Embedded newform 22.8.c.a.3.2

• Label: 22.8.c.a.3.2
• Level: $22$
• Weight: $8$
• Character: 22.3
• Analytic conductor: $6.872$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87247056065\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 2*x^11 + 718*x^10 - 5886*x^9 + 371698*x^8 + 6317266*x^7 + 171079947*x^6 + 3212289784*x^5 + 91377152203*x^4 + 810233740626*x^3 + 7721283816133*x^2 + 66377062242562*x + 475277889155161
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4}\cdot 11^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.2
Root			\(-13.3823 - 9.72279i\) of defining polynomial
Character	\(\chi\)	\(=\)	22.3
Dual form			22.8.c.a.15.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.47214 + 4.70228i) q^{2} +(3.49129 + 10.7451i) q^{3} +(19.7771 - 60.8676i) q^{4} +(-150.057 - 109.023i) q^{5} +(-73.1224 - 53.1266i) q^{6} +(47.2510 - 145.424i) q^{7} +(158.217 + 486.941i) q^{8} +(1666.05 - 1210.46i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{2} + 44 q^{3} - 192 q^{4} + 220 q^{5} - 248 q^{6} + 534 q^{7} - 1536 q^{8} - 3745 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.47214	+	4.70228i	−0.572061	+	0.415627i
							\(3\)	3.49129	+	10.7451i	0.0746554	+	0.229766i	0.981420	−	0.191871i	\(-0.0614557\pi\)
−0.906765	+	0.421637i	\(0.861456\pi\)
\(5\)	−150.057	−	109.023i	−0.536859	−	0.390051i	0.286058	−	0.958212i	\(-0.407655\pi\)
							−0.822917	+	0.568161i	\(0.807655\pi\)
\(7\)	47.2510	−	145.424i	0.0520677	−	0.160248i	−0.921642	−	0.388042i	\(-0.873151\pi\)
							0.973709	+	0.227795i	\(0.0731514\pi\)
\(11\)	3469.33	−	2729.63i	0.785908	−	0.618344i
							\(13\)	7766.45	−	5642.66i	0.980440	−	0.712331i	0.0226328	−	0.999744i	\(-0.492795\pi\)
0.957807	+	0.287413i	\(0.0927951\pi\)
\(17\)	488.662	+	355.033i	0.0241233	+	0.0175266i	0.599782	−	0.800164i	\(-0.295254\pi\)
							−0.575658	+	0.817690i	\(0.695254\pi\)
\(19\)	−3688.57	−	11352.2i	−0.123373	−	0.379703i	0.870228	−	0.492649i	\(-0.163971\pi\)
							−0.993601	+	0.112946i	\(0.963971\pi\)
\(23\)	−73066.0			−1.25218			−0.626091	−	0.779750i	\(-0.715346\pi\)
							−0.626091	+	0.779750i	\(0.715346\pi\)
\(29\)	−192.044	+	591.050i	−0.00146220	+	0.00450019i	−0.951785	−	0.306766i	\(-0.900753\pi\)
							0.950323	+	0.311266i	\(0.100753\pi\)
\(31\)	81230.9	−	59017.7i	0.489728	−	0.355809i	−0.315351	−	0.948975i	\(-0.602122\pi\)
							0.805080	+	0.593166i	\(0.202122\pi\)
\(37\)	19928.6	−	61333.8i	0.0646800	−	0.199064i	−0.913494	−	0.406852i	\(-0.866626\pi\)
							0.978174	+	0.207788i	\(0.0666264\pi\)
\(41\)	−133137.	−	409755.i	−0.301687	−	0.928497i	−0.980893	−	0.194549i	\(-0.937676\pi\)
							0.679206	−	0.733948i	\(-0.262324\pi\)
\(43\)	369626.			0.708962			0.354481	−	0.935063i	\(-0.384658\pi\)
							0.354481	+	0.935063i	\(0.384658\pi\)
\(47\)	204462.	+	629269.i	0.287256	+	0.884084i	0.985713	+	0.168432i	\(0.0538704\pi\)
							−0.698457	+	0.715652i	\(0.746130\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.8.c.a.3.2	✓	12
11.2	odd	10		242.8.a.o.1.4		6
11.4	even	5	inner	22.8.c.a.15.2	yes	12
11.9	even	5		242.8.a.q.1.4		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.8.c.a.3.2	✓	12	1.1	even	1	trivial
22.8.c.a.15.2	yes	12	11.4	even	5	inner
242.8.a.o.1.4		6	11.2	odd	10
242.8.a.q.1.4		6	11.9	even	5