Embedded newform 304.2.bg.a.3.20

• Label: 304.2.bg.a.3.20
• Level: $304$
• Weight: $2$
• Character: 304.3
• 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			3.20
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0596919 - 1.41295i) q^{2} +(-2.91030 - 1.35709i) q^{3} +(-1.99287 - 0.168684i) q^{4} +(1.72144 + 2.45847i) q^{5} +(-2.09123 + 4.03111i) q^{6} +(0.331324 + 0.573871i) q^{7} +(-0.357301 + 2.80577i) q^{8} +(4.69976 + 5.60096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0596919	−	1.41295i	0.0422085	−	0.999109i
							\(3\)	−2.91030	−	1.35709i	−1.68026	−	0.783518i	−0.998658	−	0.0517961i	\(-0.983505\pi\)
−0.681603	−	0.731722i	\(-0.738717\pi\)
\(5\)	1.72144	+	2.45847i	0.769851	+	1.09946i	0.992220	+	0.124499i	\(0.0397323\pi\)
							−0.222369	+	0.974963i	\(0.571379\pi\)
\(7\)	0.331324	+	0.573871i	0.125229	+	0.216903i	0.921822	−	0.387613i	\(-0.126700\pi\)
							−0.796594	+	0.604515i	\(0.793367\pi\)
\(11\)	−0.250361	+	0.0670841i	−0.0754868	+	0.0202266i	−0.296365	−	0.955075i	\(-0.595774\pi\)
							0.220878	+	0.975301i	\(0.429108\pi\)
\(13\)	−1.97419	+	0.920578i	−0.547540	+	0.255322i	−0.676646	−	0.736309i	\(-0.736567\pi\)
							0.129105	+	0.991631i	\(0.458789\pi\)
\(17\)	−0.561931	−	0.471516i	−0.136288	−	0.114360i	0.572095	−	0.820187i	\(-0.306131\pi\)
							−0.708384	+	0.705828i	\(0.750575\pi\)
\(19\)	2.43180	−	3.61751i	0.557893	−	0.829913i
							\(23\)	1.47655	+	8.37391i	0.307881	+	1.74608i	0.609623	+	0.792691i	\(0.291321\pi\)
−0.301742	+	0.953390i	\(0.597568\pi\)
\(29\)	−0.193274	+	2.20913i	−0.0358901	+	0.410225i	0.956905	+	0.290401i	\(0.0937887\pi\)
							−0.992795	+	0.119824i	\(0.961767\pi\)
\(31\)	5.39660	+	9.34719i	0.969259	+	1.67881i	0.697710	+	0.716381i	\(0.254203\pi\)
							0.271549	+	0.962425i	\(0.412464\pi\)
\(37\)	1.78230	+	1.78230i	0.293008	+	0.293008i	0.838267	−	0.545259i	\(-0.183569\pi\)
							−0.545259	+	0.838267i	\(0.683569\pi\)
\(41\)	8.71047	−	3.17035i	1.36035	−	0.495126i	0.444186	−	0.895935i	\(-0.353493\pi\)
							0.916162	+	0.400809i	\(0.131271\pi\)
\(43\)	−7.43651	+	5.20710i	−1.13406	+	0.794076i	−0.980974	−	0.194138i	\(-0.937809\pi\)
							−0.153083	+	0.988213i	\(0.548920\pi\)
\(47\)	1.57901	+	1.88179i	0.230322	+	0.274487i	0.868811	−	0.495144i	\(-0.164885\pi\)
							−0.638489	+	0.769631i	\(0.720440\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.20	✓	456
16.11	odd	4	inner	304.2.bg.a.155.35	yes	456
19.13	odd	18	inner	304.2.bg.a.51.35	yes	456
304.203	even	36	inner	304.2.bg.a.203.20	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.20	✓	456	1.1	even	1	trivial
304.2.bg.a.51.35	yes	456	19.13	odd	18	inner
304.2.bg.a.155.35	yes	456	16.11	odd	4	inner
304.2.bg.a.203.20	yes	456	304.203	even	36	inner