Embedded newform 353.2.f.a.36.2

• Label: 353.2.f.a.36.2
• Level: $353$
• Weight: $2$
• Character: 353.36
• Analytic conductor: $2.819$
• Analytic rank: $0$
• Dimension: $232$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 353 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	353.f (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.81871919135\)
Analytic rank:	\(0\)
Dimension:	\(232\)
Relative dimension:	\(29\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			36.2
Character	\(\chi\)	\(=\)	353.36
Dual form			353.2.f.a.304.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.83165 + 1.83165i) q^{2} +(-3.10034 + 0.616696i) q^{3} -4.70991i q^{4} +(-0.939070 + 1.40542i) q^{5} +(4.54918 - 6.80832i) q^{6} +(1.69432 - 1.13211i) q^{7} +(4.96362 + 4.96362i) q^{8} +(6.46015 - 2.67588i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/353\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.83165	+	1.83165i	−1.29518	+	1.29518i	−0.363633	+	0.931542i	\(0.618464\pi\)
							−0.931542	+	0.363633i	\(0.881536\pi\)
\(3\)	−3.10034	+	0.616696i	−1.78998	+	0.356050i	−0.974787	−	0.223137i	\(-0.928370\pi\)
							−0.815195	+	0.579187i	\(0.803370\pi\)
\(5\)	−0.939070	+	1.40542i	−0.419965	+	0.628522i	−0.979776	−	0.200099i	\(-0.935874\pi\)
							0.559811	+	0.828620i	\(0.310874\pi\)
\(7\)	1.69432	−	1.13211i	0.640394	−	0.427898i	−0.192525	−	0.981292i	\(-0.561668\pi\)
							0.832919	+	0.553394i	\(0.186668\pi\)
\(11\)	3.98789	+	3.98789i	1.20239	+	1.20239i	0.973437	+	0.228956i	\(0.0735312\pi\)
							0.228956	+	0.973437i	\(0.426469\pi\)
\(13\)	−4.08380	+	0.812319i	−1.13264	+	0.225297i	−0.725603	−	0.688114i	\(-0.758439\pi\)
							−0.407040	+	0.913410i	\(0.633439\pi\)
\(17\)	2.77209	−	2.77209i	0.672331	−	0.672331i	−0.285922	−	0.958253i	\(-0.592300\pi\)
							0.958253	+	0.285922i	\(0.0922998\pi\)
\(19\)	2.20657	−	5.32713i	0.506222	−	1.22213i	−0.439821	−	0.898085i	\(-0.644958\pi\)
							0.946043	−	0.324042i	\(-0.105042\pi\)
\(23\)	0.527821	−	1.27427i	0.110058	−	0.265704i	−0.859249	−	0.511558i	\(-0.829068\pi\)
							0.969307	+	0.245854i	\(0.0790685\pi\)
\(29\)	0.383465	+	0.383465i	0.0712077	+	0.0712077i	0.741814	−	0.670606i	\(-0.233966\pi\)
							−0.670606	+	0.741814i	\(0.733966\pi\)
\(31\)	2.14196	+	1.43121i	0.384707	+	0.257053i	0.732853	−	0.680387i	\(-0.238188\pi\)
							−0.348146	+	0.937441i	\(0.613188\pi\)
\(37\)	5.42342	+	8.11672i	0.891605	+	1.33438i	0.941986	+	0.335651i	\(0.108956\pi\)
							−0.0503816	+	0.998730i	\(0.516044\pi\)
\(41\)	−6.94993	−	2.87876i	−1.08540	−	0.449586i	−0.232998	−	0.972477i	\(-0.574853\pi\)
							−0.852399	+	0.522891i	\(0.824853\pi\)
\(43\)	0.125635	−	0.303311i	0.0191592	−	0.0462544i	−0.914010	−	0.405691i	\(-0.867031\pi\)
							0.933170	+	0.359436i	\(0.117031\pi\)
\(47\)	−0.386361	−	0.932758i	−0.0563565	−	0.136057i	0.893193	−	0.449673i	\(-0.148459\pi\)
							−0.949550	+	0.313616i	\(0.898459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	353.2.f.a.36.2	✓	232
353.304	even	16	inner	353.2.f.a.304.2	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
353.2.f.a.36.2	✓	232	1.1	even	1	trivial
353.2.f.a.304.2	yes	232	353.304	even	16	inner