Properties

Label 43.2.e.b
Level 43
Weight 2
Character orbit 43.e
Analytic conductor 0.343
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.343356728692\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{14}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.222521 0.974928i
0.222521 + 0.974928i
0.900969 + 0.433884i
−0.623490 + 0.781831i
0.900969 0.433884i
−0.623490 0.781831i
0.500000 0.626980i −2.02446 2.53859i 0.301938 + 1.32288i 1.80194 + 0.867767i −2.60388 −1.19806 2.42543 + 1.16802i −1.67845 + 7.35376i 1.44504 0.695895i
11.1 0.500000 + 0.626980i −2.02446 + 2.53859i 0.301938 1.32288i 1.80194 0.867767i −2.60388 −1.19806 2.42543 1.16802i −1.67845 7.35376i 1.44504 + 0.695895i
16.1 0.500000 + 2.19064i 0.346011 1.51597i −2.74698 + 1.32288i −1.24698 1.56366i 3.49396 −4.24698 −1.46950 1.84270i 0.524459 + 0.252566i 2.80194 3.51352i
21.1 0.500000 + 0.240787i 0.178448 0.0859360i −1.05496 1.32288i 0.445042 + 1.94986i 0.109916 −2.55496 −0.455927 1.99755i −1.84601 + 2.31482i −0.246980 + 1.08209i
35.1 0.500000 2.19064i 0.346011 + 1.51597i −2.74698 1.32288i −1.24698 + 1.56366i 3.49396 −4.24698 −1.46950 + 1.84270i 0.524459 0.252566i 2.80194 + 3.51352i
41.1 0.500000 0.240787i 0.178448 + 0.0859360i −1.05496 + 1.32288i 0.445042 1.94986i 0.109916 −2.55496 −0.455927 + 1.99755i −1.84601 2.31482i −0.246980 1.08209i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.2.e.b 6
3.b odd 2 1 387.2.u.a 6
4.b odd 2 1 688.2.u.c 6
43.e even 7 1 inner 43.2.e.b 6
43.e even 7 1 1849.2.a.i 3
43.f odd 14 1 1849.2.a.l 3
129.l odd 14 1 387.2.u.a 6
172.k odd 14 1 688.2.u.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.e.b 6 1.a even 1 1 trivial
43.2.e.b 6 43.e even 7 1 inner
387.2.u.a 6 3.b odd 2 1
387.2.u.a 6 129.l odd 14 1
688.2.u.c 6 4.b odd 2 1
688.2.u.c 6 172.k odd 14 1
1849.2.a.i 3 43.e even 7 1
1849.2.a.l 3 43.f odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3 T_{2}^{5} + 9 T_{2}^{4} - 13 T_{2}^{3} + 11 T_{2}^{2} - 5 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 7 T^{2} - 15 T^{3} + 31 T^{4} - 49 T^{5} + 71 T^{6} - 98 T^{7} + 124 T^{8} - 120 T^{9} + 112 T^{10} - 96 T^{11} + 64 T^{12} \)
$3$ \( 1 + 3 T + 6 T^{2} + 2 T^{3} - 5 T^{4} - 21 T^{5} - 20 T^{6} - 63 T^{7} - 45 T^{8} + 54 T^{9} + 486 T^{10} + 729 T^{11} + 729 T^{12} \)
$5$ \( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} - 22 T^{5} + 139 T^{6} - 110 T^{7} - 475 T^{8} + 1500 T^{9} - 625 T^{10} - 6250 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 + 8 T + 40 T^{2} + 125 T^{3} + 280 T^{4} + 392 T^{5} + 343 T^{6} )^{2} \)
$11$ \( 1 - 14 T + 80 T^{2} - 231 T^{3} + 149 T^{4} + 2135 T^{5} - 11369 T^{6} + 23485 T^{7} + 18029 T^{8} - 307461 T^{9} + 1171280 T^{10} - 2254714 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 9 T + 12 T^{2} - 142 T^{3} - 524 T^{4} + 1141 T^{5} + 11103 T^{6} + 14833 T^{7} - 88556 T^{8} - 311974 T^{9} + 342732 T^{10} + 3341637 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 8 T + 19 T^{2} - 30 T^{3} - 55 T^{4} + 2588 T^{5} - 16885 T^{6} + 43996 T^{7} - 15895 T^{8} - 147390 T^{9} + 1586899 T^{10} - 11358856 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 4 T - 24 T^{2} - 207 T^{3} - 659 T^{4} + 2459 T^{5} + 28923 T^{6} + 46721 T^{7} - 237899 T^{8} - 1419813 T^{9} - 3127704 T^{10} + 9904396 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - T - 8 T^{2} - 60 T^{3} - 169 T^{4} + 1465 T^{5} - 1792 T^{6} + 33695 T^{7} - 89401 T^{8} - 730020 T^{9} - 2238728 T^{10} - 6436343 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 9 T + 24 T^{2} - 242 T^{3} + 1881 T^{4} - 5333 T^{5} + 16380 T^{6} - 154657 T^{7} + 1581921 T^{8} - 5902138 T^{9} + 16974744 T^{10} - 184600341 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 18 T + 125 T^{2} + 572 T^{3} + 3467 T^{4} + 20608 T^{5} + 104889 T^{6} + 638848 T^{7} + 3331787 T^{8} + 17040452 T^{9} + 115440125 T^{10} + 515324718 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 + 2 T + 26 T^{2} + 399 T^{3} + 962 T^{4} + 2738 T^{5} + 50653 T^{6} )^{2} \)
$41$ \( 1 - 23 T + 208 T^{2} - 1146 T^{3} + 6840 T^{4} - 44849 T^{5} + 273267 T^{6} - 1838809 T^{7} + 11498040 T^{8} - 78983466 T^{9} + 587758288 T^{10} - 2664692623 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 29 T + 393 T^{2} + 3221 T^{3} + 16899 T^{4} + 53621 T^{5} + 79507 T^{6} \)
$47$ \( 1 - T - 46 T^{2} + 93 T^{3} + 2069 T^{4} - 6440 T^{5} - 90803 T^{6} - 302680 T^{7} + 4570421 T^{8} + 9655539 T^{9} - 224465326 T^{10} - 229345007 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 10 T + 47 T^{2} + 528 T^{3} + 8081 T^{4} + 52826 T^{5} + 297535 T^{6} + 2799778 T^{7} + 22699529 T^{8} + 78607056 T^{9} + 370852607 T^{10} + 4181954930 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 22 T + 152 T^{2} + 201 T^{3} - 5011 T^{4} - 58697 T^{5} + 1026535 T^{6} - 3463123 T^{7} - 17443291 T^{8} + 41281179 T^{9} + 1841838872 T^{10} - 15728334578 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 19 T + 90 T^{2} + 436 T^{3} - 5311 T^{4} + 40503 T^{5} - 386744 T^{6} + 2470683 T^{7} - 19762231 T^{8} + 98963716 T^{9} + 1246125690 T^{10} - 16047329719 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 4 T - 30 T^{2} - 745 T^{3} + 2831 T^{4} + 213 T^{5} + 116347 T^{6} + 14271 T^{7} + 12708359 T^{8} - 224068435 T^{9} - 604533630 T^{10} + 5400500428 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 28 T + 293 T^{2} - 1470 T^{3} + 953 T^{4} + 112168 T^{5} - 1557515 T^{6} + 7963928 T^{7} + 4804073 T^{8} - 526129170 T^{9} + 7445622533 T^{10} - 50518421828 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 - 21 T + 277 T^{2} - 3395 T^{3} + 42226 T^{4} - 408520 T^{5} + 3543793 T^{6} - 29821960 T^{7} + 225022354 T^{8} - 1320712715 T^{9} + 7866312757 T^{10} - 43534503453 T^{11} + 151334226289 T^{12} \)
$79$ \( ( 1 - 3 T + 212 T^{2} - 391 T^{3} + 16748 T^{4} - 18723 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 + 39 T + 640 T^{2} + 6519 T^{3} + 67141 T^{4} + 848516 T^{5} + 9110429 T^{6} + 70426828 T^{7} + 462534349 T^{8} + 3727479453 T^{9} + 30373325440 T^{10} + 153622585077 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 11 T - 143 T^{2} + 3063 T^{3} - 12244 T^{4} - 185572 T^{5} + 3036725 T^{6} - 16515908 T^{7} - 96984724 T^{8} + 2159320047 T^{9} - 8972140463 T^{10} - 61424653939 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 19 T + 145 T^{2} + 793 T^{3} + 5720 T^{4} - 26012 T^{5} - 1115155 T^{6} - 2523164 T^{7} + 53819480 T^{8} + 723749689 T^{9} + 12836745745 T^{10} + 163159464883 T^{11} + 832972004929 T^{12} \)
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