Properties

Label 2415.2.a.v
Level $2415$
Weight $2$
Character orbit 2415.a
Self dual yes
Analytic conductor $19.284$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2838720881\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 16 x^{8} + 30 x^{7} + 83 x^{6} - 137 x^{5} - 164 x^{4} + 208 x^{3} + 108 x^{2} - 83 x - 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 16 x^{8} + 30 x^{7} + 83 x^{6} - 137 x^{5} - 164 x^{4} + 208 x^{3} + 108 x^{2} - 83 x - 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{9} + 21 \nu^{8} + 137 \nu^{7} - 272 \nu^{6} - 700 \nu^{5} + 855 \nu^{4} + 1584 \nu^{3} - 173 \nu^{2} - 1351 \nu - 331 \)\()/131\)
\(\beta_{5}\)\(=\)\((\)\( 11 \nu^{9} + 18 \nu^{8} - 182 \nu^{7} - 308 \nu^{6} + 972 \nu^{5} + 1706 \nu^{4} - 1805 \nu^{3} - 3311 \nu^{2} + 676 \nu + 1438 \)\()/131\)
\(\beta_{6}\)\(=\)\((\)\( -31 \nu^{9} - 15 \nu^{8} + 501 \nu^{7} + 213 \nu^{6} - 2644 \nu^{5} - 723 \nu^{4} + 5325 \nu^{3} - 363 \nu^{2} - 3751 \nu + 1116 \)\()/262\)
\(\beta_{7}\)\(=\)\((\)\( -45 \nu^{9} + 105 \nu^{8} + 685 \nu^{7} - 1491 \nu^{6} - 3238 \nu^{5} + 6109 \nu^{4} + 5431 \nu^{3} - 7153 \nu^{2} - 2825 \nu + 1358 \)\()/262\)
\(\beta_{8}\)\(=\)\((\)\( 29 \nu^{9} - 24 \nu^{8} - 456 \nu^{7} + 367 \nu^{6} + 2241 \nu^{5} - 1707 \nu^{4} - 3794 \nu^{3} + 2537 \nu^{2} + 1937 \nu - 520 \)\()/131\)
\(\beta_{9}\)\(=\)\((\)\( -61 \nu^{9} + 55 \nu^{8} + 1045 \nu^{7} - 781 \nu^{6} - 5938 \nu^{5} + 3175 \nu^{4} + 12701 \nu^{3} - 3647 \nu^{2} - 7381 \nu + 886 \)\()/262\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 8 \beta_{2} + 24\)
\(\nu^{5}\)\(=\)\(\beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 10 \beta_{3} - 2 \beta_{2} + 41 \beta_{1} - 3\)
\(\nu^{6}\)\(=\)\(14 \beta_{8} + 14 \beta_{7} + 12 \beta_{6} - 2 \beta_{5} - 13 \beta_{4} + \beta_{3} + 60 \beta_{2} - 2 \beta_{1} + 161\)
\(\nu^{7}\)\(=\)\(3 \beta_{9} + 2 \beta_{8} + 13 \beta_{7} - 14 \beta_{6} - 13 \beta_{5} - 28 \beta_{4} + 87 \beta_{3} - 27 \beta_{2} + 295 \beta_{1} - 46\)
\(\nu^{8}\)\(=\)\(\beta_{9} + 143 \beta_{8} + 144 \beta_{7} + 111 \beta_{6} - 28 \beta_{5} - 128 \beta_{4} + 14 \beta_{3} + 453 \beta_{2} - 36 \beta_{1} + 1141\)
\(\nu^{9}\)\(=\)\(48 \beta_{9} + 36 \beta_{8} + 128 \beta_{7} - 144 \beta_{6} - 125 \beta_{5} - 286 \beta_{4} + 725 \beta_{3} - 271 \beta_{2} + 2184 \beta_{1} - 504\)

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
−2.81587
−2.13650
−1.42324
−0.866400
−0.128823
0.663795
1.24544
2.24070
2.42079
2.80011
−2.81587 1.00000 5.92915 1.00000 −2.81587 −1.00000 −11.0640 1.00000 −2.81587
1.2 −2.13650 1.00000 2.56465 1.00000 −2.13650 −1.00000 −1.20638 1.00000 −2.13650
1.3 −1.42324 1.00000 0.0256085 1.00000 −1.42324 −1.00000 2.81003 1.00000 −1.42324
1.4 −0.866400 1.00000 −1.24935 1.00000 −0.866400 −1.00000 2.81524 1.00000 −0.866400
1.5 −0.128823 1.00000 −1.98340 1.00000 −0.128823 −1.00000 0.513155 1.00000 −0.128823
1.6 0.663795 1.00000 −1.55938 1.00000 0.663795 −1.00000 −2.36270 1.00000 0.663795
1.7 1.24544 1.00000 −0.448869 1.00000 1.24544 −1.00000 −3.04993 1.00000 1.24544
1.8 2.24070 1.00000 3.02074 1.00000 2.24070 −1.00000 2.28717 1.00000 2.24070
1.9 2.42079 1.00000 3.86022 1.00000 2.42079 −1.00000 4.50319 1.00000 2.42079
1.10 2.80011 1.00000 5.84064 1.00000 2.80011 −1.00000 10.7542 1.00000 2.80011
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 2415.2.a.v 10
3.b odd 2 1 7245.2.a.bu 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.v 10 1.a even 1 1 trivial
7245.2.a.bu 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):

\(T_{2}^{10} - \cdots\)
\(T_{11}^{10} + \cdots\)
\(T_{13}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -12 - 83 T + 108 T^{2} + 208 T^{3} - 164 T^{4} - 137 T^{5} + 83 T^{6} + 30 T^{7} - 16 T^{8} - 2 T^{9} + T^{10} \)
$3$ \( ( -1 + T )^{10} \)
$5$ \( ( -1 + T )^{10} \)
$7$ \( ( 1 + T )^{10} \)
$11$ \( 15712 - 6240 T - 43158 T^{2} + 49558 T^{3} - 14022 T^{4} - 3407 T^{5} + 1877 T^{6} + 29 T^{7} - 75 T^{8} + T^{9} + T^{10} \)
$13$ \( 801312 - 1083464 T + 336224 T^{2} + 105398 T^{3} - 60480 T^{4} - 1938 T^{5} + 3625 T^{6} - 72 T^{7} - 97 T^{8} + 2 T^{9} + T^{10} \)
$17$ \( -96704 - 1017920 T + 108272 T^{2} + 368384 T^{3} - 38820 T^{4} - 30636 T^{5} + 3453 T^{6} + 892 T^{7} - 107 T^{8} - 8 T^{9} + T^{10} \)
$19$ \( -857856 - 1252400 T - 49060 T^{2} + 358852 T^{3} + 1538 T^{4} - 31257 T^{5} + 1331 T^{6} + 1029 T^{7} - 75 T^{8} - 11 T^{9} + T^{10} \)
$23$ \( ( -1 + T )^{10} \)
$29$ \( 769536 + 3175936 T + 1730560 T^{2} - 184320 T^{3} - 239488 T^{4} - 12272 T^{5} + 9496 T^{6} + 604 T^{7} - 160 T^{8} - 6 T^{9} + T^{10} \)
$31$ \( -5701632 - 9816064 T - 2045312 T^{2} + 1796448 T^{3} + 171040 T^{4} - 108160 T^{5} - 1136 T^{6} + 2474 T^{7} - 100 T^{8} - 16 T^{9} + T^{10} \)
$37$ \( 16464432 - 11363576 T - 438788 T^{2} + 1739862 T^{3} - 243038 T^{4} - 66506 T^{5} + 14465 T^{6} + 536 T^{7} - 215 T^{8} + T^{10} \)
$41$ \( 117176544 - 19529424 T - 31527792 T^{2} + 3932712 T^{3} + 1667322 T^{4} - 266205 T^{5} - 18665 T^{6} + 4721 T^{7} - 63 T^{8} - 23 T^{9} + T^{10} \)
$43$ \( -377088 - 1325696 T + 506224 T^{2} + 444960 T^{3} - 141768 T^{4} - 35240 T^{5} + 10379 T^{6} + 662 T^{7} - 197 T^{8} - 4 T^{9} + T^{10} \)
$47$ \( 5128192 - 1598464 T - 3133824 T^{2} + 693312 T^{3} + 412512 T^{4} - 95488 T^{5} - 11056 T^{6} + 3192 T^{7} - 48 T^{8} - 21 T^{9} + T^{10} \)
$53$ \( 193211776 + 186513472 T + 34958688 T^{2} - 10663408 T^{3} - 2881048 T^{4} + 154056 T^{5} + 55814 T^{6} - 758 T^{7} - 406 T^{8} + T^{9} + T^{10} \)
$59$ \( 1190323968 - 552181384 T - 15355578 T^{2} + 31041136 T^{3} - 1337284 T^{4} - 618163 T^{5} + 37087 T^{6} + 5143 T^{7} - 335 T^{8} - 15 T^{9} + T^{10} \)
$61$ \( -33034000 + 30844400 T - 6518820 T^{2} - 1817004 T^{3} + 796576 T^{4} - 20483 T^{5} - 23103 T^{6} + 2371 T^{7} + 147 T^{8} - 29 T^{9} + T^{10} \)
$67$ \( -63414272 + 113747968 T - 54941952 T^{2} + 3218976 T^{3} + 2778864 T^{4} - 309558 T^{5} - 39569 T^{6} + 5678 T^{7} + 89 T^{8} - 32 T^{9} + T^{10} \)
$71$ \( -19522609152 + 789148672 T + 1144633344 T^{2} - 33318592 T^{3} - 21215168 T^{4} + 431744 T^{5} + 175848 T^{6} - 2228 T^{7} - 680 T^{8} + 4 T^{9} + T^{10} \)
$73$ \( 313816688 - 127058264 T - 43965068 T^{2} + 12202238 T^{3} + 1537560 T^{4} - 447050 T^{5} - 8619 T^{6} + 6360 T^{7} - 245 T^{8} - 18 T^{9} + T^{10} \)
$79$ \( 28602368 + 62646016 T + 38234944 T^{2} + 4560096 T^{3} - 2041696 T^{4} - 308040 T^{5} + 51440 T^{6} + 3306 T^{7} - 438 T^{8} - 8 T^{9} + T^{10} \)
$83$ \( 1263616 - 4563776 T - 2280544 T^{2} + 3229820 T^{3} - 695732 T^{4} - 63708 T^{5} + 25713 T^{6} + 282 T^{7} - 309 T^{8} - 2 T^{9} + T^{10} \)
$89$ \( -819561472 - 131575552 T + 152348544 T^{2} + 8275520 T^{3} - 6620896 T^{4} - 218992 T^{5} + 98032 T^{6} + 2164 T^{7} - 544 T^{8} - 6 T^{9} + T^{10} \)
$97$ \( -436736 + 1326848 T + 1648384 T^{2} - 400768 T^{3} - 454912 T^{4} + 18528 T^{5} + 20776 T^{6} - 260 T^{7} - 272 T^{8} + 2 T^{9} + T^{10} \)
show more
show less