Properties

Label 6003.2.a.k
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - x^{9} - 17 x^{8} + 23 x^{7} + 69 x^{6} - 88 x^{5} - 106 x^{4} + 101 x^{3} + 60 x^{2} - 23 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} - 17 x^{8} + 23 x^{7} + 69 x^{6} - 88 x^{5} - 106 x^{4} + 101 x^{3} + 60 x^{2} - 23 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 31 \nu^{9} - 14 \nu^{8} - 481 \nu^{7} + 466 \nu^{6} + 1581 \nu^{5} - 1705 \nu^{4} - 1465 \nu^{3} + 1920 \nu^{2} + 172 \nu - 357 \)\()/52\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{9} - 23 \nu^{8} - 182 \nu^{7} + 448 \nu^{6} + 587 \nu^{5} - 1710 \nu^{4} - 493 \nu^{3} + 2027 \nu^{2} - 87 \nu - 450 \)\()/26\)
\(\beta_{4}\)\(=\)\((\)\( -55 \nu^{9} + 24 \nu^{8} + 871 \nu^{7} - 810 \nu^{6} - 3065 \nu^{5} + 3051 \nu^{4} + 3297 \nu^{3} - 3466 \nu^{2} - 566 \nu + 703 \)\()/52\)
\(\beta_{5}\)\(=\)\((\)\( 33 \nu^{9} - 30 \nu^{8} - 533 \nu^{7} + 720 \nu^{6} + 1865 \nu^{5} - 2647 \nu^{4} - 1999 \nu^{3} + 2870 \nu^{2} + 272 \nu - 505 \)\()/26\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{9} + 3 \nu^{8} + 48 \nu^{7} - 70 \nu^{6} - 161 \nu^{5} + 256 \nu^{4} + 155 \nu^{3} - 279 \nu^{2} - 3 \nu + 54 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 51 \nu^{9} - 148 \nu^{8} - 871 \nu^{7} + 2694 \nu^{6} + 2809 \nu^{5} - 10007 \nu^{4} - 2645 \nu^{3} + 11238 \nu^{2} + 2 \nu - 2227 \)\()/52\)
\(\beta_{8}\)\(=\)\((\)\( -41 \nu^{9} + 42 \nu^{8} + 663 \nu^{7} - 982 \nu^{6} - 2299 \nu^{5} + 3737 \nu^{4} + 2367 \nu^{3} - 4252 \nu^{2} - 256 \nu + 785 \)\()/26\)
\(\beta_{9}\)\(=\)\((\)\( 93 \nu^{9} - 120 \nu^{8} - 1521 \nu^{7} + 2594 \nu^{6} + 5263 \nu^{5} - 9717 \nu^{4} - 5435 \nu^{3} + 10830 \nu^{2} + 490 \nu - 2033 \)\()/52\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(-3 \beta_{9} - 2 \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 7 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(14 \beta_{9} + 11 \beta_{8} - 14 \beta_{5} - 10 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 6 \beta_{1} + 23\)
\(\nu^{5}\)\(=\)\(-50 \beta_{9} - 34 \beta_{8} + 2 \beta_{7} - 12 \beta_{6} + 35 \beta_{5} + 17 \beta_{4} + 13 \beta_{3} - 27 \beta_{2} + 70 \beta_{1} - 29\)
\(\nu^{6}\)\(=\)\(186 \beta_{9} + 135 \beta_{8} - \beta_{7} + 8 \beta_{6} - 179 \beta_{5} - 114 \beta_{4} - 61 \beta_{3} + 43 \beta_{2} - 126 \beta_{1} + 241\)
\(\nu^{7}\)\(=\)\(-695 \beta_{9} - 480 \beta_{8} + 29 \beta_{7} - 136 \beta_{6} + 516 \beta_{5} + 268 \beta_{4} + 178 \beta_{3} - 324 \beta_{2} + 804 \beta_{1} - 510\)
\(\nu^{8}\)\(=\)\(2492 \beta_{9} + 1766 \beta_{8} - 31 \beta_{7} + 192 \beta_{6} - 2293 \beta_{5} - 1391 \beta_{4} - 804 \beta_{3} + 714 \beta_{2} - 2009 \beta_{1} + 2851\)
\(\nu^{9}\)\(=\)\(-9338 \beta_{9} - 6497 \beta_{8} + 349 \beta_{7} - 1579 \beta_{6} + 7263 \beta_{5} + 3936 \beta_{4} + 2480 \beta_{3} - 3957 \beta_{2} + 9887 \beta_{1} - 7726\)

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
1.19874
1.96386
−1.34161
1.92359
−1.52497
2.78422
−0.473620
−0.435319
−3.66000
0.565113
−2.74444 0 5.53194 −0.0839771 0 −2.30146 −9.69318 0 0.230470
1.2 −2.36221 0 3.58005 −3.25906 0 3.14721 −3.73241 0 7.69859
1.3 −2.08881 0 2.36311 −0.182168 0 1.65593 −0.758471 0 0.380514
1.4 −1.39479 0 −0.0545700 2.50233 0 4.61826 2.86569 0 −3.49022
1.5 −0.676519 0 −1.54232 1.80585 0 −4.44627 2.39645 0 −1.22169
1.6 −0.611391 0 −1.62620 −0.834863 0 −0.685491 2.21703 0 0.510427
1.7 −0.161383 0 −1.97396 −2.08087 0 3.66994 0.641331 0 0.335818
1.8 2.09108 0 2.37262 −4.27000 0 3.25035 0.779182 0 −8.92891
1.9 2.24431 0 3.03693 1.31370 0 −2.96709 2.32720 0 2.94835
1.10 2.70414 0 5.31240 −0.910949 0 −2.94138 8.95720 0 −2.46334
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

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 6003.2.a.k 10
3.b odd 2 1 2001.2.a.k 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.k 10 3.b odd 2 1
6003.2.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{10} + \cdots\)
\(T_{5}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -16 - 160 T - 441 T^{2} - 388 T^{3} + 89 T^{4} + 254 T^{5} + 45 T^{6} - 49 T^{7} - 14 T^{8} + 3 T^{9} + T^{10} \)
$3$ \( T^{10} \)
$5$ \( -2 - 38 T - 185 T^{2} - 180 T^{3} + 158 T^{4} + 223 T^{5} - 27 T^{6} - 69 T^{7} - 5 T^{8} + 6 T^{9} + T^{10} \)
$7$ \( -17576 - 18928 T + 18616 T^{2} + 10002 T^{3} - 5702 T^{4} - 1759 T^{5} + 761 T^{6} + 125 T^{7} - 46 T^{8} - 3 T^{9} + T^{10} \)
$11$ \( 10016 + 3672 T - 13255 T^{2} - 6183 T^{3} + 4868 T^{4} + 2866 T^{5} - 255 T^{6} - 304 T^{7} - 17 T^{8} + 9 T^{9} + T^{10} \)
$13$ \( 43153 + 51542 T - 120846 T^{2} - 20169 T^{3} + 32452 T^{4} + 5992 T^{5} - 2409 T^{6} - 581 T^{7} + 33 T^{8} + 16 T^{9} + T^{10} \)
$17$ \( -256 - 11904 T + 40480 T^{2} - 14301 T^{3} - 18089 T^{4} + 1808 T^{5} + 2046 T^{6} - 43 T^{7} - 80 T^{8} + T^{10} \)
$19$ \( -704 + 1136 T + 2228 T^{2} - 1903 T^{3} - 2053 T^{4} + 573 T^{5} + 639 T^{6} + 3 T^{7} - 47 T^{8} - T^{9} + T^{10} \)
$23$ \( ( -1 + T )^{10} \)
$29$ \( ( 1 + T )^{10} \)
$31$ \( 3195478 - 1062224 T - 2039879 T^{2} + 716323 T^{3} + 200466 T^{4} - 69394 T^{5} - 3459 T^{6} + 2000 T^{7} - 51 T^{8} - 17 T^{9} + T^{10} \)
$37$ \( 263470441 - 215604855 T + 33289367 T^{2} + 8678249 T^{3} - 2050423 T^{4} - 119864 T^{5} + 40086 T^{6} + 643 T^{7} - 332 T^{8} - T^{9} + T^{10} \)
$41$ \( -3146 - 8178 T + 185699 T^{2} - 52452 T^{3} - 62340 T^{4} + 7323 T^{5} + 5577 T^{6} - 257 T^{7} - 163 T^{8} + T^{10} \)
$43$ \( -68296 + 549980 T + 796474 T^{2} - 219359 T^{3} - 108767 T^{4} + 17981 T^{5} + 5582 T^{6} - 521 T^{7} - 124 T^{8} + 5 T^{9} + T^{10} \)
$47$ \( -4432 + 37704 T - 24092 T^{2} - 124406 T^{3} + 32800 T^{4} + 21823 T^{5} - 2186 T^{6} - 1357 T^{7} - 51 T^{8} + 15 T^{9} + T^{10} \)
$53$ \( -22759984 + 5170528 T + 22117252 T^{2} + 9571416 T^{3} + 924484 T^{4} - 236117 T^{5} - 57379 T^{6} - 2706 T^{7} + 299 T^{8} + 35 T^{9} + T^{10} \)
$59$ \( 29485024 + 49792528 T + 22606768 T^{2} + 1171308 T^{3} - 1526386 T^{4} - 361961 T^{5} - 10709 T^{6} + 5964 T^{7} + 878 T^{8} + 49 T^{9} + T^{10} \)
$61$ \( -105224032 - 291415736 T + 28342297 T^{2} + 21879154 T^{3} - 2412107 T^{4} - 438023 T^{5} + 51647 T^{6} + 3253 T^{7} - 400 T^{8} - 8 T^{9} + T^{10} \)
$67$ \( -787384 + 3097980 T - 4780955 T^{2} + 3688704 T^{3} - 1522223 T^{4} + 339020 T^{5} - 35401 T^{6} - 73 T^{7} + 392 T^{8} - 35 T^{9} + T^{10} \)
$71$ \( -1158168607 + 771272840 T - 72404392 T^{2} - 35368205 T^{3} + 3604064 T^{4} + 820822 T^{5} - 31685 T^{6} - 8455 T^{7} - 61 T^{8} + 30 T^{9} + T^{10} \)
$73$ \( -32 + 2896 T - 6736 T^{2} - 10606 T^{3} + 7850 T^{4} + 7851 T^{5} - 2036 T^{6} - 1589 T^{7} - 97 T^{8} + 15 T^{9} + T^{10} \)
$79$ \( 10721896 + 38082420 T - 16160094 T^{2} - 3700529 T^{3} + 2457918 T^{4} - 266458 T^{5} - 24756 T^{6} + 5361 T^{7} - 79 T^{8} - 24 T^{9} + T^{10} \)
$83$ \( 21233456 + 1255160 T - 13791076 T^{2} - 6241930 T^{3} - 364530 T^{4} + 247029 T^{5} + 36933 T^{6} - 1530 T^{7} - 401 T^{8} + T^{9} + T^{10} \)
$89$ \( -12055048 + 14120708 T + 7287022 T^{2} - 2081819 T^{3} - 714425 T^{4} + 115600 T^{5} + 21411 T^{6} - 2454 T^{7} - 233 T^{8} + 15 T^{9} + T^{10} \)
$97$ \( -139418176 + 725214784 T - 60054276 T^{2} - 49311716 T^{3} + 3626588 T^{4} + 1046929 T^{5} - 39725 T^{6} - 9929 T^{7} - 2 T^{8} + 35 T^{9} + T^{10} \)
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