Properties

Label 43.3.d.a
Level 43
Weight 3
Character orbit 43.d
Analytic conductor 1.172
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.17166513675\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + \beta_{3} q^{3} + ( -2 + \beta_{1} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{4} + ( -1 - \beta_{4} - \beta_{11} ) q^{5} + ( -2 - \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{6} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{7} + ( -2 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{8} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + \beta_{3} q^{3} + ( -2 + \beta_{1} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{4} + ( -1 - \beta_{4} - \beta_{11} ) q^{5} + ( -2 - \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{6} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{7} + ( -2 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{8} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{9} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{10} + ( 5 - \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} - 2 \beta_{10} ) q^{11} + ( -1 - 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{12} + ( 2 + \beta_{1} + 2 \beta_{4} - 3 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{13} + ( 2 + 6 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 5 \beta_{7} - 5 \beta_{8} + 2 \beta_{10} + 4 \beta_{11} ) q^{14} + ( 2 + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{11} ) q^{15} + ( 6 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{9} + 3 \beta_{10} - 4 \beta_{11} ) q^{16} + ( -2 - 5 \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{10} - 3 \beta_{11} ) q^{17} + ( 6 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{18} + ( 4 - 7 \beta_{1} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{7} + 3 \beta_{8} ) q^{19} + ( -7 + 3 \beta_{1} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{20} + ( -6 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} + \beta_{10} ) q^{21} + ( -4 + 3 \beta_{1} - 5 \beta_{2} + 7 \beta_{3} + \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 12 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{22} + ( -4 - 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 5 \beta_{6} + \beta_{7} + \beta_{8} - 5 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{23} + ( -3 + 2 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + 5 \beta_{10} + 2 \beta_{11} ) q^{24} + ( -1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{25} + ( -1 + 3 \beta_{1} - 12 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + \beta_{8} + 6 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{26} + ( -2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 3 \beta_{10} - 4 \beta_{11} ) q^{27} + ( -17 + \beta_{1} + 6 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 13 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} - 8 \beta_{10} + \beta_{11} ) q^{28} + ( -16 + 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} + 10 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{29} + ( 14 + \beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{6} - 5 \beta_{7} + 2 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{30} + ( -3 - 2 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} + \beta_{4} + 14 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} ) q^{31} + ( 5 + 16 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 9 \beta_{5} + 4 \beta_{6} - 14 \beta_{7} - 10 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} + 4 \beta_{11} ) q^{32} + ( -13 + 2 \beta_{1} + 6 \beta_{3} - 6 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 7 \beta_{11} ) q^{33} + ( 21 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - 14 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} ) q^{34} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{35} + ( 1 - 6 \beta_{1} + 4 \beta_{3} + \beta_{4} - 9 \beta_{5} - 8 \beta_{6} + 32 \beta_{7} + 8 \beta_{8} - 7 \beta_{10} - 6 \beta_{11} ) q^{36} + ( 8 + 10 \beta_{1} - 3 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 6 \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{37} + ( 25 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + 14 \beta_{5} + 7 \beta_{6} - 35 \beta_{7} - 9 \beta_{8} - 7 \beta_{9} + 13 \beta_{10} + 3 \beta_{11} ) q^{38} + ( 5 - \beta_{1} - \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} - 6 \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{39} + ( -9 \beta_{1} + 14 \beta_{2} - 5 \beta_{3} + \beta_{4} - 13 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{40} + ( 11 + 2 \beta_{1} - \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 7 \beta_{5} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{41} + ( -9 - 7 \beta_{1} + 8 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 20 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{42} + ( -7 \beta_{1} - 5 \beta_{2} - 6 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + \beta_{9} - 6 \beta_{10} + \beta_{11} ) q^{43} + ( -16 - 9 \beta_{1} - 13 \beta_{2} - 2 \beta_{3} + 7 \beta_{4} - \beta_{5} + 3 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{44} + ( 14 - 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} + 6 \beta_{5} - 28 \beta_{7} ) q^{45} + ( 16 + 10 \beta_{1} - 9 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} - 6 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} - 4 \beta_{11} ) q^{46} + ( 2 + 6 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} + 11 \beta_{5} + 3 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{47} + ( -7 + 9 \beta_{1} - 9 \beta_{3} + 3 \beta_{4} - \beta_{6} - 11 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{48} + ( -22 + 9 \beta_{1} - 16 \beta_{2} - 7 \beta_{3} - \beta_{4} - 6 \beta_{5} + 3 \beta_{6} + 18 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 5 \beta_{10} + \beta_{11} ) q^{49} + ( -18 - 2 \beta_{1} + 14 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} + 2 \beta_{6} + 9 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{50} + ( -1 + \beta_{1} + 12 \beta_{2} - 13 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} + 8 \beta_{7} - 14 \beta_{8} + 3 \beta_{9} + 10 \beta_{10} + 13 \beta_{11} ) q^{51} + ( -1 - 19 \beta_{1} + 13 \beta_{2} - 7 \beta_{3} - \beta_{4} + 6 \beta_{5} + 9 \beta_{6} - 50 \beta_{7} - 3 \beta_{8} + 8 \beta_{10} + 7 \beta_{11} ) q^{52} + ( -11 - 2 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - 21 \beta_{5} + 5 \beta_{6} + 12 \beta_{7} + 4 \beta_{8} - 5 \beta_{9} - 7 \beta_{10} - 6 \beta_{11} ) q^{53} + ( 11 - 3 \beta_{1} + \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} - \beta_{9} + 3 \beta_{10} - 4 \beta_{11} ) q^{54} + ( -14 - \beta_{1} + 15 \beta_{3} - \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 11 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 7 \beta_{11} ) q^{55} + ( -8 - 47 \beta_{1} + 14 \beta_{2} + 17 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} - 3 \beta_{6} + 56 \beta_{7} + 19 \beta_{8} - 11 \beta_{10} - 19 \beta_{11} ) q^{56} + ( -5 + 9 \beta_{1} - 5 \beta_{2} + 8 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} + 9 \beta_{6} + 10 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} - \beta_{11} ) q^{57} + ( -11 - 38 \beta_{1} + 13 \beta_{2} - 7 \beta_{3} - 11 \beta_{4} - 9 \beta_{5} + 10 \beta_{6} - 38 \beta_{7} + 9 \beta_{8} - \beta_{10} - 12 \beta_{11} ) q^{58} + ( 32 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} - 3 \beta_{4} - 6 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{59} + ( 3 - 8 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{10} + 2 \beta_{11} ) q^{60} + ( 20 + 7 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} - 7 \beta_{6} - 13 \beta_{7} - 7 \beta_{8} + 14 \beta_{9} + 13 \beta_{10} + 7 \beta_{11} ) q^{61} + ( 14 - 4 \beta_{1} + 12 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 12 \beta_{7} - 6 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{62} + ( -4 + 12 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + \beta_{6} - 10 \beta_{7} - 4 \beta_{8} + \beta_{9} + \beta_{10} + 7 \beta_{11} ) q^{63} + ( -46 + 19 \beta_{1} + 16 \beta_{2} + 3 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} + 2 \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{64} + ( -13 + 7 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} - 5 \beta_{4} + 9 \beta_{5} - 2 \beta_{6} + 28 \beta_{7} + 12 \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{65} + ( -33 + 4 \beta_{1} + 8 \beta_{2} - 11 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} + 28 \beta_{7} - 9 \beta_{8} + 3 \beta_{9} + 13 \beta_{10} + 8 \beta_{11} ) q^{66} + ( 18 + 4 \beta_{1} - 24 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 7 \beta_{6} - 17 \beta_{7} - 3 \beta_{8} - 7 \beta_{9} - 9 \beta_{10} - 8 \beta_{11} ) q^{67} + ( 4 + 42 \beta_{1} - 18 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} - 15 \beta_{7} - 14 \beta_{8} + 2 \beta_{10} + 6 \beta_{11} ) q^{68} + ( 46 - 2 \beta_{1} - 10 \beta_{2} + 9 \beta_{3} - 8 \beta_{4} + 7 \beta_{5} + 2 \beta_{6} - 18 \beta_{7} + 9 \beta_{8} - 4 \beta_{9} - 12 \beta_{10} - 2 \beta_{11} ) q^{69} + ( 13 + 17 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} + 6 \beta_{4} + 5 \beta_{5} - 26 \beta_{7} - 12 \beta_{8} + 8 \beta_{10} + 8 \beta_{11} ) q^{70} + ( -7 + 5 \beta_{1} + 14 \beta_{2} - 16 \beta_{3} + 15 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} - 16 \beta_{8} + 10 \beta_{9} + 22 \beta_{10} + 5 \beta_{11} ) q^{71} + ( 21 + 4 \beta_{1} - 33 \beta_{2} - 7 \beta_{3} + 9 \beta_{4} + 13 \beta_{5} - 4 \beta_{6} - 13 \beta_{7} - 7 \beta_{8} + 8 \beta_{9} + 9 \beta_{10} + 4 \beta_{11} ) q^{72} + ( 23 + 5 \beta_{1} - 16 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} + 7 \beta_{8} + 10 \beta_{9} - 6 \beta_{10} + 5 \beta_{11} ) q^{73} + ( -36 - 3 \beta_{1} - 8 \beta_{2} + 12 \beta_{3} - 5 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} + 47 \beta_{7} + 16 \beta_{8} + 4 \beta_{9} - 18 \beta_{10} - 7 \beta_{11} ) q^{74} + ( -8 - 5 \beta_{2} + 4 \beta_{3} - 11 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} + 16 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} ) q^{75} + ( 8 + 62 \beta_{1} + 6 \beta_{3} + 16 \beta_{4} + 11 \beta_{5} - 6 \beta_{6} - 25 \beta_{7} - 17 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} + 27 \beta_{11} ) q^{76} + ( 14 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} + \beta_{6} - 4 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} - \beta_{11} ) q^{77} + ( 9 - \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - 9 \beta_{5} + 4 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{78} + ( -7 - 19 \beta_{1} + 6 \beta_{2} + 14 \beta_{3} - 7 \beta_{4} + 8 \beta_{5} + 7 \beta_{6} + 33 \beta_{7} - 2 \beta_{8} - 7 \beta_{11} ) q^{79} + ( 31 - 10 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 8 \beta_{5} + 42 \beta_{7} + 8 \beta_{8} - 11 \beta_{11} ) q^{80} + ( 18 + 3 \beta_{1} - 22 \beta_{2} + 5 \beta_{3} + \beta_{4} - 21 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} + 11 \beta_{8} + 4 \beta_{9} - 20 \beta_{10} - 8 \beta_{11} ) q^{81} + ( -22 - 11 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 16 \beta_{4} - 17 \beta_{5} + 4 \beta_{6} + 40 \beta_{7} + 28 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} - 9 \beta_{11} ) q^{82} + ( 6 - 13 \beta_{1} + 40 \beta_{2} - 6 \beta_{3} + \beta_{4} + 14 \beta_{5} + 2 \beta_{6} - 15 \beta_{7} - 10 \beta_{8} - 2 \beta_{9} + 16 \beta_{10} + 7 \beta_{11} ) q^{83} + ( 14 - 10 \beta_{1} - 5 \beta_{2} - \beta_{3} - 4 \beta_{5} - \beta_{9} + 4 \beta_{10} - 5 \beta_{11} ) q^{84} + ( -27 - 7 \beta_{1} + 11 \beta_{2} - 13 \beta_{3} - 6 \beta_{4} - 7 \beta_{5} + 8 \beta_{6} + 46 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} + 4 \beta_{11} ) q^{85} + ( 34 - 11 \beta_{1} - 11 \beta_{2} - 20 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} - 78 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} + 12 \beta_{10} - 2 \beta_{11} ) q^{86} + ( -43 - 13 \beta_{1} - 6 \beta_{2} + 9 \beta_{3} - 13 \beta_{4} - 3 \beta_{5} - 9 \beta_{9} - 2 \beta_{10} - 7 \beta_{11} ) q^{87} + ( 43 - 20 \beta_{1} + 18 \beta_{2} + 21 \beta_{3} - 6 \beta_{4} + 29 \beta_{5} + 16 \beta_{6} - 102 \beta_{7} - 4 \beta_{8} - 8 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} ) q^{88} + ( -4 + 38 \beta_{1} - 17 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 7 \beta_{9} + 7 \beta_{10} ) q^{89} + ( 12 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{11} ) q^{90} + ( -10 + 4 \beta_{1} - 15 \beta_{3} - 6 \beta_{5} - 4 \beta_{7} + 6 \beta_{8} - 6 \beta_{11} ) q^{91} + ( -86 + 28 \beta_{1} - 48 \beta_{2} - 6 \beta_{3} + 13 \beta_{4} - 11 \beta_{5} - 12 \beta_{6} + 94 \beta_{7} - 5 \beta_{8} + 12 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} ) q^{92} + ( -60 - 2 \beta_{1} + 11 \beta_{2} - 8 \beta_{3} + 6 \beta_{4} + 15 \beta_{5} + 2 \beta_{6} + 25 \beta_{7} - 8 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} - 2 \beta_{11} ) q^{93} + ( -7 - 29 \beta_{1} + 15 \beta_{2} + 25 \beta_{3} - 3 \beta_{4} + 10 \beta_{5} + 2 \beta_{6} + 12 \beta_{7} + 4 \beta_{8} - \beta_{9} - 13 \beta_{10} - 14 \beta_{11} ) q^{94} + ( 3 - 36 \beta_{1} + 13 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 11 \beta_{5} - 16 \beta_{6} - 66 \beta_{7} - 13 \beta_{10} - 10 \beta_{11} ) q^{95} + ( -46 - 6 \beta_{1} + 22 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} + 46 \beta_{7} + 7 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} + 5 \beta_{11} ) q^{96} + ( -47 + 4 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} - 26 \beta_{4} + 14 \beta_{5} + 5 \beta_{9} + 7 \beta_{10} - 2 \beta_{11} ) q^{97} + ( -38 + 19 \beta_{1} - 19 \beta_{3} + 7 \beta_{4} + 18 \beta_{5} + 5 \beta_{6} - 58 \beta_{7} - 13 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} + 25 \beta_{11} ) q^{98} + ( -11 \beta_{1} + 11 \beta_{2} - 32 \beta_{3} - 2 \beta_{5} + 11 \beta_{6} + 14 \beta_{7} - 6 \beta_{8} + 11 \beta_{10} + 11 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 26q^{4} - 3q^{5} - 9q^{6} + 9q^{7} - 12q^{9} + O(q^{10}) \) \( 12q - 26q^{4} - 3q^{5} - 9q^{6} + 9q^{7} - 12q^{9} - q^{10} + 28q^{11} - 6q^{12} + 24q^{13} - 18q^{14} - 13q^{15} + 110q^{16} - 7q^{17} + 33q^{18} + 66q^{19} - 99q^{20} - 80q^{21} - 16q^{23} - 2q^{24} - 21q^{25} + 9q^{26} - 192q^{28} - 111q^{29} + 99q^{30} - 29q^{31} - 114q^{33} + 213q^{34} + 38q^{35} + 152q^{36} + 120q^{37} + 172q^{38} - 29q^{40} + 94q^{41} + 5q^{43} - 174q^{44} + 156q^{46} - 18q^{47} - 213q^{48} - 99q^{49} - 198q^{50} - 234q^{52} - 58q^{53} + 128q^{54} - 258q^{55} + 315q^{56} + 51q^{57} - 196q^{58} + 336q^{59} - 5q^{60} + 204q^{61} + 261q^{62} - 153q^{63} - 604q^{64} - 201q^{66} + 115q^{67} - 106q^{68} + 423q^{69} - 66q^{71} + 294q^{72} + 249q^{73} - 214q^{74} - 438q^{76} + 117q^{77} + 136q^{78} + 236q^{79} + 681q^{80} + 110q^{81} - 4q^{83} + 248q^{84} + 102q^{86} - 408q^{87} - 45q^{89} - 44q^{90} - 156q^{91} - 483q^{92} - 567q^{93} - 389q^{95} - 278q^{96} - 370q^{97} - 879q^{98} + 157q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 37 x^{10} + 483 x^{8} + 2718 x^{6} + 6923 x^{4} + 7253 x^{2} + 1849\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{10} + 137 \nu^{8} + 1563 \nu^{6} + 6752 \nu^{4} + 10271 \nu^{2} + 124 \nu + 3053 \)\()/248\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{10} + 137 \nu^{8} + 1563 \nu^{6} + 6752 \nu^{4} + 10271 \nu^{2} - 124 \nu + 3053 \)\()/248\)
\(\beta_{3}\)\(=\)\((\)\(-1485 \nu^{11} + 26101 \nu^{10} - 54558 \nu^{9} + 904290 \nu^{8} - 695669 \nu^{7} + 10473897 \nu^{6} - 3624075 \nu^{5} + 46125971 \nu^{4} - 7162682 \nu^{3} + 70377670 \nu^{2} - 3213627 \nu + 18940167\)\()/810464\)
\(\beta_{4}\)\(=\)\((\)\(-1485 \nu^{11} + 25327 \nu^{10} - 54558 \nu^{9} + 861118 \nu^{8} - 695669 \nu^{7} + 9649587 \nu^{6} - 3624075 \nu^{5} + 39890025 \nu^{4} - 7162682 \nu^{3} + 55263514 \nu^{2} - 3213627 \nu + 11827709\)\()/810464\)
\(\beta_{5}\)\(=\)\((\)\(-1485 \nu^{11} - 26101 \nu^{10} - 54558 \nu^{9} - 904290 \nu^{8} - 695669 \nu^{7} - 10473897 \nu^{6} - 3624075 \nu^{5} - 46125971 \nu^{4} - 7162682 \nu^{3} - 70377670 \nu^{2} - 3213627 \nu - 18940167\)\()/810464\)
\(\beta_{6}\)\(=\)\((\)\( 1370 \nu^{11} + 12857 \nu^{10} + 48798 \nu^{9} + 441352 \nu^{8} + 595576 \nu^{7} + 5030871 \nu^{6} + 2975030 \nu^{5} + 21503999 \nu^{4} + 6488098 \nu^{3} + 31612912 \nu^{2} + 5730264 \nu + 9110281 \)\()/405232\)
\(\beta_{7}\)\(=\)\((\)\( 71 \nu^{11} + 2455 \nu^{9} + 28402 \nu^{7} + 125769 \nu^{5} + 201197 \nu^{3} + 73310 \nu + 5332 \)\()/10664\)
\(\beta_{8}\)\(=\)\((\)\(12339 \nu^{11} - 387 \nu^{10} + 422014 \nu^{9} - 21586 \nu^{8} + 4765455 \nu^{7} - 412155 \nu^{6} + 19760933 \nu^{5} - 3117973 \nu^{4} + 25470362 \nu^{3} - 7557078 \nu^{2} + 1369793 \nu - 3556229\)\()/810464\)
\(\beta_{9}\)\(=\)\((\)\(-16407 \nu^{11} + 30315 \nu^{10} - 564102 \nu^{9} + 1049286 \nu^{8} - 6440307 \nu^{7} + 12125183 \nu^{6} - 27515185 \nu^{5} + 53241869 \nu^{4} - 39401330 \nu^{3} + 82695794 \nu^{2} - 7859677 \nu + 27766433\)\()/810464\)
\(\beta_{10}\)\(=\)\((\)\(16407 \nu^{11} + 30315 \nu^{10} + 564102 \nu^{9} + 1049286 \nu^{8} + 6440307 \nu^{7} + 12125183 \nu^{6} + 27515185 \nu^{5} + 53241869 \nu^{4} + 39401330 \nu^{3} + 81885330 \nu^{2} + 7859677 \nu + 22903649\)\()/810464\)
\(\beta_{11}\)\(=\)\((\)\(8946 \nu^{11} - 8643 \nu^{10} + 309330 \nu^{9} - 296356 \nu^{8} + 3567988 \nu^{7} - 3379585 \nu^{6} + 15569630 \nu^{5} - 14590717 \nu^{4} + 23282006 \nu^{3} - 22536644 \nu^{2} + 5334036 \nu - 6970343\)\()/405232\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(-\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11} + \beta_{9} + \beta_{5} + \beta_{1} - 6\)
\(\nu^{3}\)\(=\)\(-3 \beta_{11} - 2 \beta_{10} - \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 10 \beta_{2} - 13 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(-16 \beta_{11} + 3 \beta_{10} - 13 \beta_{9} - 12 \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 15 \beta_{1} + 62\)
\(\nu^{5}\)\(=\)\(52 \beta_{11} + 38 \beta_{10} + 14 \beta_{9} - 74 \beta_{8} - 46 \beta_{7} - 28 \beta_{6} + 41 \beta_{5} + 51 \beta_{4} - 34 \beta_{3} - 124 \beta_{2} + 176 \beta_{1} + 37\)
\(\nu^{6}\)\(=\)\(227 \beta_{11} - 61 \beta_{10} + 166 \beta_{9} + 154 \beta_{5} + 10 \beta_{4} + 63 \beta_{3} - 4 \beta_{2} + 223 \beta_{1} - 774\)
\(\nu^{7}\)\(=\)\(-789 \beta_{11} - 619 \beta_{10} - 170 \beta_{9} + 1148 \beta_{8} + 934 \beta_{7} + 340 \beta_{6} - 678 \beta_{5} - 744 \beta_{4} + 515 \beta_{3} + 1631 \beta_{2} - 2420 \beta_{1} - 637\)
\(\nu^{8}\)\(=\)\(-3164 \beta_{11} + 1023 \beta_{10} - 2141 \beta_{9} - 2061 \beta_{5} - 66 \beta_{4} - 1037 \beta_{3} - 172 \beta_{2} - 3336 \beta_{1} + 10235\)
\(\nu^{9}\)\(=\)\(11644 \beta_{11} + 9675 \beta_{10} + 1969 \beta_{9} - 17120 \beta_{8} - 17240 \beta_{7} - 3938 \beta_{6} + 10488 \beta_{5} + 10529 \beta_{4} - 7747 \beta_{3} - 21934 \beta_{2} + 33578 \beta_{1} + 10589\)
\(\nu^{10}\)\(=\)\(44107 \beta_{11} - 16266 \beta_{10} + 27841 \beta_{9} + 28102 \beta_{5} + 41 \beta_{4} + 15964 \beta_{3} + 5797 \beta_{2} + 49904 \beta_{1} - 138121\)
\(\nu^{11}\)\(=\)\(-170609 \beta_{11} - 148565 \beta_{10} - 22044 \beta_{9} + 252482 \beta_{8} + 298456 \beta_{7} + 44088 \beta_{6} - 158389 \beta_{5} - 148285 \beta_{4} + 116417 \beta_{3} + 298324 \beta_{2} - 468933 \beta_{1} - 171272\)

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{7}\)

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
3.53204i
1.61947i
0.604188i
1.51156i
2.15301i
3.82317i
3.82317i
2.15301i
1.51156i
0.604188i
1.61947i
3.53204i
3.53204i 0.357493 0.206399i −8.47532 2.82830 1.63292i −0.729010 1.26268i 0.813980 + 0.469951i 15.8070i −4.41480 + 7.64666i −5.76753 9.98966i
7.2 1.61947i 2.75697 1.59174i 1.37732 −6.40200 + 3.69620i −2.57777 4.46483i 1.18578 + 0.684612i 8.70840i 0.567259 0.982521i 5.98588 + 10.3678i
7.3 0.604188i −1.35447 + 0.782004i 3.63496 4.60389 2.65806i 0.472478 + 0.818356i −0.191259 0.110424i 4.61295i −3.27694 + 5.67582i −1.60597 2.78161i
7.4 1.51156i −3.66976 + 2.11873i 1.71518 −4.51092 + 2.60438i −3.20260 5.54707i 1.23619 + 0.713712i 8.63885i 4.47807 7.75625i −3.93669 6.81854i
7.5 2.15301i 2.77779 1.60376i −0.635471 −0.468965 + 0.270757i 3.45292 + 5.98063i −7.68536 4.43714i 7.24388i 0.644090 1.11560i −0.582944 1.00969i
7.6 3.82317i −0.868030 + 0.501157i −10.6167 2.44970 1.41434i −1.91601 3.31863i 9.14067 + 5.27737i 25.2966i −3.99768 + 6.92419i 5.40725 + 9.36563i
37.1 3.82317i −0.868030 0.501157i −10.6167 2.44970 + 1.41434i −1.91601 + 3.31863i 9.14067 5.27737i 25.2966i −3.99768 6.92419i 5.40725 9.36563i
37.2 2.15301i 2.77779 + 1.60376i −0.635471 −0.468965 0.270757i 3.45292 5.98063i −7.68536 + 4.43714i 7.24388i 0.644090 + 1.11560i −0.582944 + 1.00969i
37.3 1.51156i −3.66976 2.11873i 1.71518 −4.51092 2.60438i −3.20260 + 5.54707i 1.23619 0.713712i 8.63885i 4.47807 + 7.75625i −3.93669 + 6.81854i
37.4 0.604188i −1.35447 0.782004i 3.63496 4.60389 + 2.65806i 0.472478 0.818356i −0.191259 + 0.110424i 4.61295i −3.27694 5.67582i −1.60597 + 2.78161i
37.5 1.61947i 2.75697 + 1.59174i 1.37732 −6.40200 3.69620i −2.57777 + 4.46483i 1.18578 0.684612i 8.70840i 0.567259 + 0.982521i 5.98588 10.3678i
37.6 3.53204i 0.357493 + 0.206399i −8.47532 2.82830 + 1.63292i −0.729010 + 1.26268i 0.813980 0.469951i 15.8070i −4.41480 7.64666i −5.76753 + 9.98966i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.3.d.a 12
3.b odd 2 1 387.3.j.c 12
4.b odd 2 1 688.3.t.c 12
43.d odd 6 1 inner 43.3.d.a 12
129.h even 6 1 387.3.j.c 12
172.f even 6 1 688.3.t.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.d.a 12 1.a even 1 1 trivial
43.3.d.a 12 43.d odd 6 1 inner
387.3.j.c 12 3.b odd 2 1
387.3.j.c 12 129.h even 6 1
688.3.t.c 12 4.b odd 2 1
688.3.t.c 12 172.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 11 T^{2} + 59 T^{4} - 178 T^{6} + 635 T^{8} - 4155 T^{10} + 21681 T^{12} - 66480 T^{14} + 162560 T^{16} - 729088 T^{18} + 3866624 T^{20} - 11534336 T^{22} + 16777216 T^{24} \)
$3$ \( 1 + 33 T^{2} + 544 T^{4} - 3 T^{5} + 6224 T^{6} + 2205 T^{7} + 59278 T^{8} + 74400 T^{9} + 537073 T^{10} + 1154382 T^{11} + 4828798 T^{12} + 10389438 T^{13} + 43502913 T^{14} + 54237600 T^{15} + 388922958 T^{16} + 130203045 T^{17} + 3307688784 T^{18} - 14348907 T^{19} + 23417416224 T^{20} + 115063885233 T^{22} + 282429536481 T^{24} \)
$5$ \( 1 + 3 T + 90 T^{2} + 261 T^{3} + 3868 T^{4} + 16263 T^{5} + 135026 T^{6} + 750009 T^{7} + 4440152 T^{8} + 24002067 T^{9} + 135845122 T^{10} + 636745773 T^{11} + 3708044426 T^{12} + 15918644325 T^{13} + 84903201250 T^{14} + 375032296875 T^{15} + 1734434375000 T^{16} + 7324306640625 T^{17} + 32965332031250 T^{18} + 99261474609375 T^{19} + 590209960937500 T^{20} + 995635986328125 T^{21} + 8583068847656250 T^{22} + 7152557373046875 T^{23} + 59604644775390625 T^{24} \)
$7$ \( 1 - 9 T + 237 T^{2} - 1890 T^{3} + 29841 T^{4} - 211941 T^{5} + 2317384 T^{6} - 15083235 T^{7} + 123904335 T^{8} - 778414590 T^{9} + 4927316019 T^{10} - 34388074575 T^{11} + 204998231790 T^{12} - 1685015654175 T^{13} + 11830485761619 T^{14} - 91579698098910 T^{15} + 714283834312335 T^{16} - 4260640562350515 T^{17} + 32075577499002184 T^{18} - 143743276282689909 T^{19} + 991703881127463441 T^{20} - 3077701700050748610 T^{21} + 18910767112534044237 T^{22} - 35188389437246892441 T^{23} + \)\(19\!\cdots\!01\)\( T^{24} \)
$11$ \( ( 1 - 14 T + 430 T^{2} - 6047 T^{3} + 101321 T^{4} - 1177595 T^{5} + 15292736 T^{6} - 142488995 T^{7} + 1483440761 T^{8} - 10712629367 T^{9} + 92174318830 T^{10} - 363123944414 T^{11} + 3138428376721 T^{12} )^{2} \)
$13$ \( 1 - 24 T - 211 T^{2} + 7466 T^{3} + 36644 T^{4} - 1139421 T^{5} - 11023890 T^{6} + 164450517 T^{7} + 1885659940 T^{8} - 21544092258 T^{9} - 166029500453 T^{10} + 1118804167120 T^{11} + 20761074178322 T^{12} + 189077904243280 T^{13} - 4741968562438133 T^{14} - 103989218407744722 T^{15} + 1538190742417016740 T^{16} + 22670900257408335933 T^{17} - \)\(25\!\cdots\!90\)\( T^{18} - \)\(44\!\cdots\!69\)\( T^{19} + \)\(24\!\cdots\!04\)\( T^{20} + \)\(83\!\cdots\!14\)\( T^{21} - \)\(40\!\cdots\!11\)\( T^{22} - \)\(77\!\cdots\!56\)\( T^{23} + \)\(54\!\cdots\!61\)\( T^{24} \)
$17$ \( 1 + 7 T - 910 T^{2} - 8453 T^{3} + 343548 T^{4} + 3893616 T^{5} - 110794242 T^{6} - 1195653416 T^{7} + 42885403526 T^{8} + 359068799653 T^{9} - 11658350660948 T^{10} - 53405218008175 T^{11} + 2646840415582786 T^{12} - 15434108004362575 T^{13} - 973717105553037908 T^{14} + 8667047927371463557 T^{15} + \)\(29\!\cdots\!66\)\( T^{16} - \)\(24\!\cdots\!84\)\( T^{17} - \)\(64\!\cdots\!62\)\( T^{18} + \)\(65\!\cdots\!64\)\( T^{19} + \)\(16\!\cdots\!88\)\( T^{20} - \)\(11\!\cdots\!77\)\( T^{21} - \)\(36\!\cdots\!10\)\( T^{22} + \)\(82\!\cdots\!23\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \)
$19$ \( 1 - 66 T + 2608 T^{2} - 76296 T^{3} + 1836036 T^{4} - 38733966 T^{5} + 734123972 T^{6} - 14185527162 T^{7} + 292458472820 T^{8} - 6360444714264 T^{9} + 137883926411448 T^{10} - 2871978163512918 T^{11} + 56971930639346710 T^{12} - 1036784117028163398 T^{13} + 17969171173866314808 T^{14} - \)\(29\!\cdots\!84\)\( T^{15} + \)\(49\!\cdots\!20\)\( T^{16} - \)\(86\!\cdots\!62\)\( T^{17} + \)\(16\!\cdots\!92\)\( T^{18} - \)\(30\!\cdots\!86\)\( T^{19} + \)\(52\!\cdots\!16\)\( T^{20} - \)\(79\!\cdots\!36\)\( T^{21} + \)\(98\!\cdots\!08\)\( T^{22} - \)\(89\!\cdots\!26\)\( T^{23} + \)\(48\!\cdots\!21\)\( T^{24} \)
$23$ \( 1 + 16 T - 995 T^{2} - 970 T^{3} + 448208 T^{4} - 9378727 T^{5} + 41040474 T^{6} + 4400241755 T^{7} - 92523779140 T^{8} + 750281486226 T^{9} - 30397457804233 T^{10} - 745325850079516 T^{11} + 45354493455383514 T^{12} - 394277374692063964 T^{13} - 8506454989394366953 T^{14} + \)\(11\!\cdots\!14\)\( T^{15} - \)\(72\!\cdots\!40\)\( T^{16} + \)\(18\!\cdots\!95\)\( T^{17} + \)\(89\!\cdots\!54\)\( T^{18} - \)\(10\!\cdots\!43\)\( T^{19} + \)\(27\!\cdots\!88\)\( T^{20} - \)\(31\!\cdots\!30\)\( T^{21} - \)\(17\!\cdots\!95\)\( T^{22} + \)\(14\!\cdots\!64\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( 1 + 111 T + 7375 T^{2} + 362748 T^{3} + 14016537 T^{4} + 490234341 T^{5} + 16991400860 T^{6} + 618781423887 T^{7} + 22589952693419 T^{8} + 755597311945764 T^{9} + 22852874993878941 T^{10} + 643686573544215837 T^{11} + 18176844555197488294 T^{12} + \)\(54\!\cdots\!17\)\( T^{13} + \)\(16\!\cdots\!21\)\( T^{14} + \)\(44\!\cdots\!44\)\( T^{15} + \)\(11\!\cdots\!59\)\( T^{16} + \)\(26\!\cdots\!87\)\( T^{17} + \)\(60\!\cdots\!60\)\( T^{18} + \)\(14\!\cdots\!21\)\( T^{19} + \)\(35\!\cdots\!77\)\( T^{20} + \)\(76\!\cdots\!28\)\( T^{21} + \)\(13\!\cdots\!75\)\( T^{22} + \)\(16\!\cdots\!51\)\( T^{23} + \)\(12\!\cdots\!81\)\( T^{24} \)
$31$ \( 1 + 29 T - 2854 T^{2} - 122293 T^{3} + 3710916 T^{4} + 236687409 T^{5} - 1692705294 T^{6} - 282476528401 T^{7} - 2178393607912 T^{8} + 217078922116565 T^{9} + 5420523596702458 T^{10} - 78231274595814749 T^{11} - 6486151840040562614 T^{12} - 75180254886577973789 T^{13} + \)\(50\!\cdots\!18\)\( T^{14} + \)\(19\!\cdots\!65\)\( T^{15} - \)\(18\!\cdots\!92\)\( T^{16} - \)\(23\!\cdots\!01\)\( T^{17} - \)\(13\!\cdots\!34\)\( T^{18} + \)\(17\!\cdots\!89\)\( T^{19} + \)\(26\!\cdots\!96\)\( T^{20} - \)\(85\!\cdots\!13\)\( T^{21} - \)\(19\!\cdots\!54\)\( T^{22} + \)\(18\!\cdots\!69\)\( T^{23} + \)\(62\!\cdots\!21\)\( T^{24} \)
$37$ \( 1 - 120 T + 13269 T^{2} - 1016280 T^{3} + 71816674 T^{4} - 4159036083 T^{5} + 226460485550 T^{6} - 10769149077249 T^{7} + 489905206154870 T^{8} - 20187174180639120 T^{9} + 817471660147259059 T^{10} - 30887683233450226884 T^{11} + \)\(11\!\cdots\!42\)\( T^{12} - \)\(42\!\cdots\!96\)\( T^{13} + \)\(15\!\cdots\!99\)\( T^{14} - \)\(51\!\cdots\!80\)\( T^{15} + \)\(17\!\cdots\!70\)\( T^{16} - \)\(51\!\cdots\!01\)\( T^{17} + \)\(14\!\cdots\!50\)\( T^{18} - \)\(37\!\cdots\!87\)\( T^{19} + \)\(88\!\cdots\!34\)\( T^{20} - \)\(17\!\cdots\!20\)\( T^{21} + \)\(30\!\cdots\!69\)\( T^{22} - \)\(37\!\cdots\!80\)\( T^{23} + \)\(43\!\cdots\!61\)\( T^{24} \)
$41$ \( ( 1 - 47 T + 6771 T^{2} - 336496 T^{3} + 24096797 T^{4} - 992715921 T^{5} + 52240608766 T^{6} - 1668755463201 T^{7} + 68091789187517 T^{8} - 1598391076679536 T^{9} + 54065928726378291 T^{10} - 630864987577162847 T^{11} + 22563490300366186081 T^{12} )^{2} \)
$43$ \( 1 - 5 T + 235 T^{2} + 31132 T^{3} + 5312177 T^{4} + 54939337 T^{5} + 786324230 T^{6} + 101582834113 T^{7} + 18161276039777 T^{8} + 196796674441468 T^{9} + 2746727065236235 T^{10} - 108057411566421245 T^{11} + 39959630797262576401 T^{12} \)
$47$ \( ( 1 + 9 T + 7161 T^{2} - 54567 T^{3} + 24318411 T^{4} - 403881570 T^{5} + 60415020470 T^{6} - 892174388130 T^{7} + 118666088106891 T^{8} - 588189442857543 T^{9} + 170512623784870521 T^{10} + 473392190122470441 T^{11} + \)\(11\!\cdots\!41\)\( T^{12} )^{2} \)
$53$ \( 1 + 58 T - 10011 T^{2} - 489700 T^{3} + 62208382 T^{4} + 2298288439 T^{5} - 284627569714 T^{6} - 7541357242599 T^{7} + 1029571029628954 T^{8} + 16839043965386300 T^{9} - 3214742139372951389 T^{10} - 18444106212599549754 T^{11} + \)\(92\!\cdots\!42\)\( T^{12} - \)\(51\!\cdots\!86\)\( T^{13} - \)\(25\!\cdots\!09\)\( T^{14} + \)\(37\!\cdots\!00\)\( T^{15} + \)\(64\!\cdots\!94\)\( T^{16} - \)\(13\!\cdots\!51\)\( T^{17} - \)\(13\!\cdots\!74\)\( T^{18} + \)\(31\!\cdots\!91\)\( T^{19} + \)\(24\!\cdots\!22\)\( T^{20} - \)\(53\!\cdots\!00\)\( T^{21} - \)\(30\!\cdots\!11\)\( T^{22} + \)\(49\!\cdots\!22\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \)
$59$ \( ( 1 - 168 T + 29086 T^{2} - 3029785 T^{3} + 294619625 T^{4} - 21440752319 T^{5} + 1430753245424 T^{6} - 74635258822439 T^{7} + 3570012353809625 T^{8} - 127797948117497185 T^{9} + 4270710108159280606 T^{10} - 85867614554507755368 T^{11} + \)\(17\!\cdots\!81\)\( T^{12} )^{2} \)
$61$ \( 1 - 204 T + 33131 T^{2} - 3928836 T^{3} + 396166698 T^{4} - 36077954385 T^{5} + 3074787876040 T^{6} - 249044419787289 T^{7} + 19373960824471256 T^{8} - 1402697797539370524 T^{9} + 96292769362924730019 T^{10} - \)\(62\!\cdots\!62\)\( T^{11} + \)\(38\!\cdots\!74\)\( T^{12} - \)\(23\!\cdots\!02\)\( T^{13} + \)\(13\!\cdots\!79\)\( T^{14} - \)\(72\!\cdots\!64\)\( T^{15} + \)\(37\!\cdots\!36\)\( T^{16} - \)\(17\!\cdots\!89\)\( T^{17} + \)\(81\!\cdots\!40\)\( T^{18} - \)\(35\!\cdots\!85\)\( T^{19} + \)\(14\!\cdots\!78\)\( T^{20} - \)\(53\!\cdots\!16\)\( T^{21} + \)\(16\!\cdots\!31\)\( T^{22} - \)\(38\!\cdots\!84\)\( T^{23} + \)\(70\!\cdots\!41\)\( T^{24} \)
$67$ \( 1 - 115 T - 7960 T^{2} + 1601411 T^{3} + 12878544 T^{4} - 11631792126 T^{5} + 264032039310 T^{6} + 51372624149438 T^{7} - 2607440286917038 T^{8} - 136422464608902859 T^{9} + 13476394220383696006 T^{10} + \)\(18\!\cdots\!27\)\( T^{11} - \)\(61\!\cdots\!22\)\( T^{12} + \)\(84\!\cdots\!03\)\( T^{13} + \)\(27\!\cdots\!26\)\( T^{14} - \)\(12\!\cdots\!71\)\( T^{15} - \)\(10\!\cdots\!58\)\( T^{16} + \)\(93\!\cdots\!62\)\( T^{17} + \)\(21\!\cdots\!10\)\( T^{18} - \)\(42\!\cdots\!54\)\( T^{19} + \)\(21\!\cdots\!64\)\( T^{20} + \)\(11\!\cdots\!99\)\( T^{21} - \)\(26\!\cdots\!60\)\( T^{22} - \)\(17\!\cdots\!35\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \)
$71$ \( 1 + 66 T + 8383 T^{2} + 457446 T^{3} + 22253522 T^{4} + 2686038483 T^{5} - 142137724894 T^{6} + 6765531085149 T^{7} - 469326810763006 T^{8} + 20521634467403010 T^{9} + 5562036689541880365 T^{10} + \)\(10\!\cdots\!58\)\( T^{11} + \)\(64\!\cdots\!18\)\( T^{12} + \)\(54\!\cdots\!78\)\( T^{13} + \)\(14\!\cdots\!65\)\( T^{14} + \)\(26\!\cdots\!10\)\( T^{15} - \)\(30\!\cdots\!66\)\( T^{16} + \)\(22\!\cdots\!49\)\( T^{17} - \)\(23\!\cdots\!54\)\( T^{18} + \)\(22\!\cdots\!23\)\( T^{19} + \)\(92\!\cdots\!62\)\( T^{20} + \)\(96\!\cdots\!06\)\( T^{21} + \)\(88\!\cdots\!83\)\( T^{22} + \)\(35\!\cdots\!06\)\( T^{23} + \)\(26\!\cdots\!81\)\( T^{24} \)
$73$ \( 1 - 249 T + 47836 T^{2} - 6765081 T^{3} + 836510852 T^{4} - 87998072508 T^{5} + 8327492192822 T^{6} - 712665936231156 T^{7} + 56395869792302342 T^{8} - 4210361193162399351 T^{9} + \)\(30\!\cdots\!94\)\( T^{10} - \)\(21\!\cdots\!91\)\( T^{11} + \)\(15\!\cdots\!38\)\( T^{12} - \)\(11\!\cdots\!39\)\( T^{13} + \)\(85\!\cdots\!54\)\( T^{14} - \)\(63\!\cdots\!39\)\( T^{15} + \)\(45\!\cdots\!02\)\( T^{16} - \)\(30\!\cdots\!44\)\( T^{17} + \)\(19\!\cdots\!62\)\( T^{18} - \)\(10\!\cdots\!72\)\( T^{19} + \)\(54\!\cdots\!72\)\( T^{20} - \)\(23\!\cdots\!89\)\( T^{21} + \)\(88\!\cdots\!36\)\( T^{22} - \)\(24\!\cdots\!21\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \)
$79$ \( 1 - 236 T + 6211 T^{2} + 1816934 T^{3} + 15618410 T^{4} - 18039764431 T^{5} - 862854134850 T^{6} + 144012352794143 T^{7} + 12282532151020286 T^{8} - 938070438536680854 T^{9} - 93564616453312180819 T^{10} + \)\(13\!\cdots\!72\)\( T^{11} + \)\(78\!\cdots\!82\)\( T^{12} + \)\(84\!\cdots\!52\)\( T^{13} - \)\(36\!\cdots\!39\)\( T^{14} - \)\(22\!\cdots\!34\)\( T^{15} + \)\(18\!\cdots\!46\)\( T^{16} + \)\(13\!\cdots\!43\)\( T^{17} - \)\(50\!\cdots\!50\)\( T^{18} - \)\(66\!\cdots\!11\)\( T^{19} + \)\(35\!\cdots\!10\)\( T^{20} + \)\(26\!\cdots\!74\)\( T^{21} + \)\(55\!\cdots\!11\)\( T^{22} - \)\(13\!\cdots\!76\)\( T^{23} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( 1 + 4 T - 21815 T^{2} + 1032278 T^{3} + 256215014 T^{4} - 22099127125 T^{5} - 1301303064984 T^{6} + 269522473356119 T^{7} - 2312375630136352 T^{8} - 1824448248468292326 T^{9} + \)\(11\!\cdots\!01\)\( T^{10} + \)\(55\!\cdots\!10\)\( T^{11} - \)\(10\!\cdots\!42\)\( T^{12} + \)\(37\!\cdots\!90\)\( T^{13} + \)\(54\!\cdots\!21\)\( T^{14} - \)\(59\!\cdots\!94\)\( T^{15} - \)\(52\!\cdots\!32\)\( T^{16} + \)\(41\!\cdots\!31\)\( T^{17} - \)\(13\!\cdots\!24\)\( T^{18} - \)\(16\!\cdots\!25\)\( T^{19} + \)\(12\!\cdots\!34\)\( T^{20} + \)\(36\!\cdots\!02\)\( T^{21} - \)\(52\!\cdots\!15\)\( T^{22} + \)\(66\!\cdots\!56\)\( T^{23} + \)\(11\!\cdots\!21\)\( T^{24} \)
$89$ \( 1 + 45 T + 19568 T^{2} + 850185 T^{3} + 135178944 T^{4} + 15768566256 T^{5} + 877991398306 T^{6} + 230067075716784 T^{7} + 11948725877253974 T^{8} + 1772692905496764327 T^{9} + \)\(15\!\cdots\!42\)\( T^{10} + \)\(84\!\cdots\!51\)\( T^{11} + \)\(14\!\cdots\!90\)\( T^{12} + \)\(67\!\cdots\!71\)\( T^{13} + \)\(96\!\cdots\!22\)\( T^{14} + \)\(88\!\cdots\!47\)\( T^{15} + \)\(47\!\cdots\!94\)\( T^{16} + \)\(71\!\cdots\!84\)\( T^{17} + \)\(21\!\cdots\!26\)\( T^{18} + \)\(30\!\cdots\!96\)\( T^{19} + \)\(20\!\cdots\!84\)\( T^{20} + \)\(10\!\cdots\!85\)\( T^{21} + \)\(19\!\cdots\!68\)\( T^{22} + \)\(34\!\cdots\!45\)\( T^{23} + \)\(61\!\cdots\!41\)\( T^{24} \)
$97$ \( ( 1 + 185 T + 45810 T^{2} + 6039385 T^{3} + 922201559 T^{4} + 96415612878 T^{5} + 11078943083116 T^{6} + 907174501569102 T^{7} + 81641840955349079 T^{8} + 5030638631988128665 T^{9} + \)\(35\!\cdots\!10\)\( T^{10} + \)\(13\!\cdots\!65\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \)
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