# Properties

 Label 31.2.g.a Level 31 Weight 2 Character orbit 31.g Analytic conductor 0.248 Analytic rank 0 Dimension 16 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$31$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 31.g (of order $$15$$ and degree $$8$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.247536246266$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{15})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16}\mathstrut +\mathstrut$$ $$19$$ $$x^{14}\mathstrut +\mathstrut$$ $$140$$ $$x^{12}\mathstrut +\mathstrut$$ $$511$$ $$x^{10}\mathstrut +\mathstrut$$ $$979$$ $$x^{8}\mathstrut +\mathstrut$$ $$956$$ $$x^{6}\mathstrut +\mathstrut$$ $$410$$ $$x^{4}\mathstrut +\mathstrut$$ $$44$$ $$x^{2}\mathstrut +\mathstrut$$ $$1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{15} - 4 \nu^{14} + 48 \nu^{13} - 65 \nu^{12} + 458 \nu^{11} - 358 \nu^{10} + 2196 \nu^{9} - 641 \nu^{8} + 5467 \nu^{7} + 691 \nu^{6} + 6431 \nu^{5} + 3382 \nu^{4} + 2409 \nu^{3} + 2839 \nu^{2} - 360 \nu + 255$$$$)/186$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{15} + 4 \nu^{14} + 48 \nu^{13} + 65 \nu^{12} + 458 \nu^{11} + 358 \nu^{10} + 2196 \nu^{9} + 641 \nu^{8} + 5467 \nu^{7} - 691 \nu^{6} + 6431 \nu^{5} - 3382 \nu^{4} + 2409 \nu^{3} - 2839 \nu^{2} - 360 \nu - 255$$$$)/186$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{15} - 10 \nu^{14} + 144 \nu^{13} - 209 \nu^{12} + 1374 \nu^{11} - 1732 \nu^{10} + 6619 \nu^{9} - 7260 \nu^{8} + 16835 \nu^{7} - 16144 \nu^{6} + 21308 \nu^{5} - 17926 \nu^{4} + 10699 \nu^{3} - 7953 \nu^{2} + 439 \nu - 463$$$$)/186$$ $$\beta_{4}$$ $$=$$ $$($$$$17 \nu^{15} + 315 \nu^{13} + 2250 \nu^{11} + 7940 \nu^{9} + 14865 \nu^{7} + 14844 \nu^{5} + 7255 \nu^{3} + 1125 \nu + 93$$$$)/186$$ $$\beta_{5}$$ $$=$$ $$($$$$6 \nu^{15} + 10 \nu^{14} + 144 \nu^{13} + 209 \nu^{12} + 1374 \nu^{11} + 1732 \nu^{10} + 6619 \nu^{9} + 7260 \nu^{8} + 16835 \nu^{7} + 16144 \nu^{6} + 21308 \nu^{5} + 17926 \nu^{4} + 10699 \nu^{3} + 7953 \nu^{2} + 439 \nu + 463$$$$)/186$$ $$\beta_{6}$$ $$=$$ $$($$$$-6 \nu^{15} + 28 \nu^{14} - 144 \nu^{13} + 517 \nu^{12} - 1374 \nu^{11} + 3653 \nu^{10} - 6619 \nu^{9} + 12516 \nu^{8} - 16835 \nu^{7} + 21730 \nu^{6} - 21308 \nu^{5} + 18145 \nu^{4} - 10699 \nu^{3} + 6074 \nu^{2} - 532 \nu + 354$$$$)/186$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{15} + 33 \nu^{14} + 130 \nu^{13} + 575 \nu^{12} + 716 \nu^{11} + 3713 \nu^{10} + 1282 \nu^{9} + 10969 \nu^{8} - 1382 \nu^{7} + 14550 \nu^{6} - 6857 \nu^{5} + 6617 \nu^{4} - 6329 \nu^{3} - 412 \nu^{2} - 1347 \nu - 143$$$$)/186$$ $$\beta_{8}$$ $$=$$ $$($$$$6 \nu^{15} + 28 \nu^{14} + 144 \nu^{13} + 517 \nu^{12} + 1374 \nu^{11} + 3653 \nu^{10} + 6619 \nu^{9} + 12516 \nu^{8} + 16835 \nu^{7} + 21730 \nu^{6} + 21308 \nu^{5} + 18145 \nu^{4} + 10699 \nu^{3} + 6074 \nu^{2} + 532 \nu + 354$$$$)/186$$ $$\beta_{9}$$ $$=$$ $$($$$$-36 \nu^{15} + 38 \nu^{14} - 709 \nu^{13} + 695 \nu^{12} - 5485 \nu^{11} + 4827 \nu^{10} - 21362 \nu^{9} + 16025 \nu^{8} - 44466 \nu^{7} + 26249 \nu^{6} - 47899 \nu^{5} + 19734 \nu^{4} - 22747 \nu^{3} + 5719 \nu^{2} - 2510 \nu + 538$$$$)/186$$ $$\beta_{10}$$ $$=$$ $$($$$$-53 \nu^{15} + 5 \nu^{14} - 962 \nu^{13} + 89 \nu^{12} - 6619 \nu^{11} + 587 \nu^{10} - 21738 \nu^{9} + 1770 \nu^{8} - 35213 \nu^{7} + 2430 \nu^{6} - 26132 \nu^{5} + 1337 \nu^{4} - 7000 \nu^{3} + 303 \nu^{2} - 225 \nu + 61$$$$)/186$$ $$\beta_{11}$$ $$=$$ $$($$$$50 \nu^{15} - 25 \nu^{14} + 983 \nu^{13} - 445 \nu^{12} + 7575 \nu^{11} - 2966 \nu^{10} + 29263 \nu^{9} - 9222 \nu^{8} + 59919 \nu^{7} - 13483 \nu^{6} + 62350 \nu^{5} - 7987 \nu^{4} + 27117 \nu^{3} - 926 \nu^{2} + 1788 \nu + 36$$$$)/186$$ $$\beta_{12}$$ $$=$$ $$($$$$57 \nu^{15} + 21 \nu^{14} + 1058 \nu^{13} + 380 \nu^{12} + 7535 \nu^{11} + 2608 \nu^{10} + 26161 \nu^{9} + 8581 \nu^{8} + 46581 \nu^{7} + 14174 \nu^{6} + 41009 \nu^{5} + 11369 \nu^{4} + 15383 \nu^{3} + 3765 \nu^{2} + 1489 \nu + 126$$$$)/186$$ $$\beta_{13}$$ $$=$$ $$($$$$27 \nu^{15} + 53 \nu^{14} + 524 \nu^{13} + 962 \nu^{12} + 3982 \nu^{11} + 6619 \nu^{10} + 15200 \nu^{9} + 21738 \nu^{8} + 31009 \nu^{7} + 35213 \nu^{6} + 32677 \nu^{5} + 26132 \nu^{4} + 14464 \nu^{3} + 7093 \nu^{2} + 658 \nu + 597$$$$)/186$$ $$\beta_{14}$$ $$=$$ $$($$$$36 \nu^{15} + 68 \nu^{14} + 709 \nu^{13} + 1229 \nu^{12} + 5485 \nu^{11} + 8411 \nu^{10} + 21362 \nu^{9} + 27451 \nu^{8} + 44466 \nu^{7} + 44177 \nu^{6} + 47899 \nu^{5} + 32530 \nu^{4} + 22747 \nu^{3} + 8467 \nu^{2} + 2510 \nu + 470$$$$)/186$$ $$\beta_{15}$$ $$=$$ $$($$$$135 \nu^{15} + 34 \nu^{14} + 2558 \nu^{13} + 599 \nu^{12} + 18763 \nu^{11} + 3942 \nu^{10} + 67940 \nu^{9} + 12098 \nu^{8} + 128230 \nu^{7} + 17671 \nu^{6} + 121504 \nu^{5} + 11336 \nu^{4} + 48574 \nu^{3} + 2730 \nu^{2} + 3724 \nu + 359$$$$)/186$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{3}$$ $$\nu^{2}$$ $$=$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$3$$ $$\nu^{3}$$ $$=$$ $$-$$$$\beta_{15}\mathstrut +\mathstrut$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$12$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$14$$ $$\nu^{5}$$ $$=$$ $$7$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$37$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$37$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$21$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2$$ $$\nu^{6}$$ $$=$$ $$-$$$$19$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$55$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$19$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$47$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$72$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$34$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$77$$ $$\nu^{7}$$ $$=$$ $$-$$$$42$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$35$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$44$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$29$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$86$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$42$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$22$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$235$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$235$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$68$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$125$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$85$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$98$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$26$$ $$\nu^{8}$$ $$=$$ $$145$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$365$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$248$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$145$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$309$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$440$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$89$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$234$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$92$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$36$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$86$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$245$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$455$$ $$\nu^{9}$$ $$=$$ $$245$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$212$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$356$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$128$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$518$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$245$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$178$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1508$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$145$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$1508$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$391$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$50$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$781$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$513$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$630$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$234$$ $$\nu^{10}$$ $$=$$ $$-$$$$1022$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$2385$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$34$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$1625$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1022$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$2000$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2723$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$637$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$1517$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$688$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$34$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$303$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$601$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1589$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2780$$ $$\nu^{11}$$ $$=$$ $$-$$$$1432$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$1322$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$2578$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$518$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$3129$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$1432$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$1289$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$9702$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1179$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$9702$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$2357$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$756$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$4968$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3142$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4056$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1831$$ $$\nu^{12}$$ $$=$$ $$6922$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$15445$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$619$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$10601$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$6922$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$12821$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$16992$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$4298$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$9634$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4853$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$619$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2229$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$4010$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$10313$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$17280$$ $$\nu^{13}$$ $$=$$ $$8460$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$8372$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$17726$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$1817$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$19052$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$8460$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$8863$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$62396$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$8775$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$62396$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$14559$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$6779$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$31794$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$19510$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$26153$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$13340$$ $$\nu^{14}$$ $$=$$ $$-$$$$45841$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$99462$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$5217$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$68772$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$45841$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$81769$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$106633$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$28148$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$60771$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$33083$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$5217$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$15390$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$26186$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$66810$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$108434$$ $$\nu^{15}$$ $$=$$ $$-$$$$50619$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$53382$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$118538$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$4347$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$116998$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$50619$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$59269$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$400724$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$62032$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$400724$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$91111$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$51949$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$203762$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$122303$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$168575$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$93153$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 − 2.16544i 0.176392i 2.16544i − 0.176392i 1.03739i − 2.52368i − 1.14660i 0.333129i − 1.42343i 1.83925i 1.42343i − 1.83925i 1.14660i − 0.333129i − 1.03739i 2.52368i
−0.571745 1.75965i −0.488442 + 0.103822i −1.15144 + 0.836573i −0.603681 + 1.04561i 0.461954 + 0.800128i 3.41030 + 1.51837i −0.863288 0.627215i −2.51284 + 1.11879i 2.18505 + 0.464447i
7.2 0.380762 + 1.17187i −2.02963 + 0.431412i 0.389745 0.283166i 0.772811 1.33855i −1.27836 2.21419i −3.47491 1.54713i 2.47393 + 1.79742i 1.19265 0.531003i 1.86286 + 0.395962i
9.1 −0.571745 + 1.75965i −0.488442 0.103822i −1.15144 0.836573i −0.603681 1.04561i 0.461954 0.800128i 3.41030 1.51837i −0.863288 + 0.627215i −2.51284 1.11879i 2.18505 0.464447i
9.2 0.380762 1.17187i −2.02963 0.431412i 0.389745 + 0.283166i 0.772811 + 1.33855i −1.27836 + 2.21419i −3.47491 + 1.54713i 2.47393 1.79742i 1.19265 + 0.531003i 1.86286 0.395962i
10.1 −1.02470 0.744490i −0.155153 1.47618i −0.122284 0.376353i 1.90016 + 3.29117i −0.940018 + 1.62816i −2.14115 0.455117i −0.937688 + 2.88591i 0.779397 0.165666i 0.503147 4.78712i
10.2 −0.284315 0.206567i 0.302431 + 2.87744i −0.579869 1.78465i −1.48661 2.57489i 0.508398 0.880572i 1.05848 + 0.224987i −0.420982 + 1.29565i −5.25377 + 1.11672i −0.109221 + 1.03917i
14.1 −0.831304 + 2.55849i 0.949606 1.05464i −4.23677 3.07819i −0.304192 + 0.526876i 1.90889 + 3.30629i 0.180508 1.71742i 7.04481 5.11835i 0.103062 + 0.980572i −1.09513 1.21627i
14.2 0.640321 1.97070i −1.43153 + 1.58988i −1.85563 1.34820i −1.17396 + 2.03335i 2.21654 + 3.83916i 0.384094 3.65441i −0.492333 + 0.357701i −0.164841 1.56836i 3.25543 + 3.61552i
18.1 −1.86683 + 1.35633i −2.32289 + 1.03422i 1.02738 3.16196i 1.24923 + 2.16373i 2.93370 5.08132i 1.07187 + 1.19043i 0.944583 + 2.90713i 2.31884 2.57533i −5.26683 2.34494i
18.2 0.557811 0.405274i −0.824384 + 0.367040i −0.471127 + 1.44998i −1.85376 3.21080i −0.311099 + 0.538840i 0.510810 + 0.567312i 0.750969 + 2.31124i −1.46250 + 1.62427i −2.33530 1.03974i
19.1 −1.86683 1.35633i −2.32289 1.03422i 1.02738 + 3.16196i 1.24923 2.16373i 2.93370 + 5.08132i 1.07187 1.19043i 0.944583 2.90713i 2.31884 + 2.57533i −5.26683 + 2.34494i
19.2 0.557811 + 0.405274i −0.824384 0.367040i −0.471127 1.44998i −1.85376 + 3.21080i −0.311099 0.538840i 0.510810 0.567312i 0.750969 2.31124i −1.46250 1.62427i −2.33530 + 1.03974i
20.1 −0.831304 2.55849i 0.949606 + 1.05464i −4.23677 + 3.07819i −0.304192 0.526876i 1.90889 3.30629i 0.180508 + 1.71742i 7.04481 + 5.11835i 0.103062 0.980572i −1.09513 + 1.21627i
20.2 0.640321 + 1.97070i −1.43153 1.58988i −1.85563 + 1.34820i −1.17396 2.03335i 2.21654 3.83916i 0.384094 + 3.65441i −0.492333 0.357701i −0.164841 + 1.56836i 3.25543 3.61552i
28.1 −1.02470 + 0.744490i −0.155153 + 1.47618i −0.122284 + 0.376353i 1.90016 3.29117i −0.940018 1.62816i −2.14115 + 0.455117i −0.937688 2.88591i 0.779397 + 0.165666i 0.503147 + 4.78712i
28.2 −0.284315 + 0.206567i 0.302431 2.87744i −0.579869 + 1.78465i −1.48661 + 2.57489i 0.508398 + 0.880572i 1.05848 0.224987i −0.420982 1.29565i −5.25377 1.11672i −0.109221 1.03917i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.2 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
31.g Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(31, [\chi])$$.