Properties

Label 37.2.h.a
Level $37$
Weight $2$
Character orbit 37.h
Analytic conductor $0.295$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.295446487479\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \(x^{18} + 30 x^{16} + 333 x^{14} + 1826 x^{12} + 5490 x^{10} + 9432 x^{8} + 9385 x^{6} + 5316 x^{4} + 1584 x^{2} + 192\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} + 30 x^{16} + 333 x^{14} + 1826 x^{12} + 5490 x^{10} + 9432 x^{8} + 9385 x^{6} + 5316 x^{4} + 1584 x^{2} + 192\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -69 \nu^{16} - 2010 \nu^{14} - 21225 \nu^{12} - 107422 \nu^{10} - 284274 \nu^{8} - 398496 \nu^{6} - 289165 \nu^{4} - 101208 \nu^{2} + 64 \nu - 13392 \)\()/128\)
\(\beta_{3}\)\(=\)\((\)\(21 \nu^{17} - 57 \nu^{16} + 600 \nu^{15} - 1678 \nu^{14} + 6125 \nu^{13} - 18037 \nu^{12} + 29284 \nu^{11} - 93906 \nu^{10} + 70238 \nu^{9} - 259794 \nu^{8} + 82076 \nu^{7} - 390328 \nu^{6} + 41101 \nu^{5} - 313697 \nu^{4} + 4902 \nu^{3} - 125860 \nu^{2} - 1192 \nu - 19808\)\()/128\)
\(\beta_{4}\)\(=\)\((\)\(-153 \nu^{17} + 234 \nu^{16} - 4530 \nu^{15} + 6868 \nu^{14} - 49165 \nu^{13} + 73458 \nu^{12} - 259878 \nu^{11} + 379484 \nu^{10} - 735738 \nu^{9} + 1037220 \nu^{8} - 1142912 \nu^{7} + 1528384 \nu^{6} - 957761 \nu^{5} + 1190394 \nu^{4} - 399688 \nu^{3} + 454736 \nu^{2} - 64208 \nu + 65824\)\()/512\)
\(\beta_{5}\)\(=\)\((\)\( 279 \nu^{17} + 8094 \nu^{15} + 84867 \nu^{13} + 424554 \nu^{11} + 1102022 \nu^{9} + 1494432 \nu^{7} + 1024431 \nu^{5} + 326504 \nu^{3} + 37104 \nu - 256 \)\()/512\)
\(\beta_{6}\)\(=\)\((\)\(-691 \nu^{17} - 306 \nu^{16} - 20262 \nu^{15} - 9060 \nu^{14} - 216367 \nu^{13} - 98330 \nu^{12} - 1114850 \nu^{11} - 519756 \nu^{10} - 3034622 \nu^{9} - 1471476 \nu^{8} - 4443072 \nu^{7} - 2285824 \nu^{6} - 3428267 \nu^{5} - 1915522 \nu^{4} - 1292568 \nu^{3} - 799376 \nu^{2} - 185072 \nu - 128416\)\()/1024\)
\(\beta_{7}\)\(=\)\((\)\(-691 \nu^{17} + 306 \nu^{16} - 20262 \nu^{15} + 9060 \nu^{14} - 216367 \nu^{13} + 98330 \nu^{12} - 1114850 \nu^{11} + 519756 \nu^{10} - 3034622 \nu^{9} + 1471476 \nu^{8} - 4443072 \nu^{7} + 2285824 \nu^{6} - 3428267 \nu^{5} + 1915522 \nu^{4} - 1292568 \nu^{3} + 799376 \nu^{2} - 185072 \nu + 128416\)\()/1024\)
\(\beta_{8}\)\(=\)\((\)\(-127 \nu^{17} - 229 \nu^{16} - 3758 \nu^{15} - 6714 \nu^{14} - 40747 \nu^{13} - 71689 \nu^{12} - 215066 \nu^{11} - 369470 \nu^{10} - 607606 \nu^{9} - 1006898 \nu^{8} - 941568 \nu^{7} - 1479584 \nu^{6} - 788471 \nu^{5} - 1152173 \nu^{4} - 331624 \nu^{3} - 443000 \nu^{2} - 54448 \nu - 65488\)\()/256\)
\(\beta_{9}\)\(=\)\((\)\(667 \nu^{17} + 554 \nu^{16} + 19846 \nu^{15} + 16052 \nu^{14} + 217111 \nu^{13} + 167986 \nu^{12} + 1161122 \nu^{11} + 838268 \nu^{10} + 3341230 \nu^{9} + 2170596 \nu^{8} + 5301344 \nu^{7} + 2944576 \nu^{6} + 4549715 \nu^{5} + 2046138 \nu^{4} + 1942632 \nu^{3} + 690096 \nu^{2} + 319792 \nu + 91552\)\()/1024\)
\(\beta_{10}\)\(=\)\((\)\(-681 \nu^{17} + 754 \nu^{16} - 20018 \nu^{15} + 22180 \nu^{14} - 214685 \nu^{13} + 238170 \nu^{12} - 1114278 \nu^{11} + 1238668 \nu^{10} - 3071322 \nu^{9} + 3424628 \nu^{8} - 4597888 \nu^{7} + 5147392 \nu^{6} - 3690001 \nu^{5} + 4143426 \nu^{4} - 1486312 \nu^{3} + 1661904 \nu^{2} - 235344 \nu + 259232\)\()/1024\)
\(\beta_{11}\)\(=\)\((\)\(169 \nu^{17} + 343 \nu^{16} + 4958 \nu^{15} + 10070 \nu^{14} + 52997 \nu^{13} + 107763 \nu^{12} + 273634 \nu^{11} + 557282 \nu^{10} + 748082 \nu^{9} + 1526486 \nu^{8} + 1105720 \nu^{7} + 2260240 \nu^{6} + 870673 \nu^{5} + 1779567 \nu^{4} + 341428 \nu^{3} + 694720 \nu^{2} + 52064 \nu + 105104\)\()/256\)
\(\beta_{12}\)\(=\)\((\)\(169 \nu^{17} - 343 \nu^{16} + 4958 \nu^{15} - 10070 \nu^{14} + 52997 \nu^{13} - 107763 \nu^{12} + 273634 \nu^{11} - 557282 \nu^{10} + 748082 \nu^{9} - 1526486 \nu^{8} + 1105720 \nu^{7} - 2260240 \nu^{6} + 870673 \nu^{5} - 1779567 \nu^{4} + 341428 \nu^{3} - 694720 \nu^{2} + 52064 \nu - 105104\)\()/256\)
\(\beta_{13}\)\(=\)\((\)\(1037 \nu^{17} - 2286 \nu^{16} + 30218 \nu^{15} - 67068 \nu^{14} + 319281 \nu^{13} - 716870 \nu^{12} + 1617550 \nu^{11} - 3699860 \nu^{10} + 4287330 \nu^{9} - 10100748 \nu^{8} + 6021536 \nu^{7} - 14870336 \nu^{6} + 4372501 \nu^{5} - 11594014 \nu^{4} + 1526936 \nu^{3} - 4454544 \nu^{2} + 202576 \nu - 657888\)\()/1024\)
\(\beta_{14}\)\(=\)\((\)\(-691 \nu^{17} - 3330 \nu^{16} - 20262 \nu^{15} - 97604 \nu^{14} - 216367 \nu^{13} - 1041578 \nu^{12} - 1114850 \nu^{11} - 5362028 \nu^{10} - 3034622 \nu^{9} - 14581140 \nu^{8} - 4443072 \nu^{7} - 21341312 \nu^{6} - 3428267 \nu^{5} - 16510930 \nu^{4} - 1292568 \nu^{3} - 6289744 \nu^{2} - 185072 \nu - 920992\)\()/1024\)
\(\beta_{15}\)\(=\)\((\)\(2523 \nu^{17} + 1818 \nu^{16} + 73766 \nu^{15} + 53332 \nu^{14} + 783895 \nu^{13} + 569954 \nu^{12} + 4008898 \nu^{11} + 2940892 \nu^{10} + 10791022 \nu^{9} + 8026308 \nu^{8} + 15553184 \nu^{7} + 11813568 \nu^{6} + 11781779 \nu^{5} + 9213226 \nu^{4} + 4380808 \nu^{3} + 3544560 \nu^{2} + 624304 \nu + 524704\)\()/1024\)
\(\beta_{16}\)\(=\)\((\)\(837 \nu^{17} - 106 \nu^{16} + 24538 \nu^{15} - 3076 \nu^{14} + 261961 \nu^{13} - 32274 \nu^{12} + 1349598 \nu^{11} - 161708 \nu^{10} + 3675442 \nu^{9} - 421252 \nu^{8} + 5395296 \nu^{7} - 575744 \nu^{6} + 4198509 \nu^{5} - 401242 \nu^{4} + 1617784 \nu^{3} - 132512 \nu^{2} + 242064 \nu - 16224\)\()/256\)
\(\beta_{17}\)\(=\)\((\)\(-890 \nu^{17} - 601 \nu^{16} - 26096 \nu^{15} - 17578 \nu^{14} - 278666 \nu^{13} - 186909 \nu^{12} - 1436200 \nu^{11} - 956734 \nu^{10} - 3913276 \nu^{9} - 2578570 \nu^{8} - 5747896 \nu^{7} - 3721872 \nu^{6} - 4474666 \nu^{5} - 2820865 \nu^{4} - 1722444 \nu^{3} - 1046304 \nu^{2} - 256240 \nu - 148720\)\()/256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{17} - \beta_{15} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{16} - 3 \beta_{15} - \beta_{14} - \beta_{13} + 3 \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + 6 \beta_{5} + 2 \beta_{4} - \beta_{3} - 8 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(8 \beta_{17} - 3 \beta_{16} + 11 \beta_{15} - \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + \beta_{11} - 11 \beta_{10} + 11 \beta_{9} + 4 \beta_{8} + 5 \beta_{7} - 12 \beta_{6} - 14 \beta_{5} + 22 \beta_{4} + 7 \beta_{3} + 4 \beta_{2} + 9 \beta_{1} + 29\)
\(\nu^{5}\)\(=\)\(-5 \beta_{17} - 9 \beta_{16} + 40 \beta_{15} + 13 \beta_{14} + 19 \beta_{13} - 57 \beta_{12} - 21 \beta_{11} - 14 \beta_{10} - 14 \beta_{9} + 8 \beta_{8} - 5 \beta_{7} + 3 \beta_{6} - 101 \beta_{5} - 28 \beta_{4} + 22 \beta_{3} + 80 \beta_{1} - 53\)
\(\nu^{6}\)\(=\)\(-75 \beta_{17} + 54 \beta_{16} - 129 \beta_{15} + 26 \beta_{14} - 54 \beta_{13} - 68 \beta_{12} - 10 \beta_{11} + 121 \beta_{10} - 129 \beta_{9} - 70 \beta_{8} - 98 \beta_{7} + 139 \beta_{6} + 175 \beta_{5} - 250 \beta_{4} - 51 \beta_{3} - 78 \beta_{2} - 90 \beta_{1} - 278\)
\(\nu^{7}\)\(=\)\(93 \beta_{17} + 79 \beta_{16} - 514 \beta_{15} - 171 \beta_{14} - 265 \beta_{13} + 819 \beta_{12} + 311 \beta_{11} + 164 \beta_{10} + 172 \beta_{9} - 172 \beta_{8} + 23 \beta_{7} - 63 \beta_{6} + 1429 \beta_{5} + 336 \beta_{4} - 336 \beta_{3} - 921 \beta_{1} + 765\)
\(\nu^{8}\)\(=\)\(828 \beta_{17} - 772 \beta_{16} + 1600 \beta_{15} - 436 \beta_{14} + 772 \beta_{13} + 1078 \beta_{12} + 92 \beta_{11} - 1420 \beta_{10} + 1600 \beta_{9} + 990 \beta_{8} + 1444 \beta_{7} - 1656 \beta_{6} - 2192 \beta_{5} + 3020 \beta_{4} + 430 \beta_{3} + 1186 \beta_{2} + 1007 \beta_{1} + 3100\)
\(\nu^{9}\)\(=\)\(-1365 \beta_{17} - 766 \beta_{16} + 6639 \beta_{15} + 2254 \beta_{14} + 3496 \beta_{13} - 10992 \beta_{12} - 4234 \beta_{11} - 1927 \beta_{10} - 2131 \beta_{9} + 2700 \beta_{8} - 94 \beta_{7} + 999 \beta_{6} - 19235 \beta_{5} - 4058 \beta_{4} + 4627 \beta_{3} + 11339 \beta_{1} - 10402\)
\(\nu^{10}\)\(=\)\(-9974 \beta_{17} + 10377 \beta_{16} - 20351 \beta_{15} + 6319 \beta_{14} - 10377 \beta_{13} - 15207 \beta_{12} - 919 \beta_{11} + 17471 \beta_{10} - 20351 \beta_{9} - 13246 \beta_{8} - 19631 \beta_{7} + 20406 \beta_{6} + 27848 \beta_{5} - 37822 \beta_{4} - 4225 \beta_{3} - 16542 \beta_{2} - 12080 \beta_{1} - 37415\)
\(\nu^{11}\)\(=\)\(18648 \beta_{17} + 8283 \beta_{16} - 86009 \beta_{15} - 29539 \beta_{14} - 45579 \beta_{13} + 144461 \beta_{12} + 56031 \beta_{11} + 23431 \beta_{10} + 26931 \beta_{9} - 38068 \beta_{8} + 167 \beta_{7} - 14224 \beta_{6} + 253874 \beta_{5} + 50362 \beta_{4} - 61499 \beta_{3} - 143775 \beta_{1} + 138011\)
\(\nu^{12}\)\(=\)\(125127 \beta_{17} - 136617 \beta_{16} + 261744 \beta_{15} - 86255 \beta_{14} + 136617 \beta_{13} + 204829 \beta_{12} + 10119 \beta_{11} - 220796 \beta_{10} + 261744 \beta_{9} + 174000 \beta_{8} + 259615 \beta_{7} - 257539 \beta_{6} - 357413 \beta_{5} + 482540 \beta_{4} + 46796 \beta_{3} + 222160 \beta_{2} + 150664 \beta_{1} + 469379\)
\(\nu^{13}\)\(=\)\(-247697 \beta_{17} - 97158 \beta_{16} + 1115619 \beta_{15} + 385382 \beta_{14} + 592552 \beta_{13} - 1884676 \beta_{12} - 733382 \beta_{11} - 293195 \beta_{10} - 344855 \beta_{9} + 513244 \beta_{8} + 3566 \beta_{7} + 192911 \beta_{6} - 3321635 \beta_{5} - 638050 \beta_{4} + 806439 \beta_{3} + 1846742 \beta_{1} - 1810496\)
\(\nu^{14}\)\(=\)\(-1599045 \beta_{17} + 1784373 \beta_{16} - 3383418 \beta_{15} + 1146323 \beta_{14} - 1784373 \beta_{13} - 2704415 \beta_{12} - 119835 \beta_{11} + 2829226 \beta_{10} - 3383418 \beta_{9} - 2270058 \beta_{8} - 3397251 \beta_{7} + 3295781 \beta_{6} + 4613599 \beta_{5} - 6212644 \beta_{4} - 559168 \beta_{3} - 2931282 \beta_{2} - 1917777 \beta_{1} - 5996963\)
\(\nu^{15}\)\(=\)\(3250014 \beta_{17} + 1197218 \beta_{16} - 14478084 \beta_{15} - 5015426 \beta_{14} - 7697246 \beta_{13} + 24519142 \beta_{12} + 9555542 \beta_{11} + 3734348 \beta_{10} + 4447232 \beta_{9} - 6782020 \beta_{8} - 78882 \beta_{7} - 2557178 \beta_{6} + 43283486 \beta_{5} + 8181580 \beta_{4} - 10516368 \beta_{3} - 23858465 \beta_{1} + 23623192\)
\(\nu^{16}\)\(=\)\(20608451 \beta_{17} - 23229040 \beta_{16} + 43837491 \beta_{15} - 15047460 \beta_{14} + 23229040 \beta_{13} + 35387360 \beta_{12} + 1483420 \beta_{11} - 36501279 \beta_{10} + 43837491 \beta_{9} + 29534568 \beta_{8} + 44261776 \beta_{7} - 42486555 \beta_{6} - 59730319 \beta_{5} + 80338770 \beta_{4} + 6966711 \beta_{3} + 38351920 \beta_{2} + 24661531 \beta_{1} + 77276212\)
\(\nu^{17}\)\(=\)\(-42405000 \beta_{17} - 15148291 \beta_{16} + 187934249 \beta_{15} + 65190479 \beta_{14} + 99958291 \beta_{13} - 318622701 \beta_{12} - 124259615 \beta_{11} - 48023055 \beta_{10} - 57553291 \beta_{9} + 88786740 \beta_{8} + 1214869 \beta_{7} + 33530584 \beta_{6} - 562936206 \beta_{5} - 105576346 \beta_{4} + 136809795 \beta_{3} + 309047952 \beta_{1} - 307435721\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.23399i
0.885952i
3.60322i
0.660907i
0.834738i
2.47983i
1.77531i
0.752039i
1.92581i
1.23399i
0.885952i
3.60322i
0.660907i
0.834738i
2.47983i
1.77531i
0.752039i
1.92581i
−1.61954 + 1.93010i −0.968719 + 0.812851i −0.755055 4.28213i −0.681528 + 1.87248i 3.18617i 2.99976 + 1.09182i 5.12375 + 2.95820i −0.243256 + 1.37957i −2.51031 4.34798i
3.2 −0.256873 + 0.306129i −0.0473670 + 0.0397456i 0.319565 + 1.81234i 0.853756 2.34567i 0.0247100i −4.17818 1.52073i −1.32907 0.767337i −0.520281 + 2.95066i 0.498773 + 0.863900i
3.3 1.48976 1.77542i −2.36330 + 1.98304i −0.585455 3.32028i −0.253479 + 0.696428i 7.15011i −1.02732 0.373914i −2.75280 1.58933i 1.13178 6.41863i 0.858832 + 1.48754i
4.1 −2.59056 0.456785i 0.354362 + 2.00969i 4.62296 + 1.68262i 0.640888 + 0.763781i 5.36808i −1.58572 + 1.33057i −6.65125 3.84010i −1.09419 + 0.398252i −1.31137 2.27137i
4.2 −1.11764 0.197069i −0.522172 2.96139i −0.669111 0.243536i 1.89897 + 2.26310i 3.41266i 1.25550 1.05349i 2.66549 + 1.53892i −5.67807 + 2.06665i −1.67637 2.90355i
4.3 0.502458 + 0.0885970i 0.199899 + 1.13369i −1.63477 0.595008i −0.986823 1.17605i 0.587340i 0.422615 0.354616i −1.65240 0.954012i 1.57379 0.572814i −0.391643 0.678346i
21.1 −0.841146 + 2.31103i 2.14040 0.779043i −3.10124 2.60225i −2.84116 0.500973i 5.60182i 0.251492 1.42628i 4.36277 2.51884i 1.67628 1.40657i 3.54759 6.14461i
21.2 −0.491168 + 1.34947i −2.59058 + 0.942893i −0.0477438 0.0400618i 2.52515 + 0.445253i 3.95904i 0.593686 3.36696i −2.40985 + 1.39133i 3.52391 2.95691i −1.84113 + 3.18893i
21.3 0.424710 1.16688i −0.702528 + 0.255699i 0.350853 + 0.294401i −2.65578 0.468285i 0.928365i −0.231838 + 1.31482i 2.64335 1.52614i −1.86997 + 1.56909i −1.67437 + 2.90009i
25.1 −1.61954 1.93010i −0.968719 0.812851i −0.755055 + 4.28213i −0.681528 1.87248i 3.18617i 2.99976 1.09182i 5.12375 2.95820i −0.243256 1.37957i −2.51031 + 4.34798i
25.2 −0.256873 0.306129i −0.0473670 0.0397456i 0.319565 1.81234i 0.853756 + 2.34567i 0.0247100i −4.17818 + 1.52073i −1.32907 + 0.767337i −0.520281 2.95066i 0.498773 0.863900i
25.3 1.48976 + 1.77542i −2.36330 1.98304i −0.585455 + 3.32028i −0.253479 0.696428i 7.15011i −1.02732 + 0.373914i −2.75280 + 1.58933i 1.13178 + 6.41863i 0.858832 1.48754i
28.1 −2.59056 + 0.456785i 0.354362 2.00969i 4.62296 1.68262i 0.640888 0.763781i 5.36808i −1.58572 1.33057i −6.65125 + 3.84010i −1.09419 0.398252i −1.31137 + 2.27137i
28.2 −1.11764 + 0.197069i −0.522172 + 2.96139i −0.669111 + 0.243536i 1.89897 2.26310i 3.41266i 1.25550 + 1.05349i 2.66549 1.53892i −5.67807 2.06665i −1.67637 + 2.90355i
28.3 0.502458 0.0885970i 0.199899 1.13369i −1.63477 + 0.595008i −0.986823 + 1.17605i 0.587340i 0.422615 + 0.354616i −1.65240 + 0.954012i 1.57379 + 0.572814i −0.391643 + 0.678346i
30.1 −0.841146 2.31103i 2.14040 + 0.779043i −3.10124 + 2.60225i −2.84116 + 0.500973i 5.60182i 0.251492 + 1.42628i 4.36277 + 2.51884i 1.67628 + 1.40657i 3.54759 + 6.14461i
30.2 −0.491168 1.34947i −2.59058 0.942893i −0.0477438 + 0.0400618i 2.52515 0.445253i 3.95904i 0.593686 + 3.36696i −2.40985 1.39133i 3.52391 + 2.95691i −1.84113 3.18893i
30.3 0.424710 + 1.16688i −0.702528 0.255699i 0.350853 0.294401i −2.65578 + 0.468285i 0.928365i −0.231838 1.31482i 2.64335 + 1.52614i −1.86997 1.56909i −1.67437 2.90009i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.h.a 18
3.b odd 2 1 333.2.bl.d 18
4.b odd 2 1 592.2.bq.d 18
5.b even 2 1 925.2.bb.a 18
5.c odd 4 2 925.2.ba.a 36
37.f even 9 1 1369.2.b.g 18
37.h even 18 1 inner 37.2.h.a 18
37.h even 18 1 1369.2.b.g 18
37.i odd 36 2 1369.2.a.m 18
111.n odd 18 1 333.2.bl.d 18
148.o odd 18 1 592.2.bq.d 18
185.v even 18 1 925.2.bb.a 18
185.y odd 36 2 925.2.ba.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.h.a 18 1.a even 1 1 trivial
37.2.h.a 18 37.h even 18 1 inner
333.2.bl.d 18 3.b odd 2 1
333.2.bl.d 18 111.n odd 18 1
592.2.bq.d 18 4.b odd 2 1
592.2.bq.d 18 148.o odd 18 1
925.2.ba.a 36 5.c odd 4 2
925.2.ba.a 36 185.y odd 36 2
925.2.bb.a 18 5.b even 2 1
925.2.bb.a 18 185.v even 18 1
1369.2.a.m 18 37.i odd 36 2
1369.2.b.g 18 37.f even 9 1
1369.2.b.g 18 37.h even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 243 + 486 T - 81 T^{2} - 3942 T^{3} - 3537 T^{4} + 2700 T^{5} + 7687 T^{6} + 11322 T^{7} + 12558 T^{8} + 11016 T^{9} + 8052 T^{10} + 4464 T^{11} + 2024 T^{12} + 837 T^{13} + 345 T^{14} + 135 T^{15} + 42 T^{16} + 9 T^{17} + T^{18} \)
$3$ \( 64 + 1824 T + 23040 T^{2} + 72608 T^{3} + 115332 T^{4} + 124218 T^{5} + 110585 T^{6} + 75012 T^{7} + 34263 T^{8} + 7601 T^{9} - 21 T^{10} - 1341 T^{11} - 637 T^{12} + 69 T^{13} + 216 T^{14} + 122 T^{15} + 42 T^{16} + 9 T^{17} + T^{18} \)
$5$ \( 110592 + 82944 T + 207360 T^{2} + 30528 T^{3} + 136944 T^{4} + 142920 T^{5} + 108937 T^{6} + 23046 T^{7} - 5094 T^{8} - 2535 T^{9} - 1590 T^{10} - 285 T^{11} - 331 T^{12} - 102 T^{13} + 81 T^{14} + 6 T^{15} - 6 T^{16} + 3 T^{17} + T^{18} \)
$7$ \( 36864 - 55296 T + 27648 T^{2} + 82944 T^{3} + 50688 T^{4} + 64512 T^{5} + 45696 T^{6} - 864 T^{7} + 23328 T^{8} - 192 T^{9} + 5496 T^{10} + 2544 T^{11} + 1177 T^{12} - 576 T^{13} + 18 T^{14} + 4 T^{15} - 6 T^{16} + 3 T^{17} + T^{18} \)
$11$ \( 46656 - 116640 T + 431568 T^{2} - 92016 T^{3} + 950940 T^{4} - 662904 T^{5} + 917253 T^{6} - 631935 T^{7} + 625698 T^{8} - 396387 T^{9} + 237231 T^{10} - 95094 T^{11} + 33663 T^{12} - 8298 T^{13} + 2043 T^{14} - 369 T^{15} + 78 T^{16} - 9 T^{17} + T^{18} \)
$13$ \( 27 - 1134 T + 27459 T^{2} - 116721 T^{3} + 89829 T^{4} + 122715 T^{5} + 63558 T^{6} + 16983 T^{7} + 16335 T^{8} + 13365 T^{9} - 11124 T^{10} - 1701 T^{11} + 1422 T^{12} - 801 T^{13} + 531 T^{14} - 225 T^{15} + 63 T^{16} - 9 T^{17} + T^{18} \)
$17$ \( 23705163 + 9487125 T - 10313676 T^{2} + 79953876 T^{3} + 81683676 T^{4} + 29021436 T^{5} + 58382227 T^{6} + 25951134 T^{7} + 9184131 T^{8} + 1087971 T^{9} + 25878 T^{10} - 85206 T^{11} - 30823 T^{12} - 7890 T^{13} - 1134 T^{14} + 174 T^{15} + 93 T^{16} + 15 T^{17} + T^{18} \)
$19$ \( 5614272 + 17286048 T + 75186144 T^{2} + 54858384 T^{3} + 23562252 T^{4} + 18690750 T^{5} + 10598373 T^{6} + 3500550 T^{7} + 1068606 T^{8} + 629280 T^{9} + 278892 T^{10} + 14292 T^{11} + 3120 T^{12} + 2592 T^{13} + 414 T^{14} + 78 T^{15} - 33 T^{16} - 6 T^{17} + T^{18} \)
$23$ \( 180910927872 + 21004904448 T - 56720701440 T^{2} - 6680019456 T^{3} + 11766802176 T^{4} + 2065377024 T^{5} - 1280779712 T^{6} - 283458816 T^{7} + 99952464 T^{8} + 29191320 T^{9} - 3777648 T^{10} - 1668762 T^{11} + 72239 T^{12} + 68796 T^{13} + 4197 T^{14} - 864 T^{15} - 69 T^{16} + 9 T^{17} + T^{18} \)
$29$ \( 30509163 + 502324902 T + 3322605447 T^{2} + 9314511894 T^{3} + 13362103557 T^{4} + 9837630216 T^{5} + 3801591946 T^{6} + 602319195 T^{7} - 64252350 T^{8} - 25357185 T^{9} + 9647880 T^{10} + 6029109 T^{11} + 1249718 T^{12} + 98982 T^{13} - 4329 T^{14} - 1026 T^{15} + 51 T^{16} + 18 T^{17} + T^{18} \)
$31$ \( 1142154432 + 1863432864 T^{2} + 1215790236 T^{4} + 410376429 T^{6} + 77685507 T^{8} + 8348454 T^{10} + 491961 T^{12} + 14760 T^{14} + 204 T^{16} + T^{18} \)
$37$ \( 129961739795077 - 21074876723526 T - 1139182525596 T^{2} + 1259771666819 T^{3} - 117329975244 T^{4} - 3272285106 T^{5} + 6279655022 T^{6} - 675478290 T^{7} - 42475704 T^{8} + 35821187 T^{9} - 1147992 T^{10} - 493410 T^{11} + 123974 T^{12} - 1746 T^{13} - 1692 T^{14} + 491 T^{15} - 12 T^{16} - 6 T^{17} + T^{18} \)
$41$ \( 6561 + 715149 T + 22077765 T^{2} + 1694196 T^{3} + 98505396 T^{4} + 98575380 T^{5} - 71951004 T^{6} - 42423183 T^{7} + 57881304 T^{8} - 8086932 T^{9} - 2709288 T^{10} - 540351 T^{11} + 237357 T^{12} + 99036 T^{13} + 26595 T^{14} + 3675 T^{15} + 378 T^{16} + 24 T^{17} + T^{18} \)
$43$ \( 74566243008 + 1388484605088 T^{2} + 621018830988 T^{4} + 92382435225 T^{6} + 6572102724 T^{8} + 254015298 T^{10} + 5544198 T^{12} + 67113 T^{14} + 414 T^{16} + T^{18} \)
$47$ \( 2176782336 - 12516498432 T + 93329542656 T^{2} + 146044710720 T^{3} + 145154724192 T^{4} + 87166952820 T^{5} + 39916526949 T^{6} + 13976996715 T^{7} + 4137753699 T^{8} + 1039329468 T^{9} + 234521487 T^{10} + 46117998 T^{11} + 7960599 T^{12} + 1135215 T^{13} + 133002 T^{14} + 11898 T^{15} + 810 T^{16} + 36 T^{17} + T^{18} \)
$53$ \( 1450162667529 - 1757518728966 T + 45415352979315 T^{2} + 35015215967037 T^{3} + 19015293001968 T^{4} + 6790978374795 T^{5} + 1341980697744 T^{6} + 143438111172 T^{7} + 16777746027 T^{8} + 4372163415 T^{9} + 928680750 T^{10} + 129713391 T^{11} + 12127008 T^{12} + 695196 T^{13} + 32679 T^{14} + 4461 T^{15} + 600 T^{16} + 39 T^{17} + T^{18} \)
$59$ \( 1084930414272 - 5477627781216 T + 10240350689088 T^{2} - 9048841158384 T^{3} + 3957441658092 T^{4} - 149178280482 T^{5} + 205474799971 T^{6} - 37125701079 T^{7} - 6450595140 T^{8} + 545218269 T^{9} + 33550341 T^{10} + 2277456 T^{11} + 130070 T^{12} + 102999 T^{13} + 26196 T^{14} + 1749 T^{15} + 204 T^{16} + 6 T^{17} + T^{18} \)
$61$ \( 24685456563 - 35048009070 T + 19396851462 T^{2} + 12823999056 T^{3} - 5245589484 T^{4} - 7648642890 T^{5} - 359307648 T^{6} + 1889304534 T^{7} + 160045848 T^{8} - 5021820 T^{9} + 29894463 T^{10} - 10068399 T^{11} + 3258627 T^{12} - 725418 T^{13} + 111294 T^{14} - 12009 T^{15} + 885 T^{16} - 42 T^{17} + T^{18} \)
$67$ \( 1266057659543616 - 307890904267872 T + 40241580357456 T^{2} + 8483342890752 T^{3} - 3726877061808 T^{4} + 542291297994 T^{5} + 87520970829 T^{6} - 34384405536 T^{7} + 6338929509 T^{8} - 229665744 T^{9} - 62705478 T^{10} + 21024156 T^{11} + 596728 T^{12} + 19674 T^{13} + 55500 T^{14} + 520 T^{15} - 78 T^{16} + T^{18} \)
$71$ \( 20139015744 - 88832370816 T + 146280906720 T^{2} - 115900327440 T^{3} + 74319963696 T^{4} - 17672565258 T^{5} - 1868217777 T^{6} + 2443445433 T^{7} - 208466298 T^{8} - 83746791 T^{9} + 40372020 T^{10} - 7210053 T^{11} + 1700784 T^{12} - 119556 T^{13} + 11313 T^{14} + 639 T^{15} - 63 T^{16} + 9 T^{17} + T^{18} \)
$73$ \( ( 8467983 + 32865831 T + 20760624 T^{2} + 4582296 T^{3} + 225117 T^{4} - 52749 T^{5} - 6787 T^{6} - 39 T^{7} + 27 T^{8} + T^{9} )^{2} \)
$79$ \( 890793213988032 - 979029864487584 T + 506835435272112 T^{2} - 154435292890848 T^{3} + 27312118638336 T^{4} - 2021808969846 T^{5} - 283519364373 T^{6} + 96409732977 T^{7} - 7944153624 T^{8} + 102643362 T^{9} + 95892444 T^{10} - 6567345 T^{11} - 185685 T^{12} - 83295 T^{13} + 9945 T^{14} - 1455 T^{15} + 51 T^{16} + 6 T^{17} + T^{18} \)
$83$ \( 4807480896 - 19903869504 T + 36598832640 T^{2} - 18162236016 T^{3} + 6634284912 T^{4} - 4371738642 T^{5} + 328393683 T^{6} + 513621081 T^{7} + 240217569 T^{8} + 123182154 T^{9} + 42679710 T^{10} + 7476624 T^{11} + 628074 T^{12} + 53109 T^{13} + 14526 T^{14} + 2694 T^{15} + 306 T^{16} + 24 T^{17} + T^{18} \)
$89$ \( 15282295962363 + 16048302925653 T - 12144680115453 T^{2} - 6055646979582 T^{3} + 6080997848931 T^{4} - 2654829165573 T^{5} + 678918672667 T^{6} - 25921168929 T^{7} + 15579150876 T^{8} - 3174856164 T^{9} + 321455457 T^{10} - 36615789 T^{11} + 5171813 T^{12} - 687375 T^{13} + 57933 T^{14} - 4689 T^{15} + 387 T^{16} - 18 T^{17} + T^{18} \)
$97$ \( 2765473961067 - 3288352719150 T - 1364854187052 T^{2} + 3172709282400 T^{3} + 1336876976673 T^{4} - 3196597306599 T^{5} + 1502102205945 T^{6} - 159871753863 T^{7} - 45244550673 T^{8} + 7226733411 T^{9} + 1468418922 T^{10} - 102691800 T^{11} - 14052708 T^{12} + 828585 T^{13} + 104022 T^{14} - 3564 T^{15} - 369 T^{16} + 9 T^{17} + T^{18} \)
show more
show less