Properties

Label 1045.2.a.i
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.34436701122\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 9 x^{6} + 12 x^{5} + 28 x^{4} - 17 x^{3} - 28 x^{2} + 6 x + 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 9 x^{6} + 12 x^{5} + 28 x^{4} - 17 x^{3} - 28 x^{2} + 6 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 10 \nu^{3} + 21 \nu^{2} - 8 \nu - 12 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 6 \nu^{5} + 18 \nu^{4} + 11 \nu^{3} - 29 \nu^{2} - 4 \nu + 11 \)
\(\beta_{7}\)\(=\)\( -\nu^{7} + 2 \nu^{6} + 9 \nu^{5} - 13 \nu^{4} - 24 \nu^{3} + 19 \nu^{2} + 12 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 8 \beta_{2} + 9 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + 9 \beta_{3} + 19 \beta_{2} + 31 \beta_{1} + 31\)
\(\nu^{6}\)\(=\)\(2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 14 \beta_{4} + 24 \beta_{3} + 61 \beta_{2} + 71 \beta_{1} + 109\)
\(\nu^{7}\)\(=\)\(12 \beta_{7} + 13 \beta_{6} + 15 \beta_{5} + 42 \beta_{4} + 79 \beta_{3} + 160 \beta_{2} + 215 \beta_{1} + 268\)

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
−1.82030
−1.57959
−0.865980
−0.649219
0.714778
1.06639
2.25600
2.87791
−2.82030 0.978567 5.95409 1.00000 −2.75985 −4.31627 −11.1517 −2.04241 −2.82030
1.2 −2.57959 −2.53274 4.65426 1.00000 6.53343 2.87748 −6.84689 3.41479 −2.57959
1.3 −1.86598 2.01288 1.48188 1.00000 −3.75600 −2.04136 0.966800 1.05170 −1.86598
1.4 −1.64922 −2.98027 0.719922 1.00000 4.91512 −5.09280 2.11113 5.88203 −1.64922
1.5 −0.285222 −1.61587 −1.91865 1.00000 0.460881 −1.06724 1.11768 −0.388965 −0.285222
1.6 0.0663929 −0.255194 −1.99559 1.00000 −0.0169431 2.56056 −0.265279 −2.93488 0.0663929
1.7 1.25600 0.772194 −0.422456 1.00000 0.969878 −3.10169 −3.04261 −2.40372 1.25600
1.8 1.87791 −3.37956 1.52654 1.00000 −6.34651 −0.818685 −0.889109 8.42145 1.87791
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)
\(19\) \(1\)

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 1045.2.a.i 8
3.b odd 2 1 9405.2.a.bf 8
5.b even 2 1 5225.2.a.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.i 8 1.a even 1 1 trivial
5225.2.a.o 8 5.b even 2 1
9405.2.a.bf 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 11 T + 60 T^{2} + 21 T^{3} - 47 T^{4} - 28 T^{5} + 5 T^{6} + 6 T^{7} + T^{8} \)
$3$ \( -16 - 44 T + 92 T^{2} + 59 T^{3} - 67 T^{4} - 40 T^{5} + 7 T^{6} + 7 T^{7} + T^{8} \)
$5$ \( ( -1 + T )^{8} \)
$7$ \( 896 + 2384 T + 1792 T^{2} - 47 T^{3} - 489 T^{4} - 120 T^{5} + 23 T^{6} + 11 T^{7} + T^{8} \)
$11$ \( ( -1 + T )^{8} \)
$13$ \( -13392 + 25704 T + 17766 T^{2} - 1453 T^{3} - 2544 T^{4} - 329 T^{5} + 65 T^{6} + 17 T^{7} + T^{8} \)
$17$ \( 8344 + 23908 T + 22974 T^{2} + 7505 T^{3} - 434 T^{4} - 585 T^{5} - 47 T^{6} + 9 T^{7} + T^{8} \)
$19$ \( ( 1 + T )^{8} \)
$23$ \( 1504 + 88 T - 5196 T^{2} + 2571 T^{3} + 832 T^{4} - 289 T^{5} - 48 T^{6} + 8 T^{7} + T^{8} \)
$29$ \( 84152 + 84028 T - 56750 T^{2} - 5567 T^{3} + 5302 T^{4} - 73 T^{5} - 133 T^{6} + 3 T^{7} + T^{8} \)
$31$ \( -35840 + 95104 T - 35124 T^{2} - 7061 T^{3} + 3818 T^{4} + 79 T^{5} - 115 T^{6} + T^{7} + T^{8} \)
$37$ \( -280640 - 185152 T + 51526 T^{2} + 29089 T^{3} - 1995 T^{4} - 1394 T^{5} - 28 T^{6} + 17 T^{7} + T^{8} \)
$41$ \( 30440 - 50028 T - 13742 T^{2} + 10435 T^{3} + 2283 T^{4} - 531 T^{5} - 110 T^{6} + 5 T^{7} + T^{8} \)
$43$ \( -18944 + 12032 T + 15100 T^{2} - 1005 T^{3} - 2631 T^{4} - 290 T^{5} + 101 T^{6} + 21 T^{7} + T^{8} \)
$47$ \( 29984 + 132584 T + 113116 T^{2} + 30399 T^{3} - 226 T^{4} - 1153 T^{5} - 104 T^{6} + 8 T^{7} + T^{8} \)
$53$ \( 10528 - 47584 T + 16662 T^{2} + 7563 T^{3} - 2166 T^{4} - 579 T^{5} + 59 T^{6} + 19 T^{7} + T^{8} \)
$59$ \( 6694912 + 13720992 T + 5411584 T^{2} + 321439 T^{3} - 72498 T^{4} - 8117 T^{5} + 57 T^{6} + 33 T^{7} + T^{8} \)
$61$ \( 25298072 - 457052 T - 1826334 T^{2} + 32971 T^{3} + 42070 T^{4} - 499 T^{5} - 369 T^{6} + T^{7} + T^{8} \)
$67$ \( 56432 + 148348 T + 112344 T^{2} + 13061 T^{3} - 8866 T^{4} - 2124 T^{5} - 43 T^{6} + 18 T^{7} + T^{8} \)
$71$ \( -833024 - 600192 T + 11164 T^{2} + 68299 T^{3} + 3954 T^{4} - 2305 T^{5} - 106 T^{6} + 18 T^{7} + T^{8} \)
$73$ \( -22472 + 16324 T + 22750 T^{2} + 701 T^{3} - 2704 T^{4} - 410 T^{5} + 67 T^{6} + 18 T^{7} + T^{8} \)
$79$ \( -197248 - 1156976 T - 191152 T^{2} + 80581 T^{3} + 13209 T^{4} - 1402 T^{5} - 234 T^{6} + 5 T^{7} + T^{8} \)
$83$ \( 2043584 + 2611504 T + 666740 T^{2} - 115971 T^{3} - 51030 T^{4} - 3841 T^{5} + 197 T^{6} + 33 T^{7} + T^{8} \)
$89$ \( 216 + 3132 T + 9930 T^{2} + 8089 T^{3} - 1092 T^{4} - 938 T^{5} + 23 T^{6} + 20 T^{7} + T^{8} \)
$97$ \( 9664 + 352512 T - 116266 T^{2} - 130095 T^{3} + 43918 T^{4} + 556 T^{5} - 419 T^{6} + T^{8} \)
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