Properties

Label 390.2.bb.c
Level $390$
Weight $2$
Character orbit 390.bb
Analytic conductor $3.114$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.17284886784.1
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} -\beta_{6} q^{3} + ( 1 - \beta_{6} ) q^{4} -\beta_{2} q^{5} + ( -\beta_{2} - \beta_{5} ) q^{6} + ( -\beta_{1} + \beta_{4} ) q^{7} -\beta_{2} q^{8} + ( -1 + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} -\beta_{6} q^{3} + ( 1 - \beta_{6} ) q^{4} -\beta_{2} q^{5} + ( -\beta_{2} - \beta_{5} ) q^{6} + ( -\beta_{1} + \beta_{4} ) q^{7} -\beta_{2} q^{8} + ( -1 + \beta_{6} ) q^{9} -\beta_{6} q^{10} + ( 1 - \beta_{2} + \beta_{4} - \beta_{7} ) q^{11} - q^{12} + ( -2 - \beta_{2} - \beta_{5} + \beta_{7} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{14} -\beta_{5} q^{15} -\beta_{6} q^{16} + ( 4 - 4 \beta_{6} ) q^{17} + \beta_{2} q^{18} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{19} + ( -\beta_{2} - \beta_{5} ) q^{20} + ( -\beta_{2} + \beta_{3} - \beta_{7} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{22} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{23} -\beta_{5} q^{24} - q^{25} + ( -1 - \beta_{3} - 2 \beta_{5} ) q^{26} + q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{28} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{29} + ( -1 + \beta_{6} ) q^{30} + ( -4 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{31} + ( -\beta_{2} - \beta_{5} ) q^{32} + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{33} -4 \beta_{2} q^{34} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{35} + \beta_{6} q^{36} + ( 5 + \beta_{2} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{37} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{38} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{39} - q^{40} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{41} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{42} + ( 3 + 3 \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{44} + ( \beta_{2} + \beta_{5} ) q^{45} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{46} + ( -2 - \beta_{2} - 2 \beta_{3} + 4 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -1 + \beta_{6} ) q^{48} + ( 1 - \beta_{1} + 8 \beta_{2} - \beta_{3} + 3 \beta_{5} + 3 \beta_{6} ) q^{49} -\beta_{5} q^{50} -4 q^{51} + ( -1 - \beta_{1} - 2 \beta_{5} + \beta_{6} ) q^{52} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - \beta_{7} ) q^{53} + \beta_{5} q^{54} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{56} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{57} + ( 1 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{58} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{59} + \beta_{2} q^{60} + ( -4 + 2 \beta_{2} - 2 \beta_{5} + 4 \beta_{6} ) q^{61} + ( 2 - \beta_{1} + 5 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{62} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{63} - q^{64} + ( -2 + \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{65} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{66} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{67} -4 \beta_{6} q^{68} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{69} + ( -\beta_{2} + \beta_{3} - \beta_{7} ) q^{70} + ( 2 - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( \beta_{2} + \beta_{5} ) q^{72} + ( -4 - 4 \beta_{2} + 2 \beta_{3} + 8 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{74} + \beta_{6} q^{75} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{76} + ( 2 - 2 \beta_{1} + 5 \beta_{2} + \beta_{3} + 2 \beta_{4} + 12 \beta_{5} - \beta_{7} ) q^{77} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{78} + ( -4 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{79} -\beta_{5} q^{80} -\beta_{6} q^{81} + ( 2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{82} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( \beta_{1} - \beta_{4} ) q^{84} + ( -4 \beta_{2} - 4 \beta_{5} ) q^{85} + ( -4 - 4 \beta_{2} + \beta_{3} + 8 \beta_{6} - \beta_{7} ) q^{86} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{88} + ( 7 - \beta_{1} + \beta_{3} + \beta_{5} - 4 \beta_{6} ) q^{89} + q^{90} + ( -5 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} ) q^{91} + ( \beta_{3} + \beta_{7} ) q^{92} + ( 5 - \beta_{1} + \beta_{3} + \beta_{5} - 3 \beta_{6} ) q^{93} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{6} ) q^{94} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{95} + \beta_{2} q^{96} + ( -2 - 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} ) q^{97} + ( 4 - \beta_{1} + 3 \beta_{2} + \beta_{4} + 3 \beta_{5} + 4 \beta_{6} ) q^{98} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 4q^{4} - 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 4q^{4} - 4q^{9} - 4q^{10} + 6q^{11} - 8q^{12} - 12q^{13} + 4q^{14} - 4q^{16} + 16q^{17} - 6q^{19} + 2q^{22} + 4q^{23} - 8q^{25} - 12q^{26} + 8q^{27} - 8q^{29} - 4q^{30} - 6q^{33} + 2q^{35} + 4q^{36} + 30q^{37} + 6q^{39} - 8q^{40} - 2q^{42} + 14q^{43} - 6q^{46} - 4q^{48} + 14q^{49} - 32q^{51} - 6q^{52} + 16q^{53} - 2q^{55} + 2q^{56} - 6q^{58} + 24q^{59} - 16q^{61} + 4q^{62} - 8q^{64} - 6q^{65} - 4q^{66} + 24q^{67} - 16q^{68} + 4q^{69} - 12q^{71} + 10q^{74} + 4q^{75} - 6q^{76} + 16q^{77} + 6q^{78} - 20q^{79} - 4q^{81} + 4q^{82} - 8q^{87} - 2q^{88} + 42q^{89} + 8q^{90} - 10q^{91} + 8q^{92} + 30q^{93} - 8q^{94} - 24q^{97} + 48q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2225 \nu^{7} + 27304 \nu^{6} - 47714 \nu^{5} - 21770 \nu^{4} + 204731 \nu^{3} + 5488658 \nu^{2} + 258164 \nu - 320632 \)\()/20561424\)
\(\beta_{3}\)\(=\)\((\)\( -1777 \nu^{7} - 125666 \nu^{6} + 800522 \nu^{5} - 2370014 \nu^{4} - 784473 \nu^{3} - 5277648 \nu^{2} + 34618460 \nu - 62031216 \)\()/6853808\)
\(\beta_{4}\)\(=\)\((\)\( -299 \nu^{7} + 7801 \nu^{6} - 39200 \nu^{5} + 84154 \nu^{4} + 80915 \nu^{3} + 360425 \nu^{2} - 1640788 \nu + 2274116 \)\()/395412\)
\(\beta_{5}\)\(=\)\((\)\( 43733 \nu^{7} - 71918 \nu^{6} - 318186 \nu^{5} + 3350390 \nu^{4} + 3714597 \nu^{3} - 2633192 \nu^{2} - 18392236 \nu + 94417440 \)\()/20561424\)
\(\beta_{6}\)\(=\)\((\)\( -151 \nu^{7} + 822 \nu^{6} - 2018 \nu^{5} - 2554 \nu^{4} - 7135 \nu^{3} + 37828 \nu^{2} - 46500 \nu - 3328 \)\()/51792\)
\(\beta_{7}\)\(=\)\((\)\( -123974 \nu^{7} + 362933 \nu^{6} - 209286 \nu^{5} - 5793110 \nu^{4} - 10547568 \nu^{3} + 7562729 \nu^{2} + 3493156 \nu - 117816036 \)\()/10280712\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} + 9 \beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(6 \beta_{7} - 10 \beta_{6} + 26 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 18 \beta_{2} + 3 \beta_{1} - 18\)
\(\nu^{4}\)\(=\)\(23 \beta_{7} + 139 \beta_{5} + 5 \beta_{4} + 28 \beta_{3} + 67 \beta_{2} - 5 \beta_{1} - 149\)
\(\nu^{5}\)\(=\)\(250 \beta_{6} + 322 \beta_{5} - 72 \beta_{4} + 72 \beta_{3} - 76 \beta_{2} - 149 \beta_{1} - 398\)
\(\nu^{6}\)\(=\)\(-471 \beta_{7} + 1748 \beta_{6} - 619 \beta_{5} - 619 \beta_{4} - 148 \beta_{3} - 2095 \beta_{2} - 619 \beta_{1} - 255\)
\(\nu^{7}\)\(=\)\(-2986 \beta_{7} + 5302 \beta_{6} - 11002 \beta_{5} - 2714 \beta_{4} - 2714 \beta_{3} - 10118 \beta_{2} - 1493 \beta_{1} + 7530\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
3.17270 + 3.17270i
−1.80668 1.80668i
1.33404 1.33404i
−1.70006 + 1.70006i
3.17270 3.17270i
−1.80668 + 1.80668i
1.33404 + 1.33404i
−1.70006 1.70006i
−0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i −2.01141 + 1.16129i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
121.2 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i 1.14539 0.661290i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
121.3 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i −3.15637 + 1.82233i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
121.4 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i 4.02239 2.32233i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i −2.01141 1.16129i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.2 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i 1.14539 + 0.661290i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.3 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i −3.15637 1.82233i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.4 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i 4.02239 + 2.32233i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bb.c 8
3.b odd 2 1 1170.2.bs.f 8
5.b even 2 1 1950.2.bc.g 8
5.c odd 4 1 1950.2.y.j 8
5.c odd 4 1 1950.2.y.k 8
13.c even 3 1 5070.2.b.ba 8
13.e even 6 1 inner 390.2.bb.c 8
13.e even 6 1 5070.2.b.ba 8
13.f odd 12 1 5070.2.a.bz 4
13.f odd 12 1 5070.2.a.ca 4
39.h odd 6 1 1170.2.bs.f 8
65.l even 6 1 1950.2.bc.g 8
65.r odd 12 1 1950.2.y.j 8
65.r odd 12 1 1950.2.y.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 1.a even 1 1 trivial
390.2.bb.c 8 13.e even 6 1 inner
1170.2.bs.f 8 3.b odd 2 1
1170.2.bs.f 8 39.h odd 6 1
1950.2.y.j 8 5.c odd 4 1
1950.2.y.j 8 65.r odd 12 1
1950.2.y.k 8 5.c odd 4 1
1950.2.y.k 8 65.r odd 12 1
1950.2.bc.g 8 5.b even 2 1
1950.2.bc.g 8 65.l even 6 1
5070.2.a.bz 4 13.f odd 12 1
5070.2.a.ca 4 13.f odd 12 1
5070.2.b.ba 8 13.c even 3 1
5070.2.b.ba 8 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 21 T_{7}^{6} + 389 T_{7}^{4} + 504 T_{7}^{3} - 900 T_{7}^{2} - 1248 T_{7} + 2704 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( 2704 - 1248 T - 900 T^{2} + 504 T^{3} + 389 T^{4} - 21 T^{6} + T^{8} \)
$11$ \( 32761 + 22806 T - 138 T^{2} - 3780 T^{3} + 467 T^{4} + 180 T^{5} - 18 T^{6} - 6 T^{7} + T^{8} \)
$13$ \( 28561 + 26364 T + 7605 T^{2} - 312 T^{3} - 556 T^{4} - 24 T^{5} + 45 T^{6} + 12 T^{7} + T^{8} \)
$17$ \( ( 16 - 4 T + T^{2} )^{4} \)
$19$ \( 219024 - 84240 T - 15876 T^{2} + 10260 T^{3} + 2421 T^{4} - 342 T^{5} - 45 T^{6} + 6 T^{7} + T^{8} \)
$23$ \( 2704 + 832 T + 2492 T^{2} - 272 T^{3} + 1861 T^{4} + 140 T^{5} + 59 T^{6} - 4 T^{7} + T^{8} \)
$29$ \( 141376 - 55648 T + 36568 T^{2} - 244 T^{3} + 2329 T^{4} - 16 T^{5} + 103 T^{6} + 8 T^{7} + T^{8} \)
$31$ \( 913936 + 158088 T^{2} + 8777 T^{4} + 174 T^{6} + T^{8} \)
$37$ \( 644809 - 216810 T - 28698 T^{2} + 17820 T^{3} + 2459 T^{4} - 1980 T^{5} + 366 T^{6} - 30 T^{7} + T^{8} \)
$41$ \( 692224 - 159744 T - 57600 T^{2} + 16128 T^{3} + 6224 T^{4} - 84 T^{6} + T^{8} \)
$43$ \( 327184 - 478192 T + 674300 T^{2} - 51964 T^{3} + 14125 T^{4} - 1070 T^{5} + 239 T^{6} - 14 T^{7} + T^{8} \)
$47$ \( 1216609 + 421284 T^{2} + 16934 T^{4} + 228 T^{6} + T^{8} \)
$53$ \( ( 52 + 4 T - 75 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$59$ \( 80656 - 23856 T - 10996 T^{2} + 3948 T^{3} + 1821 T^{4} - 1128 T^{5} + 239 T^{6} - 24 T^{7} + T^{8} \)
$61$ \( ( 16 + 32 T + 60 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$67$ \( 123904 + 118272 T + 27776 T^{2} - 9408 T^{3} - 2256 T^{4} + 672 T^{5} + 164 T^{6} - 24 T^{7} + T^{8} \)
$71$ \( 25240576 + 19051008 T + 5817984 T^{2} + 773568 T^{3} + 31472 T^{4} - 2448 T^{5} - 156 T^{6} + 12 T^{7} + T^{8} \)
$73$ \( 20647936 + 1859072 T^{2} + 47760 T^{4} + 392 T^{6} + T^{8} \)
$79$ \( ( 3508 - 860 T - 147 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$83$ \( 25240576 + 2743296 T^{2} + 61904 T^{4} + 456 T^{6} + T^{8} \)
$89$ \( 1008016 + 349392 T - 131316 T^{2} - 59508 T^{3} + 35117 T^{4} - 7182 T^{5} + 759 T^{6} - 42 T^{7} + T^{8} \)
$97$ \( 29246464 + 23622144 T + 7289984 T^{2} + 751296 T^{3} + 48 T^{4} - 4128 T^{5} + 20 T^{6} + 24 T^{7} + T^{8} \)
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