Properties

Label 3549.2.a.be
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 11 x^{7} + 8 x^{6} + 37 x^{5} - 18 x^{4} - 41 x^{3} + 12 x^{2} + 6 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 11 x^{7} + 8 x^{6} + 37 x^{5} - 18 x^{4} - 41 x^{3} + 12 x^{2} + 6 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 8 \nu^{5} + 5 \nu^{4} + 13 \nu^{3} - 3 \nu^{2} + 2 \nu - 1 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{8} - \nu^{7} - 10 \nu^{6} + 7 \nu^{5} + 27 \nu^{4} - 11 \nu^{3} - 14 \nu^{2} + \nu - 8 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{8} + 2 \nu^{7} + 9 \nu^{6} - 15 \nu^{5} - 22 \nu^{4} + 24 \nu^{3} + 15 \nu^{2} - 3 \nu - 1 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 10 \nu^{5} + 7 \nu^{4} + 27 \nu^{3} - 11 \nu^{2} - 18 \nu + 1 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{8} - 13 \nu^{6} - \nu^{5} + 52 \nu^{4} + 4 \nu^{3} - 65 \nu^{2} - 3 \nu + 9 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{8} - 2 \nu^{7} - 21 \nu^{6} + 14 \nu^{5} + 66 \nu^{4} - 21 \nu^{3} - 68 \nu^{2} - \nu + 9 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( 2 \nu^{8} - 2 \nu^{7} - 21 \nu^{6} + 15 \nu^{5} + 65 \nu^{4} - 30 \nu^{3} - 62 \nu^{2} + 17 \nu + 4 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} - \beta_{6} + 8 \beta_{4} + 5 \beta_{3} - 7 \beta_{2} + 8 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(-7 \beta_{8} + 8 \beta_{7} - \beta_{6} - 9 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} - \beta_{2} + 29 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(-\beta_{8} + 11 \beta_{7} - 12 \beta_{6} - \beta_{5} + 57 \beta_{4} + 29 \beta_{3} - 44 \beta_{2} + 58 \beta_{1} + 41\)
\(\nu^{7}\)\(=\)\(-44 \beta_{8} + 57 \beta_{7} - 15 \beta_{6} - 60 \beta_{5} + 95 \beta_{4} + 58 \beta_{3} - 16 \beta_{2} + 186 \beta_{1} + 16\)
\(\nu^{8}\)\(=\)\(-16 \beta_{8} + 95 \beta_{7} - 101 \beta_{6} - 18 \beta_{5} + 397 \beta_{4} + 186 \beta_{3} - 274 \beta_{2} + 415 \beta_{1} + 239\)

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
2.67225
1.90042
1.43930
0.494186
0.146218
−0.374817
−1.36450
−1.60159
−2.31147
−2.67225 −1.00000 5.14091 1.32568 2.67225 1.00000 −8.39329 1.00000 −3.54256
1.2 −1.90042 −1.00000 1.61158 1.76582 1.90042 1.00000 0.738153 1.00000 −3.35579
1.3 −1.43930 −1.00000 0.0715948 2.86638 1.43930 1.00000 2.77556 1.00000 −4.12559
1.4 −0.494186 −1.00000 −1.75578 −1.64627 0.494186 1.00000 1.85605 1.00000 0.813565
1.5 −0.146218 −1.00000 −1.97862 −1.04369 0.146218 1.00000 0.581745 1.00000 0.152606
1.6 0.374817 −1.00000 −1.85951 −1.32690 −0.374817 1.00000 −1.44661 1.00000 −0.497345
1.7 1.36450 −1.00000 −0.138150 3.32059 −1.36450 1.00000 −2.91750 1.00000 4.53093
1.8 1.60159 −1.00000 0.565099 2.27787 −1.60159 1.00000 −2.29813 1.00000 3.64823
1.9 2.31147 −1.00000 3.34288 1.46052 −2.31147 1.00000 3.10401 1.00000 3.37595
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

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 3549.2.a.be 9
13.b even 2 1 3549.2.a.bf yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.be 9 1.a even 1 1 trivial
3549.2.a.bf yes 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\(T_{2}^{9} + \cdots\)
\(T_{5}^{9} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T - 12 T^{2} - 41 T^{3} + 18 T^{4} + 37 T^{5} - 8 T^{6} - 11 T^{7} + T^{8} + T^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( 169 - 131 T - 255 T^{2} + 249 T^{3} + 83 T^{4} - 141 T^{5} + 20 T^{6} + 22 T^{7} - 9 T^{8} + T^{9} \)
$7$ \( ( -1 + T )^{9} \)
$11$ \( -29 + 504 T + 197 T^{2} - 851 T^{3} - 215 T^{4} + 348 T^{5} + 10 T^{6} - 37 T^{7} + T^{8} + T^{9} \)
$13$ \( T^{9} \)
$17$ \( -2633 - 3756 T + 8935 T^{2} + 7121 T^{3} - 4589 T^{4} - 482 T^{5} + 426 T^{6} - 13 T^{7} - 11 T^{8} + T^{9} \)
$19$ \( -1861 - 7010 T - 9139 T^{2} - 3797 T^{3} + 1369 T^{4} + 1412 T^{5} + 188 T^{6} - 55 T^{7} - 7 T^{8} + T^{9} \)
$23$ \( -1861 + 24665 T - 52820 T^{2} + 14109 T^{3} + 8153 T^{4} - 3511 T^{5} + 55 T^{6} + 142 T^{7} - 22 T^{8} + T^{9} \)
$29$ \( 31424 + 1472 T - 63040 T^{2} + 32976 T^{3} + 3268 T^{4} - 5656 T^{5} + 1275 T^{6} - 58 T^{7} - 11 T^{8} + T^{9} \)
$31$ \( -8471 - 16626 T + 33889 T^{2} + 21851 T^{3} - 18989 T^{4} + 1692 T^{5} + 746 T^{6} - 99 T^{7} - 7 T^{8} + T^{9} \)
$37$ \( -136487 + 188511 T + 43481 T^{2} - 80179 T^{3} - 3663 T^{4} + 8201 T^{5} - 176 T^{6} - 194 T^{7} + T^{8} + T^{9} \)
$41$ \( 45907 - 56655 T - 44154 T^{2} + 68597 T^{3} - 15141 T^{4} - 3141 T^{5} + 1035 T^{6} - 2 T^{7} - 16 T^{8} + T^{9} \)
$43$ \( -4296704 + 3650560 T - 264704 T^{2} - 628928 T^{3} + 254976 T^{4} - 38160 T^{5} + 1248 T^{6} + 284 T^{7} - 32 T^{8} + T^{9} \)
$47$ \( 8667968 - 768704 T - 2062000 T^{2} - 23792 T^{3} + 131560 T^{4} + 6680 T^{5} - 2763 T^{6} - 209 T^{7} + 12 T^{8} + T^{9} \)
$53$ \( -455104 - 211072 T + 157968 T^{2} + 48336 T^{3} - 23744 T^{4} - 2216 T^{5} + 1413 T^{6} - 73 T^{7} - 13 T^{8} + T^{9} \)
$59$ \( -3519424 + 7154816 T - 3731600 T^{2} + 571120 T^{3} + 79392 T^{4} - 32652 T^{5} + 2477 T^{6} + 163 T^{7} - 29 T^{8} + T^{9} \)
$61$ \( -33721792 + 65881344 T - 9309152 T^{2} - 2611120 T^{3} + 335676 T^{4} + 43680 T^{5} - 3639 T^{6} - 347 T^{7} + 12 T^{8} + T^{9} \)
$67$ \( 13046272 - 4778752 T - 3267904 T^{2} + 498784 T^{3} + 212768 T^{4} - 3808 T^{5} - 3877 T^{6} - 113 T^{7} + 20 T^{8} + T^{9} \)
$71$ \( -80941127 + 25334313 T + 5232100 T^{2} - 1976311 T^{3} - 74947 T^{4} + 46039 T^{5} - T^{6} - 382 T^{7} + 2 T^{8} + T^{9} \)
$73$ \( -75156992 + 50932736 T + 2187904 T^{2} - 2926432 T^{3} - 31464 T^{4} + 55332 T^{5} + 311 T^{6} - 406 T^{7} - T^{8} + T^{9} \)
$79$ \( -29684416 + 14214720 T + 7062384 T^{2} - 1859840 T^{3} - 156260 T^{4} + 43340 T^{5} + 1185 T^{6} - 360 T^{7} - 3 T^{8} + T^{9} \)
$83$ \( 53248 + 1069056 T + 1579520 T^{2} - 684160 T^{3} - 282736 T^{4} + 9128 T^{5} + 4885 T^{6} - 135 T^{7} - 24 T^{8} + T^{9} \)
$89$ \( 296717 - 1737545 T + 2264607 T^{2} - 223427 T^{3} - 181119 T^{4} + 18221 T^{5} + 3382 T^{6} - 338 T^{7} - 11 T^{8} + T^{9} \)
$97$ \( -231232 + 642496 T - 407600 T^{2} - 70736 T^{3} + 139332 T^{4} - 47012 T^{5} + 6215 T^{6} - 197 T^{7} - 20 T^{8} + T^{9} \)
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