Embedded newform 353.2.f.a.36.16

• Label: 353.2.f.a.36.16
• 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			36.16
Character	\(\chi\)	\(=\)	353.36
Dual form			353.2.f.a.304.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.166074 - 0.166074i) q^{2} +(2.69712 - 0.536491i) q^{3} +1.94484i q^{4} +(2.33567 - 3.49558i) q^{5} +(0.358824 - 0.537018i) q^{6} +(-2.03978 + 1.36294i) q^{7} +(0.655134 + 0.655134i) q^{8} +(4.21501 - 1.74592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.166074	−	0.166074i	0.117432	−	0.117432i	−0.645949	−	0.763381i	\(-0.723538\pi\)
							0.763381	+	0.645949i	\(0.223538\pi\)
\(3\)	2.69712	−	0.536491i	1.55719	−	0.309743i	0.659954	−	0.751306i	\(-0.270576\pi\)
							0.897231	+	0.441562i	\(0.145576\pi\)
\(5\)	2.33567	−	3.49558i	1.04454	−	1.56327i	0.238746	−	0.971082i	\(-0.423264\pi\)
							0.805799	−	0.592189i	\(-0.201736\pi\)
\(7\)	−2.03978	+	1.36294i	−0.770964	+	0.515142i	−0.877738	−	0.479142i	\(-0.840948\pi\)
							0.106773	+	0.994283i	\(0.465948\pi\)
\(11\)	−0.825620	−	0.825620i	−0.248934	−	0.248934i	0.571599	−	0.820533i	\(-0.306323\pi\)
							−0.820533	+	0.571599i	\(0.806323\pi\)
\(13\)	−3.16708	+	0.629972i	−0.878390	+	0.174723i	−0.613632	−	0.789592i	\(-0.710292\pi\)
							−0.264758	+	0.964315i	\(0.585292\pi\)
\(17\)	−1.41119	+	1.41119i	−0.342263	+	0.342263i	−0.857218	−	0.514954i	\(-0.827809\pi\)
							0.514954	+	0.857218i	\(0.327809\pi\)
\(19\)	−0.413569	+	0.998444i	−0.0948793	+	0.229059i	−0.964193	−	0.265202i	\(-0.914561\pi\)
							0.869314	+	0.494261i	\(0.164561\pi\)
\(23\)	−1.00242	+	2.42006i	−0.209020	+	0.504618i	−0.993269	−	0.115827i	\(-0.963048\pi\)
							0.784250	+	0.620445i	\(0.213048\pi\)
\(29\)	5.38992	+	5.38992i	1.00088	+	1.00088i	1.00000		0.000882900i	\(0.000281036\pi\)
							0.000882900		1.00000i	\(0.499719\pi\)
\(31\)	−4.20483	−	2.80958i	−0.755209	−	0.504615i	0.117369	−	0.993088i	\(-0.462554\pi\)
							−0.872579	+	0.488474i	\(0.837554\pi\)
\(37\)	5.63550	+	8.43413i	0.926471	+	1.38656i	0.922260	+	0.386571i	\(0.126341\pi\)
							0.00421132	+	0.999991i	\(0.498659\pi\)
\(41\)	0.215780	+	0.0893792i	0.0336993	+	0.0139587i	0.399469	−	0.916747i	\(-0.369194\pi\)
							−0.365770	+	0.930705i	\(0.619194\pi\)
\(43\)	3.19021	−	7.70186i	0.486503	−	1.17452i	−0.469965	−	0.882685i	\(-0.655733\pi\)
							0.956468	−	0.291837i	\(-0.0942666\pi\)
\(47\)	−2.18237	−	5.26872i	−0.318332	−	0.768522i	−0.999343	−	0.0362482i	\(-0.988459\pi\)
							0.681011	−	0.732273i	\(-0.261541\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	353.2.f.a.36.16	✓	232
353.304	even	16	inner	353.2.f.a.304.16	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
353.2.f.a.36.16	✓	232	1.1	even	1	trivial
353.2.f.a.304.16	yes	232	353.304	even	16	inner