# 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 $$10$$, degree $$4$$, minimal)

## 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 Defining polynomial: $$x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16$$ x^8 - 2*x^6 + 4*x^4 - 8*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (2 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{4} - \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{6}+ \cdots + (\beta_{6} + 4 \beta_{5} - 4 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (2*b7 - b6 - b5 + 2*b4 + b3 - 2*b2 - b1 + 1) * q^3 + 2*b2 * q^4 + (-2*b7 - b6 - 3*b4 - b2 - 2*b1) * q^5 + (-b7 + 2*b6 + 2*b5 - 2*b4 - 2*b3 + 2*b2 + b1 - 4) * q^6 + (-2*b7 + 4*b6 + b4 - b3 + 2*b2 + b1 - 5) * q^7 + 2*b3 * q^8 + (b6 + 4*b5 - 4*b3 + b2 + 2*b1 - 1) * q^9 $$q + \beta_1 q^{2} + (2 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{4} - \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{6}+ \cdots + (33 \beta_{7} - 11 \beta_{6} - 33 \beta_{5} + 22 \beta_{4} + 22 \beta_{3} + \cdots - 33) q^{99}+O(q^{100})$$ q + b1 * q^2 + (2*b7 - b6 - b5 + 2*b4 + b3 - 2*b2 - b1 + 1) * q^3 + 2*b2 * q^4 + (-2*b7 - b6 - 3*b4 - b2 - 2*b1) * q^5 + (-b7 + 2*b6 + 2*b5 - 2*b4 - 2*b3 + 2*b2 + b1 - 4) * q^6 + (-2*b7 + 4*b6 + b4 - b3 + 2*b2 + b1 - 5) * q^7 + 2*b3 * q^8 + (b6 + 4*b5 - 4*b3 + b2 + 2*b1 - 1) * q^9 + (-b7 - 4*b6 - 3*b5 + 4*b4 - b3 - 8*b2 + 4) * q^10 + (-3*b7 - 4*b6 + 4*b5 - 2*b4 + 3*b3 + 4*b2 + 2*b1 - 1) * q^11 + (2*b7 + 2*b6 - 2*b5 - 2*b4 + 2*b3 - 4*b1 + 2) * q^12 + (10*b7 + 2*b6 - 10*b5 - 4*b4 + 5*b2 - 4*b1 + 1) * q^13 + (4*b7 - 4*b6 + b5 + 2*b4 + 2*b3 - 2*b2 - 5*b1 + 4) * q^14 + (3*b7 - 6*b5 + 10*b4 + 9*b3 - 9*b2 + 3*b1 + 10) * q^15 + 4*b4 * q^16 + (-8*b7 - 5*b6 + 6*b5 + 3*b4 - 6*b3 - b2 + 8*b1 + 6) * q^17 + (b7 + 8*b6 - 8*b4 + b3 + 4*b2 - b1) * q^18 + (-3*b6 + 11*b5 - 4*b4 - 5*b3 - 4*b2 + 11*b1 - 3) * q^19 + (-4*b7 - 8*b6 + 4*b5 - 8*b3 - 2*b2 + 4*b1 + 2) * q^20 + (-8*b7 + 9*b6 - 2*b5 - 3*b4 - 8*b3 + 12*b2 - 6) * q^21 + (-4*b7 + 2*b6 - 2*b5 + 12*b4 + 4*b3 - 2*b2 - b1 + 6) * q^22 + (4*b7 + 6*b6 - 8*b5 - 6*b4 + 12*b3 - 16*b1 - 16) * q^23 + (2*b7 - 2*b5 - 4*b2 + 2*b1 - 4) * q^24 + (8*b7 + 8*b6 + 8*b5 - 9*b4 + 4*b3 + 9*b2 - 16*b1 - 8) * q^25 + (2*b7 - 4*b5 - 20*b4 + 5*b3 + 12*b2 + b1 - 20) * q^26 + (-7*b7 - 4*b6 - 6*b5 + 5*b4 - 6*b3 - 4*b2 - 7*b1) * q^27 + (-4*b7 + 10*b6 + 2*b5 - 4*b4 - 2*b3 - 2*b2 + 4*b1 - 8) * q^28 + (-4*b7 - 36*b6 + 19*b4 - 6*b3 - 18*b2 + 6*b1 + 17) * q^29 + (-6*b6 + 10*b5 + 12*b4 - 9*b3 + 12*b2 + 10*b1 - 6) * q^30 + (2*b7 + 14*b6 + 4*b5 - 2*b3 - 3*b2 + b1 + 3) * q^31 + 4*b5 * q^32 + (6*b7 + 19*b6 - 8*b5 - 18*b4 + 16*b3 - 8*b2 - 4*b1 - 9) * q^33 + (-5*b7 - 4*b6 + 3*b5 + 4*b4 - b3 + 6*b1 + 16) * q^34 + (3*b7 - 7*b6 - 3*b5 + 14*b4 + 7*b1 + 14) * q^35 + (8*b7 + 2*b6 - 8*b5 + 4*b3 - 2) * q^36 + (2*b7 - 4*b5 + 3*b4 + 6*b3 - 6*b2 + 2*b1 + 3) * q^37 + (-3*b7 + 22*b6 - 4*b5 - 10*b4 - 4*b3 + 22*b2 - 3*b1) * q^38 + (b7 - 26*b6 + 14*b5 + 13*b4 - 14*b3 - b1 + 26) * q^39 + (-8*b7 - 8*b4 - 2*b3 + 2*b1 + 8) * q^40 + (25*b6 - 8*b5 - 12*b4 + 8*b3 - 12*b2 - 8*b1 + 25) * q^41 + (9*b7 - 20*b6 - 3*b5 + 12*b3 - 16*b2 - 6*b1 + 16) * q^42 + (4*b7 + 4*b6 - 10*b5 - 24*b4 + 4*b3 + 28*b2 - 14) * q^43 + (2*b7 - 12*b6 + 12*b5 + 16*b4 - 2*b3 - 10*b2 + 6*b1 + 8) * q^44 + (-4*b7 - 28*b6 + 8*b5 + 28*b4 - 12*b3 + 16*b1 - 3) * q^45 + (6*b7 - 8*b6 - 6*b5 + 16*b4 - 24*b2 - 16*b1 - 8) * q^46 + (2*b7 + 13*b6 - 11*b5 - 6*b4 + b3 + 6*b2 + 9*b1 - 13) * q^47 + (-4*b4 - 4*b3 + 8*b2 - 4*b1 - 4) * q^48 + (10*b7 - 25*b6 + 14*b5 + 22*b4 + 14*b3 - 25*b2 + 10*b1) * q^49 + (8*b7 + 32*b6 - 9*b5 - 8*b4 + 9*b3 - 16*b2 - 8*b1 - 16) * q^50 + (5*b7 + 30*b6 - 12*b4 + 11*b3 + 15*b2 - 11*b1 - 18) * q^51 + (-4*b6 - 20*b5 + 6*b4 + 12*b3 + 6*b2 - 20*b1 - 4) * q^52 + (-12*b7 + 34*b6 - 12*b3 + 57*b2 + 6*b1 - 57) * q^53 + (-4*b7 - 26*b6 + 5*b5 + 2*b4 - 4*b3 - 28*b2 + 14) * q^54 + (-25*b7 - 15*b6 + 15*b5 - 35*b4 - 41*b3 + 15*b2 + 13*b1 - 12) * q^55 + (10*b7 - 4*b6 - 4*b5 + 4*b4 - 2*b3 - 8*b1 + 8) * q^56 + (-26*b7 + 24*b6 + 26*b5 - 48*b4 + 11*b2 + 10*b1 - 37) * q^57 + (-36*b7 - 8*b6 + 19*b5 - 4*b4 - 18*b3 + 4*b2 + 17*b1 + 8) * q^58 + (-10*b7 + 20*b5 + 15*b4 - 33*b3 - 20*b2 - 13*b1 + 15) * q^59 + (-6*b7 + 20*b6 + 12*b5 - 18*b4 + 12*b3 + 20*b2 - 6*b1) * q^60 + (6*b7 - 19*b6 - 34*b5 + 5*b4 + 34*b3 + 9*b2 - 6*b1 + 10) * q^61 + (14*b7 + 12*b6 - 8*b4 - 3*b3 + 6*b2 + 3*b1 - 4) * q^62 + (-2*b6 - 12*b5 - b4 + 13*b3 - b2 - 12*b1 - 2) * q^63 + 8*b6 * q^64 + (18*b7 + 9*b6 + 10*b5 + 51*b4 + 18*b3 - 42*b2 + 21) * q^65 + (19*b7 - 4*b6 - 18*b5 + 20*b4 - 8*b3 + 4*b2 - 9*b1 - 12) * q^66 + (10*b7 + 4*b6 + 2*b5 - 4*b4 - 14*b3 + 4*b1 + 12) * q^67 + (-4*b7 - 4*b6 + 4*b5 + 8*b4 + 2*b2 + 16*b1 + 10) * q^68 + (-8*b7 - 30*b6 - 18*b5 + 26*b4 - 4*b3 - 26*b2 + 26*b1 + 30) * q^69 + (-7*b7 + 14*b5 - 6*b4 + 20*b2 + 14*b1 - 6) * q^70 + (19*b7 - 3*b6 - 7*b5 - 33*b4 - 7*b3 - 3*b2 + 19*b1) * q^71 + (2*b7 - 8*b4 + 16*b2 - 2*b1 - 16) * q^72 + (26*b7 + 4*b6 + 5*b4 + 22*b3 + 2*b2 - 22*b1 - 9) * q^73 + (-4*b6 + 3*b5 + 8*b4 - 6*b3 + 8*b2 + 3*b1 - 4) * q^74 + (5*b7 - 39*b6 - 37*b5 + 42*b3 - 66*b2 - 21*b1 + 66) * q^75 + (22*b7 - 14*b6 - 10*b5 - 2*b4 + 22*b3 - 12*b2 + 6) * q^76 + (40*b7 + 2*b6 - 24*b5 + 23*b4 + 4*b3 - 2*b2 - 34*b1 + 39) * q^77 + (-26*b7 + 30*b6 + 13*b5 - 30*b4 + 26*b1 - 2) * q^78 + (-24*b7 - 22*b6 + 24*b5 + 44*b4 + 15*b2 - 17*b1 + 59) * q^79 + (-16*b6 - 8*b5 + 12*b4 - 12*b2 + 8*b1 + 16) * q^80 + (-14*b7 + 28*b5 + 22*b4 - 32*b3 + 24*b2 - 4*b1 + 22) * q^81 + (25*b7 - 16*b6 - 12*b5 + 16*b4 - 12*b3 - 16*b2 + 25*b1) * q^82 + (39*b7 + 31*b6 - 33*b5 - 15*b4 + 33*b3 - b2 - 39*b1 - 30) * q^83 + (-20*b7 + 12*b6 + 6*b4 - 16*b3 + 6*b2 + 16*b1 - 18) * q^84 + (-33*b6 - 22*b5 - 22*b4 - 26*b3 - 22*b2 - 22*b1 - 33) * q^85 + (4*b7 - 12*b6 - 24*b5 + 28*b3 + 8*b2 - 14*b1 - 8) * q^86 + (8*b7 + 21*b6 + 47*b5 - 43*b4 + 8*b3 + 64*b2 - 32) * q^87 + (-12*b7 + 28*b6 + 16*b5 - 8*b4 - 10*b3 + 16*b2 + 8*b1 - 4) * q^88 + (-42*b7 + 66*b6 + 24*b5 - 66*b4 - 6*b3 + 48*b1 - 30) * q^89 + (-28*b7 + 8*b6 + 28*b5 - 16*b4 + 24*b2 - 3*b1 + 8) * q^90 + (-84*b7 + 47*b6 + 39*b5 + 3*b4 - 42*b3 - 3*b2 + 45*b1 - 47) * q^91 + (-8*b7 + 16*b5 - 12*b4 - 24*b3 - 20*b2 - 8*b1 - 12) * q^92 + (-2*b7 - 22*b6 - 10*b5 + 23*b4 - 10*b3 - 22*b2 - 2*b1) * q^93 + (13*b7 - 18*b6 - 6*b5 - 2*b4 + 6*b3 + 22*b2 - 13*b1 - 4) * q^94 + (-15*b7 - 50*b6 + 58*b4 - 29*b3 - 25*b2 + 29*b1 - 8) * q^95 + (-4*b5 - 8*b4 + 8*b3 - 8*b2 - 4*b1) * q^96 + (16*b7 - 12*b6 - 8*b5 + 24*b3 - b2 - 12*b1 + 1) * q^97 + (-25*b7 + 48*b6 + 22*b5 + 8*b4 - 25*b3 + 40*b2 - 20) * q^98 + (33*b7 - 11*b6 - 33*b5 + 22*b4 + 22*b3 + 11*b2 - 33) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{3} + 4 q^{4} + 2 q^{5} - 20 q^{6} - 30 q^{7} - 4 q^{9}+O(q^{10})$$ 8 * q - 2 * q^3 + 4 * q^4 + 2 * q^5 - 20 * q^6 - 30 * q^7 - 4 * q^9 $$8 q - 2 q^{3} + 4 q^{4} + 2 q^{5} - 20 q^{6} - 30 q^{7} - 4 q^{9} - 4 q^{11} + 24 q^{12} + 30 q^{13} + 16 q^{14} + 42 q^{15} - 8 q^{16} + 30 q^{17} + 40 q^{18} - 30 q^{19} - 4 q^{20} + 24 q^{22} - 104 q^{23} - 40 q^{24} - 12 q^{25} - 96 q^{26} - 26 q^{27} - 40 q^{28} - 10 q^{29} - 60 q^{30} + 46 q^{31} - 14 q^{33} + 112 q^{34} + 70 q^{35} - 12 q^{36} + 6 q^{37} + 108 q^{38} + 130 q^{39} + 80 q^{40} + 250 q^{41} + 56 q^{42} - 12 q^{44} - 136 q^{45} - 160 q^{46} - 54 q^{47} - 8 q^{48} - 144 q^{49} - 80 q^{50} - 30 q^{51} - 40 q^{52} - 274 q^{53} - 26 q^{55} + 48 q^{56} - 130 q^{57} + 64 q^{58} + 50 q^{59} + 116 q^{60} + 50 q^{61} + 20 q^{62} - 20 q^{63} + 16 q^{64} - 136 q^{66} + 112 q^{67} + 60 q^{68} + 76 q^{69} + 4 q^{70} + 54 q^{71} - 80 q^{72} - 70 q^{73} - 40 q^{74} + 318 q^{75} + 266 q^{77} + 104 q^{78} + 370 q^{79} + 48 q^{80} + 180 q^{81} - 96 q^{82} - 150 q^{83} - 120 q^{84} - 330 q^{85} - 72 q^{86} + 72 q^{88} + 24 q^{89} + 160 q^{90} - 294 q^{91} - 112 q^{92} - 134 q^{93} - 20 q^{94} - 330 q^{95} - 18 q^{97} - 308 q^{99}+O(q^{100})$$ 8 * q - 2 * q^3 + 4 * q^4 + 2 * q^5 - 20 * q^6 - 30 * q^7 - 4 * q^9 - 4 * q^11 + 24 * q^12 + 30 * q^13 + 16 * q^14 + 42 * q^15 - 8 * q^16 + 30 * q^17 + 40 * q^18 - 30 * q^19 - 4 * q^20 + 24 * q^22 - 104 * q^23 - 40 * q^24 - 12 * q^25 - 96 * q^26 - 26 * q^27 - 40 * q^28 - 10 * q^29 - 60 * q^30 + 46 * q^31 - 14 * q^33 + 112 * q^34 + 70 * q^35 - 12 * q^36 + 6 * q^37 + 108 * q^38 + 130 * q^39 + 80 * q^40 + 250 * q^41 + 56 * q^42 - 12 * q^44 - 136 * q^45 - 160 * q^46 - 54 * q^47 - 8 * q^48 - 144 * q^49 - 80 * q^50 - 30 * q^51 - 40 * q^52 - 274 * q^53 - 26 * q^55 + 48 * q^56 - 130 * q^57 + 64 * q^58 + 50 * q^59 + 116 * q^60 + 50 * q^61 + 20 * q^62 - 20 * q^63 + 16 * q^64 - 136 * q^66 + 112 * q^67 + 60 * q^68 + 76 * q^69 + 4 * q^70 + 54 * q^71 - 80 * q^72 - 70 * q^73 - 40 * q^74 + 318 * q^75 + 266 * q^77 + 104 * q^78 + 370 * q^79 + 48 * q^80 + 180 * q^81 - 96 * q^82 - 150 * q^83 - 120 * q^84 - 330 * q^85 - 72 * q^86 + 72 * q^88 + 24 * q^89 + 160 * q^90 - 294 * q^91 - 112 * q^92 - 134 * q^93 - 20 * q^94 - 330 * q^95 - 18 * q^97 - 308 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 4$$ (v^5) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 8$$ (v^7) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3 $$\nu^{4}$$ $$=$$ $$4\beta_{4}$$ 4*b4 $$\nu^{5}$$ $$=$$ $$4\beta_{5}$$ 4*b5 $$\nu^{6}$$ $$=$$ $$8\beta_{6}$$ 8*b6 $$\nu^{7}$$ $$=$$ $$8\beta_{7}$$ 8*b7

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.3.d.a 8
3.b odd 2 1 198.3.j.a 8
4.b odd 2 1 176.3.n.b 8
11.b odd 2 1 242.3.d.c 8
11.c even 5 1 242.3.b.d 8
11.c even 5 1 242.3.d.c 8
11.c even 5 1 242.3.d.d 8
11.c even 5 1 242.3.d.e 8
11.d odd 10 1 inner 22.3.d.a 8
11.d odd 10 1 242.3.b.d 8
11.d odd 10 1 242.3.d.d 8
11.d odd 10 1 242.3.d.e 8
33.f even 10 1 198.3.j.a 8
33.f even 10 1 2178.3.d.l 8
33.h odd 10 1 2178.3.d.l 8
44.g even 10 1 176.3.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.d.a 8 1.a even 1 1 trivial
22.3.d.a 8 11.d odd 10 1 inner
176.3.n.b 8 4.b odd 2 1
176.3.n.b 8 44.g even 10 1
198.3.j.a 8 3.b odd 2 1
198.3.j.a 8 33.f even 10 1
242.3.b.d 8 11.c even 5 1
242.3.b.d 8 11.d odd 10 1
242.3.d.c 8 11.b odd 2 1
242.3.d.c 8 11.c even 5 1
242.3.d.d 8 11.c even 5 1
242.3.d.d 8 11.d odd 10 1
242.3.d.e 8 11.c even 5 1
242.3.d.e 8 11.d odd 10 1
2178.3.d.l 8 33.f even 10 1
2178.3.d.l 8 33.h odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(22, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + \cdots + 16$$
$3$ $$T^{8} + 2 T^{7} + 13 T^{6} - 16 T^{5} + \cdots + 841$$
$5$ $$T^{8} - 2 T^{7} + 33 T^{6} + \cdots + 57121$$
$7$ $$T^{8} + 30 T^{7} + 473 T^{6} + \cdots + 192721$$
$11$ $$T^{8} + 4 T^{7} - 484 T^{5} + \cdots + 214358881$$
$13$ $$T^{8} - 30 T^{7} + \cdots + 2268521641$$
$17$ $$T^{8} - 30 T^{7} + 97 T^{6} + \cdots + 22934521$$
$19$ $$T^{8} + 30 T^{7} + \cdots + 3189877441$$
$23$ $$(T^{4} + 52 T^{3} + 124 T^{2} + \cdots - 90224)^{2}$$
$29$ $$T^{8} + 10 T^{7} + \cdots + 206760274681$$
$31$ $$T^{8} - 46 T^{7} + \cdots + 996728041$$
$37$ $$T^{8} - 6 T^{7} + 317 T^{6} + \cdots + 20079361$$
$41$ $$T^{8} - 250 T^{7} + \cdots + 405257161$$
$43$ $$T^{8} + 3632 T^{6} + \cdots + 453519616$$
$47$ $$T^{8} + 54 T^{7} + \cdots + 7428543721$$
$53$ $$T^{8} + \cdots + 189900554154481$$
$59$ $$T^{8} - 50 T^{7} + \cdots + 47196831300025$$
$61$ $$T^{8} - 50 T^{7} + \cdots + 4097365398025$$
$67$ $$(T^{4} - 56 T^{3} - 344 T^{2} + \cdots + 112576)^{2}$$
$71$ $$T^{8} - 54 T^{7} + \cdots + 14204072031241$$
$73$ $$T^{8} + 70 T^{7} + \cdots + 36635874025$$
$79$ $$T^{8} - 370 T^{7} + \cdots + 33\!\cdots\!41$$
$83$ $$T^{8} + 150 T^{7} + \cdots + 4768279282321$$
$89$ $$(T^{4} - 12 T^{3} - 19836 T^{2} + \cdots - 22808304)^{2}$$
$97$ $$T^{8} + 18 T^{7} + \cdots + 185279454481$$