Properties

Label 966.2.i.l
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1768034304.4
Defining polynomial: \(x^{8} - 2 x^{7} - x^{6} - 6 x^{5} + 14 x^{4} + 18 x^{3} - 31 x^{2} - 14 x + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - x^{6} - 6 x^{5} + 14 x^{4} + 18 x^{3} - 31 x^{2} - 14 x + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{7} - 67 \nu^{6} + 173 \nu^{5} - 208 \nu^{4} - 56 \nu^{3} - 965 \nu^{2} + 1303 \nu + 3997 \)\()/1659\)
\(\beta_{2}\)\(=\)\((\)\( 34 \nu^{7} - 33 \nu^{6} + 176 \nu^{5} - 218 \nu^{4} + 154 \nu^{3} - 1474 \nu^{2} - 522 \nu - 623 \)\()/3318\)
\(\beta_{3}\)\(=\)\((\)\( 128 \nu^{7} - 417 \nu^{6} - 1094 \nu^{5} - 40 \nu^{4} + 5264 \nu^{3} + 7918 \nu^{2} + 1548 \nu - 22351 \)\()/9954\)
\(\beta_{4}\)\(=\)\((\)\( 20 \nu^{7} - 117 \nu^{6} - 8 \nu^{5} - 184 \nu^{4} + 704 \nu^{3} + 304 \nu^{2} - 1506 \nu + 1655 \)\()/1422\)
\(\beta_{5}\)\(=\)\((\)\( -332 \nu^{7} + 615 \nu^{6} + 38 \nu^{5} + 1348 \nu^{4} - 6188 \nu^{3} - 9028 \nu^{2} + 21492 \nu + 16135 \)\()/9954\)
\(\beta_{6}\)\(=\)\((\)\( 338 \nu^{7} + 843 \nu^{6} - 1178 \nu^{5} - 1972 \nu^{4} - 7252 \nu^{3} + 11110 \nu^{2} + 666 \nu - 22393 \)\()/9954\)
\(\beta_{7}\)\(=\)\((\)\( -254 \nu^{7} - 339 \nu^{6} + 149 \nu^{5} + 3190 \nu^{4} + 3241 \nu^{3} - 1870 \nu^{2} - 8982 \nu + 4459 \)\()/4977\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + \beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(-\beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - 3 \beta_{1} + 9\)\()/2\)
\(\nu^{4}\)\(=\)\(3 \beta_{7} + 3 \beta_{6} - 3 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} + 3 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\((\)\(-4 \beta_{6} - 18 \beta_{5} - 23 \beta_{4} - 3 \beta_{3} - 12 \beta_{2} + 17 \beta_{1} - 3\)\()/2\)
\(\nu^{6}\)\(=\)\(-8 \beta_{7} - 12 \beta_{6} - 8 \beta_{5} - 15 \beta_{4} - 3 \beta_{3} - 15 \beta_{1} + 40\)
\(\nu^{7}\)\(=\)\((\)\(32 \beta_{7} + 54 \beta_{6} - 37 \beta_{4} + 59 \beta_{3} + 148 \beta_{2} + 37 \beta_{1} + 207\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
−0.885685 + 1.90520i
2.09279 + 0.185573i
0.972165 0.800426i
−1.17927 + 0.441707i
−0.885685 1.90520i
2.09279 0.185573i
0.972165 + 0.800426i
−1.17927 0.441707i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.914214 1.58346i 1.00000 −1.75255 1.98206i 1.00000 −0.500000 0.866025i −0.914214 + 1.58346i
277.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.914214 1.58346i 1.00000 2.45965 0.974732i 1.00000 −0.500000 0.866025i −0.914214 + 1.58346i
277.3 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.91421 + 3.31552i 1.00000 −1.87485 + 1.86680i 1.00000 −0.500000 0.866025i 1.91421 3.31552i
277.4 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.91421 + 3.31552i 1.00000 1.16774 2.37411i 1.00000 −0.500000 0.866025i 1.91421 3.31552i
415.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.914214 + 1.58346i 1.00000 −1.75255 + 1.98206i 1.00000 −0.500000 + 0.866025i −0.914214 1.58346i
415.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.914214 + 1.58346i 1.00000 2.45965 + 0.974732i 1.00000 −0.500000 + 0.866025i −0.914214 1.58346i
415.3 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.91421 3.31552i 1.00000 −1.87485 1.86680i 1.00000 −0.500000 + 0.866025i 1.91421 + 3.31552i
415.4 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.91421 3.31552i 1.00000 1.16774 + 2.37411i 1.00000 −0.500000 + 0.866025i 1.91421 + 3.31552i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.4
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 966.2.i.l 8
7.c even 3 1 inner 966.2.i.l 8
7.c even 3 1 6762.2.a.cp 4
7.d odd 6 1 6762.2.a.cm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.l 8 1.a even 1 1 trivial
966.2.i.l 8 7.c even 3 1 inner
6762.2.a.cm 4 7.d odd 6 1
6762.2.a.cp 4 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{4} - 2 T_{5}^{3} + 11 T_{5}^{2} + 14 T_{5} + 49 \)
\(T_{11}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{4} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( ( 49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$7$ \( 2401 - 84 T^{3} + 53 T^{4} - 12 T^{5} + T^{8} \)
$11$ \( 35344 + 29328 T + 20764 T^{2} + 5220 T^{3} + 1485 T^{4} + 198 T^{5} + 55 T^{6} + 6 T^{7} + T^{8} \)
$13$ \( ( -2 + 36 T - T^{2} - 6 T^{3} + T^{4} )^{2} \)
$17$ \( 21316 - 9928 T + 6814 T^{2} - 1900 T^{3} + 1051 T^{4} - 286 T^{5} + 85 T^{6} - 10 T^{7} + T^{8} \)
$19$ \( 4096 + 2048 T + 2944 T^{2} - 448 T^{3} + 964 T^{4} + 56 T^{5} + 46 T^{6} - 4 T^{7} + T^{8} \)
$23$ \( ( 1 + T + T^{2} )^{4} \)
$29$ \( ( 16 + 24 T - T^{2} - 6 T^{3} + T^{4} )^{2} \)
$31$ \( 35344 - 41360 T + 35804 T^{2} - 13988 T^{3} + 4237 T^{4} - 574 T^{5} + 71 T^{6} + 2 T^{7} + T^{8} \)
$37$ \( 18496 - 9792 T + 13072 T^{2} + 4176 T^{3} + 3228 T^{4} + 144 T^{5} + 58 T^{6} + T^{8} \)
$41$ \( ( 64 + 32 T - 30 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$43$ \( ( -5696 + 1376 T + 20 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$47$ \( 1817104 - 674000 T + 351100 T^{2} + 10540 T^{3} + 9277 T^{4} + 250 T^{5} + 175 T^{6} + 10 T^{7} + T^{8} \)
$53$ \( 305809 - 26544 T + 34378 T^{2} + 2784 T^{3} + 2811 T^{4} + 96 T^{5} + 58 T^{6} + T^{8} \)
$59$ \( 12544 - 24192 T + 52144 T^{2} + 9240 T^{3} + 3585 T^{4} + 138 T^{5} + 85 T^{6} + 6 T^{7} + T^{8} \)
$61$ \( 78287104 - 3397632 T + 2200192 T^{2} + 89088 T^{3} + 44976 T^{4} + 768 T^{5} + 232 T^{6} + T^{8} \)
$67$ \( 125350416 + 33453648 T + 7819740 T^{2} + 698868 T^{3} + 74781 T^{4} + 4194 T^{5} + 423 T^{6} + 18 T^{7} + T^{8} \)
$71$ \( ( 10654 - 4332 T + 647 T^{2} - 42 T^{3} + T^{4} )^{2} \)
$73$ \( 1024 - 3072 T + 7456 T^{2} - 4896 T^{3} + 2481 T^{4} - 522 T^{5} + 91 T^{6} + 6 T^{7} + T^{8} \)
$79$ \( 256 + 384 T + 592 T^{2} + 168 T^{3} + 129 T^{4} - 42 T^{5} + 37 T^{6} - 6 T^{7} + T^{8} \)
$83$ \( ( 1252 - 220 T - 91 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$89$ \( 399424 + 318528 T + 187024 T^{2} + 53424 T^{3} + 11868 T^{4} + 1008 T^{5} + 106 T^{6} + T^{8} \)
$97$ \( ( 64 - 208 T - 25 T^{2} + 10 T^{3} + T^{4} )^{2} \)
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