# Properties

 Label 23.2.c.a Level 23 Weight 2 Character orbit 23.c Analytic conductor 0.184 Analytic rank 0 Dimension 10 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$23$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 23.c (of order $$11$$ and degree $$10$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.183655924649$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{22}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-\zeta_{22}$$

## 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
 −0.841254 − 0.540641i 0.142315 + 0.989821i −0.415415 − 0.909632i −0.415415 + 0.909632i 0.142315 − 0.989821i 0.959493 − 0.281733i −0.841254 + 0.540641i 0.654861 + 0.755750i 0.654861 − 0.755750i 0.959493 + 0.281733i
−2.11435 1.35881i −0.226900 1.57812i 1.79329 + 3.92676i 1.41899 + 0.416652i −1.66463 + 3.64502i −0.804632 + 0.928595i 0.828708 5.76379i 0.439490 0.129046i −2.43409 2.80909i
3.1 −0.313607 2.18119i −1.04408 + 2.28621i −2.74024 + 0.804606i 0.809721 0.934468i 5.31408 + 1.56036i −1.99611 + 1.28282i 0.783524 + 1.71568i −2.17208 2.50672i −2.29218 1.47310i
4.1 0.198939 + 0.435615i −2.11435 + 0.620830i 1.15954 1.33818i −2.18251 1.40261i −0.691070 0.797537i 0.483568 + 3.36329i 1.73259 + 0.508735i 1.56130 1.00339i 0.176814 1.22977i
6.1 0.198939 0.435615i −2.11435 0.620830i 1.15954 + 1.33818i −2.18251 + 1.40261i −0.691070 + 0.797537i 0.483568 3.36329i 1.73259 0.508735i 1.56130 + 1.00339i 0.176814 + 1.22977i
8.1 −0.313607 + 2.18119i −1.04408 2.28621i −2.74024 0.804606i 0.809721 + 0.934468i 5.31408 1.56036i −1.99611 1.28282i 0.783524 1.71568i −2.17208 + 2.50672i −2.29218 + 1.47310i
9.1 −0.226900 + 0.0666238i −0.313607 0.361922i −1.63546 + 1.05105i −0.215370 1.49793i 0.0952700 + 0.0612263i −1.05773 + 2.31611i 0.610783 0.704881i 0.394306 2.74246i 0.148666 + 0.325532i
12.1 −2.11435 + 1.35881i −0.226900 + 1.57812i 1.79329 3.92676i 1.41899 0.416652i −1.66463 3.64502i −0.804632 0.928595i 0.828708 + 5.76379i 0.439490 + 0.129046i −2.43409 + 2.80909i
13.1 −1.04408 1.20493i 0.198939 + 0.127850i −0.0771283 + 0.536439i −1.33083 + 2.91411i −0.0536570 0.373193i 0.874908 + 0.256896i −1.95561 + 1.25679i −1.22301 2.67803i 4.90079 1.43900i
16.1 −1.04408 + 1.20493i 0.198939 0.127850i −0.0771283 0.536439i −1.33083 2.91411i −0.0536570 + 0.373193i 0.874908 0.256896i −1.95561 1.25679i −1.22301 + 2.67803i 4.90079 + 1.43900i
18.1 −0.226900 0.0666238i −0.313607 + 0.361922i −1.63546 1.05105i −0.215370 + 1.49793i 0.0952700 0.0612263i −1.05773 2.31611i 0.610783 + 0.704881i 0.394306 + 2.74246i 0.148666 0.325532i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 18.1 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
23.c Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(23, [\chi])$$.