Properties

Label 13.3.f.a
Level 13
Weight 3
Character orbit 13.f
Analytic conductor 0.354
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 13.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.354224343668\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{4} + ( -3 + \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{5} + ( 3 - 3 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( 2 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{7} + ( -4 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{8} + ( 5 + 2 \zeta_{12} - 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{4} + ( -3 + \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{5} + ( 3 - 3 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( 2 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{7} + ( -4 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{8} + ( 5 + 2 \zeta_{12} - 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{9} + ( 8 - 5 \zeta_{12} - 4 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{10} + ( -4 - 6 \zeta_{12} + 10 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{11} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{12} -13 \zeta_{12}^{2} q^{13} + ( -10 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{14} + ( -8 + \zeta_{12} + 9 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{15} + ( 8 \zeta_{12} - \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{16} + ( -2 - 15 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{17} + ( -2 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{18} + ( 1 - \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{19} + ( 8 + 8 \zeta_{12} - 3 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{20} + ( 14 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{21} + ( -2 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{22} + ( 6 + 15 \zeta_{12} - 3 \zeta_{12}^{2} - 15 \zeta_{12}^{3} ) q^{23} + ( 8 - 13 \zeta_{12} + 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{24} + ( -7 + 14 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{25} + ( 13 - 13 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{26} + ( -8 + 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{27} + ( -10 - 12 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{28} + ( -6 \zeta_{12} + \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{29} + ( -9 + 17 \zeta_{12} - 9 \zeta_{12}^{2} ) q^{30} + ( -9 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{31} + ( 1 - \zeta_{12} - 12 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{32} + ( -4 - 4 \zeta_{12} - 8 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{33} + ( -11 + 13 \zeta_{12} + 13 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{34} + ( 16 - 22 \zeta_{12} - 16 \zeta_{12}^{2} + 44 \zeta_{12}^{3} ) q^{35} + ( -18 - 16 \zeta_{12} + 9 \zeta_{12}^{2} + 16 \zeta_{12}^{3} ) q^{36} + ( -5 + 29 \zeta_{12} - 24 \zeta_{12}^{2} - 24 \zeta_{12}^{3} ) q^{37} + ( -1 + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{38} + ( -13 + 13 \zeta_{12} + 13 \zeta_{12}^{2} - 26 \zeta_{12}^{3} ) q^{39} + ( 18 - 38 \zeta_{12} + 19 \zeta_{12}^{3} ) q^{40} + ( 27 + 23 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{41} + ( -14 \zeta_{12} + 22 \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{42} + ( 30 + 30 \zeta_{12}^{2} ) q^{43} + ( 34 + 24 \zeta_{12} - 24 \zeta_{12}^{2} - 34 \zeta_{12}^{3} ) q^{44} + ( -10 + 10 \zeta_{12} + 19 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{45} + ( -3 - 3 \zeta_{12} - 9 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{46} + ( -7 - 20 \zeta_{12} - 20 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{47} + ( -25 + 9 \zeta_{12} + 25 \zeta_{12}^{2} - 18 \zeta_{12}^{3} ) q^{48} + ( -24 + 7 \zeta_{12} + 12 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{49} + ( -8 + 15 \zeta_{12} - 7 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{50} + ( 13 - 26 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{51} + ( -13 + 26 \zeta_{12}^{2} + 26 \zeta_{12}^{3} ) q^{52} + ( 32 - 54 \zeta_{12} + 27 \zeta_{12}^{3} ) q^{53} + ( 20 - 12 \zeta_{12} - 32 \zeta_{12}^{2} + 32 \zeta_{12}^{3} ) q^{54} + ( -6 \zeta_{12} - 50 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{55} + ( -14 + 48 \zeta_{12} - 14 \zeta_{12}^{2} ) q^{56} + ( -4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{57} + ( -13 + 13 \zeta_{12} + 6 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{58} + ( -60 - 60 \zeta_{12} + 38 \zeta_{12}^{2} + 22 \zeta_{12}^{3} ) q^{59} + ( -1 + 6 \zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{60} + ( -62 + 21 \zeta_{12} + 62 \zeta_{12}^{2} - 42 \zeta_{12}^{3} ) q^{61} + ( 2 + 7 \zeta_{12} - \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{62} + ( 26 - 26 \zeta_{12}^{2} - 26 \zeta_{12}^{3} ) q^{63} + ( 31 - 62 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{64} + ( -13 + 39 \zeta_{12} + 52 \zeta_{12}^{2} - 52 \zeta_{12}^{3} ) q^{65} + ( 20 - 16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{66} + ( 48 + 11 \zeta_{12} - 37 \zeta_{12}^{2} + 37 \zeta_{12}^{3} ) q^{67} + ( 19 \zeta_{12} + 36 \zeta_{12}^{2} + 19 \zeta_{12}^{3} ) q^{68} + ( 12 - 6 \zeta_{12} + 12 \zeta_{12}^{2} ) q^{69} + ( 22 - 38 \zeta_{12} + 38 \zeta_{12}^{2} - 22 \zeta_{12}^{3} ) q^{70} + ( -11 + 11 \zeta_{12} - 21 \zeta_{12}^{2} - 32 \zeta_{12}^{3} ) q^{71} + ( 17 + 17 \zeta_{12} + 2 \zeta_{12}^{2} - 19 \zeta_{12}^{3} ) q^{72} + ( 34 - 39 \zeta_{12} - 39 \zeta_{12}^{2} + 34 \zeta_{12}^{3} ) q^{73} + ( 34 - 29 \zeta_{12} - 34 \zeta_{12}^{2} + 58 \zeta_{12}^{3} ) q^{74} + ( 16 - 22 \zeta_{12} - 8 \zeta_{12}^{2} + 22 \zeta_{12}^{3} ) q^{75} + ( 10 + 3 \zeta_{12} - 13 \zeta_{12}^{2} - 13 \zeta_{12}^{3} ) q^{76} + ( -44 + 88 \zeta_{12}^{2} + 20 \zeta_{12}^{3} ) q^{77} + ( -13 + 26 \zeta_{12} - 26 \zeta_{12}^{2} + 13 \zeta_{12}^{3} ) q^{78} + ( -10 + 84 \zeta_{12} - 42 \zeta_{12}^{3} ) q^{79} + ( -57 - 13 \zeta_{12} + 44 \zeta_{12}^{2} - 44 \zeta_{12}^{3} ) q^{80} + ( -38 \zeta_{12} - \zeta_{12}^{2} - 38 \zeta_{12}^{3} ) q^{81} + ( 4 - 31 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{82} + ( -34 + 26 \zeta_{12} - 26 \zeta_{12}^{2} + 34 \zeta_{12}^{3} ) q^{83} + ( 6 - 6 \zeta_{12} - 10 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{84} + ( 49 + 49 \zeta_{12} - 50 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{85} + ( -60 + 30 \zeta_{12} + 30 \zeta_{12}^{2} - 60 \zeta_{12}^{3} ) q^{86} + ( 19 - 7 \zeta_{12} - 19 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{87} + ( -68 - 6 \zeta_{12} + 34 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{88} + ( -50 + 5 \zeta_{12} + 45 \zeta_{12}^{2} + 45 \zeta_{12}^{3} ) q^{89} + ( 10 - 20 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{90} + ( 52 - 78 \zeta_{12} - 78 \zeta_{12}^{2} + 52 \zeta_{12}^{3} ) q^{91} + ( -39 - 42 \zeta_{12} + 21 \zeta_{12}^{3} ) q^{92} + ( -2 - 8 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{93} + ( 7 \zeta_{12} + 13 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{94} + ( -13 - \zeta_{12} - 13 \zeta_{12}^{2} ) q^{95} + ( 11 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{96} + ( 35 - 35 \zeta_{12} + 19 \zeta_{12}^{2} + 54 \zeta_{12}^{3} ) q^{97} + ( 12 + 12 \zeta_{12} - 31 \zeta_{12}^{2} + 19 \zeta_{12}^{3} ) q^{98} + ( 26 + 52 \zeta_{12} + 52 \zeta_{12}^{2} + 26 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{3} - 6q^{4} - 14q^{5} + 10q^{6} + 16q^{7} - 6q^{8} + 10q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{3} - 6q^{4} - 14q^{5} + 10q^{6} + 16q^{7} - 6q^{8} + 10q^{9} + 24q^{10} + 4q^{11} - 26q^{13} - 40q^{14} - 14q^{15} - 2q^{16} - 12q^{17} - 2q^{18} + 10q^{19} + 26q^{20} + 40q^{21} - 4q^{22} + 18q^{23} + 42q^{24} + 52q^{26} - 32q^{27} - 44q^{28} + 2q^{29} - 54q^{30} - 20q^{31} - 20q^{32} - 32q^{33} - 18q^{34} + 32q^{35} - 54q^{36} - 68q^{37} - 26q^{39} + 72q^{40} + 100q^{41} + 44q^{42} + 180q^{43} + 88q^{44} - 2q^{45} - 30q^{46} - 68q^{47} - 50q^{48} - 72q^{49} - 46q^{50} + 128q^{53} + 16q^{54} - 100q^{55} - 84q^{56} - 20q^{57} - 40q^{58} - 164q^{59} + 8q^{60} - 124q^{61} + 6q^{62} + 52q^{63} + 52q^{65} + 80q^{66} + 118q^{67} + 72q^{68} + 72q^{69} + 164q^{70} - 86q^{71} + 72q^{72} + 58q^{73} + 68q^{74} + 48q^{75} + 14q^{76} - 104q^{78} - 40q^{79} - 140q^{80} - 2q^{81} + 24q^{82} - 188q^{83} + 4q^{84} + 96q^{85} - 180q^{86} + 38q^{87} - 204q^{88} - 110q^{89} + 52q^{91} - 156q^{92} - 20q^{93} + 26q^{94} - 78q^{95} + 40q^{96} + 178q^{97} - 14q^{98} + 208q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.500000 0.133975i 0.366025 + 0.633975i −3.23205 1.86603i −2.63397 + 2.63397i −0.0980762 0.366025i 5.73205 1.53590i 2.83013 + 2.83013i 4.23205 7.33013i 1.66987 0.964102i
6.1 −0.500000 1.86603i −1.36603 + 2.36603i 0.232051 0.133975i −4.36603 + 4.36603i 5.09808 + 1.36603i 2.26795 8.46410i −5.83013 5.83013i 0.767949 + 1.33013i 10.3301 + 5.96410i
7.1 −0.500000 + 0.133975i 0.366025 0.633975i −3.23205 + 1.86603i −2.63397 2.63397i −0.0980762 + 0.366025i 5.73205 + 1.53590i 2.83013 2.83013i 4.23205 + 7.33013i 1.66987 + 0.964102i
11.1 −0.500000 + 1.86603i −1.36603 2.36603i 0.232051 + 0.133975i −4.36603 4.36603i 5.09808 1.36603i 2.26795 + 8.46410i −5.83013 + 5.83013i 0.767949 1.33013i 10.3301 5.96410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.3.f.a 4
3.b odd 2 1 117.3.bd.b 4
4.b odd 2 1 208.3.bd.d 4
5.b even 2 1 325.3.t.a 4
5.c odd 4 1 325.3.w.a 4
5.c odd 4 1 325.3.w.b 4
13.b even 2 1 169.3.f.b 4
13.c even 3 1 169.3.d.a 4
13.c even 3 1 169.3.f.c 4
13.d odd 4 1 169.3.f.a 4
13.d odd 4 1 169.3.f.c 4
13.e even 6 1 169.3.d.c 4
13.e even 6 1 169.3.f.a 4
13.f odd 12 1 inner 13.3.f.a 4
13.f odd 12 1 169.3.d.a 4
13.f odd 12 1 169.3.d.c 4
13.f odd 12 1 169.3.f.b 4
39.k even 12 1 117.3.bd.b 4
52.l even 12 1 208.3.bd.d 4
65.o even 12 1 325.3.w.b 4
65.s odd 12 1 325.3.t.a 4
65.t even 12 1 325.3.w.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.3.f.a 4 1.a even 1 1 trivial
13.3.f.a 4 13.f odd 12 1 inner
117.3.bd.b 4 3.b odd 2 1
117.3.bd.b 4 39.k even 12 1
169.3.d.a 4 13.c even 3 1
169.3.d.a 4 13.f odd 12 1
169.3.d.c 4 13.e even 6 1
169.3.d.c 4 13.f odd 12 1
169.3.f.a 4 13.d odd 4 1
169.3.f.a 4 13.e even 6 1
169.3.f.b 4 13.b even 2 1
169.3.f.b 4 13.f odd 12 1
169.3.f.c 4 13.c even 3 1
169.3.f.c 4 13.d odd 4 1
208.3.bd.d 4 4.b odd 2 1
208.3.bd.d 4 52.l even 12 1
325.3.t.a 4 5.b even 2 1
325.3.t.a 4 65.s odd 12 1
325.3.w.a 4 5.c odd 4 1
325.3.w.a 4 65.t even 12 1
325.3.w.b 4 5.c odd 4 1
325.3.w.b 4 65.o even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(13, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 5 T^{2} + 12 T^{3} + 17 T^{4} + 48 T^{5} + 80 T^{6} + 128 T^{7} + 256 T^{8} \)
$3$ \( 1 + 2 T - 12 T^{2} - 4 T^{3} + 139 T^{4} - 36 T^{5} - 972 T^{6} + 1458 T^{7} + 6561 T^{8} \)
$5$ \( ( 1 + 6 T + 11 T^{2} + 150 T^{3} + 625 T^{4} )( 1 + 8 T + 39 T^{2} + 200 T^{3} + 625 T^{4} ) \)
$7$ \( 1 - 16 T + 164 T^{2} - 1236 T^{3} + 8927 T^{4} - 60564 T^{5} + 393764 T^{6} - 1882384 T^{7} + 5764801 T^{8} \)
$11$ \( 1 - 4 T + 200 T^{2} + 960 T^{3} + 17471 T^{4} + 116160 T^{5} + 2928200 T^{6} - 7086244 T^{7} + 214358881 T^{8} \)
$13$ \( ( 1 + 13 T + 169 T^{2} )^{2} \)
$17$ \( 1 + 12 T + 413 T^{2} + 4380 T^{3} + 63576 T^{4} + 1265820 T^{5} + 34494173 T^{6} + 289650828 T^{7} + 6975757441 T^{8} \)
$19$ \( 1 - 10 T + 74 T^{2} - 1752 T^{3} - 96625 T^{4} - 632472 T^{5} + 9643754 T^{6} - 470458810 T^{7} + 16983563041 T^{8} \)
$23$ \( 1 - 18 T + 968 T^{2} - 15480 T^{3} + 516891 T^{4} - 8188920 T^{5} + 270886088 T^{6} - 2664646002 T^{7} + 78310985281 T^{8} \)
$29$ \( 1 - 2 T - 1571 T^{2} + 214 T^{3} + 1769980 T^{4} + 179974 T^{5} - 1111138451 T^{6} - 1189646642 T^{7} + 500246412961 T^{8} \)
$31$ \( 1 + 20 T + 200 T^{2} + 18300 T^{3} + 1672334 T^{4} + 17586300 T^{5} + 184704200 T^{6} + 17750073620 T^{7} + 852891037441 T^{8} \)
$37$ \( 1 + 68 T + 1517 T^{2} - 93168 T^{3} - 5925028 T^{4} - 127546992 T^{5} + 2843102237 T^{6} + 174469395812 T^{7} + 3512479453921 T^{8} \)
$41$ \( 1 - 100 T + 3461 T^{2} + 7272 T^{3} - 4096804 T^{4} + 12224232 T^{5} + 9779958821 T^{6} - 475010424100 T^{7} + 7984925229121 T^{8} \)
$43$ \( ( 1 - 90 T + 4549 T^{2} - 166410 T^{3} + 3418801 T^{4} )^{2} \)
$47$ \( 1 + 68 T + 2312 T^{2} + 148716 T^{3} + 9565454 T^{4} + 328513644 T^{5} + 11281822472 T^{6} + 732986642372 T^{7} + 23811286661761 T^{8} \)
$53$ \( ( 1 - 64 T + 4455 T^{2} - 179776 T^{3} + 7890481 T^{4} )^{2} \)
$59$ \( 1 + 164 T + 6980 T^{2} - 551844 T^{3} - 68910913 T^{4} - 1920968964 T^{5} + 84579179780 T^{6} + 6917607517124 T^{7} + 146830437604321 T^{8} \)
$61$ \( 1 + 124 T + 5413 T^{2} + 312604 T^{3} + 28201432 T^{4} + 1163199484 T^{5} + 74947537333 T^{6} + 6388526420764 T^{7} + 191707312997281 T^{8} \)
$67$ \( 1 - 118 T + 10706 T^{2} - 763488 T^{3} + 51177839 T^{4} - 3427297632 T^{5} + 215737901426 T^{6} - 10674089095942 T^{7} + 406067677556641 T^{8} \)
$71$ \( 1 + 86 T + 4658 T^{2} + 258336 T^{3} + 1457087 T^{4} + 1302271776 T^{5} + 118367610098 T^{6} + 11016624417206 T^{7} + 645753531245761 T^{8} \)
$73$ \( 1 - 58 T + 1682 T^{2} - 201144 T^{3} + 20590727 T^{4} - 1071896376 T^{5} + 47765841362 T^{6} - 8777385124762 T^{7} + 806460091894081 T^{8} \)
$79$ \( ( 1 + 20 T + 7290 T^{2} + 124820 T^{3} + 38950081 T^{4} )^{2} \)
$83$ \( 1 + 188 T + 17672 T^{2} + 1935084 T^{3} + 200304482 T^{4} + 13330793676 T^{5} + 838683448712 T^{6} + 61464790193372 T^{7} + 2252292232139041 T^{8} \)
$89$ \( 1 + 110 T + 12050 T^{2} + 1194180 T^{3} + 87746159 T^{4} + 9459099780 T^{5} + 756044004050 T^{6} + 54667942005710 T^{7} + 3936588805702081 T^{8} \)
$97$ \( 1 - 178 T + 13250 T^{2} - 318828 T^{3} - 39375793 T^{4} - 2999852652 T^{5} + 1173012973250 T^{6} - 148269016877362 T^{7} + 7837433594376961 T^{8} \)
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