Properties

Label 43.2.e.a
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.

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

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
−0.777479 + 0.974928i 0.277479 + 0.347948i 0.0990311 + 0.433884i −0.500000 0.240787i −0.554958 1.35690 −2.74698 1.32288i 0.623490 2.73169i 0.623490 0.300257i
11.1 −0.777479 0.974928i 0.277479 0.347948i 0.0990311 0.433884i −0.500000 + 0.240787i −0.554958 1.35690 −2.74698 + 1.32288i 0.623490 + 2.73169i 0.623490 + 0.300257i
16.1 −0.0990311 0.433884i −0.400969 + 1.75676i 1.62349 0.781831i −0.500000 0.626980i 0.801938 −3.04892 −1.05496 1.32288i −0.222521 0.107160i −0.222521 + 0.279032i
21.1 −1.62349 0.781831i 1.12349 0.541044i 0.777479 + 0.974928i −0.500000 2.19064i −2.24698 1.69202 0.301938 + 1.32288i −0.900969 + 1.12978i −0.900969 + 3.94740i
35.1 −0.0990311 + 0.433884i −0.400969 1.75676i 1.62349 + 0.781831i −0.500000 + 0.626980i 0.801938 −3.04892 −1.05496 + 1.32288i −0.222521 + 0.107160i −0.222521 0.279032i
41.1 −1.62349 + 0.781831i 1.12349 + 0.541044i 0.777479 0.974928i −0.500000 + 2.19064i −2.24698 1.69202 0.301938 1.32288i −0.900969 1.12978i −0.900969 3.94740i
\(n\): 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.a 6
3.b odd 2 1 387.2.u.c 6
4.b odd 2 1 688.2.u.b 6
43.e even 7 1 inner 43.2.e.a 6
43.e even 7 1 1849.2.a.k 3
43.f odd 14 1 1849.2.a.j 3
129.l odd 14 1 387.2.u.c 6
172.k odd 14 1 688.2.u.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.e.a 6 1.a even 1 1 trivial
43.2.e.a 6 43.e even 7 1 inner
387.2.u.c 6 3.b odd 2 1
387.2.u.c 6 129.l odd 14 1
688.2.u.b 6 4.b odd 2 1
688.2.u.b 6 172.k odd 14 1
1849.2.a.j 3 43.f odd 14 1
1849.2.a.k 3 43.e even 7 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 9 T^{2} + 7 T^{3} + 3 T^{4} + T^{5} - T^{6} + 2 T^{7} + 12 T^{8} + 56 T^{9} + 144 T^{10} + 160 T^{11} + 64 T^{12} \)
$3$ \( 1 - 2 T + T^{2} + 4 T^{3} + 3 T^{4} - 4 T^{5} + 13 T^{6} - 12 T^{7} + 27 T^{8} + 108 T^{9} + 81 T^{10} - 486 T^{11} + 729 T^{12} \)
$5$ \( 1 + 3 T + 4 T^{2} + 18 T^{3} + 76 T^{4} + 145 T^{5} + 251 T^{6} + 725 T^{7} + 1900 T^{8} + 2250 T^{9} + 2500 T^{10} + 9375 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 + 14 T^{2} + 7 T^{3} + 98 T^{4} + 343 T^{6} )^{2} \)
$11$ \( 1 + 10 T + 26 T^{2} - 81 T^{3} - 529 T^{4} - 199 T^{5} + 3535 T^{6} - 2189 T^{7} - 64009 T^{8} - 107811 T^{9} + 380666 T^{10} + 1610510 T^{11} + 1771561 T^{12} \)
$13$ \( 1 - 5 T + 40 T^{2} - 212 T^{3} + 953 T^{4} - 4319 T^{5} + 14876 T^{6} - 56147 T^{7} + 161057 T^{8} - 465764 T^{9} + 1142440 T^{10} - 1856465 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 4 T - T^{2} - 72 T^{3} + 93 T^{4} - 42 T^{5} - 293 T^{6} - 714 T^{7} + 26877 T^{8} - 353736 T^{9} - 83521 T^{10} + 5679428 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - 2 T + 13 T^{2} + 40 T^{3} + 23 T^{4} + 1756 T^{5} - 1723 T^{6} + 33364 T^{7} + 8303 T^{8} + 274360 T^{9} + 1694173 T^{10} - 4952198 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - T - 8 T^{2} + 255 T^{3} - 57 T^{4} - 1090 T^{5} + 35133 T^{6} - 25070 T^{7} - 30153 T^{8} + 3102585 T^{9} - 2238728 T^{10} - 6436343 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 2142 T^{5} + 16003 T^{6} + 62118 T^{7} - 836795 T^{8} + 3219348 T^{9} + 4950967 T^{10} - 123066894 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 6 T - 23 T^{2} - 478 T^{3} - 1763 T^{4} + 9154 T^{5} + 96655 T^{6} + 283774 T^{7} - 1694243 T^{8} - 14240098 T^{9} - 21240983 T^{10} + 171774906 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 + 7 T + 97 T^{2} + 427 T^{3} + 3589 T^{4} + 9583 T^{5} + 50653 T^{6} )^{2} \)
$41$ \( 1 - 2 T + 19 T^{2} + 30 T^{3} - 307 T^{4} - 4018 T^{5} + 60803 T^{6} - 164738 T^{7} - 516067 T^{8} + 2067630 T^{9} + 53689459 T^{10} - 231712402 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 13 T + 141 T^{2} + 895 T^{3} + 6063 T^{4} + 24037 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 17 T + 158 T^{2} - 816 T^{3} + 3471 T^{4} - 14523 T^{5} + 108772 T^{6} - 682581 T^{7} + 7667439 T^{8} - 84719568 T^{9} + 770989598 T^{10} - 3898865119 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 2 T + 56 T^{2} + 531 T^{3} + 6599 T^{4} + 27685 T^{5} + 387913 T^{6} + 1467305 T^{7} + 18536591 T^{8} + 79053687 T^{9} + 441866936 T^{10} + 836390986 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 12 T - 20 T^{2} + 1053 T^{3} - 3791 T^{4} - 30145 T^{5} + 385979 T^{6} - 1778555 T^{7} - 13196471 T^{8} + 216264087 T^{9} - 242347220 T^{10} - 8579091588 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 12 T + 27 T^{2} - 404 T^{3} + 3229 T^{4} + 16766 T^{5} - 226185 T^{6} + 1022726 T^{7} + 12015109 T^{8} - 91700324 T^{9} + 373837707 T^{10} - 10135155612 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 18 T + 117 T^{2} + 704 T^{3} + 12085 T^{4} + 129244 T^{5} + 998865 T^{6} + 8659348 T^{7} + 54249565 T^{8} + 211737152 T^{9} + 2357681157 T^{10} + 24302251926 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 26 T + 101 T^{2} + 3378 T^{3} - 35947 T^{4} - 149374 T^{5} + 4056325 T^{6} - 10605554 T^{7} - 181208827 T^{8} + 1209023358 T^{9} + 2566579781 T^{10} - 46909963126 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 9 T - 6 T^{2} + 430 T^{3} - 1187 T^{4} - 23719 T^{5} + 416688 T^{6} - 1731487 T^{7} - 6325523 T^{8} + 167277310 T^{9} - 170389446 T^{10} + 18657644337 T^{11} + 151334226289 T^{12} \)
$79$ \( ( 1 + 10 T + 254 T^{2} + 1581 T^{3} + 20066 T^{4} + 62410 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 20 T + 72 T^{2} + 1865 T^{3} - 18531 T^{4} - 47725 T^{5} + 1743083 T^{6} - 3961175 T^{7} - 127660059 T^{8} + 1066382755 T^{9} + 3416999112 T^{10} - 78780812860 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 11 T + 81 T^{2} + 431 T^{3} + 2694 T^{4} - 97288 T^{5} + 1979249 T^{6} - 8658632 T^{7} + 21339174 T^{8} + 303841639 T^{9} + 5082121521 T^{10} - 61424653939 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 28 T + 449 T^{2} - 5950 T^{3} + 80095 T^{4} - 914662 T^{5} + 9384215 T^{6} - 88722214 T^{7} + 753613855 T^{8} - 5430404350 T^{9} + 39749647169 T^{10} - 240445527196 T^{11} + 832972004929 T^{12} \)
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