Embedded newform 335.2.w.a.2.17

• Label: 335.2.w.a.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00997829 - 0.0109765i) q^{2} +(0.109887 + 0.201243i) q^{3} +(0.190091 - 1.99072i) q^{4} +(-1.22223 - 1.87247i) q^{5} +(0.00111246 - 0.00321424i) q^{6} +(-0.560754 + 1.75419i) q^{7} +(-0.0474985 + 0.0355570i) q^{8} +(1.59350 - 2.47953i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00997829	−	0.0109765i	−0.00705572	−	0.00776155i	0.736413	−	0.676532i	\(-0.236518\pi\)
							−0.743469	+	0.668770i	\(0.766821\pi\)
\(3\)	0.109887	+	0.201243i	0.0634433	+	0.116188i	0.907476	−	0.420104i	\(-0.138006\pi\)
							−0.844033	+	0.536292i	\(0.819825\pi\)
\(5\)	−1.22223	−	1.87247i	−0.546599	−	0.837394i
							\(7\)	−0.560754	+	1.75419i	−0.211945	+	0.663023i	0.786993	+	0.616961i	\(0.211637\pi\)
−0.998939	+	0.0460611i	\(0.985333\pi\)
\(11\)	−0.921321	+	0.318872i	−0.277789	+	0.0961436i	−0.462410	−	0.886666i	\(-0.653015\pi\)
							0.184622	+	0.982810i	\(0.440894\pi\)
\(13\)	−1.05764	−	0.126456i	−0.293335	−	0.0350725i	−0.0292508	−	0.999572i	\(-0.509312\pi\)
							−0.264084	+	0.964500i	\(0.585070\pi\)
\(17\)	−5.09010	−	4.20274i	−1.23453	−	1.01931i	−0.998872	−	0.0474788i	\(-0.984881\pi\)
							−0.235659	−	0.971836i	\(-0.575725\pi\)
\(19\)	2.81286	−	5.45620i	0.645315	−	1.25174i	−0.308466	−	0.951235i	\(-0.599816\pi\)
							0.953781	−	0.300502i	\(-0.0971541\pi\)
\(23\)	0.406834	+	0.247978i	0.0848307	+	0.0517070i	0.562241	−	0.826974i	\(-0.309939\pi\)
							−0.477410	+	0.878681i	\(0.658424\pi\)
\(29\)	3.77565	+	2.17987i	0.701120	+	0.404792i	0.807764	−	0.589506i	\(-0.200677\pi\)
							−0.106645	+	0.994297i	\(0.534011\pi\)
\(31\)	5.00476	−	6.36407i	0.898882	−	1.14302i	−0.0902396	−	0.995920i	\(-0.528763\pi\)
							0.989121	−	0.147101i	\(-0.0469943\pi\)
\(37\)	−2.35813	+	8.80065i	−0.387674	+	1.44682i	0.446235	+	0.894916i	\(0.352765\pi\)
							−0.833909	+	0.551903i	\(0.813902\pi\)
\(41\)	−3.74153	+	2.66433i	−0.584329	+	0.416099i	−0.833627	−	0.552328i	\(-0.813740\pi\)
							0.249298	+	0.968427i	\(0.419800\pi\)
\(43\)	9.45306	−	3.52581i	1.44158	−	0.537681i	0.497518	−	0.867454i	\(-0.334245\pi\)
							0.944060	+	0.329773i	\(0.106972\pi\)
\(47\)	9.13752	+	0.217513i	1.33284	+	0.0317276i	0.684161	−	0.729331i	\(-0.260169\pi\)
							0.648683	+	0.761059i	\(0.275320\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.17	✓	1280
5.3	odd	4	inner	335.2.w.a.203.17	yes	1280
67.34	odd	66	inner	335.2.w.a.302.17	yes	1280
335.168	even	132	inner	335.2.w.a.168.17	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.17	✓	1280	1.1	even	1	trivial
335.2.w.a.168.17	yes	1280	335.168	even	132	inner
335.2.w.a.203.17	yes	1280	5.3	odd	4	inner
335.2.w.a.302.17	yes	1280	67.34	odd	66	inner