Embedded newform 335.2.w.a.2.9

• Label: 335.2.w.a.2.9
• Level: $335$
• Weight: $2$
• Character: 335.2
• Analytic conductor: $2.675$
• Analytic rank: $0$
• Dimension: $1280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 335 = 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	335.w (of order \(132\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.67498846771\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(32\) over \(\Q(\zeta_{132})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{132}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08050 - 1.18859i) q^{2} +(1.50186 + 2.75045i) q^{3} +(-0.0551539 + 0.577598i) q^{4} +(-0.203774 + 2.22676i) q^{5} +(1.64640 - 4.75695i) q^{6} +(-0.253409 + 0.792734i) q^{7} +(-1.82571 + 1.36671i) q^{8} +(-3.68747 + 5.73781i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/335\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(202\)
\(\chi(n)\)	\(e\left(\frac{1}{66}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08050	−	1.18859i	−0.764029	−	0.840460i	0.227030	−	0.973888i	\(-0.427099\pi\)
							−0.991059	+	0.133428i	\(0.957402\pi\)
\(3\)	1.50186	+	2.75045i	0.867099	+	1.58797i	0.807589	+	0.589745i	\(0.200772\pi\)
							0.0595093	+	0.998228i	\(0.481046\pi\)
\(5\)	−0.203774	+	2.22676i	−0.0911306	+	0.995839i
							\(7\)	−0.253409	+	0.792734i	−0.0957798	+	0.299625i	−0.988551	−	0.150884i	\(-0.951788\pi\)
0.892772	+	0.450509i	\(0.148758\pi\)
\(11\)	3.83063	−	1.32579i	1.15498	−	0.399742i	0.318598	−	0.947890i	\(-0.396788\pi\)
							0.836381	+	0.548148i	\(0.184667\pi\)
\(13\)	−6.11355	−	0.730965i	−1.69559	−	0.202733i	−0.785999	−	0.618228i	\(-0.787851\pi\)
							−0.909596	+	0.415495i	\(0.863608\pi\)
\(17\)	−0.823504	−	0.679943i	−0.199729	−	0.164910i	0.531464	−	0.847081i	\(-0.321642\pi\)
							−0.731193	+	0.682171i	\(0.761036\pi\)
\(19\)	−0.213401	+	0.413941i	−0.0489576	+	0.0949646i	−0.912045	−	0.410090i	\(-0.865497\pi\)
							0.863087	+	0.505054i	\(0.168528\pi\)
\(23\)	0.499502	+	0.304462i	0.104153	+	0.0634847i	0.571576	−	0.820549i	\(-0.306332\pi\)
							−0.467422	+	0.884034i	\(0.654817\pi\)
\(29\)	6.10945	+	3.52729i	1.13450	+	0.655002i	0.945062	−	0.326891i	\(-0.106001\pi\)
							0.189435	+	0.981893i	\(0.439334\pi\)
\(31\)	−0.682339	+	0.867664i	−0.122552	+	0.155837i	−0.843458	−	0.537196i	\(-0.819484\pi\)
							0.720906	+	0.693033i	\(0.243726\pi\)
\(37\)	−2.05576	+	7.67221i	−0.337965	+	1.26130i	0.562654	+	0.826693i	\(0.309780\pi\)
							−0.900619	+	0.434610i	\(0.856886\pi\)
\(41\)	2.87022	−	2.04387i	0.448253	−	0.319199i	−0.333542	−	0.942735i	\(-0.608244\pi\)
							0.781795	+	0.623536i	\(0.214305\pi\)
\(43\)	7.71302	−	2.87681i	1.17623	−	0.438709i	0.316067	−	0.948737i	\(-0.397638\pi\)
							0.860158	+	0.510028i	\(0.170365\pi\)
\(47\)	13.0941	+	0.311698i	1.90997	+	0.0454658i	0.962025	−	0.272960i	\(-0.0880027\pi\)
							0.947948	+	0.318426i	\(0.103154\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.9	✓	1280
5.3	odd	4	inner	335.2.w.a.203.9	yes	1280
67.34	odd	66	inner	335.2.w.a.302.9	yes	1280
335.168	even	132	inner	335.2.w.a.168.9	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.9	✓	1280	1.1	even	1	trivial
335.2.w.a.168.9	yes	1280	335.168	even	132	inner
335.2.w.a.203.9	yes	1280	5.3	odd	4	inner
335.2.w.a.302.9	yes	1280	67.34	odd	66	inner