# Properties

 Label 65.2.o.a Level 65 Weight 2 Character orbit 65.o Analytic conductor 0.519 Analytic rank 0 Dimension 20 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$65 = 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 65.o (of order $$12$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + 68102 \nu^{6} + 46015 \nu^{4} + 16571 \nu^{2} + 2996 \nu - 409$$$$)/5992$$ $$\beta_{2}$$ $$=$$ $$($$$$20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} - 68102 \nu^{6} - 46015 \nu^{4} - 16571 \nu^{2} + 2996 \nu + 409$$$$)/5992$$ $$\beta_{3}$$ $$=$$ $$($$$$-69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} - 257858 \nu^{6} - 105030 \nu^{4} - 19961 \nu^{2} - 539$$$$)/1712$$ $$\beta_{4}$$ $$=$$ $$($$$$321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + 869910 \nu^{13} - 548254 \nu^{12} + 3425391 \nu^{11} - 1503061 \nu^{10} + 7807469 \nu^{9} - 2268567 \nu^{8} + 9753906 \nu^{7} - 1805006 \nu^{6} + 5800042 \nu^{5} - 735210 \nu^{4} + 1183313 \nu^{3} - 139727 \nu^{2} + 39483 \nu - 3773$$$$)/23968$$ $$\beta_{5}$$ $$=$$ $$($$$$700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + 912240 \nu^{13} + 1108682 \nu^{12} + 2625448 \nu^{11} + 3846937 \nu^{10} + 3934784 \nu^{9} + 7752327 \nu^{8} + 2192820 \nu^{7} + 8524592 \nu^{6} - 974624 \nu^{5} + 4331072 \nu^{4} - 1268316 \nu^{3} + 669321 \nu^{2} - 170688 \nu + 26961$$$$)/11984$$ $$\beta_{6}$$ $$=$$ $$($$$$-409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} - 4719229 \nu^{7} - 2594086 \nu^{5} - 562582 \nu^{3} - 48473 \nu - 2996$$$$)/5992$$ $$\beta_{7}$$ $$=$$ $$($$$$1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + 2423698 \nu^{13} - 319614 \nu^{12} + 7689981 \nu^{11} - 870831 \nu^{10} + 13910023 \nu^{9} - 1440917 \nu^{8} + 13309334 \nu^{7} - 1613590 \nu^{6} + 5445918 \nu^{5} - 1272982 \nu^{4} + 481307 \nu^{3} - 501629 \nu^{2} + 71073 \nu - 30699$$$$)/23968$$ $$\beta_{8}$$ $$=$$ $$($$$$700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + 912240 \nu^{13} - 1108682 \nu^{12} + 2625448 \nu^{11} - 3846937 \nu^{10} + 3934784 \nu^{9} - 7752327 \nu^{8} + 2192820 \nu^{7} - 8524592 \nu^{6} - 974624 \nu^{5} - 4331072 \nu^{4} - 1268316 \nu^{3} - 669321 \nu^{2} - 170688 \nu - 26961$$$$)/11984$$ $$\beta_{9}$$ $$=$$ $$($$$$-419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} - 1064734 \nu^{13} + 7212374 \nu^{12} - 4108073 \nu^{11} + 23752795 \nu^{10} - 9110155 \nu^{9} + 45500681 \nu^{8} - 10820986 \nu^{7} + 47979462 \nu^{6} - 5625126 \nu^{5} + 23917926 \nu^{4} - 518719 \nu^{3} + 3775361 \nu^{2} + 159919 \nu + 83703$$$$)/23968$$ $$\beta_{10}$$ $$=$$ $$($$$$-3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} - 5941306 \nu^{13} - 198098 \nu^{12} - 19648615 \nu^{11} + 414413 \nu^{10} - 37985157 \nu^{9} + 3110895 \nu^{8} - 40898326 \nu^{7} + 5871486 \nu^{6} - 21497042 \nu^{5} + 4145562 \nu^{4} - 4030089 \nu^{3} + 861543 \nu^{2} - 166551 \nu + 81509$$$$)/23968$$ $$\beta_{11}$$ $$=$$ $$($$$$3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + 5941306 \nu^{13} + 869910 \nu^{12} + 19648615 \nu^{11} + 3425391 \nu^{10} + 37985157 \nu^{9} + 7807469 \nu^{8} + 40898326 \nu^{7} + 9753906 \nu^{6} + 21497042 \nu^{5} + 5800042 \nu^{4} + 4030089 \nu^{3} + 1183313 \nu^{2} + 154567 \nu + 39483$$$$)/23968$$ $$\beta_{12}$$ $$=$$ $$($$$$3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + 5941306 \nu^{13} - 869910 \nu^{12} + 19648615 \nu^{11} - 3425391 \nu^{10} + 37985157 \nu^{9} - 7807469 \nu^{8} + 40898326 \nu^{7} - 9753906 \nu^{6} + 21497042 \nu^{5} - 5800042 \nu^{4} + 4030089 \nu^{3} - 1183313 \nu^{2} + 154567 \nu - 39483$$$$)/23968$$ $$\beta_{13}$$ $$=$$ $$($$$$-2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} - 4279914 \nu^{13} - 1404094 \nu^{12} - 14010639 \nu^{11} - 4705845 \nu^{10} - 26598933 \nu^{9} - 9214959 \nu^{8} - 27637370 \nu^{7} - 9971998 \nu^{6} - 13384022 \nu^{5} - 5100714 \nu^{4} - 1923401 \nu^{3} - 798859 \nu^{2} - 7127 \nu - 15221$$$$)/11984$$ $$\beta_{14}$$ $$=$$ $$($$$$2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + 4279914 \nu^{13} - 1404094 \nu^{12} + 14010639 \nu^{11} - 4705845 \nu^{10} + 26598933 \nu^{9} - 9214959 \nu^{8} + 27637370 \nu^{7} - 9971998 \nu^{6} + 13384022 \nu^{5} - 5100714 \nu^{4} + 1923401 \nu^{3} - 798859 \nu^{2} + 7127 \nu - 15221$$$$)/11984$$ $$\beta_{15}$$ $$=$$ $$($$$$-6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} - 11139534 \nu^{13} + 3217222 \nu^{12} - 37111487 \nu^{11} + 10951069 \nu^{10} - 72399061 \nu^{9} + 21535831 \nu^{8} - 78924566 \nu^{7} + 22849098 \nu^{6} - 42433790 \nu^{5} + 10804742 \nu^{4} - 8585045 \nu^{3} + 1254563 \nu^{2} - 504763 \nu + 1285$$$$)/23968$$ $$\beta_{16}$$ $$=$$ $$($$$$8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + 13551722 \nu^{13} + 273770 \nu^{12} + 45488043 \nu^{11} + 1753171 \nu^{10} + 89206401 \nu^{9} + 5197801 \nu^{8} + 97040830 \nu^{7} + 7462350 \nu^{6} + 50921018 \nu^{5} + 4439554 \nu^{4} + 9271077 \nu^{3} + 591213 \nu^{2} + 419203 \nu + 23947$$$$)/23968$$ $$\beta_{17}$$ $$=$$ $$($$$$-4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} - 53105203 \nu^{9} - 58320582 \nu^{7} - 31671424 \nu^{5} - 6571905 \nu^{3} + 5992 \nu^{2} - 381235 \nu + 11984$$$$)/11984$$ $$\beta_{18}$$ $$=$$ $$($$$$10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + 16210930 \nu^{13} - 548254 \nu^{12} + 53334711 \nu^{11} - 1503061 \nu^{10} + 102576749 \nu^{9} - 2268567 \nu^{8} + 110068986 \nu^{7} - 1805006 \nu^{6} + 58130790 \nu^{5} - 735210 \nu^{4} + 11443805 \nu^{3} - 127743 \nu^{2} + 679767 \nu + 20195$$$$)/23968$$ $$\beta_{19}$$ $$=$$ $$($$$$-16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} - 110586 \nu^{14} - 26151990 \nu^{13} - 548254 \nu^{12} - 87742737 \nu^{11} - 1503061 \nu^{10} - 172338195 \nu^{9} - 2268567 \nu^{8} - 188614114 \nu^{7} - 1805006 \nu^{6} - 100694134 \nu^{5} - 735210 \nu^{4} - 19274543 \nu^{3} - 139727 \nu^{2} - 820129 \nu - 15757$$$$)/23968$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} + \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{10} - \beta_{3} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 6 \beta_{2} - 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 7 \beta_{1} + 8$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + 17 \beta_{10} + 8 \beta_{9} + 10 \beta_{8} + 8 \beta_{7} + 5 \beta_{6} - 6 \beta_{5} - 18 \beta_{4} + 9 \beta_{3} + 37 \beta_{2} + 46 \beta_{1} - 2$$ $$\nu^{6}$$ $$=$$ $$-13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + 4 \beta_{13} + 38 \beta_{12} + 25 \beta_{11} + 53 \beta_{10} - 10 \beta_{9} + 37 \beta_{8} + 10 \beta_{7} - 13 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} - 53 \beta_{3} + 18 \beta_{2} + 51 \beta_{1} - 39$$ $$\nu^{7}$$ $$=$$ $$66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} - 121 \beta_{11} - 120 \beta_{10} - 52 \beta_{9} - 76 \beta_{8} - 52 \beta_{7} - 24 \beta_{6} + 28 \beta_{5} + 138 \beta_{4} - 69 \beta_{3} - 236 \beta_{2} - 304 \beta_{1} + 21$$ $$\nu^{8}$$ $$=$$ $$120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} - 56 \beta_{13} - 252 \beta_{12} - 229 \beta_{11} - 397 \beta_{10} + 76 \beta_{9} - 332 \beta_{8} - 84 \beta_{7} + 112 \beta_{6} + 100 \beta_{5} - 152 \beta_{4} + 398 \beta_{3} - 185 \beta_{2} - 368 \beta_{1} + 213$$ $$\nu^{9}$$ $$=$$ $$-462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + 818 \beta_{11} + 813 \beta_{10} + 325 \beta_{9} + 537 \beta_{8} + 325 \beta_{7} + 122 \beta_{6} - 113 \beta_{5} - 1011 \beta_{4} + 506 \beta_{3} + 1544 \beta_{2} + 2032 \beta_{1} - 170$$ $$\nu^{10}$$ $$=$$ $$-974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + 546 \beta_{13} + 1707 \beta_{12} + 1862 \beta_{11} + 2910 \beta_{10} - 539 \beta_{9} + 2648 \beta_{8} + 659 \beta_{7} - 854 \beta_{6} - 820 \beta_{5} + 1229 \beta_{4} - 2933 \beta_{3} + 1578 \beta_{2} + 2621 \beta_{1} - 1255$$ $$\nu^{11}$$ $$=$$ $$3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} - 5456 \beta_{11} - 5476 \beta_{10} - 2042 \beta_{9} - 3716 \beta_{8} - 2042 \beta_{7} - 676 \beta_{6} + 368 \beta_{5} + 7277 \beta_{4} - 3655 \beta_{3} - 10296 \beta_{2} - 13730 \beta_{1} + 1267$$ $$\nu^{12}$$ $$=$$ $$7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} - 4610 \beta_{13} - 11688 \beta_{12} - 14278 \beta_{11} - 20988 \beta_{10} + 3766 \beta_{9} - 19984 \beta_{8} - 4978 \beta_{7} + 6226 \beta_{6} + 6320 \beta_{5} - 9292 \beta_{4} + 21288 \beta_{3} - 12378 \beta_{2} - 18508 \beta_{1} + 7797$$ $$\nu^{13}$$ $$=$$ $$-22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + 60422 \beta_{12} + 36438 \beta_{11} + 37058 \beta_{10} + 13074 \beta_{9} + 25620 \beta_{8} + 13074 \beta_{7} + 4096 \beta_{6} - 528 \beta_{5} - 51934 \beta_{4} + 26218 \beta_{3} + 69645 \beta_{2} + 93629 \beta_{1} - 9130$$ $$\nu^{14}$$ $$=$$ $$-54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} - 100084 \beta_{14} + 36202 \beta_{13} + 80555 \beta_{12} + 105990 \beta_{11} + 149815 \beta_{10} - 26322 \beta_{9} + 146494 \beta_{8} + 36730 \beta_{7} - 44574 \beta_{6} - 46938 \beta_{5} + 68030 \beta_{4} - 152877 \beta_{3} + 92972 \beta_{2} + 130077 \beta_{1} - 50290$$ $$\nu^{15}$$ $$=$$ $$156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} - 412141 \beta_{12} - 245006 \beta_{11} - 252616 \beta_{10} - 85481 \beta_{9} - 176993 \beta_{8} - 85481 \beta_{7} - 26707 \beta_{6} - 6031 \beta_{5} + 368826 \beta_{4} - 187213 \beta_{3} - 476096 \beta_{2} - 643231 \beta_{1} + 64782$$ $$\nu^{16}$$ $$=$$ $$398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + 709463 \beta_{14} - 272912 \beta_{13} - 557750 \beta_{12} - 771768 \beta_{11} - 1062637 \beta_{10} + 184581 \beta_{9} - 1056441 \beta_{8} - 266881 \beta_{7} + 316597 \beta_{6} + 340947 \beta_{5} - 489763 \beta_{4} + 1090029 \beta_{3} - 681429 \beta_{2} - 912121 \beta_{1} + 333034$$ $$\nu^{17}$$ $$=$$ $$-1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + 2825568 \beta_{12} + 1660831 \beta_{11} + 1734115 \beta_{10} + 569378 \beta_{9} + 1226922 \beta_{8} + 569378 \beta_{7} + 182897 \beta_{6} + 88166 \beta_{5} - 2610994 \beta_{4} + 1332033 \beta_{3} + 3279739 \beta_{2} + 4444476 \beta_{1} - 456258$$ $$\nu^{18}$$ $$=$$ $$-2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} - 5013336 \beta_{14} + 2007386 \beta_{13} + 3875098 \beta_{12} + 5552425 \beta_{11} + 7508303 \beta_{10} - 1297848 \beta_{9} + 7545155 \beta_{8} + 1919220 \beta_{7} - 2240065 \beta_{6} - 2443653 \beta_{5} + 3492968 \beta_{4} - 7734751 \beta_{3} + 4921700 \beta_{2} + 6390257 \beta_{1} - 2246355$$ $$\nu^{19}$$ $$=$$ $$7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} - 19469181 \beta_{12} - 11344809 \beta_{11} - 11973534 \beta_{10} - 3849162 \beta_{9} - 8532880 \beta_{8} - 3849162 \beta_{7} - 1287312 \beta_{6} - 834556 \beta_{5} + 18442656 \beta_{4} - 9449577 \beta_{3} - 22718962 \beta_{2} - 30843334 \beta_{1} + 3202109$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/65\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-\beta_{11} - \beta_{12}$$ $$-\beta_{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 2.64975i 1.83163i 0.493902i 0.274809i − 1.51805i 2.08794i 1.58474i − 0.131303i − 1.02262i − 2.25081i − 2.64975i − 1.83163i − 0.493902i − 0.274809i 1.51805i − 2.08794i − 1.58474i 0.131303i 1.02262i 2.25081i
−1.32488 2.29475i −1.25278 0.335680i −2.51060 + 4.34849i −1.30391 1.81654i 0.889471 + 3.31955i 0.0972962 + 0.0561740i 8.00544 −1.14131 0.658935i −2.44100 + 5.39885i
2.2 −0.915816 1.58624i 1.91432 + 0.512942i −0.677439 + 1.17336i 1.45480 + 1.69810i −0.939520 3.50634i −3.06478 1.76945i −1.18163 0.803451 + 0.463873i 1.36126 3.86282i
2.3 −0.246951 0.427732i 0.908353 + 0.243392i 0.878030 1.52079i −2.21791 + 0.284413i −0.120212 0.448637i 3.18307 + 1.83775i −1.85513 −1.83221 1.05783i 0.669366 + 0.878433i
2.4 −0.137404 0.237991i −2.28256 0.611610i 0.962240 1.66665i 1.69883 1.45395i 0.168076 + 0.627267i −0.334376 0.193052i −1.07848 2.23793 + 1.29207i −0.579454 0.204528i
2.5 0.759023 + 1.31467i −0.653367 0.175069i −0.152233 + 0.263675i 0.600231 + 2.15400i −0.265763 0.991842i −2.24723 1.29744i 2.57390 −2.20184 1.27123i −2.37621 + 2.42404i
32.1 −1.04397 + 1.80821i 0.713171 + 2.66159i −1.17974 2.04338i −0.194361 2.22760i −5.55724 1.48906i 2.52122 1.45563i 0.750585 −3.97738 + 2.29634i 4.23088 + 1.97411i
32.2 −0.792369 + 1.37242i −0.0510678 0.190588i −0.255697 0.442881i 0.0672627 + 2.23506i 0.302032 + 0.0809291i −0.474866 + 0.274164i −2.35905 2.56436 1.48053i −3.12074 1.67868i
32.3 0.0656513 0.113711i −0.0890070 0.332179i 0.991380 + 1.71712i 0.813169 2.08297i −0.0436159 0.0116869i −2.40874 + 1.39069i 0.522947 2.49566 1.44087i −0.183472 0.229216i
32.4 0.511309 0.885613i −0.721300 2.69193i 0.477126 + 0.826407i −1.69584 + 1.45744i −2.75281 0.737614i 0.834479 0.481787i 3.02107 −4.12812 + 2.38337i 0.423625 + 2.24706i
32.5 1.12540 1.94926i 0.514229 + 1.91913i −1.53307 2.65535i −2.22228 0.247944i 4.31958 + 1.15743i −1.10607 + 0.638592i −2.39966 −0.820542 + 0.473740i −2.98427 + 4.05275i
33.1 −1.32488 + 2.29475i −1.25278 + 0.335680i −2.51060 4.34849i −1.30391 + 1.81654i 0.889471 3.31955i 0.0972962 0.0561740i 8.00544 −1.14131 + 0.658935i −2.44100 5.39885i
33.2 −0.915816 + 1.58624i 1.91432 0.512942i −0.677439 1.17336i 1.45480 1.69810i −0.939520 + 3.50634i −3.06478 + 1.76945i −1.18163 0.803451 0.463873i 1.36126 + 3.86282i
33.3 −0.246951 + 0.427732i 0.908353 0.243392i 0.878030 + 1.52079i −2.21791 0.284413i −0.120212 + 0.448637i 3.18307 1.83775i −1.85513 −1.83221 + 1.05783i 0.669366 0.878433i
33.4 −0.137404 + 0.237991i −2.28256 + 0.611610i 0.962240 + 1.66665i 1.69883 + 1.45395i 0.168076 0.627267i −0.334376 + 0.193052i −1.07848 2.23793 1.29207i −0.579454 + 0.204528i
33.5 0.759023 1.31467i −0.653367 + 0.175069i −0.152233 0.263675i 0.600231 2.15400i −0.265763 + 0.991842i −2.24723 + 1.29744i 2.57390 −2.20184 + 1.27123i −2.37621 2.42404i
63.1 −1.04397 1.80821i 0.713171 2.66159i −1.17974 + 2.04338i −0.194361 + 2.22760i −5.55724 + 1.48906i 2.52122 + 1.45563i 0.750585 −3.97738 2.29634i 4.23088 1.97411i
63.2 −0.792369 1.37242i −0.0510678 + 0.190588i −0.255697 + 0.442881i 0.0672627 2.23506i 0.302032 0.0809291i −0.474866 0.274164i −2.35905 2.56436 + 1.48053i −3.12074 + 1.67868i
63.3 0.0656513 + 0.113711i −0.0890070 + 0.332179i 0.991380 1.71712i 0.813169 + 2.08297i −0.0436159 + 0.0116869i −2.40874 1.39069i 0.522947 2.49566 + 1.44087i −0.183472 + 0.229216i
63.4 0.511309 + 0.885613i −0.721300 + 2.69193i 0.477126 0.826407i −1.69584 1.45744i −2.75281 + 0.737614i 0.834479 + 0.481787i 3.02107 −4.12812 2.38337i 0.423625 2.24706i
63.5 1.12540 + 1.94926i 0.514229 1.91913i −1.53307 + 2.65535i −2.22228 + 0.247944i 4.31958 1.15743i −1.10607 0.638592i −2.39966 −0.820542 0.473740i −2.98427 4.05275i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
65.o Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(65, [\chi])$$.