Properties

Label 1334.2.a.k
Level $1334$
Weight $2$
Character orbit 1334.a
Self dual yes
Analytic conductor $10.652$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6520436296\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 5 x^{9} - 8 x^{8} + 60 x^{7} + 13 x^{6} - 241 x^{5} + 6 x^{4} + 346 x^{3} + 16 x^{2} - 64 x - 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 - \beta_{1} ) q^{3} + q^{4} -\beta_{5} q^{5} + ( 1 - \beta_{1} ) q^{6} + \beta_{4} q^{7} + q^{8} + ( 2 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 - \beta_{1} ) q^{3} + q^{4} -\beta_{5} q^{5} + ( 1 - \beta_{1} ) q^{6} + \beta_{4} q^{7} + q^{8} + ( 2 - \beta_{1} + \beta_{2} ) q^{9} -\beta_{5} q^{10} + \beta_{8} q^{11} + ( 1 - \beta_{1} ) q^{12} + ( 1 - \beta_{3} - \beta_{6} ) q^{13} + \beta_{4} q^{14} + ( -\beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{15} + q^{16} + ( \beta_{6} - \beta_{7} ) q^{17} + ( 2 - \beta_{1} + \beta_{2} ) q^{18} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{19} -\beta_{5} q^{20} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{21} + \beta_{8} q^{22} + q^{23} + ( 1 - \beta_{1} ) q^{24} + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{25} + ( 1 - \beta_{3} - \beta_{6} ) q^{26} + ( 3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{27} + \beta_{4} q^{28} + q^{29} + ( -\beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{30} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{31} + q^{32} + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{33} + ( \beta_{6} - \beta_{7} ) q^{34} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} - \beta_{9} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{37} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{38} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} ) q^{39} -\beta_{5} q^{40} + ( -1 + \beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{41} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{42} + ( 2 \beta_{2} + 2 \beta_{4} + \beta_{8} ) q^{43} + \beta_{8} q^{44} + ( -3 + \beta_{1} - \beta_{2} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{45} + q^{46} + ( \beta_{2} - \beta_{5} + \beta_{7} ) q^{47} + ( 1 - \beta_{1} ) q^{48} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{7} - \beta_{8} ) q^{49} + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{50} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{51} + ( 1 - \beta_{3} - \beta_{6} ) q^{52} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{53} + ( 3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{54} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{9} ) q^{55} + \beta_{4} q^{56} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{57} + q^{58} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{59} + ( -\beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{60} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{62} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} ) q^{63} + q^{64} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{65} + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{66} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{67} + ( \beta_{6} - \beta_{7} ) q^{68} + ( 1 - \beta_{1} ) q^{69} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} - \beta_{9} ) q^{70} + ( 1 - \beta_{3} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} ) q^{71} + ( 2 - \beta_{1} + \beta_{2} ) q^{72} + ( 1 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{73} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{74} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{75} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{76} + ( -3 - 2 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{7} - \beta_{9} ) q^{77} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} ) q^{78} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{79} -\beta_{5} q^{80} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{81} + ( -1 + \beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{82} + ( 1 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{83} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{84} + ( 2 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{85} + ( 2 \beta_{2} + 2 \beta_{4} + \beta_{8} ) q^{86} + ( 1 - \beta_{1} ) q^{87} + \beta_{8} q^{88} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{89} + ( -3 + \beta_{1} - \beta_{2} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{90} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{91} + q^{92} + ( -4 - 2 \beta_{2} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{93} + ( \beta_{2} - \beta_{5} + \beta_{7} ) q^{94} + ( -2 + 3 \beta_{1} - \beta_{3} - \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{95} + ( 1 - \beta_{1} ) q^{96} + ( 2 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{97} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{7} - \beta_{8} ) q^{98} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 10q^{2} + 5q^{3} + 10q^{4} - q^{5} + 5q^{6} + 2q^{7} + 10q^{8} + 11q^{9} + O(q^{10}) \) \( 10q + 10q^{2} + 5q^{3} + 10q^{4} - q^{5} + 5q^{6} + 2q^{7} + 10q^{8} + 11q^{9} - q^{10} + q^{11} + 5q^{12} + 9q^{13} + 2q^{14} + 5q^{15} + 10q^{16} + 11q^{18} + 20q^{19} - q^{20} + 14q^{21} + q^{22} + 10q^{23} + 5q^{24} + 23q^{25} + 9q^{26} + 23q^{27} + 2q^{28} + 10q^{29} + 5q^{30} + 21q^{31} + 10q^{32} + q^{33} + 2q^{35} + 11q^{36} - 8q^{37} + 20q^{38} + 23q^{39} - q^{40} - 4q^{41} + 14q^{42} - 3q^{43} + q^{44} - 18q^{45} + 10q^{46} - q^{47} + 5q^{48} + 18q^{49} + 23q^{50} - 6q^{51} + 9q^{52} - 13q^{53} + 23q^{54} + 13q^{55} + 2q^{56} - 22q^{57} + 10q^{58} + 14q^{59} + 5q^{60} + 12q^{61} + 21q^{62} - 26q^{63} + 10q^{64} - 25q^{65} + q^{66} + 6q^{67} + 5q^{69} + 2q^{70} + 8q^{71} + 11q^{72} + 16q^{73} - 8q^{74} + 8q^{75} + 20q^{76} - 16q^{77} + 23q^{78} + 7q^{79} - q^{80} - 2q^{81} - 4q^{82} + 18q^{83} + 14q^{84} + 8q^{85} - 3q^{86} + 5q^{87} + q^{88} + 2q^{89} - 18q^{90} + 8q^{91} + 10q^{92} - 41q^{93} - q^{94} - 16q^{95} + 5q^{96} + 6q^{97} + 18q^{98} - 22q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 5 x^{9} - 8 x^{8} + 60 x^{7} + 13 x^{6} - 241 x^{5} + 6 x^{4} + 346 x^{3} + 16 x^{2} - 64 x - 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} - 6 \nu^{8} - 4 \nu^{7} + 70 \nu^{6} - 29 \nu^{5} - 276 \nu^{4} + 132 \nu^{3} + 392 \nu^{2} - 80 \nu - 52 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{9} + 5 \nu^{8} + 8 \nu^{7} - 61 \nu^{6} - 8 \nu^{5} + 246 \nu^{4} - 51 \nu^{3} - 342 \nu^{2} + 84 \nu + 48 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{9} + 11 \nu^{8} + 11 \nu^{7} - 127 \nu^{6} + 29 \nu^{5} + 488 \nu^{4} - 206 \nu^{3} - 664 \nu^{2} + 196 \nu + 88 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{9} + 16 \nu^{8} + 20 \nu^{7} - 192 \nu^{6} + 13 \nu^{5} + 768 \nu^{4} - 230 \nu^{3} - 1080 \nu^{2} + 228 \nu + 148 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{9} - 21 \nu^{8} - 25 \nu^{7} + 239 \nu^{6} - 23 \nu^{5} - 890 \nu^{4} + 296 \nu^{3} + 1148 \nu^{2} - 308 \nu - 148 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( -2 \nu^{9} + 11 \nu^{8} + 11 \nu^{7} - 126 \nu^{6} + 26 \nu^{5} + 479 \nu^{4} - 181 \nu^{3} - 646 \nu^{2} + 150 \nu + 90 \)\()/2\)
\(\beta_{9}\)\(=\)\((\)\( 11 \nu^{9} - 57 \nu^{8} - 75 \nu^{7} + 665 \nu^{6} - 10 \nu^{5} - 2560 \nu^{4} + 682 \nu^{3} + 3432 \nu^{2} - 736 \nu - 448 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(\beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} + 11 \beta_{1} + 28\)
\(\nu^{5}\)\(=\)\(\beta_{9} + 2 \beta_{8} + 14 \beta_{6} + \beta_{5} - 12 \beta_{4} + 17 \beta_{3} + 16 \beta_{2} + 47 \beta_{1} + 55\)
\(\nu^{6}\)\(=\)\(3 \beta_{9} + 17 \beta_{8} + 9 \beta_{7} + 35 \beta_{6} + 8 \beta_{5} - 29 \beta_{4} + 62 \beta_{3} + 77 \beta_{2} + 118 \beta_{1} + 243\)
\(\nu^{7}\)\(=\)\(20 \beta_{9} + 50 \beta_{8} + 2 \beta_{7} + 161 \beta_{6} + 2 \beta_{5} - 119 \beta_{4} + 221 \beta_{3} + 211 \beta_{2} + 438 \beta_{1} + 636\)
\(\nu^{8}\)\(=\)\(70 \beta_{9} + 249 \beta_{8} + 59 \beta_{7} + 462 \beta_{6} + 13 \beta_{5} - 316 \beta_{4} + 770 \beta_{3} + 836 \beta_{2} + 1285 \beta_{1} + 2376\)
\(\nu^{9}\)\(=\)\(319 \beta_{9} + 838 \beta_{8} + 8 \beta_{7} + 1792 \beta_{6} - 169 \beta_{5} - 1110 \beta_{4} + 2633 \beta_{3} + 2618 \beta_{2} + 4497 \beta_{1} + 7069\)

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
3.37864
3.05371
2.20998
2.02950
0.521310
−0.267188
−0.290163
−1.61750
−1.74474
−2.27355
1.00000 −2.37864 1.00000 0.983407 −2.37864 −3.11298 1.00000 2.65792 0.983407
1.2 1.00000 −2.05371 1.00000 −4.01245 −2.05371 −3.53542 1.00000 1.21772 −4.01245
1.3 1.00000 −1.20998 1.00000 −1.11783 −1.20998 −0.129988 1.00000 −1.53596 −1.11783
1.4 1.00000 −1.02950 1.00000 2.59754 −1.02950 4.22406 1.00000 −1.94012 2.59754
1.5 1.00000 0.478690 1.00000 −3.82976 0.478690 4.18279 1.00000 −2.77086 −3.82976
1.6 1.00000 1.26719 1.00000 1.36043 1.26719 1.67775 1.00000 −1.39423 1.36043
1.7 1.00000 1.29016 1.00000 4.10547 1.29016 −1.09974 1.00000 −1.33548 4.10547
1.8 1.00000 2.61750 1.00000 0.316103 2.61750 −4.23734 1.00000 3.85129 0.316103
1.9 1.00000 2.74474 1.00000 1.89466 2.74474 2.43465 1.00000 4.53361 1.89466
1.10 1.00000 3.27355 1.00000 −3.29757 3.27355 1.59623 1.00000 7.71610 −3.29757
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

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 1334.2.a.k 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.2.a.k 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{10} \)
$3$ \( 112 - 224 T - 224 T^{2} + 428 T^{3} + 116 T^{4} - 271 T^{5} - T^{6} + 64 T^{7} - 8 T^{8} - 5 T^{9} + T^{10} \)
$5$ \( 484 - 2112 T + 1440 T^{2} + 1870 T^{3} - 1942 T^{4} - 119 T^{5} + 433 T^{6} - 16 T^{7} - 36 T^{8} + T^{9} + T^{10} \)
$7$ \( -768 - 5632 T + 2880 T^{2} + 5456 T^{3} - 2856 T^{4} - 1078 T^{5} + 582 T^{6} + 80 T^{7} - 42 T^{8} - 2 T^{9} + T^{10} \)
$11$ \( -288 - 2384 T + 2464 T^{2} + 9096 T^{3} - 4042 T^{4} - 2013 T^{5} + 937 T^{6} + 92 T^{7} - 56 T^{8} - T^{9} + T^{10} \)
$13$ \( 1568 - 560 T - 7728 T^{2} + 5712 T^{3} + 4678 T^{4} - 2729 T^{5} - 565 T^{6} + 408 T^{7} - 26 T^{8} - 9 T^{9} + T^{10} \)
$17$ \( 11944 - 22488 T - 1448 T^{2} + 15604 T^{3} - 3740 T^{4} - 2974 T^{5} + 1136 T^{6} + 88 T^{7} - 62 T^{8} + T^{10} \)
$19$ \( -3816320 + 3910848 T - 491648 T^{2} - 658720 T^{3} + 244296 T^{4} - 5124 T^{5} - 9812 T^{6} + 1328 T^{7} + 54 T^{8} - 20 T^{9} + T^{10} \)
$23$ \( ( -1 + T )^{10} \)
$29$ \( ( -1 + T )^{10} \)
$31$ \( 665088 + 52830208 T - 28806912 T^{2} + 2055056 T^{3} + 1241160 T^{4} - 195517 T^{5} - 12883 T^{6} + 3712 T^{7} - 58 T^{8} - 21 T^{9} + T^{10} \)
$37$ \( 13526272 + 11109120 T - 2165632 T^{2} - 2202912 T^{3} - 110640 T^{4} + 94742 T^{5} + 9122 T^{6} - 1496 T^{7} - 172 T^{8} + 8 T^{9} + T^{10} \)
$41$ \( 29042496 + 34497568 T + 7913696 T^{2} - 2757888 T^{3} - 813060 T^{4} + 79582 T^{5} + 23474 T^{6} - 960 T^{7} - 264 T^{8} + 4 T^{9} + T^{10} \)
$43$ \( -3872 + 37744 T - 119936 T^{2} + 149864 T^{3} - 67554 T^{4} - 889 T^{5} + 5949 T^{6} - 280 T^{7} - 148 T^{8} + 3 T^{9} + T^{10} \)
$47$ \( -384 - 42688 T - 10784 T^{2} + 57352 T^{3} - 12140 T^{4} - 9857 T^{5} + 2751 T^{6} + 362 T^{7} - 132 T^{8} + T^{9} + T^{10} \)
$53$ \( 24284932 + 27980288 T - 3217480 T^{2} - 3926058 T^{3} + 67510 T^{4} + 186989 T^{5} + 6089 T^{6} - 3376 T^{7} - 224 T^{8} + 13 T^{9} + T^{10} \)
$59$ \( -61239296 - 143480832 T + 14284800 T^{2} + 11942912 T^{3} - 923264 T^{4} - 335800 T^{5} + 24176 T^{6} + 3716 T^{7} - 264 T^{8} - 14 T^{9} + T^{10} \)
$61$ \( -4864512 - 4186880 T + 751360 T^{2} + 976256 T^{3} - 75752 T^{4} - 69858 T^{5} + 5510 T^{6} + 1680 T^{7} - 136 T^{8} - 12 T^{9} + T^{10} \)
$67$ \( 209133512 - 195515208 T + 6151600 T^{2} + 20831836 T^{3} - 1860268 T^{4} - 591210 T^{5} + 66432 T^{6} + 3610 T^{7} - 484 T^{8} - 6 T^{9} + T^{10} \)
$71$ \( 1443840 - 18891776 T + 22210560 T^{2} + 730496 T^{3} - 2391360 T^{4} - 145840 T^{5} + 49664 T^{6} + 2136 T^{7} - 380 T^{8} - 8 T^{9} + T^{10} \)
$73$ \( -7603392 - 34393568 T - 33738944 T^{2} + 10056496 T^{3} + 1695252 T^{4} - 606774 T^{5} + 16314 T^{6} + 5914 T^{7} - 320 T^{8} - 16 T^{9} + T^{10} \)
$79$ \( 9788 - 9114216 T + 14423148 T^{2} - 9095558 T^{3} + 2841650 T^{4} - 420649 T^{5} + 12371 T^{6} + 3818 T^{7} - 326 T^{8} - 7 T^{9} + T^{10} \)
$83$ \( -16748704 + 85852064 T - 52345888 T^{2} + 6971264 T^{3} + 1838000 T^{4} - 484588 T^{5} + 3364 T^{6} + 6028 T^{7} - 282 T^{8} - 18 T^{9} + T^{10} \)
$89$ \( -541360056 + 379902664 T + 143499416 T^{2} - 11947788 T^{3} - 5659460 T^{4} + 60154 T^{5} + 79436 T^{6} + 374 T^{7} - 474 T^{8} - 2 T^{9} + T^{10} \)
$97$ \( 158407200 - 58188608 T - 19342304 T^{2} + 9670624 T^{3} - 46160 T^{4} - 381700 T^{5} + 33860 T^{6} + 3412 T^{7} - 422 T^{8} - 6 T^{9} + T^{10} \)
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