Embedded newform 402.2.n.b.11.7

• Label: 402.2.n.b.11.7
• 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			11.7
Character	\(\chi\)	\(=\)	402.11
Dual form			402.2.n.b.329.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.327068 - 0.945001i) q^{2} +(1.33877 + 1.09896i) q^{3} +(-0.786053 + 0.618159i) q^{4} +(-1.51180 + 0.443905i) q^{5} +(0.600648 - 1.62457i) q^{6} +(0.543896 - 2.82200i) q^{7} +(0.841254 + 0.540641i) q^{8} +(0.584588 + 2.94249i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 11 q^{2} - q^{3} + 11 q^{4} - q^{6} + 6 q^{7} - 22 q^{8} - q^{9}+O(q^{10}) \)

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
							\(3\)	1.33877	+	1.09896i	0.772937	+	0.634483i
\(5\)	−1.51180	+	0.443905i	−0.676098	+	0.198520i								−0.601717	−	0.798709i	\(-0.705516\pi\)
							−0.0743815	+	0.997230i	\(0.523698\pi\)
\(7\)	0.543896	−	2.82200i	0.205573	−	1.06662i	−0.721693	−	0.692213i	\(-0.756636\pi\)
							0.927266	−	0.374402i	\(-0.122152\pi\)
\(11\)	3.41791	+	3.25897i	1.03054	+	0.982617i	0.999857	−	0.0169125i	\(-0.00538367\pi\)
							0.0306818	+	0.999529i	\(0.490232\pi\)
\(13\)	3.72072	−	0.177240i	1.03194	−	0.0491575i	0.475229	−	0.879862i	\(-0.342365\pi\)
							0.556713	+	0.830705i	\(0.312062\pi\)
\(17\)	2.27730	−	2.89582i	0.552327	−	0.702340i	−0.427086	−	0.904211i	\(-0.640460\pi\)
							0.979412	+	0.201871i	\(0.0647021\pi\)
\(19\)	1.80255	−	0.347414i	0.413534	−	0.0797022i	0.0217593	−	0.999763i	\(-0.493073\pi\)
							0.391775	+	0.920061i	\(0.371861\pi\)
\(23\)	−0.483130	+	5.05957i	−0.100740	+	1.05499i	0.793853	+	0.608110i	\(0.208072\pi\)
							−0.894592	+	0.446883i	\(0.852534\pi\)
\(29\)	−1.69894	+	0.980882i	−0.315485	+	0.182145i	−0.649378	−	0.760466i	\(-0.724971\pi\)
							0.333894	+	0.942611i	\(0.391637\pi\)
\(31\)	10.7370	+	0.511466i	1.92842	+	0.0918620i	0.976636	−	0.214901i	\(-0.0689430\pi\)
							0.951786	+	0.306763i	\(0.0992461\pi\)
\(37\)	2.29867	−	3.98141i	0.377899	−	0.654540i	−0.612857	−	0.790194i	\(-0.709980\pi\)
							0.990756	+	0.135653i	\(0.0433133\pi\)
\(41\)	−5.92168	+	2.37068i	−0.924811	+	0.370239i	−0.784685	−	0.619895i	\(-0.787175\pi\)
							−0.140127	+	0.990134i	\(0.544751\pi\)
\(43\)	−8.85426	−	1.27305i	−1.35026	−	0.194138i	−0.571058	−	0.820910i	\(-0.693467\pi\)
							−0.779203	+	0.626771i	\(0.784376\pi\)
\(47\)	−7.78693	−	5.54505i	−1.13584	−	0.808829i	−0.152090	−	0.988367i	\(-0.548600\pi\)
							−0.983751	+	0.179538i	\(0.942540\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.b.11.7	yes	220
3.2	odd	2		402.2.n.a.11.5	✓	220
67.61	odd	66		402.2.n.a.329.5	yes	220
201.128	even	66	inner	402.2.n.b.329.7	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.11.5	✓	220	3.2	odd	2
402.2.n.a.329.5	yes	220	67.61	odd	66
402.2.n.b.11.7	yes	220	1.1	even	1	trivial
402.2.n.b.329.7	yes	220	201.128	even	66	inner