Properties

Label 1287.2.b.c
Level $1287$
Weight $2$
Character orbit 1287.b
Analytic conductor $10.277$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.2767467401\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \(x^{14} + 23 x^{12} + 201 x^{10} + 835 x^{8} + 1695 x^{6} + 1565 x^{4} + 511 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 429)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{5} + \beta_{6} ) q^{4} + ( \beta_{2} - \beta_{3} ) q^{5} + ( -\beta_{3} + \beta_{13} ) q^{7} + ( -2 \beta_{1} + \beta_{3} + \beta_{7} - \beta_{9} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{5} + \beta_{6} ) q^{4} + ( \beta_{2} - \beta_{3} ) q^{5} + ( -\beta_{3} + \beta_{13} ) q^{7} + ( -2 \beta_{1} + \beta_{3} + \beta_{7} - \beta_{9} ) q^{8} + ( \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{10} + \beta_{3} q^{11} + ( -\beta_{1} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{13} + ( -3 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} - 2 \beta_{12} ) q^{14} + ( 1 + \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{12} ) q^{16} + ( -1 + \beta_{5} + \beta_{6} + \beta_{11} ) q^{17} + ( \beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{19} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{13} ) q^{20} + \beta_{5} q^{22} + ( 2 + \beta_{4} - 2 \beta_{5} - \beta_{12} ) q^{23} + ( -1 - \beta_{4} - 2 \beta_{5} + 2 \beta_{11} - \beta_{12} ) q^{25} + ( 1 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{26} + ( -3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{28} + ( 2 + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{29} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{9} ) q^{31} + ( 4 \beta_{1} - 4 \beta_{3} - \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{13} ) q^{32} + ( -\beta_{2} - 5 \beta_{3} + \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{34} + ( -1 - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{11} + \beta_{12} ) q^{35} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} ) q^{37} + ( 2 - \beta_{4} + \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{38} + ( -1 - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} ) q^{40} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} ) q^{41} + ( 3 + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{11} ) q^{43} + ( \beta_{1} - \beta_{3} + \beta_{9} ) q^{44} + ( 2 \beta_{1} + 5 \beta_{3} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{13} ) q^{46} + ( -3 \beta_{1} + \beta_{3} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} ) q^{47} + ( -3 - \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} ) q^{49} + ( -\beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{13} ) q^{50} + ( 1 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{52} + ( -2 - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{11} - \beta_{12} ) q^{53} + ( 1 - \beta_{11} ) q^{55} + ( \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{10} + 4 \beta_{11} + \beta_{12} ) q^{56} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{9} + \beta_{13} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{8} + \beta_{10} ) q^{59} + ( -1 - \beta_{12} ) q^{61} + ( -2 - 5 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{62} + ( -1 + \beta_{4} - 8 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} ) q^{64} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{65} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{7} - \beta_{8} - 4 \beta_{9} + \beta_{10} ) q^{67} + ( 4 - 7 \beta_{5} + \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} ) q^{68} + ( 5 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{9} + \beta_{13} ) q^{70} + ( \beta_{1} - 5 \beta_{3} + 3 \beta_{9} - \beta_{13} ) q^{71} + ( -4 \beta_{1} - 5 \beta_{3} + 2 \beta_{7} - 2 \beta_{9} - \beta_{13} ) q^{73} + ( 1 - 4 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 4 \beta_{11} - \beta_{12} ) q^{74} + ( 4 \beta_{2} - \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - \beta_{13} ) q^{76} + ( 1 + \beta_{4} ) q^{77} + ( 1 - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{79} + ( -6 \beta_{1} + 5 \beta_{2} + \beta_{3} + 4 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} ) q^{80} + ( 4 + \beta_{4} - 6 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} ) q^{82} + ( -\beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{83} + ( -\beta_{1} + 5 \beta_{3} + \beta_{7} - \beta_{9} ) q^{85} + ( 10 \beta_{1} - \beta_{2} + 5 \beta_{3} - 6 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} ) q^{86} + ( -1 - 2 \beta_{5} + \beta_{6} - \beta_{12} ) q^{88} + ( \beta_{1} + \beta_{2} + 6 \beta_{3} - 2 \beta_{7} - \beta_{9} + \beta_{13} ) q^{89} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} ) q^{91} + ( -2 + 7 \beta_{5} + \beta_{6} - 2 \beta_{11} + 2 \beta_{12} ) q^{92} + ( 7 + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{11} - 3 \beta_{12} ) q^{94} + ( 1 + 4 \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{95} + ( 5 \beta_{1} + 3 \beta_{3} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{13} ) q^{97} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{13} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 18q^{4} + O(q^{10}) \) \( 14q - 18q^{4} - 16q^{14} + 34q^{16} - 4q^{17} + 6q^{22} + 8q^{23} - 26q^{25} + 6q^{26} + 24q^{29} + 8q^{35} + 32q^{38} - 20q^{40} + 32q^{43} - 46q^{49} + 4q^{52} - 20q^{53} + 12q^{55} + 32q^{56} - 20q^{61} - 72q^{62} - 58q^{64} - 12q^{65} + 20q^{68} + 12q^{77} + 12q^{79} + 20q^{82} - 30q^{88} + 16q^{91} + 24q^{92} + 64q^{94} + 36q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} + 23 x^{12} + 201 x^{10} + 835 x^{8} + 1695 x^{6} + 1565 x^{4} + 511 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{13} - 2 \nu^{11} + 233 \nu^{9} + 2504 \nu^{7} + 10009 \nu^{5} + 16390 \nu^{3} + 8491 \nu \)\()/784\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{13} + 6 \nu^{11} - 699 \nu^{9} - 7120 \nu^{7} - 24931 \nu^{5} - 29962 \nu^{3} - 6265 \nu \)\()/784\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{12} - 62 \nu^{10} - 449 \nu^{8} - 1280 \nu^{6} - 885 \nu^{4} + 954 \nu^{2} + 413 \)\()/56\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{12} - 186 \nu^{10} - 1375 \nu^{8} - 4288 \nu^{6} - 4951 \nu^{4} - 1114 \nu^{2} - 21 \)\()/112\)
\(\beta_{6}\)\(=\)\((\)\( -9 \nu^{12} - 186 \nu^{10} - 1375 \nu^{8} - 4288 \nu^{6} - 4951 \nu^{4} - 1002 \nu^{2} + 315 \)\()/112\)
\(\beta_{7}\)\(=\)\((\)\( -57 \nu^{13} - 1290 \nu^{11} - 11023 \nu^{9} - 44256 \nu^{7} - 84519 \nu^{5} - 66938 \nu^{3} - 8757 \nu \)\()/784\)
\(\beta_{8}\)\(=\)\((\)\( -51 \nu^{13} - 7 \nu^{12} - 1082 \nu^{11} - 210 \nu^{10} - 8305 \nu^{9} - 2485 \nu^{8} - 27528 \nu^{7} - 14224 \nu^{6} - 35989 \nu^{5} - 38521 \nu^{4} - 11418 \nu^{3} - 39914 \nu^{2} + 3213 \nu - 7399 \)\()/784\)
\(\beta_{9}\)\(=\)\((\)\( -27 \nu^{13} - 642 \nu^{11} - 5861 \nu^{9} - 25688 \nu^{7} - 54725 \nu^{5} - 48842 \nu^{3} - 9863 \nu \)\()/392\)
\(\beta_{10}\)\(=\)\((\)\( 51 \nu^{13} - 7 \nu^{12} + 1082 \nu^{11} - 210 \nu^{10} + 8305 \nu^{9} - 2485 \nu^{8} + 27528 \nu^{7} - 14224 \nu^{6} + 35989 \nu^{5} - 38521 \nu^{4} + 11418 \nu^{3} - 39914 \nu^{2} - 3213 \nu - 7399 \)\()/784\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{12} - 22 \nu^{10} - 179 \nu^{8} - 660 \nu^{6} - 1091 \nu^{4} - 714 \nu^{2} - 117 \)\()/8\)
\(\beta_{12}\)\(=\)\((\)\( 3 \nu^{12} + 62 \nu^{10} + 456 \nu^{8} + 1392 \nu^{6} + 1473 \nu^{4} + 138 \nu^{2} - 14 \)\()/14\)
\(\beta_{13}\)\(=\)\((\)\( 219 \nu^{13} + 4750 \nu^{11} + 37957 \nu^{9} + 136448 \nu^{7} + 216869 \nu^{5} + 135374 \nu^{3} + 25207 \nu \)\()/784\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} - 3\)
\(\nu^{3}\)\(=\)\(-\beta_{9} + \beta_{7} + \beta_{3} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12} - 7 \beta_{6} + 9 \beta_{5} + \beta_{4} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{13} - \beta_{10} + 11 \beta_{9} + \beta_{8} - 9 \beta_{7} - 12 \beta_{3} + 40 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-13 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} + 48 \beta_{6} - 74 \beta_{5} - 9 \beta_{4} - 87\)
\(\nu^{7}\)\(=\)\(-13 \beta_{13} + 13 \beta_{10} - 94 \beta_{9} - 13 \beta_{8} + 68 \beta_{7} + 109 \beta_{3} + 6 \beta_{2} - 275 \beta_{1}\)
\(\nu^{8}\)\(=\)\(126 \beta_{12} + 32 \beta_{11} - 32 \beta_{10} - 32 \beta_{8} - 336 \beta_{6} + 584 \beta_{5} + 68 \beta_{4} + 543\)
\(\nu^{9}\)\(=\)\(126 \beta_{13} - 132 \beta_{10} + 746 \beta_{9} + 132 \beta_{8} - 504 \beta_{7} - 890 \beta_{3} - 96 \beta_{2} + 1935 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-1094 \beta_{12} - 360 \beta_{11} + 354 \beta_{10} + 354 \beta_{8} + 2397 \beta_{6} - 4497 \beta_{5} - 504 \beta_{4} - 3545\)
\(\nu^{11}\)\(=\)\(-1094 \beta_{13} + 1218 \beta_{10} - 5747 \beta_{9} - 1218 \beta_{8} + 3765 \beta_{7} + 6895 \beta_{3} + 1068 \beta_{2} - 13856 \beta_{1}\)
\(\nu^{12}\)\(=\)\(9003 \beta_{12} + 3504 \beta_{11} - 3380 \beta_{10} - 3380 \beta_{8} - 17347 \beta_{6} + 34133 \beta_{5} + 3765 \beta_{4} + 23873\)
\(\nu^{13}\)\(=\)\(9003 \beta_{13} - 10649 \beta_{10} + 43645 \beta_{9} + 10649 \beta_{8} - 28381 \beta_{7} - 51942 \beta_{3} - 10264 \beta_{2} + 100478 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
2.73878i
2.53441i
2.15754i
1.42819i
1.36814i
0.584778i
0.409068i
0.409068i
0.584778i
1.36814i
1.42819i
2.15754i
2.53441i
2.73878i
2.73878i 0 −5.50093 2.84154i 0 3.93129i 9.58828i 0 7.78236
298.2 2.53441i 0 −4.42325 3.70100i 0 0.957295i 6.14151i 0 −9.37985
298.3 2.15754i 0 −2.65498 0.710210i 0 2.30964i 1.41315i 0 1.53231
298.4 1.42819i 0 −0.0397381 0.0606573i 0 1.70646i 2.79963i 0 −0.0866304
298.5 1.36814i 0 0.128197 2.18365i 0 4.27070i 2.91167i 0 2.98753
298.6 0.584778i 0 1.65803 1.95350i 0 1.51078i 2.13914i 0 −1.14237
298.7 0.409068i 0 1.83266 4.13953i 0 5.18273i 1.56782i 0 −1.69335
298.8 0.409068i 0 1.83266 4.13953i 0 5.18273i 1.56782i 0 −1.69335
298.9 0.584778i 0 1.65803 1.95350i 0 1.51078i 2.13914i 0 −1.14237
298.10 1.36814i 0 0.128197 2.18365i 0 4.27070i 2.91167i 0 2.98753
298.11 1.42819i 0 −0.0397381 0.0606573i 0 1.70646i 2.79963i 0 −0.0866304
298.12 2.15754i 0 −2.65498 0.710210i 0 2.30964i 1.41315i 0 1.53231
298.13 2.53441i 0 −4.42325 3.70100i 0 0.957295i 6.14151i 0 −9.37985
298.14 2.73878i 0 −5.50093 2.84154i 0 3.93129i 9.58828i 0 7.78236
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 298.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1287.2.b.c 14
3.b odd 2 1 429.2.b.b 14
13.b even 2 1 inner 1287.2.b.c 14
39.d odd 2 1 429.2.b.b 14
39.f even 4 1 5577.2.a.x 7
39.f even 4 1 5577.2.a.y 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.b.b 14 3.b odd 2 1
429.2.b.b 14 39.d odd 2 1
1287.2.b.c 14 1.a even 1 1 trivial
1287.2.b.c 14 13.b even 2 1 inner
5577.2.a.x 7 39.f even 4 1
5577.2.a.y 7 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 23 T_{2}^{12} + 201 T_{2}^{10} + 835 T_{2}^{8} + 1695 T_{2}^{6} + 1565 T_{2}^{4} + 511 T_{2}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(1287, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 49 + 511 T^{2} + 1565 T^{4} + 1695 T^{6} + 835 T^{8} + 201 T^{10} + 23 T^{12} + T^{14} \)
$3$ \( T^{14} \)
$5$ \( 64 + 17568 T^{2} + 47236 T^{4} + 28504 T^{6} + 7180 T^{8} + 860 T^{10} + 48 T^{12} + T^{14} \)
$7$ \( 246016 + 545408 T^{2} + 414608 T^{4} + 142976 T^{6} + 23888 T^{8} + 1924 T^{10} + 72 T^{12} + T^{14} \)
$11$ \( ( 1 + T^{2} )^{7} \)
$13$ \( 62748517 + 2599051 T^{2} - 913952 T^{3} + 160381 T^{4} + 43264 T^{5} - 14677 T^{6} - 2752 T^{7} - 1129 T^{8} + 256 T^{9} + 73 T^{10} - 32 T^{11} + 7 T^{12} + T^{14} \)
$17$ \( ( -1256 - 2672 T + 134 T^{2} + 698 T^{3} - 32 T^{4} - 50 T^{5} + 2 T^{6} + T^{7} )^{2} \)
$19$ \( 6400 + 144256 T^{2} + 619664 T^{4} + 293776 T^{6} + 50128 T^{8} + 3660 T^{10} + 108 T^{12} + T^{14} \)
$23$ \( ( -1024 - 704 T + 2044 T^{2} + 1448 T^{3} + 48 T^{4} - 76 T^{5} - 4 T^{6} + T^{7} )^{2} \)
$29$ \( ( -104 + 3040 T - 1170 T^{2} - 790 T^{3} + 358 T^{4} + 2 T^{5} - 12 T^{6} + T^{7} )^{2} \)
$31$ \( 153664 + 29166368 T^{2} + 24272740 T^{4} + 6155880 T^{6} + 509580 T^{8} + 16644 T^{10} + 224 T^{12} + T^{14} \)
$37$ \( 15831678976 + 7097976832 T^{2} + 1061431360 T^{4} + 70680960 T^{6} + 2272224 T^{8} + 37504 T^{10} + 308 T^{12} + T^{14} \)
$41$ \( 1580857600 + 1839908224 T^{2} + 375249296 T^{4} + 31270960 T^{6} + 1275472 T^{8} + 26460 T^{10} + 264 T^{12} + T^{14} \)
$43$ \( ( 692704 - 35104 T - 63862 T^{2} + 2962 T^{3} + 1840 T^{4} - 90 T^{5} - 16 T^{6} + T^{7} )^{2} \)
$47$ \( 14973927424 + 12350205952 T^{2} + 1930292800 T^{4} + 124195648 T^{6} + 3896032 T^{8} + 60192 T^{10} + 412 T^{12} + T^{14} \)
$53$ \( ( -11456 + 19584 T + 8048 T^{2} - 2688 T^{3} - 1248 T^{4} - 88 T^{5} + 10 T^{6} + T^{7} )^{2} \)
$59$ \( 21143486464 + 7453736960 T^{2} + 978043904 T^{4} + 62149888 T^{6} + 2094464 T^{8} + 37472 T^{10} + 324 T^{12} + T^{14} \)
$61$ \( ( 32 + 192 T + 56 T^{2} - 248 T^{3} - 116 T^{4} + 12 T^{5} + 10 T^{6} + T^{7} )^{2} \)
$67$ \( 250296087616 + 156298667808 T^{2} + 16231912036 T^{4} + 603440952 T^{6} + 10892628 T^{8} + 104076 T^{10} + 508 T^{12} + T^{14} \)
$71$ \( 352284983296 + 173172549632 T^{2} + 24636273728 T^{4} + 938186240 T^{6} + 15972800 T^{8} + 138048 T^{10} + 592 T^{12} + T^{14} \)
$73$ \( 308886962176 + 114194570240 T^{2} + 13135268240 T^{4} + 615147968 T^{6} + 13468928 T^{8} + 139508 T^{10} + 628 T^{12} + T^{14} \)
$79$ \( ( 55424 - 44976 T - 5158 T^{2} + 6730 T^{3} + 560 T^{4} - 154 T^{5} - 6 T^{6} + T^{7} )^{2} \)
$83$ \( 1284505600 + 6424477696 T^{2} + 1396183296 T^{4} + 111558656 T^{6} + 3994304 T^{8} + 64064 T^{10} + 436 T^{12} + T^{14} \)
$89$ \( 7722894400 + 4732849696 T^{2} + 985488196 T^{4} + 91864816 T^{6} + 4020264 T^{8} + 77204 T^{10} + 496 T^{12} + T^{14} \)
$97$ \( 46960623616 + 30044407808 T^{2} + 6232743488 T^{4} + 470003456 T^{6} + 13621120 T^{8} + 153584 T^{10} + 680 T^{12} + T^{14} \)
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