Embedded newform 402.2.n.a.329.1

• Label: 402.2.n.a.329.1
• Level: $402$
• Weight: $2$
• Character: 402.329
• Analytic conductor: $3.210$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• 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			329.1
Character	\(\chi\)	\(=\)	402.329
Dual form			402.2.n.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.327068 - 0.945001i) q^{2} +(-1.66618 - 0.473108i) q^{3} +(-0.786053 - 0.618159i) q^{4} +(2.76497 + 0.811870i) q^{5} +(-0.992043 + 1.41981i) q^{6} +(-0.155035 - 0.804400i) q^{7} +(-0.841254 + 0.540641i) q^{8} +(2.55234 + 1.57657i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 11 q^{2} + q^{3} + 11 q^{4} + 2 q^{6} + 6 q^{7} + 22 q^{8} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(269\)	\(337\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{66}\right)\)

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
							\(3\)	−1.66618	−	0.473108i	−0.961972	−	0.273149i
\(5\)	2.76497	+	0.811870i	1.23653	+	0.363079i								0.833712	−	0.552199i	\(-0.186211\pi\)
							0.402822	+	0.915278i	\(0.368029\pi\)
\(7\)	−0.155035	−	0.804400i	−0.0585979	−	0.304035i	0.940791	−	0.338987i	\(-0.110084\pi\)
							−0.999389	+	0.0349523i	\(0.988872\pi\)
\(11\)	1.62440	−	1.54886i	0.489775	−	0.466999i	−0.404487	−	0.914544i	\(-0.632550\pi\)
							0.894262	+	0.447544i	\(0.147701\pi\)
\(13\)	3.13312	+	0.149249i	0.868971	+	0.0413942i	0.477330	−	0.878724i	\(-0.341605\pi\)
							0.391641	+	0.920118i	\(0.371908\pi\)
\(17\)	−2.19338	−	2.78911i	−0.531974	−	0.676460i	0.443576	−	0.896237i	\(-0.353709\pi\)
							−0.975550	+	0.219777i	\(0.929467\pi\)
\(19\)	2.01695	+	0.388735i	0.462719	+	0.0891818i	0.415283	−	0.909692i	\(-0.363683\pi\)
							0.0474366	+	0.998874i	\(0.484895\pi\)
\(23\)	−0.705097	−	7.38411i	−0.147023	−	1.53969i	−0.704622	−	0.709583i	\(-0.748884\pi\)
							0.557599	−	0.830110i	\(-0.311723\pi\)
\(29\)	−0.735210	−	0.424474i	−0.136525	−	0.0788228i	0.430182	−	0.902742i	\(-0.358449\pi\)
							−0.566707	+	0.823920i	\(0.691783\pi\)
\(31\)	0.832006	−	0.0396333i	0.149433	−	0.00711836i	0.0272677	−	0.999628i	\(-0.491319\pi\)
							0.122165	+	0.992510i	\(0.461016\pi\)
\(37\)	3.36274	+	5.82443i	0.552830	+	0.957530i	0.998069	+	0.0621182i	\(0.0197856\pi\)
							−0.445238	+	0.895412i	\(0.646881\pi\)
\(41\)	−0.731352	−	0.292789i	−0.114218	−	0.0457260i	0.313850	−	0.949473i	\(-0.398381\pi\)
							−0.428068	+	0.903747i	\(0.640805\pi\)
\(43\)	−3.99850	+	0.574897i	−0.609765	+	0.0876710i	−0.440278	−	0.897861i	\(-0.645120\pi\)
							−0.169487	+	0.985532i	\(0.554211\pi\)
\(47\)	2.27505	−	1.62006i	0.331851	−	0.236310i	−0.401970	−	0.915653i	\(-0.631674\pi\)
							0.733821	+	0.679343i	\(0.237735\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.a.329.1	yes	220
3.2	odd	2		402.2.n.b.329.2	yes	220
67.11	odd	66		402.2.n.b.11.2	yes	220
201.11	even	66	inner	402.2.n.a.11.1	✓	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.11.1	✓	220	201.11	even	66	inner
402.2.n.a.329.1	yes	220	1.1	even	1	trivial
402.2.n.b.11.2	yes	220	67.11	odd	66
402.2.n.b.329.2	yes	220	3.2	odd	2