Embedded newform 304.2.bg.a.211.15

• Label: 304.2.bg.a.211.15
• Level: $304$
• Weight: $2$
• Character: 304.211
• 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			211.15
Character	\(\chi\)	\(=\)	304.211
Dual form			304.2.bg.a.219.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.509489 - 1.31925i) q^{2} +(0.00341934 + 0.0390833i) q^{3} +(-1.48084 + 1.34429i) q^{4} +(-0.335049 + 0.718516i) q^{5} +(0.0498185 - 0.0244235i) q^{6} +(-0.378432 - 0.655464i) q^{7} +(2.52792 + 1.26870i) q^{8} +(2.95291 - 0.520677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.509489	−	1.31925i	−0.360263	−	0.932851i
							\(3\)	0.00341934	+	0.0390833i	0.00197416	+	0.0225647i	0.997118	−	0.0758678i	\(-0.0241727\pi\)
−0.995144	+	0.0984325i	\(0.968617\pi\)
\(5\)	−0.335049	+	0.718516i	−0.149839	+	0.321330i	−0.966873	−	0.255260i	\(-0.917839\pi\)
							0.817034	+	0.576590i	\(0.195617\pi\)
\(7\)	−0.378432	−	0.655464i	−0.143034	−	0.247742i	0.785604	−	0.618730i	\(-0.212352\pi\)
							−0.928638	+	0.370988i	\(0.879019\pi\)
\(11\)	2.26861	−	0.607873i	0.684013	−	0.183281i	0.0999541	−	0.994992i	\(-0.468130\pi\)
							0.584059	+	0.811711i	\(0.301464\pi\)
\(13\)	0.294512	−	3.36629i	0.0816830	−	0.933641i	−0.839205	−	0.543815i	\(-0.816979\pi\)
							0.920888	−	0.389827i	\(-0.127465\pi\)
\(17\)	1.23116	−	6.98226i	0.298600	−	1.69345i	−0.353597	−	0.935398i	\(-0.615042\pi\)
							0.652198	−	0.758049i	\(-0.273847\pi\)
\(19\)	1.52234	+	4.08442i	0.349248	+	0.937030i
							\(23\)	−0.183543	−	0.0668042i	−0.0382714	−	0.0139296i	0.322814	−	0.946463i	\(-0.395371\pi\)
−0.361085	+	0.932533i	\(0.617594\pi\)
\(29\)	4.84632	+	3.39343i	0.899940	+	0.630145i	0.929260	−	0.369425i	\(-0.120445\pi\)
							−0.0293206	+	0.999570i	\(0.509334\pi\)
\(31\)	−2.47922	−	4.29413i	−0.445281	−	0.771249i	0.552791	−	0.833320i	\(-0.313563\pi\)
							−0.998072	+	0.0620712i	\(0.980229\pi\)
\(37\)	−1.19232	−	1.19232i	−0.196017	−	0.196017i	0.602273	−	0.798290i	\(-0.294262\pi\)
							−0.798290	+	0.602273i	\(0.794262\pi\)
\(41\)	1.96625	+	1.64988i	0.307077	+	0.257668i	0.783283	−	0.621666i	\(-0.213544\pi\)
							−0.476206	+	0.879334i	\(0.657988\pi\)
\(43\)	−6.24383	−	2.91155i	−0.952176	−	0.444007i	−0.116476	−	0.993194i	\(-0.537160\pi\)
							−0.835700	+	0.549187i	\(0.814938\pi\)
\(47\)	−3.65403	+	0.644304i	−0.532995	+	0.0939815i	−0.433667	−	0.901073i	\(-0.642781\pi\)
							−0.0993284	+	0.995055i	\(0.531669\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.211.15	yes	456
16.11	odd	4	inner	304.2.bg.a.59.12	✓	456
19.10	odd	18	inner	304.2.bg.a.67.12	yes	456
304.219	even	36	inner	304.2.bg.a.219.15	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.59.12	✓	456	16.11	odd	4	inner
304.2.bg.a.67.12	yes	456	19.10	odd	18	inner
304.2.bg.a.211.15	yes	456	1.1	even	1	trivial
304.2.bg.a.219.15	yes	456	304.219	even	36	inner