# 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: $$x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16$$ 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.

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

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

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

## 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 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

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4} - 8 T^{6} + 16 T^{8}$$
$3$ $$1 + 2 T - 5 T^{2} + 20 T^{3} + 45 T^{4} - 284 T^{5} + 837 T^{6} + 3350 T^{7} - 3020 T^{8} + 30150 T^{9} + 67797 T^{10} - 207036 T^{11} + 295245 T^{12} + 1180980 T^{13} - 2657205 T^{14} + 9565938 T^{15} + 43046721 T^{16}$$
$5$ $$1 - 2 T - 17 T^{2} + 156 T^{3} - 385 T^{4} - 7664 T^{5} + 29213 T^{6} + 102118 T^{7} - 966104 T^{8} + 2552950 T^{9} + 18258125 T^{10} - 119750000 T^{11} - 150390625 T^{12} + 1523437500 T^{13} - 4150390625 T^{14} - 12207031250 T^{15} + 152587890625 T^{16}$$
$7$ $$1 + 30 T + 571 T^{2} + 8260 T^{3} + 97845 T^{4} + 993580 T^{5} + 8888309 T^{6} + 71468530 T^{7} + 524056964 T^{8} + 3501957970 T^{9} + 21340829909 T^{10} + 116893693420 T^{11} + 564056953845 T^{12} + 2333245556740 T^{13} + 7903374991771 T^{14} + 20346692185470 T^{15} + 33232930569601 T^{16}$$
$11$ $$1 + 4 T - 484 T^{3} - 7986 T^{4} - 58564 T^{5} + 7086244 T^{7} + 214358881 T^{8}$$
$13$ $$1 - 30 T + 651 T^{2} - 6800 T^{3} + 48375 T^{4} + 238980 T^{5} - 3389311 T^{6} + 64452890 T^{7} - 51252296 T^{8} + 10892538410 T^{9} - 96802111471 T^{10} + 1153510814820 T^{11} + 39460973628375 T^{12} - 937437744573200 T^{13} + 15167053414735131 T^{14} - 118121291570978670 T^{15} + 665416609183179841 T^{16}$$
$17$ $$1 - 30 T + 675 T^{2} - 9160 T^{3} + 193599 T^{4} - 3443220 T^{5} + 74881625 T^{6} - 864455750 T^{7} + 15090566536 T^{8} - 249827711750 T^{9} + 6254188201625 T^{10} - 83110960332180 T^{11} + 1350499664820159 T^{12} - 18466504128112840 T^{13} + 393270010130088675 T^{14} - 5051334796782027870 T^{15} + 48661191875666868481 T^{16}$$
$19$ $$1 + 30 T + 999 T^{2} + 20440 T^{3} + 415005 T^{4} + 6646680 T^{5} + 145579961 T^{6} + 3061994690 T^{7} + 58504536724 T^{8} + 1105380083090 T^{9} + 18972126097481 T^{10} + 312698916325080 T^{11} + 7048263579830205 T^{12} + 125318994309452440 T^{13} + 2211101604147094839 T^{14} + 23970200573486523630 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$( 1 + 52 T + 2240 T^{2} + 63372 T^{3} + 1720014 T^{4} + 33523788 T^{5} + 626843840 T^{6} + 7697866228 T^{7} + 78310985281 T^{8} )^{2}$$
$29$ $$1 + 10 T + 1515 T^{2} - 62640 T^{3} - 142561 T^{4} - 111518460 T^{5} + 1681998945 T^{6} - 18301053550 T^{7} + 3803814204136 T^{8} - 15391186035550 T^{9} + 1189645895818545 T^{10} - 66333780730005660 T^{11} - 71315628878133121 T^{12} - 26353101093924590640 T^{13} +$$$$53\!\cdots\!15$$$$T^{14} +$$$$29\!\cdots\!10$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$1 - 46 T - 815 T^{2} + 85900 T^{3} - 728375 T^{4} + 2317612 T^{5} - 1257533057 T^{6} - 34712738410 T^{7} + 3877854489500 T^{8} - 33358941612010 T^{9} - 1161358186333697 T^{10} + 2056889181129772 T^{11} - 621224509396088375 T^{12} + 70406069851650805900 T^{13} -$$$$64\!\cdots\!15$$$$T^{14} -$$$$34\!\cdots\!66$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 - 6 T - 2421 T^{2} - 14136 T^{3} + 5054271 T^{4} - 6253908 T^{5} - 8619590399 T^{6} + 1373683362 T^{7} + 13772803179432 T^{8} + 1880572522578 T^{9} - 16154500161780239 T^{10} - 16045816915056372 T^{11} + 17753023042048746591 T^{12} - 67974148688498713464 T^{13} -$$$$15\!\cdots\!01$$$$T^{14} -$$$$54\!\cdots\!34$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$1 - 250 T + 27975 T^{2} - 1814420 T^{3} + 77101799 T^{4} - 2779318000 T^{5} + 139016689365 T^{6} - 8505740551010 T^{7} + 413536754728696 T^{8} - 14298149866247810 T^{9} + 392827939156731765 T^{10} - 13202050218887638000 T^{11} +$$$$61\!\cdots\!79$$$$T^{12} -$$$$24\!\cdots\!20$$$$T^{13} +$$$$63\!\cdots\!75$$$$T^{14} -$$$$94\!\cdots\!50$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$1 - 11160 T^{2} + 57008924 T^{4} - 179309232040 T^{6} + 390998083541766 T^{8} - 613022581807584040 T^{10} +$$$$66\!\cdots\!24$$$$T^{12} -$$$$44\!\cdots\!60$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 54 T - 1941 T^{2} - 123836 T^{3} + 6739581 T^{4} + 148592292 T^{5} - 27255791899 T^{6} - 305251818238 T^{7} + 49111755331252 T^{8} - 674301266487742 T^{9} - 132999569869504219 T^{10} + 1601708311697644068 T^{11} +$$$$16\!\cdots\!41$$$$T^{12} -$$$$65\!\cdots\!64$$$$T^{13} -$$$$22\!\cdots\!81$$$$T^{14} +$$$$13\!\cdots\!26$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 + 274 T + 38659 T^{2} + 3972344 T^{3} + 334778231 T^{4} + 23878670812 T^{5} + 1495056085961 T^{6} + 86289965064362 T^{7} + 4702527071102392 T^{8} + 242388511865792858 T^{9} + 11796711640209637241 T^{10} +$$$$52\!\cdots\!48$$$$T^{11} +$$$$20\!\cdots\!91$$$$T^{12} +$$$$69\!\cdots\!56$$$$T^{13} +$$$$18\!\cdots\!19$$$$T^{14} +$$$$37\!\cdots\!06$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$1 - 50 T + 1263 T^{2} + 70200 T^{3} - 2470907 T^{4} + 1076735200 T^{5} - 12586111599 T^{6} + 433998162250 T^{7} + 92677785681540 T^{8} + 1510747602792250 T^{9} - 152510457831370239 T^{10} + 45417265326048863200 T^{11} -$$$$36\!\cdots\!47$$$$T^{12} +$$$$35\!\cdots\!00$$$$T^{13} +$$$$22\!\cdots\!03$$$$T^{14} -$$$$30\!\cdots\!50$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$1 - 50 T + 10707 T^{2} - 429000 T^{3} + 36206983 T^{4} - 875034700 T^{5} - 30917305831 T^{6} + 2941568049750 T^{7} - 516150507559720 T^{8} + 10945574713119750 T^{9} - 428076100684398871 T^{10} - 45082115322865326700 T^{11} +$$$$69\!\cdots\!23$$$$T^{12} -$$$$30\!\cdots\!00$$$$T^{13} +$$$$28\!\cdots\!47$$$$T^{14} -$$$$49\!\cdots\!50$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$( 1 - 56 T + 17612 T^{2} - 732008 T^{3} + 117930870 T^{4} - 3285983912 T^{5} + 354901543052 T^{6} - 5065669401464 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$1 - 54 T - 9075 T^{2} + 1051560 T^{3} - 949335 T^{4} - 6026869752 T^{5} + 391790178883 T^{6} + 11973624475590 T^{7} - 2665107650348580 T^{8} + 60359040981449190 T^{9} + 9956047044707732323 T^{10} -$$$$77\!\cdots\!92$$$$T^{11} -$$$$61\!\cdots\!35$$$$T^{12} +$$$$34\!\cdots\!60$$$$T^{13} -$$$$14\!\cdots\!75$$$$T^{14} -$$$$44\!\cdots\!74$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16}$$
$73$ $$1 + 70 T + 14643 T^{2} + 877280 T^{3} + 86819703 T^{4} + 4301905820 T^{5} + 83475543561 T^{6} + 7156391135710 T^{7} - 896454269626600 T^{8} + 38136408362198590 T^{9} + 2370558603651276201 T^{10} +$$$$65\!\cdots\!80$$$$T^{11} +$$$$70\!\cdots\!43$$$$T^{12} +$$$$37\!\cdots\!20$$$$T^{13} +$$$$33\!\cdots\!03$$$$T^{14} +$$$$85\!\cdots\!30$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$1 - 370 T + 69785 T^{2} - 8224700 T^{3} + 599308969 T^{4} - 15803939780 T^{5} - 2371050090825 T^{6} + 423925649678930 T^{7} - 40585609775890084 T^{8} + 2645719979646202130 T^{9} - 92352593092691106825 T^{10} -$$$$38\!\cdots\!80$$$$T^{11} +$$$$90\!\cdots\!09$$$$T^{12} -$$$$77\!\cdots\!00$$$$T^{13} +$$$$41\!\cdots\!85$$$$T^{14} -$$$$13\!\cdots\!70$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$1 + 150 T + 16661 T^{2} + 847320 T^{3} + 56225865 T^{4} + 2891415120 T^{5} + 366983457739 T^{6} - 42655302510 T^{7} + 368491437191804 T^{8} - 293852378991390 T^{9} + 17416418739067396219 T^{10} +$$$$94\!\cdots\!80$$$$T^{11} +$$$$12\!\cdots\!65$$$$T^{12} +$$$$13\!\cdots\!80$$$$T^{13} +$$$$17\!\cdots\!21$$$$T^{14} +$$$$11\!\cdots\!50$$$$T^{15} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$( 1 - 12 T + 11848 T^{2} + 1140876 T^{3} + 39403230 T^{4} + 9036878796 T^{5} + 743370071368 T^{6} - 5963775491532 T^{7} + 3936588805702081 T^{8} )^{2}$$
$97$ $$1 + 18 T - 17345 T^{2} - 545820 T^{3} + 224453335 T^{4} + 2169903904 T^{5} - 2647455520643 T^{6} - 4734820613590 T^{7} + 27980585557864280 T^{8} - 44549927153268310 T^{9} -$$$$23\!\cdots\!83$$$$T^{10} +$$$$18\!\cdots\!16$$$$T^{11} +$$$$17\!\cdots\!35$$$$T^{12} -$$$$40\!\cdots\!80$$$$T^{13} -$$$$12\!\cdots\!45$$$$T^{14} +$$$$11\!\cdots\!42$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$