Properties

Label 336.2.bq.a
Level 336
Weight 2
Character orbit 336.bq
Analytic conductor 2.683
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.bq (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(1\) \(1\) \(-\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 0.258819i
0.965926 + 0.258819i
−1.22474 0.707107i −0.965926 0.258819i 1.00000 + 1.73205i 3.69798 0.990870i 1.00000 + 1.00000i 0.358719 + 2.62132i 2.82843i 0.866025 + 0.500000i −5.22973 1.40130i
37.2 1.22474 + 0.707107i 0.965926 + 0.258819i 1.00000 + 1.73205i 1.76612 0.473232i 1.00000 + 1.00000i −2.09077 1.62132i 2.82843i 0.866025 + 0.500000i 2.49768 + 0.669251i
109.1 −1.22474 + 0.707107i −0.965926 + 0.258819i 1.00000 1.73205i 3.69798 + 0.990870i 1.00000 1.00000i 0.358719 2.62132i 2.82843i 0.866025 0.500000i −5.22973 + 1.40130i
109.2 1.22474 0.707107i 0.965926 0.258819i 1.00000 1.73205i 1.76612 + 0.473232i 1.00000 1.00000i −2.09077 + 1.62132i 2.82843i 0.866025 0.500000i 2.49768 0.669251i
205.1 −1.22474 0.707107i −0.258819 + 0.965926i 1.00000 + 1.73205i −0.473232 1.76612i 1.00000 1.00000i 2.09077 + 1.62132i 2.82843i −0.866025 0.500000i −0.669251 + 2.49768i
205.2 1.22474 + 0.707107i 0.258819 0.965926i 1.00000 + 1.73205i −0.990870 3.69798i 1.00000 1.00000i −0.358719 2.62132i 2.82843i −0.866025 0.500000i 1.40130 5.22973i
277.1 −1.22474 + 0.707107i −0.258819 0.965926i 1.00000 1.73205i −0.473232 + 1.76612i 1.00000 + 1.00000i 2.09077 1.62132i 2.82843i −0.866025 + 0.500000i −0.669251 2.49768i
277.2 1.22474 0.707107i 0.258819 + 0.965926i 1.00000 1.73205i −0.990870 + 3.69798i 1.00000 + 1.00000i −0.358719 + 2.62132i 2.82843i −0.866025 + 0.500000i 1.40130 + 5.22973i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bq.a 8
7.c even 3 1 inner 336.2.bq.a 8
16.e even 4 1 inner 336.2.bq.a 8
112.w even 12 1 inner 336.2.bq.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bq.a 8 1.a even 1 1 trivial
336.2.bq.a 8 7.c even 3 1 inner
336.2.bq.a 8 16.e even 4 1 inner
336.2.bq.a 8 112.w even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \)
$3$ \( 1 - T^{4} + T^{8} \)
$5$ \( 1 - 8 T + 32 T^{2} - 64 T^{3} + 17 T^{4} + 272 T^{5} - 672 T^{6} + 376 T^{7} + 816 T^{8} + 1880 T^{9} - 16800 T^{10} + 34000 T^{11} + 10625 T^{12} - 200000 T^{13} + 500000 T^{14} - 625000 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \)
$11$ \( 1 - 4 T + 8 T^{2} + 64 T^{3} - 415 T^{4} + 1384 T^{5} - 168 T^{6} - 14524 T^{7} + 77232 T^{8} - 159764 T^{9} - 20328 T^{10} + 1842104 T^{11} - 6076015 T^{12} + 10307264 T^{13} + 14172488 T^{14} - 77948684 T^{15} + 214358881 T^{16} \)
$13$ \( ( 1 + 4 T + 8 T^{2} + 44 T^{3} + 238 T^{4} + 572 T^{5} + 1352 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 8 T + 22 T^{2} - 64 T^{3} + 387 T^{4} - 1088 T^{5} + 6358 T^{6} - 39304 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 4 T + 8 T^{2} + 120 T^{3} - 601 T^{4} + 2280 T^{5} + 2888 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( 1 + 24 T^{2} + 526 T^{4} - 24192 T^{6} - 582045 T^{8} - 12797568 T^{10} + 147196366 T^{12} + 3552861336 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 - 16 T + 128 T^{2} - 832 T^{3} + 4879 T^{4} - 24128 T^{5} + 107648 T^{6} - 390224 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 2 T - 57 T^{2} + 2 T^{3} + 2636 T^{4} + 62 T^{5} - 54777 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 4 T + 8 T^{2} - 264 T^{3} - 1897 T^{4} - 9768 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 152 T^{2} + 9106 T^{4} - 255512 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 24 T + 288 T^{2} - 2664 T^{3} + 20018 T^{4} - 114552 T^{5} + 532512 T^{6} - 1908168 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 4 T - 64 T^{2} + 56 T^{3} + 3439 T^{4} + 2632 T^{5} - 141376 T^{6} - 415292 T^{7} + 4879681 T^{8} )^{2} \)
$53$ \( 1 + 16 T + 128 T^{2} - 384 T^{3} - 14095 T^{4} - 132640 T^{5} - 244352 T^{6} + 5040144 T^{7} + 65449648 T^{8} + 267127632 T^{9} - 686384768 T^{10} - 19747045280 T^{11} - 111216329695 T^{12} - 160587069312 T^{13} + 2837038224512 T^{14} + 18795378237392 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 20 T + 200 T^{2} + 640 T^{3} - 7087 T^{4} - 104360 T^{5} - 465000 T^{6} + 1041740 T^{7} + 28323408 T^{8} + 61462660 T^{9} - 1618665000 T^{10} - 21433352440 T^{11} - 85875737407 T^{12} + 457551551360 T^{13} + 8436106728200 T^{14} + 49773029696380 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 4 T + 8 T^{2} - 344 T^{3} - 4910 T^{4} - 20804 T^{5} + 15232 T^{6} + 1637244 T^{7} + 8848707 T^{8} + 99871884 T^{9} + 56678272 T^{10} - 4722112724 T^{11} - 67983079310 T^{12} - 290541127544 T^{13} + 412162994888 T^{14} + 12570971344084 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 32 T + 512 T^{2} + 4160 T^{3} + 8974 T^{4} - 146656 T^{5} - 634880 T^{6} + 15805728 T^{7} + 232306035 T^{8} + 1058983776 T^{9} - 2849976320 T^{10} - 44108698528 T^{11} + 180836159854 T^{12} + 5616520445120 T^{13} + 46314691670528 T^{14} + 193942771370336 T^{15} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 16 T^{2} - 3966 T^{4} - 80656 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 14 T + 7 T^{2} + 434 T^{3} + 13996 T^{4} + 34286 T^{5} + 43687 T^{6} + 6902546 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 4 T + 8 T^{2} - 144 T^{3} - 11569 T^{4} - 11952 T^{5} + 55112 T^{6} + 2287148 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 64 T^{2} - 3825 T^{4} - 506944 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 6 T + 75 T^{2} + 582 T^{3} + 9409 T^{4} )^{4} \)
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