Embedded newform 304.2.bg.a.3.18

• Label: 304.2.bg.a.3.18
• 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			3.18
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.302259 + 1.38154i) q^{2} +(0.00958574 + 0.00446990i) q^{3} +(-1.81728 - 0.835162i) q^{4} +(-1.14263 - 1.63184i) q^{5} +(-0.00907270 + 0.0118920i) q^{6} +(0.290298 + 0.502811i) q^{7} +(1.70309 - 2.25820i) q^{8} +(-1.92829 - 2.29805i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.302259	+	1.38154i	−0.213729	+	0.976893i
							\(3\)	0.00958574	+	0.00446990i	0.00553433	+	0.00258070i	0.425383	−	0.905013i	\(-0.360139\pi\)
−0.419849	+	0.907594i	\(0.637917\pi\)
\(5\)	−1.14263	−	1.63184i	−0.510999	−	0.729782i	0.477893	−	0.878418i	\(-0.341401\pi\)
							−0.988892	+	0.148636i	\(0.952512\pi\)
\(7\)	0.290298	+	0.502811i	0.109722	+	0.190045i	0.915658	−	0.401959i	\(-0.131671\pi\)
							−0.805935	+	0.592003i	\(0.798337\pi\)
\(11\)	5.36066	−	1.43638i	1.61630	−	0.433086i	0.666388	−	0.745605i	\(-0.267839\pi\)
							0.949912	+	0.312518i	\(0.101173\pi\)
\(13\)	1.15036	−	0.536421i	0.319052	−	0.148776i	−0.256494	−	0.966546i	\(-0.582567\pi\)
							0.575546	+	0.817769i	\(0.304790\pi\)
\(17\)	1.73275	+	1.45395i	0.420252	+	0.352634i	0.828259	−	0.560345i	\(-0.189332\pi\)
							−0.408007	+	0.912979i	\(0.633776\pi\)
\(19\)	−0.337655	−	4.34580i	−0.0774634	−	0.996995i
							\(23\)	−1.08082	−	6.12966i	−0.225367	−	1.27812i	−0.861981	−	0.506940i	\(-0.830777\pi\)
0.636614	−	0.771183i	\(-0.280334\pi\)
\(29\)	−0.651386	+	7.44538i	−0.120959	+	1.38257i	0.656826	+	0.754042i	\(0.271899\pi\)
							−0.777785	+	0.628530i	\(0.783657\pi\)
\(31\)	−3.57096	−	6.18508i	−0.641363	−	1.11087i	−0.985129	−	0.171817i	\(-0.945036\pi\)
							0.343766	−	0.939055i	\(-0.388297\pi\)
\(37\)	2.25770	+	2.25770i	0.371164	+	0.371164i	0.867901	−	0.496737i	\(-0.165469\pi\)
							−0.496737	+	0.867901i	\(0.665469\pi\)
\(41\)	−2.45344	+	0.892979i	−0.383163	+	0.139460i	−0.526418	−	0.850226i	\(-0.676465\pi\)
							0.143255	+	0.989686i	\(0.454243\pi\)
\(43\)	−0.0665660	+	0.0466100i	−0.0101512	+	0.00710796i	−0.578641	−	0.815582i	\(-0.696417\pi\)
							0.568490	+	0.822690i	\(0.307528\pi\)
\(47\)	4.45540	+	5.30974i	0.649887	+	0.774505i	0.985897	−	0.167354i	\(-0.0535221\pi\)
							−0.336010	+	0.941858i	\(0.609078\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.18	✓	456
16.11	odd	4	inner	304.2.bg.a.155.4	yes	456
19.13	odd	18	inner	304.2.bg.a.51.4	yes	456
304.203	even	36	inner	304.2.bg.a.203.18	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.18	✓	456	1.1	even	1	trivial
304.2.bg.a.51.4	yes	456	19.13	odd	18	inner
304.2.bg.a.155.4	yes	456	16.11	odd	4	inner
304.2.bg.a.203.18	yes	456	304.203	even	36	inner