Properties

Label 31.2.g.a
Level 31
Weight 2
Character orbit 31.g
Analytic conductor 0.248
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 31.g (of order \(15\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(0.247536246266\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut +\mathstrut \) \(19\) \(x^{14}\mathstrut +\mathstrut \) \(140\) \(x^{12}\mathstrut +\mathstrut \) \(511\) \(x^{10}\mathstrut +\mathstrut \) \(979\) \(x^{8}\mathstrut +\mathstrut \) \(956\) \(x^{6}\mathstrut +\mathstrut \) \(410\) \(x^{4}\mathstrut +\mathstrut \) \(44\) \(x^{2}\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{15} - 4 \nu^{14} + 48 \nu^{13} - 65 \nu^{12} + 458 \nu^{11} - 358 \nu^{10} + 2196 \nu^{9} - 641 \nu^{8} + 5467 \nu^{7} + 691 \nu^{6} + 6431 \nu^{5} + 3382 \nu^{4} + 2409 \nu^{3} + 2839 \nu^{2} - 360 \nu + 255 \)\()/186\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{15} + 4 \nu^{14} + 48 \nu^{13} + 65 \nu^{12} + 458 \nu^{11} + 358 \nu^{10} + 2196 \nu^{9} + 641 \nu^{8} + 5467 \nu^{7} - 691 \nu^{6} + 6431 \nu^{5} - 3382 \nu^{4} + 2409 \nu^{3} - 2839 \nu^{2} - 360 \nu - 255 \)\()/186\)
\(\beta_{3}\)\(=\)\((\)\( 6 \nu^{15} - 10 \nu^{14} + 144 \nu^{13} - 209 \nu^{12} + 1374 \nu^{11} - 1732 \nu^{10} + 6619 \nu^{9} - 7260 \nu^{8} + 16835 \nu^{7} - 16144 \nu^{6} + 21308 \nu^{5} - 17926 \nu^{4} + 10699 \nu^{3} - 7953 \nu^{2} + 439 \nu - 463 \)\()/186\)
\(\beta_{4}\)\(=\)\((\)\( 17 \nu^{15} + 315 \nu^{13} + 2250 \nu^{11} + 7940 \nu^{9} + 14865 \nu^{7} + 14844 \nu^{5} + 7255 \nu^{3} + 1125 \nu + 93 \)\()/186\)
\(\beta_{5}\)\(=\)\((\)\( 6 \nu^{15} + 10 \nu^{14} + 144 \nu^{13} + 209 \nu^{12} + 1374 \nu^{11} + 1732 \nu^{10} + 6619 \nu^{9} + 7260 \nu^{8} + 16835 \nu^{7} + 16144 \nu^{6} + 21308 \nu^{5} + 17926 \nu^{4} + 10699 \nu^{3} + 7953 \nu^{2} + 439 \nu + 463 \)\()/186\)
\(\beta_{6}\)\(=\)\((\)\( -6 \nu^{15} + 28 \nu^{14} - 144 \nu^{13} + 517 \nu^{12} - 1374 \nu^{11} + 3653 \nu^{10} - 6619 \nu^{9} + 12516 \nu^{8} - 16835 \nu^{7} + 21730 \nu^{6} - 21308 \nu^{5} + 18145 \nu^{4} - 10699 \nu^{3} + 6074 \nu^{2} - 532 \nu + 354 \)\()/186\)
\(\beta_{7}\)\(=\)\((\)\( 8 \nu^{15} + 33 \nu^{14} + 130 \nu^{13} + 575 \nu^{12} + 716 \nu^{11} + 3713 \nu^{10} + 1282 \nu^{9} + 10969 \nu^{8} - 1382 \nu^{7} + 14550 \nu^{6} - 6857 \nu^{5} + 6617 \nu^{4} - 6329 \nu^{3} - 412 \nu^{2} - 1347 \nu - 143 \)\()/186\)
\(\beta_{8}\)\(=\)\((\)\( 6 \nu^{15} + 28 \nu^{14} + 144 \nu^{13} + 517 \nu^{12} + 1374 \nu^{11} + 3653 \nu^{10} + 6619 \nu^{9} + 12516 \nu^{8} + 16835 \nu^{7} + 21730 \nu^{6} + 21308 \nu^{5} + 18145 \nu^{4} + 10699 \nu^{3} + 6074 \nu^{2} + 532 \nu + 354 \)\()/186\)
\(\beta_{9}\)\(=\)\((\)\( -36 \nu^{15} + 38 \nu^{14} - 709 \nu^{13} + 695 \nu^{12} - 5485 \nu^{11} + 4827 \nu^{10} - 21362 \nu^{9} + 16025 \nu^{8} - 44466 \nu^{7} + 26249 \nu^{6} - 47899 \nu^{5} + 19734 \nu^{4} - 22747 \nu^{3} + 5719 \nu^{2} - 2510 \nu + 538 \)\()/186\)
\(\beta_{10}\)\(=\)\((\)\( -53 \nu^{15} + 5 \nu^{14} - 962 \nu^{13} + 89 \nu^{12} - 6619 \nu^{11} + 587 \nu^{10} - 21738 \nu^{9} + 1770 \nu^{8} - 35213 \nu^{7} + 2430 \nu^{6} - 26132 \nu^{5} + 1337 \nu^{4} - 7000 \nu^{3} + 303 \nu^{2} - 225 \nu + 61 \)\()/186\)
\(\beta_{11}\)\(=\)\((\)\( 50 \nu^{15} - 25 \nu^{14} + 983 \nu^{13} - 445 \nu^{12} + 7575 \nu^{11} - 2966 \nu^{10} + 29263 \nu^{9} - 9222 \nu^{8} + 59919 \nu^{7} - 13483 \nu^{6} + 62350 \nu^{5} - 7987 \nu^{4} + 27117 \nu^{3} - 926 \nu^{2} + 1788 \nu + 36 \)\()/186\)
\(\beta_{12}\)\(=\)\((\)\( 57 \nu^{15} + 21 \nu^{14} + 1058 \nu^{13} + 380 \nu^{12} + 7535 \nu^{11} + 2608 \nu^{10} + 26161 \nu^{9} + 8581 \nu^{8} + 46581 \nu^{7} + 14174 \nu^{6} + 41009 \nu^{5} + 11369 \nu^{4} + 15383 \nu^{3} + 3765 \nu^{2} + 1489 \nu + 126 \)\()/186\)
\(\beta_{13}\)\(=\)\((\)\( 27 \nu^{15} + 53 \nu^{14} + 524 \nu^{13} + 962 \nu^{12} + 3982 \nu^{11} + 6619 \nu^{10} + 15200 \nu^{9} + 21738 \nu^{8} + 31009 \nu^{7} + 35213 \nu^{6} + 32677 \nu^{5} + 26132 \nu^{4} + 14464 \nu^{3} + 7093 \nu^{2} + 658 \nu + 597 \)\()/186\)
\(\beta_{14}\)\(=\)\((\)\( 36 \nu^{15} + 68 \nu^{14} + 709 \nu^{13} + 1229 \nu^{12} + 5485 \nu^{11} + 8411 \nu^{10} + 21362 \nu^{9} + 27451 \nu^{8} + 44466 \nu^{7} + 44177 \nu^{6} + 47899 \nu^{5} + 32530 \nu^{4} + 22747 \nu^{3} + 8467 \nu^{2} + 2510 \nu + 470 \)\()/186\)
\(\beta_{15}\)\(=\)\((\)\( 135 \nu^{15} + 34 \nu^{14} + 2558 \nu^{13} + 599 \nu^{12} + 18763 \nu^{11} + 3942 \nu^{10} + 67940 \nu^{9} + 12098 \nu^{8} + 128230 \nu^{7} + 17671 \nu^{6} + 121504 \nu^{5} + 11336 \nu^{4} + 48574 \nu^{3} + 2730 \nu^{2} + 3724 \nu + 359 \)\()/186\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\)
\(\nu^{2}\)\(=\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(6\) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(8\) \(\beta_{14}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut -\mathstrut \) \(6\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(7\) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(7\) \(\beta_{15}\mathstrut -\mathstrut \) \(6\) \(\beta_{14}\mathstrut -\mathstrut \) \(4\) \(\beta_{13}\mathstrut -\mathstrut \) \(6\) \(\beta_{12}\mathstrut -\mathstrut \) \(14\) \(\beta_{11}\mathstrut -\mathstrut \) \(7\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(37\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(37\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(21\) \(\beta_{3}\mathstrut -\mathstrut \) \(14\) \(\beta_{2}\mathstrut -\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(-\)\(19\) \(\beta_{15}\mathstrut +\mathstrut \) \(55\) \(\beta_{14}\mathstrut -\mathstrut \) \(8\) \(\beta_{12}\mathstrut +\mathstrut \) \(38\) \(\beta_{11}\mathstrut -\mathstrut \) \(19\) \(\beta_{10}\mathstrut +\mathstrut \) \(47\) \(\beta_{9}\mathstrut -\mathstrut \) \(72\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(34\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(38\) \(\beta_{1}\mathstrut -\mathstrut \) \(77\)
\(\nu^{7}\)\(=\)\(-\)\(42\) \(\beta_{15}\mathstrut +\mathstrut \) \(35\) \(\beta_{14}\mathstrut +\mathstrut \) \(44\) \(\beta_{13}\mathstrut +\mathstrut \) \(29\) \(\beta_{12}\mathstrut +\mathstrut \) \(86\) \(\beta_{11}\mathstrut +\mathstrut \) \(42\) \(\beta_{10}\mathstrut -\mathstrut \) \(22\) \(\beta_{9}\mathstrut -\mathstrut \) \(235\) \(\beta_{8}\mathstrut -\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(235\) \(\beta_{6}\mathstrut +\mathstrut \) \(68\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(125\) \(\beta_{3}\mathstrut +\mathstrut \) \(85\) \(\beta_{2}\mathstrut +\mathstrut \) \(98\) \(\beta_{1}\mathstrut -\mathstrut \) \(26\)
\(\nu^{8}\)\(=\)\(145\) \(\beta_{15}\mathstrut -\mathstrut \) \(365\) \(\beta_{14}\mathstrut +\mathstrut \) \(14\) \(\beta_{12}\mathstrut -\mathstrut \) \(248\) \(\beta_{11}\mathstrut +\mathstrut \) \(145\) \(\beta_{10}\mathstrut -\mathstrut \) \(309\) \(\beta_{9}\mathstrut +\mathstrut \) \(440\) \(\beta_{8}\mathstrut +\mathstrut \) \(89\) \(\beta_{7}\mathstrut +\mathstrut \) \(234\) \(\beta_{6}\mathstrut +\mathstrut \) \(92\) \(\beta_{5}\mathstrut -\mathstrut \) \(14\) \(\beta_{4}\mathstrut -\mathstrut \) \(36\) \(\beta_{3}\mathstrut +\mathstrut \) \(86\) \(\beta_{2}\mathstrut -\mathstrut \) \(245\) \(\beta_{1}\mathstrut +\mathstrut \) \(455\)
\(\nu^{9}\)\(=\)\(245\) \(\beta_{15}\mathstrut -\mathstrut \) \(212\) \(\beta_{14}\mathstrut -\mathstrut \) \(356\) \(\beta_{13}\mathstrut -\mathstrut \) \(128\) \(\beta_{12}\mathstrut -\mathstrut \) \(518\) \(\beta_{11}\mathstrut -\mathstrut \) \(245\) \(\beta_{10}\mathstrut +\mathstrut \) \(178\) \(\beta_{9}\mathstrut +\mathstrut \) \(1508\) \(\beta_{8}\mathstrut +\mathstrut \) \(145\) \(\beta_{7}\mathstrut -\mathstrut \) \(1508\) \(\beta_{6}\mathstrut -\mathstrut \) \(391\) \(\beta_{5}\mathstrut +\mathstrut \) \(50\) \(\beta_{4}\mathstrut -\mathstrut \) \(781\) \(\beta_{3}\mathstrut -\mathstrut \) \(513\) \(\beta_{2}\mathstrut -\mathstrut \) \(630\) \(\beta_{1}\mathstrut +\mathstrut \) \(234\)
\(\nu^{10}\)\(=\)\(-\)\(1022\) \(\beta_{15}\mathstrut +\mathstrut \) \(2385\) \(\beta_{14}\mathstrut +\mathstrut \) \(34\) \(\beta_{12}\mathstrut +\mathstrut \) \(1625\) \(\beta_{11}\mathstrut -\mathstrut \) \(1022\) \(\beta_{10}\mathstrut +\mathstrut \) \(2000\) \(\beta_{9}\mathstrut -\mathstrut \) \(2723\) \(\beta_{8}\mathstrut -\mathstrut \) \(637\) \(\beta_{7}\mathstrut -\mathstrut \) \(1517\) \(\beta_{6}\mathstrut -\mathstrut \) \(688\) \(\beta_{5}\mathstrut -\mathstrut \) \(34\) \(\beta_{4}\mathstrut +\mathstrut \) \(303\) \(\beta_{3}\mathstrut -\mathstrut \) \(601\) \(\beta_{2}\mathstrut +\mathstrut \) \(1589\) \(\beta_{1}\mathstrut -\mathstrut \) \(2780\)
\(\nu^{11}\)\(=\)\(-\)\(1432\) \(\beta_{15}\mathstrut +\mathstrut \) \(1322\) \(\beta_{14}\mathstrut +\mathstrut \) \(2578\) \(\beta_{13}\mathstrut +\mathstrut \) \(518\) \(\beta_{12}\mathstrut +\mathstrut \) \(3129\) \(\beta_{11}\mathstrut +\mathstrut \) \(1432\) \(\beta_{10}\mathstrut -\mathstrut \) \(1289\) \(\beta_{9}\mathstrut -\mathstrut \) \(9702\) \(\beta_{8}\mathstrut -\mathstrut \) \(1179\) \(\beta_{7}\mathstrut +\mathstrut \) \(9702\) \(\beta_{6}\mathstrut +\mathstrut \) \(2357\) \(\beta_{5}\mathstrut -\mathstrut \) \(756\) \(\beta_{4}\mathstrut +\mathstrut \) \(4968\) \(\beta_{3}\mathstrut +\mathstrut \) \(3142\) \(\beta_{2}\mathstrut +\mathstrut \) \(4056\) \(\beta_{1}\mathstrut -\mathstrut \) \(1831\)
\(\nu^{12}\)\(=\)\(6922\) \(\beta_{15}\mathstrut -\mathstrut \) \(15445\) \(\beta_{14}\mathstrut -\mathstrut \) \(619\) \(\beta_{12}\mathstrut -\mathstrut \) \(10601\) \(\beta_{11}\mathstrut +\mathstrut \) \(6922\) \(\beta_{10}\mathstrut -\mathstrut \) \(12821\) \(\beta_{9}\mathstrut +\mathstrut \) \(16992\) \(\beta_{8}\mathstrut +\mathstrut \) \(4298\) \(\beta_{7}\mathstrut +\mathstrut \) \(9634\) \(\beta_{6}\mathstrut +\mathstrut \) \(4853\) \(\beta_{5}\mathstrut +\mathstrut \) \(619\) \(\beta_{4}\mathstrut -\mathstrut \) \(2229\) \(\beta_{3}\mathstrut +\mathstrut \) \(4010\) \(\beta_{2}\mathstrut -\mathstrut \) \(10313\) \(\beta_{1}\mathstrut +\mathstrut \) \(17280\)
\(\nu^{13}\)\(=\)\(8460\) \(\beta_{15}\mathstrut -\mathstrut \) \(8372\) \(\beta_{14}\mathstrut -\mathstrut \) \(17726\) \(\beta_{13}\mathstrut -\mathstrut \) \(1817\) \(\beta_{12}\mathstrut -\mathstrut \) \(19052\) \(\beta_{11}\mathstrut -\mathstrut \) \(8460\) \(\beta_{10}\mathstrut +\mathstrut \) \(8863\) \(\beta_{9}\mathstrut +\mathstrut \) \(62396\) \(\beta_{8}\mathstrut +\mathstrut \) \(8775\) \(\beta_{7}\mathstrut -\mathstrut \) \(62396\) \(\beta_{6}\mathstrut -\mathstrut \) \(14559\) \(\beta_{5}\mathstrut +\mathstrut \) \(6779\) \(\beta_{4}\mathstrut -\mathstrut \) \(31794\) \(\beta_{3}\mathstrut -\mathstrut \) \(19510\) \(\beta_{2}\mathstrut -\mathstrut \) \(26153\) \(\beta_{1}\mathstrut +\mathstrut \) \(13340\)
\(\nu^{14}\)\(=\)\(-\)\(45841\) \(\beta_{15}\mathstrut +\mathstrut \) \(99462\) \(\beta_{14}\mathstrut +\mathstrut \) \(5217\) \(\beta_{12}\mathstrut +\mathstrut \) \(68772\) \(\beta_{11}\mathstrut -\mathstrut \) \(45841\) \(\beta_{10}\mathstrut +\mathstrut \) \(81769\) \(\beta_{9}\mathstrut -\mathstrut \) \(106633\) \(\beta_{8}\mathstrut -\mathstrut \) \(28148\) \(\beta_{7}\mathstrut -\mathstrut \) \(60771\) \(\beta_{6}\mathstrut -\mathstrut \) \(33083\) \(\beta_{5}\mathstrut -\mathstrut \) \(5217\) \(\beta_{4}\mathstrut +\mathstrut \) \(15390\) \(\beta_{3}\mathstrut -\mathstrut \) \(26186\) \(\beta_{2}\mathstrut +\mathstrut \) \(66810\) \(\beta_{1}\mathstrut -\mathstrut \) \(108434\)
\(\nu^{15}\)\(=\)\(-\)\(50619\) \(\beta_{15}\mathstrut +\mathstrut \) \(53382\) \(\beta_{14}\mathstrut +\mathstrut \) \(118538\) \(\beta_{13}\mathstrut +\mathstrut \) \(4347\) \(\beta_{12}\mathstrut +\mathstrut \) \(116998\) \(\beta_{11}\mathstrut +\mathstrut \) \(50619\) \(\beta_{10}\mathstrut -\mathstrut \) \(59269\) \(\beta_{9}\mathstrut -\mathstrut \) \(400724\) \(\beta_{8}\mathstrut -\mathstrut \) \(62032\) \(\beta_{7}\mathstrut +\mathstrut \) \(400724\) \(\beta_{6}\mathstrut +\mathstrut \) \(91111\) \(\beta_{5}\mathstrut -\mathstrut \) \(51949\) \(\beta_{4}\mathstrut +\mathstrut \) \(203762\) \(\beta_{3}\mathstrut +\mathstrut \) \(122303\) \(\beta_{2}\mathstrut +\mathstrut \) \(168575\) \(\beta_{1}\mathstrut -\mathstrut \) \(93153\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.16544i
0.176392i
2.16544i
0.176392i
1.03739i
2.52368i
1.14660i
0.333129i
1.42343i
1.83925i
1.42343i
1.83925i
1.14660i
0.333129i
1.03739i
2.52368i
−0.571745 1.75965i −0.488442 + 0.103822i −1.15144 + 0.836573i −0.603681 + 1.04561i 0.461954 + 0.800128i 3.41030 + 1.51837i −0.863288 0.627215i −2.51284 + 1.11879i 2.18505 + 0.464447i
7.2 0.380762 + 1.17187i −2.02963 + 0.431412i 0.389745 0.283166i 0.772811 1.33855i −1.27836 2.21419i −3.47491 1.54713i 2.47393 + 1.79742i 1.19265 0.531003i 1.86286 + 0.395962i
9.1 −0.571745 + 1.75965i −0.488442 0.103822i −1.15144 0.836573i −0.603681 1.04561i 0.461954 0.800128i 3.41030 1.51837i −0.863288 + 0.627215i −2.51284 1.11879i 2.18505 0.464447i
9.2 0.380762 1.17187i −2.02963 0.431412i 0.389745 + 0.283166i 0.772811 + 1.33855i −1.27836 + 2.21419i −3.47491 + 1.54713i 2.47393 1.79742i 1.19265 + 0.531003i 1.86286 0.395962i
10.1 −1.02470 0.744490i −0.155153 1.47618i −0.122284 0.376353i 1.90016 + 3.29117i −0.940018 + 1.62816i −2.14115 0.455117i −0.937688 + 2.88591i 0.779397 0.165666i 0.503147 4.78712i
10.2 −0.284315 0.206567i 0.302431 + 2.87744i −0.579869 1.78465i −1.48661 2.57489i 0.508398 0.880572i 1.05848 + 0.224987i −0.420982 + 1.29565i −5.25377 + 1.11672i −0.109221 + 1.03917i
14.1 −0.831304 + 2.55849i 0.949606 1.05464i −4.23677 3.07819i −0.304192 + 0.526876i 1.90889 + 3.30629i 0.180508 1.71742i 7.04481 5.11835i 0.103062 + 0.980572i −1.09513 1.21627i
14.2 0.640321 1.97070i −1.43153 + 1.58988i −1.85563 1.34820i −1.17396 + 2.03335i 2.21654 + 3.83916i 0.384094 3.65441i −0.492333 + 0.357701i −0.164841 1.56836i 3.25543 + 3.61552i
18.1 −1.86683 + 1.35633i −2.32289 + 1.03422i 1.02738 3.16196i 1.24923 + 2.16373i 2.93370 5.08132i 1.07187 + 1.19043i 0.944583 + 2.90713i 2.31884 2.57533i −5.26683 2.34494i
18.2 0.557811 0.405274i −0.824384 + 0.367040i −0.471127 + 1.44998i −1.85376 3.21080i −0.311099 + 0.538840i 0.510810 + 0.567312i 0.750969 + 2.31124i −1.46250 + 1.62427i −2.33530 1.03974i
19.1 −1.86683 1.35633i −2.32289 1.03422i 1.02738 + 3.16196i 1.24923 2.16373i 2.93370 + 5.08132i 1.07187 1.19043i 0.944583 2.90713i 2.31884 + 2.57533i −5.26683 + 2.34494i
19.2 0.557811 + 0.405274i −0.824384 0.367040i −0.471127 1.44998i −1.85376 + 3.21080i −0.311099 0.538840i 0.510810 0.567312i 0.750969 2.31124i −1.46250 1.62427i −2.33530 + 1.03974i
20.1 −0.831304 2.55849i 0.949606 + 1.05464i −4.23677 + 3.07819i −0.304192 0.526876i 1.90889 3.30629i 0.180508 + 1.71742i 7.04481 + 5.11835i 0.103062 0.980572i −1.09513 + 1.21627i
20.2 0.640321 + 1.97070i −1.43153 1.58988i −1.85563 + 1.34820i −1.17396 2.03335i 2.21654 3.83916i 0.384094 + 3.65441i −0.492333 0.357701i −0.164841 + 1.56836i 3.25543 3.61552i
28.1 −1.02470 + 0.744490i −0.155153 + 1.47618i −0.122284 + 0.376353i 1.90016 3.29117i −0.940018 1.62816i −2.14115 + 0.455117i −0.937688 2.88591i 0.779397 + 0.165666i 0.503147 + 4.78712i
28.2 −0.284315 + 0.206567i 0.302431 2.87744i −0.579869 + 1.78465i −1.48661 + 2.57489i 0.508398 + 0.880572i 1.05848 0.224987i −0.420982 1.29565i −5.25377 1.11672i −0.109221 1.03917i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
31.g Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(31, [\chi])\).