Embedded newform 402.2.n.b.11.4

• Label: 402.2.n.b.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	402.11
Dual form			402.2.n.b.329.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.327068 - 0.945001i) q^{2} +(-0.663010 + 1.60013i) q^{3} +(-0.786053 + 0.618159i) q^{4} +(0.141454 - 0.0415348i) q^{5} +(1.72897 + 0.103193i) q^{6} +(-0.267898 + 1.38999i) q^{7} +(0.841254 + 0.540641i) q^{8} +(-2.12084 - 2.12180i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 11 q^{2} - q^{3} + 11 q^{4} - q^{6} + 6 q^{7} - 22 q^{8} - q^{9}+O(q^{10}) \)

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
							\(3\)	−0.663010	+	1.60013i	−0.382789	+	0.923836i
\(5\)	0.141454	−	0.0415348i	0.0632604	−	0.0185749i								−0.249949	−	0.968259i	\(-0.580414\pi\)
							0.313210	+	0.949684i	\(0.398596\pi\)
\(7\)	−0.267898	+	1.38999i	−0.101256	+	0.525367i	0.895512	+	0.445037i	\(0.146809\pi\)
							−0.996768	+	0.0803297i	\(0.974403\pi\)
\(11\)	2.22115	+	2.11786i	0.669702	+	0.638560i	0.946627	−	0.322332i	\(-0.104467\pi\)
							−0.276924	+	0.960892i	\(0.589315\pi\)
\(13\)	−4.74175	+	0.225877i	−1.31513	+	0.0626471i	−0.693380	−	0.720572i	\(-0.743879\pi\)
							−0.621745	+	0.783219i	\(0.713576\pi\)
\(17\)	0.147603	−	0.187693i	0.0357991	−	0.0455222i	−0.767824	−	0.640661i	\(-0.778660\pi\)
							0.803623	+	0.595139i	\(0.202903\pi\)
\(19\)	−6.09562	+	1.17483i	−1.39843	+	0.269526i	−0.832088	−	0.554644i	\(-0.812855\pi\)
							−0.566343	+	0.824169i	\(0.691642\pi\)
\(23\)	−0.691577	+	7.24252i	−0.144204	+	1.51017i	0.576597	+	0.817028i	\(0.304380\pi\)
							−0.720801	+	0.693142i	\(0.756226\pi\)
\(29\)	2.94107	−	1.69803i	0.546144	−	0.315316i	−0.201422	−	0.979505i	\(-0.564556\pi\)
							0.747565	+	0.664188i	\(0.231223\pi\)
\(31\)	−1.81876	−	0.0866384i	−0.326660	−	0.0155607i	−0.116388	−	0.993204i	\(-0.537132\pi\)
							−0.210271	+	0.977643i	\(0.567435\pi\)
\(37\)	−2.49075	+	4.31411i	−0.409477	+	0.709236i	−0.994831	−	0.101542i	\(-0.967622\pi\)
							0.585354	+	0.810778i	\(0.300956\pi\)
\(41\)	−6.09535	+	2.44021i	−0.951934	+	0.381097i	−0.795066	−	0.606523i	\(-0.792564\pi\)
							−0.156868	+	0.987620i	\(0.550140\pi\)
\(43\)	−0.487608	−	0.0701075i	−0.0743595	−	0.0106913i	0.105035	−	0.994469i	\(-0.466505\pi\)
							−0.179394	+	0.983777i	\(0.557414\pi\)
\(47\)	2.08385	+	1.48390i	0.303961	+	0.216449i	0.721852	−	0.692048i	\(-0.243291\pi\)
							−0.417891	+	0.908497i	\(0.637231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.b.11.4	yes	220
3.2	odd	2		402.2.n.a.11.2	✓	220
67.61	odd	66		402.2.n.a.329.2	yes	220
201.128	even	66	inner	402.2.n.b.329.4	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.11.2	✓	220	3.2	odd	2
402.2.n.a.329.2	yes	220	67.61	odd	66
402.2.n.b.11.4	yes	220	1.1	even	1	trivial
402.2.n.b.329.4	yes	220	201.128	even	66	inner