Embedded newform 304.2.bi.a.149.29

• Label: 304.2.bi.a.149.29
• Level: $304$
• Weight: $2$
• Character: 304.149
• 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			149.29
Character	\(\chi\)	\(=\)	304.149
Dual form			304.2.bi.a.253.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.02732 + 0.971907i) q^{2} +(-0.719027 + 1.54196i) q^{3} +(0.110792 + 1.99693i) q^{4} +(3.48462 - 2.43995i) q^{5} +(-2.23732 + 0.885265i) q^{6} +(1.42795 - 0.824426i) q^{7} +(-1.82701 + 2.15917i) q^{8} +(0.0677251 + 0.0807116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.02732	+	0.971907i	0.726428	+	0.687242i
							\(3\)	−0.719027	+	1.54196i	−0.415131	+	0.890251i	0.581901	+	0.813260i	\(0.302309\pi\)
−0.997032	+	0.0769910i	\(0.975469\pi\)
\(5\)	3.48462	−	2.43995i	1.55837	−	1.09118i	0.602206	−	0.798341i	\(-0.294288\pi\)
							0.956161	−	0.292840i	\(-0.0946004\pi\)
\(7\)	1.42795	−	0.824426i	0.539713	−	0.311604i	−0.205249	−	0.978710i	\(-0.565801\pi\)
							0.744963	+	0.667106i	\(0.232467\pi\)
\(11\)	1.16242	−	0.311471i	0.350484	−	0.0939119i	−0.0792820	−	0.996852i	\(-0.525263\pi\)
							0.429766	+	0.902940i	\(0.358596\pi\)
\(13\)	−1.15212	+	0.537243i	−0.319541	+	0.149005i	−0.575770	−	0.817612i	\(-0.695298\pi\)
							0.256229	+	0.966616i	\(0.417520\pi\)
\(17\)	−5.18067	−	4.34710i	−1.25650	−	1.05433i	−0.996046	−	0.0888386i	\(-0.971684\pi\)
							−0.260450	−	0.965487i	\(-0.583871\pi\)
\(19\)	−4.31729	+	0.600809i	−0.990455	+	0.137835i
							\(23\)	−4.57178	+	0.806128i	−0.953281	+	0.168089i	−0.628595	−	0.777733i	\(-0.716370\pi\)
−0.324686	+	0.945822i	\(0.605259\pi\)
\(29\)	0.370817	−	4.23846i	0.0688590	−	0.787062i	−0.880741	−	0.473598i	\(-0.842955\pi\)
							0.949600	−	0.313464i	\(-0.101490\pi\)
\(31\)	4.45519	+	7.71662i	0.800176	+	1.38595i	0.919500	+	0.393090i	\(0.128594\pi\)
							−0.119324	+	0.992855i	\(0.538073\pi\)
\(37\)	1.56058	+	1.56058i	0.256557	+	0.256557i	0.823652	−	0.567095i	\(-0.191933\pi\)
							−0.567095	+	0.823652i	\(0.691933\pi\)
\(41\)	1.99789	+	5.48915i	0.312017	+	0.857261i	0.992249	+	0.124264i	\(0.0396569\pi\)
							−0.680232	+	0.732997i	\(0.738121\pi\)
\(43\)	5.82852	−	4.08118i	0.888842	−	0.622374i	−0.0374015	−	0.999300i	\(-0.511908\pi\)
							0.926243	+	0.376927i	\(0.123019\pi\)
\(47\)	−0.111935	+	0.0939246i	−0.0163274	+	0.0137003i	−0.650915	−	0.759151i	\(-0.725615\pi\)
							0.634587	+	0.772851i	\(0.281170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.149.29	yes	456
16.13	even	4	inner	304.2.bi.a.301.32	yes	456
19.6	even	9	inner	304.2.bi.a.101.32	✓	456
304.253	even	36	inner	304.2.bi.a.253.29	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.101.32	✓	456	19.6	even	9	inner
304.2.bi.a.149.29	yes	456	1.1	even	1	trivial
304.2.bi.a.253.29	yes	456	304.253	even	36	inner
304.2.bi.a.301.32	yes	456	16.13	even	4	inner