Embedded newform 304.2.bg.a.3.7

• Label: 304.2.bg.a.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20851 - 0.734512i) q^{2} +(1.75061 + 0.816321i) q^{3} +(0.920986 + 1.77533i) q^{4} +(0.635225 + 0.907195i) q^{5} +(-1.51603 - 2.27237i) q^{6} +(0.220626 + 0.382136i) q^{7} +(0.190979 - 2.82197i) q^{8} +(0.469880 + 0.559981i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20851	−	0.734512i	−0.854545	−	0.519378i
							\(3\)	1.75061	+	0.816321i	1.01071	+	0.471303i	0.856176	−	0.516684i	\(-0.172834\pi\)
0.154537	+	0.987987i	\(0.450611\pi\)
\(5\)	0.635225	+	0.907195i	0.284081	+	0.405710i	0.935678	−	0.352856i	\(-0.114789\pi\)
							−0.651597	+	0.758566i	\(0.725900\pi\)
\(7\)	0.220626	+	0.382136i	0.0833888	+	0.144434i	0.904704	−	0.426042i	\(-0.140092\pi\)
							−0.821315	+	0.570475i	\(0.806759\pi\)
\(11\)	3.66209	−	0.981254i	1.10416	−	0.295859i	0.339703	−	0.940533i	\(-0.389673\pi\)
							0.764458	+	0.644673i	\(0.223007\pi\)
\(13\)	−0.867891	+	0.404704i	−0.240710	+	0.112245i	−0.539230	−	0.842158i	\(-0.681285\pi\)
							0.298520	+	0.954403i	\(0.403507\pi\)
\(17\)	4.12394	+	3.46040i	1.00020	+	0.839270i	0.987013	−	0.160643i	\(-0.0513569\pi\)
							0.0131900	+	0.999913i	\(0.495801\pi\)
\(19\)	−4.11312	+	1.44300i	−0.943614	+	0.331047i
							\(23\)	0.875109	+	4.96299i	0.182473	+	1.03486i	0.929159	+	0.369679i	\(0.120532\pi\)
−0.746687	+	0.665176i	\(0.768357\pi\)
\(29\)	−0.488971	+	5.58896i	−0.0907996	+	1.03784i	0.804967	+	0.593320i	\(0.202183\pi\)
							−0.895767	+	0.444525i	\(0.853373\pi\)
\(31\)	−1.05906	−	1.83434i	−0.190212	−	0.329457i	0.755108	−	0.655600i	\(-0.227584\pi\)
							−0.945320	+	0.326143i	\(0.894251\pi\)
\(37\)	−1.70013	−	1.70013i	−0.279499	−	0.279499i	0.553410	−	0.832909i	\(-0.313326\pi\)
							−0.832909	+	0.553410i	\(0.813326\pi\)
\(41\)	2.43887	−	0.887676i	0.380888	−	0.138632i	−0.144479	−	0.989508i	\(-0.546151\pi\)
							0.525366	+	0.850876i	\(0.323928\pi\)
\(43\)	1.92782	−	1.34988i	0.293990	−	0.205854i	−0.417271	−	0.908782i	\(-0.637013\pi\)
							0.711261	+	0.702928i	\(0.248124\pi\)
\(47\)	−6.67134	−	7.95059i	−0.973115	−	1.15971i	−0.987148	−	0.159812i	\(-0.948911\pi\)
							0.0140326	−	0.999902i	\(-0.495533\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.7	✓	456
16.11	odd	4	inner	304.2.bg.a.155.28	yes	456
19.13	odd	18	inner	304.2.bg.a.51.28	yes	456
304.203	even	36	inner	304.2.bg.a.203.7	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.7	✓	456	1.1	even	1	trivial
304.2.bg.a.51.28	yes	456	19.13	odd	18	inner
304.2.bg.a.155.28	yes	456	16.11	odd	4	inner
304.2.bg.a.203.7	yes	456	304.203	even	36	inner