Embedded newform 354.3.h.a.5.13

• Label: 354.3.h.a.5.13
• Level: $354$
• Weight: $3$
• Character: 354.5
• Analytic conductor: $9.646$
• Analytic rank: $0$
• Dimension: $1120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	354.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.64580135835\)
Analytic rank:	\(0\)
Dimension:	\(1120\)
Relative dimension:	\(40\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			5.13
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(1.39003 - 2.65854i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(-7.10734 - 2.83182i) q^{5} +(2.93524 + 3.06339i) q^{6} +(7.21460 - 3.33783i) q^{7} +(2.34106 - 1.58728i) q^{8} +(-5.13563 - 7.39089i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.451561	+	1.34018i	−0.225780	+	0.670092i
							\(3\)	1.39003	−	2.65854i	0.463344	−	0.886179i
\(5\)	−7.10734	−	2.83182i	−1.42147	−	0.566365i								−0.472386	−	0.881392i	\(-0.656607\pi\)
							−0.949083	+	0.315027i	\(0.897986\pi\)
\(7\)	7.21460	−	3.33783i	1.03066	−	0.476832i	0.169692	−	0.985497i	\(-0.445723\pi\)
							0.860965	+	0.508665i	\(0.169861\pi\)
\(11\)	−1.77764	−	0.493560i	−0.161604	−	0.0448691i	0.185784	−	0.982591i	\(-0.440517\pi\)
							−0.347388	+	0.937722i	\(0.612931\pi\)
\(13\)	−6.81946	+	12.8629i	−0.524574	+	0.989451i	0.469724	+	0.882813i	\(0.344353\pi\)
							−0.994298	+	0.106638i	\(0.965991\pi\)
\(17\)	−2.71642	+	5.87144i	−0.159789	+	0.345379i	−0.970889	−	0.239529i	\(-0.923007\pi\)
							0.811100	+	0.584908i	\(0.198869\pi\)
\(19\)	−2.31673	+	0.509952i	−0.121933	+	0.0268396i	−0.275518	−	0.961296i	\(-0.588849\pi\)
							0.153584	+	0.988136i	\(0.450918\pi\)
\(23\)	−44.2237	−	7.25011i	−1.92277	−	0.315222i	−0.924887	−	0.380241i	\(-0.875841\pi\)
							−0.997884	+	0.0650188i	\(0.979289\pi\)
\(29\)	−2.33111	−	6.91850i	−0.0803833	−	0.238569i	0.899924	−	0.436046i	\(-0.143622\pi\)
							−0.980307	+	0.197477i	\(0.936725\pi\)
\(31\)	−17.8623	−	3.93179i	−0.576204	−	0.126832i	−0.0826982	−	0.996575i	\(-0.526354\pi\)
							−0.493506	+	0.869743i	\(0.664285\pi\)
\(37\)	18.1366	−	26.7495i	0.490179	−	0.722960i	−0.499492	−	0.866319i	\(-0.666480\pi\)
							0.989671	+	0.143358i	\(0.0457902\pi\)
\(41\)	−12.4095	+	2.03443i	−0.302670	+	0.0496202i	−0.311203	−	0.950343i	\(-0.600732\pi\)
							0.00853348	+	0.999964i	\(0.497284\pi\)
\(43\)	16.3494	+	58.8854i	0.380220	+	1.36943i	0.868045	+	0.496486i	\(0.165377\pi\)
							−0.487825	+	0.872941i	\(0.662210\pi\)
\(47\)	−62.8292	+	25.0335i	−1.33679	+	0.532627i	−0.925455	−	0.378857i	\(-0.876317\pi\)
							−0.411337	+	0.911483i	\(0.634938\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.13	✓	1120
3.2	odd	2	inner	354.3.h.a.5.28	yes	1120
59.12	even	29	inner	354.3.h.a.71.28	yes	1120
177.71	odd	58	inner	354.3.h.a.71.13	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.13	✓	1120	1.1	even	1	trivial
354.3.h.a.5.28	yes	1120	3.2	odd	2	inner
354.3.h.a.71.13	yes	1120	177.71	odd	58	inner
354.3.h.a.71.28	yes	1120	59.12	even	29	inner