Properties

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

Related objects

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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: \(x^{7} - x^{6} - 34 x^{5} + 7 x^{4} + 295 x^{3} + 84 x^{2} - 524 x - 288\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 3 - \beta_{1} + \beta_{2} ) q^{4} + ( -6 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{5} + ( -3 + 3 \beta_{1} ) q^{6} + 7 q^{7} + ( -5 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 3 - \beta_{1} + \beta_{2} ) q^{4} + ( -6 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{5} + ( -3 + 3 \beta_{1} ) q^{6} + 7 q^{7} + ( -5 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{8} + 9 q^{9} + ( 1 - 11 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{10} + ( -20 + 3 \beta_{1} + 4 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{11} + ( 9 - 3 \beta_{1} + 3 \beta_{2} ) q^{12} + ( -9 - 7 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{13} + ( -7 + 7 \beta_{1} ) q^{14} + ( -18 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} ) q^{15} + ( 2 - 7 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{16} + ( -27 - 17 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{5} + 7 \beta_{6} ) q^{17} + ( -9 + 9 \beta_{1} ) q^{18} + ( -41 - 11 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 3 \beta_{6} ) q^{19} + ( -63 - 3 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{20} + 21 q^{21} + ( 51 - 26 \beta_{1} - \beta_{2} - 7 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{22} + 23 q^{23} + ( -15 + 3 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + 3 \beta_{5} ) q^{24} + ( 63 + 5 \beta_{1} + 13 \beta_{2} - 5 \beta_{4} - 3 \beta_{5} + 11 \beta_{6} ) q^{25} + ( -55 + 13 \beta_{1} + \beta_{2} + 11 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} ) q^{26} + 27 q^{27} + ( 21 - 7 \beta_{1} + 7 \beta_{2} ) q^{28} + ( -48 + 34 \beta_{1} - 17 \beta_{2} - \beta_{4} + 17 \beta_{5} - 10 \beta_{6} ) q^{29} + ( 3 - 33 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 9 \beta_{6} ) q^{30} + ( 43 - 5 \beta_{1} + \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 13 \beta_{6} ) q^{31} + ( -44 + \beta_{1} - 6 \beta_{2} + \beta_{3} - 6 \beta_{4} - 11 \beta_{5} - 9 \beta_{6} ) q^{32} + ( -60 + 9 \beta_{1} + 12 \beta_{4} - 9 \beta_{5} + 6 \beta_{6} ) q^{33} + ( -134 - 12 \beta_{1} - 24 \beta_{2} + 22 \beta_{3} - 10 \beta_{4} + 5 \beta_{5} - 24 \beta_{6} ) q^{34} + ( -42 - 7 \beta_{2} - 7 \beta_{4} - 7 \beta_{6} ) q^{35} + ( 27 - 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( -98 - 24 \beta_{1} + \beta_{2} + 14 \beta_{3} + 13 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{37} + ( -42 - 68 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} - 3 \beta_{5} - 6 \beta_{6} ) q^{38} + ( -27 - 21 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} ) q^{39} + ( 32 - 37 \beta_{1} + 12 \beta_{2} - 22 \beta_{3} + 7 \beta_{4} - \beta_{5} - 6 \beta_{6} ) q^{40} + ( -46 - 16 \beta_{1} + \beta_{2} - 6 \beta_{3} - 29 \beta_{4} - \beta_{5} ) q^{41} + ( -21 + 21 \beta_{1} ) q^{42} + ( -26 + 15 \beta_{1} + 27 \beta_{2} + 6 \beta_{3} - 19 \beta_{4} - 13 \beta_{5} + 5 \beta_{6} ) q^{43} + ( -200 + 50 \beta_{1} - 47 \beta_{2} + 10 \beta_{3} - 16 \beta_{4} + 7 \beta_{5} - 12 \beta_{6} ) q^{44} + ( -54 - 9 \beta_{2} - 9 \beta_{4} - 9 \beta_{6} ) q^{45} + ( -23 + 23 \beta_{1} ) q^{46} + ( -183 - 13 \beta_{1} + 16 \beta_{2} - 5 \beta_{3} + 11 \beta_{4} + 23 \beta_{5} - 3 \beta_{6} ) q^{47} + ( 6 - 21 \beta_{1} - 3 \beta_{2} - 15 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{48} + 49 q^{49} + ( 33 + 108 \beta_{1} + 10 \beta_{2} + 28 \beta_{3} - 21 \beta_{4} + 29 \beta_{5} - 30 \beta_{6} ) q^{50} + ( -81 - 51 \beta_{1} + 9 \beta_{2} - 15 \beta_{3} - 6 \beta_{5} + 21 \beta_{6} ) q^{51} + ( 316 - 35 \beta_{1} + 22 \beta_{2} - 36 \beta_{3} - 17 \beta_{4} - 14 \beta_{5} + 18 \beta_{6} ) q^{52} + ( -146 - 25 \beta_{1} + 16 \beta_{2} - 14 \beta_{3} + 14 \beta_{4} - 26 \beta_{5} - 9 \beta_{6} ) q^{53} + ( -27 + 27 \beta_{1} ) q^{54} + ( 8 - 21 \beta_{1} + 44 \beta_{2} + 2 \beta_{3} + 12 \beta_{4} + 34 \beta_{5} - 5 \beta_{6} ) q^{55} + ( -35 + 7 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} + 7 \beta_{5} ) q^{56} + ( -123 - 33 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 18 \beta_{4} + 9 \beta_{6} ) q^{57} + ( 389 - 97 \beta_{1} + 58 \beta_{2} + 7 \beta_{3} - 15 \beta_{4} - 26 \beta_{5} + 13 \beta_{6} ) q^{58} + ( -188 - 77 \beta_{1} - 6 \beta_{2} - 30 \beta_{3} - 22 \beta_{4} - 10 \beta_{5} - 7 \beta_{6} ) q^{59} + ( -189 - 9 \beta_{1} - 24 \beta_{2} + 21 \beta_{3} + 15 \beta_{4} - 6 \beta_{5} - 9 \beta_{6} ) q^{60} + ( -4 - 12 \beta_{1} + 38 \beta_{2} + 20 \beta_{3} + 16 \beta_{4} - 15 \beta_{5} - \beta_{6} ) q^{61} + ( -42 + 38 \beta_{1} + 2 \beta_{2} + 36 \beta_{3} - 34 \beta_{4} + 13 \beta_{5} - 46 \beta_{6} ) q^{62} + 63 q^{63} + ( -36 - 37 \beta_{1} - 20 \beta_{2} - 17 \beta_{3} + 50 \beta_{4} - 16 \beta_{5} + 31 \beta_{6} ) q^{64} + ( -172 - 14 \beta_{1} + 7 \beta_{2} + 34 \beta_{3} + 67 \beta_{4} - 31 \beta_{5} + 38 \beta_{6} ) q^{65} + ( 153 - 78 \beta_{1} - 3 \beta_{2} - 21 \beta_{3} + 18 \beta_{4} - 6 \beta_{5} - 9 \beta_{6} ) q^{66} + ( -162 + 53 \beta_{1} + 14 \beta_{2} + 32 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 47 \beta_{6} ) q^{67} + ( 213 - 160 \beta_{1} - 6 \beta_{2} - 83 \beta_{3} + 28 \beta_{4} + 33 \beta_{6} ) q^{68} + 69 q^{69} + ( 7 - 77 \beta_{1} - 7 \beta_{2} - 21 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} + 21 \beta_{6} ) q^{70} + ( -244 + 102 \beta_{1} + 14 \beta_{2} + 2 \beta_{3} - 38 \beta_{4} + \beta_{5} + 45 \beta_{6} ) q^{71} + ( -45 + 9 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} + 9 \beta_{5} ) q^{72} + ( -124 - 37 \beta_{1} - 50 \beta_{2} - 2 \beta_{3} + 66 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} ) q^{73} + ( -105 - 123 \beta_{1} - 4 \beta_{2} - 51 \beta_{3} + 23 \beta_{4} + 23 \beta_{6} ) q^{74} + ( 189 + 15 \beta_{1} + 39 \beta_{2} - 15 \beta_{4} - 9 \beta_{5} + 33 \beta_{6} ) q^{75} + ( -397 + 76 \beta_{1} - 80 \beta_{2} - 7 \beta_{3} - 30 \beta_{4} - 20 \beta_{5} - 15 \beta_{6} ) q^{76} + ( -140 + 21 \beta_{1} + 28 \beta_{4} - 21 \beta_{5} + 14 \beta_{6} ) q^{77} + ( -165 + 39 \beta_{1} + 3 \beta_{2} + 33 \beta_{3} - 9 \beta_{4} + 9 \beta_{6} ) q^{78} + ( 106 + 5 \beta_{1} + 10 \beta_{2} + 54 \beta_{3} + 12 \beta_{4} + 51 \beta_{5} - 10 \beta_{6} ) q^{79} + ( -19 + 211 \beta_{1} - 25 \beta_{2} + 25 \beta_{3} - 19 \beta_{4} - 7 \beta_{5} + 21 \beta_{6} ) q^{80} + 81 q^{81} + ( -141 - 53 \beta_{1} - 30 \beta_{2} + 17 \beta_{3} - 27 \beta_{4} + 24 \beta_{5} - 5 \beta_{6} ) q^{82} + ( -291 + 45 \beta_{1} - 17 \beta_{2} - 67 \beta_{3} - 38 \beta_{4} - 60 \beta_{5} + 27 \beta_{6} ) q^{83} + ( 63 - 21 \beta_{1} + 21 \beta_{2} ) q^{84} + ( -254 + 349 \beta_{1} - 10 \beta_{2} + 36 \beta_{3} + 16 \beta_{4} + 40 \beta_{5} - 33 \beta_{6} ) q^{85} + ( 186 + 63 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 57 \beta_{5} + 4 \beta_{6} ) q^{86} + ( -144 + 102 \beta_{1} - 51 \beta_{2} - 3 \beta_{4} + 51 \beta_{5} - 30 \beta_{6} ) q^{87} + ( 293 - 282 \beta_{1} + 80 \beta_{2} - 59 \beta_{3} - 54 \beta_{4} - 17 \beta_{5} + 63 \beta_{6} ) q^{88} + ( -83 + 136 \beta_{1} + 29 \beta_{2} - 27 \beta_{3} - 30 \beta_{4} - 50 \beta_{5} + 26 \beta_{6} ) q^{89} + ( 9 - 99 \beta_{1} - 9 \beta_{2} - 27 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} + 27 \beta_{6} ) q^{90} + ( -63 - 49 \beta_{1} + 14 \beta_{2} + 7 \beta_{3} + 7 \beta_{4} + 28 \beta_{5} - 14 \beta_{6} ) q^{91} + ( 69 - 23 \beta_{1} + 23 \beta_{2} ) q^{92} + ( 129 - 15 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} + 39 \beta_{6} ) q^{93} + ( 87 - 9 \beta_{1} + 20 \beta_{2} + 113 \beta_{3} - 29 \beta_{4} - 3 \beta_{5} - 19 \beta_{6} ) q^{94} + ( 52 + 115 \beta_{1} + 114 \beta_{2} + 6 \beta_{3} + 44 \beta_{4} + 50 \beta_{5} - 29 \beta_{6} ) q^{95} + ( -132 + 3 \beta_{1} - 18 \beta_{2} + 3 \beta_{3} - 18 \beta_{4} - 33 \beta_{5} - 27 \beta_{6} ) q^{96} + ( -227 + 160 \beta_{1} - 6 \beta_{2} + 21 \beta_{3} - 77 \beta_{4} + 64 \beta_{5} + 13 \beta_{6} ) q^{97} + ( -49 + 49 \beta_{1} ) q^{98} + ( -180 + 27 \beta_{1} + 36 \beta_{4} - 27 \beta_{5} + 18 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 6q^{2} + 21q^{3} + 18q^{4} - 41q^{5} - 18q^{6} + 49q^{7} - 33q^{8} + 63q^{9} + O(q^{10}) \) \( 7q - 6q^{2} + 21q^{3} + 18q^{4} - 41q^{5} - 18q^{6} + 49q^{7} - 33q^{8} + 63q^{9} + q^{10} - 126q^{11} + 54q^{12} - 87q^{13} - 42q^{14} - 123q^{15} + 2q^{16} - 204q^{17} - 54q^{18} - 286q^{19} - 418q^{20} + 147q^{21} + 329q^{22} + 161q^{23} - 99q^{24} + 440q^{25} - 360q^{26} + 189q^{27} + 126q^{28} - 329q^{29} + 3q^{30} + 296q^{31} - 270q^{32} - 378q^{33} - 919q^{34} - 287q^{35} + 162q^{36} - 691q^{37} - 367q^{38} - 261q^{39} + 138q^{40} - 343q^{41} - 126q^{42} - 171q^{43} - 1279q^{44} - 369q^{45} - 138q^{46} - 1403q^{47} + 6q^{48} + 343q^{49} + 230q^{50} - 612q^{51} + 2157q^{52} - 1024q^{53} - 162q^{54} - 158q^{55} - 231q^{56} - 858q^{57} + 2608q^{58} - 1388q^{59} - 1254q^{60} - 52q^{61} - 309q^{62} + 441q^{63} - 187q^{64} - 1067q^{65} + 987q^{66} - 1148q^{67} + 1293q^{68} + 483q^{69} + 7q^{70} - 1590q^{71} - 297q^{72} - 802q^{73} - 878q^{74} + 1320q^{75} - 2505q^{76} - 882q^{77} - 1080q^{78} + 618q^{79} + 195q^{80} + 567q^{81} - 1040q^{82} - 1818q^{83} + 378q^{84} - 1526q^{85} + 1188q^{86} - 987q^{87} + 1664q^{88} - 354q^{89} + 9q^{90} - 609q^{91} + 414q^{92} + 888q^{93} + 663q^{94} + 78q^{95} - 810q^{96} - 1575q^{97} - 294q^{98} - 1134q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 34 x^{5} + 7 x^{4} + 295 x^{3} + 84 x^{2} - 524 x - 288\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 10 \)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{6} - 4 \nu^{5} - 112 \nu^{4} + 149 \nu^{3} + 926 \nu^{2} - 1360 \nu - 824 \)\()/136\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{6} + \nu^{5} + 164 \nu^{4} + 69 \nu^{3} - 1175 \nu^{2} - 646 \nu + 784 \)\()/136\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} + 4 \nu^{5} + 112 \nu^{4} - 81 \nu^{3} - 1062 \nu^{2} + 340 \nu + 1504 \)\()/68\)
\(\beta_{6}\)\(=\)\((\)\( 11 \nu^{6} - 26 \nu^{5} - 320 \nu^{4} + 501 \nu^{3} + 2092 \nu^{2} - 1972 \nu - 1616 \)\()/136\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 10\)
\(\nu^{3}\)\(=\)\(\beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 17 \beta_{1} + 10\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 5 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 25 \beta_{2} + 35 \beta_{1} + 181\)
\(\nu^{5}\)\(=\)\(-4 \beta_{6} + 36 \beta_{5} - 16 \beta_{4} + 60 \beta_{3} + 77 \beta_{2} + 363 \beta_{1} + 382\)
\(\nu^{6}\)\(=\)\(32 \beta_{6} + 185 \beta_{5} - 96 \beta_{4} + 138 \beta_{3} + 628 \beta_{2} + 1091 \beta_{1} + 3958\)

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
−4.12074
−3.01159
−1.37660
−0.610936
1.60339
3.29766
5.21883
−5.12074 3.00000 18.2220 −15.2134 −15.3622 7.00000 −52.3444 9.00000 77.9040
1.2 −4.01159 3.00000 8.09283 9.29066 −12.0348 7.00000 −0.372383 9.00000 −37.2703
1.3 −2.37660 3.00000 −2.35176 −18.5873 −7.12981 7.00000 24.6020 9.00000 44.1746
1.4 −1.61094 3.00000 −5.40488 −4.04614 −4.83281 7.00000 21.5944 9.00000 6.51808
1.5 0.603389 3.00000 −7.63592 14.1843 1.81017 7.00000 −9.43454 9.00000 8.55866
1.6 2.29766 3.00000 −2.72078 −7.00327 6.89297 7.00000 −24.6327 9.00000 −16.0911
1.7 4.21883 3.00000 9.79849 −19.6249 12.6565 7.00000 7.58752 9.00000 −82.7940
\(n\): 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.c 7
3.b odd 2 1 1449.4.a.h 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.4.a.c 7 1.a even 1 1 trivial
1449.4.a.h 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 6 T_{2}^{6} - 19 T_{2}^{5} - 143 T_{2}^{4} - 2 T_{2}^{3} + 677 T_{2}^{2} + 388 T_{2} - 460 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(483))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -460 + 388 T + 677 T^{2} - 2 T^{3} - 143 T^{4} - 19 T^{5} + 6 T^{6} + T^{7} \)
$3$ \( ( -3 + T )^{7} \)
$5$ \( 20722732 + 7922128 T + 396923 T^{2} - 116883 T^{3} - 10719 T^{4} + 183 T^{5} + 41 T^{6} + T^{7} \)
$7$ \( ( -7 + T )^{7} \)
$11$ \( 224017353856 + 16822307664 T + 89445880 T^{2} - 20515277 T^{3} - 505278 T^{4} + 644 T^{5} + 126 T^{6} + T^{7} \)
$13$ \( 868269144 - 16404373966 T + 802684665 T^{2} + 14048647 T^{3} - 565613 T^{4} - 6755 T^{5} + 87 T^{6} + T^{7} \)
$17$ \( 12051705641600 + 777200395296 T + 7427999986 T^{2} - 220392397 T^{3} - 4092790 T^{4} - 7628 T^{5} + 204 T^{6} + T^{7} \)
$19$ \( 7302931600672 + 299753488064 T - 2686214702 T^{2} - 239806947 T^{3} - 2290352 T^{4} + 14786 T^{5} + 286 T^{6} + T^{7} \)
$23$ \( ( -23 + T )^{7} \)
$29$ \( -1013170236552340 - 11080055150124 T + 517236713393 T^{2} + 585271049 T^{3} - 34741494 T^{4} - 93102 T^{5} + 329 T^{6} + T^{7} \)
$31$ \( -623763925110016 + 23348305973404 T - 220282847696 T^{2} - 788739575 T^{3} + 18742520 T^{4} - 40662 T^{5} - 296 T^{6} + T^{7} \)
$37$ \( 701389545056212 + 32374444013932 T + 220424507947 T^{2} - 4774087767 T^{3} - 30343090 T^{4} + 72170 T^{5} + 691 T^{6} + T^{7} \)
$41$ \( -237579020750652 + 18016642748452 T + 372840249075 T^{2} - 1655508683 T^{3} - 50105358 T^{4} - 136282 T^{5} + 343 T^{6} + T^{7} \)
$43$ \( 2555644934640240 - 46634060606536 T - 385174180241 T^{2} + 8421073131 T^{3} + 2483933 T^{4} - 225505 T^{5} + 171 T^{6} + T^{7} \)
$47$ \( -142513228369482240 + 3561241818645536 T + 2564937376720 T^{2} - 103531832664 T^{3} - 218297292 T^{4} + 388214 T^{5} + 1403 T^{6} + T^{7} \)
$53$ \( -1587919831091319328 + 2615142626008036 T + 39630760294684 T^{2} - 23192377155 T^{3} - 352445616 T^{4} - 125719 T^{5} + 1024 T^{6} + T^{7} \)
$59$ \( 275405836475164896 + 9881400823024608 T + 18976320810198 T^{2} - 166817498191 T^{3} - 511144100 T^{4} + 94555 T^{5} + 1388 T^{6} + T^{7} \)
$61$ \( 18893994380750536 + 220180954375796 T - 8707152814234 T^{2} + 48740892383 T^{3} + 30048082 T^{4} - 489951 T^{5} + 52 T^{6} + T^{7} \)
$67$ \( -2170910150788401216 + 3117741223619344 T + 92532764057844 T^{2} - 27438877117 T^{3} - 719541550 T^{4} - 418277 T^{5} + 1148 T^{6} + T^{7} \)
$71$ \( -76328992428874879808 + 17538668649207688 T + 641311763381054 T^{2} + 60366635935 T^{3} - 1772481620 T^{4} - 701105 T^{5} + 1590 T^{6} + T^{7} \)
$73$ \( -2046812630975791904 + 9750198043535124 T + 185091773311976 T^{2} + 287489088311 T^{3} - 768822626 T^{4} - 1138964 T^{5} + 802 T^{6} + T^{7} \)
$79$ \( 8282270065032386560 - 49095616940735296 T - 105929696637748 T^{2} + 527961612343 T^{3} + 531733318 T^{4} - 1293984 T^{5} - 618 T^{6} + T^{7} \)
$83$ \( 757319964039325856 + 129256013401975440 T + 380648383540798 T^{2} - 934031233515 T^{3} - 2853813416 T^{4} - 841926 T^{5} + 1818 T^{6} + T^{7} \)
$89$ \( 43903068690104592 - 30139538217903388 T - 15740752953012 T^{2} + 384817367483 T^{3} + 32582348 T^{4} - 1425347 T^{5} + 354 T^{6} + T^{7} \)
$97$ \( -83575126748120323352 - 568836362474131902 T + 3304232244945739 T^{2} + 2537275327809 T^{3} - 5178916698 T^{4} - 3412484 T^{5} + 1575 T^{6} + T^{7} \)
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