Embedded newform 304.2.bi.a.157.31

• Label: 304.2.bi.a.157.31
• Level: $304$
• Weight: $2$
• Character: 304.157
• 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			157.31
Character	\(\chi\)	\(=\)	304.157
Dual form			304.2.bi.a.213.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09539 - 0.894495i) q^{2} +(0.718256 - 0.502928i) q^{3} +(0.399757 - 1.95964i) q^{4} +(2.40238 - 0.210181i) q^{5} +(0.336903 - 1.19338i) q^{6} +(-0.0989902 + 0.0571520i) q^{7} +(-1.31500 - 2.50415i) q^{8} +(-0.763105 + 2.09661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.09539	−	0.894495i	0.774557	−	0.632504i
							\(3\)	0.718256	−	0.502928i	0.414685	−	0.290366i	−0.347555	−	0.937659i	\(-0.612988\pi\)
0.762241	+	0.647294i	\(0.224099\pi\)
\(5\)	2.40238	−	0.210181i	1.07438	−	0.0939956i	0.463789	−	0.885946i	\(-0.346490\pi\)
							0.610586	+	0.791950i	\(0.290934\pi\)
\(7\)	−0.0989902	+	0.0571520i	−0.0374148	+	0.0216014i	−0.518591	−	0.855023i	\(-0.673543\pi\)
							0.481176	+	0.876624i	\(0.340210\pi\)
\(11\)	0.0164305	+	0.0613195i	0.00495398	+	0.0184885i	0.968359	−	0.249563i	\(-0.0802869\pi\)
							−0.963405	+	0.268051i	\(0.913620\pi\)
\(13\)	−2.44695	+	3.49461i	−0.678662	+	0.969230i	0.321092	+	0.947048i	\(0.395950\pi\)
							−0.999754	+	0.0221816i	\(0.992939\pi\)
\(17\)	−3.74105	+	1.36163i	−0.907337	+	0.330244i	−0.753189	−	0.657804i	\(-0.771485\pi\)
							−0.154148	+	0.988048i	\(0.549263\pi\)
\(19\)	2.92614	−	3.23075i	0.671302	−	0.741184i
							\(23\)	0.883904	+	1.05340i	0.184307	+	0.219648i	0.850284	−	0.526323i	\(-0.176430\pi\)
−0.665978	+	0.745972i	\(0.731985\pi\)
\(29\)	−0.691352	−	1.48261i	−0.128381	−	0.275314i	0.831602	−	0.555372i	\(-0.187424\pi\)
							−0.959983	+	0.280058i	\(0.909646\pi\)
\(31\)	−3.21766	−	5.57315i	−0.577909	−	1.00097i	−0.995719	−	0.0924333i	\(-0.970536\pi\)
							0.417810	−	0.908534i	\(-0.362798\pi\)
\(37\)	0.852115	−	0.852115i	0.140087	−	0.140087i	−0.633586	−	0.773673i	\(-0.718418\pi\)
							0.773673	+	0.633586i	\(0.218418\pi\)
\(41\)	9.23664	+	1.62867i	1.44252	+	0.254355i	0.839495	−	0.543367i	\(-0.182851\pi\)
							0.603025	+	0.797722i	\(0.293962\pi\)
\(43\)	2.32042	−	0.203010i	0.353860	−	0.0309587i	0.0911599	−	0.995836i	\(-0.470943\pi\)
							0.262700	+	0.964878i	\(0.415387\pi\)
\(47\)	−2.69014	−	0.979129i	−0.392397	−	0.142821i	0.138284	−	0.990393i	\(-0.455841\pi\)
							−0.530681	+	0.847572i	\(0.678064\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.157.31	yes	456
16.5	even	4	inner	304.2.bi.a.5.38	✓	456
19.4	even	9	inner	304.2.bi.a.61.38	yes	456
304.213	even	36	inner	304.2.bi.a.213.31	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.5.38	✓	456	16.5	even	4	inner
304.2.bi.a.61.38	yes	456	19.4	even	9	inner
304.2.bi.a.157.31	yes	456	1.1	even	1	trivial
304.2.bi.a.213.31	yes	456	304.213	even	36	inner