Embedded newform 304.2.bi.a.213.10

• Label: 304.2.bi.a.213.10
• Level: $304$
• Weight: $2$
• Character: 304.213
• 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.bi (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			213.10
Character	\(\chi\)	\(=\)	304.213
Dual form			304.2.bi.a.157.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07188 - 0.922537i) q^{2} +(-2.52146 - 1.76554i) q^{3} +(0.297851 + 1.97770i) q^{4} +(0.172036 + 0.0150512i) q^{5} +(1.07392 + 4.21859i) q^{6} +(3.01484 + 1.74062i) q^{7} +(1.50524 - 2.39463i) q^{8} +(2.21454 + 6.08440i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07188	−	0.922537i	−0.757933	−	0.652332i
							\(3\)	−2.52146	−	1.76554i	−1.45576	−	1.01934i	−0.990994	−	0.133905i	\(-0.957248\pi\)
−0.464769	−	0.885432i	\(-0.653863\pi\)
\(5\)	0.172036	+	0.0150512i	0.0769369	+	0.00673111i	0.125559	−	0.992086i	\(-0.459927\pi\)
							−0.0486223	+	0.998817i	\(0.515483\pi\)
\(7\)	3.01484	+	1.74062i	1.13950	+	0.657891i	0.946307	−	0.323269i	\(-0.104782\pi\)
							0.193194	+	0.981161i	\(0.438115\pi\)
\(11\)	0.624643	−	2.33120i	0.188337	−	0.702883i	−0.805555	−	0.592521i	\(-0.798133\pi\)
							0.993892	−	0.110361i	\(-0.0352008\pi\)
\(13\)	1.34675	+	1.92336i	0.373521	+	0.533443i	0.961273	−	0.275598i	\(-0.0888758\pi\)
							−0.587752	+	0.809041i	\(0.699987\pi\)
\(17\)	5.11273	+	1.86088i	1.24002	+	0.451330i	0.877018	−	0.480458i	\(-0.159530\pi\)
							0.363002	+	0.931788i	\(0.381752\pi\)
\(19\)	−4.29528	−	0.741991i	−0.985405	−	0.170224i
							\(23\)	3.02621	−	3.60649i	0.631007	−	0.752005i	−0.351914	−	0.936032i	\(-0.614469\pi\)
0.982921	+	0.184027i	\(0.0589134\pi\)
\(29\)	1.31030	−	2.80995i	0.243317	−	0.521794i	−0.746286	−	0.665626i	\(-0.768165\pi\)
							0.989602	+	0.143832i	\(0.0459424\pi\)
\(31\)	3.56156	−	6.16881i	0.639676	−	1.10795i	−0.345828	−	0.938298i	\(-0.612402\pi\)
							0.985504	−	0.169653i	\(-0.0542646\pi\)
\(37\)	7.65728	+	7.65728i	1.25885	+	1.25885i	0.951643	+	0.307207i	\(0.0993945\pi\)
							0.307207	+	0.951643i	\(0.400606\pi\)
\(41\)	1.71776	−	0.302887i	0.268269	−	0.0473031i	−0.0378954	−	0.999282i	\(-0.512065\pi\)
							0.306164	+	0.951979i	\(0.400954\pi\)
\(43\)	4.19858	+	0.367328i	0.640277	+	0.0560170i	0.402671	−	0.915345i	\(-0.368082\pi\)
							0.237606	+	0.971362i	\(0.423637\pi\)
\(47\)	−1.06796	+	0.388705i	−0.155778	+	0.0566984i	−0.418732	−	0.908110i	\(-0.637525\pi\)
							0.262954	+	0.964808i	\(0.415303\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.213.10	yes	456
16.13	even	4	inner	304.2.bi.a.61.2	yes	456
19.5	even	9	inner	304.2.bi.a.5.2	✓	456
304.157	even	36	inner	304.2.bi.a.157.10	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.5.2	✓	456	19.5	even	9	inner
304.2.bi.a.61.2	yes	456	16.13	even	4	inner
304.2.bi.a.157.10	yes	456	304.157	even	36	inner
304.2.bi.a.213.10	yes	456	1.1	even	1	trivial