Properties

Label 722.4.a.p
Level $722$
Weight $4$
Character orbit 722.a
Self dual yes
Analytic conductor $42.599$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.5993790241\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.6719782761.1
Defining polynomial: \(x^{6} - 3 x^{5} - 75 x^{4} + 135 x^{3} + 1857 x^{2} - 1425 x - 14797\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 19 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( -1 + \beta_{3} ) q^{3} + 4 q^{4} + ( -5 + \beta_{1} + \beta_{5} ) q^{5} + ( -2 + 2 \beta_{3} ) q^{6} + ( -4 - \beta_{2} - \beta_{4} ) q^{7} + 8 q^{8} + ( 4 - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{9} +O(q^{10})\) \( q + 2 q^{2} + ( -1 + \beta_{3} ) q^{3} + 4 q^{4} + ( -5 + \beta_{1} + \beta_{5} ) q^{5} + ( -2 + 2 \beta_{3} ) q^{6} + ( -4 - \beta_{2} - \beta_{4} ) q^{7} + 8 q^{8} + ( 4 - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{9} + ( -10 + 2 \beta_{1} + 2 \beta_{5} ) q^{10} + ( 4 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{11} + ( -4 + 4 \beta_{3} ) q^{12} + ( -6 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{13} + ( -8 - 2 \beta_{2} - 2 \beta_{4} ) q^{14} + ( -3 - 3 \beta_{1} + 5 \beta_{2} - 14 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{15} + 16 q^{16} + ( -18 + 7 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 14 \beta_{4} - 6 \beta_{5} ) q^{17} + ( 8 - 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{18} + ( -20 + 4 \beta_{1} + 4 \beta_{5} ) q^{20} + ( -7 - 5 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{21} + ( 8 - 10 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} ) q^{22} + ( -38 - 13 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} ) q^{23} + ( -8 + 8 \beta_{3} ) q^{24} + ( 34 - 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} - 6 \beta_{4} - 9 \beta_{5} ) q^{25} + ( -12 - 2 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{26} + ( -40 + 9 \beta_{1} - 9 \beta_{2} + \beta_{3} + 6 \beta_{4} + 9 \beta_{5} ) q^{27} + ( -16 - 4 \beta_{2} - 4 \beta_{4} ) q^{28} + ( -95 + 3 \beta_{1} - 7 \beta_{2} - 19 \beta_{3} + 17 \beta_{4} - 4 \beta_{5} ) q^{29} + ( -6 - 6 \beta_{1} + 10 \beta_{2} - 28 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{30} + ( -29 + 25 \beta_{1} + 5 \beta_{2} - 14 \beta_{3} - 13 \beta_{5} ) q^{31} + 32 q^{32} + ( -65 + 4 \beta_{1} - 10 \beta_{2} + 13 \beta_{3} - 52 \beta_{4} + 21 \beta_{5} ) q^{33} + ( -36 + 14 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} + 28 \beta_{4} - 12 \beta_{5} ) q^{34} + ( 15 - 5 \beta_{2} - 10 \beta_{3} + 12 \beta_{4} - 15 \beta_{5} ) q^{35} + ( 16 - 12 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} ) q^{36} + ( 96 - 27 \beta_{1} - 7 \beta_{2} - 14 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} ) q^{37} + ( -98 - 4 \beta_{1} + \beta_{2} + 7 \beta_{3} - 14 \beta_{4} + 10 \beta_{5} ) q^{39} + ( -40 + 8 \beta_{1} + 8 \beta_{5} ) q^{40} + ( -68 + 7 \beta_{1} + 16 \beta_{2} - 13 \beta_{3} + \beta_{4} + 14 \beta_{5} ) q^{41} + ( -14 - 10 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} - 4 \beta_{4} ) q^{42} + ( -340 - 15 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} + \beta_{4} + 11 \beta_{5} ) q^{43} + ( 16 - 20 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} - 20 \beta_{4} + 8 \beta_{5} ) q^{44} + ( -203 + \beta_{1} - 30 \beta_{2} + 57 \beta_{3} - 45 \beta_{4} + 28 \beta_{5} ) q^{45} + ( -76 - 26 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 14 \beta_{4} + 8 \beta_{5} ) q^{46} + ( 144 + 29 \beta_{1} - 2 \beta_{2} + 33 \beta_{3} - 18 \beta_{4} - 34 \beta_{5} ) q^{47} + ( -16 + 16 \beta_{3} ) q^{48} + ( -237 + 10 \beta_{1} + 11 \beta_{2} - 4 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{49} + ( 68 - 18 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} - 12 \beta_{4} - 18 \beta_{5} ) q^{50} + ( 191 + 25 \beta_{1} - 10 \beta_{2} - 7 \beta_{3} + 97 \beta_{4} - 28 \beta_{5} ) q^{51} + ( -24 - 4 \beta_{1} - 8 \beta_{2} - 12 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{52} + ( -96 - 43 \beta_{1} + 9 \beta_{2} + 10 \beta_{3} - 71 \beta_{4} + 33 \beta_{5} ) q^{53} + ( -80 + 18 \beta_{1} - 18 \beta_{2} + 2 \beta_{3} + 12 \beta_{4} + 18 \beta_{5} ) q^{54} + ( -181 + 41 \beta_{1} - 10 \beta_{2} + 58 \beta_{3} + 84 \beta_{4} + 8 \beta_{5} ) q^{55} + ( -32 - 8 \beta_{2} - 8 \beta_{4} ) q^{56} + ( -190 + 6 \beta_{1} - 14 \beta_{2} - 38 \beta_{3} + 34 \beta_{4} - 8 \beta_{5} ) q^{58} + ( -95 + 36 \beta_{1} + 54 \beta_{2} - 77 \beta_{3} - 15 \beta_{4} - 12 \beta_{5} ) q^{59} + ( -12 - 12 \beta_{1} + 20 \beta_{2} - 56 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} ) q^{60} + ( -97 + 46 \beta_{1} + 31 \beta_{2} + 30 \beta_{3} + 106 \beta_{4} + 9 \beta_{5} ) q^{61} + ( -58 + 50 \beta_{1} + 10 \beta_{2} - 28 \beta_{3} - 26 \beta_{5} ) q^{62} + ( -7 + 15 \beta_{1} + 8 \beta_{2} + 21 \beta_{3} - 22 \beta_{4} + 30 \beta_{5} ) q^{63} + 64 q^{64} + ( -86 + 7 \beta_{1} - 34 \beta_{2} + 34 \beta_{3} + 12 \beta_{4} - 14 \beta_{5} ) q^{65} + ( -130 + 8 \beta_{1} - 20 \beta_{2} + 26 \beta_{3} - 104 \beta_{4} + 42 \beta_{5} ) q^{66} + ( -320 - 19 \beta_{1} - 34 \beta_{2} + 51 \beta_{3} + 58 \beta_{4} - 5 \beta_{5} ) q^{67} + ( -72 + 28 \beta_{1} + 8 \beta_{2} + 12 \beta_{3} + 56 \beta_{4} - 24 \beta_{5} ) q^{68} + ( 16 + 12 \beta_{1} - 5 \beta_{2} - 30 \beta_{3} - 98 \beta_{4} + 7 \beta_{5} ) q^{69} + ( 30 - 10 \beta_{2} - 20 \beta_{3} + 24 \beta_{4} - 30 \beta_{5} ) q^{70} + ( -242 + 21 \beta_{1} + 15 \beta_{2} - 50 \beta_{3} + 79 \beta_{4} - 68 \beta_{5} ) q^{71} + ( 32 - 24 \beta_{3} + 24 \beta_{4} - 24 \beta_{5} ) q^{72} + ( -434 - 18 \beta_{1} - 5 \beta_{2} + 15 \beta_{3} - 63 \beta_{4} - 11 \beta_{5} ) q^{73} + ( 192 - 54 \beta_{1} - 14 \beta_{2} - 28 \beta_{3} + 8 \beta_{4} + 16 \beta_{5} ) q^{74} + ( -133 + 27 \beta_{1} - 72 \beta_{2} + 142 \beta_{3} - 66 \beta_{4} + 84 \beta_{5} ) q^{75} + ( -93 - 10 \beta_{1} - 14 \beta_{2} + 54 \beta_{3} + 36 \beta_{4} - \beta_{5} ) q^{77} + ( -196 - 8 \beta_{1} + 2 \beta_{2} + 14 \beta_{3} - 28 \beta_{4} + 20 \beta_{5} ) q^{78} + ( -155 + 18 \beta_{1} + 52 \beta_{2} - 21 \beta_{3} + 54 \beta_{4} - 35 \beta_{5} ) q^{79} + ( -80 + 16 \beta_{1} + 16 \beta_{5} ) q^{80} + ( -209 - 27 \beta_{1} + 72 \beta_{2} - 51 \beta_{3} - 39 \beta_{4} + 21 \beta_{5} ) q^{81} + ( -136 + 14 \beta_{1} + 32 \beta_{2} - 26 \beta_{3} + 2 \beta_{4} + 28 \beta_{5} ) q^{82} + ( -132 - 30 \beta_{1} - 43 \beta_{2} + 58 \beta_{3} - 121 \beta_{4} + 17 \beta_{5} ) q^{83} + ( -28 - 20 \beta_{1} + 12 \beta_{2} - 20 \beta_{3} - 8 \beta_{4} ) q^{84} + ( 113 - 94 \beta_{1} + 23 \beta_{2} - 23 \beta_{3} - 204 \beta_{4} + 44 \beta_{5} ) q^{85} + ( -680 - 30 \beta_{1} - 22 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} + 22 \beta_{5} ) q^{86} + ( -513 + 28 \beta_{1} + 15 \beta_{2} - 49 \beta_{3} + 7 \beta_{4} + 14 \beta_{5} ) q^{87} + ( 32 - 40 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} - 40 \beta_{4} + 16 \beta_{5} ) q^{88} + ( 307 - 67 \beta_{1} - 11 \beta_{2} + 139 \beta_{3} + 104 \beta_{4} + 91 \beta_{5} ) q^{89} + ( -406 + 2 \beta_{1} - 60 \beta_{2} + 114 \beta_{3} - 90 \beta_{4} + 56 \beta_{5} ) q^{90} + ( 225 + 32 \beta_{1} + 24 \beta_{2} + 17 \beta_{3} + 2 \beta_{4} + 23 \beta_{5} ) q^{91} + ( -152 - 52 \beta_{1} - 12 \beta_{2} + 8 \beta_{3} - 28 \beta_{4} + 16 \beta_{5} ) q^{92} + ( -194 - 65 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} + 147 \beta_{4} + 53 \beta_{5} ) q^{93} + ( 288 + 58 \beta_{1} - 4 \beta_{2} + 66 \beta_{3} - 36 \beta_{4} - 68 \beta_{5} ) q^{94} + ( -32 + 32 \beta_{3} ) q^{96} + ( 499 - 72 \beta_{1} + 126 \beta_{2} - \beta_{3} - 167 \beta_{4} + 4 \beta_{5} ) q^{97} + ( -474 + 20 \beta_{1} + 22 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} ) q^{98} + ( 52 - 53 \beta_{1} + 47 \beta_{2} - 185 \beta_{3} + 31 \beta_{4} - 55 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 12q^{2} - 9q^{3} + 24q^{4} - 27q^{5} - 18q^{6} - 21q^{7} + 48q^{8} + 33q^{9} + O(q^{10}) \) \( 6q + 12q^{2} - 9q^{3} + 24q^{4} - 27q^{5} - 18q^{6} - 21q^{7} + 48q^{8} + 33q^{9} - 54q^{10} + 9q^{11} - 36q^{12} - 24q^{13} - 42q^{14} + 96q^{16} - 102q^{17} + 66q^{18} - 108q^{20} - 51q^{21} + 18q^{22} - 264q^{23} - 72q^{24} + 177q^{25} - 48q^{26} - 189q^{27} - 84q^{28} - 483q^{29} - 72q^{31} + 192q^{32} - 387q^{33} - 204q^{34} + 135q^{35} + 132q^{36} + 558q^{37} - 624q^{39} - 216q^{40} - 396q^{41} - 102q^{42} - 2064q^{43} + 36q^{44} - 1296q^{45} - 528q^{46} + 858q^{47} - 144q^{48} - 1413q^{49} + 354q^{50} + 1272q^{51} - 96q^{52} - 762q^{53} - 378q^{54} - 1107q^{55} - 168q^{56} - 966q^{58} - 393q^{59} - 627q^{61} - 144q^{62} - 84q^{63} + 384q^{64} - 495q^{65} - 774q^{66} - 2028q^{67} - 408q^{68} + 237q^{69} + 270q^{70} - 1284q^{71} + 264q^{72} - 2688q^{73} + 1116q^{74} - 927q^{75} - 708q^{77} - 1248q^{78} - 969q^{79} - 432q^{80} - 1398q^{81} - 792q^{82} - 927q^{83} - 204q^{84} + 396q^{85} - 4128q^{86} - 2892q^{87} + 72q^{88} + 1257q^{89} - 2592q^{90} + 1323q^{91} - 1056q^{92} - 1368q^{93} + 1716q^{94} - 288q^{96} + 2403q^{97} - 2826q^{98} + 567q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 75 x^{4} + 135 x^{3} + 1857 x^{2} - 1425 x - 14797\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 58 \nu^{5} - 1339 \nu^{4} - 1900 \nu^{3} + 77454 \nu^{2} + 13148 \nu - 1014210 \)\()/12329\)
\(\beta_{2}\)\(=\)\((\)\( -225 \nu^{5} + 2431 \nu^{4} + 5245 \nu^{3} - 99803 \nu^{2} - 16569 \nu + 1017564 \)\()/12329\)
\(\beta_{3}\)\(=\)\((\)\( 258 \nu^{5} - 2130 \nu^{4} - 9302 \nu^{3} + 89454 \nu^{2} + 64863 \nu - 924172 \)\()/12329\)
\(\beta_{4}\)\(=\)\((\)\( 501 \nu^{5} - 3276 \nu^{4} - 22364 \nu^{3} + 141021 \nu^{2} + 244514 \nu - 1477213 \)\()/12329\)
\(\beta_{5}\)\(=\)\((\)\( -597 \nu^{5} + 4642 \nu^{4} + 22958 \nu^{3} - 183768 \nu^{2} - 222487 \nu + 1712137 \)\()/12329\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(4 \beta_{4} - 13 \beta_{3} - 6 \beta_{2}\)\()/19\)
\(\nu^{2}\)\(=\)\((\)\(19 \beta_{5} + 17 \beta_{4} - 3 \beta_{3} - 16 \beta_{2} + 494\)\()/19\)
\(\nu^{3}\)\(=\)\((\)\(114 \beta_{5} + 159 \beta_{4} - 265 \beta_{3} - 267 \beta_{2} - 57 \beta_{1} + 703\)\()/19\)
\(\nu^{4}\)\(=\)\((\)\(1292 \beta_{5} + 1162 \beta_{4} - 176 \beta_{3} - 1116 \beta_{2} - 285 \beta_{1} + 15276\)\()/19\)
\(\nu^{5}\)\(=\)\((\)\(8189 \beta_{5} + 8426 \beta_{4} - 5791 \beta_{3} - 11784 \beta_{2} - 4408 \beta_{1} + 48241\)\()/19\)

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
4.11442
5.47825
7.04755
−4.99381
−4.13096
−4.51546
2.00000 −8.87319 4.00000 −20.0362 −17.7464 −9.42541 8.00000 51.7336 −40.0725
1.2 2.00000 −5.78366 4.00000 6.19832 −11.5673 11.2225 8.00000 6.45071 12.3966
1.3 2.00000 −4.98337 4.00000 2.34989 −9.96675 2.84316 8.00000 −2.16599 4.69977
1.4 2.00000 0.235037 4.00000 11.0362 0.470075 −18.5336 8.00000 −26.9448 22.0725
1.5 2.00000 3.82555 4.00000 −15.1983 7.65110 1.61327 8.00000 −12.3652 −30.3966
1.6 2.00000 6.57964 4.00000 −11.3499 13.1593 −8.71985 8.00000 16.2917 −22.6998
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

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 722.4.a.p 6
19.b odd 2 1 722.4.a.o 6
19.e even 9 2 38.4.e.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.e.a 12 19.e even 9 2
722.4.a.o 6 19.b odd 2 1
722.4.a.p 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\):

\( T_{3}^{6} + 9 T_{3}^{5} - 57 T_{3}^{4} - 531 T_{3}^{3} + 597 T_{3}^{2} + 6327 T_{3} - 1513 \)
\( T_{5}^{6} + 27 T_{5}^{5} - 99 T_{5}^{4} - 5427 T_{5}^{3} + 1539 T_{5}^{2} + 263169 T_{5} - 555579 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{6} \)
$3$ \( -1513 + 6327 T + 597 T^{2} - 531 T^{3} - 57 T^{4} + 9 T^{5} + T^{6} \)
$5$ \( -555579 + 263169 T + 1539 T^{2} - 5427 T^{3} - 99 T^{4} + 27 T^{5} + T^{6} \)
$7$ \( -78409 + 61626 T - 2922 T^{2} - 3087 T^{3} - 102 T^{4} + 21 T^{5} + T^{6} \)
$11$ \( 236331747 - 80169822 T + 4587750 T^{2} + 55943 T^{3} - 4386 T^{4} - 9 T^{5} + T^{6} \)
$13$ \( -50265519 - 7820352 T + 1198959 T^{2} - 7286 T^{3} - 2208 T^{4} + 24 T^{5} + T^{6} \)
$17$ \( 6425825733 - 279740862 T - 49892877 T^{2} - 1466092 T^{3} - 9522 T^{4} + 102 T^{5} + T^{6} \)
$19$ \( T^{6} \)
$23$ \( 1288970568 + 4136856768 T - 76305294 T^{2} - 1906489 T^{3} + 9099 T^{4} + 264 T^{5} + T^{6} \)
$29$ \( -655147831257 + 40899318471 T - 451487709 T^{2} - 6310257 T^{3} + 42669 T^{4} + 483 T^{5} + T^{6} \)
$31$ \( -3586719447 + 810034263 T + 604768608 T^{2} - 9551087 T^{3} - 89202 T^{4} + 72 T^{5} + T^{6} \)
$37$ \( -20292669795592 - 333342861012 T + 873439542 T^{2} + 27776061 T^{3} - 5829 T^{4} - 558 T^{5} + T^{6} \)
$41$ \( -6229329066963 - 318952710990 T - 5597061471 T^{2} - 37463158 T^{3} - 42618 T^{4} + 396 T^{5} + T^{6} \)
$43$ \( 711315853703439 + 18396963074322 T + 166699330383 T^{2} + 735259660 T^{3} + 1721544 T^{4} + 2064 T^{5} + T^{6} \)
$47$ \( -108448526272959 - 6635874882402 T - 20401533765 T^{2} + 216805256 T^{3} - 119490 T^{4} - 858 T^{5} + T^{6} \)
$53$ \( 32216889916737 + 118342752156 T - 13247521947 T^{2} - 129402764 T^{3} - 83196 T^{4} + 762 T^{5} + T^{6} \)
$59$ \( -19047267094980753 + 48639075854490 T + 232880038974 T^{2} - 279275059 T^{3} - 864582 T^{4} + 393 T^{5} + T^{6} \)
$61$ \( 16867331361800893 + 141476034856740 T + 144034102698 T^{2} - 636245971 T^{3} - 925284 T^{4} + 627 T^{5} + T^{6} \)
$67$ \( 2154139523420568 - 11839997109384 T - 145470288786 T^{2} - 14934233 T^{3} + 1162005 T^{4} + 2028 T^{5} + T^{6} \)
$71$ \( 9092682876775752 + 32405993057160 T - 148782824118 T^{2} - 567139535 T^{3} - 41445 T^{4} + 1284 T^{5} + T^{6} \)
$73$ \( -4558088077043392 - 2166454842384 T + 248773667712 T^{2} + 1299286701 T^{3} + 2752893 T^{4} + 2688 T^{5} + T^{6} \)
$79$ \( -7937755936451317 + 39757958790189 T + 68305747083 T^{2} - 375670719 T^{3} - 348831 T^{4} + 969 T^{5} + T^{6} \)
$83$ \( 6552608983449957 + 49787418227184 T - 87565666734 T^{2} - 785550743 T^{3} - 661224 T^{4} + 927 T^{5} + T^{6} \)
$89$ \( 896266768522805613 - 2850036607226565 T + 577944380961 T^{2} + 4645212713 T^{3} - 3011373 T^{4} - 1257 T^{5} + T^{6} \)
$97$ \( -1074803648205286817 - 4635379275422334 T + 1416423740796 T^{2} + 6812560721 T^{3} - 2229828 T^{4} - 2403 T^{5} + T^{6} \)
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