Embedded newform 353.2.f.a.36.4

• Label: 353.2.f.a.36.4
• 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			36.4
Character	\(\chi\)	\(=\)	353.36
Dual form			353.2.f.a.304.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60440 + 1.60440i) q^{2} +(2.28136 - 0.453790i) q^{3} -3.14821i q^{4} +(1.28352 - 1.92092i) q^{5} +(-2.93215 + 4.38827i) q^{6} +(1.74145 - 1.16360i) q^{7} +(1.84218 + 1.84218i) q^{8} +(2.22703 - 0.922464i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60440	+	1.60440i	−1.13448	+	1.13448i	−0.145060	+	0.989423i	\(0.546338\pi\)
							−0.989423	+	0.145060i	\(0.953662\pi\)
\(3\)	2.28136	−	0.453790i	1.31714	−	0.261996i	0.514003	−	0.857788i	\(-0.328162\pi\)
							0.803139	+	0.595792i	\(0.203162\pi\)
\(5\)	1.28352	−	1.92092i	0.574007	−	0.859062i	−0.424927	−	0.905228i	\(-0.639700\pi\)
							0.998934	+	0.0461655i	\(0.0147002\pi\)
\(7\)	1.74145	−	1.16360i	0.658207	−	0.439800i	−0.181095	−	0.983466i	\(-0.557964\pi\)
							0.839302	+	0.543666i	\(0.182964\pi\)
\(11\)	−2.45507	−	2.45507i	−0.740231	−	0.740231i	0.232392	−	0.972622i	\(-0.425345\pi\)
							−0.972622	+	0.232392i	\(0.925345\pi\)
\(13\)	−3.05854	+	0.608381i	−0.848285	+	0.168734i	−0.600047	−	0.799965i	\(-0.704852\pi\)
							−0.248238	+	0.968699i	\(0.579852\pi\)
\(17\)	3.25064	−	3.25064i	0.788395	−	0.788395i	−0.192836	−	0.981231i	\(-0.561769\pi\)
							0.981231	+	0.192836i	\(0.0617686\pi\)
\(19\)	0.426806	−	1.03040i	0.0979159	−	0.236390i	−0.867330	−	0.497734i	\(-0.834166\pi\)
							0.965246	+	0.261344i	\(0.0841657\pi\)
\(23\)	0.0973175	−	0.234945i	0.0202921	−	0.0489895i	−0.913409	−	0.407044i	\(-0.866560\pi\)
							0.933701	+	0.358054i	\(0.116560\pi\)
\(29\)	−4.57451	−	4.57451i	−0.849464	−	0.849464i	0.140602	−	0.990066i	\(-0.455096\pi\)
							−0.990066	+	0.140602i	\(0.955096\pi\)
\(31\)	4.58133	+	3.06115i	0.822832	+	0.549799i	0.894208	−	0.447651i	\(-0.147739\pi\)
							−0.0713766	+	0.997449i	\(0.522739\pi\)
\(37\)	4.34977	+	6.50989i	0.715098	+	1.07022i	0.993944	+	0.109884i	\(0.0350479\pi\)
							−0.278846	+	0.960336i	\(0.589952\pi\)
\(41\)	5.84150	+	2.41963i	0.912289	+	0.377882i	0.788932	−	0.614480i	\(-0.210634\pi\)
							0.123356	+	0.992362i	\(0.460634\pi\)
\(43\)	−2.68145	+	6.47359i	−0.408917	+	0.987214i	0.576506	+	0.817093i	\(0.304416\pi\)
							−0.985423	+	0.170121i	\(0.945584\pi\)
\(47\)	3.63023	+	8.76415i	0.529523	+	1.27838i	0.931836	+	0.362880i	\(0.118206\pi\)
							−0.402313	+	0.915502i	\(0.631794\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	353.2.f.a.36.4	✓	232
353.304	even	16	inner	353.2.f.a.304.4	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
353.2.f.a.36.4	✓	232	1.1	even	1	trivial
353.2.f.a.304.4	yes	232	353.304	even	16	inner