Embedded newform 402.2.n.a.101.5

• Label: 402.2.n.a.101.5
• Level: $402$
• Weight: $2$
• Character: 402.101
• 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			101.5
Character	\(\chi\)	\(=\)	402.101
Dual form			402.2.n.a.203.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0475819 - 0.998867i) q^{2} +(-0.521792 - 1.65158i) q^{3} +(-0.995472 + 0.0950560i) q^{4} +(-1.35611 - 1.56504i) q^{5} +(-1.62489 + 0.599787i) q^{6} +(2.04761 - 3.97182i) q^{7} +(0.142315 + 0.989821i) q^{8} +(-2.45547 + 1.72357i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 11 q^{2} + q^{3} + 11 q^{4} + 2 q^{6} + 6 q^{7} + 22 q^{8} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(269\)	\(337\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{65}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0475819	−	0.998867i	−0.0336455	−	0.706306i
							\(3\)	−0.521792	−	1.65158i	−0.301257	−	0.953543i
\(5\)	−1.35611	−	1.56504i	−0.606471	−	0.699905i								0.366608	−	0.930375i	\(-0.380519\pi\)
							−0.973079	+	0.230470i	\(0.925974\pi\)
\(7\)	2.04761	−	3.97182i	0.773926	−	1.50121i	−0.0890868	−	0.996024i	\(-0.528395\pi\)
							0.863013	−	0.505183i	\(-0.168575\pi\)
\(11\)	0.348711	−	1.00753i	0.105140	−	0.303783i	−0.880144	−	0.474707i	\(-0.842554\pi\)
							0.985284	+	0.170924i	\(0.0546753\pi\)
\(13\)	0.277771	−	0.353214i	0.0770398	−	0.0979641i	−0.746006	−	0.665940i	\(-0.768031\pi\)
							0.823045	+	0.567976i	\(0.192273\pi\)
\(17\)	−0.631347	+	6.61176i	−0.153124	+	1.60359i	0.513430	+	0.858131i	\(0.328375\pi\)
							−0.666554	+	0.745456i	\(0.732232\pi\)
\(19\)	1.75863	−	0.906634i	0.403456	−	0.207996i	−0.244541	−	0.969639i	\(-0.578637\pi\)
							0.647997	+	0.761643i	\(0.275607\pi\)
\(23\)	−5.57059	−	1.35141i	−1.16155	−	0.281789i	−0.391747	−	0.920073i	\(-0.628129\pi\)
							−0.769801	+	0.638284i	\(0.779644\pi\)
\(29\)	−1.17482	+	0.678280i	−0.218158	+	0.125953i	−0.605097	−	0.796152i	\(-0.706866\pi\)
							0.386939	+	0.922105i	\(0.373532\pi\)
\(31\)	−0.967058	−	1.22971i	−0.173689	−	0.220863i	0.691413	−	0.722460i	\(-0.256989\pi\)
							−0.865102	+	0.501597i	\(0.832746\pi\)
\(37\)	−2.96783	+	5.14043i	−0.487908	+	0.845082i	−0.999903	−	0.0139067i	\(-0.995573\pi\)
							0.511995	+	0.858988i	\(0.328907\pi\)
\(41\)	3.52393	−	4.94867i	0.550346	−	0.772853i	−0.441910	−	0.897059i	\(-0.645699\pi\)
							0.992256	+	0.124206i	\(0.0396385\pi\)
\(43\)	10.1539	−	4.63715i	1.54846	−	0.707159i	0.556168	−	0.831070i	\(-0.312271\pi\)
							0.992293	+	0.123911i	\(0.0395436\pi\)
\(47\)	−5.31836	+	5.57774i	−0.775763	+	0.813597i	−0.986468	−	0.163953i	\(-0.947576\pi\)
							0.210705	+	0.977550i	\(0.432424\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.a.101.5	✓	220
3.2	odd	2		402.2.n.b.101.5	yes	220
67.2	odd	66		402.2.n.b.203.5	yes	220
201.2	even	66	inner	402.2.n.a.203.5	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.101.5	✓	220	1.1	even	1	trivial
402.2.n.a.203.5	yes	220	201.2	even	66	inner
402.2.n.b.101.5	yes	220	3.2	odd	2
402.2.n.b.203.5	yes	220	67.2	odd	66