Properties

Label 546.2.p.d
Level $546$
Weight $2$
Character orbit 546.p
Analytic conductor $4.360$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} + 8 x^{18} - 20 x^{17} + 56 x^{16} - 140 x^{15} + 288 x^{14} - 532 x^{13} + 1065 x^{12} - 2080 x^{11} + 3712 x^{10} - 6240 x^{9} + 9585 x^{8} - 14364 x^{7} + 23328 x^{6} - 34020 x^{5} + 40824 x^{4} - 43740 x^{3} + 52488 x^{2} - 78732 x + 59049\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} + 8 x^{18} - 20 x^{17} + 56 x^{16} - 140 x^{15} + 288 x^{14} - 532 x^{13} + 1065 x^{12} - 2080 x^{11} + 3712 x^{10} - 6240 x^{9} + 9585 x^{8} - 14364 x^{7} + 23328 x^{6} - 34020 x^{5} + 40824 x^{4} - 43740 x^{3} + 52488 x^{2} - 78732 x + 59049\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\nu^{19} - 4 \nu^{18} + 8 \nu^{17} - 20 \nu^{16} + 56 \nu^{15} - 140 \nu^{14} + 288 \nu^{13} - 532 \nu^{12} + 1065 \nu^{11} - 2080 \nu^{10} + 3712 \nu^{9} - 6240 \nu^{8} + 9585 \nu^{7} - 14364 \nu^{6} + 23328 \nu^{5} - 34020 \nu^{4} + 40824 \nu^{3} - 43740 \nu^{2} + 52488 \nu - 78732\)\()/19683\)
\(\beta_{3}\)\(=\)\((\)\(-591283 \nu^{19} + 1812757 \nu^{18} - 1243709 \nu^{17} + 8796287 \nu^{16} - 22183049 \nu^{15} + 49528031 \nu^{14} - 96430077 \nu^{13} + 163122697 \nu^{12} - 377618736 \nu^{11} + 700523074 \nu^{10} - 1226120692 \nu^{9} + 1951118562 \nu^{8} - 2911598613 \nu^{7} + 4228091973 \nu^{6} - 8595608949 \nu^{5} + 9631351899 \nu^{4} - 9909953829 \nu^{3} + 13461848865 \nu^{2} - 14639907033 \nu + 31565725515\)\()/ 922620942 \)
\(\beta_{4}\)\(=\)\((\)\(-1145710 \nu^{19} + 2718859 \nu^{18} - 4435910 \nu^{17} + 14742083 \nu^{16} - 39475514 \nu^{15} + 91907789 \nu^{14} - 165573378 \nu^{13} + 316347085 \nu^{12} - 655162428 \nu^{11} + 1207343404 \nu^{10} - 2081305228 \nu^{9} + 3365474952 \nu^{8} - 4835271474 \nu^{7} + 7518209373 \nu^{6} - 12869580186 \nu^{5} + 15725937699 \nu^{4} - 16944262182 \nu^{3} + 17447577633 \nu^{2} - 28858414158 \nu + 35608574349\)\()/ 922620942 \)
\(\beta_{5}\)\(=\)\((\)\(1809103 \nu^{19} - 3799282 \nu^{18} + 6316247 \nu^{17} - 22874330 \nu^{16} + 57083519 \nu^{15} - 134847878 \nu^{14} + 245298297 \nu^{13} - 465722662 \nu^{12} + 977653440 \nu^{11} - 1797446956 \nu^{10} + 3093360124 \nu^{9} - 5044887036 \nu^{8} + 7243827399 \nu^{7} - 11480141070 \nu^{6} + 19648126665 \nu^{5} - 22936943502 \nu^{4} + 26677007775 \nu^{3} - 28297378674 \nu^{2} + 42613465365 \nu - 55859054922\)\()/ 922620942 \)
\(\beta_{6}\)\(=\)\((\)\(207109 \nu^{19} - 525530 \nu^{18} + 908013 \nu^{17} - 2742694 \nu^{16} + 7610179 \nu^{15} - 18265678 \nu^{14} + 32574515 \nu^{13} - 62779858 \nu^{12} + 130637044 \nu^{11} - 241285588 \nu^{10} + 420417272 \nu^{9} - 682928764 \nu^{8} + 993196563 \nu^{7} - 1539726966 \nu^{6} + 2583457389 \nu^{5} - 3314244762 \nu^{4} + 3629530863 \nu^{3} - 3475290258 \nu^{2} + 6066162819 \nu - 7517003310\)\()/ 102513438 \)
\(\beta_{7}\)\(=\)\((\)\(3600752 \nu^{19} - 7719209 \nu^{18} + 14871178 \nu^{17} - 43755853 \nu^{16} + 113996236 \nu^{15} - 280118989 \nu^{14} + 490538010 \nu^{13} - 942541547 \nu^{12} + 1922005830 \nu^{11} - 3540629210 \nu^{10} + 6124264532 \nu^{9} - 9974073090 \nu^{8} + 13997000556 \nu^{7} - 22141668087 \nu^{6} + 36623065734 \nu^{5} - 45473045139 \nu^{4} + 52647881364 \nu^{3} - 45174739797 \nu^{2} + 83350747170 \nu - 101791598967\)\()/ 922620942 \)
\(\beta_{8}\)\(=\)\((\)\(-6137750 \nu^{19} + 11934278 \nu^{18} - 23850313 \nu^{17} + 72754033 \nu^{16} - 189994162 \nu^{15} + 456946786 \nu^{14} - 796311297 \nu^{13} + 1568419823 \nu^{12} - 3198906456 \nu^{11} + 5905808318 \nu^{10} - 10122436646 \nu^{9} + 16545669270 \nu^{8} - 23286524094 \nu^{7} + 37684471428 \nu^{6} - 61702135317 \nu^{5} + 75069094059 \nu^{4} - 86923334142 \nu^{3} + 78811309368 \nu^{2} - 148550336937 \nu + 160632077265\)\()/ 922620942 \)
\(\beta_{9}\)\(=\)\((\)\(-6219053 \nu^{19} + 16297241 \nu^{18} - 25628572 \nu^{17} + 87485407 \nu^{16} - 226745719 \nu^{15} + 546222811 \nu^{14} - 1000573218 \nu^{13} + 1870517609 \nu^{12} - 3958976274 \nu^{11} + 7269391790 \nu^{10} - 12657643124 \nu^{9} + 20557329630 \nu^{8} - 30043821447 \nu^{7} + 46048004037 \nu^{6} - 79390694106 \nu^{5} + 97514241525 \nu^{4} - 110437162803 \nu^{3} + 112341087729 \nu^{2} - 169649850276 \nu + 245514384909\)\()/ 922620942 \)
\(\beta_{10}\)\(=\)\((\)\(8160955 \nu^{19} - 14230570 \nu^{18} + 29484806 \nu^{17} - 91668161 \nu^{16} + 238751381 \nu^{15} - 572551214 \nu^{14} + 979514682 \nu^{13} - 1952694169 \nu^{12} + 3986157606 \nu^{11} - 7378067032 \nu^{10} + 12576040006 \nu^{9} - 20557049262 \nu^{8} + 28585745865 \nu^{7} - 47364385338 \nu^{6} + 77325343956 \nu^{5} - 92529283149 \nu^{4} + 107955544743 \nu^{3} - 96190169274 \nu^{2} + 191918277936 \nu - 196877298249\)\()/ 922620942 \)
\(\beta_{11}\)\(=\)\((\)\(1401858 \nu^{19} - 2805743 \nu^{18} + 5555663 \nu^{17} - 17079982 \nu^{16} + 44704246 \nu^{15} - 107928967 \nu^{14} + 188540353 \nu^{13} - 370866366 \nu^{12} + 762301298 \nu^{11} - 1406765706 \nu^{10} + 2417098970 \nu^{9} - 3949312184 \nu^{8} + 5608685508 \nu^{7} - 9053255187 \nu^{6} + 14859684549 \nu^{5} - 18182685762 \nu^{4} + 21072652848 \nu^{3} - 19289765007 \nu^{2} + 35845028415 \nu - 40269777750\)\()/ 102513438 \)
\(\beta_{12}\)\(=\)\((\)\(12764248 \nu^{19} - 32968507 \nu^{18} + 55041872 \nu^{17} - 177316112 \nu^{16} + 461647082 \nu^{15} - 1132385207 \nu^{14} + 2051470158 \nu^{13} - 3846558796 \nu^{12} + 8107972296 \nu^{11} - 14901403606 \nu^{10} + 25969269982 \nu^{9} - 42245052504 \nu^{8} + 61482360132 \nu^{7} - 94587787719 \nu^{6} + 161223451704 \nu^{5} - 200684011590 \nu^{4} + 230059852452 \nu^{3} - 220782301749 \nu^{2} + 348951351456 \nu - 503863699824\)\()/ 922620942 \)
\(\beta_{13}\)\(=\)\((\)\(-14967076 \nu^{19} + 32476501 \nu^{18} - 60283367 \nu^{17} + 191012801 \nu^{16} - 492502406 \nu^{15} + 1194139403 \nu^{14} - 2126160861 \nu^{13} + 4096932529 \nu^{12} - 8480479290 \nu^{11} + 15609420130 \nu^{10} - 26934548428 \nu^{9} + 43974219426 \nu^{8} - 62868993030 \nu^{7} + 99623908881 \nu^{6} - 165877709517 \nu^{5} + 202410264879 \nu^{4} - 235608345810 \nu^{3} + 221225221737 \nu^{2} - 377550481455 \nu + 469129641165\)\()/ 922620942 \)
\(\beta_{14}\)\(=\)\((\)\(-16405789 \nu^{19} + 40407640 \nu^{18} - 67932887 \nu^{17} + 223729739 \nu^{16} - 572088509 \nu^{15} + 1400929526 \nu^{14} - 2542697307 \nu^{13} + 4767005767 \nu^{12} - 10035743556 \nu^{11} + 18429100294 \nu^{10} - 32045519230 \nu^{9} + 52195233648 \nu^{8} - 75640778703 \nu^{7} + 116936126496 \nu^{6} - 199352694537 \nu^{5} + 244642780845 \nu^{4} - 283947641073 \nu^{3} + 275112593370 \nu^{2} - 426842054109 \nu + 614553274503\)\()/ 922620942 \)
\(\beta_{15}\)\(=\)\((\)\(-22008049 \nu^{19} + 50526346 \nu^{18} - 89059982 \nu^{17} + 285844295 \nu^{16} - 740590265 \nu^{15} + 1807660472 \nu^{14} - 3228128322 \nu^{13} + 6142923817 \nu^{12} - 12845965920 \nu^{11} + 23651480950 \nu^{10} - 40988799724 \nu^{9} + 66770982618 \nu^{8} - 95957394957 \nu^{7} + 150629797800 \nu^{6} - 254122535574 \nu^{5} + 312275207637 \nu^{4} - 361083418299 \nu^{3} + 340455280734 \nu^{2} - 568586875698 \nu + 757656557703\)\()/ 922620942 \)
\(\beta_{16}\)\(=\)\((\)\(22182449 \nu^{19} - 48929219 \nu^{18} + 88744186 \nu^{17} - 282283006 \nu^{16} + 733999765 \nu^{15} - 1784719441 \nu^{14} + 3163841238 \nu^{13} - 6075890744 \nu^{12} + 12654713910 \nu^{11} - 23322070364 \nu^{10} + 40306249970 \nu^{9} - 65675532234 \nu^{8} + 94070430369 \nu^{7} - 148887620343 \nu^{6} + 249558807654 \nu^{5} - 306037622898 \nu^{4} + 353585946357 \nu^{3} - 331359392457 \nu^{2} + 571404405294 \nu - 726777611424\)\()/ 922620942 \)
\(\beta_{17}\)\(=\)\((\)\(-9829197 \nu^{19} + 21546844 \nu^{18} - 39629719 \nu^{17} + 125443901 \nu^{16} - 323611943 \nu^{15} + 790047680 \nu^{14} - 1403067887 \nu^{13} + 2691078003 \nu^{12} - 5599878604 \nu^{11} + 10308804456 \nu^{10} - 17821601710 \nu^{9} + 29068716970 \nu^{8} - 41597505453 \nu^{7} + 65859809976 \nu^{6} - 110070068589 \nu^{5} + 134794242531 \nu^{4} - 157087185489 \nu^{3} + 146069719794 \nu^{2} - 249990115401 \nu + 320861998035\)\()/ 307540314 \)
\(\beta_{18}\)\(=\)\((\)\(12501950 \nu^{19} - 29001470 \nu^{18} + 51438895 \nu^{17} - 163953493 \nu^{16} + 424488796 \nu^{15} - 1036729930 \nu^{14} + 1855119417 \nu^{13} - 3530868755 \nu^{12} + 7375086990 \nu^{11} - 13568359388 \nu^{10} + 23519747714 \nu^{9} - 38329918236 \nu^{8} + 55164606012 \nu^{7} - 86427077166 \nu^{6} + 145479539781 \nu^{5} - 179124391359 \nu^{4} + 207392260842 \nu^{3} - 195523866738 \nu^{2} + 325027052055 \nu - 433184428467\)\()/ 307540314 \)
\(\beta_{19}\)\(=\)\((\)\(38492941 \nu^{19} - 87947617 \nu^{18} + 156364490 \nu^{17} - 502678874 \nu^{16} + 1298071811 \nu^{15} - 3167240945 \nu^{14} + 5662985592 \nu^{13} - 10793859646 \nu^{12} + 22566210714 \nu^{11} - 41527419520 \nu^{10} + 71931354142 \nu^{9} - 117229552668 \nu^{8} + 168641891631 \nu^{7} - 265040419653 \nu^{6} + 446073934248 \nu^{5} - 547162246098 \nu^{4} + 634610200473 \nu^{3} - 600430984443 \nu^{2} + 999051645006 \nu - 1324865603718\)\()/ 922620942 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{19} - \beta_{18} - \beta_{17} - 2 \beta_{15} - \beta_{13} - \beta_{10} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-\beta_{19} + 2 \beta_{18} + \beta_{16} + \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{19} + 5 \beta_{18} + 3 \beta_{17} + \beta_{16} + 4 \beta_{15} + 4 \beta_{14} - \beta_{12} - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 5 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_{1}\)
\(\nu^{5}\)\(=\)\(4 \beta_{19} + 10 \beta_{18} + 6 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} - 4 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 7 \beta_{3} - 10 \beta_{1} + 13\)
\(\nu^{6}\)\(=\)\(5 \beta_{19} + 16 \beta_{18} + 6 \beta_{17} + 14 \beta_{15} + \beta_{14} + 9 \beta_{13} - 12 \beta_{12} + \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 7 \beta_{8} + \beta_{7} - \beta_{6} - 6 \beta_{5} + \beta_{4} - 15 \beta_{3} + 2 \beta_{2} - 10 \beta_{1} + 14\)
\(\nu^{7}\)\(=\)\(-15 \beta_{19} - 17 \beta_{18} - 8 \beta_{17} + 14 \beta_{16} - 20 \beta_{15} - 8 \beta_{14} - 20 \beta_{13} + 4 \beta_{12} - 14 \beta_{10} + 14 \beta_{9} + 10 \beta_{8} - 9 \beta_{7} + 14 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} + 3 \beta_{3} - 29 \beta_{2} + 22 \beta_{1} - 9\)
\(\nu^{8}\)\(=\)\(-5 \beta_{19} + 3 \beta_{18} + 4 \beta_{17} + 28 \beta_{16} - 24 \beta_{15} + 7 \beta_{14} - 17 \beta_{13} - 34 \beta_{12} - 8 \beta_{11} - 42 \beta_{10} + 20 \beta_{9} + 13 \beta_{8} - 14 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} - 55 \beta_{4} + 7 \beta_{3} - 25 \beta_{2} - 14 \beta_{1} + 16\)
\(\nu^{9}\)\(=\)\(46 \beta_{19} + 53 \beta_{18} + 30 \beta_{17} + 20 \beta_{16} + 55 \beta_{15} + 48 \beta_{14} + 13 \beta_{13} - 55 \beta_{12} - 24 \beta_{11} - 41 \beta_{10} + 6 \beta_{9} - 10 \beta_{8} + 38 \beta_{7} + 10 \beta_{6} + 29 \beta_{5} + 37 \beta_{4} - 43 \beta_{3} - 13 \beta_{2} - 42 \beta_{1} + 18\)
\(\nu^{10}\)\(=\)\(-12 \beta_{19} + 21 \beta_{18} + 32 \beta_{17} - 47 \beta_{16} - 30 \beta_{15} + 7 \beta_{14} - 25 \beta_{13} + 17 \beta_{12} + 73 \beta_{11} + 39 \beta_{10} + 27 \beta_{9} + 178 \beta_{8} + 75 \beta_{7} + 27 \beta_{6} + 138 \beta_{5} + 9 \beta_{4} + 17 \beta_{3} - 99 \beta_{2} + 65 \beta_{1} + 15\)
\(\nu^{11}\)\(=\)\(17 \beta_{19} + 147 \beta_{18} - 8 \beta_{17} - 73 \beta_{16} + 110 \beta_{15} - 5 \beta_{14} + 113 \beta_{13} - 100 \beta_{12} - 3 \beta_{11} + 49 \beta_{10} - 32 \beta_{9} - 24 \beta_{8} - 45 \beta_{7} + 18 \beta_{6} - 96 \beta_{5} - 81 \beta_{4} - 42 \beta_{3} - 34 \beta_{2} - 86 \beta_{1} + 180\)
\(\nu^{12}\)\(=\)\(-69 \beta_{19} - 59 \beta_{18} - 221 \beta_{17} + 16 \beta_{16} + 180 \beta_{15} + 32 \beta_{14} - 28 \beta_{13} + 195 \beta_{12} - 179 \beta_{11} + 125 \beta_{10} - 52 \beta_{9} - 74 \beta_{8} - 71 \beta_{7} + 173 \beta_{6} + 63 \beta_{5} + 157 \beta_{4} - 55 \beta_{3} + 59 \beta_{2} + 300 \beta_{1} + 394\)
\(\nu^{13}\)\(=\)\(-143 \beta_{19} - 185 \beta_{18} - 115 \beta_{17} - 69 \beta_{16} - 360 \beta_{15} + 17 \beta_{14} - 282 \beta_{13} + 175 \beta_{12} - 40 \beta_{11} + 36 \beta_{10} + 265 \beta_{9} + 376 \beta_{8} - 157 \beta_{7} + 82 \beta_{6} + 239 \beta_{5} - 667 \beta_{4} + 224 \beta_{3} - 17 \beta_{2} + 833 \beta_{1} + 132\)
\(\nu^{14}\)\(=\)\(-111 \beta_{19} + 603 \beta_{18} - 472 \beta_{17} - 193 \beta_{16} - 50 \beta_{15} + 495 \beta_{14} + 238 \beta_{13} - 822 \beta_{12} - 408 \beta_{11} - 748 \beta_{10} - 297 \beta_{9} - 1004 \beta_{8} - 703 \beta_{7} - 173 \beta_{6} - 990 \beta_{5} - 455 \beta_{4} - 747 \beta_{3} + 407 \beta_{2} - 460 \beta_{1} + 293\)
\(\nu^{15}\)\(=\)\(-964 \beta_{19} + 1029 \beta_{18} - 486 \beta_{17} + 354 \beta_{16} + 574 \beta_{15} + 816 \beta_{14} - 1100 \beta_{13} + 100 \beta_{12} - 640 \beta_{11} + 294 \beta_{10} + 206 \beta_{9} + 1721 \beta_{8} - 216 \beta_{7} + 732 \beta_{6} + 933 \beta_{5} + 577 \beta_{4} - 885 \beta_{3} - 161 \beta_{2} + 493 \beta_{1} + 624\)
\(\nu^{16}\)\(=\)\(1579 \beta_{19} + 4376 \beta_{18} + 2728 \beta_{17} - \beta_{16} + 2526 \beta_{15} + 2276 \beta_{14} + 634 \beta_{13} - 1442 \beta_{12} - 1053 \beta_{11} - 120 \beta_{10} + 2568 \beta_{9} + 109 \beta_{8} + 62 \beta_{7} - 282 \beta_{6} + 3426 \beta_{5} - 537 \beta_{4} - 1396 \beta_{3} - 993 \beta_{2} - 2536 \beta_{1} + 499\)
\(\nu^{17}\)\(=\)\(3875 \beta_{19} + 9049 \beta_{18} + 1817 \beta_{17} - 1996 \beta_{16} + 8600 \beta_{15} + 4560 \beta_{14} + 6889 \beta_{13} - 3279 \beta_{12} - 619 \beta_{11} - 2099 \beta_{10} + 547 \beta_{9} - 7356 \beta_{8} + 749 \beta_{7} - 4003 \beta_{6} + 1814 \beta_{5} + 1357 \beta_{4} - 7411 \beta_{3} + 4139 \beta_{2} - 9022 \beta_{1} + 9755\)
\(\nu^{18}\)\(=\)\(1799 \beta_{19} + 3898 \beta_{18} + 1953 \beta_{17} - 513 \beta_{16} + 6992 \beta_{15} - 1440 \beta_{14} + 184 \beta_{13} - 332 \beta_{12} + 1965 \beta_{11} + 2549 \beta_{10} + 6753 \beta_{9} + 7899 \beta_{8} + 3107 \beta_{7} + 4710 \beta_{6} + 4187 \beta_{5} - 4451 \beta_{4} - 5106 \beta_{3} - 110 \beta_{2} + 4713 \beta_{1} + 5875\)
\(\nu^{19}\)\(=\)\(-6914 \beta_{19} - 7282 \beta_{18} - 5584 \beta_{17} + 4891 \beta_{16} - 15402 \beta_{15} - 5939 \beta_{14} - 5717 \beta_{13} - 7184 \beta_{12} - 1869 \beta_{11} - 11635 \beta_{10} + 7658 \beta_{9} - 2514 \beta_{8} - 10618 \beta_{7} + 10304 \beta_{6} - 4366 \beta_{5} - 14896 \beta_{4} + 3383 \beta_{3} - 13990 \beta_{2} + 6136 \beta_{1} - 13187\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
1.47393 0.909692i
1.72893 + 0.103975i
−0.962473 1.44002i
−1.25702 1.19160i
−1.39758 + 1.02312i
0.0557032 + 1.73115i
0.831607 + 1.51935i
−1.01577 1.40293i
1.72939 + 0.0958811i
0.813276 1.52924i
1.47393 + 0.909692i
1.72893 0.103975i
−0.962473 + 1.44002i
−1.25702 + 1.19160i
−1.39758 1.02312i
0.0557032 1.73115i
0.831607 1.51935i
−1.01577 + 1.40293i
1.72939 0.0958811i
0.813276 + 1.52924i
−0.707107 0.707107i −1.68547 + 0.398975i 1.00000i 0.559062 + 0.559062i 1.47393 + 0.909692i −0.707107 0.707107i 0.707107 0.707107i 2.68164 1.34492i 0.790633i
239.2 −0.707107 0.707107i −1.14901 + 1.29606i 1.00000i −0.237140 0.237140i 1.72893 0.103975i −0.707107 0.707107i 0.707107 0.707107i −0.359531 2.97838i 0.335367i
239.3 −0.707107 0.707107i −0.337674 1.69882i 1.00000i 2.19168 + 2.19168i −0.962473 + 1.44002i −0.707107 0.707107i 0.707107 0.707107i −2.77195 + 1.14729i 3.09950i
239.4 −0.707107 0.707107i 0.0462556 1.73143i 1.00000i −1.43062 1.43062i −1.25702 + 1.19160i −0.707107 0.707107i 0.707107 0.707107i −2.99572 0.160177i 2.02321i
239.5 −0.707107 0.707107i 1.71169 0.264783i 1.00000i −0.790081 0.790081i −1.39758 1.02312i −0.707107 0.707107i 0.707107 0.707107i 2.85978 0.906454i 1.11734i
239.6 0.707107 + 0.707107i −1.18472 1.26350i 1.00000i −2.80148 2.80148i 0.0557032 1.73115i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.192862 + 2.99379i 3.96189i
239.7 0.707107 + 0.707107i −0.486309 1.66238i 1.00000i 2.72859 + 2.72859i 0.831607 1.51935i 0.707107 + 0.707107i −0.707107 + 0.707107i −2.52701 + 1.61686i 3.85881i
239.8 0.707107 + 0.707107i 0.273767 + 1.71028i 1.00000i 1.75112 + 1.75112i −1.01577 + 1.40293i 0.707107 + 0.707107i −0.707107 + 0.707107i −2.85010 + 0.936434i 2.47645i
239.9 0.707107 + 0.707107i 1.15507 1.29067i 1.00000i −0.616653 0.616653i 1.72939 0.0958811i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.331633 2.98161i 0.872079i
239.10 0.707107 + 0.707107i 1.65641 + 0.506265i 1.00000i 0.645532 + 0.645532i 0.813276 + 1.52924i 0.707107 + 0.707107i −0.707107 + 0.707107i 2.48739 + 1.67717i 0.912921i
281.1 −0.707107 + 0.707107i −1.68547 0.398975i 1.00000i 0.559062 0.559062i 1.47393 0.909692i −0.707107 + 0.707107i 0.707107 + 0.707107i 2.68164 + 1.34492i 0.790633i
281.2 −0.707107 + 0.707107i −1.14901 1.29606i 1.00000i −0.237140 + 0.237140i 1.72893 + 0.103975i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.359531 + 2.97838i 0.335367i
281.3 −0.707107 + 0.707107i −0.337674 + 1.69882i 1.00000i 2.19168 2.19168i −0.962473 1.44002i −0.707107 + 0.707107i 0.707107 + 0.707107i −2.77195 1.14729i 3.09950i
281.4 −0.707107 + 0.707107i 0.0462556 + 1.73143i 1.00000i −1.43062 + 1.43062i −1.25702 1.19160i −0.707107 + 0.707107i 0.707107 + 0.707107i −2.99572 + 0.160177i 2.02321i
281.5 −0.707107 + 0.707107i 1.71169 + 0.264783i 1.00000i −0.790081 + 0.790081i −1.39758 + 1.02312i −0.707107 + 0.707107i 0.707107 + 0.707107i 2.85978 + 0.906454i 1.11734i
281.6 0.707107 0.707107i −1.18472 + 1.26350i 1.00000i −2.80148 + 2.80148i 0.0557032 + 1.73115i 0.707107 0.707107i −0.707107 0.707107i −0.192862 2.99379i 3.96189i
281.7 0.707107 0.707107i −0.486309 + 1.66238i 1.00000i 2.72859 2.72859i 0.831607 + 1.51935i 0.707107 0.707107i −0.707107 0.707107i −2.52701 1.61686i 3.85881i
281.8 0.707107 0.707107i 0.273767 1.71028i 1.00000i 1.75112 1.75112i −1.01577 1.40293i 0.707107 0.707107i −0.707107 0.707107i −2.85010 0.936434i 2.47645i
281.9 0.707107 0.707107i 1.15507 + 1.29067i 1.00000i −0.616653 + 0.616653i 1.72939 + 0.0958811i 0.707107 0.707107i −0.707107 0.707107i −0.331633 + 2.98161i 0.872079i
281.10 0.707107 0.707107i 1.65641 0.506265i 1.00000i 0.645532 0.645532i 0.813276 1.52924i 0.707107 0.707107i −0.707107 0.707107i 2.48739 1.67717i 0.912921i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 281.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.p.d yes 20
3.b odd 2 1 546.2.p.c 20
13.d odd 4 1 546.2.p.c 20
39.f even 4 1 inner 546.2.p.d yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.p.c 20 3.b odd 2 1
546.2.p.c 20 13.d odd 4 1
546.2.p.d yes 20 1.a even 1 1 trivial
546.2.p.d yes 20 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{20} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{5} \)
$3$ \( 59049 + 26244 T^{2} - 17496 T^{3} - 7776 T^{5} + 1134 T^{6} + 585 T^{8} + 264 T^{9} - 40 T^{10} + 88 T^{11} + 65 T^{12} + 14 T^{14} - 32 T^{15} - 8 T^{17} + 4 T^{18} + T^{20} \)
$5$ \( 3136 + 10752 T + 18432 T^{2} - 16512 T^{3} + 23328 T^{4} + 44224 T^{5} + 57984 T^{6} - 39008 T^{7} + 31892 T^{8} + 42512 T^{9} + 44192 T^{10} - 18896 T^{11} + 4360 T^{12} + 1344 T^{13} + 2176 T^{14} - 1064 T^{15} + 261 T^{16} + 4 T^{17} + 8 T^{18} - 4 T^{19} + T^{20} \)
$7$ \( ( 1 + T^{4} )^{5} \)
$11$ \( 1024 + 81920 T + 3276800 T^{2} - 26042368 T^{3} + 98110656 T^{4} - 211429376 T^{5} + 286302208 T^{6} - 236432640 T^{7} + 126800129 T^{8} - 44535312 T^{9} + 11620480 T^{10} - 3096784 T^{11} + 1110371 T^{12} - 354912 T^{13} + 80128 T^{14} - 12768 T^{15} + 2339 T^{16} - 592 T^{17} + 128 T^{18} - 16 T^{19} + T^{20} \)
$13$ \( 137858491849 - 42417997492 T - 4894384326 T^{2} + 7780816108 T^{3} - 1974164881 T^{4} - 252479240 T^{5} + 245738844 T^{6} - 49511592 T^{7} - 402896 T^{8} + 2991560 T^{9} - 957612 T^{10} + 230120 T^{11} - 2384 T^{12} - 22536 T^{13} + 8604 T^{14} - 680 T^{15} - 409 T^{16} + 124 T^{17} - 6 T^{18} - 4 T^{19} + T^{20} \)
$17$ \( ( 170048 + 450816 T + 22624 T^{2} - 118368 T^{3} - 14420 T^{4} + 11032 T^{5} + 1624 T^{6} - 432 T^{7} - 69 T^{8} + 6 T^{9} + T^{10} )^{2} \)
$19$ \( 470369344 + 3279225600 T + 11430720000 T^{2} + 6259912448 T^{3} + 332138528 T^{4} - 3496389248 T^{5} + 9208126208 T^{6} - 259795456 T^{7} + 57832404 T^{8} - 210790288 T^{9} + 160307360 T^{10} - 25359920 T^{11} + 2197832 T^{12} - 303616 T^{13} + 234080 T^{14} - 38672 T^{15} + 3253 T^{16} - 92 T^{17} + 72 T^{18} - 12 T^{19} + T^{20} \)
$23$ \( ( 13712 - 51520 T + 47192 T^{2} + 10296 T^{3} - 19119 T^{4} - 818 T^{5} + 2319 T^{6} + 76 T^{7} - 89 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$29$ \( 84953531416576 + 98234701709312 T^{2} + 32072967028736 T^{4} + 4752153034752 T^{6} + 374791065856 T^{8} + 17090254592 T^{10} + 471365792 T^{12} + 7955696 T^{14} + 80001 T^{16} + 438 T^{18} + T^{20} \)
$31$ \( 27561312256 + 285133144064 T + 1474909995008 T^{2} + 2742279274496 T^{3} + 2896737183744 T^{4} + 1871859079168 T^{5} + 774645587968 T^{6} + 197212745472 T^{7} + 34453106000 T^{8} + 7361778720 T^{9} + 2895672352 T^{10} + 724391672 T^{11} + 92622833 T^{12} + 1561584 T^{13} + 605248 T^{14} + 151680 T^{15} + 19106 T^{16} + 80 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$37$ \( 13705040929024 + 46843439790080 T + 80054771916800 T^{2} + 26144522589184 T^{3} + 3490453302560 T^{4} - 958828678528 T^{5} + 1271485018112 T^{6} + 278716596480 T^{7} + 34335550849 T^{8} - 11363896288 T^{9} + 2005160448 T^{10} + 180101088 T^{11} + 39555811 T^{12} - 12844352 T^{13} + 1590272 T^{14} - 42304 T^{15} + 13699 T^{16} - 3808 T^{17} + 512 T^{18} - 32 T^{19} + T^{20} \)
$41$ \( 2379590078464 - 8715521392640 T + 15960798003200 T^{2} + 42485509382144 T^{3} + 42645764694784 T^{4} + 23289224050688 T^{5} + 7930394181632 T^{6} + 1662471100416 T^{7} + 215213323056 T^{8} + 20757685120 T^{9} + 4746760192 T^{10} + 1083655712 T^{11} + 136469345 T^{12} + 4565656 T^{13} + 617504 T^{14} + 169704 T^{15} + 23250 T^{16} + 88 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$43$ \( 28202500096 + 105549660160 T^{2} + 91497693184 T^{4} + 32908148736 T^{6} + 6073467904 T^{8} + 625207296 T^{10} + 36976064 T^{12} + 1264736 T^{14} + 24433 T^{16} + 246 T^{18} + T^{20} \)
$47$ \( 77352165720064 + 57670485737472 T + 21498331004928 T^{2} - 18064289734656 T^{3} + 7491004737536 T^{4} + 178890403840 T^{5} + 160717916160 T^{6} - 136441507072 T^{7} + 55311984208 T^{8} - 1229384896 T^{9} + 216756224 T^{10} - 188147840 T^{11} + 85787137 T^{12} - 2218112 T^{13} + 18432 T^{14} - 16000 T^{15} + 18290 T^{16} - 192 T^{17} + T^{20} \)
$53$ \( 38225235214336 + 192066910126080 T^{2} + 71287695503360 T^{4} + 10364871487488 T^{6} + 771354171136 T^{8} + 32811511680 T^{10} + 834546064 T^{12} + 12744096 T^{14} + 112904 T^{16} + 528 T^{18} + T^{20} \)
$59$ \( 2548339707904 - 4176363331584 T + 3422230290432 T^{2} - 775279747072 T^{3} + 6262658560 T^{4} - 7876141056 T^{5} + 122429010944 T^{6} - 13725167872 T^{7} - 971208048 T^{8} + 1982250784 T^{9} + 1942530848 T^{10} + 180423056 T^{11} + 9078836 T^{12} + 1889216 T^{13} + 2542048 T^{14} + 245456 T^{15} + 11952 T^{16} + 216 T^{17} + 200 T^{18} + 20 T^{19} + T^{20} \)
$61$ \( ( 933154 + 4092164 T + 2707810 T^{2} - 2126960 T^{3} - 620581 T^{4} + 83702 T^{5} + 22981 T^{6} - 892 T^{7} - 277 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$67$ \( 69751627841536 + 62225704288256 T + 27755898994688 T^{2} - 22340077355008 T^{3} + 15036897705984 T^{4} + 3074494259200 T^{5} + 336758947840 T^{6} - 157940872192 T^{7} + 219224306692 T^{8} + 53731682072 T^{9} + 6445483592 T^{10} - 8007204 T^{11} + 490941625 T^{12} + 121291720 T^{13} + 14686416 T^{14} + 788864 T^{15} + 43862 T^{16} + 5424 T^{17} + 648 T^{18} + 36 T^{19} + T^{20} \)
$71$ \( 4294967296 - 438086664192 T + 22342419873792 T^{2} + 616059371520 T^{3} + 166983630848 T^{4} - 8169765994496 T^{5} + 8850291818496 T^{6} - 4375506845696 T^{7} + 1311830196224 T^{8} - 252221620224 T^{9} + 34045460480 T^{10} - 4077617152 T^{11} + 666607104 T^{12} - 112128000 T^{13} + 13634560 T^{14} - 1008768 T^{15} + 51332 T^{16} - 3320 T^{17} + 392 T^{18} - 28 T^{19} + T^{20} \)
$73$ \( 3768381677824 + 88647524019200 T + 1042673511680000 T^{2} + 1533522697452032 T^{3} + 1158124771226720 T^{4} + 360305416841024 T^{5} + 63712419788288 T^{6} + 7252837816896 T^{7} + 2205986977553 T^{8} + 621169097000 T^{9} + 105466458400 T^{10} + 8691597464 T^{11} + 728699683 T^{12} + 121567888 T^{13} + 19923008 T^{14} + 1608560 T^{15} + 71539 T^{16} + 1960 T^{17} + 288 T^{18} + 24 T^{19} + T^{20} \)
$79$ \( ( -379639352 - 211194640 T - 10416868 T^{2} + 14224864 T^{3} + 2791094 T^{4} - 82628 T^{5} - 66983 T^{6} - 5116 T^{7} + 152 T^{8} + 32 T^{9} + T^{10} )^{2} \)
$83$ \( 101765807113216 - 383187984297984 T + 721426162065408 T^{2} + 435345575499776 T^{3} + 157049942183488 T^{4} - 49943714401024 T^{5} + 5903547353600 T^{6} + 1762403399296 T^{7} + 1917079758368 T^{8} - 814473422656 T^{9} + 160712057984 T^{10} - 11796624608 T^{11} + 817856052 T^{12} - 143122976 T^{13} + 26357120 T^{14} - 2095168 T^{15} + 89060 T^{16} - 1856 T^{17} + 288 T^{18} - 24 T^{19} + T^{20} \)
$89$ \( 113653519143141376 + 212992986097647616 T + 199580323023945728 T^{2} + 85931323180449792 T^{3} + 19631882009837568 T^{4} + 1348359576879104 T^{5} + 538038168584192 T^{6} + 211765886369792 T^{7} + 45782370223104 T^{8} + 2811366938624 T^{9} + 45743884288 T^{10} + 2558512128 T^{11} + 4677535360 T^{12} + 147392000 T^{13} + 1174272 T^{14} - 367808 T^{15} + 135716 T^{16} + 1256 T^{17} + 8 T^{18} - 4 T^{19} + T^{20} \)
$97$ \( 1715634956210176 - 242037256454144 T + 17072989011968 T^{2} - 281369169936384 T^{3} + 500325913105152 T^{4} - 220490939977728 T^{5} + 49200016564224 T^{6} - 1172832806912 T^{7} + 567858873136 T^{8} - 202940730752 T^{9} + 35851016192 T^{10} - 1586457024 T^{11} + 191932113 T^{12} - 51277088 T^{13} + 8169984 T^{14} - 462944 T^{15} + 24002 T^{16} - 3360 T^{17} + 512 T^{18} - 32 T^{19} + T^{20} \)
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