Embedded newform 402.2.n.b.11.9

• Label: 402.2.n.b.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	402.11
Dual form			402.2.n.b.329.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.327068 - 0.945001i) q^{2} +(1.46270 - 0.927637i) q^{3} +(-0.786053 + 0.618159i) q^{4} +(1.96820 - 0.577915i) q^{5} +(-1.35502 - 1.07885i) q^{6} +(-0.624160 + 3.23845i) q^{7} +(0.841254 + 0.540641i) q^{8} +(1.27898 - 2.71371i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 11 q^{2} - q^{3} + 11 q^{4} - q^{6} + 6 q^{7} - 22 q^{8} - q^{9}+O(q^{10}) \)

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
							\(3\)	1.46270	−	0.927637i	0.844490	−	0.535572i
\(5\)	1.96820	−	0.577915i	0.880204	−	0.258451i								0.189755	−	0.981832i	\(-0.439231\pi\)
							0.690450	+	0.723380i	\(0.257413\pi\)
\(7\)	−0.624160	+	3.23845i	−0.235910	+	1.22402i	0.649917	+	0.760005i	\(0.274803\pi\)
							−0.885828	+	0.464014i	\(0.846409\pi\)
\(11\)	−2.40996	−	2.29789i	−0.726630	−	0.692840i	0.233512	−	0.972354i	\(-0.424978\pi\)
							−0.960142	+	0.279514i	\(0.909827\pi\)
\(13\)	6.35028	−	0.302501i	1.76125	−	0.0838988i	0.858170	−	0.513366i	\(-0.171602\pi\)
							0.903082	+	0.429468i	\(0.141299\pi\)
\(17\)	4.70156	−	5.97853i	1.14030	−	1.45001i	0.266421	−	0.963857i	\(-0.414159\pi\)
							0.873876	−	0.486149i	\(-0.161599\pi\)
\(19\)	−5.03383	+	0.970190i	−1.15484	+	0.222577i	−0.730491	−	0.682922i	\(-0.760709\pi\)
							−0.424348	+	0.905499i	\(0.639497\pi\)
\(23\)	−0.485467	+	5.08404i	−0.101227	+	1.06010i	0.792010	+	0.610508i	\(0.209035\pi\)
							−0.893237	+	0.449587i	\(0.851571\pi\)
\(29\)	−4.10421	+	2.36957i	−0.762132	+	0.440017i	−0.830061	−	0.557673i	\(-0.811694\pi\)
							0.0679284	+	0.997690i	\(0.478361\pi\)
\(31\)	2.55928	+	0.121914i	0.459660	+	0.0218963i	0.276135	−	0.961119i	\(-0.410946\pi\)
							0.183525	+	0.983015i	\(0.441249\pi\)
\(37\)	2.04707	−	3.54562i	0.336536	−	0.582897i	−0.647243	−	0.762284i	\(-0.724078\pi\)
							0.983779	+	0.179387i	\(0.0574114\pi\)
\(41\)	−4.77061	+	1.90986i	−0.745044	+	0.298271i	−0.712951	−	0.701214i	\(-0.752642\pi\)
							−0.0320933	+	0.999485i	\(0.510217\pi\)
\(43\)	−2.97444	−	0.427660i	−0.453598	−	0.0652176i	−0.0882705	−	0.996097i	\(-0.528134\pi\)
							−0.365328	+	0.930879i	\(0.619043\pi\)
\(47\)	−1.16749	−	0.831365i	−0.170296	−	0.121267i	0.491731	−	0.870747i	\(-0.336364\pi\)
							−0.662027	+	0.749480i	\(0.730304\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.b.11.9	yes	220
3.2	odd	2		402.2.n.a.11.10	✓	220
67.61	odd	66		402.2.n.a.329.10	yes	220
201.128	even	66	inner	402.2.n.b.329.9	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.11.10	✓	220	3.2	odd	2
402.2.n.a.329.10	yes	220	67.61	odd	66
402.2.n.b.11.9	yes	220	1.1	even	1	trivial
402.2.n.b.329.9	yes	220	201.128	even	66	inner