Properties

Label 210.3.o.a
Level 210
Weight 3
Character orbit 210.o
Analytic conductor 5.722
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.72208555157\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.3
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{4} ) q^{2} + ( 1 + \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} ) q^{4} + \beta_{1} q^{5} + ( \beta_{2} + 2 \beta_{4} ) q^{6} + ( \beta_{1} + \beta_{2} + 3 \beta_{4} + 2 \beta_{6} ) q^{7} -2 \beta_{2} q^{8} + 3 \beta_{3} q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{4} ) q^{2} + ( 1 + \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} ) q^{4} + \beta_{1} q^{5} + ( \beta_{2} + 2 \beta_{4} ) q^{6} + ( \beta_{1} + \beta_{2} + 3 \beta_{4} + 2 \beta_{6} ) q^{7} -2 \beta_{2} q^{8} + 3 \beta_{3} q^{9} -\beta_{7} q^{10} + ( -1 - 2 \beta_{1} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{11} + ( -4 + 2 \beta_{3} ) q^{12} + ( 2 - \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( -6 + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{14} + ( \beta_{1} + \beta_{6} ) q^{15} -4 \beta_{3} q^{16} + ( 7 - 2 \beta_{2} + 7 \beta_{3} - \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{17} + 3 \beta_{4} q^{18} + ( 18 - 2 \beta_{1} - 7 \beta_{2} - 9 \beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{20} + ( -\beta_{1} - \beta_{2} + 4 \beta_{4} + 5 \beta_{6} ) q^{21} + ( -6 + 4 \beta_{1} - \beta_{2} - \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{22} + ( 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} ) q^{23} + ( -4 \beta_{2} - 2 \beta_{4} ) q^{24} + ( 5 - 5 \beta_{3} ) q^{25} + ( -16 + 2 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{26} + ( -3 + 6 \beta_{3} ) q^{27} + ( -6 \beta_{1} - 6 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{28} + ( 9 - \beta_{1} + 13 \beta_{2} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( -\beta_{5} - 2 \beta_{7} ) q^{30} + ( -11 - 2 \beta_{2} - 11 \beta_{3} - \beta_{4} - 2 \beta_{6} + 5 \beta_{7} ) q^{31} -4 \beta_{4} q^{32} + ( -2 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{33} + ( 2 + 4 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} + 14 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{34} + ( 15 - 5 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{35} -6 q^{36} + ( -5 \beta_{1} + 2 \beta_{2} - 24 \beta_{3} + 2 \beta_{4} + \beta_{5} + 10 \beta_{6} + 2 \beta_{7} ) q^{37} + ( -14 + 18 \beta_{2} - 14 \beta_{3} + 9 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{38} + ( 6 - 2 \beta_{1} - 6 \beta_{3} + 12 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} -2 \beta_{5} q^{40} + ( -3 + 19 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 19 \beta_{6} + 4 \beta_{7} ) q^{41} + ( -8 - 2 \beta_{3} - 5 \beta_{5} - 4 \beta_{7} ) q^{42} + ( -14 + 9 \beta_{1} - 5 \beta_{2} - 4 \beta_{5} + 9 \beta_{6} + 4 \beta_{7} ) q^{43} + ( 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} ) q^{44} + 3 \beta_{6} q^{45} + ( 2 - 8 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{46} + ( -4 - 8 \beta_{1} + 20 \beta_{2} + 2 \beta_{3} - 20 \beta_{4} + 2 \beta_{5} ) q^{47} + ( 4 - 8 \beta_{3} ) q^{48} + ( 15 + 9 \beta_{3} - 16 \beta_{5} - 10 \beta_{7} ) q^{49} + 5 \beta_{2} q^{50} + ( -3 \beta_{1} - 3 \beta_{2} + 21 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{51} + ( 4 - 16 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{52} + ( 8 + 8 \beta_{1} - 8 \beta_{3} - 26 \beta_{4} + 16 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} ) q^{53} + ( -3 \beta_{2} + 3 \beta_{4} ) q^{54} + ( -5 - \beta_{1} - 10 \beta_{2} + 10 \beta_{3} - 20 \beta_{4} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{55} + ( 8 - 12 \beta_{3} - 2 \beta_{5} + 4 \beta_{7} ) q^{56} + ( 27 - 2 \beta_{1} - 21 \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{57} + ( 4 \beta_{1} + 9 \beta_{2} + 26 \beta_{3} + 9 \beta_{4} + \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 11 + 14 \beta_{2} + 11 \beta_{3} + 7 \beta_{4} - 21 \beta_{6} - 8 \beta_{7} ) q^{59} + ( -4 \beta_{1} + 2 \beta_{6} ) q^{60} + ( 16 - 16 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} + 16 \beta_{5} ) q^{61} + ( 2 + 10 \beta_{1} - 11 \beta_{2} - 4 \beta_{3} - 22 \beta_{4} + 2 \beta_{5} - 10 \beta_{6} + 2 \beta_{7} ) q^{62} + ( -6 \beta_{1} - 6 \beta_{2} + 3 \beta_{4} + 9 \beta_{6} ) q^{63} + 8 q^{64} + ( 2 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} ) q^{65} + ( -6 - 2 \beta_{2} - 6 \beta_{3} - \beta_{4} + 12 \beta_{6} + 3 \beta_{7} ) q^{66} + ( -30 - 6 \beta_{1} + 30 \beta_{3} - 9 \beta_{4} + 16 \beta_{5} + 3 \beta_{6} + 8 \beta_{7} ) q^{67} + ( -28 - 6 \beta_{1} + 2 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{68} + ( -3 + 9 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 9 \beta_{6} - 6 \beta_{7} ) q^{69} + ( -6 \beta_{1} + 15 \beta_{2} + 10 \beta_{4} + 2 \beta_{6} ) q^{70} + ( 1 - 7 \beta_{1} + 29 \beta_{2} - 6 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} ) q^{71} + ( -6 \beta_{2} - 6 \beta_{4} ) q^{72} + ( 2 - 20 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} - 25 \beta_{6} + 15 \beta_{7} ) q^{73} + ( -4 + 4 \beta_{1} + 4 \beta_{3} - 24 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} ) q^{74} + ( 10 - 5 \beta_{3} ) q^{75} + ( -18 + 4 \beta_{1} - 14 \beta_{2} + 36 \beta_{3} - 28 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{76} + ( -31 - 7 \beta_{1} + 7 \beta_{2} + 8 \beta_{3} - 42 \beta_{4} - 8 \beta_{5} + 21 \beta_{6} + 2 \beta_{7} ) q^{77} + ( -24 + 2 \beta_{1} + 6 \beta_{2} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{78} + ( 2 \beta_{1} - 13 \beta_{2} + 3 \beta_{3} - 13 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} - 38 \beta_{7} ) q^{79} -4 \beta_{6} q^{80} + ( -9 + 9 \beta_{3} ) q^{81} + ( 4 + 8 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 19 \beta_{5} ) q^{82} + ( -3 + 7 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} - 30 \beta_{4} + 34 \beta_{5} - 7 \beta_{6} + 34 \beta_{7} ) q^{83} + ( -8 \beta_{1} - 8 \beta_{2} - 10 \beta_{4} - 2 \beta_{6} ) q^{84} + ( 15 + 7 \beta_{1} - 10 \beta_{2} - \beta_{5} + 7 \beta_{6} + \beta_{7} ) q^{85} + ( 8 \beta_{1} - 14 \beta_{2} - 10 \beta_{3} - 14 \beta_{4} - 9 \beta_{5} - 16 \beta_{6} - 18 \beta_{7} ) q^{86} + ( 9 + 26 \beta_{2} + 9 \beta_{3} + 13 \beta_{4} - 3 \beta_{6} + 6 \beta_{7} ) q^{87} + ( 12 - 16 \beta_{1} - 12 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} ) q^{88} + ( 82 - 19 \beta_{1} + 11 \beta_{2} - 41 \beta_{3} - 11 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -3 \beta_{5} - 3 \beta_{7} ) q^{90} + ( -42 + 12 \beta_{1} + 5 \beta_{2} + 7 \beta_{3} - 13 \beta_{4} - 21 \beta_{5} - 4 \beta_{6} - 14 \beta_{7} ) q^{91} + ( -6 + 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{92} + ( 2 \beta_{1} - 3 \beta_{2} - 33 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - 4 \beta_{6} + 10 \beta_{7} ) q^{93} + ( 40 - 4 \beta_{2} + 40 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} + 8 \beta_{7} ) q^{94} + ( -10 + 18 \beta_{1} + 10 \beta_{3} + 5 \beta_{4} - 14 \beta_{5} - 9 \beta_{6} - 7 \beta_{7} ) q^{95} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{96} + ( 36 + 8 \beta_{1} + 20 \beta_{2} - 72 \beta_{3} + 40 \beta_{4} + 18 \beta_{5} - 8 \beta_{6} + 18 \beta_{7} ) q^{97} + ( -20 \beta_{1} + 15 \beta_{2} + 24 \beta_{4} - 12 \beta_{6} ) q^{98} + ( -3 - 3 \beta_{1} - 9 \beta_{2} + 6 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 12q^{3} - 8q^{4} + 12q^{9} + O(q^{10}) \) \( 8q + 12q^{3} - 8q^{4} + 12q^{9} - 4q^{11} - 24q^{12} - 40q^{14} - 16q^{16} + 84q^{17} + 108q^{19} - 48q^{22} + 12q^{23} + 20q^{25} - 96q^{26} + 72q^{29} - 132q^{31} - 12q^{33} + 100q^{35} - 48q^{36} - 96q^{37} - 168q^{38} + 24q^{39} - 72q^{42} - 112q^{43} - 8q^{44} + 8q^{46} - 24q^{47} + 156q^{49} + 84q^{51} + 48q^{52} + 32q^{53} + 16q^{56} + 216q^{57} + 104q^{58} + 132q^{59} + 96q^{61} + 64q^{64} + 20q^{65} - 72q^{66} - 120q^{67} - 168q^{68} + 8q^{71} + 24q^{73} - 16q^{74} + 60q^{75} - 216q^{77} - 192q^{78} + 12q^{79} - 36q^{81} + 24q^{82} + 120q^{85} - 40q^{86} + 108q^{87} + 48q^{88} + 492q^{89} - 308q^{91} - 48q^{92} - 132q^{93} + 480q^{94} - 40q^{95} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 14 \nu^{4} + 7 \nu^{2} - 36 \)\()/63\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 144 \)\()/63\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} - 7 \nu^{5} - 35 \nu^{3} + 180 \nu \)\()/189\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/21\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{6} + 14 \nu^{4} + 7 \nu^{2} + 162 \)\()/63\)
\(\beta_{7}\)\(=\)\((\)\( 19 \nu^{7} - 49 \nu^{5} + 133 \nu^{3} - 684 \nu \)\()/189\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 2 \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{5} + 7 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 19 \beta_{4}\)\()/2\)
\(\nu^{6}\)\(=\)\(-7 \beta_{6} + 7 \beta_{1} + 22\)
\(\nu^{7}\)\(=\)\((\)\(29 \beta_{5} - 13 \beta_{4} - 13 \beta_{2}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.01575 1.40294i
−1.72286 + 0.178197i
−1.01575 + 1.40294i
1.72286 0.178197i
1.01575 + 1.40294i
−1.72286 0.178197i
−1.01575 1.40294i
1.72286 + 0.178197i
−0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i −1.93649 + 1.11803i 2.44949i −5.10237 4.79227i 2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
31.2 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 1.93649 1.11803i 2.44949i 6.51658 2.55620i 2.82843 1.50000 + 2.59808i −2.73861 1.58114i
31.3 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i −1.93649 + 1.11803i 2.44949i −6.51658 + 2.55620i −2.82843 1.50000 + 2.59808i −2.73861 1.58114i
31.4 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 1.93649 1.11803i 2.44949i 5.10237 + 4.79227i −2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
61.1 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i −1.93649 1.11803i 2.44949i −5.10237 + 4.79227i 2.82843 1.50000 2.59808i 2.73861 1.58114i
61.2 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 1.93649 + 1.11803i 2.44949i 6.51658 + 2.55620i 2.82843 1.50000 2.59808i −2.73861 + 1.58114i
61.3 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i −1.93649 1.11803i 2.44949i −6.51658 2.55620i −2.82843 1.50000 2.59808i −2.73861 + 1.58114i
61.4 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 1.93649 + 1.11803i 2.44949i 5.10237 4.79227i −2.82843 1.50000 2.59808i 2.73861 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.o.a 8
3.b odd 2 1 630.3.v.b 8
5.b even 2 1 1050.3.p.b 8
5.c odd 4 2 1050.3.q.c 16
7.c even 3 1 1470.3.f.a 8
7.d odd 6 1 inner 210.3.o.a 8
7.d odd 6 1 1470.3.f.a 8
21.g even 6 1 630.3.v.b 8
35.i odd 6 1 1050.3.p.b 8
35.k even 12 2 1050.3.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a 8 1.a even 1 1 trivial
210.3.o.a 8 7.d odd 6 1 inner
630.3.v.b 8 3.b odd 2 1
630.3.v.b 8 21.g even 6 1
1050.3.p.b 8 5.b even 2 1
1050.3.p.b 8 35.i odd 6 1
1050.3.q.c 16 5.c odd 4 2
1050.3.q.c 16 35.k even 12 2
1470.3.f.a 8 7.c even 3 1
1470.3.f.a 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{11}^{8} + \cdots\) acting on \(S_{3}^{\mathrm{new}}(210, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )^{4} \)
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ \( 1 - 78 T^{2} + 5243 T^{4} - 187278 T^{6} + 5764801 T^{8} \)
$11$ \( 1 + 4 T - 168 T^{2} + 1928 T^{3} + 16250 T^{4} - 341700 T^{5} + 2353408 T^{6} + 32486668 T^{7} - 296735661 T^{8} + 3930886828 T^{9} + 34456246528 T^{10} - 605342393700 T^{11} + 3483331816250 T^{12} + 50007354630728 T^{13} - 527255967289128 T^{14} + 1518999334332964 T^{15} + 45949729863572161 T^{16} \)
$13$ \( 1 - 860 T^{2} + 357498 T^{4} - 95438800 T^{6} + 18541152803 T^{8} - 2725827566800 T^{10} + 291622101296058 T^{12} - 20036353205333660 T^{14} + 665416609183179841 T^{16} \)
$17$ \( 1 - 84 T + 4208 T^{2} - 155904 T^{3} + 4746250 T^{4} - 123749124 T^{5} + 2818194464 T^{6} - 56792172564 T^{7} + 1020121281523 T^{8} - 16412937870996 T^{9} + 235378419827744 T^{10} - 2987003019239556 T^{11} + 33108688754346250 T^{12} - 314301513055600896 T^{13} + 2451674374262834288 T^{14} - 14143737430989678036 T^{15} + 48661191875666868481 T^{16} \)
$19$ \( 1 - 108 T + 6142 T^{2} - 243432 T^{3} + 7548345 T^{4} - 198492408 T^{5} + 4660679966 T^{6} - 100240172052 T^{7} + 1986343374068 T^{8} - 36186702110772 T^{9} + 607384473849086 T^{10} - 9338250206171448 T^{11} + 128197793162717145 T^{12} - 1492497721269013032 T^{13} + 13594180232904360862 T^{14} - 86292722064551485068 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} \)
$23$ \( 1 - 12 T - 1512 T^{2} + 15144 T^{3} + 1340890 T^{4} - 9662580 T^{5} - 850652928 T^{6} + 2479269564 T^{7} + 454065720819 T^{8} + 1311533599356 T^{9} - 238047566024448 T^{10} - 1430408620333620 T^{11} + 105006417053440090 T^{12} + 627363085819500456 T^{13} - 33134912141214725352 T^{14} - \)\(13\!\cdots\!08\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( ( 1 - 36 T + 2904 T^{2} - 82956 T^{3} + 3490070 T^{4} - 69765996 T^{5} + 2053944024 T^{6} - 21413639556 T^{7} + 500246412961 T^{8} )^{2} \)
$31$ \( 1 + 132 T + 11278 T^{2} + 722040 T^{3} + 38602473 T^{4} + 1778529960 T^{5} + 72524765390 T^{6} + 2637318955548 T^{7} + 86278260959732 T^{8} + 2534463516281628 T^{9} + 66978143857738190 T^{10} + 1578451886268782760 T^{11} + 32923703244758191593 T^{12} + \)\(59\!\cdots\!40\)\( T^{13} + \)\(88\!\cdots\!58\)\( T^{14} + \)\(99\!\cdots\!72\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( 1 + 96 T + 1110 T^{2} - 46464 T^{3} + 7652401 T^{4} + 360085920 T^{5} - 3532027338 T^{6} + 170035213440 T^{7} + 29111429055396 T^{8} + 232778207199360 T^{9} - 6619587887813418 T^{10} + 923881954453061280 T^{11} + 26878901285664514321 T^{12} - \)\(22\!\cdots\!36\)\( T^{13} + \)\(73\!\cdots\!10\)\( T^{14} + \)\(86\!\cdots\!44\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \)
$41$ \( 1 - 5456 T^{2} + 19963884 T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 140783428358562736 T^{10} + \)\(15\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!36\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( ( 1 + 56 T + 5082 T^{2} + 180688 T^{3} + 11078435 T^{4} + 334092112 T^{5} + 17374346682 T^{6} + 353996330744 T^{7} + 11688200277601 T^{8} )^{2} \)
$47$ \( 1 + 24 T + 3580 T^{2} + 81312 T^{3} + 3430986 T^{4} + 200307384 T^{5} + 5424417392 T^{6} + 942687331128 T^{7} + 25778360177363 T^{8} + 2082396314461752 T^{9} + 26469426483811952 T^{10} + 2159156424124689336 T^{11} + 81696191178488726346 T^{12} + \)\(42\!\cdots\!88\)\( T^{13} + \)\(41\!\cdots\!80\)\( T^{14} + \)\(61\!\cdots\!56\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 - 32 T - 3572 T^{2} + 570944 T^{3} - 366070 T^{4} - 2018388320 T^{5} + 123180800432 T^{6} + 3809526158624 T^{7} - 445561676340941 T^{8} + 10700958979574816 T^{9} + 971955765373487792 T^{10} - 44736287623035613280 T^{11} - 22791404868886921270 T^{12} + \)\(99\!\cdots\!56\)\( T^{13} - \)\(17\!\cdots\!52\)\( T^{14} - \)\(44\!\cdots\!08\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \)
$59$ \( 1 - 132 T + 15632 T^{2} - 1296768 T^{3} + 88851370 T^{4} - 4629146772 T^{5} + 218386477856 T^{6} - 7743785317668 T^{7} + 393482793133843 T^{8} - 26956116690802308 T^{9} + 2646267789699658016 T^{10} - \)\(19\!\cdots\!52\)\( T^{11} + \)\(13\!\cdots\!70\)\( T^{12} - \)\(66\!\cdots\!68\)\( T^{13} + \)\(27\!\cdots\!92\)\( T^{14} - \)\(81\!\cdots\!52\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( 1 - 96 T + 11380 T^{2} - 797568 T^{3} + 60840138 T^{4} - 3862327392 T^{5} + 197844384848 T^{6} - 13588740778464 T^{7} + 649379857320947 T^{8} - 50563704436664544 T^{9} + 2739321895348217168 T^{10} - \)\(19\!\cdots\!12\)\( T^{11} + \)\(11\!\cdots\!78\)\( T^{12} - \)\(56\!\cdots\!68\)\( T^{13} + \)\(30\!\cdots\!80\)\( T^{14} - \)\(94\!\cdots\!36\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( 1 + 120 T - 4522 T^{2} - 749040 T^{3} + 46360561 T^{4} + 4026690720 T^{5} - 240439443082 T^{6} - 2848051223400 T^{7} + 1781423144538724 T^{8} - 12784901941842600 T^{9} - 4845124310717994922 T^{10} + \)\(36\!\cdots\!80\)\( T^{11} + \)\(18\!\cdots\!01\)\( T^{12} - \)\(13\!\cdots\!60\)\( T^{13} - \)\(37\!\cdots\!42\)\( T^{14} + \)\(44\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \)
$71$ \( ( 1 - 4 T + 13176 T^{2} - 338828 T^{3} + 79144886 T^{4} - 1708031948 T^{5} + 334824308856 T^{6} - 512401135684 T^{7} + 645753531245761 T^{8} )^{2} \)
$73$ \( 1 - 24 T + 9630 T^{2} - 226512 T^{3} + 47274361 T^{4} + 2709573216 T^{5} - 18252715698 T^{6} + 34150744827912 T^{7} - 1287593588654412 T^{8} + 181989319187943048 T^{9} - 518345019296287218 T^{10} + \)\(41\!\cdots\!24\)\( T^{11} + \)\(38\!\cdots\!41\)\( T^{12} - \)\(97\!\cdots\!88\)\( T^{13} + \)\(22\!\cdots\!30\)\( T^{14} - \)\(29\!\cdots\!16\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \)
$79$ \( 1 - 12 T - 2418 T^{2} + 386472 T^{3} - 58917335 T^{4} - 324907032 T^{5} + 66125106126 T^{6} - 10581169939908 T^{7} + 1991860414252788 T^{8} - 66037081594965828 T^{9} + 2575578239741296206 T^{10} - 78980823689760123672 T^{11} - \)\(89\!\cdots\!35\)\( T^{12} + \)\(36\!\cdots\!72\)\( T^{13} - \)\(14\!\cdots\!38\)\( T^{14} - \)\(44\!\cdots\!72\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} \)
$83$ \( 1 - 2384 T^{2} + 54393900 T^{4} - 170213772976 T^{6} + 3204332319622694 T^{8} - 8078059876516133296 T^{10} + \)\(12\!\cdots\!00\)\( T^{12} - \)\(25\!\cdots\!24\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( 1 - 492 T + 137248 T^{2} - 27827520 T^{3} + 4512312618 T^{4} - 615183153660 T^{5} + 72837035721440 T^{6} - 7634393359602348 T^{7} + 716618506839255347 T^{8} - 60472029801410198508 T^{9} + \)\(45\!\cdots\!40\)\( T^{10} - \)\(30\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!58\)\( T^{12} - \)\(86\!\cdots\!20\)\( T^{13} + \)\(33\!\cdots\!08\)\( T^{14} - \)\(96\!\cdots\!72\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} \)
$97$ \( 1 - 35880 T^{2} + 724155484 T^{4} - 9811041724440 T^{6} + 103976225413471686 T^{8} - \)\(86\!\cdots\!40\)\( T^{10} + \)\(56\!\cdots\!24\)\( T^{12} - \)\(24\!\cdots\!80\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \)
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