Properties

Label 483.4.a.d
Level $483$
Weight $4$
Character orbit 483.a
Self dual yes
Analytic conductor $28.498$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4979225328\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 37 x^{5} + 71 x^{4} + 312 x^{3} - 629 x^{2} + 112 x + 180\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -3 q^{3} + ( 3 + \beta_{4} + \beta_{5} ) q^{4} + ( -2 + \beta_{2} + \beta_{4} - \beta_{6} ) q^{5} + 3 \beta_{1} q^{6} -7 q^{7} + ( 3 - 5 \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -3 q^{3} + ( 3 + \beta_{4} + \beta_{5} ) q^{4} + ( -2 + \beta_{2} + \beta_{4} - \beta_{6} ) q^{5} + 3 \beta_{1} q^{6} -7 q^{7} + ( 3 - 5 \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{8} + 9 q^{9} + ( 2 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{10} + ( -1 + 8 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{11} + ( -9 - 3 \beta_{4} - 3 \beta_{5} ) q^{12} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{13} + 7 \beta_{1} q^{14} + ( 6 - 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{15} + ( 14 + 7 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{16} + ( -11 - 5 \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{6} ) q^{17} -9 \beta_{1} q^{18} + ( 1 + 2 \beta_{1} - 5 \beta_{2} - \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} ) q^{19} + ( 9 + 10 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 9 \beta_{6} ) q^{20} + 21 q^{21} + ( -70 + 9 \beta_{1} - 6 \beta_{3} - 14 \beta_{4} - 11 \beta_{5} - 2 \beta_{6} ) q^{22} + 23 q^{23} + ( -9 + 15 \beta_{1} + 6 \beta_{2} + 3 \beta_{4} + 6 \beta_{5} ) q^{24} + ( -10 + 6 \beta_{1} - 7 \beta_{2} - 9 \beta_{4} - 15 \beta_{5} + 11 \beta_{6} ) q^{25} + ( 38 + \beta_{1} - 5 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 4 \beta_{6} ) q^{26} -27 q^{27} + ( -21 - 7 \beta_{4} - 7 \beta_{5} ) q^{28} + ( -40 + 25 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} ) q^{29} + ( -6 + 6 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{30} + ( 35 + \beta_{2} - 9 \beta_{3} - 6 \beta_{4} - 19 \beta_{5} + 7 \beta_{6} ) q^{31} + ( -81 + 18 \beta_{1} - 8 \beta_{2} - 7 \beta_{4} + 7 \beta_{5} - 8 \beta_{6} ) q^{32} + ( 3 - 24 \beta_{1} - 6 \beta_{2} + 6 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} ) q^{33} + ( -14 + 31 \beta_{1} + 9 \beta_{2} + 11 \beta_{3} + 20 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} ) q^{34} + ( 14 - 7 \beta_{2} - 7 \beta_{4} + 7 \beta_{6} ) q^{35} + ( 27 + 9 \beta_{4} + 9 \beta_{5} ) q^{36} + ( -86 + 21 \beta_{1} + 8 \beta_{2} + 7 \beta_{3} + 12 \beta_{4} + 15 \beta_{5} - 21 \beta_{6} ) q^{37} + ( -60 + 31 \beta_{1} + 19 \beta_{2} + 15 \beta_{3} + 20 \beta_{4} + 15 \beta_{5} + 9 \beta_{6} ) q^{38} + ( -15 + 6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} ) q^{39} + ( -102 - 19 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 12 \beta_{5} - 19 \beta_{6} ) q^{40} + ( 120 + 21 \beta_{1} - 16 \beta_{2} - \beta_{3} + 10 \beta_{4} + 23 \beta_{5} + 15 \beta_{6} ) q^{41} -21 \beta_{1} q^{42} + ( -153 + 12 \beta_{1} - \beta_{2} + 8 \beta_{3} + 7 \beta_{4} + 21 \beta_{5} - 19 \beta_{6} ) q^{43} + ( -119 + 114 \beta_{1} + 32 \beta_{2} - 10 \beta_{3} + 11 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{44} + ( -18 + 9 \beta_{2} + 9 \beta_{4} - 9 \beta_{6} ) q^{45} -23 \beta_{1} q^{46} + ( 25 - \beta_{1} - 7 \beta_{2} + 22 \beta_{3} + 28 \beta_{4} - 6 \beta_{5} + 26 \beta_{6} ) q^{47} + ( -42 - 21 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} - 12 \beta_{4} - 6 \beta_{5} ) q^{48} + 49 q^{49} + ( -168 + 88 \beta_{1} + 14 \beta_{2} + 3 \beta_{3} + 13 \beta_{4} + 14 \beta_{5} - 11 \beta_{6} ) q^{50} + ( 33 + 15 \beta_{2} - 3 \beta_{3} + 12 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{51} + ( -62 - 22 \beta_{1} + 11 \beta_{2} - 2 \beta_{3} + 21 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} ) q^{52} + ( -175 + 15 \beta_{1} - 23 \beta_{2} - 11 \beta_{3} - 41 \beta_{4} - 24 \beta_{5} + 16 \beta_{6} ) q^{53} + 27 \beta_{1} q^{54} + ( -47 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + 23 \beta_{4} + 40 \beta_{5} - 18 \beta_{6} ) q^{55} + ( -21 + 35 \beta_{1} + 14 \beta_{2} + 7 \beta_{4} + 14 \beta_{5} ) q^{56} + ( -3 - 6 \beta_{1} + 15 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 15 \beta_{5} + 21 \beta_{6} ) q^{57} + ( -327 + 26 \beta_{1} + 28 \beta_{2} + \beta_{3} - 22 \beta_{4} - 25 \beta_{5} + 21 \beta_{6} ) q^{58} + ( 135 + 47 \beta_{1} - 19 \beta_{2} + 5 \beta_{3} + \beta_{4} + 50 \beta_{5} - 8 \beta_{6} ) q^{59} + ( -27 - 30 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} - 27 \beta_{6} ) q^{60} + ( -162 + 9 \beta_{1} + 41 \beta_{2} - 3 \beta_{3} - 31 \beta_{4} + 15 \beta_{5} + 36 \beta_{6} ) q^{61} + ( -108 + 29 \beta_{1} + 31 \beta_{2} - 27 \beta_{3} - 22 \beta_{4} - \beta_{5} + 11 \beta_{6} ) q^{62} -63 q^{63} + ( -271 + 39 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 11 \beta_{4} - 18 \beta_{5} + 8 \beta_{6} ) q^{64} + ( -50 + 21 \beta_{1} - 12 \beta_{2} - 11 \beta_{3} + 2 \beta_{4} + 29 \beta_{5} - 25 \beta_{6} ) q^{65} + ( 210 - 27 \beta_{1} + 18 \beta_{3} + 42 \beta_{4} + 33 \beta_{5} + 6 \beta_{6} ) q^{66} + ( -55 - 61 \beta_{1} - 31 \beta_{2} + 23 \beta_{3} + 9 \beta_{4} - 4 \beta_{5} + 18 \beta_{6} ) q^{67} + ( -197 - 82 \beta_{1} - 39 \beta_{2} - \beta_{3} - 21 \beta_{4} - 38 \beta_{5} - 27 \beta_{6} ) q^{68} -69 q^{69} + ( -14 + 14 \beta_{2} + 7 \beta_{3} + 21 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} ) q^{70} + ( 36 - 139 \beta_{1} + 5 \beta_{2} - 9 \beta_{3} + 13 \beta_{4} - 29 \beta_{5} + 14 \beta_{6} ) q^{71} + ( 27 - 45 \beta_{1} - 18 \beta_{2} - 9 \beta_{4} - 18 \beta_{5} ) q^{72} + ( -67 + 22 \beta_{1} - 52 \beta_{2} - 32 \beta_{3} - 66 \beta_{4} - 11 \beta_{5} + 40 \beta_{6} ) q^{73} + ( -117 + 40 \beta_{1} - 32 \beta_{2} + 19 \beta_{3} - 22 \beta_{4} - 21 \beta_{5} + 7 \beta_{6} ) q^{74} + ( 30 - 18 \beta_{1} + 21 \beta_{2} + 27 \beta_{4} + 45 \beta_{5} - 33 \beta_{6} ) q^{75} + ( -237 - 56 \beta_{1} - 73 \beta_{2} - 9 \beta_{3} - 71 \beta_{4} - 24 \beta_{5} + 17 \beta_{6} ) q^{76} + ( 7 - 56 \beta_{1} - 14 \beta_{2} + 14 \beta_{4} - 7 \beta_{5} - 14 \beta_{6} ) q^{77} + ( -114 - 3 \beta_{1} + 15 \beta_{2} + 12 \beta_{3} + 9 \beta_{4} + 12 \beta_{6} ) q^{78} + ( -395 - 88 \beta_{1} + 30 \beta_{2} - 32 \beta_{3} + 10 \beta_{4} - 37 \beta_{5} + 22 \beta_{6} ) q^{79} + ( 96 + 90 \beta_{1} - 7 \beta_{3} + 27 \beta_{4} + 11 \beta_{5} - 47 \beta_{6} ) q^{80} + 81 q^{81} + ( -203 - 338 \beta_{1} - 16 \beta_{2} + 15 \beta_{3} - 55 \beta_{5} - 13 \beta_{6} ) q^{82} + ( -79 + 66 \beta_{1} - 17 \beta_{2} + 7 \beta_{3} - 10 \beta_{4} + 47 \beta_{5} + 21 \beta_{6} ) q^{83} + ( 63 + 21 \beta_{4} + 21 \beta_{5} ) q^{84} + ( -377 - 31 \beta_{1} + 11 \beta_{2} + 5 \beta_{3} + 9 \beta_{4} + 24 \beta_{5} - 20 \beta_{6} ) q^{85} + ( -4 + 95 \beta_{1} - 18 \beta_{2} + 37 \beta_{3} + 19 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} ) q^{86} + ( 120 - 75 \beta_{1} + 12 \beta_{2} + 21 \beta_{3} - 6 \beta_{4} + 15 \beta_{5} + 21 \beta_{6} ) q^{87} + ( -619 + 17 \beta_{1} - 22 \beta_{2} - 34 \beta_{3} - 127 \beta_{4} - 85 \beta_{5} + 38 \beta_{6} ) q^{88} + ( -48 - 143 \beta_{1} - 18 \beta_{2} - 28 \beta_{3} - 19 \beta_{4} - 17 \beta_{5} + 23 \beta_{6} ) q^{89} + ( 18 - 18 \beta_{2} - 9 \beta_{3} - 27 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} ) q^{90} + ( -35 + 14 \beta_{1} - 14 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} - 14 \beta_{6} ) q^{91} + ( 69 + 23 \beta_{4} + 23 \beta_{5} ) q^{92} + ( -105 - 3 \beta_{2} + 27 \beta_{3} + 18 \beta_{4} + 57 \beta_{5} - 21 \beta_{6} ) q^{93} + ( -141 - 147 \beta_{1} - 141 \beta_{2} + 32 \beta_{3} + 12 \beta_{4} + 4 \beta_{5} - 70 \beta_{6} ) q^{94} + ( 53 - 83 \beta_{1} + 33 \beta_{2} + 23 \beta_{3} + \beta_{4} - 96 \beta_{5} + 20 \beta_{6} ) q^{95} + ( 243 - 54 \beta_{1} + 24 \beta_{2} + 21 \beta_{4} - 21 \beta_{5} + 24 \beta_{6} ) q^{96} + ( -228 - 119 \beta_{1} - 29 \beta_{2} + 30 \beta_{3} + 28 \beta_{4} + 27 \beta_{5} + 74 \beta_{6} ) q^{97} -49 \beta_{1} q^{98} + ( -9 + 72 \beta_{1} + 18 \beta_{2} - 18 \beta_{4} + 9 \beta_{5} + 18 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 2q^{2} - 21q^{3} + 22q^{4} - 11q^{5} + 6q^{6} - 49q^{7} + 15q^{8} + 63q^{9} + O(q^{10}) \) \( 7q - 2q^{2} - 21q^{3} + 22q^{4} - 11q^{5} + 6q^{6} - 49q^{7} + 15q^{8} + 63q^{9} + 11q^{10} - 6q^{11} - 66q^{12} + 17q^{13} + 14q^{14} + 33q^{15} + 106q^{16} - 78q^{17} - 18q^{18} + 44q^{19} + 44q^{20} + 147q^{21} - 477q^{22} + 161q^{23} - 45q^{24} - 80q^{25} + 288q^{26} - 189q^{27} - 154q^{28} - 185q^{29} - 33q^{30} + 238q^{31} - 512q^{32} + 18q^{33} - 27q^{34} + 77q^{35} + 198q^{36} - 511q^{37} - 413q^{38} - 51q^{39} - 686q^{40} + 867q^{41} - 42q^{42} - 1003q^{43} - 629q^{44} - 99q^{45} - 46q^{46} + 149q^{47} - 318q^{48} + 343q^{49} - 986q^{50} + 234q^{51} - 439q^{52} - 1244q^{53} + 54q^{54} - 270q^{55} - 105q^{56} - 132q^{57} - 2376q^{58} + 1048q^{59} - 132q^{60} - 1380q^{61} - 809q^{62} - 441q^{63} - 1835q^{64} - 223q^{65} + 1431q^{66} - 500q^{67} - 1387q^{68} - 483q^{69} - 77q^{70} - 14q^{71} + 135q^{72} - 530q^{73} - 738q^{74} + 240q^{75} - 1785q^{76} + 42q^{77} - 864q^{78} - 2978q^{79} + 1043q^{80} + 567q^{81} - 1986q^{82} - 524q^{83} + 462q^{84} - 2674q^{85} + 194q^{86} + 555q^{87} - 4504q^{88} - 648q^{89} + 99q^{90} - 119q^{91} + 506q^{92} - 714q^{93} - 801q^{94} + 154q^{95} + 1536q^{96} - 1999q^{97} - 98q^{98} - 54q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 37 x^{5} + 71 x^{4} + 312 x^{3} - 629 x^{2} + 112 x + 180\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} + \nu^{5} + 38 \nu^{4} - 29 \nu^{3} - 349 \nu^{2} + 200 \nu + 212 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 37 \nu^{4} + 31 \nu^{3} + 319 \nu^{2} - 249 \nu - 90 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + \nu^{5} + 38 \nu^{4} - 33 \nu^{3} - 341 \nu^{2} + 284 \nu + 112 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 38 \nu^{4} + 33 \nu^{3} + 345 \nu^{2} - 284 \nu - 156 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{6} - 7 \nu^{5} - 342 \nu^{4} + 225 \nu^{3} + 3099 \nu^{2} - 1968 \nu - 1524 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + 11\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} + \beta_{4} + 2 \beta_{2} + 21 \beta_{1} - 3\)
\(\nu^{4}\)\(=\)\(26 \beta_{5} + 28 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 7 \beta_{1} + 214\)
\(\nu^{5}\)\(=\)\(8 \beta_{6} + 57 \beta_{5} + 39 \beta_{4} + 72 \beta_{2} + 462 \beta_{1} - 15\)
\(\nu^{6}\)\(=\)\(8 \beta_{6} + 638 \beta_{5} + 725 \beta_{4} + 152 \beta_{3} + 158 \beta_{2} + 319 \beta_{1} + 4577\)

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
5.03765
3.61105
1.18720
1.10401
−0.422009
−3.75958
−4.75832
−5.03765 −3.00000 17.3779 6.20452 15.1129 −7.00000 −47.2425 9.00000 −31.2562
1.2 −3.61105 −3.00000 5.03969 −11.1551 10.8332 −7.00000 10.6898 9.00000 40.2815
1.3 −1.18720 −3.00000 −6.59056 −20.2258 3.56160 −7.00000 17.3219 9.00000 24.0121
1.4 −1.10401 −3.00000 −6.78116 14.0332 3.31203 −7.00000 16.3186 9.00000 −15.4928
1.5 0.422009 −3.00000 −7.82191 0.947423 −1.26603 −7.00000 −6.67699 9.00000 0.399821
1.6 3.75958 −3.00000 6.13441 3.12160 −11.2787 −7.00000 −7.01383 9.00000 11.7359
1.7 4.75832 −3.00000 14.6416 −3.92582 −14.2750 −7.00000 31.6031 9.00000 −18.6803
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

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

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 483.4.a.d 7
3.b odd 2 1 1449.4.a.g 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.4.a.d 7 1.a even 1 1 trivial
1449.4.a.g 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 2 T_{2}^{6} - 37 T_{2}^{5} - 71 T_{2}^{4} + 312 T_{2}^{3} + 629 T_{2}^{2} + 112 T_{2} - 180 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(483))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -180 + 112 T + 629 T^{2} + 312 T^{3} - 71 T^{4} - 37 T^{5} + 2 T^{6} + T^{7} \)
$3$ \( ( 3 + T )^{7} \)
$5$ \( 228084 - 277000 T + 17313 T^{2} + 24233 T^{3} - 1921 T^{4} - 337 T^{5} + 11 T^{6} + T^{7} \)
$7$ \( ( 7 + T )^{7} \)
$11$ \( 5071866368 - 707551232 T - 58179904 T^{2} + 3936403 T^{3} + 64866 T^{4} - 4796 T^{5} + 6 T^{6} + T^{7} \)
$13$ \( -323309364 + 238744220 T - 47324203 T^{2} + 1249909 T^{3} + 94123 T^{4} - 3413 T^{5} - 17 T^{6} + T^{7} \)
$17$ \( 111964950432 - 14165468460 T + 257914752 T^{2} + 17016839 T^{3} - 343606 T^{4} - 6308 T^{5} + 78 T^{6} + T^{7} \)
$19$ \( 135809928192 - 104038669808 T + 2310927324 T^{2} + 141877161 T^{3} + 193200 T^{4} - 22554 T^{5} - 44 T^{6} + T^{7} \)
$23$ \( ( -23 + T )^{7} \)
$29$ \( -319608557445276 - 3436375144984 T + 166907385693 T^{2} + 779125957 T^{3} - 14284330 T^{4} - 72030 T^{5} + 185 T^{6} + T^{7} \)
$31$ \( 273519093450528 + 9996000831000 T - 486075629862 T^{2} + 1589246025 T^{3} + 21906164 T^{4} - 87634 T^{5} - 238 T^{6} + T^{7} \)
$37$ \( -13996222497396 + 1319087115576 T + 102109748519 T^{2} + 382858321 T^{3} - 27896498 T^{4} - 28554 T^{5} + 511 T^{6} + T^{7} \)
$41$ \( -42817809179724828 + 1098861219310576 T - 4035793986811 T^{2} - 24913877799 T^{3} + 131487338 T^{4} + 38214 T^{5} - 867 T^{6} + T^{7} \)
$43$ \( 447690400483584 + 65130751903744 T - 69085426853 T^{2} - 11061434569 T^{3} - 23400163 T^{4} + 248943 T^{5} + 1003 T^{6} + T^{7} \)
$47$ \( 1413708152160685824 - 12732658744096064 T - 19826254421808 T^{2} + 174889585248 T^{3} + 98287404 T^{4} - 746676 T^{5} - 149 T^{6} + T^{7} \)
$53$ \( 165333669299312784 + 1820212004734676 T + 253949837840 T^{2} - 52753939019 T^{3} - 124819388 T^{4} + 314445 T^{5} + 1244 T^{6} + T^{7} \)
$59$ \( 851274441254408544 + 3700253618653248 T - 29584366481498 T^{2} - 58141377963 T^{3} + 407135328 T^{4} - 160745 T^{5} - 1048 T^{6} + T^{7} \)
$61$ \( -395697941471390760 - 20124164485795836 T + 133424913173762 T^{2} + 45833292091 T^{3} - 832905822 T^{4} - 337123 T^{5} + 1380 T^{6} + T^{7} \)
$67$ \( 1385844516283301888 + 18686707175439360 T + 74895417961036 T^{2} + 5177269431 T^{3} - 553728750 T^{4} - 805449 T^{5} + 500 T^{6} + T^{7} \)
$71$ \( 1778009171552876160 - 19133203484263728 T - 161231746370 T^{2} + 238703189059 T^{3} - 37812952 T^{4} - 910541 T^{5} + 14 T^{6} + T^{7} \)
$73$ \( -26186334848101818576 - 31794337281155076 T + 288873469790496 T^{2} + 413171456607 T^{3} - 854440362 T^{4} - 1365780 T^{5} + 530 T^{6} + T^{7} \)
$79$ \( -7179069883733058560 - 263162985752008384 T - 1528189282899816 T^{2} - 3189359749301 T^{3} - 1853968882 T^{4} + 2041468 T^{5} + 2978 T^{6} + T^{7} \)
$83$ \( 3051910731275712 + 186936026046176 T - 7134412902360 T^{2} + 56977098725 T^{3} - 30879220 T^{4} - 529798 T^{5} + 524 T^{6} + T^{7} \)
$89$ \( 21562106462475848 + 3704921638951732 T + 37104749060654 T^{2} - 106228512749 T^{3} - 926096838 T^{4} - 1060607 T^{5} + 648 T^{6} + T^{7} \)
$97$ \( \)\(12\!\cdots\!64\)\( + 972296068853298564 T + 1538972705122827 T^{2} - 1514790063563 T^{3} - 4321944078 T^{4} - 1119994 T^{5} + 1999 T^{6} + T^{7} \)
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