Embedded newform 335.2.w.a.2.7

• Label: 335.2.w.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11051 - 1.22160i) q^{2} +(0.752678 + 1.37843i) q^{3} +(-0.0689651 + 0.722235i) q^{4} +(1.55695 - 1.60496i) q^{5} +(0.848029 - 2.45022i) q^{6} +(0.531865 - 1.66382i) q^{7} +(-1.68438 + 1.26091i) q^{8} +(0.288386 - 0.448737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11051	−	1.22160i	−0.785246	−	0.863799i	0.208202	−	0.978086i	\(-0.433239\pi\)
							−0.993448	+	0.114286i	\(0.963542\pi\)
\(3\)	0.752678	+	1.37843i	0.434559	+	0.795835i	0.999572	−	0.0292466i	\(-0.00931081\pi\)
							−0.565013	+	0.825082i	\(0.691129\pi\)
\(5\)	1.55695	−	1.60496i	0.696290	−	0.717761i
							\(7\)	0.531865	−	1.66382i	0.201026	−	0.628864i	−0.798609	−	0.601850i	\(-0.794430\pi\)
0.999635	−	0.0270139i	\(-0.00859983\pi\)
\(11\)	0.630295	−	0.218147i	0.190041	−	0.0657738i	−0.230390	−	0.973098i	\(-0.574000\pi\)
							0.420431	+	0.907325i	\(0.361879\pi\)
\(13\)	−0.819919	−	0.0980333i	−0.227405	−	0.0271895i	0.00423581	−	0.999991i	\(-0.498652\pi\)
							−0.231640	+	0.972801i	\(0.574409\pi\)
\(17\)	−1.36866	−	1.13006i	−0.331949	−	0.274081i	0.455792	−	0.890086i	\(-0.349356\pi\)
							−0.787742	+	0.616006i	\(0.788750\pi\)
\(19\)	0.272859	−	0.529273i	0.0625982	−	0.121423i	−0.855460	−	0.517868i	\(-0.826726\pi\)
							0.918059	+	0.396445i	\(0.129756\pi\)
\(23\)	1.31622	+	0.802277i	0.274450	+	0.167286i	0.650928	−	0.759139i	\(-0.274380\pi\)
							−0.376478	+	0.926426i	\(0.622865\pi\)
\(29\)	−2.36074	−	1.36297i	−0.438378	−	0.253098i	0.264531	−	0.964377i	\(-0.414783\pi\)
							−0.702909	+	0.711279i	\(0.748116\pi\)
\(31\)	6.34021	−	8.06223i	1.13873	−	1.44802i	0.263280	−	0.964719i	\(-0.415196\pi\)
							0.875455	−	0.483300i	\(-0.160562\pi\)
\(37\)	−0.239973	+	0.895591i	−0.0394513	+	0.147234i	−0.982842	−	0.184449i	\(-0.940950\pi\)
							0.943391	+	0.331683i	\(0.107617\pi\)
\(41\)	−2.85804	+	2.03520i	−0.446350	+	0.317845i	−0.781037	−	0.624485i	\(-0.785309\pi\)
							0.334687	+	0.942329i	\(0.391370\pi\)
\(43\)	−4.22321	+	1.57518i	−0.644034	+	0.240212i	−0.650193	−	0.759769i	\(-0.725312\pi\)
							0.00615926	+	0.999981i	\(0.498039\pi\)
\(47\)	−7.37294	−	0.175509i	−1.07545	−	0.0256006i	−0.517665	−	0.855583i	\(-0.673199\pi\)
							−0.557789	+	0.829983i	\(0.688350\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.7	✓	1280
5.3	odd	4	inner	335.2.w.a.203.7	yes	1280
67.34	odd	66	inner	335.2.w.a.302.7	yes	1280
335.168	even	132	inner	335.2.w.a.168.7	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.7	✓	1280	1.1	even	1	trivial
335.2.w.a.168.7	yes	1280	335.168	even	132	inner
335.2.w.a.203.7	yes	1280	5.3	odd	4	inner
335.2.w.a.302.7	yes	1280	67.34	odd	66	inner