Properties

Label 722.4.a.o
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.

\(f(q)\) \(=\) \( q -2 q^{2} + ( 1 + \beta_{1} ) q^{3} + 4 q^{4} + ( -4 - \beta_{3} - \beta_{5} ) q^{5} + ( -2 - 2 \beta_{1} ) q^{6} + ( -4 + \beta_{2} + \beta_{4} ) q^{7} -8 q^{8} + ( 4 + 3 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} ) q^{9} +O(q^{10})\) \( q -2 q^{2} + ( 1 + \beta_{1} ) q^{3} + 4 q^{4} + ( -4 - \beta_{3} - \beta_{5} ) q^{5} + ( -2 - 2 \beta_{1} ) q^{6} + ( -4 + \beta_{2} + \beta_{4} ) q^{7} -8 q^{8} + ( 4 + 3 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} ) q^{9} + ( 8 + 2 \beta_{3} + 2 \beta_{5} ) q^{10} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{11} + ( 4 + 4 \beta_{1} ) q^{12} + ( 7 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{13} + ( 8 - 2 \beta_{2} - 2 \beta_{4} ) q^{14} + ( 6 - 14 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{15} + 16 q^{16} + ( -11 - 3 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 14 \beta_{4} + 6 \beta_{5} ) q^{17} + ( -8 - 6 \beta_{1} + 6 \beta_{4} - 6 \beta_{5} ) q^{18} + ( -16 - 4 \beta_{3} - 4 \beta_{5} ) q^{20} + ( 12 - 5 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{21} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} ) q^{22} + ( -51 - 2 \beta_{1} + 3 \beta_{2} + 13 \beta_{3} + 7 \beta_{4} - 4 \beta_{5} ) q^{23} + ( -8 - 8 \beta_{1} ) q^{24} + ( 25 + 9 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 6 \beta_{4} + 9 \beta_{5} ) q^{25} + ( -14 + 6 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{26} + ( 31 + \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 6 \beta_{4} + 9 \beta_{5} ) q^{27} + ( -16 + 4 \beta_{2} + 4 \beta_{4} ) q^{28} + ( 92 - 19 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} + 17 \beta_{4} - 4 \beta_{5} ) q^{29} + ( -12 + 28 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{30} + ( 4 - 14 \beta_{1} + 5 \beta_{2} + 25 \beta_{3} - 13 \beta_{5} ) q^{31} -32 q^{32} + ( 61 + 13 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} - 52 \beta_{4} + 21 \beta_{5} ) q^{33} + ( 22 + 6 \beta_{1} + 4 \beta_{2} + 14 \beta_{3} + 28 \beta_{4} - 12 \beta_{5} ) q^{34} + ( 15 + 10 \beta_{1} + 5 \beta_{2} - 12 \beta_{4} + 15 \beta_{5} ) q^{35} + ( 16 + 12 \beta_{1} - 12 \beta_{4} + 12 \beta_{5} ) q^{36} + ( -69 - 14 \beta_{1} - 7 \beta_{2} - 27 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} ) q^{37} + ( -102 - 7 \beta_{1} - \beta_{2} + 4 \beta_{3} + 14 \beta_{4} - 10 \beta_{5} ) q^{39} + ( 32 + 8 \beta_{3} + 8 \beta_{5} ) q^{40} + ( 61 - 13 \beta_{1} + 16 \beta_{2} + 7 \beta_{3} + \beta_{4} + 14 \beta_{5} ) q^{41} + ( -24 + 10 \beta_{1} - 6 \beta_{2} + 10 \beta_{3} + 4 \beta_{4} ) q^{42} + ( -355 - 4 \beta_{1} + 11 \beta_{2} + 15 \beta_{3} - \beta_{4} - 11 \beta_{5} ) q^{43} + ( -4 + 8 \beta_{1} - 8 \beta_{2} + 20 \beta_{3} + 20 \beta_{4} - 8 \beta_{5} ) q^{44} + ( -202 - 57 \beta_{1} + 30 \beta_{2} - \beta_{3} + 45 \beta_{4} - 28 \beta_{5} ) q^{45} + ( 102 + 4 \beta_{1} - 6 \beta_{2} - 26 \beta_{3} - 14 \beta_{4} + 8 \beta_{5} ) q^{46} + ( 173 - 33 \beta_{1} + 2 \beta_{2} - 29 \beta_{3} + 18 \beta_{4} + 34 \beta_{5} ) q^{47} + ( 16 + 16 \beta_{1} ) q^{48} + ( -227 + 4 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{49} + ( -50 - 18 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} - 12 \beta_{4} - 18 \beta_{5} ) q^{50} + ( -216 - 7 \beta_{1} - 10 \beta_{2} + 25 \beta_{3} + 97 \beta_{4} - 28 \beta_{5} ) q^{51} + ( 28 - 12 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{52} + ( 139 + 10 \beta_{1} + 9 \beta_{2} - 43 \beta_{3} - 71 \beta_{4} + 33 \beta_{5} ) q^{53} + ( -62 - 2 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} - 12 \beta_{4} - 18 \beta_{5} ) q^{54} + ( -140 - 58 \beta_{1} + 10 \beta_{2} - 41 \beta_{3} - 84 \beta_{4} - 8 \beta_{5} ) q^{55} + ( 32 - 8 \beta_{2} - 8 \beta_{4} ) q^{56} + ( -184 + 38 \beta_{1} + 14 \beta_{2} - 6 \beta_{3} - 34 \beta_{4} + 8 \beta_{5} ) q^{58} + ( 59 - 77 \beta_{1} + 54 \beta_{2} + 36 \beta_{3} - 15 \beta_{4} - 12 \beta_{5} ) q^{59} + ( 24 - 56 \beta_{1} + 20 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} ) q^{60} + ( -51 - 30 \beta_{1} - 31 \beta_{2} - 46 \beta_{3} - 106 \beta_{4} - 9 \beta_{5} ) q^{61} + ( -8 + 28 \beta_{1} - 10 \beta_{2} - 50 \beta_{3} + 26 \beta_{5} ) q^{62} + ( 8 - 21 \beta_{1} - 8 \beta_{2} - 15 \beta_{3} + 22 \beta_{4} - 30 \beta_{5} ) q^{63} + 64 q^{64} + ( 79 + 34 \beta_{1} - 34 \beta_{2} + 7 \beta_{3} + 12 \beta_{4} - 14 \beta_{5} ) q^{65} + ( -122 - 26 \beta_{1} + 20 \beta_{2} - 8 \beta_{3} + 104 \beta_{4} - 42 \beta_{5} ) q^{66} + ( 339 + 51 \beta_{1} - 34 \beta_{2} - 19 \beta_{3} + 58 \beta_{4} - 5 \beta_{5} ) q^{67} + ( -44 - 12 \beta_{1} - 8 \beta_{2} - 28 \beta_{3} - 56 \beta_{4} + 24 \beta_{5} ) q^{68} + ( -28 - 30 \beta_{1} - 5 \beta_{2} + 12 \beta_{3} - 98 \beta_{4} + 7 \beta_{5} ) q^{69} + ( -30 - 20 \beta_{1} - 10 \beta_{2} + 24 \beta_{4} - 30 \beta_{5} ) q^{70} + ( 221 - 50 \beta_{1} + 15 \beta_{2} + 21 \beta_{3} + 79 \beta_{4} - 68 \beta_{5} ) q^{71} + ( -32 - 24 \beta_{1} + 24 \beta_{4} - 24 \beta_{5} ) q^{72} + ( -452 - 15 \beta_{1} + 5 \beta_{2} + 18 \beta_{3} + 63 \beta_{4} + 11 \beta_{5} ) q^{73} + ( 138 + 28 \beta_{1} + 14 \beta_{2} + 54 \beta_{3} - 8 \beta_{4} - 16 \beta_{5} ) q^{74} + ( 106 + 142 \beta_{1} - 72 \beta_{2} + 27 \beta_{3} - 66 \beta_{4} + 84 \beta_{5} ) q^{75} + ( -103 - 54 \beta_{1} + 14 \beta_{2} + 10 \beta_{3} - 36 \beta_{4} + \beta_{5} ) q^{77} + ( 204 + 14 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 28 \beta_{4} + 20 \beta_{5} ) q^{78} + ( 137 - 21 \beta_{1} + 52 \beta_{2} + 18 \beta_{3} + 54 \beta_{4} - 35 \beta_{5} ) q^{79} + ( -64 - 16 \beta_{3} - 16 \beta_{5} ) q^{80} + ( -236 + 51 \beta_{1} - 72 \beta_{2} + 27 \beta_{3} + 39 \beta_{4} - 21 \beta_{5} ) q^{81} + ( -122 + 26 \beta_{1} - 32 \beta_{2} - 14 \beta_{3} - 2 \beta_{4} - 28 \beta_{5} ) q^{82} + ( -162 - 58 \beta_{1} + 43 \beta_{2} + 30 \beta_{3} + 121 \beta_{4} - 17 \beta_{5} ) q^{83} + ( 48 - 20 \beta_{1} + 12 \beta_{2} - 20 \beta_{3} - 8 \beta_{4} ) q^{84} + ( 19 + 23 \beta_{1} - 23 \beta_{2} + 94 \beta_{3} + 204 \beta_{4} - 44 \beta_{5} ) q^{85} + ( 710 + 8 \beta_{1} - 22 \beta_{2} - 30 \beta_{3} + 2 \beta_{4} + 22 \beta_{5} ) q^{86} + ( -485 + 49 \beta_{1} - 15 \beta_{2} - 28 \beta_{3} - 7 \beta_{4} - 14 \beta_{5} ) q^{87} + ( 8 - 16 \beta_{1} + 16 \beta_{2} - 40 \beta_{3} - 40 \beta_{4} + 16 \beta_{5} ) q^{88} + ( -240 + 139 \beta_{1} - 11 \beta_{2} - 67 \beta_{3} + 104 \beta_{4} + 91 \beta_{5} ) q^{89} + ( 404 + 114 \beta_{1} - 60 \beta_{2} + 2 \beta_{3} - 90 \beta_{4} + 56 \beta_{5} ) q^{90} + ( -257 + 17 \beta_{1} + 24 \beta_{2} + 32 \beta_{3} + 2 \beta_{4} + 23 \beta_{5} ) q^{91} + ( -204 - 8 \beta_{1} + 12 \beta_{2} + 52 \beta_{3} + 28 \beta_{4} - 16 \beta_{5} ) q^{92} + ( -259 - 7 \beta_{1} + 4 \beta_{2} + 65 \beta_{3} - 147 \beta_{4} - 53 \beta_{5} ) q^{93} + ( -346 + 66 \beta_{1} - 4 \beta_{2} + 58 \beta_{3} - 36 \beta_{4} - 68 \beta_{5} ) q^{94} + ( -32 - 32 \beta_{1} ) q^{96} + ( -427 - \beta_{1} + 126 \beta_{2} - 72 \beta_{3} - 167 \beta_{4} + 4 \beta_{5} ) q^{97} + ( 454 - 8 \beta_{1} + 22 \beta_{2} + 20 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} ) q^{98} + ( -1 + 185 \beta_{1} - 47 \beta_{2} + 53 \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 \(x^{6} - 3 x^{5} - 75 x^{4} + 135 x^{3} + 1857 x^{2} - 1425 x - 14797\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -129 \nu^{5} + 1065 \nu^{4} + 4651 \nu^{3} - 44727 \nu^{2} - 50925 \nu + 474415 \)\()/12329\)
\(\beta_{2}\)\(=\)\((\)\( 354 \nu^{5} - 3496 \nu^{4} - 9896 \nu^{3} + 144530 \nu^{2} + 30507 \nu - 1467321 \)\()/12329\)
\(\beta_{3}\)\(=\)\((\)\( 430 \nu^{5} - 3550 \nu^{4} - 19613 \nu^{3} + 173748 \nu^{2} + 219066 \nu - 2029337 \)\()/12329\)
\(\beta_{4}\)\(=\)\((\)\( -501 \nu^{5} + 3276 \nu^{4} + 22364 \nu^{3} - 141021 \nu^{2} - 244514 \nu + 1477213 \)\()/12329\)
\(\beta_{5}\)\(=\)\((\)\( -969 \nu^{5} + 6853 \nu^{4} + 40671 \nu^{3} - 280062 \nu^{2} - 428405 \nu + 2727264 \)\()/12329\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{4} - 3 \beta_{2} - 16 \beta_{1} + 19\)\()/19\)
\(\nu^{2}\)\(=\)\((\)\(19 \beta_{5} - 28 \beta_{4} + 4 \beta_{2} - 23 \beta_{1} + 513\)\()/19\)
\(\nu^{3}\)\(=\)\((\)\(114 \beta_{5} - 144 \beta_{4} - 57 \beta_{3} - 12 \beta_{2} - 520 \beta_{1} + 1235\)\()/19\)
\(\nu^{4}\)\(=\)\((\)\(1292 \beta_{5} - 2165 \beta_{4} - 285 \beta_{3} + 255 \beta_{2} - 1547 \beta_{1} + 16568\)\()/19\)
\(\nu^{5}\)\(=\)\((\)\(8189 \beta_{5} - 14147 \beta_{4} - 4408 \beta_{3} + 1470 \beta_{2} - 19045 \beta_{1} + 65816\)\()/19\)

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
7.04755
5.47825
4.11442
−4.51546
−4.13096
−4.99381
−2.00000 −6.57964 4.00000 −11.3499 13.1593 −8.71985 −8.00000 16.2917 22.6998
1.2 −2.00000 −3.82555 4.00000 −15.1983 7.65110 1.61327 −8.00000 −12.3652 30.3966
1.3 −2.00000 −0.235037 4.00000 11.0362 0.470075 −18.5336 −8.00000 −26.9448 −22.0725
1.4 −2.00000 4.98337 4.00000 2.34989 −9.96675 2.84316 −8.00000 −2.16599 −4.69977
1.5 −2.00000 5.78366 4.00000 6.19832 −11.5673 11.2225 −8.00000 6.45071 −12.3966
1.6 −2.00000 8.87319 4.00000 −20.0362 −17.7464 −9.42541 −8.00000 51.7336 40.0725
\(n\): 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.o 6
19.b odd 2 1 722.4.a.p 6
19.f odd 18 2 38.4.e.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.e.a 12 19.f odd 18 2
722.4.a.o 6 1.a even 1 1 trivial
722.4.a.p 6 19.b odd 2 1

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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