Properties

Label 8034.2.a.s
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 4 x^{9} - 15 x^{8} + 72 x^{7} - 27 x^{6} - 115 x^{5} + 54 x^{4} + 68 x^{3} - 15 x^{2} - 15 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 4 x^{9} - 15 x^{8} + 72 x^{7} - 27 x^{6} - 115 x^{5} + 54 x^{4} + 68 x^{3} - 15 x^{2} - 15 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 20 \nu^{9} - 168 \nu^{8} - 115 \nu^{7} + 3250 \nu^{6} - 3919 \nu^{5} - 8561 \nu^{4} + 7648 \nu^{3} + 7872 \nu^{2} - 3054 \nu - 1728 \)\()/163\)
\(\beta_{3}\)\(=\)\((\)\( -54 \nu^{9} - 68 \nu^{8} + 1696 \nu^{7} + 1005 \nu^{6} - 14211 \nu^{5} + 2593 \nu^{4} + 18177 \nu^{3} - 1890 \nu^{2} - 5055 \nu - 94 \)\()/163\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{9} + 369 \nu^{8} - 804 \nu^{7} - 6760 \nu^{6} + 17110 \nu^{5} + 11815 \nu^{4} - 27696 \nu^{3} - 9932 \nu^{2} + 10486 \nu + 3027 \)\()/163\)
\(\beta_{5}\)\(=\)\((\)\( 114 \nu^{9} - 599 \nu^{8} - 1226 \nu^{7} + 10538 \nu^{6} - 11564 \nu^{5} - 12628 \nu^{4} + 16340 \nu^{3} + 6598 \nu^{2} - 4922 \nu - 1830 \)\()/163\)
\(\beta_{6}\)\(=\)\((\)\( 159 \nu^{9} - 651 \nu^{8} - 2259 \nu^{7} + 11412 \nu^{6} - 6323 \nu^{5} - 13349 \nu^{4} + 7957 \nu^{3} + 5402 \nu^{2} - 1280 \nu - 665 \)\()/163\)
\(\beta_{7}\)\(=\)\((\)\( 251 \nu^{9} - 1163 \nu^{8} - 3114 \nu^{7} + 20331 \nu^{6} - 18189 \nu^{5} - 22542 \nu^{4} + 26903 \nu^{3} + 9111 \nu^{2} - 9167 \nu - 2322 \)\()/163\)
\(\beta_{8}\)\(=\)\((\)\( -338 \nu^{9} + 1307 \nu^{8} + 5122 \nu^{7} - 23303 \nu^{6} + 8252 \nu^{5} + 33629 \nu^{4} - 17531 \nu^{3} - 14601 \nu^{2} + 6103 \nu + 1591 \)\()/163\)
\(\beta_{9}\)\(=\)\((\)\( 481 \nu^{9} - 1791 \nu^{8} - 7452 \nu^{7} + 31789 \nu^{6} - 8734 \nu^{5} - 43813 \nu^{4} + 18685 \nu^{3} + 17813 \nu^{2} - 5657 \nu - 2145 \)\()/163\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-3 \beta_{9} - 3 \beta_{8} + 3 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 12 \beta_{1} - 5\)
\(\nu^{4}\)\(=\)\(-\beta_{9} - \beta_{8} - 18 \beta_{7} + 6 \beta_{6} + 29 \beta_{5} - 21 \beta_{4} - 18 \beta_{3} - 38 \beta_{2} - 6 \beta_{1} + 67\)
\(\nu^{5}\)\(=\)\(-50 \beta_{9} - 47 \beta_{8} - \beta_{7} + 70 \beta_{6} - 35 \beta_{5} - 47 \beta_{4} - 38 \beta_{3} - 60 \beta_{2} + 182 \beta_{1} - 106\)
\(\nu^{6}\)\(=\)\(-12 \beta_{9} - 3 \beta_{8} - 299 \beta_{7} + 157 \beta_{6} + 419 \beta_{5} - 388 \beta_{4} - 309 \beta_{3} - 655 \beta_{2} - 129 \beta_{1} + 1040\)
\(\nu^{7}\)\(=\)\(-807 \beta_{9} - 728 \beta_{8} - 31 \beta_{7} + 1326 \beta_{6} - 734 \beta_{5} - 910 \beta_{4} - 674 \beta_{3} - 1093 \beta_{2} + 2916 \beta_{1} - 1863\)
\(\nu^{8}\)\(=\)\(-176 \beta_{9} + 60 \beta_{8} - 4970 \beta_{7} + 3126 \beta_{6} + 6429 \beta_{5} - 6818 \beta_{4} - 5256 \beta_{3} - 11136 \beta_{2} - 2220 \beta_{1} + 16758\)
\(\nu^{9}\)\(=\)\(-13247 \beta_{9} - 11685 \beta_{8} - 846 \beta_{7} + 23508 \beta_{6} - 13154 \beta_{5} - 16494 \beta_{4} - 11806 \beta_{3} - 19470 \beta_{2} + 47740 \beta_{1} - 31006\)

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
−0.297898
4.13381
−0.671383
0.757747
−0.207472
1.27214
−0.894486
2.73145
1.22324
−4.04714
−1.00000 −1.00000 1.00000 −4.09900 1.00000 −3.00074 −1.00000 1.00000 4.09900
1.2 −1.00000 −1.00000 1.00000 −2.24210 1.00000 −4.80624 −1.00000 1.00000 2.24210
1.3 −1.00000 −1.00000 1.00000 −1.82988 1.00000 1.45707 −1.00000 1.00000 1.82988
1.4 −1.00000 −1.00000 1.00000 −0.284726 1.00000 −1.57147 −1.00000 1.00000 0.284726
1.5 −1.00000 −1.00000 1.00000 0.448250 1.00000 1.73996 −1.00000 1.00000 −0.448250
1.6 −1.00000 −1.00000 1.00000 1.40775 1.00000 −4.52824 −1.00000 1.00000 −1.40775
1.7 −1.00000 −1.00000 1.00000 2.08098 1.00000 1.85452 −1.00000 1.00000 −2.08098
1.8 −1.00000 −1.00000 1.00000 2.66497 1.00000 2.36859 −1.00000 1.00000 −2.66497
1.9 −1.00000 −1.00000 1.00000 3.54892 1.00000 0.812393 −1.00000 1.00000 −3.54892
1.10 −1.00000 −1.00000 1.00000 4.30484 1.00000 −3.32584 −1.00000 1.00000 −4.30484
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(13\) \(1\)
\(103\) \(1\)

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 8034.2.a.s 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{10} - \cdots\)
\(T_{7}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{10} \)
$3$ \( ( 1 + T )^{10} \)
$5$ \( 256 + 112 T - 2392 T^{2} + 1671 T^{3} + 1013 T^{4} - 945 T^{5} - 49 T^{6} + 151 T^{7} - 17 T^{8} - 6 T^{9} + T^{10} \)
$7$ \( -3088 + 5417 T + 628 T^{2} - 4323 T^{3} + 540 T^{4} + 1207 T^{5} - 123 T^{6} - 162 T^{7} + 9 T^{9} + T^{10} \)
$11$ \( 4057 + 15121 T + 10147 T^{2} - 11724 T^{3} - 7450 T^{4} + 1659 T^{5} + 1175 T^{6} - 69 T^{7} - 62 T^{8} + T^{9} + T^{10} \)
$13$ \( ( 1 + T )^{10} \)
$17$ \( -520 - 100 T + 1898 T^{2} + 645 T^{3} - 1950 T^{4} - 849 T^{5} + 444 T^{6} + 148 T^{7} - 43 T^{8} - 5 T^{9} + T^{10} \)
$19$ \( 310484 + 180984 T - 209991 T^{2} - 100925 T^{3} + 27586 T^{4} + 16609 T^{5} + 165 T^{6} - 714 T^{7} - 58 T^{8} + 9 T^{9} + T^{10} \)
$23$ \( 236641 + 2053717 T + 1326503 T^{2} - 218266 T^{3} - 181746 T^{4} + 5699 T^{5} + 8333 T^{6} + 13 T^{7} - 154 T^{8} - T^{9} + T^{10} \)
$29$ \( 4098880 - 468944 T - 1324443 T^{2} + 21470 T^{3} + 153670 T^{4} + 9534 T^{5} - 7281 T^{6} - 929 T^{7} + 96 T^{8} + 22 T^{9} + T^{10} \)
$31$ \( 10520 - 42880 T - 156283 T^{2} - 86505 T^{3} + 28386 T^{4} + 16947 T^{5} - 893 T^{6} - 896 T^{7} - 34 T^{8} + 13 T^{9} + T^{10} \)
$37$ \( -47782309 - 23354850 T + 13126405 T^{2} + 3292738 T^{3} - 716753 T^{4} - 125257 T^{5} + 17463 T^{6} + 1881 T^{7} - 209 T^{8} - 10 T^{9} + T^{10} \)
$41$ \( -3167460 + 4941987 T + 3600175 T^{2} - 2028741 T^{3} - 177860 T^{4} + 124950 T^{5} + 3998 T^{6} - 2592 T^{7} - 100 T^{8} + 18 T^{9} + T^{10} \)
$43$ \( 64389274 - 41331027 T - 17321913 T^{2} + 9185349 T^{3} - 79265 T^{4} - 318128 T^{5} + 21738 T^{6} + 3226 T^{7} - 285 T^{8} - 10 T^{9} + T^{10} \)
$47$ \( 45377510 + 22334339 T - 14616844 T^{2} - 3209627 T^{3} + 1911999 T^{4} - 95344 T^{5} - 40303 T^{6} + 4762 T^{7} + 55 T^{8} - 28 T^{9} + T^{10} \)
$53$ \( 192209152 - 505474936 T + 70992640 T^{2} + 21093333 T^{3} - 3279622 T^{4} - 329075 T^{5} + 53685 T^{6} + 2285 T^{7} - 382 T^{8} - 6 T^{9} + T^{10} \)
$59$ \( 352672 - 3717544 T - 1087964 T^{2} + 1290985 T^{3} + 32048 T^{4} - 120126 T^{5} + 11970 T^{6} + 1909 T^{7} - 235 T^{8} - 7 T^{9} + T^{10} \)
$61$ \( 27180500 + 8336293 T - 30193095 T^{2} - 5930161 T^{3} + 1580425 T^{4} + 382468 T^{5} - 6204 T^{6} - 5270 T^{7} - 169 T^{8} + 20 T^{9} + T^{10} \)
$67$ \( -5548352 - 9258208 T - 2046786 T^{2} + 2682433 T^{3} + 1314073 T^{4} + 117092 T^{5} - 26819 T^{6} - 4942 T^{7} - 102 T^{8} + 21 T^{9} + T^{10} \)
$71$ \( 425984 + 316416 T - 822240 T^{2} - 875131 T^{3} - 59060 T^{4} + 106115 T^{5} + 2701 T^{6} - 2625 T^{7} - 95 T^{8} + 19 T^{9} + T^{10} \)
$73$ \( 8301920 - 21531768 T + 2089428 T^{2} + 8863749 T^{3} - 2021057 T^{4} - 196195 T^{5} + 53132 T^{6} + 1255 T^{7} - 421 T^{8} - 3 T^{9} + T^{10} \)
$79$ \( -9716480 - 8784448 T + 4907350 T^{2} + 2705951 T^{3} - 1329243 T^{4} + 7535 T^{5} + 43935 T^{6} - 2152 T^{7} - 361 T^{8} + 11 T^{9} + T^{10} \)
$83$ \( -15802272 - 38207832 T - 4076548 T^{2} + 6665575 T^{3} + 75976 T^{4} - 288884 T^{5} + 5807 T^{6} + 4357 T^{7} - 165 T^{8} - 20 T^{9} + T^{10} \)
$89$ \( -361792 + 3430768 T - 7788837 T^{2} - 1866554 T^{3} + 1708764 T^{4} - 199845 T^{5} - 23059 T^{6} + 5012 T^{7} - 96 T^{8} - 22 T^{9} + T^{10} \)
$97$ \( 3124648 + 5836264 T - 9165061 T^{2} - 4673828 T^{3} - 101715 T^{4} + 181871 T^{5} + 13859 T^{6} - 2352 T^{7} - 219 T^{8} + 10 T^{9} + T^{10} \)
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