 10325 T^{8} - 4603 T^{9} - 222 T^{10} + 17 T^{11} + T^{12} \)
$67$ \( 2048 + 48128 T + 219136 T^{2} + 78080 T^{3} - 533762 T^{4} - 241134 T^{5} + 191013 T^{6} + 121168 T^{7} + 12884 T^{8} - 2108 T^{9} - 241 T^{10} + 9 T^{11} + T^{12} \)
$71$ \( -68516352 - 13823712 T + 89302104 T^{2} - 693085 T^{3} - 30272951 T^{4} + 6035335 T^{5} + 1544628 T^{6} - 532998 T^{7} + 29538 T^{8} + 3975 T^{9} - 376 T^{10} - 7 T^{11} + T^{12} \)
$73$ \( 30528 + 484752 T + 556768 T^{2} - 368252 T^{3} - 502376 T^{4} + 58856 T^{5} + 125532 T^{6} + 13073 T^{7} - 9428 T^{8} - 2565 T^{9} - 167 T^{10} + 9 T^{11} + T^{12} \)
$79$ \( -33564676 + 380442330 T + 343605092 T^{2} - 301692881 T^{3} - 136308476 T^{4} + 83104541 T^{5} - 4117560 T^{6} - 1357377 T^{7} + 91465 T^{8} + 7553 T^{9} - 538 T^{10} - 14 T^{11} + T^{12} \)
$83$ \( 407213568 - 351516672 T - 361781928 T^{2} + 546237468 T^{3} - 239892300 T^{4} + 40793144 T^{5} - 165022 T^{6} - 728329 T^{7} + 54279 T^{8} + 3642 T^{9} - 431 T^{10} - 5 T^{11} + T^{12} \)
$89$ \( -79609014 - 184652190 T + 7692507 T^{2} + 90720204 T^{3} - 2859816 T^{4} - 9553209 T^{5} - 75560 T^{6} + 335944 T^{7} + 7575 T^{8} - 4643 T^{9} - 157 T^{10} + 22 T^{11} + T^{12} \)
$97$ \( 9231761664 - 9179318016 T + 2507661216 T^{2} + 281626544 T^{3} - 235529968 T^{4} + 23400952 T^{5} + 4785786 T^{6} - 997181 T^{7} + 10843 T^{8} + 8259 T^{9} - 390 T^{10} - 17 T^{11} + T^{12} \)
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