Embedded newform 354.3.f.a.13.14

• Label: 354.3.f.a.13.14
• Level: $354$
• Weight: $3$
• Character: 354.13
• Analytic conductor: $9.646$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.64580135835\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(20\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			13.14
Character	\(\chi\)	\(=\)	354.13
Dual form			354.3.f.a.109.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07786 - 0.915542i) q^{2} +(-1.37887 + 1.04819i) q^{3} +(0.323564 - 1.97365i) q^{4} +(-0.150737 + 0.284320i) q^{5} +(-0.526568 + 2.39222i) q^{6} +(-7.95376 + 0.865024i) q^{7} +(-1.45821 - 2.42356i) q^{8} +(0.802585 - 2.89065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} - 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{45}{58}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.07786	−	0.915542i	0.538930	−	0.457771i
							\(3\)	−1.37887	+	1.04819i	−0.459625	+	0.349397i
\(5\)	−0.150737	+	0.284320i	−0.0301474	+	0.0568641i								−0.898128	−	0.439734i	\(-0.855073\pi\)
							0.867981	+	0.496598i	\(0.165418\pi\)
\(7\)	−7.95376	+	0.865024i	−1.13625	+	0.123575i	−0.656850	−	0.754021i	\(-0.728112\pi\)
							−0.479401	+	0.877596i	\(0.659146\pi\)
\(11\)	6.38205	+	18.9412i	0.580186	+	1.72193i	0.686054	+	0.727551i	\(0.259341\pi\)
							−0.105868	+	0.994380i	\(0.533762\pi\)
\(13\)	23.5704	−	6.54428i	1.81310	−	0.503406i	0.815905	−	0.578187i	\(-0.196239\pi\)
							0.997200	+	0.0747807i	\(0.0238257\pi\)
\(17\)	12.8702	+	1.39972i	0.757072	+	0.0823366i	0.478511	−	0.878082i	\(-0.341177\pi\)
							0.278562	+	0.960418i	\(0.410142\pi\)
\(19\)	1.05823	+	19.5180i	0.0556966	+	1.02726i	0.884149	+	0.467204i	\(0.154739\pi\)
							−0.828453	+	0.560059i	\(0.810779\pi\)
\(23\)	−7.85450	+	16.9772i	−0.341500	+	0.738141i	−0.999894	−	0.0145327i	\(-0.995374\pi\)
							0.658394	+	0.752673i	\(0.271236\pi\)
\(29\)	13.8339	−	16.2865i	0.477030	−	0.561603i	−0.470034	−	0.882648i	\(-0.655758\pi\)
							0.947064	+	0.321045i	\(0.104034\pi\)
\(31\)	−19.4452	−	1.05429i	−0.627264	−	0.0340093i	−0.262233	−	0.965005i	\(-0.584459\pi\)
							−0.365031	+	0.930995i	\(0.618942\pi\)
\(37\)	−12.0518	+	20.0302i	−0.325723	+	0.541357i	−0.975842	−	0.218476i	\(-0.929891\pi\)
							0.650119	+	0.759832i	\(0.274719\pi\)
\(41\)	−33.6145	+	15.5517i	−0.819865	+	0.379310i	−0.784550	−	0.620066i	\(-0.787106\pi\)
							−0.0353157	+	0.999376i	\(0.511244\pi\)
\(43\)	2.18103	−	6.47306i	0.0507216	−	0.150536i	−0.919299	−	0.393561i	\(-0.871243\pi\)
							0.970020	+	0.243025i	\(0.0781396\pi\)
\(47\)	57.7778	−	30.6319i	1.22932	−	0.651742i	0.278184	−	0.960528i	\(-0.410268\pi\)
							0.951131	+	0.308786i	\(0.0999228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.f.a.13.14	✓	560
59.50	odd	58	inner	354.3.f.a.109.14	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.f.a.13.14	✓	560	1.1	even	1	trivial
354.3.f.a.109.14	yes	560	59.50	odd	58	inner