Embedded newform 304.2.bg.a.219.24

• Label: 304.2.bg.a.219.24
• Level: $304$
• Weight: $2$
• Character: 304.219
• 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			219.24
Character	\(\chi\)	\(=\)	304.219
Dual form			304.2.bg.a.211.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.401433 + 1.35604i) q^{2} +(0.242708 - 2.77416i) q^{3} +(-1.67770 + 1.08872i) q^{4} +(-0.947648 - 2.03224i) q^{5} +(3.85931 - 0.784518i) q^{6} +(-0.0320165 + 0.0554542i) q^{7} +(-2.14984 - 1.83799i) q^{8} +(-4.68264 - 0.825676i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.401433	+	1.35604i	0.283856	+	0.958867i
							\(3\)	0.242708	−	2.77416i	0.140127	−	1.60166i	−0.522450	−	0.852670i	\(-0.674982\pi\)
0.662578	−	0.748993i	\(-0.269462\pi\)
\(5\)	−0.947648	−	2.03224i	−0.423801	−	0.908845i	−0.996052	−	0.0887663i	\(-0.971708\pi\)
							0.572251	−	0.820078i	\(-0.306070\pi\)
\(7\)	−0.0320165	+	0.0554542i	−0.0121011	+	0.0209597i	−0.872013	−	0.489484i	\(-0.837185\pi\)
							0.859911	+	0.510443i	\(0.170519\pi\)
\(11\)	−4.04848	−	1.08479i	−1.22066	−	0.327076i	−0.409726	−	0.912209i	\(-0.634376\pi\)
							−0.810938	+	0.585133i	\(0.801042\pi\)
\(13\)	−0.161592	−	1.84700i	−0.0448175	−	0.512266i	−0.985113	−	0.171907i	\(-0.945007\pi\)
							0.940296	−	0.340359i	\(-0.110549\pi\)
\(17\)	1.04297	+	5.91496i	0.252957	+	1.43459i	0.801264	+	0.598312i	\(0.204161\pi\)
							−0.548307	+	0.836277i	\(0.684727\pi\)
\(19\)	−1.50896	−	4.08938i	−0.346180	−	0.938168i
							\(23\)	7.10810	−	2.58714i	1.48214	−	0.539455i	0.530774	−	0.847513i	\(-0.321901\pi\)
0.951368	+	0.308058i	\(0.0996791\pi\)
\(29\)	2.17175	−	1.52067i	0.403283	−	0.282382i	−0.354285	−	0.935137i	\(-0.615276\pi\)
							0.757569	+	0.652755i	\(0.226387\pi\)
\(31\)	2.86851	−	4.96840i	0.515199	−	0.892351i	−0.484645	−	0.874711i	\(-0.661051\pi\)
							0.999844	−	0.0176405i	\(-0.00561544\pi\)
\(37\)	−2.41985	+	2.41985i	−0.397820	+	0.397820i	−0.877464	−	0.479643i	\(-0.840766\pi\)
							0.479643	+	0.877464i	\(0.340766\pi\)
\(41\)	−4.08505	+	3.42777i	−0.637978	+	0.535327i	−0.903397	−	0.428806i	\(-0.858934\pi\)
							0.265418	+	0.964133i	\(0.414490\pi\)
\(43\)	8.35744	−	3.89714i	1.27450	−	0.594308i	0.336740	−	0.941598i	\(-0.390676\pi\)
							0.937758	+	0.347290i	\(0.112898\pi\)
\(47\)	−1.34512	−	0.237180i	−0.196205	−	0.0345963i	0.0746817	−	0.997207i	\(-0.476206\pi\)
							−0.270887	+	0.962611i	\(0.587317\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.219.24	yes	456
16.3	odd	4	inner	304.2.bg.a.67.20	yes	456
19.2	odd	18	inner	304.2.bg.a.59.20	✓	456
304.211	even	36	inner	304.2.bg.a.211.24	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.59.20	✓	456	19.2	odd	18	inner
304.2.bg.a.67.20	yes	456	16.3	odd	4	inner
304.2.bg.a.211.24	yes	456	304.211	even	36	inner
304.2.bg.a.219.24	yes	456	1.1	even	1	trivial