Properties

Label 105.3.c.a
Level 105
Weight 3
Character orbit 105.c
Analytic conductor 2.861
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 105.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{4} ) q^{3} + ( -2 + \beta_{7} - \beta_{8} ) q^{4} + \beta_{9} q^{5} + ( -3 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} - \beta_{9} + \beta_{13} - \beta_{15} ) q^{6} -\beta_{8} q^{7} + ( -1 - 3 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{9} - \beta_{11} + \beta_{13} ) q^{8} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{4} ) q^{3} + ( -2 + \beta_{7} - \beta_{8} ) q^{4} + \beta_{9} q^{5} + ( -3 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} - \beta_{9} + \beta_{13} - \beta_{15} ) q^{6} -\beta_{8} q^{7} + ( -1 - 3 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{9} - \beta_{11} + \beta_{13} ) q^{8} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{9} + ( -1 - \beta_{4} + \beta_{5} ) q^{10} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{9} + 2 \beta_{11} + 2 \beta_{12} ) q^{11} + ( -3 \beta_{1} + 2 \beta_{4} - \beta_{5} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{14} ) q^{12} + ( 1 + \beta_{2} - \beta_{4} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{13} + ( -2 \beta_{1} - \beta_{2} - \beta_{11} ) q^{14} + ( \beta_{5} + \beta_{9} + \beta_{13} - \beta_{15} ) q^{15} + ( 4 + 3 \beta_{1} + \beta_{2} + 2 \beta_{4} + 4 \beta_{5} + \beta_{6} + 6 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{16} + ( 2 + \beta_{2} - 2 \beta_{6} + \beta_{7} - 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{17} + ( -4 + \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{18} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{19} + ( -\beta_{2} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{20} + ( -1 - \beta_{7} + \beta_{14} ) q^{21} + ( 1 - 2 \beta_{1} + \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{11} - 3 \beta_{13} + 2 \beta_{14} ) q^{22} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{7} + 3 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{23} + ( 9 + 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + 4 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} - 3 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{24} -5 q^{25} + ( -2 + 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} + 4 \beta_{9} + 4 \beta_{11} + 4 \beta_{12} - 4 \beta_{15} ) q^{26} + ( -6 - 7 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 4 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{27} + ( 6 + 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{28} + ( 1 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{29} + ( 4 + 3 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} ) q^{30} + ( -4 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - 3 \beta_{10} - \beta_{11} - \beta_{12} + 4 \beta_{13} - \beta_{14} ) q^{31} + ( 2 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 8 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{32} + ( 1 + 3 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - 5 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} + 7 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} ) q^{33} + ( -8 - 5 \beta_{1} - \beta_{2} - 4 \beta_{4} - 8 \beta_{5} - \beta_{6} + 3 \beta_{7} - 9 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} + \beta_{11} - \beta_{12} + 3 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{34} + ( \beta_{12} - \beta_{13} ) q^{35} + ( -2 - 10 \beta_{1} - 2 \beta_{2} + \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 9 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{36} + ( 3 - 4 \beta_{1} + \beta_{3} - 7 \beta_{4} - \beta_{6} - 7 \beta_{7} - 9 \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{13} + 2 \beta_{15} ) q^{37} + ( 1 - 3 \beta_{2} - 4 \beta_{4} + 3 \beta_{6} + \beta_{7} - 10 \beta_{9} + \beta_{10} + 7 \beta_{12} - 5 \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{38} + ( 4 + 9 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 4 \beta_{9} - \beta_{10} + 3 \beta_{12} + \beta_{13} - \beta_{15} ) q^{39} + ( -5 - \beta_{1} - \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{40} + ( 4 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{41} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{42} + ( 15 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{4} + 3 \beta_{6} - \beta_{7} + 5 \beta_{8} - 4 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{15} ) q^{43} + ( 2 - 5 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 4 \beta_{7} + 9 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} - 3 \beta_{12} + \beta_{13} + 4 \beta_{14} + 9 \beta_{15} ) q^{44} + ( 3 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{45} + ( 6 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} - 12 \beta_{4} + 12 \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} - 4 \beta_{10} - 3 \beta_{11} - \beta_{12} + 7 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{46} + ( 2 - 14 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + 14 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{47} + ( -7 + 11 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 10 \beta_{9} + 7 \beta_{10} + 5 \beta_{11} + 5 \beta_{12} - 5 \beta_{13} - 5 \beta_{14} ) q^{48} + 7 q^{49} -5 \beta_{1} q^{50} + ( 4 - 8 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 6 \beta_{8} - 12 \beta_{9} + 7 \beta_{10} - 7 \beta_{11} + 4 \beta_{12} - 8 \beta_{13} - 2 \beta_{14} ) q^{51} + ( -12 + 4 \beta_{1} - 2 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{52} + ( 3 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 4 \beta_{6} - 9 \beta_{9} - 7 \beta_{11} - 6 \beta_{12} - \beta_{13} + 6 \beta_{15} ) q^{53} + ( 9 - 13 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 6 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + 9 \beta_{13} - \beta_{14} - \beta_{15} ) q^{54} + ( 7 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{7} + 7 \beta_{8} - \beta_{9} + 6 \beta_{10} - \beta_{11} + 3 \beta_{12} - 5 \beta_{13} + 2 \beta_{15} ) q^{55} + ( 2 + 9 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - 4 \beta_{9} + \beta_{10} - \beta_{11} - 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{56} + ( -1 + 4 \beta_{1} + 7 \beta_{3} + 5 \beta_{4} - \beta_{6} - \beta_{7} + 15 \beta_{8} + 6 \beta_{9} + 8 \beta_{10} - \beta_{11} + 2 \beta_{12} - 11 \beta_{13} - 4 \beta_{15} ) q^{57} + ( -11 + 6 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 11 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} - \beta_{11} - 6 \beta_{12} + 9 \beta_{13} + 4 \beta_{14} - 8 \beta_{15} ) q^{58} + ( -10 + 14 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} + 9 \beta_{4} + \beta_{6} + \beta_{7} - 13 \beta_{9} + \beta_{10} + 5 \beta_{11} + 3 \beta_{12} + 8 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{59} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 6 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + \beta_{15} ) q^{60} + ( 2 - 9 \beta_{1} + 3 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + \beta_{6} - 5 \beta_{7} + 15 \beta_{8} + 2 \beta_{9} + 8 \beta_{10} + 5 \beta_{11} + 3 \beta_{12} - 13 \beta_{13} + 2 \beta_{14} + 5 \beta_{15} ) q^{61} + ( -8 + 8 \beta_{1} - \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 3 \beta_{6} - \beta_{7} + 15 \beta_{9} - \beta_{10} + 9 \beta_{11} + \beta_{12} + 10 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{62} + ( -6 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{63} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 8 \beta_{5} + 5 \beta_{7} + 3 \beta_{8} + 4 \beta_{10} + 2 \beta_{12} - 4 \beta_{13} + 2 \beta_{15} ) q^{64} + ( -3 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{6} - 3 \beta_{11} - \beta_{12} + 3 \beta_{13} - \beta_{15} ) q^{65} + ( -18 - 11 \beta_{1} - 6 \beta_{2} - 5 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 21 \beta_{8} + 13 \beta_{9} - 7 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + 7 \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{66} + ( -13 + 8 \beta_{1} - 5 \beta_{3} - \beta_{4} + 16 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + \beta_{8} - 6 \beta_{9} + 4 \beta_{10} - 5 \beta_{11} + \beta_{13} + 4 \beta_{14} - 6 \beta_{15} ) q^{67} + ( -11 - 32 \beta_{1} - 16 \beta_{2} + 6 \beta_{3} + 11 \beta_{4} + 23 \beta_{9} - 5 \beta_{11} + 11 \beta_{13} - 6 \beta_{15} ) q^{68} + ( -9 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 10 \beta_{5} + \beta_{6} + 3 \beta_{7} + 13 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{69} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{70} + ( 2 - 12 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 14 \beta_{12} + 12 \beta_{13} + 2 \beta_{15} ) q^{71} + ( 21 + 9 \beta_{1} - 8 \beta_{2} - \beta_{3} - 3 \beta_{4} + 8 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} + 13 \beta_{8} - 20 \beta_{9} + 8 \beta_{10} - \beta_{11} + 10 \beta_{12} + \beta_{13} - 6 \beta_{14} - 4 \beta_{15} ) q^{72} + ( -14 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} - 10 \beta_{8} - \beta_{9} + 5 \beta_{10} - 3 \beta_{11} + 5 \beta_{12} - 2 \beta_{13} - 5 \beta_{14} + 4 \beta_{15} ) q^{73} + ( 2 + 3 \beta_{1} - 11 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 21 \beta_{9} + 2 \beta_{10} - 9 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - 7 \beta_{15} ) q^{74} + ( -5 + 5 \beta_{4} ) q^{75} + ( -6 + 3 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} + 27 \beta_{4} - 16 \beta_{5} + 12 \beta_{7} + 15 \beta_{8} - \beta_{9} - 9 \beta_{10} + 6 \beta_{11} - 8 \beta_{12} + 3 \beta_{13} + 7 \beta_{14} - 9 \beta_{15} ) q^{76} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{6} + 4 \beta_{9} + \beta_{11} - \beta_{13} + 3 \beta_{15} ) q^{77} + ( -18 + 22 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} - 2 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} + 18 \beta_{9} - 4 \beta_{10} + 10 \beta_{11} + 2 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{78} + ( -19 + 13 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 16 \beta_{5} + \beta_{6} + 6 \beta_{7} - 6 \beta_{9} + 2 \beta_{10} - 9 \beta_{11} + 3 \beta_{12} + 7 \beta_{13} - 4 \beta_{14} - 3 \beta_{15} ) q^{79} + ( 3 - 14 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{7} + 3 \beta_{9} + \beta_{10} - 7 \beta_{11} - 7 \beta_{12} - \beta_{14} + 5 \beta_{15} ) q^{80} + ( 28 - 10 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} - 11 \beta_{8} - 3 \beta_{10} - 11 \beta_{11} - 4 \beta_{12} + 4 \beta_{15} ) q^{81} + ( -18 - 6 \beta_{1} - \beta_{2} + 2 \beta_{3} - 13 \beta_{4} + \beta_{6} + 3 \beta_{7} - 22 \beta_{8} + 5 \beta_{9} - 5 \beta_{10} + 3 \beta_{11} - \beta_{12} + 2 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{82} + ( 8 - 10 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} - 13 \beta_{4} + 5 \beta_{6} + \beta_{7} + 7 \beta_{9} + \beta_{10} - 9 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{83} + ( -1 + 6 \beta_{1} + 3 \beta_{2} - \beta_{3} - 9 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 3 \beta_{8} - 8 \beta_{9} + 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{84} + ( 11 - 4 \beta_{1} - 2 \beta_{2} - 8 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{85} + ( 8 + 37 \beta_{1} + 11 \beta_{2} - 5 \beta_{3} - 8 \beta_{4} - 9 \beta_{9} + 3 \beta_{11} + \beta_{12} - 9 \beta_{13} + 5 \beta_{15} ) q^{86} + ( -25 + 7 \beta_{1} + 3 \beta_{2} - 10 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - \beta_{6} + 11 \beta_{7} + 8 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 5 \beta_{11} + 8 \beta_{12} - 8 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} ) q^{87} + ( 31 - 10 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 37 \beta_{4} + 16 \beta_{5} - 3 \beta_{6} - 15 \beta_{7} - \beta_{8} + 4 \beta_{9} + 8 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} - 5 \beta_{13} - 4 \beta_{14} + 10 \beta_{15} ) q^{88} + ( 22 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} - 7 \beta_{4} + 7 \beta_{6} + \beta_{7} + 11 \beta_{9} + \beta_{10} + 17 \beta_{11} - \beta_{12} + 8 \beta_{13} - \beta_{14} - 14 \beta_{15} ) q^{89} + ( -6 + 3 \beta_{1} + 4 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 11 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} - \beta_{14} + \beta_{15} ) q^{90} + ( -3 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + \beta_{6} + 4 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} - \beta_{11} + 3 \beta_{12} - 5 \beta_{13} + \beta_{15} ) q^{91} + ( -9 + 32 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} - 39 \beta_{9} + 4 \beta_{10} + 15 \beta_{11} + 16 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - 10 \beta_{15} ) q^{92} + ( 6 - 4 \beta_{1} - 5 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 5 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} - 8 \beta_{10} - 4 \beta_{11} + 6 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{93} + ( 68 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 9 \beta_{4} + 2 \beta_{5} - 10 \beta_{7} - \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{94} + ( 2 - 11 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 4 \beta_{6} - \beta_{9} - 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{95} + ( 7 - 9 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} + 5 \beta_{4} - 14 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} - 16 \beta_{8} - 17 \beta_{9} + 6 \beta_{10} + 4 \beta_{12} - 15 \beta_{13} - \beta_{15} ) q^{96} + ( 43 + 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 6 \beta_{5} + 5 \beta_{6} - 6 \beta_{7} - 12 \beta_{8} - \beta_{9} - 6 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{97} + 7 \beta_{1} q^{98} + ( -15 - 17 \beta_{1} - 13 \beta_{2} + \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + 6 \beta_{7} + 5 \beta_{8} + 33 \beta_{9} - \beta_{10} + 6 \beta_{11} + 9 \beta_{12} - 5 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{3} - 28q^{4} - 28q^{6} + 22q^{9} + O(q^{10}) \) \( 16q + 8q^{3} - 28q^{4} - 28q^{6} + 22q^{9} - 20q^{10} + 12q^{12} + 10q^{15} + 92q^{16} - 52q^{18} - 16q^{19} - 14q^{21} + 16q^{22} + 128q^{24} - 80q^{25} - 148q^{27} + 112q^{28} + 80q^{30} - 72q^{31} - 4q^{33} - 176q^{34} - 76q^{36} - 40q^{37} + 90q^{39} - 60q^{40} + 280q^{43} + 40q^{45} + 72q^{46} - 172q^{48} + 112q^{49} + 38q^{51} - 88q^{52} + 208q^{54} + 80q^{55} - 36q^{57} - 24q^{58} - 80q^{60} - 56q^{61} - 56q^{63} - 44q^{64} - 260q^{66} - 120q^{67} + 60q^{69} + 376q^{72} - 208q^{73} - 40q^{75} + 144q^{76} - 228q^{78} - 204q^{79} + 458q^{81} - 384q^{82} - 84q^{84} + 100q^{85} - 324q^{87} + 168q^{88} - 160q^{90} - 28q^{91} + 108q^{93} + 984q^{94} + 40q^{96} + 728q^{97} - 166q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 46 x^{14} + 823 x^{12} + 7252 x^{10} + 32831 x^{8} + 71486 x^{6} + 60809 x^{4} + 15680 x^{2} + 576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-19 \nu^{15} + 346 \nu^{14} - 314 \nu^{13} + 11114 \nu^{12} + 6979 \nu^{11} + 96196 \nu^{10} + 199188 \nu^{9} - 172404 \nu^{8} + 1667083 \nu^{7} - 5356966 \nu^{6} + 5855246 \nu^{5} - 19319222 \nu^{4} + 7881413 \nu^{3} - 16640728 \nu^{2} + 328536 \nu - 1400928\)\()/468672\)
\(\beta_{3}\)\(=\)\((\)\(-22 \nu^{15} - 87 \nu^{14} - 749 \nu^{13} - 2851 \nu^{12} - 6822 \nu^{11} - 26756 \nu^{10} + 24310 \nu^{9} + 936 \nu^{8} + 703126 \nu^{7} + 1044327 \nu^{6} + 3811631 \nu^{5} + 4040059 \nu^{4} + 7361646 \nu^{3} + 4003172 \nu^{2} + 3163664 \nu + 884592\)\()/117168\)
\(\beta_{4}\)\(=\)\((\)\(355 \nu^{15} - 362 \nu^{14} + 14860 \nu^{13} - 13434 \nu^{12} + 235239 \nu^{11} - 173056 \nu^{10} + 1782656 \nu^{9} - 883956 \nu^{8} + 6908345 \nu^{7} - 1385434 \nu^{6} + 13831052 \nu^{5} + 315822 \nu^{4} + 13937181 \nu^{3} - 1164668 \nu^{2} + 5755144 \nu - 480768\)\()/468672\)
\(\beta_{5}\)\(=\)\((\)\(355 \nu^{15} - 454 \nu^{14} + 14860 \nu^{13} - 17010 \nu^{12} + 235239 \nu^{11} - 222000 \nu^{10} + 1782656 \nu^{9} - 1143564 \nu^{8} + 6908345 \nu^{7} - 1576670 \nu^{6} + 13831052 \nu^{5} + 2390046 \nu^{4} + 13937181 \nu^{3} + 2361972 \nu^{2} + 5755144 \nu + 357696\)\()/468672\)
\(\beta_{6}\)\(=\)\((\)\(-9 \nu^{15} - 536 \nu^{14} + 1136 \nu^{13} - 19136 \nu^{12} + 51355 \nu^{11} - 226568 \nu^{10} + 720276 \nu^{9} - 882084 \nu^{8} + 4021041 \nu^{7} + 703220 \nu^{6} + 7443556 \nu^{5} + 8395940 \nu^{4} - 2062147 \nu^{3} + 6841676 \nu^{2} - 5906280 \nu + 1054080\)\()/234336\)
\(\beta_{7}\)\(=\)\((\)\( 187 \nu^{14} + 7587 \nu^{12} + 114130 \nu^{10} + 791734 \nu^{8} + 2620631 \nu^{6} + 3968375 \nu^{4} + 2637324 \nu^{2} + 780032 \)\()/78112\)
\(\beta_{8}\)\(=\)\((\)\( 187 \nu^{14} + 7587 \nu^{12} + 114130 \nu^{10} + 791734 \nu^{8} + 2620631 \nu^{6} + 3968375 \nu^{4} + 2559212 \nu^{2} + 311360 \)\()/78112\)
\(\beta_{9}\)\(=\)\((\)\( -321 \nu^{15} - 14812 \nu^{13} - 265971 \nu^{11} - 2352364 \nu^{9} - 10668555 \nu^{7} - 23042624 \nu^{5} - 18482577 \nu^{3} - 3269960 \nu \)\()/234336\)
\(\beta_{10}\)\(=\)\((\)\(-48 \nu^{15} - 1040 \nu^{14} - 4519 \nu^{13} - 43396 \nu^{12} - 130499 \nu^{11} - 680212 \nu^{10} - 1650778 \nu^{9} - 5011884 \nu^{8} - 9843186 \nu^{7} - 17945092 \nu^{6} - 26446439 \nu^{5} - 28644464 \nu^{4} - 26992363 \nu^{3} - 14503832 \nu^{2} - 8320424 \nu - 455520\)\()/234336\)
\(\beta_{11}\)\(=\)\((\)\(1141 \nu^{15} - 346 \nu^{14} + 45836 \nu^{13} - 11114 \nu^{12} + 677801 \nu^{11} - 96196 \nu^{10} + 4551216 \nu^{9} + 172404 \nu^{8} + 14056703 \nu^{7} + 5356966 \nu^{6} + 17955004 \nu^{5} + 19319222 \nu^{4} + 7473859 \nu^{3} + 16640728 \nu^{2} + 602280 \nu + 1400928\)\()/468672\)
\(\beta_{12}\)\(=\)\((\)\(695 \nu^{15} + 354 \nu^{14} + 32427 \nu^{13} + 12274 \nu^{12} + 589430 \nu^{11} + 134626 \nu^{10} + 5278382 \nu^{9} + 355776 \nu^{8} + 24214099 \nu^{7} - 1985766 \nu^{6} + 53028927 \nu^{5} - 9817522 \nu^{4} + 44727856 \nu^{3} - 7738030 \nu^{2} + 11118040 \nu - 225744\)\()/234336\)
\(\beta_{13}\)\(=\)\((\)\(695 \nu^{15} + 354 \nu^{14} + 29986 \nu^{13} + 12274 \nu^{12} + 494231 \nu^{11} + 134626 \nu^{10} + 3935832 \nu^{9} + 355776 \nu^{8} + 15909817 \nu^{7} - 1985766 \nu^{6} + 30959846 \nu^{5} - 9817522 \nu^{4} + 23366665 \nu^{3} - 7738030 \nu^{2} + 3599760 \nu - 225744\)\()/234336\)
\(\beta_{14}\)\(=\)\((\)\(1823 \nu^{15} - 956 \nu^{14} + 86142 \nu^{13} - 40980 \nu^{12} + 1587409 \nu^{11} - 664816 \nu^{10} + 14396428 \nu^{9} - 5104704 \nu^{8} + 66456433 \nu^{7} - 18986740 \nu^{6} + 143530734 \nu^{5} - 30319692 \nu^{4} + 110530943 \nu^{3} - 12555680 \nu^{2} + 18251624 \nu + 1631712\)\()/468672\)
\(\beta_{15}\)\(=\)\((\)\(511 \nu^{15} - 87 \nu^{14} + 22834 \nu^{13} - 2851 \nu^{12} + 393902 \nu^{11} - 26756 \nu^{10} + 3318976 \nu^{9} + 936 \nu^{8} + 14253143 \nu^{7} + 1044327 \nu^{6} + 29074142 \nu^{5} + 4040059 \nu^{4} + 22147396 \nu^{3} + 4003172 \nu^{2} + 4671320 \nu + 884592\)\()/117168\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{8} + \beta_{7} - 6\)
\(\nu^{3}\)\(=\)\(\beta_{13} - \beta_{11} + \beta_{9} + \beta_{4} - 2 \beta_{2} - 11 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{15} - \beta_{13} + \beta_{12} - \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 18 \beta_{8} - 12 \beta_{7} + \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + \beta_{2} + 3 \beta_{1} + 60\)
\(\nu^{5}\)\(=\)\(2 \beta_{15} - 2 \beta_{14} - 22 \beta_{13} + 2 \beta_{12} + 16 \beta_{11} + 2 \beta_{10} - 24 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 16 \beta_{4} - 2 \beta_{3} + 34 \beta_{2} + 133 \beta_{1} + 18\)
\(\nu^{6}\)\(=\)\(22 \beta_{15} + 16 \beta_{13} - 18 \beta_{12} + 20 \beta_{11} - 36 \beta_{10} + 40 \beta_{9} - 261 \beta_{8} + 149 \beta_{7} - 20 \beta_{6} - 88 \beta_{5} - 34 \beta_{4} + 2 \beta_{3} - 18 \beta_{2} - 62 \beta_{1} - 692\)
\(\nu^{7}\)\(=\)\(-34 \beta_{15} + 48 \beta_{14} + 353 \beta_{13} - 46 \beta_{12} - 225 \beta_{11} - 48 \beta_{10} + 433 \beta_{9} - 48 \beta_{7} + 40 \beta_{6} + 227 \beta_{4} + 42 \beta_{3} - 484 \beta_{2} - 1701 \beta_{1} - 267\)
\(\nu^{8}\)\(=\)\(-343 \beta_{15} - 12 \beta_{14} - 217 \beta_{13} + 271 \beta_{12} - 313 \beta_{11} + 530 \beta_{10} - 614 \beta_{9} + 3598 \beta_{8} - 1932 \beta_{7} + 313 \beta_{6} + 1468 \beta_{5} + 416 \beta_{4} - 42 \beta_{3} + 271 \beta_{2} + 969 \beta_{1} + 8626\)
\(\nu^{9}\)\(=\)\(380 \beta_{15} - 854 \beta_{14} - 5158 \beta_{13} + 860 \beta_{12} + 3116 \beta_{11} + 854 \beta_{10} - 6924 \beta_{9} + 854 \beta_{7} - 590 \beta_{6} - 3118 \beta_{4} - 644 \beta_{3} + 6560 \beta_{2} + 22447 \beta_{1} + 3708\)
\(\nu^{10}\)\(=\)\(4724 \beta_{15} + 480 \beta_{14} + 2808 \beta_{13} - 3924 \beta_{12} + 4560 \beta_{11} - 7368 \beta_{10} + 8648 \beta_{9} - 48769 \beta_{8} + 25729 \beta_{7} - 4568 \beta_{6} - 22320 \beta_{5} - 4268 \beta_{4} + 636 \beta_{3} - 3924 \beta_{2} - 13852 \beta_{1} - 112226\)
\(\nu^{11}\)\(=\)\(-2828 \beta_{15} + 13672 \beta_{14} + 72729 \beta_{13} - 14956 \beta_{12} - 43313 \beta_{11} - 13672 \beta_{10} + 104577 \beta_{9} - 13672 \beta_{7} + 7688 \beta_{6} + 42397 \beta_{4} + 8812 \beta_{3} - 87414 \beta_{2} - 301335 \beta_{1} - 50085\)
\(\nu^{12}\)\(=\)\(-61357 \beta_{15} - 12128 \beta_{14} - 35657 \beta_{13} + 56229 \beta_{12} - 64673 \beta_{11} + 100330 \beta_{10} - 117586 \beta_{9} + 657138 \beta_{8} - 347836 \beta_{7} + 65041 \beta_{6} + 326308 \beta_{5} + 35262 \beta_{4} - 8444 \beta_{3} + 56229 \beta_{2} + 191071 \beta_{1} + 1494128\)
\(\nu^{13}\)\(=\)\(-1122 \beta_{15} - 208722 \beta_{14} - 1010382 \beta_{13} + 248790 \beta_{12} + 604952 \beta_{11} + 208722 \beta_{10} - 1535136 \beta_{9} + 208722 \beta_{7} - 93330 \beta_{6} - 574932 \beta_{4} - 114270 \beta_{3} + 1157822 \beta_{2} + 4083745 \beta_{1} + 668262\)
\(\nu^{14}\)\(=\)\(771362 \beta_{15} + 249912 \beta_{14} + 448648 \beta_{13} - 802782 \beta_{12} + 907004 \beta_{11} - 1355652 \beta_{10} + 1574144 \beta_{9} - 8840469 \beta_{8} + 4742237 \beta_{7} - 917052 \beta_{6} - 4683640 \beta_{5} - 153062 \beta_{4} + 104222 \beta_{3} - 802782 \beta_{2} - 2595418 \beta_{1} - 20142648\)
\(\nu^{15}\)\(=\)\(596674 \beta_{15} + 3109496 \beta_{14} + 13949913 \beta_{13} - 4004922 \beta_{12} - 8474417 \beta_{11} - 3109496 \beta_{10} + 22201169 \beta_{9} - 3109496 \beta_{7} + 1074320 \beta_{6} + 7796351 \beta_{4} + 1438502 \beta_{3} - 15309912 \beta_{2} - 55647953 \beta_{1} - 8870671\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
3.73696i
3.57278i
2.62785i
2.60953i
2.02253i
1.02879i
0.601965i
0.209282i
0.209282i
0.601965i
1.02879i
2.02253i
2.60953i
2.62785i
3.57278i
3.73696i
3.73696i 1.47033 2.61498i −9.96484 2.23607i −9.77207 5.49456i −2.64575 22.2903i −4.67625 7.68978i 8.35609
71.2 3.57278i −2.99481 + 0.176471i −8.76473 2.23607i 0.630492 + 10.6998i −2.64575 17.0233i 8.93772 1.05699i −7.98897
71.3 2.62785i −0.217001 2.99214i −2.90558 2.23607i −7.86289 + 0.570245i 2.64575 2.87597i −8.90582 + 1.29859i −5.87604
71.4 2.60953i 2.93192 + 0.635503i −2.80964 2.23607i 1.65836 7.65092i −2.64575 3.10627i 8.19227 + 3.72648i −5.83508
71.5 2.02253i 2.91626 0.703870i −0.0906451 2.23607i −1.42360 5.89823i 2.64575 7.90680i 8.00914 4.10533i 4.52252
71.6 1.02879i −2.94860 + 0.552947i 2.94159 2.23607i 0.568867 + 3.03349i 2.64575 7.14144i 8.38850 3.26084i −2.30045
71.7 0.601965i 0.926467 + 2.85336i 3.63764 2.23607i 1.71762 0.557701i 2.64575 4.59759i −7.28332 + 5.28709i −1.34603
71.8 0.209282i 1.91543 + 2.30892i 3.95620 2.23607i 0.483215 0.400865i −2.64575 1.66509i −1.66223 + 8.84517i 0.467968
71.9 0.209282i 1.91543 2.30892i 3.95620 2.23607i 0.483215 + 0.400865i −2.64575 1.66509i −1.66223 8.84517i 0.467968
71.10 0.601965i 0.926467 2.85336i 3.63764 2.23607i 1.71762 + 0.557701i 2.64575 4.59759i −7.28332 5.28709i −1.34603
71.11 1.02879i −2.94860 0.552947i 2.94159 2.23607i 0.568867 3.03349i 2.64575 7.14144i 8.38850 + 3.26084i −2.30045
71.12 2.02253i 2.91626 + 0.703870i −0.0906451 2.23607i −1.42360 + 5.89823i 2.64575 7.90680i 8.00914 + 4.10533i 4.52252
71.13 2.60953i 2.93192 0.635503i −2.80964 2.23607i 1.65836 + 7.65092i −2.64575 3.10627i 8.19227 3.72648i −5.83508
71.14 2.62785i −0.217001 + 2.99214i −2.90558 2.23607i −7.86289 0.570245i 2.64575 2.87597i −8.90582 1.29859i −5.87604
71.15 3.57278i −2.99481 0.176471i −8.76473 2.23607i 0.630492 10.6998i −2.64575 17.0233i 8.93772 + 1.05699i −7.98897
71.16 3.73696i 1.47033 + 2.61498i −9.96484 2.23607i −9.77207 + 5.49456i −2.64575 22.2903i −4.67625 + 7.68978i 8.35609
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.c.a 16
3.b odd 2 1 inner 105.3.c.a 16
4.b odd 2 1 1680.3.l.a 16
5.b even 2 1 525.3.c.b 16
5.c odd 4 2 525.3.f.b 32
12.b even 2 1 1680.3.l.a 16
15.d odd 2 1 525.3.c.b 16
15.e even 4 2 525.3.f.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.c.a 16 1.a even 1 1 trivial
105.3.c.a 16 3.b odd 2 1 inner
525.3.c.b 16 5.b even 2 1
525.3.c.b 16 15.d odd 2 1
525.3.f.b 32 5.c odd 4 2
525.3.f.b 32 15.e even 4 2
1680.3.l.a 16 4.b odd 2 1
1680.3.l.a 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(105, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 18 T^{2} + 167 T^{4} - 1116 T^{6} + 6143 T^{8} - 30562 T^{10} + 147193 T^{12} - 678704 T^{14} + 2863008 T^{16} - 10859264 T^{18} + 37681408 T^{20} - 125181952 T^{22} + 402587648 T^{24} - 1170210816 T^{26} + 2801795072 T^{28} - 4831838208 T^{30} + 4294967296 T^{32} \)
$3$ \( 1 - 8 T + 21 T^{2} + 52 T^{3} - 630 T^{4} + 2116 T^{5} - 1381 T^{6} - 17424 T^{7} + 83754 T^{8} - 156816 T^{9} - 111861 T^{10} + 1542564 T^{11} - 4133430 T^{12} + 3070548 T^{13} + 11160261 T^{14} - 38263752 T^{15} + 43046721 T^{16} \)
$5$ \( ( 1 + 5 T^{2} )^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{8} \)
$11$ \( 1 - 634 T^{2} + 247245 T^{4} - 65949338 T^{6} + 13712984198 T^{8} - 2284687822518 T^{10} + 326907972407363 T^{12} - 41733915025260470 T^{14} + 5125952848251765522 T^{16} - \)\(61\!\cdots\!70\)\( T^{18} + \)\(70\!\cdots\!03\)\( T^{20} - \)\(71\!\cdots\!78\)\( T^{22} + \)\(63\!\cdots\!78\)\( T^{24} - \)\(44\!\cdots\!38\)\( T^{26} + \)\(24\!\cdots\!45\)\( T^{28} - \)\(91\!\cdots\!54\)\( T^{30} + \)\(21\!\cdots\!21\)\( T^{32} \)
$13$ \( ( 1 + 745 T^{2} + 1452 T^{3} + 296782 T^{4} + 900988 T^{5} + 78625591 T^{6} + 264119992 T^{7} + 15370753026 T^{8} + 44636278648 T^{9} + 2245625504551 T^{10} + 4348896987292 T^{11} + 242094194839822 T^{12} + 200170530164748 T^{13} + 17357073416248345 T^{14} + 665416609183179841 T^{16} )^{2} \)
$17$ \( 1 - 1918 T^{2} + 1941581 T^{4} - 1392408966 T^{6} + 790147542694 T^{8} - 373036090829946 T^{10} + 150674753302219043 T^{12} - 52956031786998379138 T^{14} + \)\(16\!\cdots\!82\)\( T^{16} - \)\(44\!\cdots\!98\)\( T^{18} + \)\(10\!\cdots\!63\)\( T^{20} - \)\(21\!\cdots\!06\)\( T^{22} + \)\(38\!\cdots\!14\)\( T^{24} - \)\(56\!\cdots\!66\)\( T^{26} + \)\(65\!\cdots\!01\)\( T^{28} - \)\(54\!\cdots\!38\)\( T^{30} + \)\(23\!\cdots\!61\)\( T^{32} \)
$19$ \( ( 1 + 8 T + 860 T^{2} + 16312 T^{3} + 704036 T^{4} + 10691240 T^{5} + 381169188 T^{6} + 5670939160 T^{7} + 156781324406 T^{8} + 2047209036760 T^{9} + 49674349749348 T^{10} + 502978804782440 T^{11} + 11957039789133476 T^{12} + 100009952797249912 T^{13} + 1903450830396898460 T^{14} + 6392053486263072968 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \)
$23$ \( 1 - 4480 T^{2} + 10302200 T^{4} - 16115718144 T^{6} + 19153465500700 T^{8} - 18300056706334464 T^{10} + 14511541318853927624 T^{12} - \)\(97\!\cdots\!40\)\( T^{14} + \)\(55\!\cdots\!30\)\( T^{16} - \)\(27\!\cdots\!40\)\( T^{18} + \)\(11\!\cdots\!44\)\( T^{20} - \)\(40\!\cdots\!44\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{24} - \)\(27\!\cdots\!44\)\( T^{26} + \)\(49\!\cdots\!00\)\( T^{28} - \)\(60\!\cdots\!80\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \)
$29$ \( 1 - 5378 T^{2} + 14503181 T^{4} - 26572841642 T^{6} + 38033992302502 T^{8} - 46226481721014678 T^{10} + 49944592354476620099 T^{12} - \)\(48\!\cdots\!42\)\( T^{14} + \)\(43\!\cdots\!90\)\( T^{16} - \)\(34\!\cdots\!02\)\( T^{18} + \)\(24\!\cdots\!39\)\( T^{20} - \)\(16\!\cdots\!98\)\( T^{22} + \)\(95\!\cdots\!42\)\( T^{24} - \)\(47\!\cdots\!42\)\( T^{26} + \)\(18\!\cdots\!61\)\( T^{28} - \)\(47\!\cdots\!58\)\( T^{30} + \)\(62\!\cdots\!41\)\( T^{32} \)
$31$ \( ( 1 + 36 T + 5588 T^{2} + 98172 T^{3} + 10424196 T^{4} - 32178444 T^{5} + 7586949868 T^{6} - 323190800532 T^{7} + 3540198882934 T^{8} - 310586359311252 T^{9} + 7006707529045228 T^{10} - 28558487498852364 T^{11} + 8890703340928322436 T^{12} + 80464548189479195772 T^{13} + \)\(44\!\cdots\!68\)\( T^{14} + \)\(27\!\cdots\!56\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \)
$37$ \( ( 1 + 20 T + 4860 T^{2} + 56172 T^{3} + 13001540 T^{4} + 138784868 T^{5} + 25923379716 T^{6} + 250888407260 T^{7} + 39435511353270 T^{8} + 343466229538940 T^{9} + 48584587251918276 T^{10} + 356084000997179012 T^{11} + 45667642119332038340 T^{12} + \)\(27\!\cdots\!28\)\( T^{13} + \)\(31\!\cdots\!60\)\( T^{14} + \)\(18\!\cdots\!80\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \)
$41$ \( 1 - 17544 T^{2} + 152048344 T^{4} - 866284444952 T^{6} + 3637613347314524 T^{8} - 11949543415657926792 T^{10} + \)\(31\!\cdots\!80\)\( T^{12} - \)\(69\!\cdots\!84\)\( T^{14} + \)\(12\!\cdots\!02\)\( T^{16} - \)\(19\!\cdots\!24\)\( T^{18} + \)\(25\!\cdots\!80\)\( T^{20} - \)\(26\!\cdots\!52\)\( T^{22} + \)\(23\!\cdots\!84\)\( T^{24} - \)\(15\!\cdots\!52\)\( T^{26} + \)\(77\!\cdots\!84\)\( T^{28} - \)\(25\!\cdots\!24\)\( T^{30} + \)\(40\!\cdots\!81\)\( T^{32} \)
$43$ \( ( 1 - 140 T + 17296 T^{2} - 1328244 T^{3} + 93523420 T^{4} - 4955438268 T^{5} + 256598382768 T^{6} - 10987637195140 T^{7} + 504821370222726 T^{8} - 20316141173813860 T^{9} + 877258807605621168 T^{10} - 31325124358935759132 T^{11} + \)\(10\!\cdots\!20\)\( T^{12} - \)\(28\!\cdots\!56\)\( T^{13} + \)\(69\!\cdots\!96\)\( T^{14} - \)\(10\!\cdots\!60\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( 1 - 16494 T^{2} + 133830909 T^{4} - 714029015638 T^{6} + 2855140329276550 T^{8} - 9328199881448227946 T^{10} + \)\(26\!\cdots\!07\)\( T^{12} - \)\(67\!\cdots\!70\)\( T^{14} + \)\(15\!\cdots\!66\)\( T^{16} - \)\(33\!\cdots\!70\)\( T^{18} + \)\(63\!\cdots\!27\)\( T^{20} - \)\(10\!\cdots\!86\)\( T^{22} + \)\(16\!\cdots\!50\)\( T^{24} - \)\(19\!\cdots\!38\)\( T^{26} + \)\(18\!\cdots\!29\)\( T^{28} - \)\(10\!\cdots\!34\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$53$ \( 1 - 26408 T^{2} + 323575832 T^{4} - 2478395195256 T^{6} + 13692147525400668 T^{8} - 60596495283388870952 T^{10} + \)\(23\!\cdots\!48\)\( T^{12} - \)\(79\!\cdots\!44\)\( T^{14} + \)\(23\!\cdots\!78\)\( T^{16} - \)\(62\!\cdots\!64\)\( T^{18} + \)\(14\!\cdots\!28\)\( T^{20} - \)\(29\!\cdots\!32\)\( T^{22} + \)\(53\!\cdots\!28\)\( T^{24} - \)\(75\!\cdots\!56\)\( T^{26} + \)\(78\!\cdots\!92\)\( T^{28} - \)\(50\!\cdots\!88\)\( T^{30} + \)\(15\!\cdots\!41\)\( T^{32} \)
$59$ \( 1 - 14800 T^{2} + 138468568 T^{4} - 955392890736 T^{6} + 5479095487372636 T^{8} - 27019630879601870800 T^{10} + \)\(12\!\cdots\!12\)\( T^{12} - \)\(48\!\cdots\!04\)\( T^{14} + \)\(17\!\cdots\!62\)\( T^{16} - \)\(58\!\cdots\!44\)\( T^{18} + \)\(17\!\cdots\!52\)\( T^{20} - \)\(48\!\cdots\!00\)\( T^{22} + \)\(11\!\cdots\!76\)\( T^{24} - \)\(24\!\cdots\!36\)\( T^{26} + \)\(43\!\cdots\!48\)\( T^{28} - \)\(56\!\cdots\!00\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \)
$61$ \( ( 1 + 28 T + 12104 T^{2} + 279684 T^{3} + 87089292 T^{4} + 2698559212 T^{5} + 465312999544 T^{6} + 14439842194996 T^{7} + 1856098427255974 T^{8} + 53730652807580116 T^{9} + 6442649806919296504 T^{10} + \)\(13\!\cdots\!32\)\( T^{11} + \)\(16\!\cdots\!52\)\( T^{12} + \)\(19\!\cdots\!84\)\( T^{13} + \)\(32\!\cdots\!84\)\( T^{14} + \)\(27\!\cdots\!48\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \)
$67$ \( ( 1 + 60 T + 16740 T^{2} + 1235332 T^{3} + 154240932 T^{4} + 12326585708 T^{5} + 1033583040796 T^{6} + 78720265804692 T^{7} + 5329117931496566 T^{8} + 353375273197262388 T^{9} + 20827856918628132316 T^{10} + \)\(11\!\cdots\!52\)\( T^{11} + \)\(62\!\cdots\!12\)\( T^{12} + \)\(22\!\cdots\!68\)\( T^{13} + \)\(13\!\cdots\!40\)\( T^{14} + \)\(22\!\cdots\!40\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} )^{2} \)
$71$ \( 1 - 17240 T^{2} + 219765336 T^{4} - 2041434479624 T^{6} + 16624899469572444 T^{8} - \)\(11\!\cdots\!96\)\( T^{10} + \)\(74\!\cdots\!40\)\( T^{12} - \)\(43\!\cdots\!72\)\( T^{14} + \)\(22\!\cdots\!98\)\( T^{16} - \)\(10\!\cdots\!32\)\( T^{18} + \)\(48\!\cdots\!40\)\( T^{20} - \)\(19\!\cdots\!36\)\( T^{22} + \)\(69\!\cdots\!24\)\( T^{24} - \)\(21\!\cdots\!24\)\( T^{26} + \)\(59\!\cdots\!16\)\( T^{28} - \)\(11\!\cdots\!40\)\( T^{30} + \)\(17\!\cdots\!41\)\( T^{32} \)
$73$ \( ( 1 + 104 T + 24348 T^{2} + 2586552 T^{3} + 336158388 T^{4} + 30976570216 T^{5} + 3043939907492 T^{6} + 241340462677368 T^{7} + 19104376400195478 T^{8} + 1286103325607694072 T^{9} + 86442539082475521572 T^{10} + \)\(46\!\cdots\!24\)\( T^{11} + \)\(27\!\cdots\!28\)\( T^{12} + \)\(11\!\cdots\!48\)\( T^{13} + \)\(55\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!36\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} )^{2} \)
$79$ \( ( 1 + 102 T + 30141 T^{2} + 2523638 T^{3} + 463628630 T^{4} + 32866476410 T^{5} + 4655301343539 T^{6} + 284411378115370 T^{7} + 33720462322717682 T^{8} + 1775011410818024170 T^{9} + \)\(18\!\cdots\!59\)\( T^{10} + \)\(79\!\cdots\!10\)\( T^{11} + \)\(70\!\cdots\!30\)\( T^{12} + \)\(23\!\cdots\!38\)\( T^{13} + \)\(17\!\cdots\!81\)\( T^{14} + \)\(37\!\cdots\!62\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} )^{2} \)
$83$ \( 1 - 42240 T^{2} + 975418872 T^{4} - 15454190107264 T^{6} + 185929829730979228 T^{8} - \)\(17\!\cdots\!28\)\( T^{10} + \)\(14\!\cdots\!28\)\( T^{12} - \)\(10\!\cdots\!48\)\( T^{14} + \)\(75\!\cdots\!98\)\( T^{16} - \)\(51\!\cdots\!08\)\( T^{18} + \)\(33\!\cdots\!48\)\( T^{20} - \)\(19\!\cdots\!08\)\( T^{22} + \)\(94\!\cdots\!68\)\( T^{24} - \)\(37\!\cdots\!64\)\( T^{26} + \)\(11\!\cdots\!12\)\( T^{28} - \)\(22\!\cdots\!40\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{32} \)
$89$ \( 1 - 33032 T^{2} + 706967320 T^{4} - 12081015642520 T^{6} + 168525587092572636 T^{8} - \)\(20\!\cdots\!04\)\( T^{10} + \)\(21\!\cdots\!16\)\( T^{12} - \)\(20\!\cdots\!76\)\( T^{14} + \)\(16\!\cdots\!54\)\( T^{16} - \)\(12\!\cdots\!16\)\( T^{18} + \)\(84\!\cdots\!96\)\( T^{20} - \)\(50\!\cdots\!84\)\( T^{22} + \)\(26\!\cdots\!96\)\( T^{24} - \)\(11\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!20\)\( T^{28} - \)\(12\!\cdots\!92\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \)
$97$ \( ( 1 - 364 T + 112297 T^{2} - 23728036 T^{3} + 4389819350 T^{4} - 662305688748 T^{5} + 89325314408919 T^{6} - 10317093604637892 T^{7} + 1074975098720446610 T^{8} - 97073533726037925828 T^{9} + \)\(79\!\cdots\!39\)\( T^{10} - \)\(55\!\cdots\!92\)\( T^{11} + \)\(34\!\cdots\!50\)\( T^{12} - \)\(17\!\cdots\!64\)\( T^{13} + \)\(77\!\cdots\!77\)\( T^{14} - \)\(23\!\cdots\!16\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \)
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