Embedded newform 353.2.f.a.36.20

• Label: 353.2.f.a.36.20
• 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			36.20
Character	\(\chi\)	\(=\)	353.36
Dual form			353.2.f.a.304.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.778358 - 0.778358i) q^{2} +(-1.28403 + 0.255410i) q^{3} +0.788317i q^{4} +(1.89695 - 2.83899i) q^{5} +(-0.800637 + 1.19824i) q^{6} +(1.01666 - 0.679308i) q^{7} +(2.17031 + 2.17031i) q^{8} +(-1.18813 + 0.492142i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.778358	−	0.778358i	0.550382	−	0.550382i	−0.376169	−	0.926551i	\(-0.622759\pi\)
							0.926551	+	0.376169i	\(0.122759\pi\)
\(3\)	−1.28403	+	0.255410i	−0.741336	+	0.147461i	−0.551289	−	0.834314i	\(-0.685864\pi\)
							−0.190047	+	0.981775i	\(0.560864\pi\)
\(5\)	1.89695	−	2.83899i	0.848342	−	1.26963i	−0.112811	−	0.993616i	\(-0.535985\pi\)
							0.961153	−	0.276017i	\(-0.0890146\pi\)
\(7\)	1.01666	−	0.679308i	0.384260	−	0.256754i	−0.348405	−	0.937344i	\(-0.613276\pi\)
							0.732665	+	0.680590i	\(0.238276\pi\)
\(11\)	−1.64491	−	1.64491i	−0.495958	−	0.495958i	0.414219	−	0.910177i	\(-0.364055\pi\)
							−0.910177	+	0.414219i	\(0.864055\pi\)
\(13\)	4.66020	−	0.926971i	1.29251	−	0.257095i	0.499503	−	0.866312i	\(-0.333516\pi\)
							0.793004	+	0.609217i	\(0.208516\pi\)
\(17\)	4.42856	−	4.42856i	1.07408	−	1.07408i	0.0770573	−	0.997027i	\(-0.475448\pi\)
							0.997027	−	0.0770573i	\(-0.0245524\pi\)
\(19\)	−0.592616	+	1.43070i	−0.135956	+	0.328226i	−0.977164	−	0.212485i	\(-0.931844\pi\)
							0.841209	+	0.540710i	\(0.181844\pi\)
\(23\)	−0.119315	+	0.288053i	−0.0248790	+	0.0600632i	−0.935831	−	0.352450i	\(-0.885349\pi\)
							0.910952	+	0.412513i	\(0.135349\pi\)
\(29\)	2.50186	+	2.50186i	0.464584	+	0.464584i	0.900154	−	0.435571i	\(-0.143453\pi\)
							−0.435571	+	0.900154i	\(0.643453\pi\)
\(31\)	−7.87716	−	5.26335i	−1.41478	−	0.945326i	−0.999363	−	0.0356949i	\(-0.988636\pi\)
							−0.415417	−	0.909631i	\(-0.636364\pi\)
\(37\)	0.752994	+	1.12693i	0.123791	+	0.185267i	0.888189	−	0.459478i	\(-0.151964\pi\)
							−0.764398	+	0.644745i	\(0.776964\pi\)
\(41\)	3.05715	+	1.26631i	0.477446	+	0.197765i	0.608411	−	0.793622i	\(-0.291807\pi\)
							−0.130964	+	0.991387i	\(0.541807\pi\)
\(43\)	−3.19977	+	7.72492i	−0.487960	+	1.17804i	0.467785	+	0.883842i	\(0.345052\pi\)
							−0.955745	+	0.294197i	\(0.904948\pi\)
\(47\)	1.65582	+	3.99751i	0.241527	+	0.583097i	0.997435	−	0.0715809i	\(-0.0228044\pi\)
							−0.755908	+	0.654678i	\(0.772804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	353.2.f.a.36.20	✓	232
353.304	even	16	inner	353.2.f.a.304.20	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
353.2.f.a.36.20	✓	232	1.1	even	1	trivial
353.2.f.a.304.20	yes	232	353.304	even	16	inner