Embedded newform 354.3.h.a.71.13

• Label: 354.3.h.a.71.13
• Level: $354$
• Weight: $3$
• Character: 354.71
• 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			71.13
Character	\(\chi\)	\(=\)	354.71
Dual form			354.3.h.a.5.13

$q$-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 \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.13	✓	1120	177.5	odd	58	inner
354.3.h.a.5.28	yes	1120	59.5	even	29	inner
354.3.h.a.71.13	yes	1120	1.1	even	1	trivial
354.3.h.a.71.28	yes	1120	3.2	odd	2	inner