Embedded newform 304.2.bg.a.3.15

• Label: 304.2.bg.a.3.15
• 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			3.15
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.463438 - 1.33612i) q^{2} +(2.94375 + 1.37269i) q^{3} +(-1.57045 + 1.23842i) q^{4} +(-1.38143 - 1.97288i) q^{5} +(0.469841 - 4.56937i) q^{6} +(1.46995 + 2.54603i) q^{7} +(2.38249 + 1.52438i) q^{8} +(4.85302 + 5.78360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.463438	−	1.33612i	−0.327700	−	0.944782i
							\(3\)	2.94375	+	1.37269i	1.69958	+	0.792525i	0.996697	+	0.0812157i	\(0.0258803\pi\)
0.702879	+	0.711309i	\(0.251898\pi\)
\(5\)	−1.38143	−	1.97288i	−0.617793	−	0.882299i	0.381319	−	0.924444i	\(-0.375470\pi\)
							−0.999112	+	0.0421443i	\(0.986581\pi\)
\(7\)	1.46995	+	2.54603i	0.555589	+	0.962308i	0.997857	+	0.0654257i	\(0.0208405\pi\)
							−0.442268	+	0.896883i	\(0.645826\pi\)
\(11\)	3.39074	−	0.908546i	1.02235	−	0.273937i	0.291567	−	0.956550i	\(-0.405823\pi\)
							0.730780	+	0.682613i	\(0.239157\pi\)
\(13\)	−2.05651	+	0.958964i	−0.570372	+	0.265969i	−0.686336	−	0.727285i	\(-0.740782\pi\)
							0.115964	+	0.993253i	\(0.463004\pi\)
\(17\)	−5.21968	−	4.37983i	−1.26596	−	1.06226i	−0.995021	−	0.0996672i	\(-0.968222\pi\)
							−0.270937	−	0.962597i	\(-0.587333\pi\)
\(19\)	3.40551	+	2.72076i	0.781277	+	0.624184i
							\(23\)	−0.997546	−	5.65736i	−0.208003	−	1.17964i	−0.892644	−	0.450762i	\(-0.851152\pi\)
0.684642	−	0.728880i	\(-0.259959\pi\)
\(29\)	−0.213231	+	2.43725i	−0.0395961	+	0.452585i	0.950354	+	0.311170i	\(0.100721\pi\)
							−0.989950	+	0.141415i	\(0.954835\pi\)
\(31\)	0.381793	+	0.661284i	0.0685720	+	0.118770i	0.898273	−	0.439438i	\(-0.144822\pi\)
							−0.829701	+	0.558208i	\(0.811489\pi\)
\(37\)	−5.50108	−	5.50108i	−0.904372	−	0.904372i	0.0914384	−	0.995811i	\(-0.470854\pi\)
							−0.995811	+	0.0914384i	\(0.970854\pi\)
\(41\)	−0.138065	+	0.0502516i	−0.0215622	+	0.00784798i	−0.352779	−	0.935707i	\(-0.614763\pi\)
							0.331217	+	0.943555i	\(0.392541\pi\)
\(43\)	−4.17928	+	2.92637i	−0.637335	+	0.446267i	−0.847063	−	0.531493i	\(-0.821631\pi\)
							0.209728	+	0.977760i	\(0.432742\pi\)
\(47\)	2.52575	+	3.01007i	0.368418	+	0.439063i	0.918123	−	0.396295i	\(-0.129704\pi\)
							−0.549705	+	0.835359i	\(0.685260\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.15	✓	456
16.11	odd	4	inner	304.2.bg.a.155.36	yes	456
19.13	odd	18	inner	304.2.bg.a.51.36	yes	456
304.203	even	36	inner	304.2.bg.a.203.15	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.15	✓	456	1.1	even	1	trivial
304.2.bg.a.51.36	yes	456	19.13	odd	18	inner
304.2.bg.a.155.36	yes	456	16.11	odd	4	inner
304.2.bg.a.203.15	yes	456	304.203	even	36	inner