Embedded newform 304.2.bg.a.3.2

• Label: 304.2.bg.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38797 + 0.271180i) q^{2} +(2.35855 + 1.09981i) q^{3} +(1.85292 - 0.752781i) q^{4} +(1.12887 + 1.61220i) q^{5} +(-3.57184 - 0.886910i) q^{6} +(0.814974 + 1.41158i) q^{7} +(-2.36766 + 1.54731i) q^{8} +(2.42481 + 2.88978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38797	+	0.271180i	−0.981443	+	0.191754i
							\(3\)	2.35855	+	1.09981i	1.36171	+	0.634975i	0.959795	−	0.280702i	\(-0.0905673\pi\)
0.401914	+	0.915677i	\(0.368345\pi\)
\(5\)	1.12887	+	1.61220i	0.504848	+	0.720997i	0.987985	−	0.154547i	\(-0.0493918\pi\)
							−0.483138	+	0.875544i	\(0.660503\pi\)
\(7\)	0.814974	+	1.41158i	0.308031	+	0.533525i	0.977932	−	0.208925i	\(-0.0669966\pi\)
							−0.669901	+	0.742451i	\(0.733663\pi\)
\(11\)	−2.18860	+	0.586433i	−0.659887	+	0.176816i	−0.573195	−	0.819419i	\(-0.694296\pi\)
							−0.0866921	+	0.996235i	\(0.527630\pi\)
\(13\)	0.0883852	−	0.0412147i	0.0245136	−	0.0114309i	−0.410323	−	0.911940i	\(-0.634584\pi\)
							0.434836	+	0.900509i	\(0.356806\pi\)
\(17\)	−3.10369	−	2.60431i	−0.752757	−	0.631638i	0.183474	−	0.983025i	\(-0.441266\pi\)
							−0.936230	+	0.351387i	\(0.885710\pi\)
\(19\)	0.322435	−	4.34696i	0.0739716	−	0.997260i
							\(23\)	−0.445610	−	2.52718i	−0.0929161	−	0.526953i	−0.995366	−	0.0961585i	\(-0.969344\pi\)
0.902450	−	0.430795i	\(-0.141767\pi\)
\(29\)	−0.215572	+	2.46400i	−0.0400307	+	0.457553i	0.949554	+	0.313603i	\(0.101536\pi\)
							−0.989585	+	0.143950i	\(0.954020\pi\)
\(31\)	0.251483	+	0.435582i	0.0451677	+	0.0782328i	0.887725	−	0.460373i	\(-0.152284\pi\)
							−0.842558	+	0.538606i	\(0.818951\pi\)
\(37\)	5.31265	+	5.31265i	0.873394	+	0.873394i	0.992841	−	0.119446i	\(-0.0381119\pi\)
							−0.119446	+	0.992841i	\(0.538112\pi\)
\(41\)	8.54153	−	3.10886i	1.33396	−	0.485523i	0.426056	−	0.904697i	\(-0.359903\pi\)
							0.907906	+	0.419174i	\(0.137680\pi\)
\(43\)	−6.82234	+	4.77706i	−1.04040	+	0.728494i	−0.963272	−	0.268527i	\(-0.913463\pi\)
							−0.0771253	+	0.997021i	\(0.524574\pi\)
\(47\)	4.02260	+	4.79395i	0.586757	+	0.699270i	0.974979	−	0.222296i	\(-0.0713552\pi\)
							−0.388222	+	0.921566i	\(0.626911\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.2	✓	456
16.11	odd	4	inner	304.2.bg.a.155.20	yes	456
19.13	odd	18	inner	304.2.bg.a.51.20	yes	456
304.203	even	36	inner	304.2.bg.a.203.2	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.2	✓	456	1.1	even	1	trivial
304.2.bg.a.51.20	yes	456	19.13	odd	18	inner
304.2.bg.a.155.20	yes	456	16.11	odd	4	inner
304.2.bg.a.203.2	yes	456	304.203	even	36	inner