Properties

Label 1859.2.a.o
Level $1859$
Weight $2$
Character orbit 1859.a
Self dual yes
Analytic conductor $14.844$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8441897358\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 8 x^{6} + 18 x^{5} + 7 x^{4} - 22 x^{3} - 3 x^{2} + 6 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 8 x^{6} + 18 x^{5} + 7 x^{4} - 22 x^{3} - 3 x^{2} + 6 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 7 \nu^{3} + 3 \nu^{2} + 5 \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{6} - \nu^{5} - 8 \nu^{4} + 10 \nu^{3} + 9 \nu^{2} - 10 \nu - 2 \)
\(\beta_{4}\)\(=\)\( \nu^{7} - \nu^{6} - 8 \nu^{5} + 10 \nu^{4} + 9 \nu^{3} - 10 \nu^{2} - 2 \nu + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{7} - 2 \nu^{6} - 7 \nu^{5} + 17 \nu^{4} - \nu^{3} - 13 \nu^{2} + 5 \nu + 1 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 2 \nu^{6} - 7 \nu^{5} + 18 \nu^{4} - 19 \nu^{2} + 3 \nu + 4 \)
\(\beta_{7}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 7 \nu^{5} + 25 \nu^{4} - 3 \nu^{3} - 24 \nu^{2} + 4 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{4} + \beta_{3} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(6 \beta_{7} - 6 \beta_{6} - 7 \beta_{5} + 7 \beta_{4} - \beta_{3} + 6 \beta_{2} + 3 \beta_{1} + 10\)
\(\nu^{5}\)\(=\)\(-3 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} - 10 \beta_{4} + 7 \beta_{3} - 2 \beta_{2} + 27 \beta_{1} - 12\)
\(\nu^{6}\)\(=\)\(36 \beta_{7} - 39 \beta_{6} - 44 \beta_{5} + 47 \beta_{4} - 10 \beta_{3} + 37 \beta_{2} + 2 \beta_{1} + 62\)
\(\nu^{7}\)\(=\)\(-38 \beta_{7} + 82 \beta_{6} + 40 \beta_{5} - 83 \beta_{4} + 47 \beta_{3} - 29 \beta_{2} + 155 \beta_{1} - 106\)

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
2.20621
2.07742
1.37460
0.681121
−0.170890
−0.549080
−0.920076
−2.69931
−2.20621 2.48500 2.86737 −1.68345 −5.48243 0.336274 −1.91361 3.17521 3.71405
1.2 −2.07742 −1.13599 2.31569 −4.26888 2.35994 −3.46221 −0.655817 −1.70952 8.86827
1.3 −1.37460 −2.08494 −0.110482 2.34290 2.86595 −1.29593 2.90106 1.34697 −3.22054
1.4 −0.681121 −0.0549966 −1.53607 −3.68496 0.0374594 0.234733 2.40849 −2.99698 2.50991
1.5 0.170890 2.94878 −1.97080 −1.29847 0.503917 −3.65280 −0.678569 5.69533 −0.221895
1.6 0.549080 −1.45990 −1.69851 −0.775548 −0.801604 1.41094 −2.03078 −0.868681 −0.425838
1.7 0.920076 0.898842 −1.15346 2.76396 0.827003 −3.49246 −2.90142 −2.19208 2.54305
1.8 2.69931 −1.59679 5.28626 −1.39555 −4.31023 −4.07855 8.87064 −0.450256 −3.76701
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(13\) \(-1\)

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 1859.2.a.o 8
13.b even 2 1 1859.2.a.p 8
13.f odd 12 2 143.2.j.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.j.b 16 13.f odd 12 2
1859.2.a.o 8 1.a even 1 1 trivial
1859.2.a.p 8 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):

\(T_{2}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T - 3 T^{2} + 22 T^{3} + 7 T^{4} - 18 T^{5} - 8 T^{6} + 2 T^{7} + T^{8} \)
$3$ \( -2 - 38 T - 27 T^{2} + 52 T^{3} + 48 T^{4} - 8 T^{5} - 13 T^{6} + T^{8} \)
$5$ \( 241 + 744 T + 719 T^{2} + 104 T^{3} - 201 T^{4} - 88 T^{5} + 6 T^{6} + 8 T^{7} + T^{8} \)
$7$ \( -26 + 158 T - 111 T^{2} - 374 T^{3} - 76 T^{4} + 118 T^{5} + 69 T^{6} + 14 T^{7} + T^{8} \)
$11$ \( ( -1 + T )^{8} \)
$13$ \( T^{8} \)
$17$ \( -3119 + 22092 T - 2527 T^{2} - 4776 T^{3} + 725 T^{4} + 306 T^{5} - 50 T^{6} - 6 T^{7} + T^{8} \)
$19$ \( -2018 - 6394 T + 775 T^{2} + 3022 T^{3} + 399 T^{4} - 280 T^{5} - 56 T^{6} + 4 T^{7} + T^{8} \)
$23$ \( 33844 + 55856 T + 26393 T^{2} - 340 T^{3} - 3162 T^{4} - 666 T^{5} + 9 T^{6} + 14 T^{7} + T^{8} \)
$29$ \( -2063 - 1716 T + 1752 T^{2} + 1502 T^{3} - 149 T^{4} - 240 T^{5} - 11 T^{6} + 10 T^{7} + T^{8} \)
$31$ \( -50588 - 271876 T - 224067 T^{2} - 69768 T^{3} - 6465 T^{4} + 1244 T^{5} + 359 T^{6} + 32 T^{7} + T^{8} \)
$37$ \( 61 - 628 T - 1482 T^{2} - 16 T^{3} + 1167 T^{4} + 764 T^{5} + 203 T^{6} + 24 T^{7} + T^{8} \)
$41$ \( 1711933 - 376862 T - 237335 T^{2} + 27542 T^{3} + 11157 T^{4} - 512 T^{5} - 194 T^{6} + 2 T^{7} + T^{8} \)
$43$ \( -3656 - 620 T + 9533 T^{2} + 304 T^{3} - 5025 T^{4} + 1470 T^{5} - 60 T^{6} - 14 T^{7} + T^{8} \)
$47$ \( 54286 + 167522 T + 143715 T^{2} + 22970 T^{3} - 5775 T^{4} - 1516 T^{5} - 4 T^{6} + 18 T^{7} + T^{8} \)
$53$ \( 1610773 - 576322 T - 197893 T^{2} + 50858 T^{3} + 10095 T^{4} - 1182 T^{5} - 186 T^{6} + 8 T^{7} + T^{8} \)
$59$ \( -773474 - 178002 T + 222779 T^{2} + 63870 T^{3} - 8455 T^{4} - 2976 T^{5} - 92 T^{6} + 18 T^{7} + T^{8} \)
$61$ \( 514561 - 250118 T - 170230 T^{2} + 36634 T^{3} + 10533 T^{4} - 816 T^{5} - 195 T^{6} + 4 T^{7} + T^{8} \)
$67$ \( -27643106 + 11012950 T + 189753 T^{2} - 427584 T^{3} + 22212 T^{4} + 4450 T^{5} - 301 T^{6} - 14 T^{7} + T^{8} \)
$71$ \( -12355388 + 5134240 T + 162301 T^{2} - 268488 T^{3} + 16954 T^{4} + 3320 T^{5} - 267 T^{6} - 12 T^{7} + T^{8} \)
$73$ \( 4343872 + 4405504 T + 1137440 T^{2} - 28064 T^{3} - 46412 T^{4} - 5296 T^{5} - 7 T^{6} + 26 T^{7} + T^{8} \)
$79$ \( 2764672 - 2050176 T + 429248 T^{2} + 24736 T^{3} - 19080 T^{4} + 1624 T^{5} + 138 T^{6} - 26 T^{7} + T^{8} \)
$83$ \( -14622626 - 6723358 T + 332467 T^{2} + 384088 T^{3} + 16023 T^{4} - 4636 T^{5} - 266 T^{6} + 16 T^{7} + T^{8} \)
$89$ \( -424748 + 5332000 T + 398577 T^{2} - 517144 T^{3} + 44678 T^{4} + 4004 T^{5} - 447 T^{6} - 8 T^{7} + T^{8} \)
$97$ \( -3157100 - 1577200 T + 313173 T^{2} + 118164 T^{3} - 4977 T^{4} - 2728 T^{5} - 49 T^{6} + 20 T^{7} + T^{8} \)
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