Properties

Label 37.2.f.b
Level 37
Weight 2
Character orbit 37.f
Analytic conductor 0.295
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.295446487479\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 0.642788i
0.939693 0.342020i
1.26604 0.460802i −1.76604 0.642788i −0.141559 + 0.118782i 0.233956 + 1.32683i −2.53209 −0.120615 0.684040i −1.47178 + 2.54920i 0.407604 + 0.342020i 0.907604 + 1.57202i
9.1 −0.439693 2.49362i −0.0603074 + 0.342020i −4.14543 + 1.50881i 1.93969 + 1.62760i 0.879385 −2.34730 1.96962i 3.05303 + 5.28801i 2.70574 + 0.984808i 3.20574 5.55250i
12.1 0.673648 0.565258i −1.17365 0.984808i −0.213011 + 1.20805i 0.826352 0.300767i −1.34730 −3.53209 + 1.28558i 1.41875 + 2.45734i −0.113341 0.642788i 0.386659 0.669713i
16.1 1.26604 + 0.460802i −1.76604 + 0.642788i −0.141559 0.118782i 0.233956 1.32683i −2.53209 −0.120615 + 0.684040i −1.47178 2.54920i 0.407604 0.342020i 0.907604 1.57202i
33.1 −0.439693 + 2.49362i −0.0603074 0.342020i −4.14543 1.50881i 1.93969 1.62760i 0.879385 −2.34730 + 1.96962i 3.05303 5.28801i 2.70574 0.984808i 3.20574 + 5.55250i
34.1 0.673648 + 0.565258i −1.17365 + 0.984808i −0.213011 1.20805i 0.826352 + 0.300767i −1.34730 −3.53209 1.28558i 1.41875 2.45734i −0.113341 + 0.642788i 0.386659 + 0.669713i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.f.b 6
3.b odd 2 1 333.2.x.a 6
4.b odd 2 1 592.2.bc.c 6
5.b even 2 1 925.2.p.a 6
5.c odd 4 2 925.2.bc.b 12
37.f even 9 1 inner 37.2.f.b 6
37.f even 9 1 1369.2.a.i 3
37.h even 18 1 1369.2.a.l 3
37.i odd 36 2 1369.2.b.e 6
111.p odd 18 1 333.2.x.a 6
148.p odd 18 1 592.2.bc.c 6
185.x even 18 1 925.2.p.a 6
185.bd odd 36 2 925.2.bc.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.f.b 6 1.a even 1 1 trivial
37.2.f.b 6 37.f even 9 1 inner
333.2.x.a 6 3.b odd 2 1
333.2.x.a 6 111.p odd 18 1
592.2.bc.c 6 4.b odd 2 1
592.2.bc.c 6 148.p odd 18 1
925.2.p.a 6 5.b even 2 1
925.2.p.a 6 185.x even 18 1
925.2.bc.b 12 5.c odd 4 2
925.2.bc.b 12 185.bd odd 36 2
1369.2.a.i 3 37.f even 9 1
1369.2.a.l 3 37.h even 18 1
1369.2.b.e 6 37.i odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3 T_{2}^{5} + 9 T_{2}^{4} - 24 T_{2}^{3} + 36 T_{2}^{2} - 27 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 9 T^{2} - 18 T^{3} + 36 T^{4} - 57 T^{5} + 91 T^{6} - 114 T^{7} + 144 T^{8} - 144 T^{9} + 144 T^{10} - 96 T^{11} + 64 T^{12} \)
$3$ \( 1 + 6 T + 15 T^{2} + 19 T^{3} + 3 T^{4} - 51 T^{5} - 134 T^{6} - 153 T^{7} + 27 T^{8} + 513 T^{9} + 1215 T^{10} + 1458 T^{11} + 729 T^{12} \)
$5$ \( 1 - 6 T + 18 T^{2} - 45 T^{3} + 81 T^{4} - 87 T^{5} + 109 T^{6} - 435 T^{7} + 2025 T^{8} - 5625 T^{9} + 11250 T^{10} - 18750 T^{11} + 15625 T^{12} \)
$7$ \( 1 + 12 T + 60 T^{2} + 152 T^{3} + 108 T^{4} - 702 T^{5} - 3051 T^{6} - 4914 T^{7} + 5292 T^{8} + 52136 T^{9} + 144060 T^{10} + 201684 T^{11} + 117649 T^{12} \)
$11$ \( 1 - 9 T + 30 T^{2} - 45 T^{3} + 177 T^{4} - 1548 T^{5} + 7099 T^{6} - 17028 T^{7} + 21417 T^{8} - 59895 T^{9} + 439230 T^{10} - 1449459 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 12 T + 78 T^{2} + 359 T^{3} + 1485 T^{4} + 6129 T^{5} + 23841 T^{6} + 79677 T^{7} + 250965 T^{8} + 788723 T^{9} + 2227758 T^{10} + 4455516 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 3 T - 72 T^{3} - 117 T^{4} + 1275 T^{5} + 1369 T^{6} + 21675 T^{7} - 33813 T^{8} - 353736 T^{9} - 4259571 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 9 T + 18 T^{2} - 142 T^{3} - 837 T^{4} + 513 T^{5} + 15087 T^{6} + 9747 T^{7} - 302157 T^{8} - 973978 T^{9} + 2345778 T^{10} + 22284891 T^{11} + 47045881 T^{12} \)
$23$ \( ( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} )^{3} \)
$29$ \( 1 - 9 T - 6 T^{2} + 207 T^{3} + 591 T^{4} - 5220 T^{5} + 6181 T^{6} - 151380 T^{7} + 497031 T^{8} + 5048523 T^{9} - 4243686 T^{10} - 184600341 T^{11} + 594823321 T^{12} \)
$31$ \( ( 1 + 9 T + 99 T^{2} + 505 T^{3} + 3069 T^{4} + 8649 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 + 12 T - 30 T^{2} - 803 T^{3} - 1110 T^{4} + 16428 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 15 T + 72 T^{2} + 162 T^{3} - 2889 T^{4} + 6123 T^{5} + 21493 T^{6} + 251043 T^{7} - 4856409 T^{8} + 11165202 T^{9} + 203454792 T^{10} - 1737843015 T^{11} + 4750104241 T^{12} \)
$43$ \( ( 1 + 9 T + 153 T^{2} + 793 T^{3} + 6579 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \)
$47$ \( 1 - 9 T - 24 T^{2} + 765 T^{3} - 1605 T^{4} - 20826 T^{5} + 241567 T^{6} - 978822 T^{7} - 3545445 T^{8} + 79424595 T^{9} - 117112344 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 21 T + 180 T^{2} - 756 T^{3} + 4158 T^{4} - 67395 T^{5} + 688951 T^{6} - 3571935 T^{7} + 11679822 T^{8} - 112551012 T^{9} + 1420286580 T^{10} - 8782105353 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 + 6 T + 18 T^{2} + 180 T^{3} - 972 T^{4} - 38640 T^{5} - 291923 T^{6} - 2279760 T^{7} - 3383532 T^{8} + 36968220 T^{9} + 218112498 T^{10} + 4289545794 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 9 T + 45 T^{2} + 335 T^{3} + 2538 T^{4} + 11988 T^{5} - 65103 T^{6} + 731268 T^{7} + 9443898 T^{8} + 76038635 T^{9} + 623062845 T^{10} + 7601366709 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 24 T + 258 T^{2} - 1711 T^{3} + 603 T^{4} + 133227 T^{5} - 1538559 T^{6} + 8926209 T^{7} + 2706867 T^{8} - 514605493 T^{9} + 5198989218 T^{10} - 32403002568 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 12 T + 126 T^{2} + 1692 T^{3} + 16038 T^{4} + 158646 T^{5} + 1508509 T^{6} + 11263866 T^{7} + 80847558 T^{8} + 605585412 T^{9} + 3201871806 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \)
$73$ \( ( 1 + 180 T^{2} - 89 T^{3} + 13140 T^{4} + 389017 T^{6} )^{2} \)
$79$ \( 1 + 3 T + 114 T^{2} + 656 T^{3} + 8793 T^{4} + 56367 T^{5} + 695565 T^{6} + 4452993 T^{7} + 54877113 T^{8} + 323433584 T^{9} + 4440309234 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 3 T - 117 T^{2} + 639 T^{3} - 1080 T^{4} - 30306 T^{5} + 685621 T^{6} - 2515398 T^{7} - 7440120 T^{8} + 365371893 T^{9} - 5552623557 T^{10} - 11817121929 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 24 T + 207 T^{2} + 225 T^{3} - 6615 T^{4} + 1455 T^{5} + 482158 T^{6} + 129495 T^{7} - 52397415 T^{8} + 158618025 T^{9} + 12987643887 T^{10} + 134017426776 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 27 T + 234 T^{2} + 2171 T^{3} + 45891 T^{4} + 431748 T^{5} + 2525169 T^{6} + 41879556 T^{7} + 431788419 T^{8} + 1981413083 T^{9} + 20715851754 T^{10} + 231858186939 T^{11} + 832972004929 T^{12} \)
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