Properties

Label 1849.2.a.n
Level $1849$
Weight $2$
Character orbit 1849.a
Self dual yes
Analytic conductor $14.764$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7643393337\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} - 3223 x^{10} - 8851 x^{9} + 9909 x^{8} + 10400 x^{7} - 14483 x^{6} - 5118 x^{5} + 9945 x^{4} - 27 x^{3} - 2493 x^{2} + 513 x - 27\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} - 3223 x^{10} - 8851 x^{9} + 9909 x^{8} + 10400 x^{7} - 14483 x^{6} - 5118 x^{5} + 9945 x^{4} - 27 x^{3} - 2493 x^{2} + 513 x - 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(13275 \nu^{17} + 49216 \nu^{16} - 638270 \nu^{15} - 817698 \nu^{14} + 10040797 \nu^{13} + 4293380 \nu^{12} - 76161915 \nu^{11} - 2151884 \nu^{10} + 314282184 \nu^{9} - 52495987 \nu^{8} - 726303448 \nu^{7} + 175925085 \nu^{6} + 916736762 \nu^{5} - 215589719 \nu^{4} - 580027476 \nu^{3} + 104938560 \nu^{2} + 140920983 \nu - 16390890\)\()/1312164\)
\(\beta_{3}\)\(=\)\((\)\(20710 \nu^{17} - 38599 \nu^{16} - 543260 \nu^{15} + 949666 \nu^{14} + 5791002 \nu^{13} - 9559915 \nu^{12} - 31639730 \nu^{11} + 50308357 \nu^{10} + 90932936 \nu^{9} - 146365724 \nu^{8} - 120576325 \nu^{7} + 228076724 \nu^{6} + 27615381 \nu^{5} - 167553770 \nu^{4} + 64429719 \nu^{3} + 40814856 \nu^{2} - 32487966 \nu - 7389\)\()/1312164\)
\(\beta_{4}\)\(=\)\((\)\(-44003 \nu^{17} + 173548 \nu^{16} + 771312 \nu^{15} - 3533264 \nu^{14} - 4677973 \nu^{13} + 28270866 \nu^{12} + 8945071 \nu^{11} - 112470798 \nu^{10} + 16877036 \nu^{9} + 233856473 \nu^{8} - 86897862 \nu^{7} - 253491883 \nu^{6} + 100861486 \nu^{5} + 150145347 \nu^{4} - 26028240 \nu^{3} - 48095238 \nu^{2} - 7993449 \nu + 401490\)\()/1312164\)
\(\beta_{5}\)\(=\)\((\)\(-55583 \nu^{17} + 218229 \nu^{16} + 1056380 \nu^{15} - 4620818 \nu^{14} - 7782617 \nu^{13} + 39149731 \nu^{12} + 28825423 \nu^{11} - 169440829 \nu^{10} - 64968220 \nu^{9} + 397852707 \nu^{8} + 127874509 \nu^{7} - 500960689 \nu^{6} - 228333277 \nu^{5} + 317741963 \nu^{4} + 221535819 \nu^{3} - 89122374 \nu^{2} - 73666089 \nu + 9437373\)\()/1312164\)
\(\beta_{6}\)\(=\)\((\)\(69430 \nu^{17} - 284265 \nu^{16} - 1359184 \nu^{15} + 6326890 \nu^{14} + 10180162 \nu^{13} - 57120029 \nu^{12} - 35687486 \nu^{11} + 268538099 \nu^{10} + 55159664 \nu^{9} - 702822648 \nu^{8} - 19113311 \nu^{7} + 1015190912 \nu^{6} - 24846673 \nu^{5} - 746262274 \nu^{4} + 11271585 \nu^{3} + 218833080 \nu^{2} + 3726342 \nu - 4659615\)\()/1312164\)
\(\beta_{7}\)\(=\)\((\)\(-76293 \nu^{17} + 256828 \nu^{16} + 1599640 \nu^{15} - 5570484 \nu^{14} - 13573619 \nu^{13} + 48709646 \nu^{12} + 60465153 \nu^{11} - 219749186 \nu^{10} - 155901156 \nu^{9} + 544218431 \nu^{8} + 248450834 \nu^{7} - 729037413 \nu^{6} - 255948658 \nu^{5} + 485295733 \nu^{4} + 157106100 \nu^{3} - 128625066 \nu^{2} - 41178123 \nu + 5508270\)\()/1312164\)
\(\beta_{8}\)\(=\)\((\)\(59046 \nu^{17} - 244357 \nu^{16} - 1095040 \nu^{15} + 5288955 \nu^{14} + 7351847 \nu^{13} - 46037891 \nu^{12} - 18893697 \nu^{11} + 206093975 \nu^{10} - 6203235 \nu^{9} - 504355352 \nu^{8} + 123911218 \nu^{7} + 663208641 \nu^{6} - 230450510 \nu^{5} - 424393738 \nu^{4} + 170060541 \nu^{3} + 94679412 \nu^{2} - 43668981 \nu + 4297068\)\()/656082\)
\(\beta_{9}\)\(=\)\((\)\(124637 \nu^{17} - 455245 \nu^{16} - 2516574 \nu^{15} + 9987848 \nu^{14} + 19725175 \nu^{13} - 88235805 \nu^{12} - 74360329 \nu^{11} + 401793495 \nu^{10} + 131050912 \nu^{9} - 1004438171 \nu^{8} - 64409787 \nu^{7} + 1360900177 \nu^{6} - 94828159 \nu^{5} - 915839985 \nu^{4} + 130684977 \nu^{3} + 231376572 \nu^{2} - 43334415 \nu + 2059911\)\()/1312164\)
\(\beta_{10}\)\(=\)\((\)\(141246 \nu^{17} - 576685 \nu^{16} - 2561026 \nu^{15} + 12268074 \nu^{14} + 16443476 \nu^{13} - 104501855 \nu^{12} - 36379716 \nu^{11} + 454974713 \nu^{10} - 48490122 \nu^{9} - 1073642906 \nu^{8} + 361330573 \nu^{7} + 1347595896 \nu^{6} - 608721671 \nu^{5} - 817237726 \nu^{4} + 426518757 \nu^{3} + 172094370 \nu^{2} - 109231326 \nu + 8750907\)\()/1312164\)
\(\beta_{11}\)\(=\)\((\)\(167003 \nu^{17} - 649353 \nu^{16} - 3148820 \nu^{15} + 13928240 \nu^{14} + 21852299 \nu^{13} - 119904103 \nu^{12} - 61900753 \nu^{11} + 529610989 \nu^{10} + 16191022 \nu^{9} - 1276405047 \nu^{8} + 277767581 \nu^{7} + 1655953135 \nu^{6} - 561275045 \nu^{5} - 1061265551 \nu^{4} + 411372357 \nu^{3} + 250284606 \nu^{2} - 99718389 \nu + 6009885\)\()/1312164\)
\(\beta_{12}\)\(=\)\((\)\(94434 \nu^{17} - 367211 \nu^{16} - 1822472 \nu^{15} + 8017242 \nu^{14} + 13158388 \nu^{13} - 70696261 \nu^{12} - 40633854 \nu^{11} + 322516021 \nu^{10} + 25631112 \nu^{9} - 810758590 \nu^{8} + 147391919 \nu^{7} + 1105999782 \nu^{6} - 361418209 \nu^{5} - 742219478 \nu^{4} + 311539797 \nu^{3} + 175378248 \nu^{2} - 91873764 \nu + 9088893\)\()/656082\)
\(\beta_{13}\)\(=\)\((\)\(204816 \nu^{17} - 823249 \nu^{16} - 3821242 \nu^{15} + 17751510 \nu^{14} + 25899626 \nu^{13} - 153706883 \nu^{12} - 68256762 \nu^{11} + 683247641 \nu^{10} - 11491806 \nu^{9} - 1657489664 \nu^{8} + 404042737 \nu^{7} + 2161286394 \nu^{6} - 747127031 \nu^{5} - 1383174472 \nu^{4} + 526887669 \nu^{3} + 318068358 \nu^{2} - 126336348 \nu + 11305827\)\()/1312164\)
\(\beta_{14}\)\(=\)\((\)\(245597 \nu^{17} - 1067329 \nu^{16} - 4433508 \nu^{15} + 23271404 \nu^{14} + 27818161 \nu^{13} - 204864951 \nu^{12} - 52751011 \nu^{11} + 932572533 \nu^{10} - 152533502 \nu^{9} - 2337456797 \nu^{8} + 859608609 \nu^{7} + 3175200109 \nu^{6} - 1460957353 \nu^{5} - 2117529921 \nu^{4} + 1061084985 \nu^{3} + 497629530 \nu^{2} - 274838571 \nu + 24053769\)\()/1312164\)
\(\beta_{15}\)\(=\)\((\)\(-253199 \nu^{17} + 944792 \nu^{16} + 4923820 \nu^{15} - 20322290 \nu^{14} - 36421791 \nu^{13} + 175651034 \nu^{12} + 122991985 \nu^{11} - 780467606 \nu^{10} - 157879042 \nu^{9} + 1898220181 \nu^{8} - 84082444 \nu^{7} - 2498571613 \nu^{6} + 382011948 \nu^{5} + 1642014991 \nu^{4} - 290653932 \nu^{3} - 412418418 \nu^{2} + 59409549 \nu - 2620152\)\()/1312164\)
\(\beta_{16}\)\(=\)\((\)\(160886 \nu^{17} - 603273 \nu^{16} - 3199685 \nu^{15} + 13198295 \nu^{14} + 24565100 \nu^{13} - 116487784 \nu^{12} - 89543356 \nu^{11} + 531312130 \nu^{10} + 146650858 \nu^{9} - 1335019437 \nu^{8} - 46035718 \nu^{7} + 1826041525 \nu^{6} - 138676475 \nu^{5} - 1248308096 \nu^{4} + 145462917 \nu^{3} + 327551883 \nu^{2} - 38129769 \nu - 572175\)\()/656082\)
\(\beta_{17}\)\(=\)\((\)\(249517 \nu^{17} - 1016292 \nu^{16} - 4644175 \nu^{15} + 21964198 \nu^{14} + 31393966 \nu^{13} - 190931687 \nu^{12} - 82343444 \nu^{11} + 853971125 \nu^{10} - 15885937 \nu^{9} - 2090623248 \nu^{8} + 495031120 \nu^{7} + 2759880713 \nu^{6} - 915641593 \nu^{5} - 1791688513 \nu^{4} + 651327492 \nu^{3} + 420139959 \nu^{2} - 160192611 \nu + 13820985\)\()/656082\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{5} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{17} + \beta_{16} + \beta_{15} - \beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{16} + \beta_{13} + \beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{9} + 8 \beta_{7} - 2 \beta_{6} - 8 \beta_{5} - \beta_{4} + 7 \beta_{3} + \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(7 \beta_{17} + 10 \beta_{16} + 7 \beta_{15} + \beta_{12} - 9 \beta_{11} - 2 \beta_{10} - 9 \beta_{9} - 7 \beta_{8} + 18 \beta_{7} - 18 \beta_{6} - 11 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} + 6 \beta_{2} + 28 \beta_{1} - 7\)
\(\nu^{6}\)\(=\)\(2 \beta_{17} + 15 \beta_{16} + \beta_{15} + 7 \beta_{13} + 9 \beta_{12} - 23 \beta_{11} - 9 \beta_{10} - 12 \beta_{9} - 4 \beta_{8} + 58 \beta_{7} - 25 \beta_{6} - 56 \beta_{5} - 11 \beta_{4} + 46 \beta_{3} - \beta_{2} + 16 \beta_{1} + 60\)
\(\nu^{7}\)\(=\)\(45 \beta_{17} + 82 \beta_{16} + 42 \beta_{15} - \beta_{14} - 3 \beta_{13} + 14 \beta_{12} - 73 \beta_{11} - 24 \beta_{10} - 67 \beta_{9} - 46 \beta_{8} + 142 \beta_{7} - 138 \beta_{6} - 98 \beta_{5} - 56 \beta_{4} + 72 \beta_{3} + 29 \beta_{2} + 167 \beta_{1} - 43\)
\(\nu^{8}\)\(=\)\(30 \beta_{17} + 152 \beta_{16} + 10 \beta_{15} + 33 \beta_{13} + 69 \beta_{12} - 206 \beta_{11} - 73 \beta_{10} - 108 \beta_{9} - 52 \beta_{8} + 424 \beta_{7} - 240 \beta_{6} - 394 \beta_{5} - 94 \beta_{4} + 306 \beta_{3} - 14 \beta_{2} + 161 \beta_{1} + 278\)
\(\nu^{9}\)\(=\)\(296 \beta_{17} + 635 \beta_{16} + 240 \beta_{15} - 10 \beta_{14} - 55 \beta_{13} + 136 \beta_{12} - 579 \beta_{11} - 221 \beta_{10} - 472 \beta_{9} - 315 \beta_{8} + 1085 \beta_{7} - 1031 \beta_{6} - 808 \beta_{5} - 387 \beta_{4} + 556 \beta_{3} + 121 \beta_{2} + 1042 \beta_{1} - 271\)
\(\nu^{10}\)\(=\)\(316 \beta_{17} + 1342 \beta_{16} + 69 \beta_{15} + 10 \beta_{14} + 78 \beta_{13} + 510 \beta_{12} - 1692 \beta_{11} - 590 \beta_{10} - 868 \beta_{9} - 503 \beta_{8} + 3135 \beta_{7} - 2093 \beta_{6} - 2827 \beta_{5} - 749 \beta_{4} + 2078 \beta_{3} - 146 \beta_{2} + 1385 \beta_{1} + 1229\)
\(\nu^{11}\)\(=\)\(2009 \beta_{17} + 4827 \beta_{16} + 1330 \beta_{15} - 40 \beta_{14} - 677 \beta_{13} + 1147 \beta_{12} - 4541 \beta_{11} - 1862 \beta_{10} - 3259 \beta_{9} - 2253 \beta_{8} + 8199 \beta_{7} - 7724 \beta_{6} - 6419 \beta_{5} - 2710 \beta_{4} + 4214 \beta_{3} + 374 \beta_{2} + 6742 \beta_{1} - 1846\)
\(\nu^{12}\)\(=\)\(2884 \beta_{17} + 11118 \beta_{16} + 381 \beta_{15} + 236 \beta_{14} - 629 \beta_{13} + 3722 \beta_{12} - 13363 \beta_{11} - 4774 \beta_{10} - 6582 \beta_{9} - 4401 \beta_{8} + 23307 \beta_{7} - 17419 \beta_{6} - 20616 \beta_{5} - 5820 \beta_{4} + 14367 \beta_{3} - 1373 \beta_{2} + 11104 \beta_{1} + 4724\)
\(\nu^{13}\)\(=\)\(13986 \beta_{17} + 36498 \beta_{16} + 7132 \beta_{15} + 284 \beta_{14} - 7029 \beta_{13} + 9034 \beta_{12} - 35326 \beta_{11} - 15080 \beta_{10} - 22338 \beta_{9} - 16633 \beta_{8} + 61687 \beta_{7} - 58350 \beta_{6} - 49990 \beta_{5} - 19326 \beta_{4} + 31561 \beta_{3} - 93 \beta_{2} + 44941 \beta_{1} - 13667\)
\(\nu^{14}\)\(=\)\(24429 \beta_{17} + 89053 \beta_{16} + 1512 \beta_{15} + 3507 \beta_{14} - 12885 \beta_{13} + 27005 \beta_{12} - 103453 \beta_{11} - 38486 \beta_{10} - 48265 \beta_{9} - 36816 \beta_{8} + 173685 \beta_{7} - 141204 \beta_{6} - 152080 \beta_{5} - 44738 \beta_{4} + 100734 \beta_{3} - 12230 \beta_{2} + 85859 \beta_{1} + 10825\)
\(\nu^{15}\)\(=\)\(99222 \beta_{17} + 275782 \beta_{16} + 36415 \beta_{15} + 8100 \beta_{14} - 66561 \beta_{13} + 68573 \beta_{12} - 273052 \beta_{11} - 119674 \beta_{10} - 152784 \beta_{9} - 125404 \beta_{8} + 463302 \beta_{7} - 444135 \beta_{6} - 384896 \beta_{5} - 140118 \beta_{4} + 234512 \beta_{3} - 16495 \beta_{2} + 306917 \beta_{1} - 107636\)
\(\nu^{16}\)\(=\)\(198097 \beta_{17} + 699601 \beta_{16} + 156 \beta_{15} + 42115 \beta_{14} - 150414 \beta_{13} + 195355 \beta_{12} - 792366 \beta_{11} - 308182 \beta_{10} - 346727 \beta_{9} - 300668 \beta_{8} + 1296038 \beta_{7} - 1126248 \beta_{6} - 1130729 \beta_{5} - 342050 \beta_{4} + 713907 \beta_{3} - 105095 \beta_{2} + 650835 \beta_{1} - 58659\)
\(\nu^{17}\)\(=\)\(713751 \beta_{17} + 2085523 \beta_{16} + 170041 \beta_{15} + 112366 \beta_{14} - 595897 \beta_{13} + 509658 \beta_{12} - 2099810 \beta_{11} - 938905 \beta_{10} - 1045038 \beta_{9} - 958176 \beta_{8} + 3477606 \beta_{7} - 3398743 \beta_{6} - 2943361 \beta_{5} - 1029194 \beta_{4} + 1733440 \beta_{3} - 210984 \beta_{2} + 2137240 \beta_{1} - 877416\)

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
2.75930
2.61223
2.31259
2.16300
1.78421
1.69598
1.10272
1.04477
0.847401
0.131407
0.0953887
−0.800597
−1.09100
−1.18520
−1.70036
−2.17399
−2.27976
−2.31808
−2.75930 −1.44592 5.61371 0.916508 3.98973 1.94308 −9.97129 −0.909309 −2.52892
1.2 −2.61223 1.92275 4.82373 −2.98756 −5.02267 0.679767 −7.37622 0.696981 7.80417
1.3 −2.31259 −2.69989 3.34809 −2.53973 6.24374 2.18729 −3.11758 4.28939 5.87337
1.4 −2.16300 −2.91800 2.67858 2.73005 6.31164 −2.02167 −1.46776 5.51474 −5.90511
1.5 −1.78421 2.96672 1.18339 −0.596316 −5.29325 −4.36308 1.45700 5.80145 1.06395
1.6 −1.69598 −1.71311 0.876333 0.385406 2.90540 −3.49171 1.90571 −0.0652397 −0.653640
1.7 −1.10272 −0.645986 −0.784019 −3.32240 0.712338 −2.46546 3.06998 −2.58270 3.66366
1.8 −1.04477 2.52412 −0.908449 −0.499809 −2.63714 3.30668 3.03867 3.37119 0.522187
1.9 −0.847401 0.0176385 −1.28191 3.43657 −0.0149469 0.269043 2.78110 −2.99969 −2.91216
1.10 −0.131407 1.32082 −1.98273 1.78074 −0.173564 0.0149856 0.523357 −1.25544 −0.234000
1.11 −0.0953887 −2.36384 −1.99090 −3.81314 0.225484 −2.61962 0.380687 2.58774 0.363731
1.12 0.800597 −0.744601 −1.35904 1.40135 −0.596125 4.75856 −2.68924 −2.44557 1.12192
1.13 1.09100 1.45952 −0.809726 1.35087 1.59233 0.216325 −3.06540 −0.869803 1.47379
1.14 1.18520 −2.30227 −0.595300 −3.24275 −2.72866 2.76837 −3.07595 2.30047 −3.84331
1.15 1.70036 1.15587 0.891224 −1.76341 1.96539 0.594565 −1.88532 −1.66397 −2.99843
1.16 2.17399 0.327650 2.72622 −0.0263127 0.712306 −3.12927 1.57880 −2.89265 −0.0572034
1.17 2.27976 −3.20997 3.19730 −0.137674 −7.31797 −1.86944 2.72957 7.30393 −0.313864
1.18 2.31808 1.34851 3.37350 −4.07240 3.12595 −2.77841 3.18389 −1.18153 −9.44015
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(1\)

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 1849.2.a.n 18
43.b odd 2 1 1849.2.a.o 18
43.g even 21 2 43.2.g.a 36
129.o odd 42 2 387.2.y.c 36
172.o odd 42 2 688.2.bg.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.g.a 36 43.g even 21 2
387.2.y.c 36 129.o odd 42 2
688.2.bg.c 36 172.o odd 42 2
1849.2.a.n 18 1.a even 1 1 trivial
1849.2.a.o 18 43.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{18} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1849))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -27 - 513 T - 2493 T^{2} + 27 T^{3} + 9945 T^{4} + 5118 T^{5} - 14483 T^{6} - 10400 T^{7} + 9909 T^{8} + 8851 T^{9} - 3223 T^{10} - 3855 T^{11} + 364 T^{12} + 897 T^{13} + 47 T^{14} - 106 T^{15} - 15 T^{16} + 5 T^{17} + T^{18} \)
$3$ \( -41 + 2378 T - 2703 T^{2} - 19006 T^{3} + 7362 T^{4} + 43163 T^{5} - 8785 T^{6} - 44518 T^{7} + 4519 T^{8} + 24790 T^{9} - 451 T^{10} - 7908 T^{11} - 434 T^{12} + 1429 T^{13} + 166 T^{14} - 134 T^{15} - 22 T^{16} + 5 T^{17} + T^{18} \)
$5$ \( -27 - 1215 T - 6894 T^{2} + 12483 T^{3} + 63333 T^{4} - 24411 T^{5} - 144101 T^{6} + 25945 T^{7} + 120642 T^{8} - 4364 T^{9} - 48510 T^{10} - 4630 T^{11} + 9617 T^{12} + 2016 T^{13} - 802 T^{14} - 266 T^{15} + 11 T^{16} + 11 T^{17} + T^{18} \)
$7$ \( 211 - 16259 T + 152200 T^{2} - 458062 T^{3} + 252206 T^{4} + 698794 T^{5} - 500658 T^{6} - 563949 T^{7} + 229698 T^{8} + 236940 T^{9} - 37024 T^{10} - 51731 T^{11} + 284 T^{12} + 5807 T^{13} + 481 T^{14} - 310 T^{15} - 42 T^{16} + 6 T^{17} + T^{18} \)
$11$ \( 4079943 - 14637483 T + 4749129 T^{2} + 29137311 T^{3} - 24693249 T^{4} - 12315093 T^{5} + 16151227 T^{6} + 1580760 T^{7} - 4617545 T^{8} + 89759 T^{9} + 717487 T^{10} - 40350 T^{11} - 64800 T^{12} + 3760 T^{13} + 3369 T^{14} - 145 T^{15} - 92 T^{16} + 2 T^{17} + T^{18} \)
$13$ \( 5432407 + 26640969 T + 48459597 T^{2} + 34109359 T^{3} - 7418534 T^{4} - 26092773 T^{5} - 12018399 T^{6} + 2920734 T^{7} + 4024857 T^{8} + 613334 T^{9} - 463188 T^{10} - 163497 T^{11} + 17011 T^{12} + 14127 T^{13} + 603 T^{14} - 529 T^{15} - 57 T^{16} + 7 T^{17} + T^{18} \)
$17$ \( 6460371 - 40192551 T + 59737365 T^{2} + 41073777 T^{3} - 148228083 T^{4} + 70752078 T^{5} + 46820329 T^{6} - 38474330 T^{7} - 4824435 T^{8} + 7515652 T^{9} + 200084 T^{10} - 747144 T^{11} - 15578 T^{12} + 40107 T^{13} + 1766 T^{14} - 1072 T^{15} - 75 T^{16} + 11 T^{17} + T^{18} \)
$19$ \( 6136831 + 146806601 T + 400984109 T^{2} + 302069486 T^{3} - 268695444 T^{4} - 669771961 T^{5} - 519199841 T^{6} - 164734412 T^{7} + 20203719 T^{8} + 36019728 T^{9} + 11899678 T^{10} + 867261 T^{11} - 510378 T^{12} - 164759 T^{13} - 17525 T^{14} + 609 T^{15} + 333 T^{16} + 31 T^{17} + T^{18} \)
$23$ \( -189 - 129465 T + 3216591 T^{2} + 6092352 T^{3} - 23525943 T^{4} - 49258020 T^{5} - 17948363 T^{6} + 16932416 T^{7} + 12815489 T^{8} - 159138 T^{9} - 2174131 T^{10} - 416812 T^{11} + 111012 T^{12} + 38510 T^{13} - 596 T^{14} - 1169 T^{15} - 69 T^{16} + 11 T^{17} + T^{18} \)
$29$ \( 505748853 + 16187972067 T + 5194883403 T^{2} - 26982361332 T^{3} - 19809447075 T^{4} + 3461085936 T^{5} + 6300759150 T^{6} + 871693029 T^{7} - 629665704 T^{8} - 187035984 T^{9} + 15205983 T^{10} + 11594346 T^{11} + 829948 T^{12} - 238441 T^{13} - 41961 T^{14} - 318 T^{15} + 425 T^{16} + 37 T^{17} + T^{18} \)
$31$ \( 765689239 + 1714442331 T - 1027287499 T^{2} - 3413697259 T^{3} + 365330280 T^{4} + 2168672135 T^{5} - 164731644 T^{6} - 552011243 T^{7} + 28277431 T^{8} + 69064359 T^{9} - 1037218 T^{10} - 4521342 T^{11} - 102668 T^{12} + 150937 T^{13} + 8101 T^{14} - 2262 T^{15} - 164 T^{16} + 12 T^{17} + T^{18} \)
$37$ \( 21918373 - 2379209006 T + 2622354060 T^{2} + 5766817709 T^{3} - 10612699753 T^{4} + 4426443191 T^{5} + 1470772980 T^{6} - 1599184580 T^{7} + 276716459 T^{8} + 98162982 T^{9} - 39669096 T^{10} + 1057491 T^{11} + 1412028 T^{12} - 181823 T^{13} - 12810 T^{14} + 3531 T^{15} - 77 T^{16} - 19 T^{17} + T^{18} \)
$41$ \( -40423514319 + 300688988940 T - 454536811251 T^{2} + 209907933354 T^{3} + 51954187134 T^{4} - 72673297932 T^{5} + 12800719384 T^{6} + 5842607266 T^{7} - 2067443518 T^{8} - 122417580 T^{9} + 109336397 T^{10} - 2321948 T^{11} - 2877033 T^{12} + 124459 T^{13} + 41617 T^{14} - 1639 T^{15} - 318 T^{16} + 7 T^{17} + T^{18} \)
$43$ \( T^{18} \)
$47$ \( 7697721249 - 888986471436 T + 1537421255262 T^{2} + 1211246248446 T^{3} - 955487742228 T^{4} - 124448348175 T^{5} + 133180030000 T^{6} + 3492369122 T^{7} - 8423516626 T^{8} + 37855934 T^{9} + 291089692 T^{10} - 3457900 T^{11} - 5825809 T^{12} + 60440 T^{13} + 66826 T^{14} - 421 T^{15} - 405 T^{16} + T^{17} + T^{18} \)
$53$ \( -85602704687163 - 897164279820 T + 40456599824952 T^{2} + 2907848217312 T^{3} - 6363742772220 T^{4} - 534872008215 T^{5} + 492525580186 T^{6} + 39110319852 T^{7} - 21727818000 T^{8} - 1450415124 T^{9} + 582077550 T^{10} + 29278218 T^{11} - 9580385 T^{12} - 316842 T^{13} + 93330 T^{14} + 1647 T^{15} - 483 T^{16} - 3 T^{17} + T^{18} \)
$59$ \( 3722978673 - 13332217275 T - 46386872856 T^{2} - 14386656465 T^{3} + 29078854398 T^{4} + 6781621362 T^{5} - 8584837583 T^{6} - 358872403 T^{7} + 1185735006 T^{8} - 122433139 T^{9} - 60424803 T^{10} + 11419852 T^{11} + 1021706 T^{12} - 339945 T^{13} + 1739 T^{14} + 4082 T^{15} - 172 T^{16} - 17 T^{17} + T^{18} \)
$61$ \( -78208816261967 - 54539019480663 T + 11813676981290 T^{2} + 16216362273535 T^{3} + 1379695114083 T^{4} - 1555212144570 T^{5} - 306873769847 T^{6} + 62148821752 T^{7} + 20086152802 T^{8} - 669437411 T^{9} - 625365793 T^{10} - 25317438 T^{11} + 9501052 T^{12} + 857138 T^{13} - 54862 T^{14} - 8964 T^{15} - 79 T^{16} + 28 T^{17} + T^{18} \)
$67$ \( 3811996587673 + 3055910237888 T - 10364776794071 T^{2} - 925905306863 T^{3} + 6785087481584 T^{4} - 1555765521367 T^{5} - 711446461932 T^{6} + 208680404181 T^{7} + 23190417906 T^{8} - 9291592461 T^{9} - 192487204 T^{10} + 189422837 T^{11} - 3298474 T^{12} - 1925594 T^{13} + 71329 T^{14} + 9481 T^{15} - 459 T^{16} - 18 T^{17} + T^{18} \)
$71$ \( -6746599476063 - 37371528667395 T - 33419093205897 T^{2} + 16668136457829 T^{3} + 43528862999559 T^{4} + 31472105865330 T^{5} + 12499688383156 T^{6} + 3010286198889 T^{7} + 409837886056 T^{8} + 13005320091 T^{9} - 6263096939 T^{10} - 1305594311 T^{11} - 117505922 T^{12} - 3056353 T^{13} + 455312 T^{14} + 57591 T^{15} + 3103 T^{16} + 86 T^{17} + T^{18} \)
$73$ \( 41464581495793 + 43795684111438 T - 47272295524978 T^{2} - 30285849478340 T^{3} + 23829008071520 T^{4} + 1287881612701 T^{5} - 2453924282150 T^{6} + 134629255583 T^{7} + 94480435018 T^{8} - 9459522021 T^{9} - 1651481370 T^{10} + 228006952 T^{11} + 12289268 T^{12} - 2581098 T^{13} - 8996 T^{14} + 13692 T^{15} - 307 T^{16} - 27 T^{17} + T^{18} \)
$79$ \( 6058161830533 - 18634395562703 T - 18830053098343 T^{2} + 12918524805371 T^{3} + 4886775879621 T^{4} - 2753084139070 T^{5} - 325665877569 T^{6} + 218431092434 T^{7} + 6748956473 T^{8} - 8330104108 T^{9} + 78329326 T^{10} + 166811929 T^{11} - 5283008 T^{12} - 1751618 T^{13} + 79141 T^{14} + 8892 T^{15} - 476 T^{16} - 17 T^{17} + T^{18} \)
$83$ \( 36277283457297 + 19566704318742 T - 19804057157079 T^{2} - 8257134822849 T^{3} + 3177848657355 T^{4} + 1318812515049 T^{5} - 198710443208 T^{6} - 99518359677 T^{7} + 4343849419 T^{8} + 3919900078 T^{9} + 44133746 T^{10} - 84201898 T^{11} - 3556444 T^{12} + 978095 T^{13} + 59921 T^{14} - 5627 T^{15} - 416 T^{16} + 12 T^{17} + T^{18} \)
$89$ \( -1411268878683 + 2088334360701 T + 4209203540115 T^{2} - 137053494225 T^{3} - 1964623192875 T^{4} - 361153235082 T^{5} + 294149907571 T^{6} + 81626449377 T^{7} - 14316180539 T^{8} - 6068922110 T^{9} - 9617980 T^{10} + 173634125 T^{11} + 15126935 T^{12} - 1308796 T^{13} - 248125 T^{14} - 7355 T^{15} + 667 T^{16} + 51 T^{17} + T^{18} \)
$97$ \( 77389866919 - 374697555796 T - 500445016360 T^{2} + 1429597741769 T^{3} + 142996780134 T^{4} - 797677209475 T^{5} - 176208134555 T^{6} + 133665954313 T^{7} + 59343404805 T^{8} + 4810482903 T^{9} - 1269144530 T^{10} - 222914028 T^{11} + 4319040 T^{12} + 2646634 T^{13} + 69508 T^{14} - 12177 T^{15} - 537 T^{16} + 19 T^{17} + T^{18} \)
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