Embedded newform 335.2.w.a.2.2

• Label: 335.2.w.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71233 - 1.88362i) q^{2} +(0.923973 + 1.69213i) q^{3} +(-0.425859 + 4.45980i) q^{4} +(1.82956 + 1.28558i) q^{5} +(1.60519 - 4.63789i) q^{6} +(-0.501820 + 1.56983i) q^{7} +(5.05407 - 3.78343i) q^{8} +(-0.387654 + 0.603201i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71233	−	1.88362i	−1.21080	−	1.33192i	−0.927927	−	0.372763i	\(-0.878410\pi\)
							−0.282870	−	0.959158i	\(-0.591287\pi\)
\(3\)	0.923973	+	1.69213i	0.533456	+	0.976951i	0.995914	+	0.0903101i	\(0.0287858\pi\)
							−0.462458	+	0.886641i	\(0.653032\pi\)
\(5\)	1.82956	+	1.28558i	0.818202	+	0.574930i
							\(7\)	−0.501820	+	1.56983i	−0.189670	+	0.593340i	0.810303	+	0.586011i	\(0.199303\pi\)
−0.999973	+	0.00732890i	\(0.997667\pi\)
\(11\)	−5.29563	+	1.83284i	−1.59669	+	0.552621i	−0.973350	−	0.229326i	\(-0.926348\pi\)
							−0.623344	+	0.781947i	\(0.714227\pi\)
\(13\)	3.07511	+	0.367675i	0.852883	+	0.101975i	0.533654	−	0.845703i	\(-0.320819\pi\)
							0.319229	+	0.947678i	\(0.396576\pi\)
\(17\)	−3.08267	−	2.54526i	−0.747656	−	0.617317i	0.183042	−	0.983105i	\(-0.441406\pi\)
							−0.930698	+	0.365788i	\(0.880800\pi\)
\(19\)	−3.53018	+	6.84760i	−0.809880	+	1.57095i	0.0110643	+	0.999939i	\(0.496478\pi\)
							−0.820944	+	0.571009i	\(0.806552\pi\)
\(23\)	1.42236	+	0.866973i	0.296582	+	0.180776i	0.660850	−	0.750518i	\(-0.270196\pi\)
							−0.364268	+	0.931294i	\(0.618681\pi\)
\(29\)	−1.62860	−	0.940271i	−0.302423	−	0.174604i	0.341108	−	0.940024i	\(-0.389198\pi\)
							−0.643531	+	0.765420i	\(0.722531\pi\)
\(31\)	−1.38891	+	1.76614i	−0.249455	+	0.317208i	−0.894614	−	0.446841i	\(-0.852549\pi\)
							0.645158	+	0.764049i	\(0.276791\pi\)
\(37\)	1.96149	−	7.32037i	0.322467	−	1.20346i	−0.594367	−	0.804194i	\(-0.702597\pi\)
							0.916834	−	0.399269i	\(-0.130736\pi\)
\(41\)	−0.471558	+	0.335795i	−0.0736450	+	0.0524423i	−0.616286	−	0.787522i	\(-0.711364\pi\)
							0.542641	+	0.839964i	\(0.317424\pi\)
\(43\)	0.371439	−	0.138540i	0.0566440	−	0.0211271i	−0.320982	−	0.947085i	\(-0.604013\pi\)
							0.377626	+	0.925958i	\(0.376740\pi\)
\(47\)	12.4280	+	0.295842i	1.81281	+	0.0431530i	0.916446	−	0.400159i	\(-0.131045\pi\)
							0.896368	+	0.443312i	\(0.146197\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.2	✓	1280
5.3	odd	4	inner	335.2.w.a.203.2	yes	1280
67.34	odd	66	inner	335.2.w.a.302.2	yes	1280
335.168	even	132	inner	335.2.w.a.168.2	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.2	✓	1280	1.1	even	1	trivial
335.2.w.a.168.2	yes	1280	335.168	even	132	inner
335.2.w.a.203.2	yes	1280	5.3	odd	4	inner
335.2.w.a.302.2	yes	1280	67.34	odd	66	inner