Embedded newform 335.2.w.a.2.4

• Label: 335.2.w.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58453 - 1.74304i) q^{2} +(-1.25172 - 2.29235i) q^{3} +(-0.337346 + 3.53285i) q^{4} +(-1.49883 + 1.65937i) q^{5} +(-2.01227 + 5.81408i) q^{6} +(-0.00628661 + 0.0196662i) q^{7} +(2.92090 - 2.18656i) q^{8} +(-2.06614 + 3.21497i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58453	−	1.74304i	−1.12043	−	1.23252i	−0.969794	−	0.243926i	\(-0.921564\pi\)
							−0.150637	−	0.988589i	\(-0.548133\pi\)
\(3\)	−1.25172	−	2.29235i	−0.722679	−	1.32349i	−0.937894	−	0.346921i	\(-0.887227\pi\)
							0.215216	−	0.976567i	\(-0.430954\pi\)
\(5\)	−1.49883	+	1.65937i	−0.670296	+	0.742094i
							\(7\)	−0.00628661	+	0.0196662i	−0.00237612	+	0.00743314i	−0.953697	−	0.300768i	\(-0.902757\pi\)
0.951321	+	0.308201i	\(0.0997268\pi\)
\(11\)	−1.32229	+	0.457650i	−0.398686	+	0.137987i	−0.519051	−	0.854743i	\(-0.673715\pi\)
							0.120365	+	0.992730i	\(0.461593\pi\)
\(13\)	3.00398	+	0.359169i	0.833153	+	0.0996157i	0.524342	−	0.851508i	\(-0.324311\pi\)
							0.308811	+	0.951123i	\(0.400069\pi\)
\(17\)	1.74277	+	1.43895i	0.422684	+	0.348997i	0.823653	−	0.567094i	\(-0.191933\pi\)
							−0.400969	+	0.916092i	\(0.631326\pi\)
\(19\)	−2.48510	+	4.82042i	−0.570121	+	1.10588i	0.410846	+	0.911705i	\(0.365233\pi\)
							−0.980967	+	0.194175i	\(0.937797\pi\)
\(23\)	−2.49892	−	1.52317i	−0.521060	−	0.317603i	0.235120	−	0.971966i	\(-0.424452\pi\)
							−0.756180	+	0.654364i	\(0.772937\pi\)
\(29\)	3.44248	+	1.98752i	0.639252	+	0.369072i	0.784326	−	0.620348i	\(-0.213009\pi\)
							−0.145074	+	0.989421i	\(0.546342\pi\)
\(31\)	2.40675	−	3.06043i	0.432266	−	0.549670i	−0.520308	−	0.853979i	\(-0.674183\pi\)
							0.952574	+	0.304308i	\(0.0984253\pi\)
\(37\)	−2.37967	+	8.88104i	−0.391215	+	1.46003i	0.436918	+	0.899501i	\(0.356070\pi\)
							−0.828133	+	0.560532i	\(0.810597\pi\)
\(41\)	−7.96316	+	5.67054i	−1.24364	+	0.885590i	−0.996517	−	0.0833951i	\(-0.973424\pi\)
							−0.247120	+	0.968985i	\(0.579484\pi\)
\(43\)	10.6955	−	3.98920i	1.63104	−	0.608347i	0.643464	−	0.765476i	\(-0.277497\pi\)
							0.987578	+	0.157129i	\(0.0502239\pi\)
\(47\)	−4.43486	−	0.105569i	−0.646891	−	0.0153989i	−0.300928	−	0.953647i	\(-0.597296\pi\)
							−0.345963	+	0.938248i	\(0.612448\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.4	✓	1280
5.3	odd	4	inner	335.2.w.a.203.4	yes	1280
67.34	odd	66	inner	335.2.w.a.302.4	yes	1280
335.168	even	132	inner	335.2.w.a.168.4	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.4	✓	1280	1.1	even	1	trivial
335.2.w.a.168.4	yes	1280	335.168	even	132	inner
335.2.w.a.203.4	yes	1280	5.3	odd	4	inner
335.2.w.a.302.4	yes	1280	67.34	odd	66	inner