Properties

Label 3525.2.a.bg
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 9 x^{8} + 29 x^{7} + 25 x^{6} - 91 x^{5} - 21 x^{4} + 101 x^{3} + 6 x^{2} - 30 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 705)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 3 x^{9} - 9 x^{8} + 29 x^{7} + 25 x^{6} - 91 x^{5} - 21 x^{4} + 101 x^{3} + 6 x^{2} - 30 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{9} + 3 \nu^{8} + 22 \nu^{7} - 23 \nu^{6} - 82 \nu^{5} + 43 \nu^{4} + 110 \nu^{3} - 3 \nu^{2} - 40 \nu - 12 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{9} - 3 \nu^{8} - 22 \nu^{7} + 23 \nu^{6} + 84 \nu^{5} - 47 \nu^{4} - 118 \nu^{3} + 19 \nu^{2} + 40 \nu + 4 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{9} - 3 \nu^{8} - 22 \nu^{7} + 23 \nu^{6} + 84 \nu^{5} - 45 \nu^{4} - 122 \nu^{3} + 11 \nu^{2} + 54 \nu + 6 \)\()/2\)
\(\beta_{7}\)\(=\)\( \nu^{9} - \nu^{8} - 12 \nu^{7} + 7 \nu^{6} + 49 \nu^{5} - 9 \nu^{4} - 73 \nu^{3} - 9 \nu^{2} + 29 \nu + 8 \)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{9} - 5 \nu^{8} - 33 \nu^{7} + 43 \nu^{6} + 121 \nu^{5} - 105 \nu^{4} - 155 \nu^{3} + 63 \nu^{2} + 48 \nu + 2 \)\()/2\)
\(\beta_{9}\)\(=\)\((\)\( -3 \nu^{9} + 4 \nu^{8} + 35 \nu^{7} - 32 \nu^{6} - 139 \nu^{5} + 66 \nu^{4} + 201 \nu^{3} - 20 \nu^{2} - 78 \nu - 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{6} + \beta_{5} + 2 \beta_{3} + 6 \beta_{2} + \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(-6 \beta_{6} + 7 \beta_{5} + \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(\beta_{9} + \beta_{8} + \beta_{7} - 8 \beta_{6} + 9 \beta_{5} + 2 \beta_{4} + 17 \beta_{3} + 34 \beta_{2} + 11 \beta_{1} + 65\)
\(\nu^{7}\)\(=\)\(4 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 32 \beta_{6} + 42 \beta_{5} + 10 \beta_{4} + 52 \beta_{3} + 56 \beta_{2} + 89 \beta_{1} + 78\)
\(\nu^{8}\)\(=\)\(17 \beta_{9} + 13 \beta_{8} + 17 \beta_{7} - 51 \beta_{6} + 64 \beta_{5} + 24 \beta_{4} + 115 \beta_{3} + 197 \beta_{2} + 90 \beta_{1} + 351\)
\(\nu^{9}\)\(=\)\(58 \beta_{9} + 30 \beta_{8} + 47 \beta_{7} - 167 \beta_{6} + 244 \beta_{5} + 81 \beta_{4} + 319 \beta_{3} + 375 \beta_{2} + 471 \beta_{1} + 551\)

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
−2.04563
−1.88719
−1.16553
−0.563403
−0.138546
0.801361
1.49126
1.55802
2.38580
2.56385
−2.04563 −1.00000 2.18458 0 2.04563 1.36348 −0.377587 1.00000 0
1.2 −1.88719 −1.00000 1.56149 0 1.88719 1.44822 0.827559 1.00000 0
1.3 −1.16553 −1.00000 −0.641537 0 1.16553 −4.15990 3.07879 1.00000 0
1.4 −0.563403 −1.00000 −1.68258 0 0.563403 1.09545 2.07478 1.00000 0
1.5 −0.138546 −1.00000 −1.98081 0 0.138546 3.39422 0.551524 1.00000 0
1.6 0.801361 −1.00000 −1.35782 0 −0.801361 −0.0381502 −2.69083 1.00000 0
1.7 1.49126 −1.00000 0.223857 0 −1.49126 −1.33606 −2.64869 1.00000 0
1.8 1.55802 −1.00000 0.427425 0 −1.55802 3.87455 −2.45010 1.00000 0
1.9 2.38580 −1.00000 3.69205 0 −2.38580 −1.19234 4.03690 1.00000 0
1.10 2.56385 −1.00000 4.57334 0 −2.56385 −4.44946 6.59766 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(47\) \(-1\)

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 3525.2.a.bg 10
5.b even 2 1 3525.2.a.bf 10
5.c odd 4 2 705.2.c.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.b 20 5.c odd 4 2
3525.2.a.bf 10 5.b even 2 1
3525.2.a.bg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\(T_{2}^{10} - \cdots\)
\(T_{7}^{10} - \cdots\)
\(T_{11}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 - 30 T + 6 T^{2} + 101 T^{3} - 21 T^{4} - 91 T^{5} + 25 T^{6} + 29 T^{7} - 9 T^{8} - 3 T^{9} + T^{10} \)
$3$ \( ( 1 + T )^{10} \)
$5$ \( T^{10} \)
$7$ \( -32 - 812 T + 745 T^{2} + 1084 T^{3} - 980 T^{4} - 388 T^{5} + 382 T^{6} + 20 T^{7} - 36 T^{8} + T^{10} \)
$11$ \( 8192 - 9888 T - 11148 T^{2} + 10088 T^{3} + 6776 T^{4} - 1832 T^{5} - 1652 T^{6} - 172 T^{7} + 65 T^{8} + 16 T^{9} + T^{10} \)
$13$ \( -188 - 4090 T + 12334 T^{2} + 12403 T^{3} - 7809 T^{4} - 2933 T^{5} + 1501 T^{6} + 119 T^{7} - 73 T^{8} - T^{9} + T^{10} \)
$17$ \( -13912 + 44704 T - 10314 T^{2} - 43718 T^{3} + 30725 T^{4} - 3148 T^{5} - 2297 T^{6} + 560 T^{7} + 15 T^{8} - 14 T^{9} + T^{10} \)
$19$ \( -1782512 - 777808 T + 674528 T^{2} + 413332 T^{3} - 14365 T^{4} - 51680 T^{5} - 11419 T^{6} - 270 T^{7} + 201 T^{8} + 26 T^{9} + T^{10} \)
$23$ \( -7036 - 29045 T + 48475 T^{2} + 31002 T^{3} - 19750 T^{4} - 5940 T^{5} + 2196 T^{6} + 406 T^{7} - 78 T^{8} - 7 T^{9} + T^{10} \)
$29$ \( -25147124 + 25960080 T - 1313247 T^{2} - 3397370 T^{3} + 131624 T^{4} + 174498 T^{5} + 5326 T^{6} - 2822 T^{7} - 164 T^{8} + 14 T^{9} + T^{10} \)
$31$ \( 891008 - 935440 T - 642920 T^{2} + 232208 T^{3} + 152500 T^{4} - 1224 T^{5} - 9608 T^{6} - 1148 T^{7} + 88 T^{8} + 22 T^{9} + T^{10} \)
$37$ \( -33248 - 32024 T + 255684 T^{2} + 180368 T^{3} - 232328 T^{4} - 11052 T^{5} + 13168 T^{6} + 290 T^{7} - 215 T^{8} - 2 T^{9} + T^{10} \)
$41$ \( -74416 + 210144 T - 143064 T^{2} - 47968 T^{3} + 52117 T^{4} + 7890 T^{5} - 4869 T^{6} - 792 T^{7} + 91 T^{8} + 22 T^{9} + T^{10} \)
$43$ \( 16405888 + 525728 T - 4978208 T^{2} - 962240 T^{3} + 304336 T^{4} + 92560 T^{5} - 1376 T^{6} - 2310 T^{7} - 148 T^{8} + 11 T^{9} + T^{10} \)
$47$ \( ( -1 + T )^{10} \)
$53$ \( -361232 + 1133756 T - 844514 T^{2} - 103324 T^{3} + 200469 T^{4} - 20104 T^{5} - 10083 T^{6} + 1854 T^{7} + 33 T^{8} - 22 T^{9} + T^{10} \)
$59$ \( 7796972 + 20094518 T + 15795534 T^{2} + 2741787 T^{3} - 1593975 T^{4} - 740907 T^{5} - 111271 T^{6} - 5257 T^{7} + 289 T^{8} + 37 T^{9} + T^{10} \)
$61$ \( -507918704 - 315641312 T - 26603870 T^{2} + 22057219 T^{3} + 6171845 T^{4} + 395811 T^{5} - 48799 T^{6} - 7263 T^{7} - 89 T^{8} + 25 T^{9} + T^{10} \)
$67$ \( -1106944 - 1398880 T + 3593736 T^{2} + 1613336 T^{3} - 1341156 T^{4} + 4432 T^{5} + 37020 T^{6} - 612 T^{7} - 340 T^{8} + 4 T^{9} + T^{10} \)
$71$ \( -17743288 - 19722556 T + 8975546 T^{2} + 10933891 T^{3} + 2767861 T^{4} + 105581 T^{5} - 47623 T^{6} - 5765 T^{7} - 3 T^{8} + 27 T^{9} + T^{10} \)
$73$ \( 102689648 - 49648536 T - 7322344 T^{2} + 6791188 T^{3} - 433724 T^{4} - 252200 T^{5} + 36080 T^{6} + 1392 T^{7} - 362 T^{8} - T^{9} + T^{10} \)
$79$ \( -182722976 + 327198948 T - 197093324 T^{2} + 46351704 T^{3} - 2006440 T^{4} - 735632 T^{5} + 74384 T^{6} + 3457 T^{7} - 501 T^{8} - 5 T^{9} + T^{10} \)
$83$ \( -1600000 - 16640000 T - 20348000 T^{2} - 8728000 T^{3} - 1147200 T^{4} + 139904 T^{5} + 38664 T^{6} - 168 T^{7} - 362 T^{8} - 2 T^{9} + T^{10} \)
$89$ \( -11488 + 91760 T + 42904 T^{2} - 101236 T^{3} - 62964 T^{4} + 216 T^{5} + 5100 T^{6} + 378 T^{7} - 130 T^{8} - 9 T^{9} + T^{10} \)
$97$ \( 1969378304 - 533157312 T - 319900592 T^{2} + 5075456 T^{3} + 10428112 T^{4} + 480744 T^{5} - 108304 T^{6} - 8860 T^{7} + 233 T^{8} + 40 T^{9} + T^{10} \)
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