Embedded newform 335.2.w.a.2.12

• Label: 335.2.w.a.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.775382 - 0.852949i) q^{2} +(-0.958841 - 1.75599i) q^{3} +(0.0638075 - 0.668222i) q^{4} +(-2.11931 - 0.713096i) q^{5} +(-0.754298 + 2.17940i) q^{6} +(1.22621 - 3.83592i) q^{7} +(-2.46502 + 1.84529i) q^{8} +(-0.542187 + 0.843660i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.775382	−	0.852949i	−0.548278	−	0.603126i	0.401574	−	0.915827i	\(-0.368463\pi\)
							−0.949852	+	0.312701i	\(0.898766\pi\)
\(3\)	−0.958841	−	1.75599i	−0.553587	−	1.01382i	−0.993287	−	0.115674i	\(-0.963097\pi\)
							0.439700	−	0.898145i	\(-0.355085\pi\)
\(5\)	−2.11931	−	0.713096i	−0.947786	−	0.318906i
							\(7\)	1.22621	−	3.83592i	0.463464	−	1.44984i	−0.386149	−	0.922436i	\(-0.626195\pi\)
0.849613	−	0.527406i	\(-0.176835\pi\)
\(11\)	5.49547	−	1.90200i	1.65695	−	0.573474i	0.671135	−	0.741335i	\(-0.265807\pi\)
							0.985811	+	0.167861i	\(0.0536860\pi\)
\(13\)	−5.75647	−	0.688271i	−1.59656	−	0.190892i	−0.727680	−	0.685917i	\(-0.759401\pi\)
							−0.868879	+	0.495025i	\(0.835159\pi\)
\(17\)	1.88633	+	1.55749i	0.457502	+	0.377746i	0.836782	−	0.547536i	\(-0.184434\pi\)
							−0.379280	+	0.925282i	\(0.623828\pi\)
\(19\)	−0.932715	+	1.80922i	−0.213980	+	0.415062i	−0.971258	−	0.238030i	\(-0.923498\pi\)
							0.757278	+	0.653092i	\(0.226529\pi\)
\(23\)	5.41992	+	3.30362i	1.13013	+	0.688852i	0.955268	−	0.295741i	\(-0.0955665\pi\)
							0.174864	+	0.984593i	\(0.444051\pi\)
\(29\)	1.22370	+	0.706501i	0.227235	+	0.131194i	0.609296	−	0.792943i	\(-0.291452\pi\)
							−0.382061	+	0.924137i	\(0.624786\pi\)
\(31\)	3.90593	−	4.96680i	0.701526	−	0.892063i	−0.296489	−	0.955036i	\(-0.595816\pi\)
							0.998015	+	0.0629729i	\(0.0200582\pi\)
\(37\)	0.844342	−	3.15113i	0.138809	−	0.518042i	−0.861144	−	0.508361i	\(-0.830252\pi\)
							0.999953	−	0.00968126i	\(-0.00308169\pi\)
\(41\)	−5.54943	+	3.95173i	−0.866675	+	0.617157i	−0.924505	−	0.381170i	\(-0.875521\pi\)
							0.0578298	+	0.998326i	\(0.481582\pi\)
\(43\)	−2.78559	+	1.03897i	−0.424799	+	0.158442i	−0.552766	−	0.833337i	\(-0.686428\pi\)
							0.127967	+	0.991778i	\(0.459155\pi\)
\(47\)	0.844415	+	0.0201008i	0.123171	+	0.00293201i	0.0853378	−	0.996352i	\(-0.472803\pi\)
							0.0378328	+	0.999284i	\(0.487955\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.12	✓	1280
5.3	odd	4	inner	335.2.w.a.203.12	yes	1280
67.34	odd	66	inner	335.2.w.a.302.12	yes	1280
335.168	even	132	inner	335.2.w.a.168.12	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.12	✓	1280	1.1	even	1	trivial
335.2.w.a.168.12	yes	1280	335.168	even	132	inner
335.2.w.a.203.12	yes	1280	5.3	odd	4	inner
335.2.w.a.302.12	yes	1280	67.34	odd	66	inner