Embedded newform 402.2.n.b.11.10

• Label: 402.2.n.b.11.10
• Level: $402$
• Weight: $2$
• Character: 402.11
• Analytic conductor: $3.210$
• Analytic rank: $0$
• Dimension: $220$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	402.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.20998616126\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			11.10
Character	\(\chi\)	\(=\)	402.11
Dual form			402.2.n.b.329.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.327068 - 0.945001i) q^{2} +(1.63470 + 0.572487i) q^{3} +(-0.786053 + 0.618159i) q^{4} +(-3.32765 + 0.977087i) q^{5} +(0.00634093 - 1.73204i) q^{6} +(-0.598560 + 3.10562i) q^{7} +(0.841254 + 0.540641i) q^{8} +(2.34452 + 1.87169i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 11 q^{2} - q^{3} + 11 q^{4} - q^{6} + 6 q^{7} - 22 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/402\mathbb{Z}\right)^\times\).

\(n\)	\(269\)	\(337\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{59}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.327068	−	0.945001i	−0.231272	−	0.668216i
							\(3\)	1.63470	+	0.572487i	0.943797	+	0.330525i
\(5\)	−3.32765	+	0.977087i	−1.48817	+	0.436967i								−0.921957	−	0.387291i	\(-0.873411\pi\)
							−0.566214	+	0.824258i	\(0.691593\pi\)
\(7\)	−0.598560	+	3.10562i	−0.226234	+	1.17382i	0.674108	+	0.738633i	\(0.264528\pi\)
							−0.900342	+	0.435183i	\(0.856684\pi\)
\(11\)	−4.30870	−	4.10834i	−1.29912	−	1.23871i	−0.953877	−	0.300198i	\(-0.902947\pi\)
							−0.345245	−	0.938512i	\(-0.612204\pi\)
\(13\)	−3.17852	+	0.151412i	−0.881564	+	0.0419941i	−0.483481	−	0.875355i	\(-0.660628\pi\)
							−0.398084	+	0.917349i	\(0.630325\pi\)
\(17\)	−3.30110	+	4.19770i	−0.800635	+	1.01809i	0.198642	+	0.980072i	\(0.436347\pi\)
							−0.999277	+	0.0380185i	\(0.987895\pi\)
\(19\)	0.991550	−	0.191106i	0.227477	−	0.0438426i	−0.0742388	−	0.997240i	\(-0.523653\pi\)
							0.301716	+	0.953398i	\(0.402441\pi\)
\(23\)	−0.781662	+	8.18593i	−0.162988	+	1.70689i	0.430951	+	0.902375i	\(0.358178\pi\)
							−0.593939	+	0.804510i	\(0.702428\pi\)
\(29\)	5.07675	−	2.93106i	0.942728	−	0.544284i	0.0519138	−	0.998652i	\(-0.483468\pi\)
							0.890815	+	0.454367i	\(0.150135\pi\)
\(31\)	−2.05909	−	0.0980865i	−0.369823	−	0.0176169i	−0.138152	−	0.990411i	\(-0.544116\pi\)
							−0.231671	+	0.972794i	\(0.574419\pi\)
\(37\)	−1.31386	+	2.27567i	−0.215997	+	0.374118i	−0.953581	−	0.301138i	\(-0.902634\pi\)
							0.737583	+	0.675256i	\(0.235967\pi\)
\(41\)	8.91393	−	3.56860i	1.39212	−	0.557321i	0.450131	−	0.892962i	\(-0.351377\pi\)
							0.941990	+	0.335641i	\(0.108953\pi\)
\(43\)	4.54379	+	0.653298i	0.692921	+	0.0996270i	0.479773	−	0.877393i	\(-0.340719\pi\)
							0.213148	+	0.977020i	\(0.431628\pi\)
\(47\)	−0.631966	−	0.450021i	−0.0921817	−	0.0656423i	0.533037	−	0.846092i	\(-0.321051\pi\)
							−0.625218	+	0.780450i	\(0.714990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.b.11.10	yes	220
3.2	odd	2		402.2.n.a.11.6	✓	220
67.61	odd	66		402.2.n.a.329.6	yes	220
201.128	even	66	inner	402.2.n.b.329.10	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.11.6	✓	220	3.2	odd	2
402.2.n.a.329.6	yes	220	67.61	odd	66
402.2.n.b.11.10	yes	220	1.1	even	1	trivial
402.2.n.b.329.10	yes	220	201.128	even	66	inner