Properties

Label 48.3.i.b
Level $48$
Weight $3$
Character orbit 48.i
Analytic conductor $1.308$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13} \)
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.

\(f(q)\) \(=\) \( q -\beta_{15} q^{2} + \beta_{4} q^{3} + ( \beta_{1} + \beta_{8} - \beta_{12} ) q^{4} + ( \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{5} + ( -1 - \beta_{1} - \beta_{3} - \beta_{10} - \beta_{11} + \beta_{18} ) q^{6} + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{14} ) q^{7} + ( \beta_{9} - \beta_{13} - \beta_{15} - \beta_{18} ) q^{8} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{12} - \beta_{14} - \beta_{19} ) q^{9} +O(q^{10})\) \( q -\beta_{15} q^{2} + \beta_{4} q^{3} + ( \beta_{1} + \beta_{8} - \beta_{12} ) q^{4} + ( \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{5} + ( -1 - \beta_{1} - \beta_{3} - \beta_{10} - \beta_{11} + \beta_{18} ) q^{6} + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{14} ) q^{7} + ( \beta_{9} - \beta_{13} - \beta_{15} - \beta_{18} ) q^{8} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{12} - \beta_{14} - \beta_{19} ) q^{9} + ( 1 - 3 \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} + \beta_{13} ) q^{10} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} + 2 \beta_{19} ) q^{11} + ( -5 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{12} + ( 5 + 5 \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{8} + \beta_{12} - \beta_{14} ) q^{13} + ( -2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} + 2 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{14} + ( -6 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{17} - \beta_{19} ) q^{15} + ( -1 + 7 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} ) q^{16} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{10} - \beta_{11} - \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{12} - \beta_{13} + 2 \beta_{16} - 2 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{18} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{19} + ( -2 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{16} + 2 \beta_{18} - 2 \beta_{19} ) q^{20} + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} - 3 \beta_{15} - \beta_{16} + 2 \beta_{17} - 5 \beta_{18} - \beta_{19} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{8} - \beta_{9} - \beta_{13} - 4 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} ) q^{22} + ( \beta_{4} - \beta_{5} - 6 \beta_{10} - \beta_{12} + \beta_{14} + 6 \beta_{15} + 6 \beta_{18} ) q^{23} + ( -1 + 5 \beta_{1} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{13} - 4 \beta_{14} + 5 \beta_{15} - \beta_{18} + 2 \beta_{19} ) q^{24} + ( -5 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} - \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} ) q^{25} + ( -\beta_{2} - \beta_{3} - \beta_{9} - 4 \beta_{11} + \beta_{13} - 5 \beta_{15} - 3 \beta_{18} ) q^{26} + ( -3 \beta_{4} + 3 \beta_{5} - 6 \beta_{7} - 3 \beta_{10} + 3 \beta_{11} - \beta_{12} - 6 \beta_{15} ) q^{27} + ( 9 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} ) q^{28} + ( -2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} - 6 \beta_{10} - 5 \beta_{11} - 2 \beta_{13} + \beta_{15} + 2 \beta_{16} - 3 \beta_{18} + 2 \beta_{19} ) q^{29} + ( 6 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 4 \beta_{8} - 2 \beta_{11} + 2 \beta_{12} + \beta_{13} + 6 \beta_{14} + 6 \beta_{15} ) q^{30} + ( -5 + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + \beta_{6} + 6 \beta_{7} + 3 \beta_{8} - 2 \beta_{12} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{31} + ( \beta_{2} + \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + \beta_{9} + 4 \beta_{11} - \beta_{13} + 4 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 6 \beta_{18} ) q^{32} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + \beta_{10} - 5 \beta_{11} - 2 \beta_{12} - 8 \beta_{15} - \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{33} + ( 4 - 6 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{12} - 4 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} ) q^{34} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{7} + 5 \beta_{10} - 7 \beta_{11} - 4 \beta_{13} - 12 \beta_{15} + \beta_{16} - 3 \beta_{17} + 2 \beta_{18} + 4 \beta_{19} ) q^{35} + ( 3 - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 4 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{13} - 8 \beta_{14} - 2 \beta_{17} ) q^{36} + ( -11 + 9 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} + 2 \beta_{9} - 3 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} ) q^{37} + ( 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} + 3 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} + 8 \beta_{12} + \beta_{13} - 8 \beta_{14} + 6 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - 4 \beta_{19} ) q^{38} + ( 1 + 10 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - 3 \beta_{19} ) q^{39} + ( 6 + 2 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 8 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{13} ) q^{40} + ( 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 4 \beta_{9} - \beta_{10} + 11 \beta_{11} + \beta_{12} + 4 \beta_{13} - \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} + 4 \beta_{18} ) q^{41} + ( 15 + 7 \beta_{1} - 2 \beta_{2} + \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 7 \beta_{8} - \beta_{9} + 8 \beta_{11} - 3 \beta_{12} + \beta_{13} + 8 \beta_{14} + \beta_{15} - 2 \beta_{16} + 5 \beta_{18} + 2 \beta_{19} ) q^{42} + ( 10 - 14 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} - 2 \beta_{9} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{16} + \beta_{17} ) q^{43} + ( \beta_{2} + \beta_{3} - 8 \beta_{4} + 8 \beta_{5} - 2 \beta_{7} + \beta_{9} + 10 \beta_{10} - 6 \beta_{11} - 6 \beta_{12} + \beta_{13} + 6 \beta_{14} - 8 \beta_{15} - 2 \beta_{16} - 2 \beta_{19} ) q^{44} + ( 1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} + \beta_{9} + 5 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} + 5 \beta_{15} - \beta_{16} + 4 \beta_{17} + 9 \beta_{18} - \beta_{19} ) q^{45} + ( -18 + 14 \beta_{1} + \beta_{2} - \beta_{3} + 6 \beta_{4} + 6 \beta_{7} - 6 \beta_{8} - \beta_{9} + 6 \beta_{12} - \beta_{13} ) q^{46} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{7} + 2 \beta_{9} + 6 \beta_{11} + 2 \beta_{15} + 2 \beta_{17} - 10 \beta_{18} - 2 \beta_{19} ) q^{47} + ( 5 - 11 \beta_{1} + \beta_{4} - 8 \beta_{5} + 3 \beta_{7} - 7 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} - 3 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{19} ) q^{48} + ( -15 - \beta_{2} + \beta_{3} - 2 \beta_{4} + 8 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} - 8 \beta_{8} + 7 \beta_{12} - \beta_{14} + 3 \beta_{16} - 3 \beta_{17} ) q^{49} + ( \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{7} - \beta_{9} - 4 \beta_{10} - 4 \beta_{11} - 3 \beta_{13} - 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + 3 \beta_{18} + 4 \beta_{19} ) q^{50} + ( 8 + 10 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 9 \beta_{5} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} - \beta_{16} + \beta_{17} - 6 \beta_{18} ) q^{51} + ( -15 - 19 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} - 5 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} ) q^{52} + ( 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{7} - \beta_{10} - 6 \beta_{11} - 7 \beta_{12} - 4 \beta_{13} + 7 \beta_{14} + 5 \beta_{15} + 4 \beta_{16} + 7 \beta_{18} + 4 \beta_{19} ) q^{53} + ( 1 + 29 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 6 \beta_{8} - 5 \beta_{10} - 7 \beta_{11} - 6 \beta_{12} - \beta_{13} - \beta_{15} ) q^{54} + ( -8 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 5 \beta_{5} - 4 \beta_{6} + 10 \beta_{7} - 10 \beta_{8} - 4 \beta_{9} + 7 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} ) q^{55} + ( -\beta_{2} - \beta_{3} - 10 \beta_{4} + 10 \beta_{5} - 3 \beta_{9} - 16 \beta_{10} + 4 \beta_{11} + 3 \beta_{13} - 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} ) q^{56} + ( 2 - 18 \beta_{1} + 5 \beta_{3} - 12 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{9} - 5 \beta_{10} - 5 \beta_{11} + 2 \beta_{12} + 10 \beta_{14} + 14 \beta_{15} - 2 \beta_{17} - 4 \beta_{18} + 3 \beta_{19} ) q^{57} + ( -33 + \beta_{1} + \beta_{2} - \beta_{3} - 11 \beta_{4} - 8 \beta_{5} - 4 \beta_{6} + \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + \beta_{12} - 2 \beta_{13} - 4 \beta_{14} ) q^{58} + ( -4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{7} + 12 \beta_{10} + 7 \beta_{12} + 4 \beta_{13} - 7 \beta_{14} + 12 \beta_{15} - 4 \beta_{16} - 12 \beta_{18} - 4 \beta_{19} ) q^{59} + ( -10 - 16 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{7} - 6 \beta_{8} - 12 \beta_{10} + 6 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} + 2 \beta_{17} + 10 \beta_{18} - 4 \beta_{19} ) q^{60} + ( -7 - 5 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 7 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} - 4 \beta_{16} + 4 \beta_{17} ) q^{61} + ( 2 \beta_{7} + \beta_{9} + 12 \beta_{11} - 10 \beta_{12} - 3 \beta_{13} + 10 \beta_{14} + \beta_{15} - 2 \beta_{16} - 4 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{62} + ( 15 - 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 6 \beta_{7} - 9 \beta_{8} + 12 \beta_{10} - 2 \beta_{11} + 11 \beta_{12} - 6 \beta_{14} + 10 \beta_{15} - \beta_{16} - 3 \beta_{17} + 14 \beta_{18} ) q^{63} + ( 2 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} - 6 \beta_{5} - 2 \beta_{7} + 8 \beta_{12} + 8 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} ) q^{64} + ( 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 5 \beta_{10} + 7 \beta_{11} + \beta_{12} - \beta_{14} + 12 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} ) q^{65} + ( -21 - 5 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} + 10 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} + 13 \beta_{8} + 7 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} - \beta_{12} + 2 \beta_{13} - 6 \beta_{15} - 6 \beta_{16} + 4 \beta_{17} - 6 \beta_{18} - 4 \beta_{19} ) q^{66} + ( 16 + 12 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + 2 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + 4 \beta_{16} - 4 \beta_{17} ) q^{67} + ( -2 \beta_{2} - 2 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} + 4 \beta_{7} + 4 \beta_{10} - 12 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} - 6 \beta_{15} + 4 \beta_{16} + 10 \beta_{18} + 4 \beta_{19} ) q^{68} + ( 3 + 2 \beta_{1} - \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} - \beta_{6} + 12 \beta_{7} - 7 \beta_{9} + 12 \beta_{11} + \beta_{12} - \beta_{14} + 5 \beta_{16} - 6 \beta_{17} - 12 \beta_{18} + 5 \beta_{19} ) q^{69} + ( -26 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 10 \beta_{8} + 2 \beta_{9} - 18 \beta_{12} + 2 \beta_{13} - 8 \beta_{14} - 6 \beta_{16} + 6 \beta_{17} ) q^{70} + ( -4 \beta_{2} - 4 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 6 \beta_{7} + 2 \beta_{9} - 4 \beta_{10} + 7 \beta_{12} + 4 \beta_{13} - 7 \beta_{14} - 2 \beta_{16} + 4 \beta_{17} + 8 \beta_{18} - 6 \beta_{19} ) q^{71} + ( -26 + 16 \beta_{1} - \beta_{2} + \beta_{3} - 8 \beta_{4} + 12 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} + 18 \beta_{10} - 6 \beta_{11} + 4 \beta_{13} - 8 \beta_{14} - 11 \beta_{15} - 2 \beta_{16} + 4 \beta_{17} - 5 \beta_{18} - 6 \beta_{19} ) q^{72} + ( 4 + 20 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} + 10 \beta_{8} - 10 \beta_{12} ) q^{73} + ( 5 \beta_{2} + 5 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} + 4 \beta_{7} + \beta_{9} - 4 \beta_{10} - 8 \beta_{11} - 4 \beta_{12} - 5 \beta_{13} + 4 \beta_{14} + 7 \beta_{15} - 4 \beta_{17} - 5 \beta_{18} + 4 \beta_{19} ) q^{74} + ( -10 + 6 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 4 \beta_{9} - 7 \beta_{10} + 3 \beta_{11} - 2 \beta_{13} - 7 \beta_{14} - 6 \beta_{15} + 4 \beta_{18} + 2 \beta_{19} ) q^{75} + ( 18 + 12 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} + 9 \beta_{9} + 10 \beta_{12} + 9 \beta_{13} + 10 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} ) q^{76} + ( 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} + 2 \beta_{10} - 6 \beta_{11} + 4 \beta_{13} - 8 \beta_{15} - 4 \beta_{16} - 20 \beta_{18} - 4 \beta_{19} ) q^{77} + ( 4 - 22 \beta_{1} + \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 8 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + \beta_{9} - 2 \beta_{10} - 12 \beta_{11} - 6 \beta_{13} + 4 \beta_{14} - 7 \beta_{15} + 2 \beta_{16} - 3 \beta_{18} - 2 \beta_{19} ) q^{78} + ( 21 - 4 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + 9 \beta_{8} - 6 \beta_{12} + 3 \beta_{14} - 3 \beta_{16} + 3 \beta_{17} ) q^{79} + ( -6 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 16 \beta_{10} - 8 \beta_{11} - 6 \beta_{15} - 4 \beta_{16} - 4 \beta_{17} + 14 \beta_{18} ) q^{80} + ( -9 - \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 12 \beta_{7} + 10 \beta_{8} - 6 \beta_{9} - 12 \beta_{10} - 12 \beta_{11} - 9 \beta_{12} + 9 \beta_{14} - 12 \beta_{15} - \beta_{16} + 3 \beta_{17} + 12 \beta_{18} + 6 \beta_{19} ) q^{81} + ( 32 + 42 \beta_{1} - \beta_{2} + \beta_{3} + 12 \beta_{4} + 12 \beta_{5} - 2 \beta_{8} + \beta_{9} - 6 \beta_{12} + \beta_{13} - 8 \beta_{14} + 4 \beta_{16} - 4 \beta_{17} ) q^{82} + ( \beta_{2} + \beta_{3} - 8 \beta_{7} + 11 \beta_{10} + 3 \beta_{11} + 8 \beta_{13} - 8 \beta_{15} - \beta_{16} + 7 \beta_{17} + 10 \beta_{18} - 8 \beta_{19} ) q^{83} + ( 33 + 9 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} - 7 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} + 12 \beta_{11} - 3 \beta_{12} - 4 \beta_{13} - 10 \beta_{14} - 8 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 10 \beta_{18} ) q^{84} + ( 12 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} + 10 \beta_{12} + 10 \beta_{14} + 4 \beta_{16} - 4 \beta_{17} ) q^{85} + ( -\beta_{2} - \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + 8 \beta_{7} - 6 \beta_{9} - 2 \beta_{10} + 10 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} + 6 \beta_{16} - 2 \beta_{17} + 10 \beta_{18} + 8 \beta_{19} ) q^{86} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} - \beta_{4} + 5 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} + \beta_{8} - 5 \beta_{9} + 18 \beta_{10} - 10 \beta_{11} + \beta_{12} - 10 \beta_{13} - 5 \beta_{14} - 6 \beta_{15} + 5 \beta_{16} - 6 \beta_{17} - 30 \beta_{18} + 7 \beta_{19} ) q^{87} + ( 32 - 48 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} + 12 \beta_{8} - 4 \beta_{9} - 8 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} ) q^{88} + ( -5 \beta_{2} - 5 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} - 2 \beta_{7} + 8 \beta_{9} + 8 \beta_{10} + 4 \beta_{11} - 7 \beta_{12} - 6 \beta_{13} + 7 \beta_{14} - 28 \beta_{15} + 3 \beta_{16} + 5 \beta_{17} + 12 \beta_{18} - 2 \beta_{19} ) q^{89} + ( 35 + 5 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + 9 \beta_{7} - 7 \beta_{8} + \beta_{9} + 8 \beta_{10} + 12 \beta_{11} + 7 \beta_{12} - 5 \beta_{13} + 4 \beta_{14} + 6 \beta_{16} - 8 \beta_{18} - 2 \beta_{19} ) q^{90} + ( 6 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} + 6 \beta_{9} - \beta_{12} + 6 \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} ) q^{91} + ( -4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} - 8 \beta_{9} - 4 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} + 24 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} - 8 \beta_{18} + 4 \beta_{19} ) q^{92} + ( -7 - 9 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 14 \beta_{5} - 10 \beta_{6} + 6 \beta_{7} - 10 \beta_{8} - 2 \beta_{9} + 8 \beta_{10} + 15 \beta_{11} + 5 \beta_{12} - 6 \beta_{13} - 5 \beta_{14} + 7 \beta_{15} + 4 \beta_{16} - 12 \beta_{17} + 15 \beta_{18} + 4 \beta_{19} ) q^{93} + ( 6 + 30 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + 4 \beta_{6} - 10 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} + 6 \beta_{12} + 2 \beta_{13} + 12 \beta_{14} ) q^{94} + ( -4 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 24 \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} - 26 \beta_{15} - 4 \beta_{16} - 4 \beta_{17} - 22 \beta_{18} ) q^{95} + ( -4 - 46 \beta_{1} + 3 \beta_{2} - \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 2 \beta_{8} - 7 \beta_{9} + 20 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - \beta_{13} - 4 \beta_{14} - 6 \beta_{15} - 4 \beta_{16} + 4 \beta_{18} - 4 \beta_{19} ) q^{96} + ( 10 + \beta_{2} - \beta_{3} - 8 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 11 \beta_{12} - 9 \beta_{14} - 7 \beta_{16} + 7 \beta_{17} ) q^{97} + ( -3 \beta_{2} - 3 \beta_{3} - 14 \beta_{4} + 14 \beta_{5} - 8 \beta_{7} - 3 \beta_{9} - 4 \beta_{10} - 12 \beta_{11} + 12 \beta_{12} + 11 \beta_{13} - 12 \beta_{14} + 19 \beta_{15} - 6 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - 8 \beta_{19} ) q^{98} + ( -24 - 28 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + 13 \beta_{4} - 13 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} - 6 \beta_{8} - 4 \beta_{9} - 20 \beta_{10} + 16 \beta_{11} + 3 \beta_{12} - 8 \beta_{13} - 3 \beta_{14} + 36 \beta_{15} + 5 \beta_{16} - 7 \beta_{17} + 4 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 6q^{3} + 4q^{4} - 12q^{6} + O(q^{10}) \) \( 20q - 6q^{3} + 4q^{4} - 12q^{6} + 32q^{10} - 88q^{12} + 92q^{13} - 116q^{15} - 16q^{16} + 4q^{18} - 52q^{19} + 48q^{21} + 24q^{22} - 8q^{24} + 18q^{27} + 56q^{28} + 28q^{30} - 80q^{31} + 60q^{33} + 104q^{34} + 92q^{36} - 116q^{37} + 88q^{40} + 304q^{42} + 172q^{43} + 60q^{45} - 424q^{46} + 176q^{48} - 364q^{49} + 128q^{51} - 208q^{52} + 40q^{54} - 512q^{58} - 240q^{60} - 244q^{61} + 296q^{63} + 88q^{64} - 492q^{66} + 356q^{67} - 20q^{69} + 200q^{70} - 472q^{72} - 146q^{75} + 328q^{76} + 84q^{78} + 384q^{79} - 188q^{81} + 560q^{82} + 816q^{84} + 48q^{85} + 416q^{88} + 616q^{90} + 136q^{91} - 132q^{93} + 32q^{94} - 24q^{96} + 472q^{97} - 452q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 33 \nu^{18} + 42 \nu^{16} - 1090 \nu^{14} + 528 \nu^{12} + 11816 \nu^{10} - 6496 \nu^{8} + 64512 \nu^{6} - 1111040 \nu^{4} - 1482752 \nu^{2} + 16842752 \)\()/10158080\)
\(\beta_{2}\)\(=\)\((\)\(-25 \nu^{19} + 68 \nu^{18} - 2270 \nu^{17} - 1448 \nu^{16} + 7850 \nu^{15} - 10920 \nu^{14} + 2360 \nu^{13} + 33248 \nu^{12} - 93800 \nu^{11} + 41376 \nu^{10} - 8000 \nu^{9} + 361984 \nu^{8} - 926080 \nu^{7} - 25088 \nu^{6} - 404480 \nu^{5} - 10813440 \nu^{4} + 29163520 \nu^{3} + 12976128 \nu^{2} - 177602560 \nu + 120061952\)\()/40632320\)
\(\beta_{3}\)\(=\)\((\)\(-25 \nu^{19} - 68 \nu^{18} - 2270 \nu^{17} + 1448 \nu^{16} + 7850 \nu^{15} + 10920 \nu^{14} + 2360 \nu^{13} - 33248 \nu^{12} - 93800 \nu^{11} - 41376 \nu^{10} - 8000 \nu^{9} - 361984 \nu^{8} - 926080 \nu^{7} + 25088 \nu^{6} - 404480 \nu^{5} + 10813440 \nu^{4} + 29163520 \nu^{3} - 12976128 \nu^{2} - 177602560 \nu - 120061952\)\()/40632320\)
\(\beta_{4}\)\(=\)\((\)\(-4 \nu^{19} + 112 \nu^{18} - 101 \nu^{17} - 542 \nu^{16} - 410 \nu^{15} + 980 \nu^{14} + 1946 \nu^{13} + 5372 \nu^{12} - 368 \nu^{11} - 13216 \nu^{10} + 24248 \nu^{9} + 64976 \nu^{8} - 17696 \nu^{7} - 29632 \nu^{6} - 398080 \nu^{5} - 750080 \nu^{4} + 1181696 \nu^{3} + 7708672 \nu^{2} + 753664 \nu - 18481152\)\()/10158080\)
\(\beta_{5}\)\(=\)\((\)\(4 \nu^{19} + 112 \nu^{18} + 101 \nu^{17} - 542 \nu^{16} + 410 \nu^{15} + 980 \nu^{14} - 1946 \nu^{13} + 5372 \nu^{12} + 368 \nu^{11} - 13216 \nu^{10} - 24248 \nu^{9} + 64976 \nu^{8} + 17696 \nu^{7} - 29632 \nu^{6} + 398080 \nu^{5} - 750080 \nu^{4} - 1181696 \nu^{3} + 7708672 \nu^{2} - 753664 \nu - 18481152\)\()/10158080\)
\(\beta_{6}\)\(=\)\((\)\(4 \nu^{19} - 149 \nu^{18} + 101 \nu^{17} - 836 \nu^{16} + 410 \nu^{15} + 790 \nu^{14} - 1946 \nu^{13} - 12204 \nu^{12} + 368 \nu^{11} - 36008 \nu^{10} - 24248 \nu^{9} + 8848 \nu^{8} + 17696 \nu^{7} + 83904 \nu^{6} + 398080 \nu^{5} - 53760 \nu^{4} - 1181696 \nu^{3} - 2801664 \nu^{2} - 753664 \nu - 63307776\)\()/10158080\)
\(\beta_{7}\)\(=\)\((\)\(4 \nu^{19} + 145 \nu^{18} + 101 \nu^{17} - 500 \nu^{16} + 410 \nu^{15} - 110 \nu^{14} - 1946 \nu^{13} + 5900 \nu^{12} + 368 \nu^{11} - 1400 \nu^{10} - 24248 \nu^{9} + 58480 \nu^{8} + 17696 \nu^{7} + 34880 \nu^{6} + 398080 \nu^{5} - 1861120 \nu^{4} - 1181696 \nu^{3} + 16384000 \nu^{2} - 753664 \nu - 1638400\)\()/10158080\)
\(\beta_{8}\)\(=\)\((\)\(-79 \nu^{19} - 160 \nu^{18} + 34 \nu^{17} - 2040 \nu^{16} + 2030 \nu^{15} + 2960 \nu^{14} + 4896 \nu^{13} + 31600 \nu^{12} + 12392 \nu^{11} - 109440 \nu^{10} - 68512 \nu^{9} - 209600 \nu^{8} - 78336 \nu^{7} - 1826560 \nu^{6} - 10240 \nu^{5} - 460800 \nu^{4} + 6332416 \nu^{3} + 37683200 \nu^{2} - 10878976 \nu - 82575360\)\()/40632320\)
\(\beta_{9}\)\(=\)\((\)\(51 \nu^{19} - 448 \nu^{18} - 1606 \nu^{17} - 32 \nu^{16} + 2450 \nu^{15} + 12480 \nu^{14} + 56 \nu^{13} + 17472 \nu^{12} + 24312 \nu^{11} + 2304 \nu^{10} - 2752 \nu^{9} - 248064 \nu^{8} - 1475456 \nu^{7} - 571392 \nu^{6} - 353280 \nu^{5} + 4403200 \nu^{4} + 17760256 \nu^{3} + 31260672 \nu^{2} - 25296896 \nu - 70254592\)\()/40632320\)
\(\beta_{10}\)\(=\)\((\)\( -33 \nu^{19} - 42 \nu^{17} + 1090 \nu^{15} - 528 \nu^{13} - 11816 \nu^{11} + 6496 \nu^{9} - 64512 \nu^{7} + 1111040 \nu^{5} + 1482752 \nu^{3} - 6684672 \nu \)\()/10158080\)
\(\beta_{11}\)\(=\)\((\)\( 33 \nu^{19} + 42 \nu^{17} - 1090 \nu^{15} + 528 \nu^{13} + 11816 \nu^{11} - 6496 \nu^{9} + 64512 \nu^{7} - 1111040 \nu^{5} - 1482752 \nu^{3} + 27000832 \nu \)\()/10158080\)
\(\beta_{12}\)\(=\)\((\)\(-79 \nu^{19} + 592 \nu^{18} + 34 \nu^{17} - 3112 \nu^{16} + 2030 \nu^{15} + 2320 \nu^{14} + 4896 \nu^{13} + 18832 \nu^{12} + 12392 \nu^{11} - 77056 \nu^{10} - 68512 \nu^{9} + 518336 \nu^{8} - 78336 \nu^{7} - 1806592 \nu^{6} - 10240 \nu^{5} - 8714240 \nu^{4} + 6332416 \nu^{3} + 46989312 \nu^{2} - 10878976 \nu - 96468992\)\()/40632320\)
\(\beta_{13}\)\(=\)\((\)\(-51 \nu^{19} - 448 \nu^{18} + 1606 \nu^{17} - 32 \nu^{16} - 2450 \nu^{15} + 12480 \nu^{14} - 56 \nu^{13} + 17472 \nu^{12} - 24312 \nu^{11} + 2304 \nu^{10} + 2752 \nu^{9} - 248064 \nu^{8} + 1475456 \nu^{7} - 571392 \nu^{6} + 353280 \nu^{5} + 4403200 \nu^{4} - 17760256 \nu^{3} + 31260672 \nu^{2} + 25296896 \nu - 70254592\)\()/40632320\)
\(\beta_{14}\)\(=\)\((\)\(79 \nu^{19} + 592 \nu^{18} - 34 \nu^{17} - 3112 \nu^{16} - 2030 \nu^{15} + 2320 \nu^{14} - 4896 \nu^{13} + 18832 \nu^{12} - 12392 \nu^{11} - 77056 \nu^{10} + 68512 \nu^{9} + 518336 \nu^{8} + 78336 \nu^{7} - 1806592 \nu^{6} + 10240 \nu^{5} - 8714240 \nu^{4} - 6332416 \nu^{3} + 46989312 \nu^{2} + 10878976 \nu - 96468992\)\()/40632320\)
\(\beta_{15}\)\(=\)\((\)\( \nu^{19} - 2 \nu^{17} + 6 \nu^{15} - 24 \nu^{13} - 24 \nu^{11} + 1216 \nu^{9} - 384 \nu^{7} - 6144 \nu^{5} + 24576 \nu^{3} - 131072 \nu \)\()/262144\)
\(\beta_{16}\)\(=\)\((\)\(211 \nu^{19} - 468 \nu^{18} + 334 \nu^{17} + 1008 \nu^{16} - 2470 \nu^{15} + 2520 \nu^{14} + 4496 \nu^{13} - 14768 \nu^{12} - 57928 \nu^{11} - 136096 \nu^{10} + 99168 \nu^{9} - 296384 \nu^{8} - 77056 \nu^{7} + 1475328 \nu^{6} - 5427200 \nu^{5} - 1546240 \nu^{4} + 13950976 \nu^{3} - 13959168 \nu^{2} + 10092544 \nu - 122683392\)\()/40632320\)
\(\beta_{17}\)\(=\)\((\)\(211 \nu^{19} + 468 \nu^{18} + 334 \nu^{17} - 1008 \nu^{16} - 2470 \nu^{15} - 2520 \nu^{14} + 4496 \nu^{13} + 14768 \nu^{12} - 57928 \nu^{11} + 136096 \nu^{10} + 99168 \nu^{9} + 296384 \nu^{8} - 77056 \nu^{7} - 1475328 \nu^{6} - 5427200 \nu^{5} + 1546240 \nu^{4} + 13950976 \nu^{3} + 13959168 \nu^{2} + 10092544 \nu + 122683392\)\()/40632320\)
\(\beta_{18}\)\(=\)\((\)\( 257 \nu^{19} - 1042 \nu^{17} + 870 \nu^{15} + 11272 \nu^{13} - 14616 \nu^{11} + 123456 \nu^{9} + 5248 \nu^{7} - 2611200 \nu^{5} + 24092672 \nu^{3} - 9961472 \nu \)\()/40632320\)
\(\beta_{19}\)\(=\)\((\)\(-157 \nu^{19} - 140 \nu^{18} + 982 \nu^{17} + 240 \nu^{16} - 1110 \nu^{15} + 2600 \nu^{14} - 8192 \nu^{13} + 2160 \nu^{12} + 27896 \nu^{11} + 36000 \nu^{10} - 116576 \nu^{9} - 46400 \nu^{8} + 467712 \nu^{7} - 546560 \nu^{6} + 2037760 \nu^{5} + 3348480 \nu^{4} - 12480512 \nu^{3} - 5079040 \nu^{2} + 35028992 \nu + 14745600\)\()/10158080\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + \beta_{10}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{5} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{19} + 2 \beta_{18} + \beta_{16} - \beta_{13} - \beta_{11} + \beta_{7} - \beta_{5} + \beta_{4}\)
\(\nu^{4}\)\(=\)\(\beta_{17} - \beta_{16} - \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} - 7 \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(6 \beta_{18} - \beta_{17} - \beta_{16} - 6 \beta_{15} + 4 \beta_{14} - \beta_{13} - 4 \beta_{12} - 7 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{17} + 2 \beta_{16} - 10 \beta_{14} - 4 \beta_{12} - 6 \beta_{8} + 6 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 12 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} + 4\)
\(\nu^{7}\)\(=\)\(10 \beta_{19} + 28 \beta_{18} - 12 \beta_{17} - 2 \beta_{16} + 8 \beta_{15} + 4 \beta_{14} + 6 \beta_{13} - 4 \beta_{12} + 12 \beta_{11} - 2 \beta_{10} - 16 \beta_{9} + 10 \beta_{7} - 10 \beta_{5} + 10 \beta_{4} + 8 \beta_{3} + 8 \beta_{2}\)
\(\nu^{8}\)\(=\)\(-2 \beta_{17} + 2 \beta_{16} - 10 \beta_{14} + 2 \beta_{13} + 44 \beta_{12} + 2 \beta_{9} - 54 \beta_{8} - 10 \beta_{7} - 8 \beta_{6} - 18 \beta_{4} - 24 \beta_{3} + 24 \beta_{2} - 146 \beta_{1} + 6\)
\(\nu^{9}\)\(=\)\(4 \beta_{19} - 44 \beta_{18} - 2 \beta_{17} + 2 \beta_{16} + 180 \beta_{15} - 24 \beta_{14} + 2 \beta_{13} + 24 \beta_{12} - 2 \beta_{11} + 58 \beta_{10} - 6 \beta_{9} + 4 \beta_{7} - 82 \beta_{5} + 82 \beta_{4} - 12 \beta_{3} - 12 \beta_{2}\)
\(\nu^{10}\)\(=\)\(80 \beta_{17} - 80 \beta_{16} - 64 \beta_{14} + 76 \beta_{13} + 16 \beta_{12} + 76 \beta_{9} - 80 \beta_{8} + 8 \beta_{7} + 52 \beta_{6} - 32 \beta_{5} + 28 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 276 \beta_{1} - 660\)
\(\nu^{11}\)\(=\)\(68 \beta_{19} + 128 \beta_{18} - 116 \beta_{17} - 48 \beta_{16} + 296 \beta_{15} - 200 \beta_{14} - 112 \beta_{13} + 200 \beta_{12} - 204 \beta_{11} - 360 \beta_{10} + 44 \beta_{9} + 68 \beta_{7} - 72 \beta_{5} + 72 \beta_{4} - 56 \beta_{3} - 56 \beta_{2}\)
\(\nu^{12}\)\(=\)\(-44 \beta_{17} + 44 \beta_{16} - 76 \beta_{14} + 52 \beta_{13} - 312 \beta_{12} + 52 \beta_{9} + 236 \beta_{8} - 524 \beta_{7} - 312 \beta_{6} + 928 \beta_{5} + 92 \beta_{4} - 232 \beta_{3} + 232 \beta_{2} - 116 \beta_{1} - 1348\)
\(\nu^{13}\)\(=\)\(-1120 \beta_{19} - 328 \beta_{18} + 236 \beta_{17} - 884 \beta_{16} - 1720 \beta_{15} - 832 \beta_{14} + 972 \beta_{13} + 832 \beta_{12} - 660 \beta_{11} - 132 \beta_{10} + 148 \beta_{9} - 1120 \beta_{7} + 580 \beta_{5} - 580 \beta_{4} - 536 \beta_{3} - 536 \beta_{2}\)
\(\nu^{14}\)\(=\)\(-344 \beta_{17} + 344 \beta_{16} - 56 \beta_{14} + 1424 \beta_{13} + 1648 \beta_{12} + 1424 \beta_{9} - 1704 \beta_{8} - 2024 \beta_{7} - 56 \beta_{6} + 2272 \beta_{5} + 192 \beta_{4} + 472 \beta_{3} - 472 \beta_{2} + 3664 \beta_{1} + 7840\)
\(\nu^{15}\)\(=\)\(-1928 \beta_{19} - 4240 \beta_{18} - 1928 \beta_{16} + 6592 \beta_{15} - 5808 \beta_{14} + 2616 \beta_{13} + 5808 \beta_{12} + 7248 \beta_{11} + 5096 \beta_{10} - 688 \beta_{9} - 1928 \beta_{7} + 3928 \beta_{5} - 3928 \beta_{4} + 576 \beta_{3} + 576 \beta_{2}\)
\(\nu^{16}\)\(=\)\(-3928 \beta_{17} + 3928 \beta_{16} + 2152 \beta_{14} - 72 \beta_{13} + 6608 \beta_{12} - 72 \beta_{9} - 4456 \beta_{8} + 2376 \beta_{7} - 7808 \beta_{6} - 4608 \beta_{5} - 10040 \beta_{4} - 13304 \beta_{1} - 17656\)
\(\nu^{17}\)\(=\)\(-1424 \beta_{19} - 7568 \beta_{18} + 9960 \beta_{17} + 8536 \beta_{16} + 6512 \beta_{15} - 15328 \beta_{14} - 680 \beta_{13} + 15328 \beta_{12} - 22200 \beta_{11} - 5736 \beta_{10} + 2104 \beta_{9} - 1424 \beta_{7} + 7080 \beta_{5} - 7080 \beta_{4} - 7760 \beta_{3} - 7760 \beta_{2}\)
\(\nu^{18}\)\(=\)\(2880 \beta_{17} - 2880 \beta_{16} + 3456 \beta_{14} - 14192 \beta_{13} - 5568 \beta_{12} - 14192 \beta_{9} + 9024 \beta_{8} + 544 \beta_{7} - 11024 \beta_{6} + 24768 \beta_{5} + 14288 \beta_{4} + 19920 \beta_{3} - 19920 \beta_{2} + 47280 \beta_{1} - 11376\)
\(\nu^{19}\)\(=\)\(-42128 \beta_{19} + 57472 \beta_{18} + 14480 \beta_{17} - 27648 \beta_{16} - 51232 \beta_{15} + 34720 \beta_{14} + 21888 \beta_{13} - 34720 \beta_{12} - 54480 \beta_{11} - 53472 \beta_{10} + 20240 \beta_{9} - 42128 \beta_{7} + 129376 \beta_{5} - 129376 \beta_{4} - 27808 \beta_{3} - 27808 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−1.96139 0.391068i
−1.85381 0.750590i
−1.28499 + 1.53258i
−1.21144 1.59136i
−0.312316 + 1.97546i
0.312316 1.97546i
1.21144 + 1.59136i
1.28499 1.53258i
1.85381 + 0.750590i
1.96139 + 0.391068i
−1.96139 + 0.391068i
−1.85381 + 0.750590i
−1.28499 1.53258i
−1.21144 + 1.59136i
−0.312316 1.97546i
0.312316 + 1.97546i
1.21144 1.59136i
1.28499 + 1.53258i
1.85381 0.750590i
1.96139 0.391068i
−1.96139 + 0.391068i −2.99548 + 0.164573i 3.69413 1.53408i 3.61305 3.61305i 5.81096 1.49423i 12.2792i −6.64572 + 4.45358i 8.94583 0.985948i −5.67366 + 8.49955i
5.2 −1.85381 + 0.750590i 1.50491 + 2.59524i 2.87323 2.78290i −2.59897 + 2.59897i −4.73777 3.68151i 7.30027i −3.23761 + 7.31559i −4.47050 + 7.81118i 2.86723 6.76875i
5.3 −1.28499 1.53258i −2.06336 + 2.17774i −0.697601 + 3.93870i −3.17955 + 3.17955i 5.98896 + 0.363879i 6.03979i 6.93278 3.99206i −0.485128 8.98692i 8.95859 + 0.787223i
5.4 −1.21144 + 1.59136i 1.14944 2.77106i −1.06484 3.85566i 4.80434 4.80434i 3.01728 + 5.18614i 7.36187i 7.42573 + 2.97634i −6.35757 6.37035i 1.82527 + 13.4656i
5.5 −0.312316 1.97546i −1.18505 2.75602i −3.80492 + 1.23394i −0.00985921 + 0.00985921i −5.07432 + 3.20176i 6.42277i 3.62594 + 7.13110i −6.19134 + 6.53203i 0.0225557 + 0.0163973i
5.6 0.312316 + 1.97546i 2.75602 + 1.18505i −3.80492 + 1.23394i 0.00985921 0.00985921i −1.48026 + 5.81454i 6.42277i −3.62594 7.13110i 6.19134 + 6.53203i 0.0225557 + 0.0163973i
5.7 1.21144 1.59136i 2.77106 1.14944i −1.06484 3.85566i −4.80434 + 4.80434i 1.52779 5.80223i 7.36187i −7.42573 2.97634i 6.35757 6.37035i 1.82527 + 13.4656i
5.8 1.28499 + 1.53258i −2.17774 + 2.06336i −0.697601 + 3.93870i 3.17955 3.17955i −5.96063 0.686173i 6.03979i −6.93278 + 3.99206i 0.485128 8.98692i 8.95859 + 0.787223i
5.9 1.85381 0.750590i −2.59524 1.50491i 2.87323 2.78290i 2.59897 2.59897i −5.94065 0.841858i 7.30027i 3.23761 7.31559i 4.47050 + 7.81118i 2.86723 6.76875i
5.10 1.96139 0.391068i −0.164573 + 2.99548i 3.69413 1.53408i −3.61305 + 3.61305i 0.848646 + 5.93968i 12.2792i 6.64572 4.45358i −8.94583 0.985948i −5.67366 + 8.49955i
29.1 −1.96139 0.391068i −2.99548 0.164573i 3.69413 + 1.53408i 3.61305 + 3.61305i 5.81096 + 1.49423i 12.2792i −6.64572 4.45358i 8.94583 + 0.985948i −5.67366 8.49955i
29.2 −1.85381 0.750590i 1.50491 2.59524i 2.87323 + 2.78290i −2.59897 2.59897i −4.73777 + 3.68151i 7.30027i −3.23761 7.31559i −4.47050 7.81118i 2.86723 + 6.76875i
29.3 −1.28499 + 1.53258i −2.06336 2.17774i −0.697601 3.93870i −3.17955 3.17955i 5.98896 0.363879i 6.03979i 6.93278 + 3.99206i −0.485128 + 8.98692i 8.95859 0.787223i
29.4 −1.21144 1.59136i 1.14944 + 2.77106i −1.06484 + 3.85566i 4.80434 + 4.80434i 3.01728 5.18614i 7.36187i 7.42573 2.97634i −6.35757 + 6.37035i 1.82527 13.4656i
29.5 −0.312316 + 1.97546i −1.18505 + 2.75602i −3.80492 1.23394i −0.00985921 0.00985921i −5.07432 3.20176i 6.42277i 3.62594 7.13110i −6.19134 6.53203i 0.0225557 0.0163973i
29.6 0.312316 1.97546i 2.75602 1.18505i −3.80492 1.23394i 0.00985921 + 0.00985921i −1.48026 5.81454i 6.42277i −3.62594 + 7.13110i 6.19134 6.53203i 0.0225557 0.0163973i
29.7 1.21144 + 1.59136i 2.77106 + 1.14944i −1.06484 + 3.85566i −4.80434 4.80434i 1.52779 + 5.80223i 7.36187i −7.42573 + 2.97634i 6.35757 + 6.37035i 1.82527 13.4656i
29.8 1.28499 1.53258i −2.17774 2.06336i −0.697601 3.93870i 3.17955 + 3.17955i −5.96063 + 0.686173i 6.03979i −6.93278 3.99206i 0.485128 + 8.98692i 8.95859 0.787223i
29.9 1.85381 + 0.750590i −2.59524 + 1.50491i 2.87323 + 2.78290i 2.59897 + 2.59897i −5.94065 + 0.841858i 7.30027i 3.23761 + 7.31559i 4.47050 7.81118i 2.86723 + 6.76875i
29.10 1.96139 + 0.391068i −0.164573 2.99548i 3.69413 + 1.53408i −3.61305 3.61305i 0.848646 5.93968i 12.2792i 6.64572 + 4.45358i −8.94583 + 0.985948i −5.67366 8.49955i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.i.b 20
3.b odd 2 1 inner 48.3.i.b 20
4.b odd 2 1 192.3.i.b 20
8.b even 2 1 384.3.i.d 20
8.d odd 2 1 384.3.i.c 20
12.b even 2 1 192.3.i.b 20
16.e even 4 1 inner 48.3.i.b 20
16.e even 4 1 384.3.i.d 20
16.f odd 4 1 192.3.i.b 20
16.f odd 4 1 384.3.i.c 20
24.f even 2 1 384.3.i.c 20
24.h odd 2 1 384.3.i.d 20
48.i odd 4 1 inner 48.3.i.b 20
48.i odd 4 1 384.3.i.d 20
48.k even 4 1 192.3.i.b 20
48.k even 4 1 384.3.i.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.b 20 1.a even 1 1 trivial
48.3.i.b 20 3.b odd 2 1 inner
48.3.i.b 20 16.e even 4 1 inner
48.3.i.b 20 48.i odd 4 1 inner
192.3.i.b 20 4.b odd 2 1
192.3.i.b 20 12.b even 2 1
192.3.i.b 20 16.f odd 4 1
192.3.i.b 20 48.k even 4 1
384.3.i.c 20 8.d odd 2 1
384.3.i.c 20 16.f odd 4 1
384.3.i.c 20 24.f even 2 1
384.3.i.c 20 48.k even 4 1
384.3.i.d 20 8.b even 2 1
384.3.i.d 20 16.e even 4 1
384.3.i.d 20 24.h odd 2 1
384.3.i.d 20 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 3404 T_{5}^{16} + 3190384 T_{5}^{12} + 1068787520 T_{5}^{8} + 108375444480 T_{5}^{4} + 4096 \) acting on \(S_{3}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1048576 - 131072 T^{2} + 24576 T^{4} - 6144 T^{6} - 384 T^{8} + 1216 T^{10} - 24 T^{12} - 24 T^{14} + 6 T^{16} - 2 T^{18} + T^{20} \)
$3$ \( 3486784401 + 2324522934 T + 774840978 T^{2} + 143489070 T^{3} + 34543665 T^{4} + 9920232 T^{5} + 1889568 T^{6} + 1137240 T^{7} + 913518 T^{8} + 578988 T^{9} + 225180 T^{10} + 64332 T^{11} + 11278 T^{12} + 1560 T^{13} + 288 T^{14} + 168 T^{15} + 65 T^{16} + 30 T^{17} + 18 T^{18} + 6 T^{19} + T^{20} \)
$5$ \( 4096 + 108375444480 T^{4} + 1068787520 T^{8} + 3190384 T^{12} + 3404 T^{16} + T^{20} \)
$7$ \( ( 655360000 + 62587904 T^{2} + 2308480 T^{4} + 40676 T^{6} + 336 T^{8} + T^{10} )^{2} \)
$11$ \( 2240545423360000 + 8039082785243136 T^{4} + 204108821294144 T^{8} + 12405334576 T^{12} + 207308 T^{16} + T^{20} \)
$13$ \( ( 33620000 + 46477600 T + 32126224 T^{2} + 960768 T^{3} - 352112 T^{4} - 191600 T^{5} + 85320 T^{6} - 12736 T^{7} + 1058 T^{8} - 46 T^{9} + T^{10} )^{2} \)
$17$ \( ( 15510536192 + 1356865536 T^{2} + 34106368 T^{4} + 297760 T^{6} + 952 T^{8} + T^{10} )^{2} \)
$19$ \( ( 23975244288 + 21827527680 T + 9936102400 T^{2} + 2439206208 T^{3} + 353223624 T^{4} + 25536424 T^{5} + 942404 T^{6} + 3952 T^{7} + 338 T^{8} + 26 T^{9} + T^{10} )^{2} \)
$23$ \( ( -2157878476800 + 60056123392 T^{2} - 495575104 T^{4} + 1632624 T^{6} - 2236 T^{8} + T^{10} )^{2} \)
$29$ \( \)\(14\!\cdots\!00\)\( + \)\(63\!\cdots\!56\)\( T^{4} + 979627163704944448 T^{8} + 14592942824560 T^{12} + 8250700 T^{16} + T^{20} \)
$31$ \( ( 6473680 + 937016 T - 7516 T^{2} - 2750 T^{3} + 20 T^{4} + T^{5} )^{4} \)
$37$ \( ( 93878430976800 - 17120760301920 T + 1561170283024 T^{2} - 60745982336 T^{3} + 998599952 T^{4} - 6429040 T^{5} + 4219528 T^{6} - 130144 T^{7} + 1682 T^{8} + 58 T^{9} + T^{10} )^{2} \)
$41$ \( ( -89172136396800 + 4759931137024 T^{2} - 24298249280 T^{4} + 24250736 T^{6} - 8644 T^{8} + T^{10} )^{2} \)
$43$ \( ( 398518394892800 - 94334096030720 T + 11165007422464 T^{2} - 692164733120 T^{3} + 26089764040 T^{4} - 583167128 T^{5} + 8195716 T^{6} - 111760 T^{7} + 3698 T^{8} - 86 T^{9} + T^{10} )^{2} \)
$47$ \( ( 2199023255552 + 146867748864 T^{2} + 2561671168 T^{4} + 6660224 T^{6} + 4944 T^{8} + T^{10} )^{2} \)
$53$ \( \)\(51\!\cdots\!00\)\( + \)\(21\!\cdots\!76\)\( T^{4} + \)\(31\!\cdots\!08\)\( T^{8} + 185846408858736 T^{12} + 33387084 T^{16} + T^{20} \)
$59$ \( \)\(55\!\cdots\!00\)\( + \)\(78\!\cdots\!96\)\( T^{4} + \)\(17\!\cdots\!92\)\( T^{8} + 2609014014580272 T^{12} + 96029644 T^{16} + T^{20} \)
$61$ \( ( 7907544324449568 + 1330247085982368 T + 111890445197584 T^{2} + 5334667962752 T^{3} + 157007740304 T^{4} + 2668194448 T^{5} + 26687560 T^{6} + 245344 T^{7} + 7442 T^{8} + 122 T^{9} + T^{10} )^{2} \)
$67$ \( ( 1442777382684800 + 245738443228160 T + 20927477518336 T^{2} - 934037268512 T^{3} + 18053629992 T^{4} - 143429896 T^{5} + 16045188 T^{6} - 702440 T^{7} + 15842 T^{8} - 178 T^{9} + T^{10} )^{2} \)
$71$ \( ( -6303938155520000 + 49273291498496 T^{2} - 91544227136 T^{4} + 55549616 T^{6} - 12876 T^{8} + T^{10} )^{2} \)
$73$ \( ( 900192010240000 + 76360353726464 T^{2} + 137600509952 T^{4} + 77003072 T^{6} + 16160 T^{8} + T^{10} )^{2} \)
$79$ \( ( -147403248 + 3773624 T + 263044 T^{2} - 3534 T^{3} - 96 T^{4} + T^{5} )^{4} \)
$83$ \( \)\(53\!\cdots\!00\)\( + \)\(82\!\cdots\!96\)\( T^{4} + \)\(34\!\cdots\!04\)\( T^{8} + 36252904681183792 T^{12} + 433330892 T^{16} + T^{20} \)
$89$ \( ( -15016301280992460800 + 15202762675533824 T^{2} - 5223733537856 T^{4} + 780108464 T^{6} - 49740 T^{8} + T^{10} )^{2} \)
$97$ \( ( -2657552000 + 30283520 T + 1148408 T^{2} - 11780 T^{3} - 118 T^{4} + T^{5} )^{4} \)
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