Properties

Label 65.2.t.a
Level 65
Weight 2
Character orbit 65.t
Analytic conductor 0.519
Analytic rank 0
Dimension 20
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 65 = 5 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 65.t (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + 68102 \nu^{6} + 46015 \nu^{4} + 16571 \nu^{2} + 2996 \nu - 409 \)\()/5992\)
\(\beta_{2}\)\(=\)\((\)\( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} - 68102 \nu^{6} - 46015 \nu^{4} - 16571 \nu^{2} + 2996 \nu + 409 \)\()/5992\)
\(\beta_{3}\)\(=\)\((\)\( -69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} - 257858 \nu^{6} - 105030 \nu^{4} - 19961 \nu^{2} - 539 \)\()/1712\)
\(\beta_{4}\)\(=\)\((\)\(321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + 869910 \nu^{13} - 548254 \nu^{12} + 3425391 \nu^{11} - 1503061 \nu^{10} + 7807469 \nu^{9} - 2268567 \nu^{8} + 9753906 \nu^{7} - 1805006 \nu^{6} + 5800042 \nu^{5} - 735210 \nu^{4} + 1183313 \nu^{3} - 139727 \nu^{2} + 39483 \nu - 3773\)\()/23968\)
\(\beta_{5}\)\(=\)\((\)\(700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + 912240 \nu^{13} + 1108682 \nu^{12} + 2625448 \nu^{11} + 3846937 \nu^{10} + 3934784 \nu^{9} + 7752327 \nu^{8} + 2192820 \nu^{7} + 8524592 \nu^{6} - 974624 \nu^{5} + 4331072 \nu^{4} - 1268316 \nu^{3} + 669321 \nu^{2} - 170688 \nu + 26961\)\()/11984\)
\(\beta_{6}\)\(=\)\((\)\( -409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} - 4719229 \nu^{7} - 2594086 \nu^{5} - 562582 \nu^{3} - 48473 \nu - 2996 \)\()/5992\)
\(\beta_{7}\)\(=\)\((\)\(1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + 2423698 \nu^{13} - 319614 \nu^{12} + 7689981 \nu^{11} - 870831 \nu^{10} + 13910023 \nu^{9} - 1440917 \nu^{8} + 13309334 \nu^{7} - 1613590 \nu^{6} + 5445918 \nu^{5} - 1272982 \nu^{4} + 481307 \nu^{3} - 501629 \nu^{2} + 71073 \nu - 30699\)\()/23968\)
\(\beta_{8}\)\(=\)\((\)\(700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + 912240 \nu^{13} - 1108682 \nu^{12} + 2625448 \nu^{11} - 3846937 \nu^{10} + 3934784 \nu^{9} - 7752327 \nu^{8} + 2192820 \nu^{7} - 8524592 \nu^{6} - 974624 \nu^{5} - 4331072 \nu^{4} - 1268316 \nu^{3} - 669321 \nu^{2} - 170688 \nu - 26961\)\()/11984\)
\(\beta_{9}\)\(=\)\((\)\(-419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} - 1064734 \nu^{13} + 7212374 \nu^{12} - 4108073 \nu^{11} + 23752795 \nu^{10} - 9110155 \nu^{9} + 45500681 \nu^{8} - 10820986 \nu^{7} + 47979462 \nu^{6} - 5625126 \nu^{5} + 23917926 \nu^{4} - 518719 \nu^{3} + 3775361 \nu^{2} + 159919 \nu + 83703\)\()/23968\)
\(\beta_{10}\)\(=\)\((\)\(-3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} - 5941306 \nu^{13} - 198098 \nu^{12} - 19648615 \nu^{11} + 414413 \nu^{10} - 37985157 \nu^{9} + 3110895 \nu^{8} - 40898326 \nu^{7} + 5871486 \nu^{6} - 21497042 \nu^{5} + 4145562 \nu^{4} - 4030089 \nu^{3} + 861543 \nu^{2} - 166551 \nu + 81509\)\()/23968\)
\(\beta_{11}\)\(=\)\((\)\(3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + 5941306 \nu^{13} + 869910 \nu^{12} + 19648615 \nu^{11} + 3425391 \nu^{10} + 37985157 \nu^{9} + 7807469 \nu^{8} + 40898326 \nu^{7} + 9753906 \nu^{6} + 21497042 \nu^{5} + 5800042 \nu^{4} + 4030089 \nu^{3} + 1183313 \nu^{2} + 154567 \nu + 39483\)\()/23968\)
\(\beta_{12}\)\(=\)\((\)\(3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + 5941306 \nu^{13} - 869910 \nu^{12} + 19648615 \nu^{11} - 3425391 \nu^{10} + 37985157 \nu^{9} - 7807469 \nu^{8} + 40898326 \nu^{7} - 9753906 \nu^{6} + 21497042 \nu^{5} - 5800042 \nu^{4} + 4030089 \nu^{3} - 1183313 \nu^{2} + 154567 \nu - 39483\)\()/23968\)
\(\beta_{13}\)\(=\)\((\)\(-2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} - 4279914 \nu^{13} - 1404094 \nu^{12} - 14010639 \nu^{11} - 4705845 \nu^{10} - 26598933 \nu^{9} - 9214959 \nu^{8} - 27637370 \nu^{7} - 9971998 \nu^{6} - 13384022 \nu^{5} - 5100714 \nu^{4} - 1923401 \nu^{3} - 798859 \nu^{2} - 7127 \nu - 15221\)\()/11984\)
\(\beta_{14}\)\(=\)\((\)\(2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + 4279914 \nu^{13} - 1404094 \nu^{12} + 14010639 \nu^{11} - 4705845 \nu^{10} + 26598933 \nu^{9} - 9214959 \nu^{8} + 27637370 \nu^{7} - 9971998 \nu^{6} + 13384022 \nu^{5} - 5100714 \nu^{4} + 1923401 \nu^{3} - 798859 \nu^{2} + 7127 \nu - 15221\)\()/11984\)
\(\beta_{15}\)\(=\)\((\)\(-6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} - 11139534 \nu^{13} + 3217222 \nu^{12} - 37111487 \nu^{11} + 10951069 \nu^{10} - 72399061 \nu^{9} + 21535831 \nu^{8} - 78924566 \nu^{7} + 22849098 \nu^{6} - 42433790 \nu^{5} + 10804742 \nu^{4} - 8585045 \nu^{3} + 1254563 \nu^{2} - 504763 \nu + 1285\)\()/23968\)
\(\beta_{16}\)\(=\)\((\)\(8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + 13551722 \nu^{13} + 273770 \nu^{12} + 45488043 \nu^{11} + 1753171 \nu^{10} + 89206401 \nu^{9} + 5197801 \nu^{8} + 97040830 \nu^{7} + 7462350 \nu^{6} + 50921018 \nu^{5} + 4439554 \nu^{4} + 9271077 \nu^{3} + 591213 \nu^{2} + 419203 \nu + 23947\)\()/23968\)
\(\beta_{17}\)\(=\)\((\)\(-4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} - 53105203 \nu^{9} - 58320582 \nu^{7} - 31671424 \nu^{5} - 6571905 \nu^{3} + 5992 \nu^{2} - 381235 \nu + 11984\)\()/11984\)
\(\beta_{18}\)\(=\)\((\)\(10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + 16210930 \nu^{13} - 548254 \nu^{12} + 53334711 \nu^{11} - 1503061 \nu^{10} + 102576749 \nu^{9} - 2268567 \nu^{8} + 110068986 \nu^{7} - 1805006 \nu^{6} + 58130790 \nu^{5} - 735210 \nu^{4} + 11443805 \nu^{3} - 127743 \nu^{2} + 679767 \nu + 20195\)\()/23968\)
\(\beta_{19}\)\(=\)\((\)\(-16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} - 110586 \nu^{14} - 26151990 \nu^{13} - 548254 \nu^{12} - 87742737 \nu^{11} - 1503061 \nu^{10} - 172338195 \nu^{9} - 2268567 \nu^{8} - 188614114 \nu^{7} - 1805006 \nu^{6} - 100694134 \nu^{5} - 735210 \nu^{4} - 19274543 \nu^{3} - 139727 \nu^{2} - 820129 \nu - 15757\)\()/23968\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{12} + \beta_{10} - \beta_{3} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 6 \beta_{2} - 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 7 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(-9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + 17 \beta_{10} + 8 \beta_{9} + 10 \beta_{8} + 8 \beta_{7} + 5 \beta_{6} - 6 \beta_{5} - 18 \beta_{4} + 9 \beta_{3} + 37 \beta_{2} + 46 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + 4 \beta_{13} + 38 \beta_{12} + 25 \beta_{11} + 53 \beta_{10} - 10 \beta_{9} + 37 \beta_{8} + 10 \beta_{7} - 13 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} - 53 \beta_{3} + 18 \beta_{2} + 51 \beta_{1} - 39\)
\(\nu^{7}\)\(=\)\(66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} - 121 \beta_{11} - 120 \beta_{10} - 52 \beta_{9} - 76 \beta_{8} - 52 \beta_{7} - 24 \beta_{6} + 28 \beta_{5} + 138 \beta_{4} - 69 \beta_{3} - 236 \beta_{2} - 304 \beta_{1} + 21\)
\(\nu^{8}\)\(=\)\(120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} - 56 \beta_{13} - 252 \beta_{12} - 229 \beta_{11} - 397 \beta_{10} + 76 \beta_{9} - 332 \beta_{8} - 84 \beta_{7} + 112 \beta_{6} + 100 \beta_{5} - 152 \beta_{4} + 398 \beta_{3} - 185 \beta_{2} - 368 \beta_{1} + 213\)
\(\nu^{9}\)\(=\)\(-462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + 818 \beta_{11} + 813 \beta_{10} + 325 \beta_{9} + 537 \beta_{8} + 325 \beta_{7} + 122 \beta_{6} - 113 \beta_{5} - 1011 \beta_{4} + 506 \beta_{3} + 1544 \beta_{2} + 2032 \beta_{1} - 170\)
\(\nu^{10}\)\(=\)\(-974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + 546 \beta_{13} + 1707 \beta_{12} + 1862 \beta_{11} + 2910 \beta_{10} - 539 \beta_{9} + 2648 \beta_{8} + 659 \beta_{7} - 854 \beta_{6} - 820 \beta_{5} + 1229 \beta_{4} - 2933 \beta_{3} + 1578 \beta_{2} + 2621 \beta_{1} - 1255\)
\(\nu^{11}\)\(=\)\(3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} - 5456 \beta_{11} - 5476 \beta_{10} - 2042 \beta_{9} - 3716 \beta_{8} - 2042 \beta_{7} - 676 \beta_{6} + 368 \beta_{5} + 7277 \beta_{4} - 3655 \beta_{3} - 10296 \beta_{2} - 13730 \beta_{1} + 1267\)
\(\nu^{12}\)\(=\)\(7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} - 4610 \beta_{13} - 11688 \beta_{12} - 14278 \beta_{11} - 20988 \beta_{10} + 3766 \beta_{9} - 19984 \beta_{8} - 4978 \beta_{7} + 6226 \beta_{6} + 6320 \beta_{5} - 9292 \beta_{4} + 21288 \beta_{3} - 12378 \beta_{2} - 18508 \beta_{1} + 7797\)
\(\nu^{13}\)\(=\)\(-22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + 60422 \beta_{12} + 36438 \beta_{11} + 37058 \beta_{10} + 13074 \beta_{9} + 25620 \beta_{8} + 13074 \beta_{7} + 4096 \beta_{6} - 528 \beta_{5} - 51934 \beta_{4} + 26218 \beta_{3} + 69645 \beta_{2} + 93629 \beta_{1} - 9130\)
\(\nu^{14}\)\(=\)\(-54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} - 100084 \beta_{14} + 36202 \beta_{13} + 80555 \beta_{12} + 105990 \beta_{11} + 149815 \beta_{10} - 26322 \beta_{9} + 146494 \beta_{8} + 36730 \beta_{7} - 44574 \beta_{6} - 46938 \beta_{5} + 68030 \beta_{4} - 152877 \beta_{3} + 92972 \beta_{2} + 130077 \beta_{1} - 50290\)
\(\nu^{15}\)\(=\)\(156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} - 412141 \beta_{12} - 245006 \beta_{11} - 252616 \beta_{10} - 85481 \beta_{9} - 176993 \beta_{8} - 85481 \beta_{7} - 26707 \beta_{6} - 6031 \beta_{5} + 368826 \beta_{4} - 187213 \beta_{3} - 476096 \beta_{2} - 643231 \beta_{1} + 64782\)
\(\nu^{16}\)\(=\)\(398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + 709463 \beta_{14} - 272912 \beta_{13} - 557750 \beta_{12} - 771768 \beta_{11} - 1062637 \beta_{10} + 184581 \beta_{9} - 1056441 \beta_{8} - 266881 \beta_{7} + 316597 \beta_{6} + 340947 \beta_{5} - 489763 \beta_{4} + 1090029 \beta_{3} - 681429 \beta_{2} - 912121 \beta_{1} + 333034\)
\(\nu^{17}\)\(=\)\(-1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + 2825568 \beta_{12} + 1660831 \beta_{11} + 1734115 \beta_{10} + 569378 \beta_{9} + 1226922 \beta_{8} + 569378 \beta_{7} + 182897 \beta_{6} + 88166 \beta_{5} - 2610994 \beta_{4} + 1332033 \beta_{3} + 3279739 \beta_{2} + 4444476 \beta_{1} - 456258\)
\(\nu^{18}\)\(=\)\(-2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} - 5013336 \beta_{14} + 2007386 \beta_{13} + 3875098 \beta_{12} + 5552425 \beta_{11} + 7508303 \beta_{10} - 1297848 \beta_{9} + 7545155 \beta_{8} + 1919220 \beta_{7} - 2240065 \beta_{6} - 2443653 \beta_{5} + 3492968 \beta_{4} - 7734751 \beta_{3} + 4921700 \beta_{2} + 6390257 \beta_{1} - 2246355\)
\(\nu^{19}\)\(=\)\(7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} - 19469181 \beta_{12} - 11344809 \beta_{11} - 11973534 \beta_{10} - 3849162 \beta_{9} - 8532880 \beta_{8} - 3849162 \beta_{7} - 1287312 \beta_{6} - 834556 \beta_{5} + 18442656 \beta_{4} - 9449577 \beta_{3} - 22718962 \beta_{2} - 30843334 \beta_{1} + 3202109\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(\beta_{11} + \beta_{12}\) \(-\beta_{11}\)

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
2.64975i
1.83163i
0.493902i
0.274809i
1.51805i
2.64975i
1.83163i
0.493902i
0.274809i
1.51805i
2.25081i
1.02262i
0.131303i
1.58474i
2.08794i
2.25081i
1.02262i
0.131303i
1.58474i
2.08794i
−2.29475 1.32488i 0.335680 + 1.25278i 2.51060 + 4.34849i 1.81654 1.30391i 0.889471 3.31955i 0.0561740 + 0.0972962i 8.00544i 1.14131 0.658935i −5.89604 + 0.585458i
7.2 −1.58624 0.915816i −0.512942 1.91432i 0.677439 + 1.17336i −1.69810 + 1.45480i −0.939520 + 3.50634i −1.76945 3.06478i 1.18163i −0.803451 + 0.463873i 4.02593 0.752519i
7.3 −0.427732 0.246951i −0.243392 0.908353i −0.878030 1.52079i −0.284413 2.21791i −0.120212 + 0.448637i 1.83775 + 3.18307i 1.85513i 1.83221 1.05783i −0.426062 + 1.01890i
7.4 −0.237991 0.137404i 0.611610 + 2.28256i −0.962240 1.66665i 1.45395 + 1.69883i 0.168076 0.627267i −0.193052 0.334376i 1.07848i −2.23793 + 1.29207i −0.112600 0.604086i
7.5 1.31467 + 0.759023i 0.175069 + 0.653367i 0.152233 + 0.263675i −2.15400 + 0.600231i −0.265763 + 0.991842i −1.29744 2.24723i 2.57390i 2.20184 1.27123i −3.28738 0.845834i
28.1 −2.29475 + 1.32488i 0.335680 1.25278i 2.51060 4.34849i 1.81654 + 1.30391i 0.889471 + 3.31955i 0.0561740 0.0972962i 8.00544i 1.14131 + 0.658935i −5.89604 0.585458i
28.2 −1.58624 + 0.915816i −0.512942 + 1.91432i 0.677439 1.17336i −1.69810 1.45480i −0.939520 3.50634i −1.76945 + 3.06478i 1.18163i −0.803451 0.463873i 4.02593 + 0.752519i
28.3 −0.427732 + 0.246951i −0.243392 + 0.908353i −0.878030 + 1.52079i −0.284413 + 2.21791i −0.120212 0.448637i 1.83775 3.18307i 1.85513i 1.83221 + 1.05783i −0.426062 1.01890i
28.4 −0.237991 + 0.137404i 0.611610 2.28256i −0.962240 + 1.66665i 1.45395 1.69883i 0.168076 + 0.627267i −0.193052 + 0.334376i 1.07848i −2.23793 1.29207i −0.112600 + 0.604086i
28.5 1.31467 0.759023i 0.175069 0.653367i 0.152233 0.263675i −2.15400 0.600231i −0.265763 0.991842i −1.29744 + 2.24723i 2.57390i 2.20184 + 1.27123i −3.28738 + 0.845834i
37.1 −1.94926 + 1.12540i −1.91913 0.514229i 1.53307 2.65535i 0.247944 2.22228i 4.31958 1.15743i 0.638592 1.10607i 2.39966i 0.820542 + 0.473740i 2.01765 + 4.61083i
37.2 −0.885613 + 0.511309i 2.69193 + 0.721300i −0.477126 + 0.826407i −1.45744 1.69584i −2.75281 + 0.737614i −0.481787 + 0.834479i 3.02107i 4.12812 + 2.38337i 2.15782 + 0.756660i
37.3 −0.113711 + 0.0656513i 0.332179 + 0.0890070i −0.991380 + 1.71712i 2.08297 + 0.813169i −0.0436159 + 0.0116869i 1.39069 2.40874i 0.522947i −2.49566 1.44087i −0.290243 + 0.0442830i
37.4 1.37242 0.792369i 0.190588 + 0.0510678i 0.255697 0.442881i −2.23506 + 0.0672627i 0.302032 0.0809291i 0.274164 0.474866i 2.35905i −2.56436 1.48053i −3.01415 + 1.86330i
37.5 1.80821 1.04397i −2.66159 0.713171i 1.17974 2.04338i 2.22760 0.194361i −5.55724 + 1.48906i −1.45563 + 2.52122i 0.750585i 3.97738 + 2.29634i 3.82506 2.67700i
58.1 −1.94926 1.12540i −1.91913 + 0.514229i 1.53307 + 2.65535i 0.247944 + 2.22228i 4.31958 + 1.15743i 0.638592 + 1.10607i 2.39966i 0.820542 0.473740i 2.01765 4.61083i
58.2 −0.885613 0.511309i 2.69193 0.721300i −0.477126 0.826407i −1.45744 + 1.69584i −2.75281 0.737614i −0.481787 0.834479i 3.02107i 4.12812 2.38337i 2.15782 0.756660i
58.3 −0.113711 0.0656513i 0.332179 0.0890070i −0.991380 1.71712i 2.08297 0.813169i −0.0436159 0.0116869i 1.39069 + 2.40874i 0.522947i −2.49566 + 1.44087i −0.290243 0.0442830i
58.4 1.37242 + 0.792369i 0.190588 0.0510678i 0.255697 + 0.442881i −2.23506 0.0672627i 0.302032 + 0.0809291i 0.274164 + 0.474866i 2.35905i −2.56436 + 1.48053i −3.01415 1.86330i
58.5 1.80821 + 1.04397i −2.66159 + 0.713171i 1.17974 + 2.04338i 2.22760 + 0.194361i −5.55724 1.48906i −1.45563 2.52122i 0.750585i 3.97738 2.29634i 3.82506 + 2.67700i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
65.t Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(65, [\chi])\).