Embedded newform 304.2.bg.a.3.13

• Label: 304.2.bg.a.3.13
• 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			3.13
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.610064 + 1.27586i) q^{2} +(0.759059 + 0.353955i) q^{3} +(-1.25564 - 1.55671i) q^{4} +(2.32234 + 3.31664i) q^{5} +(-0.914672 + 0.752519i) q^{6} +(0.950093 + 1.64561i) q^{7} +(2.75217 - 0.652334i) q^{8} +(-1.47748 - 1.76079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.610064	+	1.27586i	−0.431380	+	0.902170i
							\(3\)	0.759059	+	0.353955i	0.438243	+	0.204356i	0.629202	−	0.777242i	\(-0.283382\pi\)
−0.190959	+	0.981598i	\(0.561160\pi\)
\(5\)	2.32234	+	3.31664i	1.03858	+	1.48325i	0.867147	+	0.498052i	\(0.165951\pi\)
							0.171434	+	0.985196i	\(0.445160\pi\)
\(7\)	0.950093	+	1.64561i	0.359102	+	0.621982i	0.987811	−	0.155658i	\(-0.0497498\pi\)
							−0.628709	+	0.777640i	\(0.716417\pi\)
\(11\)	3.02972	−	0.811812i	0.913496	−	0.244770i	0.228693	−	0.973499i	\(-0.426555\pi\)
							0.684803	+	0.728728i	\(0.259888\pi\)
\(13\)	0.913247	−	0.425854i	0.253289	−	0.118111i	−0.291827	−	0.956471i	\(-0.594263\pi\)
							0.545116	+	0.838360i	\(0.316485\pi\)
\(17\)	−5.79533	−	4.86286i	−1.40558	−	1.17942i	−0.958560	−	0.284891i	\(-0.908043\pi\)
							−0.447015	−	0.894526i	\(-0.647513\pi\)
\(19\)	0.157301	+	4.35606i	0.0360874	+	0.999349i
							\(23\)	0.966884	+	5.48347i	0.201609	+	1.14338i	0.902687	+	0.430298i	\(0.141592\pi\)
−0.701077	+	0.713085i	\(0.747297\pi\)
\(29\)	0.387662	−	4.43099i	0.0719870	−	0.822815i	−0.871282	−	0.490784i	\(-0.836711\pi\)
							0.943268	−	0.332031i	\(-0.107734\pi\)
\(31\)	0.152383	+	0.263936i	0.0273688	+	0.0474042i	0.879385	−	0.476111i	\(-0.157954\pi\)
							−0.852017	+	0.523515i	\(0.824620\pi\)
\(37\)	−1.87350	−	1.87350i	−0.308001	−	0.308001i	0.536132	−	0.844134i	\(-0.319885\pi\)
							−0.844134	+	0.536132i	\(0.819885\pi\)
\(41\)	6.24647	−	2.27353i	0.975535	−	0.355066i	0.195433	−	0.980717i	\(-0.437389\pi\)
							0.780103	+	0.625651i	\(0.215167\pi\)
\(43\)	−3.73710	+	2.61674i	−0.569902	+	0.399050i	−0.822705	−	0.568468i	\(-0.807536\pi\)
							0.252804	+	0.967518i	\(0.418647\pi\)
\(47\)	−6.82286	−	8.13117i	−0.995217	−	1.18605i	−0.982524	−	0.186133i	\(-0.940404\pi\)
							−0.0126921	−	0.999919i	\(-0.504040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.13	✓	456
16.11	odd	4	inner	304.2.bg.a.155.9	yes	456
19.13	odd	18	inner	304.2.bg.a.51.9	yes	456
304.203	even	36	inner	304.2.bg.a.203.13	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.13	✓	456	1.1	even	1	trivial
304.2.bg.a.51.9	yes	456	19.13	odd	18	inner
304.2.bg.a.155.9	yes	456	16.11	odd	4	inner
304.2.bg.a.203.13	yes	456	304.203	even	36	inner