Embedded newform 402.2.n.a.11.6

• Label: 402.2.n.a.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	402.11
Dual form			402.2.n.a.329.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.327068 + 0.945001i) q^{2} +(0.637847 - 1.61033i) q^{3} +(-0.786053 + 0.618159i) q^{4} +(3.32765 - 0.977087i) q^{5} +(1.73038 + 0.0760800i) q^{6} +(-0.598560 + 3.10562i) q^{7} +(-0.841254 - 0.540641i) q^{8} +(-2.18630 - 2.05428i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 11 q^{2} + q^{3} + 11 q^{4} + 2 q^{6} + 6 q^{7} + 22 q^{8} - q^{9}+O(q^{10}) \)

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
							\(3\)	0.637847	−	1.61033i	0.368261	−	0.929722i
\(5\)	3.32765	−	0.977087i	1.48817	−	0.436967i								0.566214	−	0.824258i	\(-0.308407\pi\)
							0.921957	+	0.387291i	\(0.126589\pi\)
\(7\)	−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	\(0.0970528\pi\)
							0.345245	+	0.938512i	\(0.387796\pi\)
\(13\)	−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	\(-0.563653\pi\)
							0.999277	−	0.0380185i	\(-0.0121046\pi\)
\(19\)	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	\(-0.641822\pi\)
							0.593939	−	0.804510i	\(-0.297572\pi\)
\(29\)	−5.07675	+	2.93106i	−0.942728	+	0.544284i	−0.890815	−	0.454367i	\(-0.849865\pi\)
							−0.0519138	+	0.998652i	\(0.516532\pi\)
\(31\)	−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	−0.941990	−	0.335641i	\(-0.891047\pi\)
							−0.450131	+	0.892962i	\(0.648623\pi\)
\(43\)	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	0.625218	−	0.780450i	\(-0.285010\pi\)
							−0.533037	+	0.846092i	\(0.678949\pi\)

(See \(a_n\) instead)

Twists

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

    

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