Embedded newform 335.2.w.a.2.1

• Label: 335.2.w.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87425 - 2.06175i) q^{2} +(-0.728004 - 1.33324i) q^{3} +(-0.547868 + 5.73753i) q^{4} +(1.29736 - 1.82122i) q^{5} +(-1.38434 + 3.99979i) q^{6} +(0.706985 - 2.21164i) q^{7} +(8.39504 - 6.28445i) q^{8} +(0.374384 - 0.582553i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87425	−	2.06175i	−1.32530	−	1.45788i	−0.789839	−	0.613314i	\(-0.789836\pi\)
							−0.535458	−	0.844562i	\(-0.679861\pi\)
\(3\)	−0.728004	−	1.33324i	−0.420313	−	0.769746i	0.578636	−	0.815586i	\(-0.303585\pi\)
							−0.998949	+	0.0458400i	\(0.985404\pi\)
\(5\)	1.29736	−	1.82122i	0.580197	−	0.814476i
							\(7\)	0.706985	−	2.21164i	0.267215	−	0.835922i	−0.722311	−	0.691568i	\(-0.756920\pi\)
0.989526	−	0.144353i	\(-0.0461102\pi\)
\(11\)	3.19109	−	1.10445i	0.962150	−	0.333003i	0.199560	−	0.979886i	\(-0.436049\pi\)
							0.762590	+	0.646882i	\(0.223928\pi\)
\(13\)	2.06330	+	0.246698i	0.572258	+	0.0684218i	0.399812	−	0.916597i	\(-0.369075\pi\)
							0.172446	+	0.985019i	\(0.444833\pi\)
\(17\)	−3.10925	−	2.56721i	−0.754104	−	0.622641i	0.178334	−	0.983970i	\(-0.442929\pi\)
							−0.932438	+	0.361329i	\(0.882323\pi\)
\(19\)	0.795271	−	1.54261i	0.182448	−	0.353899i	−0.779820	−	0.626004i	\(-0.784689\pi\)
							0.962268	+	0.272105i	\(0.0877198\pi\)
\(23\)	6.87175	+	4.18855i	1.43286	+	0.873373i	0.999611	−	0.0279024i	\(-0.00888275\pi\)
							0.433247	+	0.901275i	\(0.357368\pi\)
\(29\)	5.19014	+	2.99653i	0.963785	+	0.556441i	0.897336	−	0.441349i	\(-0.145500\pi\)
							0.0664489	+	0.997790i	\(0.478833\pi\)
\(31\)	−5.11825	+	6.50838i	−0.919265	+	1.16894i	0.0659856	+	0.997821i	\(0.478981\pi\)
							−0.985250	+	0.171120i	\(0.945262\pi\)
\(37\)	−0.509811	+	1.90264i	−0.0838124	+	0.312792i	−0.995087	−	0.0990078i	\(-0.968433\pi\)
							0.911274	+	0.411800i	\(0.135100\pi\)
\(41\)	1.75484	−	1.24962i	0.274061	−	0.195158i	−0.434760	−	0.900546i	\(-0.643167\pi\)
							0.708821	+	0.705389i	\(0.249227\pi\)
\(43\)	−5.06826	+	1.89036i	−0.772902	+	0.288278i	−0.704805	−	0.709401i	\(-0.748966\pi\)
							−0.0680973	+	0.997679i	\(0.521693\pi\)
\(47\)	−4.42140	−	0.105249i	−0.644928	−	0.0153522i	−0.299938	−	0.953959i	\(-0.596966\pi\)
							−0.344990	+	0.938606i	\(0.612118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.1	✓	1280
5.3	odd	4	inner	335.2.w.a.203.1	yes	1280
67.34	odd	66	inner	335.2.w.a.302.1	yes	1280
335.168	even	132	inner	335.2.w.a.168.1	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.1	✓	1280	1.1	even	1	trivial
335.2.w.a.168.1	yes	1280	335.168	even	132	inner
335.2.w.a.203.1	yes	1280	5.3	odd	4	inner
335.2.w.a.302.1	yes	1280	67.34	odd	66	inner