Embedded newform 58.7.f.b.11.6

• Label: 58.7.f.b.11.6
• Level: $58$
• Weight: $7$
• Character: 58.11
• Analytic conductor: $13.343$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	58.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.3431368500\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(8\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	58.11
Dual form			58.7.f.b.37.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.33941 + 1.86834i) q^{2} +(2.20529 + 19.5725i) q^{3} +(25.0186 - 19.9517i) q^{4} +(-4.56263 + 9.47440i) q^{5} +(-48.3430 - 100.385i) q^{6} +(151.066 - 189.431i) q^{7} +(-96.3081 + 153.273i) q^{8} +(332.504 - 75.8920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 64 q^{2} - 28 q^{3} - 588 q^{5} - 610 q^{7} + 2048 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{25}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.33941	+	1.86834i	−0.667426	+	0.233543i
							\(3\)	2.20529	+	19.5725i	0.0816773	+	0.724906i	0.966091	+	0.258202i	\(0.0831300\pi\)
−0.884414	+	0.466704i	\(0.845441\pi\)
\(5\)	−4.56263	+	9.47440i	−0.0365010	+	0.0757952i	−0.918422	−	0.395602i	\(-0.870536\pi\)
							0.881921	+	0.471397i	\(0.156250\pi\)
\(7\)	151.066	−	189.431i	0.440425	−	0.552276i	−0.511230	−	0.859444i	\(-0.670810\pi\)
							0.951655	+	0.307168i	\(0.0993814\pi\)
\(11\)	−1163.12	−	1851.09i	−0.873869	−	1.39075i	−0.920042	−	0.391821i	\(-0.871845\pi\)
							0.0461729	−	0.998933i	\(-0.485297\pi\)
\(13\)	2018.96	+	460.814i	0.918961	+	0.209747i	0.655738	−	0.754988i	\(-0.272358\pi\)
							0.263222	+	0.964735i	\(0.415215\pi\)
\(17\)	−3342.25	+	3342.25i	−0.680287	+	0.680287i	−0.960065	−	0.279778i	\(-0.909739\pi\)
							0.279778	+	0.960065i	\(0.409739\pi\)
\(19\)	7149.59	+	805.565i	1.04237	+	0.117446i	0.616512	−	0.787345i	\(-0.288545\pi\)
							0.425854	+	0.904792i	\(0.359974\pi\)
\(23\)	19895.7	−	9581.28i	1.63522	−	0.787481i	0.635337	−	0.772235i	\(-0.280861\pi\)
							0.999884	−	0.0152459i	\(-0.00485311\pi\)
\(29\)	24077.0	+	3888.42i	0.987209	+	0.159433i
							\(31\)	−17529.6	+	6133.86i	−0.588418	+	0.205896i	−0.608034	−	0.793911i	\(-0.708041\pi\)
0.0196157	+	0.999808i	\(0.493756\pi\)
\(37\)	12773.1	−	20328.2i	0.252168	−	0.401323i	−0.696520	−	0.717538i	\(-0.745269\pi\)
							0.948688	+	0.316215i	\(0.102412\pi\)
\(41\)	−51023.0	−	51023.0i	−0.740311	−	0.740311i	0.232327	−	0.972638i	\(-0.425366\pi\)
							−0.972638	+	0.232327i	\(0.925366\pi\)
\(43\)	−28106.3	+	80323.2i	−0.353507	+	1.01027i	0.620590	+	0.784135i	\(0.286893\pi\)
							−0.974097	+	0.226130i	\(0.927393\pi\)
\(47\)	127512.	−	80121.0i	1.22817	−	0.771708i	0.247784	−	0.968815i	\(-0.420298\pi\)
							0.980382	+	0.197108i	\(0.0631549\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.7.f.b.11.6	✓	96
29.8	odd	28	inner	58.7.f.b.37.6	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.7.f.b.11.6	✓	96	1.1	even	1	trivial
58.7.f.b.37.6	yes	96	29.8	odd	28	inner