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: \(\Q(\zeta_{14})\)
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.

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

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
−0.777479 + 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

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6} \)
$3$ \( 1 - 4 T + 9 T^{2} - 8 T^{3} + 4 T^{4} - 2 T^{5} + T^{6} \)
$5$ \( 1 + 5 T + 11 T^{2} + 13 T^{3} + 9 T^{4} + 3 T^{5} + T^{6} \)
$7$ \( ( 7 - 7 T + T^{3} )^{2} \)
$11$ \( 169 + 208 T + 186 T^{2} + 62 T^{3} + 37 T^{4} + 10 T^{5} + T^{6} \)
$13$ \( 1681 + 1271 T + 121 T^{2} - 69 T^{3} + 53 T^{4} - 5 T^{5} + T^{6} \)
$17$ \( 64 + 128 T + 144 T^{2} + 64 T^{3} + 16 T^{4} + 4 T^{5} + T^{6} \)
$19$ \( 5041 - 2272 T + 289 T^{2} - 36 T^{3} + 32 T^{4} - 2 T^{5} + T^{6} \)
$23$ \( 6889 + 83 T + 127 T^{2} - 113 T^{3} + 15 T^{4} - T^{5} + T^{6} \)
$29$ \( 46656 - 7776 T + 1296 T^{2} - 216 T^{3} + 36 T^{4} - 6 T^{5} + T^{6} \)
$31$ \( 299209 - 15863 T + 5398 T^{2} + 545 T^{3} + 8 T^{4} + 6 T^{5} + T^{6} \)
$37$ \( ( -91 - 14 T + 7 T^{2} + T^{3} )^{2} \)
$41$ \( 107584 + 38048 T + 5392 T^{2} + 440 T^{3} + 60 T^{4} - 2 T^{5} + T^{6} \)
$43$ \( 79507 + 24037 T + 6063 T^{2} + 895 T^{3} + 141 T^{4} + 13 T^{5} + T^{6} \)
$47$ \( 94249 + 14429 T + 3189 T^{2} - 1427 T^{3} + 205 T^{4} - 17 T^{5} + T^{6} \)
$53$ \( 5041 + 284 T - 26 T^{2} - 370 T^{3} + 109 T^{4} + 2 T^{5} + T^{6} \)
$59$ \( 1 + 4 T + 44 T^{2} + 50 T^{3} + 39 T^{4} - 12 T^{5} + T^{6} \)
$61$ \( 64 - 192 T + 240 T^{2} - 160 T^{3} + 88 T^{4} - 12 T^{5} + T^{6} \)
$67$ \( 10816 + 9984 T + 5184 T^{2} + 1240 T^{3} + 184 T^{4} + 18 T^{5} + T^{6} \)
$71$ \( 10816 - 416 T + 3600 T^{2} + 680 T^{3} + 172 T^{4} - 26 T^{5} + T^{6} \)
$73$ \( 5041 - 213 T + 2025 T^{2} - 811 T^{3} + 67 T^{4} + 9 T^{5} + T^{6} \)
$79$ \( ( 1 + 17 T + 10 T^{2} + T^{3} )^{2} \)
$83$ \( 6889 - 9960 T + 9108 T^{2} - 874 T^{3} + 155 T^{4} - 20 T^{5} + T^{6} \)
$89$ \( 49729 + 34788 T + 17112 T^{2} - 2773 T^{3} + 170 T^{4} - 11 T^{5} + T^{6} \)
$97$ \( 247009 - 142639 T + 44884 T^{2} - 6629 T^{3} + 546 T^{4} - 28 T^{5} + T^{6} \)
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