# Properties

 Label 19.3.f.a Level 19 Weight 3 Character orbit 19.f Analytic conductor 0.518 Analytic rank 0 Dimension 12 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$19$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 19.f (of order $$18$$ and degree $$6$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.517712502285$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{10} q^{2}$$ $$+ ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{3}$$ $$+ ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{4}$$ $$+ ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{5}$$ $$+ ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{6}$$ $$+ ( -1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{7}$$ $$+ ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} + \beta_{10} ) q^{8}$$ $$+ ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{10} q^{2}$$ $$+ ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{3}$$ $$+ ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{4}$$ $$+ ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{5}$$ $$+ ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{6}$$ $$+ ( -1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{7}$$ $$+ ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} + \beta_{10} ) q^{8}$$ $$+ ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{9}$$ $$+ ( 5 - 5 \beta_{1} - \beta_{2} - 5 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{10}$$ $$+ ( 4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{11}$$ $$+ ( 4 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{12}$$ $$+ ( 5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} + 2 \beta_{10} + 3 \beta_{11} ) q^{13}$$ $$+ ( -\beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 7 \beta_{11} ) q^{14}$$ $$+ ( 5 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} + 7 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{15}$$ $$+ ( 4 + 6 \beta_{1} - \beta_{2} + 5 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 7 \beta_{11} ) q^{16}$$ $$+ ( -7 + 3 \beta_{1} - \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - 7 \beta_{6} + 4 \beta_{7} + \beta_{8} + 4 \beta_{9} - 9 \beta_{11} ) q^{17}$$ $$+ ( -7 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 9 \beta_{6} + 5 \beta_{9} - 5 \beta_{10} + 10 \beta_{11} ) q^{18}$$ $$+ ( -8 - 3 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} + 9 \beta_{4} - 8 \beta_{6} - \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 7 \beta_{11} ) q^{19}$$ $$+ ( -6 - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{20}$$ $$+ ( 7 + 2 \beta_{1} + 4 \beta_{3} - 15 \beta_{4} + 7 \beta_{5} + 12 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} - 5 \beta_{9} - 7 \beta_{10} + 9 \beta_{11} ) q^{21}$$ $$+ ( -11 + 10 \beta_{1} + 4 \beta_{2} - 7 \beta_{4} - 10 \beta_{5} + 15 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 6 \beta_{11} ) q^{22}$$ $$+ ( -4 - 16 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} - 4 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 10 \beta_{11} ) q^{23}$$ $$+ ( -7 + 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 9 \beta_{11} ) q^{24}$$ $$+ ( -15 - 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 11 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 4 \beta_{11} ) q^{25}$$ $$+ ( 5 + 19 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 18 \beta_{4} - \beta_{5} - 16 \beta_{6} + 3 \beta_{9} + 3 \beta_{10} + 8 \beta_{11} ) q^{26}$$ $$+ ( 3 - 8 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} + 11 \beta_{4} + 12 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} - 7 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{27}$$ $$+ ( -2 - 4 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 17 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 5 \beta_{11} ) q^{28}$$ $$+ ( 13 - 5 \beta_{1} + \beta_{2} - 12 \beta_{4} + 3 \beta_{5} + 13 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{29}$$ $$+ ( 4 + 10 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} - 5 \beta_{7} - 10 \beta_{8} - 4 \beta_{9} + 6 \beta_{10} - 3 \beta_{11} ) q^{30}$$ $$+ ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} - \beta_{6} + 10 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} - 8 \beta_{11} ) q^{31}$$ $$+ ( 20 + 2 \beta_{1} + 3 \beta_{2} - 17 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 6 \beta_{8} - 3 \beta_{10} + 17 \beta_{11} ) q^{32}$$ $$+ ( 2 - 11 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 12 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + 7 \beta_{9} - 2 \beta_{10} ) q^{33}$$ $$+ ( 16 - 5 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} ) q^{34}$$ $$+ ( 4 + 2 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} + 27 \beta_{4} - 13 \beta_{5} - 9 \beta_{6} - 2 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} + 10 \beta_{10} - 3 \beta_{11} ) q^{35}$$ $$+ ( 11 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 12 \beta_{5} + 13 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} + 3 \beta_{10} - 17 \beta_{11} ) q^{36}$$ $$+ ( 23 + 18 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} + 9 \beta_{4} - \beta_{5} + 8 \beta_{6} - 9 \beta_{9} + 9 \beta_{10} + 28 \beta_{11} ) q^{37}$$ $$+ ( 3 - 30 \beta_{1} - 13 \beta_{2} - \beta_{3} - 15 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 7 \beta_{9} - 9 \beta_{10} - 10 \beta_{11} ) q^{38}$$ $$+ ( -11 + 12 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 14 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{39}$$ $$+ ( -10 - 8 \beta_{1} - 4 \beta_{3} - 24 \beta_{4} + 14 \beta_{5} + 9 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + \beta_{10} + 6 \beta_{11} ) q^{40}$$ $$+ ( -23 + 23 \beta_{1} - 9 \beta_{2} - \beta_{3} - 10 \beta_{4} - 29 \beta_{5} + 12 \beta_{6} - 7 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} - 32 \beta_{11} ) q^{41}$$ $$+ ( -15 - 7 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} + 23 \beta_{4} + 12 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} + 8 \beta_{9} + 10 \beta_{10} + 6 \beta_{11} ) q^{42}$$ $$+ ( 3 - 7 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 7 \beta_{4} + \beta_{5} - 10 \beta_{6} - 3 \beta_{7} - 8 \beta_{8} - 5 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{43}$$ $$+ ( -4 - 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + 11 \beta_{4} - 16 \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 12 \beta_{11} ) q^{44}$$ $$+ ( -3 + 9 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 11 \beta_{5} - 11 \beta_{6} + 12 \beta_{7} + 6 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{45}$$ $$+ ( -20 + 14 \beta_{1} + 16 \beta_{2} - 14 \beta_{3} + 16 \beta_{4} + 14 \beta_{5} - 28 \beta_{6} + 8 \beta_{7} + 16 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{46}$$ $$+ ( -14 - 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 24 \beta_{4} - 9 \beta_{5} + 8 \beta_{6} + 3 \beta_{7} + 17 \beta_{8} + 3 \beta_{9} - 17 \beta_{10} + 10 \beta_{11} ) q^{47}$$ $$+ ( -4 - 31 \beta_{1} - 3 \beta_{2} + 8 \beta_{4} + 14 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 8 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{48}$$ $$+ ( -11 - 4 \beta_{1} - 11 \beta_{2} + 13 \beta_{3} + 7 \beta_{4} + 17 \beta_{5} + 13 \beta_{6} + 6 \beta_{7} + 12 \beta_{8} + 11 \beta_{9} - 24 \beta_{10} + 5 \beta_{11} ) q^{49}$$ $$+ ( 9 - 8 \beta_{1} - 6 \beta_{2} + 16 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} - 4 \beta_{8} - 10 \beta_{9} - 16 \beta_{10} + \beta_{11} ) q^{50}$$ $$+ ( -3 + 12 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 10 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} + 10 \beta_{8} - 3 \beta_{10} + 4 \beta_{11} ) q^{51}$$ $$+ ( -4 - 15 \beta_{1} - \beta_{2} - 10 \beta_{3} - \beta_{4} + 14 \beta_{5} + \beta_{6} + 10 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} - \beta_{10} - 14 \beta_{11} ) q^{52}$$ $$+ ( 13 + 15 \beta_{1} + 24 \beta_{2} - 3 \beta_{3} + 16 \beta_{4} - 9 \beta_{5} - 18 \beta_{6} - 15 \beta_{7} - 24 \beta_{8} + 3 \beta_{9} + 9 \beta_{10} + 15 \beta_{11} ) q^{53}$$ $$+ ( 8 + 16 \beta_{1} - \beta_{2} - 14 \beta_{3} - 26 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 18 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 17 \beta_{11} ) q^{54}$$ $$+ ( 15 - 16 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} - 5 \beta_{4} + 15 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} - 6 \beta_{8} - 11 \beta_{9} - 6 \beta_{10} - 11 \beta_{11} ) q^{55}$$ $$+ ( -9 - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - 15 \beta_{5} - 10 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} - 5 \beta_{9} + 5 \beta_{10} - 28 \beta_{11} ) q^{56}$$ $$+ ( -12 + \beta_{1} - 15 \beta_{2} + 7 \beta_{3} + 27 \beta_{4} - 4 \beta_{5} - 21 \beta_{6} - 9 \beta_{7} + 8 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 13 \beta_{11} ) q^{57}$$ $$+ ( -11 + 4 \beta_{1} + 7 \beta_{4} - 7 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 11 \beta_{9} + 11 \beta_{10} ) q^{58}$$ $$+ ( 2 + 20 \beta_{1} - 13 \beta_{3} + 13 \beta_{4} - 14 \beta_{5} - 22 \beta_{6} - 18 \beta_{7} - 13 \beta_{8} + 18 \beta_{9} + 22 \beta_{10} + 6 \beta_{11} ) q^{59}$$ $$+ ( 2 - 22 \beta_{1} + 3 \beta_{2} - \beta_{3} + 6 \beta_{4} + 16 \beta_{5} + 13 \beta_{7} + 14 \beta_{8} + 14 \beta_{9} + 25 \beta_{11} ) q^{60}$$ $$+ ( 20 + 6 \beta_{1} + 9 \beta_{2} - 11 \beta_{3} - 31 \beta_{4} + 3 \beta_{5} + 15 \beta_{6} - \beta_{7} + 9 \beta_{8} - 11 \beta_{9} + 10 \beta_{10} + 4 \beta_{11} ) q^{61}$$ $$+ ( 13 + 10 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} - 33 \beta_{4} - 11 \beta_{5} + 40 \beta_{6} + 7 \beta_{7} - 4 \beta_{8} - 11 \beta_{9} + 3 \beta_{10} - 11 \beta_{11} ) q^{62}$$ $$+ ( 23 + 8 \beta_{1} + 14 \beta_{2} + 10 \beta_{3} - 30 \beta_{4} + 43 \beta_{6} + 9 \beta_{7} - 5 \beta_{8} - 20 \beta_{9} + 9 \beta_{10} - 20 \beta_{11} ) q^{63}$$ $$+ ( 24 - \beta_{1} - 6 \beta_{2} + 11 \beta_{3} + 10 \beta_{4} + 11 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} - 17 \beta_{9} + 11 \beta_{10} + 7 \beta_{11} ) q^{64}$$ $$+ ( 18 - 34 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 10 \beta_{4} - 6 \beta_{5} + 40 \beta_{6} - 11 \beta_{7} - 8 \beta_{9} - 4 \beta_{10} + 5 \beta_{11} ) q^{65}$$ $$+ ( 23 + 6 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} + 28 \beta_{4} + 16 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} - 8 \beta_{8} - 9 \beta_{9} + 8 \beta_{10} - 5 \beta_{11} ) q^{66}$$ $$+ ( 6 + 37 \beta_{1} + 13 \beta_{2} - 5 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} + 13 \beta_{7} - 7 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} + \beta_{11} ) q^{67}$$ $$+ ( 4 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 8 \beta_{4} - 10 \beta_{5} - 14 \beta_{6} - 5 \beta_{7} - 10 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + 8 \beta_{11} ) q^{68}$$ $$+ ( -4 + 4 \beta_{1} + 4 \beta_{2} - 20 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - 16 \beta_{6} - 14 \beta_{8} + 16 \beta_{9} + 20 \beta_{10} - 12 \beta_{11} ) q^{69}$$ $$+ ( 10 + 5 \beta_{1} - \beta_{2} - 20 \beta_{3} - 11 \beta_{4} - 12 \beta_{5} - 19 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} - 9 \beta_{11} ) q^{70}$$ $$+ ( -8 + 2 \beta_{1} - 20 \beta_{2} + 16 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 16 \beta_{7} - 6 \beta_{8} - 10 \beta_{9} - 20 \beta_{10} + 4 \beta_{11} ) q^{71}$$ $$+ ( -29 + 7 \beta_{1} - 2 \beta_{2} - 16 \beta_{3} - 13 \beta_{4} + 9 \beta_{5} + 2 \beta_{8} + 16 \beta_{9} - 2 \beta_{10} - 16 \beta_{11} ) q^{72}$$ $$+ ( -19 - 36 \beta_{1} - 15 \beta_{2} + 13 \beta_{3} - 10 \beta_{4} + 33 \beta_{5} + 14 \beta_{6} + \beta_{7} + 14 \beta_{8} - 14 \beta_{9} - 30 \beta_{10} - 21 \beta_{11} ) q^{73}$$ $$+ ( -11 + 7 \beta_{1} + 4 \beta_{2} - 17 \beta_{3} - 9 \beta_{4} - 36 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} + 17 \beta_{8} + 8 \beta_{9} + 2 \beta_{10} + 33 \beta_{11} ) q^{74}$$ $$+ ( 2 + \beta_{2} + \beta_{3} + 9 \beta_{4} + 8 \beta_{5} - \beta_{6} + 23 \beta_{7} + 23 \beta_{8} + 9 \beta_{9} - 9 \beta_{10} + 22 \beta_{11} ) q^{75}$$ $$+ ( 11 + 18 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} + 25 \beta_{6} - 5 \beta_{7} - \beta_{8} + 12 \beta_{9} + 9 \beta_{10} + 18 \beta_{11} ) q^{76}$$ $$+ ( 16 - 16 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} + 28 \beta_{4} + 24 \beta_{5} - 36 \beta_{6} + 12 \beta_{9} + 12 \beta_{10} ) q^{77}$$ $$+ ( -26 - 21 \beta_{1} + 11 \beta_{3} + 14 \beta_{4} + 5 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} + 11 \beta_{8} + 5 \beta_{9} - 5 \beta_{10} - 16 \beta_{11} ) q^{78}$$ $$+ ( 2 - 19 \beta_{1} - 4 \beta_{2} + 13 \beta_{3} + 44 \beta_{4} + 9 \beta_{5} - 55 \beta_{6} + 20 \beta_{7} + 7 \beta_{8} + 7 \beta_{9} + 15 \beta_{11} ) q^{79}$$ $$+ ( -19 + 27 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 24 \beta_{4} - 29 \beta_{5} - 34 \beta_{6} + 9 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} - 11 \beta_{10} - 29 \beta_{11} ) q^{80}$$ $$+ ( -3 - 27 \beta_{1} + 11 \beta_{2} + 4 \beta_{3} + 19 \beta_{4} + 42 \beta_{5} - 15 \beta_{6} + 4 \beta_{7} + 11 \beta_{8} + 7 \beta_{9} - 11 \beta_{10} + 42 \beta_{11} ) q^{81}$$ $$+ ( -24 - 3 \beta_{1} - 22 \beta_{2} - 9 \beta_{3} + 14 \beta_{4} - 29 \beta_{6} - 22 \beta_{7} + 18 \beta_{9} - 22 \beta_{10} + 5 \beta_{11} ) q^{82}$$ $$+ ( -32 - 28 \beta_{1} - 4 \beta_{3} - 25 \beta_{4} + 3 \beta_{5} + 24 \beta_{6} + 12 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - 28 \beta_{11} ) q^{83}$$ $$+ ( 5 + 23 \beta_{1} - 23 \beta_{2} - 6 \beta_{3} - 48 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} - \beta_{7} - 23 \beta_{9} - 29 \beta_{10} - 9 \beta_{11} ) q^{84}$$ $$+ ( -31 + 8 \beta_{1} + 8 \beta_{2} - 36 \beta_{3} - 32 \beta_{4} - 5 \beta_{6} - 8 \beta_{7} - 27 \beta_{8} + 18 \beta_{9} + 27 \beta_{10} + \beta_{11} ) q^{85}$$ $$+ ( -10 + 26 \beta_{1} + 8 \beta_{2} + 14 \beta_{4} - 9 \beta_{5} - 10 \beta_{6} + 8 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} ) q^{86}$$ $$+ ( 5 + 17 \beta_{1} + 5 \beta_{2} - 10 \beta_{3} + 8 \beta_{4} - 23 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} - 24 \beta_{8} - 5 \beta_{9} + 15 \beta_{10} - 14 \beta_{11} ) q^{87}$$ $$+ ( -13 + 24 \beta_{1} + 13 \beta_{2} + 6 \beta_{3} + 8 \beta_{4} - 32 \beta_{5} + 30 \beta_{6} + 12 \beta_{8} - 19 \beta_{9} - 6 \beta_{10} + 32 \beta_{11} ) q^{88}$$ $$+ ( -57 - 33 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} + 36 \beta_{4} + 52 \beta_{5} + 30 \beta_{6} + 2 \beta_{7} - 5 \beta_{8} - 2 \beta_{10} - 27 \beta_{11} ) q^{89}$$ $$+ ( 9 + 8 \beta_{1} + \beta_{2} + 25 \beta_{3} + 25 \beta_{4} + \beta_{5} + 34 \beta_{6} - 25 \beta_{7} - 8 \beta_{8} - 17 \beta_{9} + \beta_{10} - \beta_{11} ) q^{90}$$ $$+ ( -37 - 21 \beta_{1} - 15 \beta_{2} + 12 \beta_{3} - 49 \beta_{4} - 6 \beta_{5} + 32 \beta_{6} + 11 \beta_{7} + 15 \beta_{8} - 12 \beta_{9} - 4 \beta_{10} - 20 \beta_{11} ) q^{91}$$ $$+ ( 26 + 20 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} - 46 \beta_{4} - 14 \beta_{5} + 12 \beta_{6} - 8 \beta_{7} + 12 \beta_{8} - 12 \beta_{9} + 8 \beta_{10} + 16 \beta_{11} ) q^{92}$$ $$+ ( 2 - 23 \beta_{1} - 12 \beta_{2} - 11 \beta_{3} - 2 \beta_{4} - 17 \beta_{5} - 19 \beta_{6} + 9 \beta_{7} + 11 \beta_{8} + 9 \beta_{9} - 6 \beta_{10} + 28 \beta_{11} ) q^{93}$$ $$+ ( -14 - 26 \beta_{1} + 19 \beta_{2} + 19 \beta_{3} - 35 \beta_{4} + 37 \beta_{5} + 46 \beta_{6} - 29 \beta_{7} - 29 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} - 46 \beta_{11} ) q^{94}$$ $$+ ( 7 + 23 \beta_{1} + 29 \beta_{2} - 10 \beta_{3} - 14 \beta_{4} + 37 \beta_{5} + 31 \beta_{7} + 8 \beta_{8} + 19 \beta_{9} + 11 \beta_{10} + 53 \beta_{11} ) q^{95}$$ $$+ ( 39 + 14 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} - 21 \beta_{4} - 24 \beta_{5} + 31 \beta_{6} + 20 \beta_{7} - 20 \beta_{8} - 7 \beta_{9} - 7 \beta_{10} ) q^{96}$$ $$+ ( 33 + 9 \beta_{1} + 3 \beta_{3} + 81 \beta_{4} - 47 \beta_{5} - 27 \beta_{6} + 24 \beta_{7} + 3 \beta_{8} - 24 \beta_{9} + \beta_{10} - 38 \beta_{11} ) q^{97}$$ $$+ ( 70 - 17 \beta_{1} + 3 \beta_{2} - 14 \beta_{3} - 7 \beta_{4} + 71 \beta_{5} - 13 \beta_{6} - 13 \beta_{7} + \beta_{8} + \beta_{9} + 20 \beta_{11} ) q^{98}$$ $$+ ( 40 + 5 \beta_{1} - 19 \beta_{2} + 3 \beta_{3} - 37 \beta_{4} - 24 \beta_{5} + 4 \beta_{6} - 7 \beta_{7} - 19 \beta_{8} + 3 \beta_{9} - 12 \beta_{10} + 7 \beta_{11} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q$$ $$\mathstrut -\mathstrut 6q^{2}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut -\mathstrut 36q^{6}$$ $$\mathstrut +\mathstrut 6q^{7}$$ $$\mathstrut -\mathstrut 9q^{8}$$ $$\mathstrut -\mathstrut 24q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$12q$$ $$\mathstrut -\mathstrut 6q^{2}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut -\mathstrut 36q^{6}$$ $$\mathstrut +\mathstrut 6q^{7}$$ $$\mathstrut -\mathstrut 9q^{8}$$ $$\mathstrut -\mathstrut 24q^{9}$$ $$\mathstrut +\mathstrut 51q^{10}$$ $$\mathstrut -\mathstrut 18q^{11}$$ $$\mathstrut +\mathstrut 63q^{12}$$ $$\mathstrut +\mathstrut 21q^{13}$$ $$\mathstrut +\mathstrut 9q^{14}$$ $$\mathstrut +\mathstrut 63q^{15}$$ $$\mathstrut -\mathstrut 12q^{16}$$ $$\mathstrut -\mathstrut 3q^{17}$$ $$\mathstrut -\mathstrut 24q^{19}$$ $$\mathstrut -\mathstrut 90q^{20}$$ $$\mathstrut +\mathstrut 30q^{21}$$ $$\mathstrut -\mathstrut 78q^{22}$$ $$\mathstrut -\mathstrut 102q^{23}$$ $$\mathstrut -\mathstrut 12q^{24}$$ $$\mathstrut -\mathstrut 156q^{25}$$ $$\mathstrut +\mathstrut 21q^{26}$$ $$\mathstrut -\mathstrut 27q^{27}$$ $$\mathstrut +\mathstrut 12q^{28}$$ $$\mathstrut +\mathstrut 147q^{29}$$ $$\mathstrut +\mathstrut 24q^{30}$$ $$\mathstrut +\mathstrut 99q^{31}$$ $$\mathstrut +\mathstrut 165q^{32}$$ $$\mathstrut +\mathstrut 84q^{33}$$ $$\mathstrut +\mathstrut 132q^{34}$$ $$\mathstrut +\mathstrut 96q^{35}$$ $$\mathstrut +\mathstrut 63q^{36}$$ $$\mathstrut +\mathstrut 72q^{38}$$ $$\mathstrut -\mathstrut 108q^{39}$$ $$\mathstrut -\mathstrut 138q^{40}$$ $$\mathstrut -\mathstrut 144q^{41}$$ $$\mathstrut -\mathstrut 237q^{42}$$ $$\mathstrut -\mathstrut 27q^{43}$$ $$\mathstrut -\mathstrut 123q^{44}$$ $$\mathstrut -\mathstrut 3q^{45}$$ $$\mathstrut -\mathstrut 54q^{46}$$ $$\mathstrut -\mathstrut 99q^{47}$$ $$\mathstrut -\mathstrut 51q^{48}$$ $$\mathstrut -\mathstrut 24q^{49}$$ $$\mathstrut +\mathstrut 72q^{50}$$ $$\mathstrut -\mathstrut 42q^{51}$$ $$\mathstrut +\mathstrut 93q^{52}$$ $$\mathstrut +\mathstrut 111q^{53}$$ $$\mathstrut +\mathstrut 21q^{54}$$ $$\mathstrut +\mathstrut 162q^{55}$$ $$\mathstrut -\mathstrut 168q^{57}$$ $$\mathstrut -\mathstrut 132q^{58}$$ $$\mathstrut +\mathstrut 3q^{59}$$ $$\mathstrut -\mathstrut 30q^{60}$$ $$\mathstrut +\mathstrut 150q^{61}$$ $$\mathstrut +\mathstrut 108q^{62}$$ $$\mathstrut +\mathstrut 234q^{63}$$ $$\mathstrut +\mathstrut 27q^{64}$$ $$\mathstrut +\mathstrut 126q^{65}$$ $$\mathstrut +\mathstrut 168q^{66}$$ $$\mathstrut +\mathstrut 135q^{67}$$ $$\mathstrut -\mathstrut 30q^{68}$$ $$\mathstrut +\mathstrut 72q^{69}$$ $$\mathstrut +\mathstrut 225q^{70}$$ $$\mathstrut -\mathstrut 168q^{71}$$ $$\mathstrut -\mathstrut 102q^{72}$$ $$\mathstrut -\mathstrut 90q^{73}$$ $$\mathstrut -\mathstrut 231q^{74}$$ $$\mathstrut +\mathstrut 42q^{76}$$ $$\mathstrut +\mathstrut 246q^{77}$$ $$\mathstrut -\mathstrut 189q^{78}$$ $$\mathstrut -\mathstrut 75q^{79}$$ $$\mathstrut +\mathstrut 21q^{80}$$ $$\mathstrut -\mathstrut 159q^{81}$$ $$\mathstrut -\mathstrut 117q^{82}$$ $$\mathstrut -\mathstrut 156q^{83}$$ $$\mathstrut +\mathstrut 99q^{84}$$ $$\mathstrut -\mathstrut 300q^{85}$$ $$\mathstrut -\mathstrut 144q^{86}$$ $$\mathstrut +\mathstrut 69q^{87}$$ $$\mathstrut -\mathstrut 405q^{88}$$ $$\mathstrut -\mathstrut 558q^{89}$$ $$\mathstrut -\mathstrut 66q^{90}$$ $$\mathstrut -\mathstrut 453q^{91}$$ $$\mathstrut +\mathstrut 48q^{92}$$ $$\mathstrut -\mathstrut 57q^{93}$$ $$\mathstrut -\mathstrut 69q^{95}$$ $$\mathstrut +\mathstrut 558q^{96}$$ $$\mathstrut +\mathstrut 465q^{97}$$ $$\mathstrut +\mathstrut 777q^{98}$$ $$\mathstrut +\mathstrut 462q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12}\mathstrut +\mathstrut$$ $$24$$ $$x^{10}\mathstrut +\mathstrut$$ $$216$$ $$x^{8}\mathstrut +\mathstrut$$ $$905$$ $$x^{6}\mathstrut +\mathstrut$$ $$1770$$ $$x^{4}\mathstrut +\mathstrut$$ $$1395$$ $$x^{2}\mathstrut +\mathstrut$$ $$361$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{10} - 27 \nu^{9} + 28 \nu^{8} - 277 \nu^{7} + 38 \nu^{6} - 1304 \nu^{5} - 774 \nu^{4} - 2566 \nu^{3} - 2742 \nu^{2} - 1177 \nu - 1520$$$$)/304$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} - 5 \nu^{10} - \nu^{9} - 89 \nu^{8} + 239 \nu^{7} - 437 \nu^{6} + 2078 \nu^{5} - 22 \nu^{4} + 5308 \nu^{3} + 2960 \nu^{2} + 2697 \nu + 1881$$$$)/304$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} + 5 \nu^{10} - \nu^{9} + 89 \nu^{8} + 239 \nu^{7} + 437 \nu^{6} + 2078 \nu^{5} + 22 \nu^{4} + 5308 \nu^{3} - 2960 \nu^{2} + 2697 \nu - 1881$$$$)/304$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} - 2 \nu^{10} - 27 \nu^{9} - 28 \nu^{8} - 277 \nu^{7} - 38 \nu^{6} - 1304 \nu^{5} + 774 \nu^{4} - 2566 \nu^{3} + 2742 \nu^{2} - 1177 \nu + 1520$$$$)/304$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{11} + 17 \nu^{10} + 55 \nu^{9} + 333 \nu^{8} + 315 \nu^{7} + 2185 \nu^{6} + 530 \nu^{5} + 5334 \nu^{4} - 24 \nu^{3} + 3464 \nu^{2} + 417 \nu + 171$$$$)/608$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{11} - 17 \nu^{10} + 55 \nu^{9} - 333 \nu^{8} + 315 \nu^{7} - 2185 \nu^{6} + 530 \nu^{5} - 5334 \nu^{4} - 24 \nu^{3} - 3464 \nu^{2} + 417 \nu - 171$$$$)/608$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{10} + 21 \nu^{8} + 153 \nu^{6} + 454 \nu^{4} + 504 \nu^{2} + 8 \nu + 171$$$$)/16$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{10} - 21 \nu^{8} - 153 \nu^{6} - 454 \nu^{4} - 504 \nu^{2} + 8 \nu - 171$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{11} + \nu^{10} - 56 \nu^{9} + 14 \nu^{8} - 329 \nu^{7} + 19 \nu^{6} - 549 \nu^{5} - 387 \nu^{4} + 411 \nu^{3} - 1295 \nu^{2} + 650 \nu - 380$$$$)/152$$ $$\beta_{10}$$ $$=$$ $$($$$$17 \nu^{11} + 17 \nu^{10} + 333 \nu^{9} + 333 \nu^{8} + 2185 \nu^{7} + 2185 \nu^{6} + 5334 \nu^{5} + 5334 \nu^{4} + 3464 \nu^{3} + 3768 \nu^{2} + 171 \nu + 1083$$$$)/608$$ $$\beta_{11}$$ $$=$$ $$($$$$9 \nu^{11} + 197 \nu^{9} + 1545 \nu^{7} + 5238 \nu^{5} + 7304 \nu^{3} + 2979 \nu - 152$$$$)/304$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$4$$ $$\nu^{3}$$ $$=$$ $$-$$$$5$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-$$$$7$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$23$$ $$\nu^{5}$$ $$=$$ $$2$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$32$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$32$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$18$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$23$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$26$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2$$ $$\nu^{6}$$ $$=$$ $$50$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$50$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$30$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$20$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$102$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$23$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$52$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$153$$ $$\nu^{7}$$ $$=$$ $$-$$$$38$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$44$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$44$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$226$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$226$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$158$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$114$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$200$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$130$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$130$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$244$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$25$$ $$\nu^{8}$$ $$=$$ $$-$$$$384$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$384$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$149$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$149$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$334$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$50$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$888$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$200$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$200$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$504$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1104$$ $$\nu^{9}$$ $$=$$ $$494$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$483$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$483$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1688$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$1688$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$1400$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$917$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1589$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1186$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1186$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2072$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$236$$ $$\nu^{10}$$ $$=$$ $$3088$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$3088$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1433$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1433$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$3332$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$244$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$7532$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1589$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1589$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$4444$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$8372$$ $$\nu^{11}$$ $$=$$ $$-$$$$5420$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$4765$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$4765$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$13049$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$13049$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$12374$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$7609$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$12211$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$10398$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$10398$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$16976$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2055$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/19\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{1} - \beta_{5}$$

## 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
 2.88811i − 0.918492i − 2.01431i 0.728740i − 2.88811i 0.918492i 2.01431i − 0.728740i − 2.57727i 1.89323i 2.57727i − 1.89323i
−2.84423 0.501515i 3.28392 3.91363i 4.07936 + 1.48477i −3.00117 + 1.09234i −11.3030 + 9.48432i 3.87208 + 6.70664i −0.853313 0.492661i −2.96949 16.8408i 9.08386 1.60173i
2.2 0.904538 + 0.159494i −0.464845 + 0.553981i −2.96602 1.07954i 0.295437 0.107530i −0.508827 + 0.426957i −0.328846 0.569578i −5.69245 3.28654i 1.47202 + 8.34824i 0.284385 0.0501447i
3.1 −1.29478 + 1.54305i 0.621128 + 1.70654i −0.00997859 0.0565914i 1.24445 7.05761i −3.43750 1.25115i 0.422527 + 0.731838i −6.87755 3.97075i 4.36793 3.66513i 9.27900 + 11.0583i
3.2 0.468425 0.558247i −1.14207 3.13782i 0.602375 + 3.41624i −1.13111 + 6.41483i −2.28665 0.832274i −5.47943 9.49065i 4.71370 + 2.72146i −1.64718 + 1.38215i 3.05122 + 3.63630i
10.1 −2.84423 + 0.501515i 3.28392 + 3.91363i 4.07936 1.48477i −3.00117 1.09234i −11.3030 9.48432i 3.87208 6.70664i −0.853313 + 0.492661i −2.96949 + 16.8408i 9.08386 + 1.60173i
10.2 0.904538 0.159494i −0.464845 0.553981i −2.96602 + 1.07954i 0.295437 + 0.107530i −0.508827 0.426957i −0.328846 + 0.569578i −5.69245 + 3.28654i 1.47202 8.34824i 0.284385 + 0.0501447i
13.1 −1.29478 1.54305i 0.621128 1.70654i −0.00997859 + 0.0565914i 1.24445 + 7.05761i −3.43750 + 1.25115i 0.422527 0.731838i −6.87755 + 3.97075i 4.36793 + 3.66513i 9.27900 11.0583i
13.2 0.468425 + 0.558247i −1.14207 + 3.13782i 0.602375 3.41624i −1.13111 6.41483i −2.28665 + 0.832274i −5.47943 + 9.49065i 4.71370 2.72146i −1.64718 1.38215i 3.05122 3.63630i
14.1 −0.881480 2.42185i −0.384565 0.0678091i −2.02415 + 1.69847i 1.73199 + 1.45331i 0.174763 + 0.991129i 5.72163 + 9.91015i −3.03027 1.74952i −8.31394 3.02603i 1.99298 5.47566i
14.2 0.647524 + 1.77906i −1.91357 0.337414i 0.318417 0.267183i −2.13959 1.79533i −0.638803 3.62283i −1.20796 2.09224i 7.23987 + 4.17994i −4.90934 1.78685i 1.80856 4.96897i
15.1 −0.881480 + 2.42185i −0.384565 + 0.0678091i −2.02415 1.69847i 1.73199 1.45331i 0.174763 0.991129i 5.72163 9.91015i −3.03027 + 1.74952i −8.31394 + 3.02603i 1.99298 + 5.47566i
15.2 0.647524 1.77906i −1.91357 + 0.337414i 0.318417 + 0.267183i −2.13959 + 1.79533i −0.638803 + 3.62283i −1.20796 + 2.09224i 7.23987 4.17994i −4.90934 + 1.78685i 1.80856 + 4.96897i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 15.2 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
19.f Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(19, [\chi])$$.