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

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

$p$ $F_p(T)$
$2$ \( 1 - 5 T + 11 T^{2} - 13 T^{3} + 9 T^{4} - 3 T^{5} + T^{6} \)
$3$ \( 1 - 9 T + 25 T^{2} - T^{3} + 9 T^{4} + 3 T^{5} + T^{6} \)
$5$ \( 64 - 32 T + 16 T^{2} - 8 T^{3} + 4 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( ( 13 + 19 T + 8 T^{2} + T^{3} )^{2} \)
$11$ \( 49 - 98 T + 490 T^{2} - 308 T^{3} + 91 T^{4} - 14 T^{5} + T^{6} \)
$13$ \( 1 - 3 T + 9 T^{2} + T^{3} + 25 T^{4} + 9 T^{5} + T^{6} \)
$17$ \( 64 - 64 T + 64 T^{2} - 64 T^{3} + 36 T^{4} - 8 T^{5} + T^{6} \)
$19$ \( 841 + 464 T + 956 T^{2} + 78 T^{3} - 5 T^{4} + 4 T^{5} + T^{6} \)
$23$ \( 1681 + 1189 T + 337 T^{2} + 55 T^{3} + 15 T^{4} - T^{5} + T^{6} \)
$29$ \( 169 - 403 T + 373 T^{2} - 155 T^{3} + 53 T^{4} - 9 T^{5} + T^{6} \)
$31$ \( 9409 + 6596 T + 2909 T^{2} + 820 T^{3} + 156 T^{4} + 18 T^{5} + T^{6} \)
$37$ \( ( 251 - 85 T + 2 T^{2} + T^{3} )^{2} \)
$41$ \( 32761 - 18281 T + 6799 T^{2} - 1597 T^{3} + 249 T^{4} - 23 T^{5} + T^{6} \)
$43$ \( 79507 + 53621 T + 16899 T^{2} + 3221 T^{3} + 393 T^{4} + 29 T^{5} + T^{6} \)
$47$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
$53$ \( 841 + 1363 T + 1138 T^{2} + 475 T^{3} + 100 T^{4} + 10 T^{5} + T^{6} \)
$59$ \( 113569 + 20894 T + 5314 T^{2} - 1156 T^{3} + 211 T^{4} - 22 T^{5} + T^{6} \)
$61$ \( 6889 - 2075 T + 1521 T^{2} - 601 T^{3} + 151 T^{4} - 19 T^{5} + T^{6} \)
$67$ \( 169 - 390 T + 620 T^{2} + 260 T^{3} + 37 T^{4} + 4 T^{5} + T^{6} \)
$71$ \( 3136 - 3136 T + 7840 T^{2} - 2464 T^{3} + 364 T^{4} - 28 T^{5} + T^{6} \)
$73$ \( 90601 - 58996 T + 19012 T^{2} - 3395 T^{3} + 350 T^{4} - 21 T^{5} + T^{6} \)
$79$ \( ( 83 - 25 T - 3 T^{2} + T^{3} )^{2} \)
$83$ \( 851929 + 314743 T + 68303 T^{2} + 8345 T^{3} + 723 T^{4} + 39 T^{5} + T^{6} \)
$89$ \( 1413721 + 102254 T + 53260 T^{2} - 1387 T^{3} - 54 T^{4} - 11 T^{5} + T^{6} \)
$97$ \( 21077281 + 1111022 T + 89140 T^{2} + 4479 T^{3} + 242 T^{4} + 19 T^{5} + T^{6} \)
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