Properties

Label 37.2.f.a
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: \(\Z[a_1, a_2]\)
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.

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

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
−1.76604 + 0.642788i 1.26604 + 0.460802i 1.17365 0.984808i 0.532089 + 3.01763i −2.53209 −0.613341 3.47843i 0.439693 0.761570i −0.907604 0.761570i −2.87939 4.98724i
9.1 −0.0603074 0.342020i −0.439693 + 2.49362i 1.76604 0.642788i −2.87939 2.41609i 0.879385 −0.0923963 0.0775297i −0.673648 1.16679i −3.20574 1.16679i −0.652704 + 1.13052i
12.1 −1.17365 + 0.984808i 0.673648 + 0.565258i 0.0603074 0.342020i −0.652704 + 0.237565i −1.34730 2.20574 0.802823i −1.26604 2.19285i −0.386659 2.19285i 0.532089 0.921605i
16.1 −1.76604 0.642788i 1.26604 0.460802i 1.17365 + 0.984808i 0.532089 3.01763i −2.53209 −0.613341 + 3.47843i 0.439693 + 0.761570i −0.907604 + 0.761570i −2.87939 + 4.98724i
33.1 −0.0603074 + 0.342020i −0.439693 2.49362i 1.76604 + 0.642788i −2.87939 + 2.41609i 0.879385 −0.0923963 + 0.0775297i −0.673648 + 1.16679i −3.20574 + 1.16679i −0.652704 1.13052i
34.1 −1.17365 0.984808i 0.673648 0.565258i 0.0603074 + 0.342020i −0.652704 0.237565i −1.34730 2.20574 + 0.802823i −1.26604 + 2.19285i −0.386659 + 2.19285i 0.532089 + 0.921605i
\(n\): 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.a 6
3.b odd 2 1 333.2.x.c 6
4.b odd 2 1 592.2.bc.a 6
5.b even 2 1 925.2.p.b 6
5.c odd 4 2 925.2.bc.a 12
37.f even 9 1 inner 37.2.f.a 6
37.f even 9 1 1369.2.a.j 3
37.h even 18 1 1369.2.a.k 3
37.i odd 36 2 1369.2.b.f 6
111.p odd 18 1 333.2.x.c 6
148.p odd 18 1 592.2.bc.a 6
185.x even 18 1 925.2.p.b 6
185.bd odd 36 2 925.2.bc.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.f.a 6 1.a even 1 1 trivial
37.2.f.a 6 37.f even 9 1 inner
333.2.x.c 6 3.b odd 2 1
333.2.x.c 6 111.p odd 18 1
592.2.bc.a 6 4.b odd 2 1
592.2.bc.a 6 148.p odd 18 1
925.2.p.b 6 5.b even 2 1
925.2.p.b 6 185.x even 18 1
925.2.bc.a 12 5.c odd 4 2
925.2.bc.a 12 185.bd odd 36 2
1369.2.a.j 3 37.f even 9 1
1369.2.a.k 3 37.h even 18 1
1369.2.b.f 6 37.i odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 6 T_{2}^{5} + 15 T_{2}^{4} + 19 T_{2}^{3} + 12 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 15 T^{2} + 19 T^{3} + 6 T^{4} - 27 T^{5} - 61 T^{6} - 54 T^{7} + 24 T^{8} + 152 T^{9} + 240 T^{10} + 192 T^{11} + 64 T^{12} \)
$3$ \( 1 - 3 T + 9 T^{2} - 15 T^{3} + 36 T^{4} - 54 T^{5} + 117 T^{6} - 162 T^{7} + 324 T^{8} - 405 T^{9} + 729 T^{10} - 729 T^{11} + 729 T^{12} \)
$5$ \( 1 + 6 T + 24 T^{2} + 64 T^{3} + 132 T^{4} + 252 T^{5} + 479 T^{6} + 1260 T^{7} + 3300 T^{8} + 8000 T^{9} + 15000 T^{10} + 18750 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 3 T + 12 T^{2} - 4 T^{3} + 81 T^{4} - 135 T^{5} + 1023 T^{6} - 945 T^{7} + 3969 T^{8} - 1372 T^{9} + 28812 T^{10} - 50421 T^{11} + 117649 T^{12} \)
$11$ \( 1 - 30 T^{2} - 2 T^{3} + 570 T^{4} + 30 T^{5} - 7237 T^{6} + 330 T^{7} + 68970 T^{8} - 2662 T^{9} - 439230 T^{10} + 1771561 T^{12} \)
$13$ \( 1 + 9 T + 27 T^{2} - 11 T^{3} - 198 T^{4} + 360 T^{5} + 4161 T^{6} + 4680 T^{7} - 33462 T^{8} - 24167 T^{9} + 771147 T^{10} + 3341637 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 3 T + 36 T^{2} + 90 T^{4} + 3489 T^{5} - 8513 T^{6} + 59313 T^{7} + 26010 T^{8} + 3006756 T^{10} - 4259571 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - 3 T - 36 T^{2} + 150 T^{3} + 45 T^{4} - 1767 T^{5} + 10583 T^{6} - 33573 T^{7} + 16245 T^{8} + 1028850 T^{9} - 4691556 T^{10} - 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 3 T - 60 T^{2} - 67 T^{3} + 2763 T^{4} + 1518 T^{5} - 71161 T^{6} + 34914 T^{7} + 1461627 T^{8} - 815189 T^{9} - 16790460 T^{10} + 19309029 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 15 T + 75 T^{2} + 396 T^{3} + 4245 T^{4} + 21165 T^{5} + 66382 T^{6} + 613785 T^{7} + 3570045 T^{8} + 9658044 T^{9} + 53046075 T^{10} + 307667235 T^{11} + 594823321 T^{12} \)
$31$ \( ( 1 - 3 T + 84 T^{2} - 167 T^{3} + 2604 T^{4} - 2883 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 + 6 T + 96 T^{2} + 371 T^{3} + 3552 T^{4} + 8214 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 9 T^{2} - 9 T^{3} + 369 T^{4} + 11889 T^{5} - 26774 T^{6} + 487449 T^{7} + 620289 T^{8} - 620289 T^{9} + 25431849 T^{10} + 4750104241 T^{12} \)
$43$ \( ( 1 + 3 T + 69 T^{2} + 187 T^{3} + 2967 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2} \)
$47$ \( 1 - 15 T + 21 T^{2} - 126 T^{3} + 10023 T^{4} - 44763 T^{5} - 93242 T^{6} - 2103861 T^{7} + 22140807 T^{8} - 13081698 T^{9} + 102473301 T^{10} - 3440175105 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 24 T + 228 T^{2} - 866 T^{3} - 3012 T^{4} + 75276 T^{5} - 701113 T^{6} + 3989628 T^{7} - 8460708 T^{8} - 128927482 T^{9} + 1799029668 T^{10} - 10036691832 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 6 T - 12 T^{2} + 503 T^{3} - 3207 T^{4} - 32067 T^{5} + 445277 T^{6} - 1891953 T^{7} - 11163567 T^{8} + 103305637 T^{9} - 145408332 T^{10} - 4289545794 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 27 T + 324 T^{2} + 1924 T^{3} - 2538 T^{4} - 194049 T^{5} - 2164749 T^{6} - 11836989 T^{7} - 9443898 T^{8} + 436711444 T^{9} + 4486052484 T^{10} + 22804100127 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 18 T + 144 T^{2} + 821 T^{3} + 2133 T^{4} - 39825 T^{5} - 605283 T^{6} - 2668275 T^{7} + 9575037 T^{8} + 246926423 T^{9} + 2901761424 T^{10} + 24302251926 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 33 T + 459 T^{2} - 3015 T^{3} + 3528 T^{4} + 104358 T^{5} - 1167947 T^{6} + 7409418 T^{7} + 17784648 T^{8} - 1079101665 T^{9} + 11663961579 T^{10} - 59539568583 T^{11} + 128100283921 T^{12} \)
$73$ \( ( 1 - 6 T + 168 T^{2} - 767 T^{3} + 12264 T^{4} - 31974 T^{5} + 389017 T^{6} )^{2} \)
$79$ \( 1 - 33 T + 516 T^{2} - 4424 T^{3} + 9234 T^{4} + 298269 T^{5} - 4310931 T^{6} + 23563251 T^{7} + 57629394 T^{8} - 2181204536 T^{9} + 20098241796 T^{10} - 101542861167 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 21 T + 144 T^{2} + 936 T^{3} - 15894 T^{4} - 5853 T^{5} + 1245277 T^{6} - 485799 T^{7} - 109493766 T^{8} + 535192632 T^{9} + 6833998224 T^{10} - 82719853503 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 12 T + 159 T^{2} - 2483 T^{3} + 27897 T^{4} - 256761 T^{5} + 2868014 T^{6} - 22851729 T^{7} + 220972137 T^{8} - 1750438027 T^{9} + 9976016319 T^{10} - 67008713388 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 18 T - 36 T^{2} + 646 T^{3} + 30546 T^{4} - 194922 T^{5} - 1099401 T^{6} - 18907434 T^{7} + 287407314 T^{8} + 589586758 T^{9} - 3187054116 T^{10} - 154572124626 T^{11} + 832972004929 T^{12} \)
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