Embedded newform 304.2.bg.a.3.4

• Label: 304.2.bg.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27934 - 0.602732i) q^{2} +(-2.12965 - 0.993074i) q^{3} +(1.27343 + 1.54220i) q^{4} +(-0.856388 - 1.22305i) q^{5} +(2.12600 + 2.55409i) q^{6} +(2.11839 + 3.66917i) q^{7} +(-0.699613 - 2.74054i) q^{8} +(1.62087 + 1.93167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27934	−	0.602732i	−0.904631	−	0.426196i
							\(3\)	−2.12965	−	0.993074i	−1.22956	−	0.573351i	−0.304257	−	0.952590i	\(-0.598408\pi\)
−0.925299	+	0.379239i	\(0.876186\pi\)
\(5\)	−0.856388	−	1.22305i	−0.382988	−	0.546964i	0.580657	−	0.814148i	\(-0.302796\pi\)
							−0.963645	+	0.267184i	\(0.913907\pi\)
\(7\)	2.11839	+	3.66917i	0.800678	+	1.38681i	0.919171	+	0.393859i	\(0.128860\pi\)
							−0.118493	+	0.992955i	\(0.537806\pi\)
\(11\)	4.58870	−	1.22954i	1.38355	−	0.370720i	0.511138	−	0.859499i	\(-0.329224\pi\)
							0.872408	+	0.488779i	\(0.162558\pi\)
\(13\)	−0.905909	+	0.422432i	−0.251254	+	0.117162i	−0.544166	−	0.838978i	\(-0.683154\pi\)
							0.292911	+	0.956140i	\(0.405376\pi\)
\(17\)	−3.06489	−	2.57175i	−0.743345	−	0.623741i	0.190388	−	0.981709i	\(-0.439025\pi\)
							−0.933734	+	0.357968i	\(0.883470\pi\)
\(19\)	1.38711	−	4.13230i	0.318225	−	0.948015i
							\(23\)	0.282296	+	1.60098i	0.0588627	+	0.333827i	0.999991	−	0.00422232i	\(-0.00134401\pi\)
−0.941128	+	0.338049i	\(0.890233\pi\)
\(29\)	0.670071	−	7.65894i	0.124429	−	1.42223i	−0.635438	−	0.772152i	\(-0.719181\pi\)
							0.759867	−	0.650078i	\(-0.225264\pi\)
\(31\)	−1.24784	−	2.16131i	−0.224118	−	0.388184i	0.731937	−	0.681373i	\(-0.238617\pi\)
							−0.956054	+	0.293189i	\(0.905283\pi\)
\(37\)	−1.64790	−	1.64790i	−0.270914	−	0.270914i	0.558554	−	0.829468i	\(-0.311356\pi\)
							−0.829468	+	0.558554i	\(0.811356\pi\)
\(41\)	9.32609	−	3.39442i	1.45649	−	0.530119i	0.512094	−	0.858929i	\(-0.328870\pi\)
							0.944396	+	0.328810i	\(0.106648\pi\)
\(43\)	8.28506	−	5.80126i	1.26346	−	0.884684i	0.266595	−	0.963809i	\(-0.414101\pi\)
							0.996865	+	0.0791247i	\(0.0252125\pi\)
\(47\)	1.97395	+	2.35246i	0.287930	+	0.343142i	0.890549	−	0.454887i	\(-0.150320\pi\)
							−0.602619	+	0.798029i	\(0.705876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.4	✓	456
16.11	odd	4	inner	304.2.bg.a.155.27	yes	456
19.13	odd	18	inner	304.2.bg.a.51.27	yes	456
304.203	even	36	inner	304.2.bg.a.203.4	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.4	✓	456	1.1	even	1	trivial
304.2.bg.a.51.27	yes	456	19.13	odd	18	inner
304.2.bg.a.155.27	yes	456	16.11	odd	4	inner
304.2.bg.a.203.4	yes	456	304.203	even	36	inner