# Properties

 Label 22.3.b.a Level $22$ Weight $3$ Character orbit 22.b Analytic conductor $0.599$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 22.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.599456581593$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} - 2 q^{4} - q^{5} + \beta q^{6} - 6 \beta q^{7} - 2 \beta q^{8} - 8 q^{9} +O(q^{10})$$ q + b * q^2 + q^3 - 2 * q^4 - q^5 + b * q^6 - 6*b * q^7 - 2*b * q^8 - 8 * q^9 $$q + \beta q^{2} + q^{3} - 2 q^{4} - q^{5} + \beta q^{6} - 6 \beta q^{7} - 2 \beta q^{8} - 8 q^{9} - \beta q^{10} + (6 \beta + 7) q^{11} - 2 q^{12} + 6 \beta q^{13} + 12 q^{14} - q^{15} + 4 q^{16} + 18 \beta q^{17} - 8 \beta q^{18} - 18 \beta q^{19} + 2 q^{20} - 6 \beta q^{21} + (7 \beta - 12) q^{22} + 17 q^{23} - 2 \beta q^{24} - 24 q^{25} - 12 q^{26} - 17 q^{27} + 12 \beta q^{28} - 24 \beta q^{29} - \beta q^{30} + 17 q^{31} + 4 \beta q^{32} + (6 \beta + 7) q^{33} - 36 q^{34} + 6 \beta q^{35} + 16 q^{36} + 47 q^{37} + 36 q^{38} + 6 \beta q^{39} + 2 \beta q^{40} + 6 \beta q^{41} + 12 q^{42} + 12 \beta q^{43} + ( - 12 \beta - 14) q^{44} + 8 q^{45} + 17 \beta q^{46} - 58 q^{47} + 4 q^{48} - 23 q^{49} - 24 \beta q^{50} + 18 \beta q^{51} - 12 \beta q^{52} + 2 q^{53} - 17 \beta q^{54} + ( - 6 \beta - 7) q^{55} - 24 q^{56} - 18 \beta q^{57} + 48 q^{58} - 55 q^{59} + 2 q^{60} - 60 \beta q^{61} + 17 \beta q^{62} + 48 \beta q^{63} - 8 q^{64} - 6 \beta q^{65} + (7 \beta - 12) q^{66} + 89 q^{67} - 36 \beta q^{68} + 17 q^{69} - 12 q^{70} - 7 q^{71} + 16 \beta q^{72} + 90 \beta q^{73} + 47 \beta q^{74} - 24 q^{75} + 36 \beta q^{76} + ( - 42 \beta + 72) q^{77} - 12 q^{78} - 24 \beta q^{79} - 4 q^{80} + 55 q^{81} - 12 q^{82} - 24 \beta q^{83} + 12 \beta q^{84} - 18 \beta q^{85} - 24 q^{86} - 24 \beta q^{87} + ( - 14 \beta + 24) q^{88} - 97 q^{89} + 8 \beta q^{90} + 72 q^{91} - 34 q^{92} + 17 q^{93} - 58 \beta q^{94} + 18 \beta q^{95} + 4 \beta q^{96} - 121 q^{97} - 23 \beta q^{98} + ( - 48 \beta - 56) q^{99} +O(q^{100})$$ q + b * q^2 + q^3 - 2 * q^4 - q^5 + b * q^6 - 6*b * q^7 - 2*b * q^8 - 8 * q^9 - b * q^10 + (6*b + 7) * q^11 - 2 * q^12 + 6*b * q^13 + 12 * q^14 - q^15 + 4 * q^16 + 18*b * q^17 - 8*b * q^18 - 18*b * q^19 + 2 * q^20 - 6*b * q^21 + (7*b - 12) * q^22 + 17 * q^23 - 2*b * q^24 - 24 * q^25 - 12 * q^26 - 17 * q^27 + 12*b * q^28 - 24*b * q^29 - b * q^30 + 17 * q^31 + 4*b * q^32 + (6*b + 7) * q^33 - 36 * q^34 + 6*b * q^35 + 16 * q^36 + 47 * q^37 + 36 * q^38 + 6*b * q^39 + 2*b * q^40 + 6*b * q^41 + 12 * q^42 + 12*b * q^43 + (-12*b - 14) * q^44 + 8 * q^45 + 17*b * q^46 - 58 * q^47 + 4 * q^48 - 23 * q^49 - 24*b * q^50 + 18*b * q^51 - 12*b * q^52 + 2 * q^53 - 17*b * q^54 + (-6*b - 7) * q^55 - 24 * q^56 - 18*b * q^57 + 48 * q^58 - 55 * q^59 + 2 * q^60 - 60*b * q^61 + 17*b * q^62 + 48*b * q^63 - 8 * q^64 - 6*b * q^65 + (7*b - 12) * q^66 + 89 * q^67 - 36*b * q^68 + 17 * q^69 - 12 * q^70 - 7 * q^71 + 16*b * q^72 + 90*b * q^73 + 47*b * q^74 - 24 * q^75 + 36*b * q^76 + (-42*b + 72) * q^77 - 12 * q^78 - 24*b * q^79 - 4 * q^80 + 55 * q^81 - 12 * q^82 - 24*b * q^83 + 12*b * q^84 - 18*b * q^85 - 24 * q^86 - 24*b * q^87 + (-14*b + 24) * q^88 - 97 * q^89 + 8*b * q^90 + 72 * q^91 - 34 * q^92 + 17 * q^93 - 58*b * q^94 + 18*b * q^95 + 4*b * q^96 - 121 * q^97 - 23*b * q^98 + (-48*b - 56) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{4} - 2 q^{5} - 16 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 4 * q^4 - 2 * q^5 - 16 * q^9 $$2 q + 2 q^{3} - 4 q^{4} - 2 q^{5} - 16 q^{9} + 14 q^{11} - 4 q^{12} + 24 q^{14} - 2 q^{15} + 8 q^{16} + 4 q^{20} - 24 q^{22} + 34 q^{23} - 48 q^{25} - 24 q^{26} - 34 q^{27} + 34 q^{31} + 14 q^{33} - 72 q^{34} + 32 q^{36} + 94 q^{37} + 72 q^{38} + 24 q^{42} - 28 q^{44} + 16 q^{45} - 116 q^{47} + 8 q^{48} - 46 q^{49} + 4 q^{53} - 14 q^{55} - 48 q^{56} + 96 q^{58} - 110 q^{59} + 4 q^{60} - 16 q^{64} - 24 q^{66} + 178 q^{67} + 34 q^{69} - 24 q^{70} - 14 q^{71} - 48 q^{75} + 144 q^{77} - 24 q^{78} - 8 q^{80} + 110 q^{81} - 24 q^{82} - 48 q^{86} + 48 q^{88} - 194 q^{89} + 144 q^{91} - 68 q^{92} + 34 q^{93} - 242 q^{97} - 112 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 4 * q^4 - 2 * q^5 - 16 * q^9 + 14 * q^11 - 4 * q^12 + 24 * q^14 - 2 * q^15 + 8 * q^16 + 4 * q^20 - 24 * q^22 + 34 * q^23 - 48 * q^25 - 24 * q^26 - 34 * q^27 + 34 * q^31 + 14 * q^33 - 72 * q^34 + 32 * q^36 + 94 * q^37 + 72 * q^38 + 24 * q^42 - 28 * q^44 + 16 * q^45 - 116 * q^47 + 8 * q^48 - 46 * q^49 + 4 * q^53 - 14 * q^55 - 48 * q^56 + 96 * q^58 - 110 * q^59 + 4 * q^60 - 16 * q^64 - 24 * q^66 + 178 * q^67 + 34 * q^69 - 24 * q^70 - 14 * q^71 - 48 * q^75 + 144 * q^77 - 24 * q^78 - 8 * q^80 + 110 * q^81 - 24 * q^82 - 48 * q^86 + 48 * q^88 - 194 * q^89 + 144 * q^91 - 68 * q^92 + 34 * q^93 - 242 * q^97 - 112 * q^99

## Character values

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

 $$n$$ $$13$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1
 − 1.41421i 1.41421i
1.41421i 1.00000 −2.00000 −1.00000 1.41421i 8.48528i 2.82843i −8.00000 1.41421i
21.2 1.41421i 1.00000 −2.00000 −1.00000 1.41421i 8.48528i 2.82843i −8.00000 1.41421i
 $$n$$: e.g. 2-40 or 990-1000 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.3.b.a 2
3.b odd 2 1 198.3.d.b 2
4.b odd 2 1 176.3.h.c 2
5.b even 2 1 550.3.d.a 2
5.c odd 4 2 550.3.c.a 4
7.b odd 2 1 1078.3.d.a 2
8.b even 2 1 704.3.h.d 2
8.d odd 2 1 704.3.h.e 2
11.b odd 2 1 inner 22.3.b.a 2
11.c even 5 4 242.3.d.b 8
11.d odd 10 4 242.3.d.b 8
12.b even 2 1 1584.3.j.d 2
33.d even 2 1 198.3.d.b 2
44.c even 2 1 176.3.h.c 2
55.d odd 2 1 550.3.d.a 2
55.e even 4 2 550.3.c.a 4
77.b even 2 1 1078.3.d.a 2
88.b odd 2 1 704.3.h.d 2
88.g even 2 1 704.3.h.e 2
132.d odd 2 1 1584.3.j.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.b.a 2 1.a even 1 1 trivial
22.3.b.a 2 11.b odd 2 1 inner
176.3.h.c 2 4.b odd 2 1
176.3.h.c 2 44.c even 2 1
198.3.d.b 2 3.b odd 2 1
198.3.d.b 2 33.d even 2 1
242.3.d.b 8 11.c even 5 4
242.3.d.b 8 11.d odd 10 4
550.3.c.a 4 5.c odd 4 2
550.3.c.a 4 55.e even 4 2
550.3.d.a 2 5.b even 2 1
550.3.d.a 2 55.d odd 2 1
704.3.h.d 2 8.b even 2 1
704.3.h.d 2 88.b odd 2 1
704.3.h.e 2 8.d odd 2 1
704.3.h.e 2 88.g even 2 1
1078.3.d.a 2 7.b odd 2 1
1078.3.d.a 2 77.b even 2 1
1584.3.j.d 2 12.b even 2 1
1584.3.j.d 2 132.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 72$$
$11$ $$T^{2} - 14T + 121$$
$13$ $$T^{2} + 72$$
$17$ $$T^{2} + 648$$
$19$ $$T^{2} + 648$$
$23$ $$(T - 17)^{2}$$
$29$ $$T^{2} + 1152$$
$31$ $$(T - 17)^{2}$$
$37$ $$(T - 47)^{2}$$
$41$ $$T^{2} + 72$$
$43$ $$T^{2} + 288$$
$47$ $$(T + 58)^{2}$$
$53$ $$(T - 2)^{2}$$
$59$ $$(T + 55)^{2}$$
$61$ $$T^{2} + 7200$$
$67$ $$(T - 89)^{2}$$
$71$ $$(T + 7)^{2}$$
$73$ $$T^{2} + 16200$$
$79$ $$T^{2} + 1152$$
$83$ $$T^{2} + 1152$$
$89$ $$(T + 97)^{2}$$
$97$ $$(T + 121)^{2}$$