Properties

Label 322.2.e.d
Level $322$
Weight $2$
Character orbit 322.e
Analytic conductor $2.571$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.6498455769.2
Defining polynomial: \(x^{8} - x^{7} + 6 x^{6} + 3 x^{5} + 25 x^{4} - 3 x^{3} + 6 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 6 x^{6} + 3 x^{5} + 25 x^{4} - 3 x^{3} + 6 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -14 \nu^{7} + 36 \nu^{6} - 126 \nu^{5} + 105 \nu^{4} - 396 \nu^{3} + 504 \nu^{2} - 672 \nu + 90 \)\()/119\)
\(\beta_{3}\)\(=\)\((\)\( -15 \nu^{7} + 7 \nu^{6} - 84 \nu^{5} - 66 \nu^{4} - 434 \nu^{3} - 21 \nu^{2} - 6 \nu + 315 \)\()/119\)
\(\beta_{4}\)\(=\)\((\)\( 20 \nu^{7} - 15 \nu^{6} + 112 \nu^{5} + 88 \nu^{4} + 522 \nu^{3} + 28 \nu^{2} + 8 \nu + 22 \)\()/119\)
\(\beta_{5}\)\(=\)\((\)\( -22 \nu^{7} + 42 \nu^{6} - 147 \nu^{5} + 46 \nu^{4} - 462 \nu^{3} + 588 \nu^{2} - 104 \nu + 105 \)\()/119\)
\(\beta_{6}\)\(=\)\((\)\( 24 \nu^{7} - 69 \nu^{6} + 182 \nu^{5} - 180 \nu^{4} + 402 \nu^{3} - 1085 \nu^{2} + 81 \nu + 6 \)\()/119\)
\(\beta_{7}\)\(=\)\((\)\( 105 \nu^{7} - 83 \nu^{6} + 588 \nu^{5} + 462 \nu^{4} + 2579 \nu^{3} + 147 \nu^{2} + 42 \nu + 209 \)\()/119\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + 6 \beta_{4} + \beta_{3} - 2\)
\(\nu^{4}\)\(=\)\(-6 \beta_{6} - 10 \beta_{5} + 6 \beta_{3} - \beta_{2} - 10 \beta_{1} - 6\)
\(\nu^{5}\)\(=\)\(6 \beta_{7} - 10 \beta_{6} - 13 \beta_{5} - 38 \beta_{4} - 6 \beta_{2} - 38 \beta_{1} + 13\)
\(\nu^{6}\)\(=\)\(10 \beta_{7} - 81 \beta_{4} - 38 \beta_{3} + 98\)
\(\nu^{7}\)\(=\)\(81 \beta_{6} + 114 \beta_{5} - 81 \beta_{3} + 38 \beta_{2} + 255 \beta_{1} + 81\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
1.33821 + 2.31784i
0.271028 + 0.469434i
−0.186817 0.323577i
−0.922415 1.59767i
1.33821 2.31784i
0.271028 0.469434i
−0.186817 + 0.323577i
−0.922415 + 1.59767i
0.500000 0.866025i −1.33821 2.31784i −0.500000 0.866025i 1.93020 3.34320i −2.67641 −2.21184 1.45181i −1.00000 −2.08159 + 3.60541i −1.93020 3.34320i
93.2 0.500000 0.866025i −0.271028 0.469434i −0.500000 0.866025i 0.298300 0.516670i −0.542055 −2.61586 + 0.396592i −1.00000 1.35309 2.34362i −0.298300 0.516670i
93.3 0.500000 0.866025i 0.186817 + 0.323577i −0.500000 0.866025i −1.58159 + 2.73939i 0.373635 2.36323 + 1.18960i −1.00000 1.43020 2.47718i 1.58159 + 2.73939i
93.4 0.500000 0.866025i 0.922415 + 1.59767i −0.500000 0.866025i 1.85309 3.20964i 1.84483 0.964471 + 2.46370i −1.00000 −0.201700 + 0.349355i −1.85309 3.20964i
277.1 0.500000 + 0.866025i −1.33821 + 2.31784i −0.500000 + 0.866025i 1.93020 + 3.34320i −2.67641 −2.21184 + 1.45181i −1.00000 −2.08159 3.60541i −1.93020 + 3.34320i
277.2 0.500000 + 0.866025i −0.271028 + 0.469434i −0.500000 + 0.866025i 0.298300 + 0.516670i −0.542055 −2.61586 0.396592i −1.00000 1.35309 + 2.34362i −0.298300 + 0.516670i
277.3 0.500000 + 0.866025i 0.186817 0.323577i −0.500000 + 0.866025i −1.58159 2.73939i 0.373635 2.36323 1.18960i −1.00000 1.43020 + 2.47718i 1.58159 2.73939i
277.4 0.500000 + 0.866025i 0.922415 1.59767i −0.500000 + 0.866025i 1.85309 + 3.20964i 1.84483 0.964471 2.46370i −1.00000 −0.201700 0.349355i −1.85309 + 3.20964i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.e.d 8
7.c even 3 1 inner 322.2.e.d 8
7.c even 3 1 2254.2.a.u 4
7.d odd 6 1 2254.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.e.d 8 1.a even 1 1 trivial
322.2.e.d 8 7.c even 3 1 inner
2254.2.a.r 4 7.d odd 6 1
2254.2.a.u 4 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( 1 - T + 6 T^{2} + 3 T^{3} + 25 T^{4} - 3 T^{5} + 6 T^{6} + T^{7} + T^{8} \)
$5$ \( 729 - 1377 T + 2412 T^{2} - 627 T^{3} + 331 T^{4} - 67 T^{5} + 32 T^{6} - 5 T^{7} + T^{8} \)
$7$ \( 2401 + 1029 T - 196 T^{2} - 21 T^{3} + 57 T^{4} - 3 T^{5} - 4 T^{6} + 3 T^{7} + T^{8} \)
$11$ \( 729 + 1701 T + 4941 T^{2} - 2160 T^{3} + 1395 T^{4} - 54 T^{5} + 40 T^{6} - 2 T^{7} + T^{8} \)
$13$ \( ( -9 + 38 T - 7 T^{3} + T^{4} )^{2} \)
$17$ \( 81 + 162 T + 306 T^{2} + 162 T^{3} + 121 T^{4} - 50 T^{5} + 47 T^{6} - 7 T^{7} + T^{8} \)
$19$ \( 9 + 57 T + 310 T^{2} + 317 T^{3} + 273 T^{4} + 55 T^{5} + 18 T^{6} - T^{7} + T^{8} \)
$23$ \( ( 1 - T + T^{2} )^{4} \)
$29$ \( ( 147 - 196 T - 39 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$31$ \( 36481 - 28841 T + 17835 T^{2} - 5454 T^{3} + 1471 T^{4} - 198 T^{5} + 42 T^{6} - 4 T^{7} + T^{8} \)
$37$ \( 301401 + 21411 T + 39951 T^{2} - 2730 T^{3} + 4351 T^{4} - 78 T^{5} + 70 T^{6} + T^{8} \)
$41$ \( ( -7647 - 2042 T - 78 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$43$ \( ( -547 - 187 T + 20 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$47$ \( 2424249 - 761373 T + 268704 T^{2} - 37419 T^{3} + 9253 T^{4} - 1263 T^{5} + 206 T^{6} - 15 T^{7} + T^{8} \)
$53$ \( 531441 - 19683 T + 86022 T^{2} + 33777 T^{3} + 13851 T^{4} + 2403 T^{5} + 324 T^{6} + 21 T^{7} + T^{8} \)
$59$ \( 1565001 - 1807695 T + 1647673 T^{2} - 428576 T^{3} + 76413 T^{4} - 8374 T^{5} + 672 T^{6} - 32 T^{7} + T^{8} \)
$61$ \( 729 - 270 T + 1234 T^{2} + 582 T^{3} + 1707 T^{4} + 146 T^{5} + 51 T^{6} - 3 T^{7} + T^{8} \)
$67$ \( 9 - 156 T + 2806 T^{2} + 1690 T^{3} + 1829 T^{4} - 338 T^{5} + 203 T^{6} + 13 T^{7} + T^{8} \)
$71$ \( ( 807 - 800 T - 162 T^{2} + 7 T^{3} + T^{4} )^{2} \)
$73$ \( 1750329 + 750141 T + 392931 T^{2} + 11718 T^{3} + 10665 T^{4} - 270 T^{5} + 310 T^{6} - 16 T^{7} + T^{8} \)
$79$ \( 1495729 + 234816 T + 183624 T^{2} - 30378 T^{3} + 12601 T^{4} - 744 T^{5} + 129 T^{6} + 3 T^{7} + T^{8} \)
$83$ \( ( 9 + 28 T + 9 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$89$ \( 13461561 - 7290303 T + 2513590 T^{2} - 534763 T^{3} + 83641 T^{4} - 8929 T^{5} + 698 T^{6} - 33 T^{7} + T^{8} \)
$97$ \( ( -199 - 3000 T - 243 T^{2} + 12 T^{3} + T^{4} )^{2} \)
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