Embedded newform 304.2.bg.a.3.16

• Label: 304.2.bg.a.3.16
• 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			3.16
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.443631 + 1.34283i) q^{2} +(2.49171 + 1.16190i) q^{3} +(-1.60638 - 1.19144i) q^{4} +(-0.636818 - 0.909471i) q^{5} +(-2.66564 + 2.83049i) q^{6} +(1.63512 + 2.83211i) q^{7} +(2.31255 - 1.62854i) q^{8} +(2.93025 + 3.49213i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.443631	+	1.34283i	−0.313695	+	0.949524i
							\(3\)	2.49171	+	1.16190i	1.43859	+	0.670826i	0.976403	−	0.215956i	\(-0.0692868\pi\)
0.462188	+	0.886782i	\(0.347065\pi\)
\(5\)	−0.636818	−	0.909471i	−0.284794	−	0.406728i	0.651110	−	0.758983i	\(-0.274304\pi\)
							−0.935904	+	0.352256i	\(0.885415\pi\)
\(7\)	1.63512	+	2.83211i	0.618018	+	1.07044i	0.989847	+	0.142137i	\(0.0453975\pi\)
							−0.371829	+	0.928301i	\(0.621269\pi\)
\(11\)	−5.36350	+	1.43715i	−1.61716	+	0.433316i	−0.950165	−	0.311749i	\(-0.899085\pi\)
							−0.666992	+	0.745064i	\(0.732419\pi\)
\(13\)	2.49521	−	1.16354i	0.692048	−	0.322707i	−0.0445798	−	0.999006i	\(-0.514195\pi\)
							0.736628	+	0.676299i	\(0.236417\pi\)
\(17\)	5.30713	+	4.45321i	1.28717	+	1.08006i	0.992213	+	0.124551i	\(0.0397490\pi\)
							0.294955	+	0.955511i	\(0.404695\pi\)
\(19\)	−2.08579	+	3.82746i	−0.478514	+	0.878080i
							\(23\)	−0.991533	−	5.62326i	−0.206749	−	1.17253i	−0.894664	−	0.446740i	\(-0.852585\pi\)
0.687915	−	0.725791i	\(-0.258526\pi\)
\(29\)	0.783695	−	8.95767i	0.145528	−	1.66340i	−0.475579	−	0.879673i	\(-0.657761\pi\)
							0.621108	−	0.783725i	\(-0.286683\pi\)
\(31\)	0.847934	+	1.46866i	0.152293	+	0.263780i	0.932070	−	0.362278i	\(-0.118001\pi\)
							−0.779777	+	0.626058i	\(0.784667\pi\)
\(37\)	−1.62097	−	1.62097i	−0.266487	−	0.266487i	0.561196	−	0.827683i	\(-0.310341\pi\)
							−0.827683	+	0.561196i	\(0.810341\pi\)
\(41\)	−2.15992	+	0.786146i	−0.337323	+	0.122775i	−0.505127	−	0.863045i	\(-0.668554\pi\)
							0.167805	+	0.985820i	\(0.446332\pi\)
\(43\)	3.17380	−	2.22232i	0.483999	−	0.338900i	−0.305975	−	0.952039i	\(-0.598983\pi\)
							0.789975	+	0.613139i	\(0.210094\pi\)
\(47\)	−3.48503	−	4.15330i	−0.508344	−	0.605821i	0.449439	−	0.893311i	\(-0.351624\pi\)
							−0.957784	+	0.287489i	\(0.907179\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.16	✓	456
16.11	odd	4	inner	304.2.bg.a.155.7	yes	456
19.13	odd	18	inner	304.2.bg.a.51.7	yes	456
304.203	even	36	inner	304.2.bg.a.203.16	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.16	✓	456	1.1	even	1	trivial
304.2.bg.a.51.7	yes	456	19.13	odd	18	inner
304.2.bg.a.155.7	yes	456	16.11	odd	4	inner
304.2.bg.a.203.16	yes	456	304.203	even	36	inner