Properties

Label 3856.2.a.n
Level $3856$
Weight $2$
Character orbit 3856.a
Self dual yes
Analytic conductor $30.790$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3856 = 2^{4} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7903150194\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
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.

\(f(q)\) \(=\) \( q -\beta_{8} q^{3} + ( 1 - \beta_{9} ) q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} + ( 1 - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{9} +O(q^{10})\) \( q -\beta_{8} q^{3} + ( 1 - \beta_{9} ) q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} + ( 1 - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{9} + ( -2 + \beta_{1} - \beta_{3} + \beta_{8} ) q^{11} + ( -\beta_{2} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{13} + ( -1 + \beta_{2} + \beta_{4} + \beta_{10} - \beta_{11} ) q^{15} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{17} + ( 1 - \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{19} + ( -2 - 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{21} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{23} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{11} ) q^{25} + ( -1 + 2 \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{27} + ( -1 - \beta_{1} + \beta_{3} - \beta_{6} + \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{29} + ( -\beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{31} + ( -3 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{8} - \beta_{11} ) q^{33} + ( -2 + \beta_{1} + \beta_{2} - \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{35} + ( -1 + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{37} + ( -3 - 2 \beta_{1} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{39} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{11} ) q^{41} + ( -\beta_{1} - \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{43} + ( -2 + \beta_{2} - \beta_{5} + 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{45} + ( -2 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{47} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} ) q^{49} + ( 2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{10} - 2 \beta_{11} ) q^{51} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} ) q^{53} + ( -1 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{55} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{57} + ( -4 - 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} + \beta_{8} - 3 \beta_{10} + 2 \beta_{11} ) q^{59} + ( -1 + \beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{61} + ( 3 - \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{63} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{65} + ( -3 + \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{67} + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{69} + ( -9 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{71} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{9} + 3 \beta_{11} ) q^{73} + ( 1 - 4 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{10} - 3 \beta_{11} ) q^{75} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{7} + \beta_{8} ) q^{77} + ( 2 + \beta_{2} + 2 \beta_{5} - 2 \beta_{7} + 4 \beta_{10} ) q^{79} + ( -1 + 2 \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{11} ) q^{81} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{85} + ( 4 - \beta_{2} - 5 \beta_{3} - \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{10} - \beta_{11} ) q^{87} + ( -1 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} ) q^{89} + ( 2 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{91} + ( 4 - 5 \beta_{3} + \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{93} + ( -3 - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{95} + ( -4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - q^{3} + 6q^{5} - 3q^{7} + 15q^{9} + O(q^{10}) \) \( 12q - q^{3} + 6q^{5} - 3q^{7} + 15q^{9} - 22q^{11} - 5q^{13} - 13q^{15} - 4q^{17} + 6q^{19} - 14q^{21} - 32q^{23} + 4q^{25} + 5q^{27} + 6q^{29} - 8q^{31} - 24q^{33} - 15q^{35} - 8q^{37} - 31q^{39} - q^{41} + 2q^{43} - 15q^{45} - 34q^{47} - 9q^{49} + 3q^{51} + 5q^{53} + 3q^{55} - 22q^{57} - 26q^{59} - 26q^{61} + 4q^{63} - 25q^{65} - 6q^{67} - 2q^{69} - 94q^{71} - 22q^{73} - 7q^{77} - 9q^{79} + 4q^{81} + 8q^{83} + 4q^{85} - 4q^{87} - 3q^{89} + 20q^{91} + 12q^{93} - 33q^{95} - 29q^{97} - 36q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} + \nu^{10} - 23 \nu^{9} - 13 \nu^{8} + 184 \nu^{7} + 54 \nu^{6} - 611 \nu^{5} - 94 \nu^{4} + 768 \nu^{3} + 94 \nu^{2} - 213 \nu - 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{11} + 9 \nu^{10} + 57 \nu^{9} - 119 \nu^{8} - 273 \nu^{7} + 488 \nu^{6} + 538 \nu^{5} - 663 \nu^{4} - 422 \nu^{3} + 168 \nu^{2} + 34 \nu + 7 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{11} - 10 \nu^{10} - 130 \nu^{9} + 172 \nu^{8} + 869 \nu^{7} - 1046 \nu^{6} - 2491 \nu^{5} + 2625 \nu^{4} + 2774 \nu^{3} - 2230 \nu^{2} - 745 \nu + 85 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{11} + 48 \nu^{10} + 160 \nu^{9} - 690 \nu^{8} - 569 \nu^{7} + 3330 \nu^{6} + 597 \nu^{5} - 6481 \nu^{4} - 470 \nu^{3} + 4634 \nu^{2} + 951 \nu - 169 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 13 \nu^{11} - 44 \nu^{10} - 180 \nu^{9} + 662 \nu^{8} + 829 \nu^{7} - 3426 \nu^{6} - 1629 \nu^{5} + 7333 \nu^{4} + 1798 \nu^{3} - 5698 \nu^{2} - 1335 \nu + 141 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( 11 \nu^{11} - 30 \nu^{10} - 158 \nu^{9} + 432 \nu^{8} + 773 \nu^{7} - 2086 \nu^{6} - 1631 \nu^{5} + 4025 \nu^{4} + 1654 \nu^{3} - 2750 \nu^{2} - 741 \nu + 93 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -11 \nu^{11} + 34 \nu^{10} + 150 \nu^{9} - 492 \nu^{8} - 669 \nu^{7} + 2398 \nu^{6} + 1223 \nu^{5} - 4717 \nu^{4} - 1178 \nu^{3} + 3354 \nu^{2} + 737 \nu - 109 \)\()/8\)
\(\beta_{10}\)\(=\)\((\)\( 25 \nu^{11} - 76 \nu^{10} - 340 \nu^{9} + 1086 \nu^{8} + 1513 \nu^{7} - 5178 \nu^{6} - 2777 \nu^{5} + 9809 \nu^{4} + 2718 \nu^{3} - 6602 \nu^{2} - 1643 \nu + 201 \)\()/16\)
\(\beta_{11}\)\(=\)\((\)\( 31 \nu^{11} - 90 \nu^{10} - 450 \nu^{9} + 1340 \nu^{8} + 2237 \nu^{7} - 6822 \nu^{6} - 4819 \nu^{5} + 14249 \nu^{4} + 5014 \nu^{3} - 10758 \nu^{2} - 2497 \nu + 381 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{11} + \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 8 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(11 \beta_{11} - 9 \beta_{10} + 10 \beta_{9} - \beta_{7} - 8 \beta_{6} - 11 \beta_{5} - 9 \beta_{4} + 9 \beta_{3} + \beta_{2} + 29 \beta_{1} - 10\)
\(\nu^{6}\)\(=\)\(20 \beta_{11} + 2 \beta_{10} + 10 \beta_{9} - 14 \beta_{8} - 19 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} - 2 \beta_{4} + 11 \beta_{3} + 56 \beta_{2} + 88\)
\(\nu^{7}\)\(=\)\(93 \beta_{11} - 67 \beta_{10} + 78 \beta_{9} - 3 \beta_{8} - 12 \beta_{7} - 54 \beta_{6} - 94 \beta_{5} - 68 \beta_{4} + 71 \beta_{3} + 11 \beta_{2} + 181 \beta_{1} - 77\)
\(\nu^{8}\)\(=\)\(163 \beta_{11} + 24 \beta_{10} + 76 \beta_{9} - 138 \beta_{8} - 146 \beta_{7} - 78 \beta_{6} - 71 \beta_{5} - 30 \beta_{4} + 98 \beta_{3} + 379 \beta_{2} + 4 \beta_{1} + 558\)
\(\nu^{9}\)\(=\)\(720 \beta_{11} - 478 \beta_{10} + 557 \beta_{9} - 59 \beta_{8} - 106 \beta_{7} - 355 \beta_{6} - 733 \beta_{5} - 499 \beta_{4} + 542 \beta_{3} + 88 \beta_{2} + 1179 \beta_{1} - 544\)
\(\nu^{10}\)\(=\)\(1256 \beta_{11} + 191 \beta_{10} + 522 \beta_{9} - 1189 \beta_{8} - 1056 \beta_{7} - 566 \beta_{6} - 561 \beta_{5} - 323 \beta_{4} + 822 \beta_{3} + 2542 \beta_{2} + 76 \beta_{1} + 3665\)
\(\nu^{11}\)\(=\)\(5372 \beta_{11} - 3384 \beta_{10} + 3821 \beta_{9} - 748 \beta_{8} - 845 \beta_{7} - 2351 \beta_{6} - 5486 \beta_{5} - 3655 \beta_{4} + 4093 \beta_{3} + 630 \beta_{2} + 7881 \beta_{1} - 3713\)

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
−2.02418
1.54879
−0.342147
0.0822506
2.49073
2.01020
−1.28632
1.63125
−2.59703
−1.32986
2.70063
0.115670
0 −2.93498 0 1.44091 0 −0.381245 0 5.61411 0
1.2 0 −2.81087 0 0.334961 0 4.24623 0 4.90098 0
1.3 0 −2.18519 0 −0.548903 0 −1.82459 0 1.77508 0
1.4 0 −1.81824 0 4.31963 0 −0.690569 0 0.306010 0
1.5 0 −1.22208 0 −3.14843 0 −0.136122 0 −1.50653 0
1.6 0 −0.500591 0 1.92585 0 0.852319 0 −2.74941 0
1.7 0 0.126224 0 0.612768 0 −1.03110 0 −2.98407 0
1.8 0 1.16790 0 1.75438 0 −5.06139 0 −1.63601 0
1.9 0 1.20534 0 3.49051 0 0.744578 0 −1.54716 0
1.10 0 2.18147 0 −3.40432 0 3.83334 0 1.75880 0
1.11 0 2.50808 0 0.533570 0 −0.354992 0 3.29045 0
1.12 0 3.28295 0 −1.31091 0 −3.19647 0 7.77775 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
\(2\) \(-1\)
\(241\) \(-1\)

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 3856.2.a.n 12
4.b odd 2 1 241.2.a.b 12
12.b even 2 1 2169.2.a.h 12
20.d odd 2 1 6025.2.a.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.a.b 12 4.b odd 2 1
2169.2.a.h 12 12.b even 2 1
3856.2.a.n 12 1.a even 1 1 trivial
6025.2.a.h 12 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3856))\):

\(T_{3}^{12} + \cdots\)
\(T_{5}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 64 - 400 T - 992 T^{2} + 960 T^{3} + 1540 T^{4} - 725 T^{5} - 888 T^{6} + 210 T^{7} + 224 T^{8} - 25 T^{9} - 25 T^{10} + T^{11} + T^{12} \)
$5$ \( 62 - 347 T + 339 T^{2} + 1071 T^{3} - 2193 T^{4} + 497 T^{5} + 1301 T^{6} - 797 T^{7} - 68 T^{8} + 134 T^{9} - 14 T^{10} - 6 T^{11} + T^{12} \)
$7$ \( 4 + 53 T + 200 T^{2} + 131 T^{3} - 588 T^{4} - 855 T^{5} + 263 T^{6} + 854 T^{7} + 245 T^{8} - 96 T^{9} - 33 T^{10} + 3 T^{11} + T^{12} \)
$11$ \( 128 + 1460 T + 5900 T^{2} + 9672 T^{3} + 3100 T^{4} - 9811 T^{5} - 12739 T^{6} - 5545 T^{7} - 215 T^{8} + 553 T^{9} + 177 T^{10} + 22 T^{11} + T^{12} \)
$13$ \( -52672 - 441248 T - 90512 T^{2} + 288472 T^{3} + 69802 T^{4} - 64645 T^{5} - 15049 T^{6} + 6470 T^{7} + 1425 T^{8} - 296 T^{9} - 62 T^{10} + 5 T^{11} + T^{12} \)
$17$ \( 154144 - 302576 T - 261264 T^{2} + 295792 T^{3} + 159474 T^{4} - 87027 T^{5} - 35221 T^{6} + 9972 T^{7} + 2997 T^{8} - 370 T^{9} - 97 T^{10} + 4 T^{11} + T^{12} \)
$19$ \( -3556280 + 3617301 T + 903711 T^{2} - 1614089 T^{3} + 58969 T^{4} + 247081 T^{5} - 28947 T^{6} - 16891 T^{7} + 2538 T^{8} + 524 T^{9} - 86 T^{10} - 6 T^{11} + T^{12} \)
$23$ \( -116949436 - 276824423 T - 182523158 T^{2} - 31479187 T^{3} + 11455889 T^{4} + 5191210 T^{5} + 372702 T^{6} - 138011 T^{7} - 28306 T^{8} - 627 T^{9} + 304 T^{10} + 32 T^{11} + T^{12} \)
$29$ \( 58109390 - 179077009 T + 164527164 T^{2} - 50865568 T^{3} - 3169769 T^{4} + 4236578 T^{5} - 355685 T^{6} - 116722 T^{7} + 15216 T^{8} + 1375 T^{9} - 213 T^{10} - 6 T^{11} + T^{12} \)
$31$ \( -318193616 - 468437780 T + 77270368 T^{2} + 108211396 T^{3} + 841016 T^{4} - 8192365 T^{5} - 688338 T^{6} + 208450 T^{7} + 22930 T^{8} - 2167 T^{9} - 262 T^{10} + 8 T^{11} + T^{12} \)
$37$ \( 50796928 - 18033984 T - 29177888 T^{2} + 5067040 T^{3} + 4698614 T^{4} - 619307 T^{5} - 323567 T^{6} + 36249 T^{7} + 10466 T^{8} - 928 T^{9} - 159 T^{10} + 8 T^{11} + T^{12} \)
$41$ \( -63338 - 4034251 T - 930334 T^{2} + 9341574 T^{3} - 976432 T^{4} - 2920817 T^{5} - 471938 T^{6} + 92737 T^{7} + 21111 T^{8} - 708 T^{9} - 262 T^{10} + T^{11} + T^{12} \)
$43$ \( 12503272 + 1488169 T - 25360675 T^{2} + 4519982 T^{3} + 6500883 T^{4} - 657869 T^{5} - 569920 T^{6} + 18272 T^{7} + 18808 T^{8} + 26 T^{9} - 237 T^{10} - 2 T^{11} + T^{12} \)
$47$ \( 53297792 - 37689968 T - 69239488 T^{2} - 11362328 T^{3} + 11628792 T^{4} + 4772881 T^{5} + 233524 T^{6} - 179952 T^{7} - 34508 T^{8} - 851 T^{9} + 332 T^{10} + 34 T^{11} + T^{12} \)
$53$ \( -3014 + 298853 T + 6874728 T^{2} - 11787807 T^{3} + 1527793 T^{4} + 1538756 T^{5} - 270515 T^{6} - 65448 T^{7} + 12170 T^{8} + 1019 T^{9} - 195 T^{10} - 5 T^{11} + T^{12} \)
$59$ \( -25476160 - 96501648 T - 59605584 T^{2} + 36571744 T^{3} + 47289076 T^{4} + 17302787 T^{5} + 2412075 T^{6} - 67258 T^{7} - 57444 T^{8} - 5338 T^{9} + 22 T^{10} + 26 T^{11} + T^{12} \)
$61$ \( 10893274 - 29191311 T + 16419914 T^{2} + 10617016 T^{3} - 8091392 T^{4} - 1560634 T^{5} + 801476 T^{6} + 117122 T^{7} - 22505 T^{8} - 3955 T^{9} + 20 T^{10} + 26 T^{11} + T^{12} \)
$67$ \( 4538509504 - 4714883120 T - 678403520 T^{2} + 669419464 T^{3} + 75258180 T^{4} - 29459511 T^{5} - 3555172 T^{6} + 474596 T^{7} + 62307 T^{8} - 2947 T^{9} - 429 T^{10} + 6 T^{11} + T^{12} \)
$71$ \( -12017198348 - 39886445545 T - 42976926619 T^{2} - 21246049133 T^{3} - 5606812300 T^{4} - 801669175 T^{5} - 46063490 T^{6} + 3647013 T^{7} + 918997 T^{8} + 79770 T^{9} + 3737 T^{10} + 94 T^{11} + T^{12} \)
$73$ \( 2219968 + 25741920 T + 64034288 T^{2} - 76860088 T^{3} - 83908922 T^{4} - 7168229 T^{5} + 3607447 T^{6} + 533877 T^{7} - 28097 T^{8} - 7860 T^{9} - 208 T^{10} + 22 T^{11} + T^{12} \)
$79$ \( -1277319040 - 3418562576 T + 1017318496 T^{2} + 2331826216 T^{3} + 163831840 T^{4} - 90986869 T^{5} - 7904508 T^{6} + 1163461 T^{7} + 109307 T^{8} - 5783 T^{9} - 581 T^{10} + 9 T^{11} + T^{12} \)
$83$ \( 98860915136 + 10813065520 T - 10895951696 T^{2} - 1151271176 T^{3} + 452061900 T^{4} + 42544271 T^{5} - 9197429 T^{6} - 669374 T^{7} + 100342 T^{8} + 4386 T^{9} - 548 T^{10} - 8 T^{11} + T^{12} \)
$89$ \( -1500609440 - 3427797584 T - 829552176 T^{2} + 490376448 T^{3} + 124536598 T^{4} - 21479633 T^{5} - 5233031 T^{6} + 305681 T^{7} + 79002 T^{8} - 1663 T^{9} - 479 T^{10} + 3 T^{11} + T^{12} \)
$97$ \( 107861318 - 194920563 T - 162786822 T^{2} + 64432913 T^{3} + 85302025 T^{4} + 27963948 T^{5} + 3390903 T^{6} - 118070 T^{7} - 70766 T^{8} - 5577 T^{9} + 85 T^{10} + 29 T^{11} + T^{12} \)
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