Embedded newform 304.2.bg.a.243.28

• Label: 304.2.bg.a.243.28
• Level: $304$
• Weight: $2$
• Character: 304.243
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $456$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(456\)
Relative dimension:	\(38\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			243.28
Character	\(\chi\)	\(=\)	304.243
Dual form			304.2.bg.a.299.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.948468 - 1.04900i) q^{2} +(0.578080 - 0.825584i) q^{3} +(-0.200816 - 1.98989i) q^{4} +(0.127295 - 1.45499i) q^{5} +(-0.317750 - 1.38945i) q^{6} +(0.743996 - 1.28864i) q^{7} +(-2.27787 - 1.67669i) q^{8} +(0.678648 + 1.86457i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.948468	−	1.04900i	0.670668	−	0.741757i
							\(3\)	0.578080	−	0.825584i	0.333755	−	0.476651i	−0.616839	−	0.787089i	\(-0.711587\pi\)
0.950593	+	0.310438i	\(0.100476\pi\)
\(5\)	0.127295	−	1.45499i	0.0569281	−	0.650691i	−0.913106	−	0.407722i	\(-0.866323\pi\)
							0.970034	−	0.242969i	\(-0.0781213\pi\)
\(7\)	0.743996	−	1.28864i	0.281204	−	0.487060i	−0.690478	−	0.723354i	\(-0.742600\pi\)
							0.971682	+	0.236294i	\(0.0759329\pi\)
\(11\)	−0.782370	+	2.91985i	−0.235894	+	0.880367i	0.741851	+	0.670565i	\(0.233948\pi\)
							−0.977744	+	0.209801i	\(0.932718\pi\)
\(13\)	0.820888	+	1.17235i	0.227673	+	0.325151i	0.916694	−	0.399591i	\(-0.130848\pi\)
							−0.689020	+	0.724742i	\(0.741959\pi\)
\(17\)	−0.707656	−	0.257566i	−0.171632	−	0.0624689i	0.254775	−	0.967000i	\(-0.417999\pi\)
							−0.426407	+	0.904531i	\(0.640221\pi\)
\(19\)	−3.27574	−	2.87568i	−0.751506	−	0.659726i
							\(23\)	2.43347	+	2.04192i	0.507413	+	0.425770i	0.860218	−	0.509927i	\(-0.170328\pi\)
−0.352805	+	0.935697i	\(0.614772\pi\)
\(29\)	0.970002	−	2.08018i	0.180125	−	0.386279i	−0.795427	−	0.606050i	\(-0.792753\pi\)
							0.975552	+	0.219771i	\(0.0705310\pi\)
\(31\)	3.54359	−	6.13768i	0.636448	−	1.10236i	−0.349759	−	0.936840i	\(-0.613736\pi\)
							0.986206	−	0.165520i	\(-0.0529302\pi\)
\(37\)	4.41680	+	4.41680i	0.726118	+	0.726118i	0.969844	−	0.243726i	\(-0.0783698\pi\)
							−0.243726	+	0.969844i	\(0.578370\pi\)
\(41\)	0.593109	+	3.36369i	0.0926281	+	0.525320i	0.995448	+	0.0953020i	\(0.0303817\pi\)
							−0.902820	+	0.430018i	\(0.858507\pi\)
\(43\)	−11.2933	−	0.988040i	−1.72222	−	0.150675i	−0.817287	−	0.576230i	\(-0.804523\pi\)
							−0.904932	+	0.425556i	\(0.860079\pi\)
\(47\)	0.0302093	+	0.0829994i	0.00440648	+	0.0121067i	0.941876	−	0.335960i	\(-0.109061\pi\)
							−0.937470	+	0.348066i	\(0.886838\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.243.28	yes	456
16.11	odd	4	inner	304.2.bg.a.91.19	✓	456
19.14	odd	18	inner	304.2.bg.a.147.19	yes	456
304.299	even	36	inner	304.2.bg.a.299.28	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.91.19	✓	456	16.11	odd	4	inner
304.2.bg.a.147.19	yes	456	19.14	odd	18	inner
304.2.bg.a.243.28	yes	456	1.1	even	1	trivial
304.2.bg.a.299.28	yes	456	304.299	even	36	inner