Properties

Label 845.2.k.d
Level $845$
Weight $2$
Character orbit 845.k
Analytic conductor $6.747$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} -\beta_{10} q^{3} + ( -\beta_{3} + \beta_{4} + \beta_{8} ) q^{4} + ( 1 + \beta_{2} - \beta_{3} + \beta_{10} + \beta_{14} ) q^{5} + ( \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{16} - \beta_{19} ) q^{6} + ( \beta_{3} + \beta_{11} ) q^{7} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} + \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{8} + ( \beta_{3} + \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} -\beta_{10} q^{3} + ( -\beta_{3} + \beta_{4} + \beta_{8} ) q^{4} + ( 1 + \beta_{2} - \beta_{3} + \beta_{10} + \beta_{14} ) q^{5} + ( \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{16} - \beta_{19} ) q^{6} + ( \beta_{3} + \beta_{11} ) q^{7} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} + \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{8} + ( \beta_{3} + \beta_{8} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{15} - \beta_{16} ) q^{11} + ( \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{11} + 2 \beta_{13} - \beta_{15} + \beta_{17} + \beta_{19} ) q^{12} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{18} + \beta_{19} ) q^{14} + ( -\beta_{2} + \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{15} - \beta_{19} ) q^{15} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{12} + 2 \beta_{13} + 2 \beta_{15} - \beta_{18} + 3 \beta_{19} ) q^{16} + ( 1 + \beta_{3} - \beta_{4} + \beta_{10} + \beta_{13} - \beta_{16} ) q^{17} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{18} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{19} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{8} - \beta_{11} + 2 \beta_{15} - \beta_{17} - \beta_{18} ) q^{20} + ( -1 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{19} ) q^{21} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{22} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{23} + ( 1 + 2 \beta_{2} + \beta_{7} - \beta_{11} + \beta_{14} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{24} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{17} + \beta_{18} ) q^{25} + ( 3 \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{13} - \beta_{16} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{27} + ( -2 - \beta_{2} + \beta_{3} + \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{28} + ( -1 - \beta_{2} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{14} + 3 \beta_{15} + \beta_{17} + \beta_{18} ) q^{29} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{6} - \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{30} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{14} - \beta_{16} - 2 \beta_{17} ) q^{31} + ( -1 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} - 2 \beta_{13} - 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} - 3 \beta_{19} ) q^{32} + ( -1 + \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} - \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{33} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} - \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} + 2 \beta_{19} ) q^{34} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} + \beta_{14} - \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{35} + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{36} + ( 1 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{15} - \beta_{18} ) q^{37} + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 5 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - 4 \beta_{13} - 3 \beta_{15} + \beta_{17} + 2 \beta_{18} - 4 \beta_{19} ) q^{38} + ( -1 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} - 4 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{15} + 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{40} + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} + 3 \beta_{13} - \beta_{14} - \beta_{16} ) q^{41} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{8} + 3 \beta_{9} - \beta_{10} - 4 \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} - 3 \beta_{19} ) q^{42} + ( 1 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - 4 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} - 3 \beta_{13} - 3 \beta_{15} - 2 \beta_{16} - 2 \beta_{19} ) q^{43} + ( -2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{44} + ( -2 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} - 2 \beta_{15} + \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{45} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{46} + ( -2 + \beta_{1} - \beta_{2} + 5 \beta_{3} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} - 4 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - 5 \beta_{13} - \beta_{14} - 3 \beta_{15} + \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{47} + ( 5 + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{48} + ( 3 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{49} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{15} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{50} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{9} - 2 \beta_{10} + \beta_{12} - 4 \beta_{13} + \beta_{15} - \beta_{18} ) q^{51} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{53} + ( 3 - 4 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{13} - \beta_{15} + \beta_{17} + 3 \beta_{18} - 3 \beta_{19} ) q^{54} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{55} + ( -2 \beta_{1} + \beta_{3} - \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{13} - 2 \beta_{19} ) q^{56} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{18} + 3 \beta_{19} ) q^{57} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 5 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{58} + ( -2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} - \beta_{16} - 2 \beta_{17} - 2 \beta_{19} ) q^{59} + ( 7 + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{11} + 2 \beta_{14} + \beta_{15} - \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{60} + ( 3 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{12} - \beta_{18} + \beta_{19} ) q^{61} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{62} + ( -3 + \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{12} - \beta_{14} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{63} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + 5 \beta_{8} - 5 \beta_{9} + 3 \beta_{10} - 3 \beta_{12} - \beta_{13} - \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{64} + ( -3 - 5 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{14} - \beta_{17} ) q^{66} + ( -5 + 3 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} - 2 \beta_{17} - \beta_{18} + 3 \beta_{19} ) q^{67} + ( -5 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} - 4 \beta_{13} - \beta_{15} - 2 \beta_{19} ) q^{68} + ( -3 - 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{18} ) q^{69} + ( -1 - \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} + \beta_{19} ) q^{70} + ( 6 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - 5 \beta_{7} + \beta_{11} + \beta_{14} - 2 \beta_{15} - 2 \beta_{16} ) q^{71} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{10} + 2 \beta_{11} - 3 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{72} + ( -4 - 4 \beta_{1} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - 4 \beta_{13} + 2 \beta_{14} - 4 \beta_{15} - 2 \beta_{16} - 4 \beta_{19} ) q^{73} + ( 2 - \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{74} + ( 4 - \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 6 \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{17} - 2 \beta_{18} - 3 \beta_{19} ) q^{75} + ( -2 + 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - \beta_{6} + 7 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - \beta_{11} + 2 \beta_{13} + \beta_{14} + 4 \beta_{15} + \beta_{16} - 3 \beta_{18} + 2 \beta_{19} ) q^{76} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{19} ) q^{77} + ( -3 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{79} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + 4 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} - 5 \beta_{13} + \beta_{16} + \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{80} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{81} + ( 5 + 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 5 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} + 3 \beta_{19} ) q^{82} + ( -1 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{9} + 2 \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{83} + ( -2 + 2 \beta_{1} - \beta_{2} + 4 \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} + 5 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} + \beta_{18} + 4 \beta_{19} ) q^{84} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} + 3 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{85} + ( -2 + \beta_{1} + 7 \beta_{2} - 5 \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{14} - \beta_{15} - 2 \beta_{16} - 3 \beta_{17} - 5 \beta_{18} + \beta_{19} ) q^{86} + ( -1 + 2 \beta_{2} - 5 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 8 \beta_{8} - 3 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} + 3 \beta_{13} - \beta_{14} + 5 \beta_{15} + 2 \beta_{16} + \beta_{17} + 3 \beta_{19} ) q^{87} + ( \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{13} - \beta_{15} - 2 \beta_{17} - 2 \beta_{18} ) q^{88} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 3 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} - 2 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{19} ) q^{89} + ( -1 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - \beta_{12} + 5 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + 3 \beta_{19} ) q^{90} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{8} - \beta_{14} + 2 \beta_{16} + 2 \beta_{18} + \beta_{19} ) q^{92} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{18} - \beta_{19} ) q^{93} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} - 4 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{17} + \beta_{18} ) q^{94} + ( -4 + \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{95} + ( -9 - 3 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{15} + \beta_{16} + 3 \beta_{17} ) q^{96} + ( -4 + 5 \beta_{1} - 2 \beta_{3} + 4 \beta_{5} + 4 \beta_{7} + 6 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} - 2 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} + \beta_{17} - 2 \beta_{18} + 5 \beta_{19} ) q^{97} + ( 1 - 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 5 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + 3 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} + 3 \beta_{18} - 4 \beta_{19} ) q^{98} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + 5 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 8q^{2} + 4q^{3} + 12q^{4} + 6q^{5} - 4q^{6} - 12q^{8} + O(q^{10}) \) \( 20q - 8q^{2} + 4q^{3} + 12q^{4} + 6q^{5} - 4q^{6} - 12q^{8} + 8q^{10} - 8q^{11} + 24q^{12} + 24q^{15} + 4q^{16} + 14q^{17} + 4q^{19} + 22q^{20} - 4q^{21} - 32q^{22} - 8q^{23} - 4q^{24} - 18q^{25} + 4q^{27} - 40q^{30} - 12q^{32} - 36q^{33} + 2q^{34} - 20q^{35} + 8q^{38} - 16q^{40} + 38q^{41} + 16q^{42} + 32q^{43} - 36q^{44} + 20q^{45} - 4q^{46} + 28q^{48} + 36q^{49} + 52q^{50} - 10q^{53} + 36q^{54} - 16q^{55} + 12q^{57} + 8q^{59} + 92q^{60} + 32q^{61} - 4q^{62} - 64q^{63} - 20q^{64} - 32q^{66} - 116q^{67} - 50q^{68} - 32q^{69} - 32q^{70} + 40q^{71} - 72q^{73} - 4q^{75} + 16q^{76} + 28q^{77} + 34q^{80} + 28q^{81} + 34q^{82} + 8q^{84} - 60q^{86} - 28q^{87} + 32q^{88} + 12q^{89} - 46q^{90} - 8q^{92} - 40q^{95} - 56q^{96} - 44q^{97} + 8q^{98} + 60q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 69 \nu^{18} + 1647 \nu^{16} + 15798 \nu^{14} + 78322 \nu^{12} + 214723 \nu^{10} + 324081 \nu^{8} + 257858 \nu^{6} + 105030 \nu^{4} + 19961 \nu^{2} + 539 \)\()/1712\)
\(\beta_{3}\)\(=\)\((\)\(-2051 \nu^{19} - 1625 \nu^{18} - 52213 \nu^{17} - 40407 \nu^{16} - 546574 \nu^{15} - 409402 \nu^{14} - 3057250 \nu^{13} - 2186822 \nu^{12} - 9933301 \nu^{11} - 6647231 \nu^{10} - 19090911 \nu^{9} - 11566109 \nu^{8} - 21191198 \nu^{7} - 10986282 \nu^{6} - 12715822 \nu^{5} - 5054354 \nu^{4} - 3614387 \nu^{3} - 839001 \nu^{2} - 321657 \nu - 11323\)\()/23968\)
\(\beta_{4}\)\(=\)\((\)\( -1625 \nu^{18} - 40407 \nu^{16} - 409402 \nu^{14} - 2186822 \nu^{12} - 6647231 \nu^{10} - 11566109 \nu^{8} - 10986282 \nu^{6} - 5054354 \nu^{4} - 850985 \nu^{2} - 35291 \)\()/11984\)
\(\beta_{5}\)\(=\)\((\)\(2137 \nu^{19} + 321 \nu^{18} + 55159 \nu^{17} + 9951 \nu^{16} + 586250 \nu^{15} + 127330 \nu^{14} + 3327942 \nu^{13} + 869910 \nu^{12} + 10919607 \nu^{11} + 3425391 \nu^{10} + 20857925 \nu^{9} + 7807469 \nu^{8} + 22021410 \nu^{7} + 9753906 \nu^{6} + 11120698 \nu^{5} + 5800042 \nu^{4} + 1779761 \nu^{3} + 1183313 \nu^{2} - 39325 \nu + 27499\)\()/23968\)
\(\beta_{6}\)\(=\)\((\)\(-908 \nu^{19} - 41 \nu^{18} - 22604 \nu^{17} - 11 \nu^{16} - 229460 \nu^{15} + 13762 \nu^{14} - 1229016 \nu^{13} + 175078 \nu^{12} - 3747108 \nu^{11} + 958737 \nu^{10} - 6525228 \nu^{9} + 2689731 \nu^{8} - 6133752 \nu^{7} + 3838246 \nu^{6} - 2660328 \nu^{5} + 2440386 \nu^{4} - 298224 \nu^{3} + 488651 \nu^{2} + 33412 \nu + 12681\)\()/11984\)
\(\beta_{7}\)\(=\)\((\)\(-2137 \nu^{19} + 321 \nu^{18} - 55159 \nu^{17} + 9951 \nu^{16} - 586250 \nu^{15} + 127330 \nu^{14} - 3327942 \nu^{13} + 869910 \nu^{12} - 10919607 \nu^{11} + 3425391 \nu^{10} - 20857925 \nu^{9} + 7807469 \nu^{8} - 22021410 \nu^{7} + 9753906 \nu^{6} - 11120698 \nu^{5} + 5800042 \nu^{4} - 1779761 \nu^{3} + 1183313 \nu^{2} + 39325 \nu + 51467\)\()/23968\)
\(\beta_{8}\)\(=\)\((\)\(-2051 \nu^{19} + 1625 \nu^{18} - 52213 \nu^{17} + 40407 \nu^{16} - 546574 \nu^{15} + 409402 \nu^{14} - 3057250 \nu^{13} + 2186822 \nu^{12} - 9933301 \nu^{11} + 6647231 \nu^{10} - 19090911 \nu^{9} + 11566109 \nu^{8} - 21191198 \nu^{7} + 10986282 \nu^{6} - 12715822 \nu^{5} + 5054354 \nu^{4} - 3614387 \nu^{3} + 839001 \nu^{2} - 321657 \nu + 11323\)\()/23968\)
\(\beta_{9}\)\(=\)\((\)\(-2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} - 4279914 \nu^{13} - 1404094 \nu^{12} - 14010639 \nu^{11} - 4705845 \nu^{10} - 26598933 \nu^{9} - 9214959 \nu^{8} - 27637370 \nu^{7} - 9971998 \nu^{6} - 13384022 \nu^{5} - 5100714 \nu^{4} - 1923401 \nu^{3} - 798859 \nu^{2} - 7127 \nu - 15221\)\()/11984\)
\(\beta_{10}\)\(=\)\((\)\(-2753 \nu^{19} + 867 \nu^{18} - 71035 \nu^{17} + 22593 \nu^{16} - 754642 \nu^{15} + 243222 \nu^{14} - 4279914 \nu^{13} + 1404094 \nu^{12} - 14010639 \nu^{11} + 4705845 \nu^{10} - 26598933 \nu^{9} + 9214959 \nu^{8} - 27637370 \nu^{7} + 9971998 \nu^{6} - 13384022 \nu^{5} + 5100714 \nu^{4} - 1923401 \nu^{3} + 798859 \nu^{2} - 7127 \nu + 15221\)\()/11984\)
\(\beta_{11}\)\(=\)\((\)\(-6161 \nu^{19} + 1625 \nu^{18} - 162487 \nu^{17} + 40407 \nu^{16} - 1774122 \nu^{15} + 409402 \nu^{14} - 10413446 \nu^{13} + 2186822 \nu^{12} - 35607463 \nu^{11} + 6647231 \nu^{10} - 71565685 \nu^{9} + 11566109 \nu^{8} - 80587002 \nu^{7} + 10986282 \nu^{6} - 44718762 \nu^{5} + 5054354 \nu^{4} - 9152481 \nu^{3} + 839001 \nu^{2} - 389603 \nu + 11323\)\()/23968\)
\(\beta_{12}\)\(=\)\((\)\(-5315 \nu^{19} + 3731 \nu^{18} - 137269 \nu^{17} + 93877 \nu^{16} - 1459766 \nu^{15} + 967694 \nu^{14} - 8288618 \nu^{13} + 5300554 \nu^{12} - 27174293 \nu^{11} + 16710141 \nu^{10} - 51713519 \nu^{9} + 30620807 \nu^{8} - 53993518 \nu^{7} + 31181654 \nu^{6} - 26501742 \nu^{5} + 15528646 \nu^{4} - 4113323 \nu^{3} + 2692795 \nu^{2} - 164305 \nu + 89369\)\()/23968\)
\(\beta_{13}\)\(=\)\((\)\( 539 \nu^{19} + 13945 \nu^{17} + 148734 \nu^{15} + 848758 \nu^{13} + 2806945 \nu^{11} + 5426451 \nu^{9} + 5842618 \nu^{7} + 3071006 \nu^{5} + 575727 \nu^{3} + 22081 \nu \)\()/1712\)
\(\beta_{14}\)\(=\)\((\)\(-10681 \nu^{19} - 2701 \nu^{18} - 278863 \nu^{17} - 65979 \nu^{16} - 3007858 \nu^{15} - 654010 \nu^{14} - 17403014 \nu^{13} - 3402286 \nu^{12} - 58543111 \nu^{11} - 10032411 \nu^{10} - 115640453 \nu^{9} - 16941601 \nu^{8} - 128222802 \nu^{7} - 15900242 \nu^{6} - 70857090 \nu^{5} - 7770066 \nu^{4} - 15238809 \nu^{3} - 1647861 \nu^{2} - 929939 \nu - 76919\)\()/23968\)
\(\beta_{15}\)\(=\)\((\)\(11655 \nu^{19} - 191 \nu^{18} + 301777 \nu^{17} - 4801 \nu^{16} + 3219622 \nu^{15} - 49518 \nu^{14} + 18357066 \nu^{13} - 271210 \nu^{12} + 60507937 \nu^{11} - 846985 \nu^{10} + 115985219 \nu^{9} - 1484347 \nu^{8} + 122403526 \nu^{7} - 1281222 \nu^{6} + 61246374 \nu^{5} - 266302 \nu^{4} + 9938775 \nu^{3} + 242553 \nu^{2} + 302197 \nu + 54179\)\()/23968\)
\(\beta_{16}\)\(=\)\((\)\(-5223 \nu^{19} + 1330 \nu^{18} - 135929 \nu^{17} + 34482 \nu^{16} - 1460102 \nu^{15} + 368340 \nu^{14} - 8401898 \nu^{13} + 2101764 \nu^{12} - 28052013 \nu^{11} + 6924666 \nu^{10} - 54799999 \nu^{9} + 13234102 \nu^{8} - 59649554 \nu^{7} + 13862912 \nu^{6} - 31730962 \nu^{5} + 6844404 \nu^{4} - 6110435 \nu^{3} + 1102710 \nu^{2} - 270729 \nu + 34314\)\()/11984\)
\(\beta_{17}\)\(=\)\((\)\(1767 \nu^{19} + 203 \nu^{18} + 46090 \nu^{17} + 5334 \nu^{16} + 496321 \nu^{15} + 57876 \nu^{14} + 2863199 \nu^{13} + 336154 \nu^{12} + 9578886 \nu^{11} + 1129079 \nu^{10} + 18718426 \nu^{9} + 2198945 \nu^{8} + 20288293 \nu^{7} + 2330006 \nu^{6} + 10613664 \nu^{5} + 1119517 \nu^{4} + 1933903 \nu^{3} + 133539 \nu^{2} + 77817 \nu - 455\)\()/2996\)
\(\beta_{18}\)\(=\)\((\)\(-14121 \nu^{19} + 1997 \nu^{18} - 366743 \nu^{17} + 48691 \nu^{16} - 3929786 \nu^{15} + 481250 \nu^{14} - 22550278 \nu^{13} + 2492366 \nu^{12} - 75069959 \nu^{11} + 7298451 \nu^{10} - 146282469 \nu^{9} + 12190889 \nu^{8} - 159130354 \nu^{7} + 11237658 \nu^{6} - 85133882 \nu^{5} + 5327218 \nu^{4} - 16903569 \nu^{3} + 1095077 \nu^{2} - 835507 \nu + 34959\)\()/23968\)
\(\beta_{19}\)\(=\)\((\)\(-17385 \nu^{19} - 3359 \nu^{18} - 451799 \nu^{17} - 85593 \nu^{16} - 4842978 \nu^{15} - 895846 \nu^{14} - 27781646 \nu^{13} - 4995010 \nu^{12} - 92310951 \nu^{11} - 16058921 \nu^{10} - 178905077 \nu^{9} - 29996027 \nu^{8} - 191932674 \nu^{7} - 30930278 \nu^{6} - 98919802 \nu^{5} - 15255782 \nu^{4} - 17402505 \nu^{3} - 2436719 \nu^{2} - 702123 \nu - 41765\)\()/23968\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{8} - \beta_{4} + \beta_{3} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{19} + \beta_{18} + \beta_{17} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{3} - \beta_{2} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{19} - \beta_{18} + 2 \beta_{15} + 2 \beta_{13} - \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + 8 \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + 5 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + 3 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(-8 \beta_{19} - 9 \beta_{18} - 8 \beta_{17} + \beta_{15} + 8 \beta_{14} - 15 \beta_{13} + 9 \beta_{12} - 7 \beta_{11} - 7 \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 11 \beta_{1} - 3\)
\(\nu^{6}\)\(=\)\(-32 \beta_{19} + 13 \beta_{18} + 2 \beta_{17} + 2 \beta_{16} - 19 \beta_{15} - 19 \beta_{13} + 13 \beta_{12} - 23 \beta_{10} + 25 \beta_{9} - 61 \beta_{8} + 10 \beta_{7} - 21 \beta_{6} + 10 \beta_{5} - 25 \beta_{4} + 48 \beta_{3} - 24 \beta_{2} - 32 \beta_{1} - 50\)
\(\nu^{7}\)\(=\)\(51 \beta_{19} + 68 \beta_{18} + 54 \beta_{17} - 15 \beta_{15} - 54 \beta_{14} + 99 \beta_{13} - 68 \beta_{12} + 44 \beta_{11} + 40 \beta_{10} - 10 \beta_{9} - 17 \beta_{8} - 22 \beta_{7} + 15 \beta_{6} + 22 \beta_{5} - 15 \beta_{4} + 45 \beta_{3} - 54 \beta_{2} - 48 \beta_{1} + 36\)
\(\nu^{8}\)\(=\)\(267 \beta_{19} - 120 \beta_{18} - 28 \beta_{17} - 24 \beta_{16} + 147 \beta_{15} - 4 \beta_{14} + 147 \beta_{13} - 120 \beta_{12} + 199 \beta_{10} - 227 \beta_{9} + 452 \beta_{8} - 80 \beta_{7} + 171 \beta_{6} - 80 \beta_{5} + 132 \beta_{4} - 328 \beta_{3} + 206 \beta_{2} + 267 \beta_{1} + 304\)
\(\nu^{9}\)\(=\)\(-308 \beta_{19} - 488 \beta_{18} - 352 \beta_{17} + 154 \beta_{15} + 352 \beta_{14} - 648 \beta_{13} + 488 \beta_{12} - 277 \beta_{11} - 214 \beta_{10} + 76 \beta_{9} + 181 \beta_{8} + 187 \beta_{7} - 154 \beta_{6} - 187 \beta_{5} + 154 \beta_{4} - 294 \beta_{3} + 352 \beta_{2} + 242 \beta_{1} - 323\)
\(\nu^{10}\)\(=\)\(-2044 \beta_{19} + 974 \beta_{18} + 274 \beta_{17} + 214 \beta_{16} - 1070 \beta_{15} + 60 \beta_{14} - 1070 \beta_{13} + 974 \beta_{12} - 1556 \beta_{10} + 1830 \beta_{9} - 3280 \beta_{8} + 599 \beta_{7} - 1284 \beta_{6} + 599 \beta_{5} - 733 \beta_{4} + 2246 \beta_{3} - 1576 \beta_{2} - 2044 \beta_{1} - 1940\)
\(\nu^{11}\)\(=\)\(1854 \beta_{19} + 3434 \beta_{18} + 2299 \beta_{17} - 1356 \beta_{15} - 2299 \beta_{14} + 4298 \beta_{13} - 3434 \beta_{12} + 1793 \beta_{11} + 1104 \beta_{10} - 529 \beta_{9} - 1613 \beta_{8} - 1455 \beta_{7} + 1356 \beta_{6} + 1455 \beta_{5} - 1356 \beta_{4} + 1981 \beta_{3} - 2299 \beta_{2} - 1353 \beta_{1} + 2590\)
\(\nu^{12}\)\(=\)\(15044 \beta_{19} - 7438 \beta_{18} - 2330 \beta_{17} - 1724 \beta_{16} + 7606 \beta_{15} - 606 \beta_{14} + 7606 \beta_{13} - 7438 \beta_{12} + 11610 \beta_{10} - 13940 \beta_{9} + 23486 \beta_{8} - 4360 \beta_{7} + 9330 \beta_{6} - 4360 \beta_{5} + 4238 \beta_{4} - 15442 \beta_{3} + 11490 \beta_{2} + 15044 \beta_{1} + 12755\)
\(\nu^{13}\)\(=\)\(-11342 \beta_{19} - 23984 \beta_{18} - 15204 \beta_{17} + 11014 \beta_{15} + 15204 \beta_{14} - 28946 \beta_{13} + 23984 \beta_{12} - 11940 \beta_{11} - 5532 \beta_{10} + 3576 \beta_{9} + 13144 \beta_{8} + 10862 \beta_{7} - 11014 \beta_{6} - 10862 \beta_{5} + 11014 \beta_{4} - 13672 \beta_{3} + 15204 \beta_{2} + 8101 \beta_{1} - 19642\)
\(\nu^{14}\)\(=\)\(-108504 \beta_{19} + 54982 \beta_{18} + 18452 \beta_{17} + 13248 \beta_{16} - 53522 \beta_{15} + 5204 \beta_{14} - 53522 \beta_{13} + 54982 \beta_{12} - 84520 \beta_{10} + 102972 \beta_{9} - 166807 \beta_{8} + 31270 \beta_{7} - 66770 \beta_{6} + 31270 \beta_{5} - 25317 \beta_{4} + 106621 \beta_{3} - 81960 \beta_{2} - 108504 \beta_{1} - 85534\)
\(\nu^{15}\)\(=\)\(71003 \beta_{19} + 167135 \beta_{18} + 101945 \beta_{17} - 85268 \beta_{15} - 101945 \beta_{14} + 197430 \beta_{13} - 167135 \beta_{12} + 81223 \beta_{11} + 26885 \beta_{10} - 24040 \beta_{9} - 101732 \beta_{8} - 79260 \beta_{7} + 85268 \beta_{6} + 79260 \beta_{5} - 85268 \beta_{4} + 95701 \beta_{3} - 101945 \beta_{2} - 50745 \beta_{1} + 144450\)
\(\nu^{16}\)\(=\)\(773909 \beta_{19} - 398897 \beta_{18} - 140250 \beta_{17} - 99100 \beta_{16} + 375012 \beta_{15} - 41150 \beta_{14} + 375012 \beta_{13} - 398897 \beta_{12} + 606774 \beta_{10} - 747024 \beta_{9} + 1179002 \beta_{8} - 222355 \beta_{7} + 474112 \beta_{6} - 222355 \beta_{5} + 155477 \beta_{4} - 738955 \beta_{3} + 578736 \beta_{2} + 773909 \beta_{1} + 581415\)
\(\nu^{17}\)\(=\)\(-455154 \beta_{19} - 1164737 \beta_{18} - 691774 \beta_{17} + 640259 \beta_{15} + 691774 \beta_{14} - 1359313 \beta_{13} + 1164737 \beta_{12} - 560265 \beta_{11} - 125581 \beta_{10} + 162277 \beta_{9} + 762655 \beta_{8} + 570482 \beta_{7} - 640259 \beta_{6} - 570482 \beta_{5} + 640259 \beta_{4} - 674489 \beta_{3} + 691774 \beta_{2} + 327815 \beta_{1} - 1043445\)
\(\nu^{18}\)\(=\)\(-5484892 \beta_{19} + 2861437 \beta_{18} + 1039156 \beta_{17} + 728470 \beta_{16} - 2623455 \beta_{15} + 310686 \beta_{14} - 2623455 \beta_{13} + 2861437 \beta_{12} - 4320155 \beta_{10} + 5359311 \beta_{9} - 8309477 \beta_{8} + 1572942 \beta_{7} - 3351925 \beta_{6} + 1572942 \beta_{5} - 978069 \beta_{4} + 5137354 \beta_{3} - 4067694 \beta_{2} - 5484892 \beta_{1} - 3989746\)
\(\nu^{19}\)\(=\)\(2980455 \beta_{19} + 8124372 \beta_{18} + 4738502 \beta_{17} - 4711075 \beta_{15} - 4738502 \beta_{14} + 9420783 \beta_{13} - 8124372 \beta_{12} + 3897152 \beta_{11} + 551860 \beta_{10} - 1103390 \beta_{9} - 5600415 \beta_{8} - 4070816 \beta_{7} + 4711075 \beta_{6} + 4070816 \beta_{5} - 4711075 \beta_{4} + 4765859 \beta_{3} - 4738502 \beta_{2} - 2164558 \beta_{1} + 7456686\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(\beta_{13}\) \(-\beta_{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
268.1
2.64975i
2.08794i
1.83163i
1.58474i
0.493902i
0.274809i
0.131303i
1.02262i
1.51805i
2.25081i
2.64975i
2.08794i
1.83163i
1.58474i
0.493902i
0.274809i
0.131303i
1.02262i
1.51805i
2.25081i
−2.64975 0.917096 + 0.917096i 5.02120 1.30391 1.81654i −2.43008 2.43008i 0.112348i −8.00544 1.31787i −3.45504 + 4.81339i
268.2 −2.08794 1.94842 + 1.94842i 2.35949 0.194361 2.22760i −4.06818 4.06818i 2.91126i −0.750585 4.59268i −0.405815 + 4.65110i
268.3 −1.83163 −1.40138 1.40138i 1.35488 −1.45480 + 1.69810i 2.56682 + 2.56682i 3.53890i 1.18163 0.927746i 2.66466 3.11030i
268.4 −1.58474 −0.139520 0.139520i 0.511395 −0.0672627 + 2.23506i 0.221103 + 0.221103i 0.548328i 2.35905 2.96107i 0.106594 3.54198i
268.5 −0.493902 −0.664960 0.664960i −1.75606 2.21791 + 0.284413i 0.328425 + 0.328425i 3.67549i 1.85513 2.11566i −1.09543 0.140472i
268.6 −0.274809 1.67095 + 1.67095i −1.92448 −1.69883 1.45395i −0.459191 0.459191i 0.386104i 1.07848 2.58414i 0.466854 + 0.399558i
268.7 0.131303 −0.243172 0.243172i −1.98276 −0.813169 2.08297i −0.0319291 0.0319291i 2.78137i −0.522947 2.88174i −0.106771 0.273499i
268.8 1.02262 −1.97063 1.97063i −0.954253 1.69584 + 1.45744i −2.01520 2.01520i 0.963574i −3.02107 4.76674i 1.73420 + 1.49040i
268.9 1.51805 0.478298 + 0.478298i 0.304465 −0.600231 + 2.15400i 0.726078 + 0.726078i 2.59488i −2.57390 2.54246i −0.911178 + 3.26987i
268.10 2.25081 1.40490 + 1.40490i 3.06613 2.22228 0.247944i 3.16216 + 3.16216i 1.27718i 2.39966 0.947480i 5.00192 0.558075i
577.1 −2.64975 0.917096 0.917096i 5.02120 1.30391 + 1.81654i −2.43008 + 2.43008i 0.112348i −8.00544 1.31787i −3.45504 4.81339i
577.2 −2.08794 1.94842 1.94842i 2.35949 0.194361 + 2.22760i −4.06818 + 4.06818i 2.91126i −0.750585 4.59268i −0.405815 4.65110i
577.3 −1.83163 −1.40138 + 1.40138i 1.35488 −1.45480 1.69810i 2.56682 2.56682i 3.53890i 1.18163 0.927746i 2.66466 + 3.11030i
577.4 −1.58474 −0.139520 + 0.139520i 0.511395 −0.0672627 2.23506i 0.221103 0.221103i 0.548328i 2.35905 2.96107i 0.106594 + 3.54198i
577.5 −0.493902 −0.664960 + 0.664960i −1.75606 2.21791 0.284413i 0.328425 0.328425i 3.67549i 1.85513 2.11566i −1.09543 + 0.140472i
577.6 −0.274809 1.67095 1.67095i −1.92448 −1.69883 + 1.45395i −0.459191 + 0.459191i 0.386104i 1.07848 2.58414i 0.466854 0.399558i
577.7 0.131303 −0.243172 + 0.243172i −1.98276 −0.813169 + 2.08297i −0.0319291 + 0.0319291i 2.78137i −0.522947 2.88174i −0.106771 + 0.273499i
577.8 1.02262 −1.97063 + 1.97063i −0.954253 1.69584 1.45744i −2.01520 + 2.01520i 0.963574i −3.02107 4.76674i 1.73420 1.49040i
577.9 1.51805 0.478298 0.478298i 0.304465 −0.600231 2.15400i 0.726078 0.726078i 2.59488i −2.57390 2.54246i −0.911178 3.26987i
577.10 2.25081 1.40490 1.40490i 3.06613 2.22228 + 0.247944i 3.16216 3.16216i 1.27718i 2.39966 0.947480i 5.00192 + 0.558075i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.k.d 20
5.c odd 4 1 845.2.f.d 20
13.b even 2 1 845.2.k.e 20
13.c even 3 1 845.2.o.f 20
13.c even 3 1 845.2.o.g 20
13.d odd 4 1 845.2.f.d 20
13.d odd 4 1 845.2.f.e 20
13.e even 6 1 65.2.o.a 20
13.e even 6 1 845.2.o.e 20
13.f odd 12 1 65.2.t.a yes 20
13.f odd 12 1 845.2.t.e 20
13.f odd 12 1 845.2.t.f 20
13.f odd 12 1 845.2.t.g 20
39.h odd 6 1 585.2.cf.a 20
39.k even 12 1 585.2.dp.a 20
65.f even 4 1 845.2.k.e 20
65.h odd 4 1 845.2.f.e 20
65.k even 4 1 inner 845.2.k.d 20
65.l even 6 1 325.2.s.b 20
65.o even 12 1 325.2.s.b 20
65.o even 12 1 845.2.o.f 20
65.o even 12 1 845.2.o.g 20
65.q odd 12 1 845.2.t.e 20
65.q odd 12 1 845.2.t.g 20
65.r odd 12 1 65.2.t.a yes 20
65.r odd 12 1 325.2.x.b 20
65.r odd 12 1 845.2.t.f 20
65.s odd 12 1 325.2.x.b 20
65.t even 12 1 65.2.o.a 20
65.t even 12 1 845.2.o.e 20
195.bc odd 12 1 585.2.cf.a 20
195.bf even 12 1 585.2.dp.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.o.a 20 13.e even 6 1
65.2.o.a 20 65.t even 12 1
65.2.t.a yes 20 13.f odd 12 1
65.2.t.a yes 20 65.r odd 12 1
325.2.s.b 20 65.l even 6 1
325.2.s.b 20 65.o even 12 1
325.2.x.b 20 65.r odd 12 1
325.2.x.b 20 65.s odd 12 1
585.2.cf.a 20 39.h odd 6 1
585.2.cf.a 20 195.bc odd 12 1
585.2.dp.a 20 39.k even 12 1
585.2.dp.a 20 195.bf even 12 1
845.2.f.d 20 5.c odd 4 1
845.2.f.d 20 13.d odd 4 1
845.2.f.e 20 13.d odd 4 1
845.2.f.e 20 65.h odd 4 1
845.2.k.d 20 1.a even 1 1 trivial
845.2.k.d 20 65.k even 4 1 inner
845.2.k.e 20 13.b even 2 1
845.2.k.e 20 65.f even 4 1
845.2.o.e 20 13.e even 6 1
845.2.o.e 20 65.t even 12 1
845.2.o.f 20 13.c even 3 1
845.2.o.f 20 65.o even 12 1
845.2.o.g 20 13.c even 3 1
845.2.o.g 20 65.o even 12 1
845.2.t.e 20 13.f odd 12 1
845.2.t.e 20 65.q odd 12 1
845.2.t.f 20 13.f odd 12 1
845.2.t.f 20 65.r odd 12 1
845.2.t.g 20 13.f odd 12 1
845.2.t.g 20 65.q odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T - 37 T^{2} - 52 T^{3} + 51 T^{4} + 80 T^{5} - 9 T^{6} - 34 T^{7} - 5 T^{8} + 4 T^{9} + T^{10} )^{2} \)
$3$ \( 16 + 144 T + 648 T^{2} + 796 T^{3} - 55 T^{4} - 1540 T^{5} + 8168 T^{6} - 912 T^{7} + 364 T^{8} - 3772 T^{9} + 8128 T^{10} - 3800 T^{11} + 958 T^{12} - 300 T^{13} + 616 T^{14} - 304 T^{15} + 76 T^{16} - 4 T^{17} + 8 T^{18} - 4 T^{19} + T^{20} \)
$5$ \( 9765625 - 11718750 T + 10546875 T^{2} - 8906250 T^{3} + 6140625 T^{4} - 3887500 T^{5} + 2295000 T^{6} - 1264500 T^{7} + 649950 T^{8} - 316320 T^{9} + 146954 T^{10} - 63264 T^{11} + 25998 T^{12} - 10116 T^{13} + 3672 T^{14} - 1244 T^{15} + 393 T^{16} - 114 T^{17} + 27 T^{18} - 6 T^{19} + T^{20} \)
$7$ \( 64 + 5856 T^{2} + 65209 T^{4} + 240068 T^{6} + 346004 T^{8} + 223932 T^{10} + 70494 T^{12} + 11852 T^{14} + 1092 T^{16} + 52 T^{18} + T^{20} \)
$11$ \( 256 + 10880 T + 231200 T^{2} - 3289192 T^{3} + 21680561 T^{4} - 35793616 T^{5} + 28952192 T^{6} + 3793488 T^{7} + 1643724 T^{8} - 3067632 T^{9} + 3098560 T^{10} + 672160 T^{11} + 77654 T^{12} - 15024 T^{13} + 20096 T^{14} + 4720 T^{15} + 556 T^{16} - 16 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$13$ \( T^{20} \)
$17$ \( 1168561 - 22378862 T + 214286402 T^{2} - 363680406 T^{3} + 326215005 T^{4} - 171802520 T^{5} + 62490568 T^{6} - 22597312 T^{7} + 12747202 T^{8} - 6398084 T^{9} + 2157676 T^{10} - 493836 T^{11} + 164914 T^{12} - 75512 T^{13} + 25288 T^{14} - 4464 T^{15} + 797 T^{16} - 286 T^{17} + 98 T^{18} - 14 T^{19} + T^{20} \)
$19$ \( 1583721616 + 3809113936 T + 4580776328 T^{2} - 1385600468 T^{3} - 1601671 T^{4} + 228365048 T^{5} + 1160020608 T^{6} - 710002592 T^{7} + 215427604 T^{8} - 4612008 T^{9} + 4621936 T^{10} - 3572040 T^{11} + 1341742 T^{12} + 27944 T^{13} + 9600 T^{14} - 6464 T^{15} + 2228 T^{16} + 56 T^{17} + 8 T^{18} - 4 T^{19} + T^{20} \)
$23$ \( 144 - 720 T + 1800 T^{2} - 12348 T^{3} + 164113 T^{4} - 984056 T^{5} + 3398288 T^{6} - 6910716 T^{7} + 8275668 T^{8} - 4484152 T^{9} + 1012328 T^{10} + 177028 T^{11} + 149174 T^{12} - 57128 T^{13} + 12032 T^{14} + 2764 T^{15} + 676 T^{16} - 152 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$29$ \( 206213167449 + 2551463464506 T^{2} + 1641823315357 T^{4} + 408594473912 T^{6} + 50679284690 T^{8} + 3524388668 T^{10} + 144311074 T^{12} + 3491704 T^{14} + 48149 T^{16} + 346 T^{18} + T^{20} \)
$31$ \( 2166784 - 14602240 T + 49203200 T^{2} + 45220864 T^{3} + 317008128 T^{4} - 1365233664 T^{5} + 2473766912 T^{6} - 2175176448 T^{7} + 1058401664 T^{8} - 187005312 T^{9} + 8238080 T^{10} + 315200 T^{11} + 3495856 T^{12} - 416544 T^{13} + 5408 T^{14} + 12992 T^{15} + 6072 T^{16} + 104 T^{17} + T^{20} \)
$37$ \( 4508182449 + 20710012470 T^{2} + 25007298181 T^{4} + 12792399560 T^{6} + 3106978954 T^{8} + 370061380 T^{10} + 23952906 T^{12} + 888968 T^{14} + 18901 T^{16} + 214 T^{18} + T^{20} \)
$41$ \( 3748255729 + 10823859062 T + 15628059218 T^{2} - 3650191222 T^{3} + 15790909405 T^{4} + 52094424312 T^{5} + 86371744824 T^{6} - 37685140344 T^{7} + 7450771906 T^{8} - 1066655788 T^{9} + 693030764 T^{10} - 270411476 T^{11} + 59495714 T^{12} - 8376424 T^{13} + 1310520 T^{14} - 294424 T^{15} + 59725 T^{16} - 8154 T^{17} + 722 T^{18} - 38 T^{19} + T^{20} \)
$43$ \( 1370772640000 + 13065800176000 T + 62269675239200 T^{2} - 54606039937720 T^{3} + 23237213308361 T^{4} - 3285847748268 T^{5} + 732318801352 T^{6} - 350615201288 T^{7} + 116671646236 T^{8} - 19786484868 T^{9} + 2498898544 T^{10} - 440759528 T^{11} + 109143454 T^{12} - 18486340 T^{13} + 2096584 T^{14} - 203576 T^{15} + 30348 T^{16} - 4716 T^{17} + 512 T^{18} - 32 T^{19} + T^{20} \)
$47$ \( 807469056 + 11345362944 T^{2} + 44521738240 T^{4} + 54003212288 T^{6} + 17345956864 T^{8} + 2242437632 T^{10} + 133025920 T^{12} + 3738752 T^{14} + 52960 T^{16} + 368 T^{18} + T^{20} \)
$53$ \( 2978634160384 + 1638501455872 T + 450657394688 T^{2} + 451854140672 T^{3} + 938935450496 T^{4} + 662260749312 T^{5} + 256514576512 T^{6} + 53765212096 T^{7} + 8410884016 T^{8} + 2137889376 T^{9} + 822713344 T^{10} + 169978000 T^{11} + 19301176 T^{12} + 1286456 T^{13} + 475072 T^{14} + 100768 T^{15} + 10713 T^{16} + 78 T^{17} + 50 T^{18} + 10 T^{19} + T^{20} \)
$59$ \( 33856 + 20786112 T + 6380884512 T^{2} - 6574250696 T^{3} + 3389034033 T^{4} + 705437560 T^{5} + 432397472 T^{6} - 474586176 T^{7} + 298674988 T^{8} + 113152504 T^{9} + 36052352 T^{10} - 11552304 T^{11} + 2675222 T^{12} + 647592 T^{13} + 260000 T^{14} - 84672 T^{15} + 14412 T^{16} + 296 T^{17} + 32 T^{18} - 8 T^{19} + T^{20} \)
$61$ \( ( 909097 - 168224 T - 755207 T^{2} + 162032 T^{3} + 145926 T^{4} - 31536 T^{5} - 7114 T^{6} + 2096 T^{7} - 63 T^{8} - 16 T^{9} + T^{10} )^{2} \)
$67$ \( ( 3934324 - 22189316 T - 32642851 T^{2} - 16676546 T^{3} - 3787524 T^{4} - 251338 T^{5} + 57550 T^{6} + 14246 T^{7} + 1320 T^{8} + 58 T^{9} + T^{10} )^{2} \)
$71$ \( 112163628352758544 - 91546094624318288 T + 37359202640124488 T^{2} - 9576416034716972 T^{3} + 2317098069156521 T^{4} - 735852942828004 T^{5} + 237629606472984 T^{6} - 57549573157988 T^{7} + 10168663564372 T^{8} - 1446803743524 T^{9} + 212220304024 T^{10} - 35358247644 T^{11} + 5435994526 T^{12} - 631053772 T^{13} + 51639528 T^{14} - 2950508 T^{15} + 148052 T^{16} - 10204 T^{17} + 800 T^{18} - 40 T^{19} + T^{20} \)
$73$ \( ( 253113232 + 309584640 T + 145220576 T^{2} + 29843280 T^{3} + 1070440 T^{4} - 651712 T^{5} - 115932 T^{6} - 5932 T^{7} + 249 T^{8} + 36 T^{9} + T^{10} )^{2} \)
$79$ \( 7586504765095936 + 5385471164325888 T^{2} + 1241871864168704 T^{4} + 134326970823168 T^{6} + 7680125227520 T^{8} + 243260666240 T^{10} + 4371796960 T^{12} + 45175136 T^{14} + 264160 T^{16} + 808 T^{18} + T^{20} \)
$83$ \( 11512611864576 + 13783820697600 T^{2} + 5842988808448 T^{4} + 1204294956032 T^{6} + 135217417088 T^{8} + 8634255104 T^{10} + 316478512 T^{12} + 6614080 T^{14} + 76568 T^{16} + 448 T^{18} + T^{20} \)
$89$ \( 329648222500 + 2100183285000 T + 6690116205000 T^{2} + 12239557692900 T^{3} + 13945249003841 T^{4} + 9400242302920 T^{5} + 4096129094400 T^{6} + 1175553008768 T^{7} + 225768577636 T^{8} + 29888544088 T^{9} + 4064260592 T^{10} + 872332616 T^{11} + 177210598 T^{12} + 19661784 T^{13} + 996608 T^{14} + 45152 T^{15} + 24144 T^{16} + 2560 T^{17} + 72 T^{18} - 12 T^{19} + T^{20} \)
$97$ \( ( 169592356 - 326697620 T - 215226235 T^{2} - 28046342 T^{3} + 4256820 T^{4} + 1028066 T^{5} + 4942 T^{6} - 8926 T^{7} - 312 T^{8} + 22 T^{9} + T^{10} )^{2} \)
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