Embedded newform 304.2.bg.a.243.18

• Label: 304.2.bg.a.243.18
• 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			243.18
Character	\(\chi\)	\(=\)	304.243
Dual form			304.2.bg.a.299.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.255652 + 1.39091i) q^{2} +(1.53085 - 2.18628i) q^{3} +(-1.86928 - 0.711181i) q^{4} +(0.112166 - 1.28206i) q^{5} +(2.64956 + 2.68820i) q^{6} +(-0.169261 + 0.293169i) q^{7} +(1.46708 - 2.41820i) q^{8} +(-1.41025 - 3.87464i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.255652	+	1.39091i	−0.180774	+	0.983525i
							\(3\)	1.53085	−	2.18628i	0.883835	−	1.26225i	−0.0802823	−	0.996772i	\(-0.525582\pi\)
0.964118	−	0.265475i	\(-0.0855289\pi\)
\(5\)	0.112166	−	1.28206i	0.0501619	−	0.573354i	−0.928974	−	0.370146i	\(-0.879308\pi\)
							0.979136	−	0.203208i	\(-0.0651367\pi\)
\(7\)	−0.169261	+	0.293169i	−0.0639748	+	0.110808i	−0.896239	−	0.443572i	\(-0.853711\pi\)
							0.832264	+	0.554380i	\(0.187044\pi\)
\(11\)	0.0132048	−	0.0492810i	0.00398140	−	0.0148588i	−0.963907	−	0.266241i	\(-0.914218\pi\)
							0.967888	+	0.251382i	\(0.0808850\pi\)
\(13\)	−1.98293	−	2.83192i	−0.549966	−	0.785432i	0.443894	−	0.896079i	\(-0.353597\pi\)
							−0.993860	+	0.110647i	\(0.964708\pi\)
\(17\)	3.23502	+	1.17745i	0.784607	+	0.285574i	0.703093	−	0.711098i	\(-0.251802\pi\)
							0.0815149	+	0.996672i	\(0.474024\pi\)
\(19\)	0.512186	−	4.32870i	0.117503	−	0.993072i
							\(23\)	−2.55433	−	2.14334i	−0.532615	−	0.446917i	0.336389	−	0.941723i	\(-0.390794\pi\)
−0.869003	+	0.494807i	\(0.835239\pi\)
\(29\)	1.60013	−	3.43149i	0.297136	−	0.637211i	−0.699892	−	0.714249i	\(-0.746768\pi\)
							0.997028	+	0.0770379i	\(0.0245462\pi\)
\(31\)	−4.05802	+	7.02870i	−0.728842	+	1.26239i	0.228531	+	0.973537i	\(0.426608\pi\)
							−0.957373	+	0.288854i	\(0.906726\pi\)
\(37\)	4.32118	+	4.32118i	0.710397	+	0.710397i	0.966618	−	0.256221i	\(-0.0824775\pi\)
							−0.256221	+	0.966618i	\(0.582477\pi\)
\(41\)	1.27230	+	7.21557i	0.198700	+	1.12688i	0.907050	+	0.421022i	\(0.138329\pi\)
							−0.708351	+	0.705861i	\(0.750560\pi\)
\(43\)	−1.24215	−	0.108674i	−0.189427	−	0.0165727i	−0.00795238	−	0.999968i	\(-0.502531\pi\)
							−0.181474	+	0.983396i	\(0.558087\pi\)
\(47\)	1.02614	+	2.81929i	0.149677	+	0.411235i	0.991759	−	0.128114i	\(-0.0408924\pi\)
							−0.842082	+	0.539349i	\(0.818670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.243.18	yes	456
16.11	odd	4	inner	304.2.bg.a.91.13	✓	456
19.14	odd	18	inner	304.2.bg.a.147.13	yes	456
304.299	even	36	inner	304.2.bg.a.299.18	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.91.13	✓	456	16.11	odd	4	inner
304.2.bg.a.147.13	yes	456	19.14	odd	18	inner
304.2.bg.a.243.18	yes	456	1.1	even	1	trivial
304.2.bg.a.299.18	yes	456	304.299	even	36	inner