Embedded newform 304.2.bi.a.245.20

• Label: 304.2.bi.a.245.20
• Level: $304$
• Weight: $2$
• Character: 304.245
• 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			245.20
Character	\(\chi\)	\(=\)	304.245
Dual form			304.2.bi.a.237.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.159796 - 1.40516i) q^{2} +(-3.21622 + 0.281383i) q^{3} +(-1.94893 - 0.449076i) q^{4} +(-2.71691 - 1.26692i) q^{5} +(-0.118551 + 4.56425i) q^{6} +(-0.223754 + 0.129184i) q^{7} +(-0.942453 + 2.66679i) q^{8} +(7.31045 - 1.28903i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.159796	−	1.40516i	0.112993	−	0.993596i
							\(3\)	−3.21622	+	0.281383i	−1.85688	+	0.162456i	−0.960035	−	0.279881i	\(-0.909705\pi\)
−0.896849	+	0.442337i	\(0.854149\pi\)
\(5\)	−2.71691	−	1.26692i	−1.21504	−	0.566583i	−0.293909	−	0.955834i	\(-0.594956\pi\)
							−0.921132	+	0.389251i	\(0.872734\pi\)
\(7\)	−0.223754	+	0.129184i	−0.0845710	+	0.0488271i	−0.541689	−	0.840579i	\(-0.682215\pi\)
							0.457118	+	0.889406i	\(0.348882\pi\)
\(11\)	1.33083	−	0.356595i	0.401260	−	0.107517i	−0.0525443	−	0.998619i	\(-0.516733\pi\)
							0.453805	+	0.891101i	\(0.350066\pi\)
\(13\)	−0.197254	+	2.25463i	−0.0547085	+	0.625320i	0.918504	+	0.395412i	\(0.129398\pi\)
							−0.973212	+	0.229908i	\(0.926157\pi\)
\(17\)	1.28116	−	7.26582i	0.310727	−	1.76222i	−0.284512	−	0.958673i	\(-0.591832\pi\)
							0.595239	−	0.803549i	\(-0.297057\pi\)
\(19\)	−2.33995	+	3.67758i	−0.536822	+	0.843695i
							\(23\)	−2.75063	+	7.55729i	−0.573546	+	1.57580i	0.225313	+	0.974286i	\(0.427659\pi\)
−0.798859	+	0.601518i	\(0.794563\pi\)
\(29\)	4.36216	+	3.05441i	0.810032	+	0.567191i	0.903578	−	0.428424i	\(-0.140931\pi\)
							−0.0935456	+	0.995615i	\(0.529820\pi\)
\(31\)	1.07611	+	1.86387i	0.193274	+	0.334761i	0.946333	−	0.323192i	\(-0.104756\pi\)
							−0.753059	+	0.657953i	\(0.771423\pi\)
\(37\)	3.78818	+	3.78818i	0.622774	+	0.622774i	0.946240	−	0.323466i	\(-0.104848\pi\)
							−0.323466	+	0.946240i	\(0.604848\pi\)
\(41\)	−2.55495	+	3.04488i	−0.399017	+	0.475530i	−0.927720	−	0.373277i	\(-0.878234\pi\)
							0.528703	+	0.848807i	\(0.322679\pi\)
\(43\)	−2.72455	−	1.27048i	−0.415490	−	0.193746i	0.203621	−	0.979050i	\(-0.434729\pi\)
							−0.619111	+	0.785304i	\(0.712507\pi\)
\(47\)	−0.508096	−	2.88155i	−0.0741134	−	0.420318i	−0.999179	−	0.0405107i	\(-0.987102\pi\)
							0.925066	−	0.379807i	\(-0.124010\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.245.20	yes	456
16.13	even	4	inner	304.2.bi.a.93.24	yes	456
19.9	even	9	inner	304.2.bi.a.85.24	✓	456
304.237	even	36	inner	304.2.bi.a.237.20	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.85.24	✓	456	19.9	even	9	inner
304.2.bi.a.93.24	yes	456	16.13	even	4	inner
304.2.bi.a.237.20	yes	456	304.237	even	36	inner
304.2.bi.a.245.20	yes	456	1.1	even	1	trivial