Properties

Label 671.2.a.c
Level $671$
Weight $2$
Character orbit 671.a
Self dual yes
Analytic conductor $5.358$
Analytic rank $0$
Dimension $19$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Defining polynomial: \(x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} - 4673 x^{11} - 15053 x^{10} + 16875 x^{9} + 20141 x^{8} - 28019 x^{7} - 11589 x^{6} + 21077 x^{5} + 1674 x^{4} - 6404 x^{3} + 241 x^{2} + 643 x - 43\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} - 4673 x^{11} - 15053 x^{10} + 16875 x^{9} + 20141 x^{8} - 28019 x^{7} - 11589 x^{6} + 21077 x^{5} + 1674 x^{4} - 6404 x^{3} + 241 x^{2} + 643 x - 43\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(8867201804 \nu^{18} - 28397403169 \nu^{17} - 153963852608 \nu^{16} + 585160018840 \nu^{15} + 420015084215 \nu^{14} - 3896315485489 \nu^{13} + 7816695786848 \nu^{12} + 4170318228153 \nu^{11} - 74752830871939 \nu^{10} + 61181608853165 \nu^{9} + 280962923414937 \nu^{8} - 283316198675975 \nu^{7} - 514007340828783 \nu^{6} + 474088918297633 \nu^{5} + 431277028203763 \nu^{4} - 300469578965604 \nu^{3} - 132758950019250 \nu^{2} + 50083912844543 \nu + 7406641729547\)\()/ 2584793613763 \)
\(\beta_{4}\)\(=\)\((\)\(-9925308116 \nu^{18} + 211585433401 \nu^{17} - 157290256641 \nu^{16} - 5438380518759 \nu^{15} + 7280102119263 \nu^{14} + 56715822193005 \nu^{13} - 79490940840104 \nu^{12} - 311772296964757 \nu^{11} + 401624582318481 \nu^{10} + 976030566478054 \nu^{9} - 1042517331908497 \nu^{8} - 1744313483302286 \nu^{7} + 1352625850873237 \nu^{6} + 1686914951359080 \nu^{5} - 761921796525769 \nu^{4} - 794326099908997 \nu^{3} + 121134977398858 \nu^{2} + 133177244880778 \nu + 4643140412416\)\()/ 2584793613763 \)
\(\beta_{5}\)\(=\)\((\)\(16270525372 \nu^{18} - 35753207242 \nu^{17} - 519467802678 \nu^{16} + 1065729788540 \nu^{15} + 6910095072255 \nu^{14} - 13113396066991 \nu^{13} - 49478643130019 \nu^{12} + 86061802478689 \nu^{11} + 204922845192456 \nu^{10} - 323580558431117 \nu^{9} - 488697894791415 \nu^{8} + 692134937669964 \nu^{7} + 625158766804376 \nu^{6} - 777549565519077 \nu^{5} - 360901712332219 \nu^{4} + 372817093364514 \nu^{3} + 64006649549778 \nu^{2} - 47424793625471 \nu + 1737797285981\)\()/ 2584793613763 \)
\(\beta_{6}\)\(=\)\((\)\(37762226457 \nu^{18} - 165903350608 \nu^{17} - 747153495518 \nu^{16} + 3973232345087 \nu^{15} + 4711323004499 \nu^{14} - 37022753494585 \nu^{13} - 4098672552276 \nu^{12} + 168673178705194 \nu^{11} - 72597790484259 \nu^{10} - 372792814215164 \nu^{9} + 311955896020022 \nu^{8} + 273746173403669 \nu^{7} - 483092267921848 \nu^{6} + 262742744285573 \nu^{5} + 256243772589666 \nu^{4} - 424688104742192 \nu^{3} - 12717902390185 \nu^{2} + 114678199755666 \nu - 2002160866471\)\()/ 2584793613763 \)
\(\beta_{7}\)\(=\)\((\)\(41025764486 \nu^{18} + 68688107954 \nu^{17} - 1546829355593 \nu^{16} - 1494342049450 \nu^{15} + 22839785265251 \nu^{14} + 12033133930515 \nu^{13} - 172638105605592 \nu^{12} - 42177686787597 \nu^{11} + 725805637025464 \nu^{10} + 41070927048001 \nu^{9} - 1708211052099464 \nu^{8} + 119332311754315 \nu^{7} + 2128704632011736 \nu^{6} - 311958245977134 \nu^{5} - 1222257229443266 \nu^{4} + 179493755356720 \nu^{3} + 237228681118952 \nu^{2} - 19813780702 \nu - 165641941058\)\()/ 2584793613763 \)
\(\beta_{8}\)\(=\)\((\)\(-44058942396 \nu^{18} + 100366607650 \nu^{17} + 1163734380318 \nu^{16} - 2544906190927 \nu^{15} - 12599484691005 \nu^{14} + 26056387605249 \nu^{13} + 72759363883607 \nu^{12} - 139981328381100 \nu^{11} - 242182034413731 \nu^{10} + 427053290996220 \nu^{9} + 464054899866586 \nu^{8} - 743045495285101 \nu^{7} - 477531549805097 \nu^{6} + 689021693044756 \nu^{5} + 222408147075718 \nu^{4} - 272000230119563 \nu^{3} - 27572301495516 \nu^{2} + 23059854361824 \nu - 246530814813\)\()/ 2584793613763 \)
\(\beta_{9}\)\(=\)\((\)\(-57136613064 \nu^{18} + 46120188886 \nu^{17} + 1573939504037 \nu^{16} - 711542003465 \nu^{15} - 18089060782894 \nu^{14} + 888172948827 \nu^{13} + 113473718291916 \nu^{12} + 41171189048558 \nu^{11} - 423076214974525 \nu^{10} - 308367411917029 \nu^{9} + 949735366352185 \nu^{8} + 935542592921969 \nu^{7} - 1236732975882466 \nu^{6} - 1317806710310478 \nu^{5} + 850923535863497 \nu^{4} + 778848775823786 \nu^{3} - 236302917332312 \nu^{2} - 127878466080227 \nu + 8443491451865\)\()/ 2584793613763 \)
\(\beta_{10}\)\(=\)\((\)\(-81254538506 \nu^{18} + 654428683551 \nu^{17} + 970372146851 \nu^{16} - 16384821425649 \nu^{15} + 5490595341552 \nu^{14} + 164287162414465 \nu^{13} - 143000610338671 \nu^{12} - 851967355105863 \nu^{11} + 907306072440698 \nu^{10} + 2447482008098315 \nu^{9} - 2686226489462764 \nu^{8} - 3851875830463548 \nu^{7} + 3918184841562988 \nu^{6} + 3081313386395895 \nu^{5} - 2559452836437059 \nu^{4} - 1114097846357755 \nu^{3} + 587551380088491 \nu^{2} + 144837697384176 \nu - 25082161671395\)\()/ 2584793613763 \)
\(\beta_{11}\)\(=\)\((\)\(-91424960248 \nu^{18} + 290362672773 \nu^{17} + 2163987038842 \nu^{16} - 7229272850781 \nu^{15} - 19906864553003 \nu^{14} + 71928744979165 \nu^{13} + 89628292185599 \nu^{12} - 369628093368947 \nu^{11} - 196989043393835 \nu^{10} + 1054469865039305 \nu^{9} + 148090368081498 \nu^{8} - 1669129966062270 \nu^{7} + 134975892121205 \nu^{6} + 1394627423083597 \nu^{5} - 249311959857695 \nu^{4} - 557432190432943 \nu^{3} + 94000023879346 \nu^{2} + 76490070141423 \nu - 5331850884506\)\()/ 2584793613763 \)
\(\beta_{12}\)\(=\)\((\)\(110220246484 \nu^{18} - 467343546422 \nu^{17} - 2253974970877 \nu^{16} + 11406225381357 \nu^{15} + 15508655859118 \nu^{14} - 110076102275048 \nu^{13} - 28689902332059 \nu^{12} + 538405030181766 \nu^{11} - 130776269174957 \nu^{10} - 1408591428559113 \nu^{9} + 700069942254179 \nu^{8} + 1883801315016804 \nu^{7} - 1141430209449476 \nu^{6} - 1082982581796351 \nu^{5} + 635720108720707 \nu^{4} + 170395081372216 \nu^{3} - 54746227366930 \nu^{2} + 3421690058101 \nu - 8426703234447\)\()/ 2584793613763 \)
\(\beta_{13}\)\(=\)\((\)\(112916661333 \nu^{18} - 420512816139 \nu^{17} - 2391672731682 \nu^{16} + 10136775407328 \nu^{15} + 17708092485247 \nu^{14} - 95838102406759 \nu^{13} - 44248321607856 \nu^{12} + 452239469595452 \nu^{11} - 80531592282880 \nu^{10} - 1103682120823352 \nu^{9} + 652563422729939 \nu^{8} + 1250741201795975 \nu^{7} - 1243226926847429 \nu^{6} - 354335113278947 \nu^{5} + 866457444779580 \nu^{4} - 237488214060000 \nu^{3} - 170724575819309 \nu^{2} + 72688955170899 \nu - 992900881709\)\()/ 2584793613763 \)
\(\beta_{14}\)\(=\)\((\)\(134524452259 \nu^{18} - 595936463557 \nu^{17} - 2794615480726 \nu^{16} + 14870979403509 \nu^{15} + 19764923687112 \nu^{14} - 148126283460976 \nu^{13} - 39968403213030 \nu^{12} + 759359884142781 \nu^{11} - 157363017924653 \nu^{10} - 2143032129859066 \nu^{9} + 934401836140697 \nu^{8} + 3296195916484018 \nu^{7} - 1683719333735166 \nu^{6} - 2591704634942325 \nu^{5} + 1136718336996770 \nu^{4} + 957988530198262 \nu^{3} - 195065901784280 \nu^{2} - 131669286419725 \nu - 6942619491638\)\()/ 2584793613763 \)
\(\beta_{15}\)\(=\)\((\)\(268063867548 \nu^{18} - 975838766429 \nu^{17} - 5844382212784 \nu^{16} + 23718172730414 \nu^{15} + 46258916319239 \nu^{14} - 227625432361619 \nu^{13} - 147385979907443 \nu^{12} + 1105726169117787 \nu^{11} + 23829240400472 \nu^{10} - 2873186468193146 \nu^{9} + 965965174706451 \nu^{8} + 3844359687237313 \nu^{7} - 2136542032244486 \nu^{6} - 2308864017778112 \nu^{5} + 1541694664772399 \nu^{4} + 490128717085397 \nu^{3} - 296741969124174 \nu^{2} - 29439287899993 \nu + 2625408551097\)\()/ 2584793613763 \)
\(\beta_{16}\)\(=\)\((\)\(-386309035457 \nu^{18} + 1365268186173 \nu^{17} + 8692512067202 \nu^{16} - 33566135421448 \nu^{15} - 73247230474246 \nu^{14} + 327363575480166 \nu^{13} + 276164257045428 \nu^{12} - 1627427645433792 \nu^{11} - 350590014826373 \nu^{10} + 4379858371730078 \nu^{9} - 537601793521600 \nu^{8} - 6212724005068246 \nu^{7} + 1859899256299250 \nu^{6} + 4173706077317173 \nu^{5} - 1430231827654368 \nu^{4} - 1122823610741571 \nu^{3} + 279954024772659 \nu^{2} + 99943970595050 \nu - 8519539396534\)\()/ 2584793613763 \)
\(\beta_{17}\)\(=\)\((\)\(-407596389827 \nu^{18} + 996406897672 \nu^{17} + 10303884149056 \nu^{16} - 24506168703039 \nu^{15} - 104725686151822 \nu^{14} + 239031289075534 \nu^{13} + 553369806506455 \nu^{12} - 1188615976491545 \nu^{11} - 1630922731137518 \nu^{10} + 3205416930871435 \nu^{9} + 2656196462753142 \nu^{8} - 4586218801226622 \nu^{7} - 2216000497099625 \nu^{6} + 3162740167704228 \nu^{5} + 818360171228219 \nu^{4} - 877910452999009 \nu^{3} - 76416261707646 \nu^{2} + 75002246004675 \nu - 7512666382563\)\()/ 2584793613763 \)
\(\beta_{18}\)\(=\)\((\)\(433257329190 \nu^{18} - 1312562946260 \nu^{17} - 10331601737881 \nu^{16} + 32317494993362 \nu^{15} + 96395574289456 \nu^{14} - 315991261600359 \nu^{13} - 447303281907149 \nu^{12} + 1578531396957927 \nu^{11} + 1065197051025780 \nu^{10} - 4290881606498135 \nu^{9} - 1147835903097151 \nu^{8} + 6222910401208218 \nu^{7} + 239835296100884 \nu^{6} - 4402846506872819 \nu^{5} + 270534203277648 \nu^{4} + 1308086020776475 \nu^{3} - 97910802478304 \nu^{2} - 115787993029379 \nu + 7601132183693\)\()/ 2584793613763 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{13} + \beta_{11} - \beta_{6} + \beta_{5} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{3} + 7 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{14} + 11 \beta_{13} - \beta_{12} + 7 \beta_{11} + 2 \beta_{10} + \beta_{9} - 9 \beta_{6} + 11 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 30 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(-\beta_{17} + 2 \beta_{16} + \beta_{15} + \beta_{14} - \beta_{13} - 11 \beta_{12} + \beta_{11} - 12 \beta_{10} - 12 \beta_{9} + 12 \beta_{8} + 12 \beta_{7} + \beta_{6} - 12 \beta_{5} - \beta_{4} - 11 \beta_{3} + 50 \beta_{2} - \beta_{1} + 88\)
\(\nu^{7}\)\(=\)\(10 \beta_{18} + 24 \beta_{17} - 15 \beta_{16} - 14 \beta_{14} + 94 \beta_{13} - 12 \beta_{12} + 46 \beta_{11} + 26 \beta_{10} + 12 \beta_{9} - \beta_{8} - 71 \beta_{6} + 97 \beta_{5} - 16 \beta_{4} + 17 \beta_{3} - 14 \beta_{2} + 200 \beta_{1} + 76\)
\(\nu^{8}\)\(=\)\(3 \beta_{18} - 15 \beta_{17} + 31 \beta_{16} + 14 \beta_{15} + 14 \beta_{14} - 18 \beta_{13} - 93 \beta_{12} + 19 \beta_{11} - 112 \beta_{10} - 109 \beta_{9} + 110 \beta_{8} + 109 \beta_{7} + 14 \beta_{6} - 112 \beta_{5} - 12 \beta_{4} - 94 \beta_{3} + 367 \beta_{2} - 23 \beta_{1} + 567\)
\(\nu^{9}\)\(=\)\(78 \beta_{18} + 218 \beta_{17} - 153 \beta_{16} + 2 \beta_{15} - 142 \beta_{14} + 746 \beta_{13} - 99 \beta_{12} + 317 \beta_{11} + 249 \beta_{10} + 111 \beta_{9} - 22 \beta_{8} + 2 \beta_{7} - 549 \beta_{6} + 799 \beta_{5} - 171 \beta_{4} + 196 \beta_{3} - 145 \beta_{2} + 1410 \beta_{1} + 534\)
\(\nu^{10}\)\(=\)\(47 \beta_{18} - 163 \beta_{17} + 334 \beta_{16} + 143 \beta_{15} + 143 \beta_{14} - 224 \beta_{13} - 724 \beta_{12} + 229 \beta_{11} - 958 \beta_{10} - 905 \beta_{9} + 915 \beta_{8} + 905 \beta_{7} + 148 \beta_{6} - 967 \beta_{5} - 107 \beta_{4} - 744 \beta_{3} + 2742 \beta_{2} - 313 \beta_{1} + 3874\)
\(\nu^{11}\)\(=\)\(570 \beta_{18} + 1806 \beta_{17} - 1356 \beta_{16} + 33 \beta_{15} - 1270 \beta_{14} + 5763 \beta_{13} - 699 \beta_{12} + 2290 \beta_{11} + 2145 \beta_{10} + 945 \beta_{9} - 306 \beta_{8} + 29 \beta_{7} - 4239 \beta_{6} + 6393 \beta_{5} - 1566 \beta_{4} + 1931 \beta_{3} - 1357 \beta_{2} + 10260 \beta_{1} + 3645\)
\(\nu^{12}\)\(=\)\(492 \beta_{18} - 1566 \beta_{17} + 3118 \beta_{16} + 1289 \beta_{15} + 1305 \beta_{14} - 2393 \beta_{13} - 5465 \beta_{12} + 2259 \beta_{11} - 7879 \beta_{10} - 7253 \beta_{9} + 7277 \beta_{8} + 7237 \beta_{7} + 1434 \beta_{6} - 8102 \beta_{5} - 862 \beta_{4} - 5721 \beta_{3} + 20716 \beta_{2} - 3451 \beta_{1} + 27441\)
\(\nu^{13}\)\(=\)\(4103 \beta_{18} + 14404 \beta_{17} - 11290 \beta_{16} + 341 \beta_{15} - 10672 \beta_{14} + 44078 \beta_{13} - 4515 \beta_{12} + 17038 \beta_{11} + 17659 \beta_{10} + 7783 \beta_{9} - 3478 \beta_{8} + 235 \beta_{7} - 32754 \beta_{6} + 50414 \beta_{5} - 13319 \beta_{4} + 17533 \beta_{3} - 12149 \beta_{2} + 76103 \beta_{1} + 24525\)
\(\nu^{14}\)\(=\)\(4351 \beta_{18} - 14133 \beta_{17} + 27117 \beta_{16} + 10875 \beta_{15} + 11326 \beta_{14} - 23509 \beta_{13} - 40789 \beta_{12} + 19984 \beta_{11} - 63529 \beta_{10} - 57279 \beta_{9} + 56578 \beta_{8} + 56857 \beta_{7} + 13359 \beta_{6} - 67001 \beta_{5} - 6621 \beta_{4} - 43553 \beta_{3} + 157618 \beta_{2} - 34216 \beta_{1} + 198716\)
\(\nu^{15}\)\(=\)\(29563 \beta_{18} + 112951 \beta_{17} - 91244 \beta_{16} + 2777 \beta_{15} - 86679 \beta_{14} + 336132 \beta_{13} - 27266 \beta_{12} + 128648 \beta_{11} + 142361 \beta_{10} + 63190 \beta_{9} - 35398 \beta_{8} + 1034 \beta_{7} - 253130 \beta_{6} + 394442 \beta_{5} - 108916 \beta_{4} + 151745 \beta_{3} - 106241 \beta_{2} + 571467 \beta_{1} + 163128\)
\(\nu^{16}\)\(=\)\(35212 \beta_{18} - 122951 \beta_{17} + 226841 \beta_{16} + 88185 \beta_{15} + 95848 \beta_{14} - 219042 \beta_{13} - 303570 \beta_{12} + 165507 \beta_{11} - 506776 \beta_{10} - 449554 \beta_{9} + 434837 \beta_{8} + 442742 \beta_{7} + 121414 \beta_{6} - 550426 \beta_{5} - 49387 \beta_{4} - 331180 \beta_{3} + 1204731 \beta_{2} - 319245 \beta_{1} + 1458972\)
\(\nu^{17}\)\(=\)\(214159 \beta_{18} + 879024 \beta_{17} - 726921 \beta_{16} + 18877 \beta_{15} - 690523 \beta_{14} + 2563542 \beta_{13} - 153522 \beta_{12} + 977051 \beta_{11} + 1136319 \beta_{10} + 509856 \beta_{9} - 336785 \beta_{8} - 4112 \beta_{7} - 1955646 \beta_{6} + 3072741 \beta_{5} - 871174 \beta_{4} + 1274242 \beta_{3} - 915300 \beta_{2} + 4326695 \beta_{1} + 1070635\)
\(\nu^{18}\)\(=\)\(270373 \beta_{18} - 1044853 \beta_{17} + 1855069 \beta_{16} + 697382 \beta_{15} + 799771 \beta_{14} - 1968542 \beta_{13} - 2261487 \beta_{12} + 1313376 \beta_{11} - 4017999 \beta_{10} - 3519120 \beta_{9} + 3323516 \beta_{8} + 3431715 \beta_{7} + 1082768 \beta_{6} - 4503238 \beta_{5} - 360082 \beta_{4} - 2526317 \beta_{3} + 9236634 \beta_{2} - 2867553 \beta_{1} + 10807509\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.80778
−2.21976
−2.08441
−1.82302
−1.62941
−0.976920
−0.556474
−0.436741
0.0681863
0.488177
0.686089
1.04858
1.31423
1.98166
2.04293
2.13634
2.34443
2.69542
2.72847
−2.80778 −0.314338 5.88364 −0.925514 0.882592 2.15050 −10.9044 −2.90119 2.59864
1.2 −2.21976 −2.99078 2.92734 −2.17512 6.63880 5.05675 −2.05847 5.94474 4.82825
1.3 −2.08441 1.49292 2.34475 2.98183 −3.11185 3.78550 −0.718591 −0.771187 −6.21535
1.4 −1.82302 3.09920 1.32341 −3.15385 −5.64991 −0.287436 1.23344 6.60505 5.74953
1.5 −1.62941 −2.82994 0.654988 2.04510 4.61114 −3.70972 2.19158 5.00854 −3.33231
1.6 −0.976920 0.347298 −1.04563 −2.45987 −0.339282 −4.72156 2.97533 −2.87938 2.40309
1.7 −0.556474 2.53857 −1.69034 0.874867 −1.41265 4.50273 2.05358 3.44434 −0.486841
1.8 −0.436741 −0.0103706 −1.80926 3.47884 0.00452927 −1.93516 1.66366 −2.99989 −1.51935
1.9 0.0681863 0.0544406 −1.99535 −3.91488 0.00371210 0.303195 −0.272428 −2.99704 −0.266941
1.10 0.488177 2.60530 −1.76168 1.92726 1.27185 0.259128 −1.83637 3.78757 0.940847
1.11 0.686089 −1.33733 −1.52928 3.77631 −0.917525 4.48234 −2.42140 −1.21156 2.59089
1.12 1.04858 −3.34590 −0.900472 −4.37157 −3.50846 −3.70774 −3.04139 8.19507 −4.58396
1.13 1.31423 −1.97692 −0.272811 −1.38757 −2.59812 2.03012 −2.98699 0.908205 −1.82358
1.14 1.98166 1.84905 1.92698 2.60057 3.66419 −2.18138 −0.144707 0.418993 5.15344
1.15 2.04293 0.810410 2.17358 0.258135 1.65561 3.70333 0.354615 −2.34324 0.527352
1.16 2.13634 2.50862 2.56396 −0.997990 5.35927 −0.526710 1.20482 3.29317 −2.13205
1.17 2.34443 −3.17986 3.49635 3.40234 −7.45497 −2.95885 3.50809 7.11153 7.97654
1.18 2.69542 −1.41452 5.26531 1.92020 −3.81272 2.04756 8.80138 −0.999141 5.17575
1.19 2.72847 2.09414 5.44452 −3.87909 5.71379 0.707409 9.39826 1.38542 −10.5840
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(61\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.a.c 19
3.b odd 2 1 6039.2.a.k 19
11.b odd 2 1 7381.2.a.i 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.c 19 1.a even 1 1 trivial
6039.2.a.k 19 3.b odd 2 1
7381.2.a.i 19 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{19} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(671))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -43 + 643 T + 241 T^{2} - 6404 T^{3} + 1674 T^{4} + 21077 T^{5} - 11589 T^{6} - 28019 T^{7} + 20141 T^{8} + 16875 T^{9} - 15053 T^{10} - 4673 T^{11} + 5755 T^{12} + 387 T^{13} - 1177 T^{14} + 78 T^{15} + 122 T^{16} - 18 T^{17} - 5 T^{18} + T^{19} \)
$3$ \( 5 + 386 T - 9247 T^{2} + 3270 T^{3} + 104314 T^{4} - 100249 T^{5} - 162716 T^{6} + 165352 T^{7} + 101767 T^{8} - 110216 T^{9} - 30917 T^{10} + 37620 T^{11} + 4807 T^{12} - 7135 T^{13} - 368 T^{14} + 762 T^{15} + 11 T^{16} - 43 T^{17} + T^{19} \)
$5$ \( 850564 - 2877604 T - 4039831 T^{2} + 8398322 T^{3} + 6386839 T^{4} - 9189577 T^{5} - 4246868 T^{6} + 5122072 T^{7} + 1385731 T^{8} - 1599267 T^{9} - 240424 T^{10} + 293013 T^{11} + 22483 T^{12} - 31983 T^{13} - 1065 T^{14} + 2039 T^{15} + 20 T^{16} - 70 T^{17} + T^{19} \)
$7$ \( -87296 + 327552 T + 1360000 T^{2} - 5263668 T^{3} - 3319069 T^{4} + 16761185 T^{5} - 3969728 T^{6} - 10558983 T^{7} + 4143144 T^{8} + 2692711 T^{9} - 1331765 T^{10} - 309373 T^{11} + 201547 T^{12} + 12551 T^{13} - 15666 T^{14} + 407 T^{15} + 602 T^{16} - 47 T^{17} - 9 T^{18} + T^{19} \)
$11$ \( ( -1 + T )^{19} \)
$13$ \( 966656 + 119988224 T - 344268800 T^{2} - 132665344 T^{3} + 817080320 T^{4} - 195598336 T^{5} - 488315136 T^{6} + 130238976 T^{7} + 134619072 T^{8} - 25631616 T^{9} - 17980256 T^{10} + 2658616 T^{11} + 1278028 T^{12} - 165480 T^{13} - 49625 T^{14} + 6118 T^{15} + 993 T^{16} - 122 T^{17} - 8 T^{18} + T^{19} \)
$17$ \( -16020390437 + 45772555282 T - 16659281601 T^{2} - 46877911055 T^{3} + 32991960221 T^{4} + 9936329531 T^{5} - 10257143230 T^{6} - 641741382 T^{7} + 1420376383 T^{8} - 28200026 T^{9} - 107083623 T^{10} + 6368925 T^{11} + 4682486 T^{12} - 382971 T^{13} - 118398 T^{14} + 11319 T^{15} + 1604 T^{16} - 168 T^{17} - 9 T^{18} + T^{19} \)
$19$ \( -7413317632 + 29270048768 T - 9938018304 T^{2} - 29598154752 T^{3} + 12335683584 T^{4} + 11987128832 T^{5} - 4381072384 T^{6} - 2392675200 T^{7} + 749820544 T^{8} + 254317952 T^{9} - 72731440 T^{10} - 14516568 T^{11} + 4230080 T^{12} + 407402 T^{13} - 143769 T^{14} - 3145 T^{15} + 2546 T^{16} - 73 T^{17} - 17 T^{18} + T^{19} \)
$23$ \( 76028297216 - 216012775424 T + 115588829184 T^{2} + 136686798848 T^{3} - 104102253568 T^{4} - 36216690176 T^{5} + 25889383168 T^{6} + 5257590912 T^{7} - 3077922496 T^{8} - 460876064 T^{9} + 203197584 T^{10} + 25296216 T^{11} - 7812868 T^{12} - 869318 T^{13} + 173337 T^{14} + 18071 T^{15} - 2051 T^{16} - 207 T^{17} + 10 T^{18} + T^{19} \)
$29$ \( 11106421323508 - 13297782264372 T - 789678129129 T^{2} + 6624838930119 T^{3} - 2036572185255 T^{4} - 785920666828 T^{5} + 470537594017 T^{6} - 5155885400 T^{7} - 36829924908 T^{8} + 5529786348 T^{9} + 1018499838 T^{10} - 305293195 T^{11} + 1292024 T^{12} + 6155500 T^{13} - 450489 T^{14} - 45458 T^{15} + 6363 T^{16} - 8 T^{17} - 27 T^{18} + T^{19} \)
$31$ \( 7338377216 - 121775955968 T + 544217755648 T^{2} - 305790015488 T^{3} - 287619135488 T^{4} + 284869393920 T^{5} - 38849148416 T^{6} - 30362833664 T^{7} + 9473019264 T^{8} + 808692448 T^{9} - 592379296 T^{10} + 17878296 T^{11} + 17444340 T^{12} - 1375490 T^{13} - 271171 T^{14} + 29155 T^{15} + 2161 T^{16} - 276 T^{17} - 7 T^{18} + T^{19} \)
$37$ \( 88495013888 + 655607652352 T + 394766589952 T^{2} - 740009529344 T^{3} - 385673860096 T^{4} + 324870096896 T^{5} + 79366326272 T^{6} - 67951731968 T^{7} - 940270976 T^{8} + 5613274432 T^{9} - 510928560 T^{10} - 185705816 T^{11} + 28611568 T^{12} + 2444704 T^{13} - 608621 T^{14} - 3474 T^{15} + 5741 T^{16} - 174 T^{17} - 20 T^{18} + T^{19} \)
$41$ \( -26396783722496 + 17588597678080 T + 29735289208832 T^{2} - 22498167209984 T^{3} - 4646084040704 T^{4} + 4852906152448 T^{5} + 54001391616 T^{6} - 430107928960 T^{7} + 30668722368 T^{8} + 18565145312 T^{9} - 2323373824 T^{10} - 390087672 T^{11} + 71542092 T^{12} + 3161262 T^{13} - 1064475 T^{14} + 10171 T^{15} + 7386 T^{16} - 273 T^{17} - 19 T^{18} + T^{19} \)
$43$ \( -25482644176 - 125779067404 T + 485979834775 T^{2} + 80647366682 T^{3} - 876897092328 T^{4} + 291735574050 T^{5} + 247220718431 T^{6} - 107403621583 T^{7} - 13143017981 T^{8} + 10503200389 T^{9} - 478340649 T^{10} - 369110994 T^{11} + 44978559 T^{12} + 4370485 T^{13} - 933312 T^{14} - 1572 T^{15} + 7403 T^{16} - 245 T^{17} - 20 T^{18} + T^{19} \)
$47$ \( -3936280228595 + 9393785108841 T + 4506841061267 T^{2} - 14051874380546 T^{3} - 2344545361209 T^{4} + 5625505608129 T^{5} + 1117298988932 T^{6} - 614143214378 T^{7} - 139605116019 T^{8} + 25302234923 T^{9} + 6983145211 T^{10} - 365210962 T^{11} - 158984704 T^{12} - 298812 T^{13} + 1765584 T^{14} + 50419 T^{15} - 9362 T^{16} - 407 T^{17} + 19 T^{18} + T^{19} \)
$53$ \( -13303893868544 + 8080416284672 T + 14484833886208 T^{2} - 16690095144960 T^{3} + 4285801632768 T^{4} + 1514621747200 T^{5} - 900292580352 T^{6} + 24556387328 T^{7} + 58828247680 T^{8} - 7459227456 T^{9} - 1812229456 T^{10} + 342063576 T^{11} + 29732888 T^{12} - 7311972 T^{13} - 274471 T^{14} + 81413 T^{15} + 1383 T^{16} - 454 T^{17} - 3 T^{18} + T^{19} \)
$59$ \( 8164905651748864 + 11201499643977728 T + 458273582993408 T^{2} - 3148612738783232 T^{3} - 196831714993152 T^{4} + 287744632301568 T^{5} + 22019325261312 T^{6} - 12570315305216 T^{7} - 1188795146880 T^{8} + 289957354688 T^{9} + 33817225232 T^{10} - 3468004440 T^{11} - 521721804 T^{12} + 17718522 T^{13} + 4313059 T^{14} + 10775 T^{15} - 17693 T^{16} - 411 T^{17} + 28 T^{18} + T^{19} \)
$61$ \( ( 1 + T )^{19} \)
$67$ \( 54909964304384 - 53485347332096 T - 178896411373568 T^{2} + 16329193523200 T^{3} + 71078403193856 T^{4} - 7639417922048 T^{5} - 9586777058048 T^{6} + 1494873112960 T^{7} + 432214722496 T^{8} - 78049227296 T^{9} - 9333072112 T^{10} + 1878611112 T^{11} + 109094340 T^{12} - 24321902 T^{13} - 703139 T^{14} + 174805 T^{15} + 2322 T^{16} - 656 T^{17} - 3 T^{18} + T^{19} \)
$71$ \( -7732360953856 + 126385057300480 T + 33460491292672 T^{2} - 101439521626112 T^{3} - 40304565837824 T^{4} + 13347683605504 T^{5} + 6344651760128 T^{6} - 667790158976 T^{7} - 403938328704 T^{8} + 12179198144 T^{9} + 12936133632 T^{10} + 74545480 T^{11} - 224054636 T^{12} - 5854672 T^{13} + 2105005 T^{14} + 81406 T^{15} - 10041 T^{16} - 471 T^{17} + 19 T^{18} + T^{19} \)
$73$ \( 38776064491520 + 175591173054464 T + 114449419927552 T^{2} - 93872437938176 T^{3} - 45575125012480 T^{4} + 27913426499072 T^{5} + 1055629218816 T^{6} - 2151097078400 T^{7} + 140644485952 T^{8} + 70328095232 T^{9} - 8523594080 T^{10} - 1051548256 T^{11} + 194238396 T^{12} + 5202850 T^{13} - 2122223 T^{14} + 32224 T^{15} + 10833 T^{16} - 404 T^{17} - 20 T^{18} + T^{19} \)
$79$ \( 113909830791606641 - 257894319988259525 T + 210843353960436404 T^{2} - 79099761798567253 T^{3} + 10280960790064975 T^{4} + 2339644057711763 T^{5} - 1068796155867930 T^{6} + 128457463273690 T^{7} + 6490519271179 T^{8} - 3256623872875 T^{9} + 258435852514 T^{10} + 12686581987 T^{11} - 3239831468 T^{12} + 135856650 T^{13} + 8517205 T^{14} - 955642 T^{15} + 17847 T^{16} + 1282 T^{17} - 69 T^{18} + T^{19} \)
$83$ \( 19193639684784128 - 4034895400501248 T - 11444469642747904 T^{2} + 2325967061745664 T^{3} + 2057012166883328 T^{4} - 370180459172352 T^{5} - 136490860344320 T^{6} + 24654205203328 T^{7} + 3775395078400 T^{8} - 728484152064 T^{9} - 51183040496 T^{10} + 11058280120 T^{11} + 360551072 T^{12} - 92418570 T^{13} - 1298999 T^{14} + 428673 T^{15} + 2116 T^{16} - 1031 T^{17} - T^{18} + T^{19} \)
$89$ \( -2170770723258368 + 1953160773754880 T + 1003034382585856 T^{2} - 1728196338382848 T^{3} + 578966598942720 T^{4} + 47521869723136 T^{5} - 57797171465472 T^{6} + 5123497906560 T^{7} + 2013025113088 T^{8} - 310342363424 T^{9} - 31708812736 T^{10} + 6712416960 T^{11} + 233191448 T^{12} - 69657870 T^{13} - 784929 T^{14} + 371975 T^{15} + 987 T^{16} - 979 T^{17} + T^{19} \)
$97$ \( 458738803300778447 - 832992879905416221 T + 458979883865219182 T^{2} - 65118190161721091 T^{3} - 20941740739880833 T^{4} + 6923462159680875 T^{5} - 22136396843458 T^{6} - 196320570680230 T^{7} + 15142042919471 T^{8} + 2312988049927 T^{9} - 310404855468 T^{10} - 9490823011 T^{11} + 2705596820 T^{12} - 24592480 T^{13} - 11770909 T^{14} + 332072 T^{15} + 25139 T^{16} - 996 T^{17} - 21 T^{18} + T^{19} \)
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