# Properties

 Label 22.3.d.a Level 22 Weight 3 Character orbit 22.d Analytic conductor 0.599 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$22 = 2 \cdot 11$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 22.d (of order $$10$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.599456581593$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.64000000.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{3}$$ $$+ 2 \beta_{2} q^{4}$$ $$+ ( -2 \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{5}$$ $$+ ( -4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{6}$$ $$+ ( -5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{7}$$ $$+ 2 \beta_{3} q^{8}$$ $$+ ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{3}$$ $$+ 2 \beta_{2} q^{4}$$ $$+ ( -2 \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{5}$$ $$+ ( -4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{6}$$ $$+ ( -5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{7}$$ $$+ 2 \beta_{3} q^{8}$$ $$+ ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{9}$$ $$+ ( 4 - 8 \beta_{2} - \beta_{3} + 4 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{10}$$ $$+ ( -1 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{11}$$ $$+ ( 2 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{12}$$ $$+ ( 1 - 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} + 10 \beta_{7} ) q^{13}$$ $$+ ( 4 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{14}$$ $$+ ( 10 + 3 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 3 \beta_{7} ) q^{15}$$ $$+ 4 \beta_{4} q^{16}$$ $$+ ( 6 + 8 \beta_{1} - \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} ) q^{17}$$ $$+ ( -\beta_{1} + 4 \beta_{2} + \beta_{3} - 8 \beta_{4} + 8 \beta_{6} + \beta_{7} ) q^{18}$$ $$+ ( -3 + 11 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} ) q^{19}$$ $$+ ( 2 + 4 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 4 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} ) q^{20}$$ $$+ ( -6 + 12 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 9 \beta_{6} - 8 \beta_{7} ) q^{21}$$ $$+ ( 6 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{22}$$ $$+ ( -16 - 16 \beta_{1} + 12 \beta_{3} - 6 \beta_{4} - 8 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{23}$$ $$+ ( -4 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} ) q^{24}$$ $$+ ( -8 - 16 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} - 9 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{25}$$ $$+ ( -20 + \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - 20 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{26}$$ $$+ ( -7 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} ) q^{27}$$ $$+ ( -8 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} ) q^{28}$$ $$+ ( 17 + 6 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} + 19 \beta_{4} - 36 \beta_{6} - 4 \beta_{7} ) q^{29}$$ $$+ ( -6 + 10 \beta_{1} + 12 \beta_{2} - 9 \beta_{3} + 12 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} ) q^{30}$$ $$+ ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} ) q^{31}$$ $$+ 4 \beta_{5} q^{32}$$ $$+ ( -9 - 4 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} - 18 \beta_{4} - 8 \beta_{5} + 19 \beta_{6} + 6 \beta_{7} ) q^{33}$$ $$+ ( 16 + 6 \beta_{1} - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} ) q^{34}$$ $$+ ( 14 + 7 \beta_{1} + 14 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} ) q^{35}$$ $$+ ( -2 + 4 \beta_{3} - 8 \beta_{5} + 2 \beta_{6} + 8 \beta_{7} ) q^{36}$$ $$+ ( 3 + 2 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{37}$$ $$+ ( -3 \beta_{1} + 22 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} + 22 \beta_{6} - 3 \beta_{7} ) q^{38}$$ $$+ ( 26 - \beta_{1} - 14 \beta_{3} + 13 \beta_{4} + 14 \beta_{5} - 26 \beta_{6} + \beta_{7} ) q^{39}$$ $$+ ( 8 + 2 \beta_{1} - 2 \beta_{3} - 8 \beta_{4} - 8 \beta_{7} ) q^{40}$$ $$+ ( 25 - 8 \beta_{1} - 12 \beta_{2} + 8 \beta_{3} - 12 \beta_{4} - 8 \beta_{5} + 25 \beta_{6} ) q^{41}$$ $$+ ( 16 - 6 \beta_{1} - 16 \beta_{2} + 12 \beta_{3} - 3 \beta_{5} - 20 \beta_{6} + 9 \beta_{7} ) q^{42}$$ $$+ ( -14 + 28 \beta_{2} + 4 \beta_{3} - 24 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{43}$$ $$+ ( 8 + 6 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} + 16 \beta_{4} + 12 \beta_{5} - 12 \beta_{6} + 2 \beta_{7} ) q^{44}$$ $$+ ( -3 + 16 \beta_{1} - 12 \beta_{3} + 28 \beta_{4} + 8 \beta_{5} - 28 \beta_{6} - 4 \beta_{7} ) q^{45}$$ $$+ ( -8 - 16 \beta_{1} - 24 \beta_{2} + 16 \beta_{4} - 6 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} ) q^{46}$$ $$+ ( -13 + 9 \beta_{1} + 6 \beta_{2} + \beta_{3} - 6 \beta_{4} - 11 \beta_{5} + 13 \beta_{6} + 2 \beta_{7} ) q^{47}$$ $$+ ( -4 - 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{48}$$ $$+ ( 10 \beta_{1} - 25 \beta_{2} + 14 \beta_{3} + 22 \beta_{4} + 14 \beta_{5} - 25 \beta_{6} + 10 \beta_{7} ) q^{49}$$ $$+ ( -16 - 8 \beta_{1} - 16 \beta_{2} + 9 \beta_{3} - 8 \beta_{4} - 9 \beta_{5} + 32 \beta_{6} + 8 \beta_{7} ) q^{50}$$ $$+ ( -18 - 11 \beta_{1} + 15 \beta_{2} + 11 \beta_{3} - 12 \beta_{4} + 30 \beta_{6} + 5 \beta_{7} ) q^{51}$$ $$+ ( -4 - 20 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} + 6 \beta_{4} - 20 \beta_{5} - 4 \beta_{6} ) q^{52}$$ $$+ ( -57 + 6 \beta_{1} + 57 \beta_{2} - 12 \beta_{3} + 34 \beta_{6} - 12 \beta_{7} ) q^{53}$$ $$+ ( 14 - 28 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 26 \beta_{6} - 4 \beta_{7} ) q^{54}$$ $$+ ( -12 + 13 \beta_{1} + 15 \beta_{2} - 41 \beta_{3} - 35 \beta_{4} + 15 \beta_{5} - 15 \beta_{6} - 25 \beta_{7} ) q^{55}$$ $$+ ( 8 - 8 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 10 \beta_{7} ) q^{56}$$ $$+ ( -37 + 10 \beta_{1} + 11 \beta_{2} - 48 \beta_{4} + 26 \beta_{5} + 24 \beta_{6} - 26 \beta_{7} ) q^{57}$$ $$+ ( 8 + 17 \beta_{1} + 4 \beta_{2} - 18 \beta_{3} - 4 \beta_{4} + 19 \beta_{5} - 8 \beta_{6} - 36 \beta_{7} ) q^{58}$$ $$+ ( 15 - 13 \beta_{1} - 20 \beta_{2} - 33 \beta_{3} + 15 \beta_{4} + 20 \beta_{5} - 10 \beta_{7} ) q^{59}$$ $$+ ( -6 \beta_{1} + 20 \beta_{2} + 12 \beta_{3} - 18 \beta_{4} + 12 \beta_{5} + 20 \beta_{6} - 6 \beta_{7} ) q^{60}$$ $$+ ( 10 - 6 \beta_{1} + 9 \beta_{2} + 34 \beta_{3} + 5 \beta_{4} - 34 \beta_{5} - 19 \beta_{6} + 6 \beta_{7} ) q^{61}$$ $$+ ( -4 + 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 12 \beta_{6} + 14 \beta_{7} ) q^{62}$$ $$+ ( -2 - 12 \beta_{1} - \beta_{2} + 13 \beta_{3} - \beta_{4} - 12 \beta_{5} - 2 \beta_{6} ) q^{63}$$ $$+ 8 \beta_{6} q^{64}$$ $$+ ( 21 - 42 \beta_{2} + 18 \beta_{3} + 51 \beta_{4} + 10 \beta_{5} + 9 \beta_{6} + 18 \beta_{7} ) q^{65}$$ $$+ ( -12 - 9 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} + 20 \beta_{4} - 18 \beta_{5} - 4 \beta_{6} + 19 \beta_{7} ) q^{66}$$ $$+ ( 12 + 4 \beta_{1} - 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 10 \beta_{7} ) q^{67}$$ $$+ ( 10 + 16 \beta_{1} + 2 \beta_{2} + 8 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{68}$$ $$+ ( 30 + 26 \beta_{1} - 26 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} - 18 \beta_{5} - 30 \beta_{6} - 8 \beta_{7} ) q^{69}$$ $$+ ( -6 + 14 \beta_{1} + 20 \beta_{2} - 6 \beta_{4} + 14 \beta_{5} - 7 \beta_{7} ) q^{70}$$ $$+ ( 19 \beta_{1} - 3 \beta_{2} - 7 \beta_{3} - 33 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} + 19 \beta_{7} ) q^{71}$$ $$+ ( -16 - 2 \beta_{1} + 16 \beta_{2} - 8 \beta_{4} + 2 \beta_{7} ) q^{72}$$ $$+ ( -9 - 22 \beta_{1} + 2 \beta_{2} + 22 \beta_{3} + 5 \beta_{4} + 4 \beta_{6} + 26 \beta_{7} ) q^{73}$$ $$+ ( -4 + 3 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{74}$$ $$+ ( 66 - 21 \beta_{1} - 66 \beta_{2} + 42 \beta_{3} - 37 \beta_{5} - 39 \beta_{6} + 5 \beta_{7} ) q^{75}$$ $$+ ( 6 - 12 \beta_{2} + 22 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} - 14 \beta_{6} + 22 \beta_{7} ) q^{76}$$ $$+ ( 39 - 34 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 23 \beta_{4} - 24 \beta_{5} + 2 \beta_{6} + 40 \beta_{7} ) q^{77}$$ $$+ ( -2 + 26 \beta_{1} - 30 \beta_{4} + 13 \beta_{5} + 30 \beta_{6} - 26 \beta_{7} ) q^{78}$$ $$+ ( 59 - 17 \beta_{1} + 15 \beta_{2} + 44 \beta_{4} + 24 \beta_{5} - 22 \beta_{6} - 24 \beta_{7} ) q^{79}$$ $$+ ( 16 + 8 \beta_{1} - 12 \beta_{2} + 12 \beta_{4} - 8 \beta_{5} - 16 \beta_{6} ) q^{80}$$ $$+ ( 22 - 4 \beta_{1} + 24 \beta_{2} - 32 \beta_{3} + 22 \beta_{4} + 28 \beta_{5} - 14 \beta_{7} ) q^{81}$$ $$+ ( 25 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} - 12 \beta_{5} - 16 \beta_{6} + 25 \beta_{7} ) q^{82}$$ $$+ ( -30 - 39 \beta_{1} - \beta_{2} + 33 \beta_{3} - 15 \beta_{4} - 33 \beta_{5} + 31 \beta_{6} + 39 \beta_{7} ) q^{83}$$ $$+ ( -18 + 16 \beta_{1} + 6 \beta_{2} - 16 \beta_{3} + 6 \beta_{4} + 12 \beta_{6} - 20 \beta_{7} ) q^{84}$$ $$+ ( -33 - 22 \beta_{1} - 22 \beta_{2} - 26 \beta_{3} - 22 \beta_{4} - 22 \beta_{5} - 33 \beta_{6} ) q^{85}$$ $$+ ( -8 - 14 \beta_{1} + 8 \beta_{2} + 28 \beta_{3} - 24 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} ) q^{86}$$ $$+ ( -32 + 64 \beta_{2} + 8 \beta_{3} - 43 \beta_{4} + 47 \beta_{5} + 21 \beta_{6} + 8 \beta_{7} ) q^{87}$$ $$+ ( -4 + 8 \beta_{1} + 16 \beta_{2} - 10 \beta_{3} - 8 \beta_{4} + 16 \beta_{5} + 28 \beta_{6} - 12 \beta_{7} ) q^{88}$$ $$+ ( -30 + 48 \beta_{1} - 6 \beta_{3} - 66 \beta_{4} + 24 \beta_{5} + 66 \beta_{6} - 42 \beta_{7} ) q^{89}$$ $$+ ( 8 - 3 \beta_{1} + 24 \beta_{2} - 16 \beta_{4} + 28 \beta_{5} + 8 \beta_{6} - 28 \beta_{7} ) q^{90}$$ $$+ ( -47 + 45 \beta_{1} - 3 \beta_{2} - 42 \beta_{3} + 3 \beta_{4} + 39 \beta_{5} + 47 \beta_{6} - 84 \beta_{7} ) q^{91}$$ $$+ ( -12 - 8 \beta_{1} - 20 \beta_{2} - 24 \beta_{3} - 12 \beta_{4} + 16 \beta_{5} - 8 \beta_{7} ) q^{92}$$ $$+ ( -2 \beta_{1} - 22 \beta_{2} - 10 \beta_{3} + 23 \beta_{4} - 10 \beta_{5} - 22 \beta_{6} - 2 \beta_{7} ) q^{93}$$ $$+ ( -4 - 13 \beta_{1} + 22 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 18 \beta_{6} + 13 \beta_{7} ) q^{94}$$ $$+ ( -8 + 29 \beta_{1} - 25 \beta_{2} - 29 \beta_{3} + 58 \beta_{4} - 50 \beta_{6} - 15 \beta_{7} ) q^{95}$$ $$+ ( -4 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{96}$$ $$+ ( 1 - 12 \beta_{1} - \beta_{2} + 24 \beta_{3} - 8 \beta_{5} - 12 \beta_{6} + 16 \beta_{7} ) q^{97}$$ $$+ ( -20 + 40 \beta_{2} - 25 \beta_{3} + 8 \beta_{4} + 22 \beta_{5} + 48 \beta_{6} - 25 \beta_{7} ) q^{98}$$ $$+ ( -33 + 11 \beta_{2} + 22 \beta_{3} + 22 \beta_{4} - 33 \beta_{5} - 11 \beta_{6} + 33 \beta_{7} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 20q^{6}$$ $$\mathstrut -\mathstrut 30q^{7}$$ $$\mathstrut -\mathstrut 4q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 20q^{6}$$ $$\mathstrut -\mathstrut 30q^{7}$$ $$\mathstrut -\mathstrut 4q^{9}$$ $$\mathstrut -\mathstrut 4q^{11}$$ $$\mathstrut +\mathstrut 24q^{12}$$ $$\mathstrut +\mathstrut 30q^{13}$$ $$\mathstrut +\mathstrut 16q^{14}$$ $$\mathstrut +\mathstrut 42q^{15}$$ $$\mathstrut -\mathstrut 8q^{16}$$ $$\mathstrut +\mathstrut 30q^{17}$$ $$\mathstrut +\mathstrut 40q^{18}$$ $$\mathstrut -\mathstrut 30q^{19}$$ $$\mathstrut -\mathstrut 4q^{20}$$ $$\mathstrut +\mathstrut 24q^{22}$$ $$\mathstrut -\mathstrut 104q^{23}$$ $$\mathstrut -\mathstrut 40q^{24}$$ $$\mathstrut -\mathstrut 12q^{25}$$ $$\mathstrut -\mathstrut 96q^{26}$$ $$\mathstrut -\mathstrut 26q^{27}$$ $$\mathstrut -\mathstrut 40q^{28}$$ $$\mathstrut -\mathstrut 10q^{29}$$ $$\mathstrut -\mathstrut 60q^{30}$$ $$\mathstrut +\mathstrut 46q^{31}$$ $$\mathstrut -\mathstrut 14q^{33}$$ $$\mathstrut +\mathstrut 112q^{34}$$ $$\mathstrut +\mathstrut 70q^{35}$$ $$\mathstrut -\mathstrut 12q^{36}$$ $$\mathstrut +\mathstrut 6q^{37}$$ $$\mathstrut +\mathstrut 108q^{38}$$ $$\mathstrut +\mathstrut 130q^{39}$$ $$\mathstrut +\mathstrut 80q^{40}$$ $$\mathstrut +\mathstrut 250q^{41}$$ $$\mathstrut +\mathstrut 56q^{42}$$ $$\mathstrut -\mathstrut 12q^{44}$$ $$\mathstrut -\mathstrut 136q^{45}$$ $$\mathstrut -\mathstrut 160q^{46}$$ $$\mathstrut -\mathstrut 54q^{47}$$ $$\mathstrut -\mathstrut 8q^{48}$$ $$\mathstrut -\mathstrut 144q^{49}$$ $$\mathstrut -\mathstrut 80q^{50}$$ $$\mathstrut -\mathstrut 30q^{51}$$ $$\mathstrut -\mathstrut 40q^{52}$$ $$\mathstrut -\mathstrut 274q^{53}$$ $$\mathstrut -\mathstrut 26q^{55}$$ $$\mathstrut +\mathstrut 48q^{56}$$ $$\mathstrut -\mathstrut 130q^{57}$$ $$\mathstrut +\mathstrut 64q^{58}$$ $$\mathstrut +\mathstrut 50q^{59}$$ $$\mathstrut +\mathstrut 116q^{60}$$ $$\mathstrut +\mathstrut 50q^{61}$$ $$\mathstrut +\mathstrut 20q^{62}$$ $$\mathstrut -\mathstrut 20q^{63}$$ $$\mathstrut +\mathstrut 16q^{64}$$ $$\mathstrut -\mathstrut 136q^{66}$$ $$\mathstrut +\mathstrut 112q^{67}$$ $$\mathstrut +\mathstrut 60q^{68}$$ $$\mathstrut +\mathstrut 76q^{69}$$ $$\mathstrut +\mathstrut 4q^{70}$$ $$\mathstrut +\mathstrut 54q^{71}$$ $$\mathstrut -\mathstrut 80q^{72}$$ $$\mathstrut -\mathstrut 70q^{73}$$ $$\mathstrut -\mathstrut 40q^{74}$$ $$\mathstrut +\mathstrut 318q^{75}$$ $$\mathstrut +\mathstrut 266q^{77}$$ $$\mathstrut +\mathstrut 104q^{78}$$ $$\mathstrut +\mathstrut 370q^{79}$$ $$\mathstrut +\mathstrut 48q^{80}$$ $$\mathstrut +\mathstrut 180q^{81}$$ $$\mathstrut -\mathstrut 96q^{82}$$ $$\mathstrut -\mathstrut 150q^{83}$$ $$\mathstrut -\mathstrut 120q^{84}$$ $$\mathstrut -\mathstrut 330q^{85}$$ $$\mathstrut -\mathstrut 72q^{86}$$ $$\mathstrut +\mathstrut 72q^{88}$$ $$\mathstrut +\mathstrut 24q^{89}$$ $$\mathstrut +\mathstrut 160q^{90}$$ $$\mathstrut -\mathstrut 294q^{91}$$ $$\mathstrut -\mathstrut 112q^{92}$$ $$\mathstrut -\mathstrut 134q^{93}$$ $$\mathstrut -\mathstrut 20q^{94}$$ $$\mathstrut -\mathstrut 330q^{95}$$ $$\mathstrut -\mathstrut 18q^{97}$$ $$\mathstrut -\mathstrut 308q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut -\mathstrut$$ $$2$$ $$x^{6}\mathstrut +\mathstrut$$ $$4$$ $$x^{4}\mathstrut -\mathstrut$$ $$8$$ $$x^{2}\mathstrut +\mathstrut$$ $$16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/4$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2$$ $$\beta_{3}$$ $$\nu^{4}$$ $$=$$ $$4$$ $$\beta_{4}$$ $$\nu^{5}$$ $$=$$ $$4$$ $$\beta_{5}$$ $$\nu^{6}$$ $$=$$ $$8$$ $$\beta_{6}$$ $$\nu^{7}$$ $$=$$ $$8$$ $$\beta_{7}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/22\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$\chi(n)$$ $$\beta_{2}$$

## 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
 −0.831254 + 1.14412i 0.831254 − 1.14412i −1.34500 − 0.437016i 1.34500 + 0.437016i −1.34500 + 0.437016i 1.34500 − 0.437016i −0.831254 − 1.14412i 0.831254 + 1.14412i
−0.831254 + 1.14412i −1.32276 + 4.07104i −0.618034 1.90211i 6.27955 4.56236i −3.55822 4.89747i −2.67724 + 0.869888i 2.68999 + 0.874032i −7.54250 5.47994i 10.9771i
7.2 0.831254 1.14412i −0.295274 + 0.908759i −0.618034 1.90211i −2.42545 + 1.76219i 0.794285 + 1.09324i −3.70473 + 1.20374i −2.68999 0.874032i 6.54250 + 4.75340i 4.23984i
13.1 −1.34500 0.437016i 2.48527 1.80565i 1.61803 + 1.17557i −0.399565 1.22973i −4.13178 + 1.34250i −6.48527 + 8.92621i −1.66251 2.28825i 0.135021 0.415553i 1.82860i
13.2 1.34500 + 0.437016i −1.86723 + 1.35662i 1.61803 + 1.17557i −2.45454 7.55429i −3.10429 + 1.00865i −2.13277 + 2.93550i 1.66251 + 2.28825i −1.13502 + 3.49324i 11.2332i
17.1 −1.34500 + 0.437016i 2.48527 + 1.80565i 1.61803 1.17557i −0.399565 + 1.22973i −4.13178 1.34250i −6.48527 8.92621i −1.66251 + 2.28825i 0.135021 + 0.415553i 1.82860i
17.2 1.34500 0.437016i −1.86723 1.35662i 1.61803 1.17557i −2.45454 + 7.55429i −3.10429 1.00865i −2.13277 2.93550i 1.66251 2.28825i −1.13502 3.49324i 11.2332i
19.1 −0.831254 1.14412i −1.32276 4.07104i −0.618034 + 1.90211i 6.27955 + 4.56236i −3.55822 + 4.89747i −2.67724 0.869888i 2.68999 0.874032i −7.54250 + 5.47994i 10.9771i
19.2 0.831254 + 1.14412i −0.295274 0.908759i −0.618034 + 1.90211i −2.42545 1.76219i 0.794285 1.09324i −3.70473 1.20374i −2.68999 + 0.874032i 6.54250 4.75340i 4.23984i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.2 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
11.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(22, [\chi])$$.