Embedded newform 304.2.bi.a.213.2

• Label: 304.2.bi.a.213.2
• 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			213.2
Character	\(\chi\)	\(=\)	304.213
Dual form			304.2.bi.a.157.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39438 + 0.235989i) q^{2} +(-2.43282 - 1.70348i) q^{3} +(1.88862 - 0.658120i) q^{4} +(-4.10037 - 0.358736i) q^{5} +(3.79428 + 1.80118i) q^{6} +(-3.20625 - 1.85113i) q^{7} +(-2.47815 + 1.36337i) q^{8} +(1.99070 + 5.46941i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39438	+	0.235989i	−0.985979	+	0.166870i
							\(3\)	−2.43282	−	1.70348i	−1.40459	−	0.983502i	−0.997540	−	0.0700947i	\(-0.977670\pi\)
−0.407046	−	0.913407i	\(-0.633441\pi\)
\(5\)	−4.10037	−	0.358736i	−1.83374	−	0.160431i	−0.882788	−	0.469771i	\(-0.844336\pi\)
							−0.950952	+	0.309340i	\(0.899892\pi\)
\(7\)	−3.20625	−	1.85113i	−1.21185	−	0.699660i	−0.248686	−	0.968584i	\(-0.579999\pi\)
							−0.963161	+	0.268924i	\(0.913332\pi\)
\(11\)	0.183519	−	0.684902i	0.0553331	−	0.206506i	−0.932725	−	0.360589i	\(-0.882576\pi\)
							0.988058	+	0.154083i	\(0.0492424\pi\)
\(13\)	−1.23405	−	1.76240i	−0.342264	−	0.488803i	0.610711	−	0.791854i	\(-0.290884\pi\)
							−0.952974	+	0.303051i	\(0.901995\pi\)
\(17\)	1.32787	+	0.483305i	0.322055	+	0.117219i	0.497989	−	0.867184i	\(-0.334072\pi\)
							−0.175933	+	0.984402i	\(0.556294\pi\)
\(19\)	−0.525546	−	4.32710i	−0.120569	−	0.992705i
							\(23\)	1.66813	−	1.98799i	0.347828	−	0.414526i	−0.563559	−	0.826076i	\(-0.690568\pi\)
0.911387	+	0.411550i	\(0.135013\pi\)
\(29\)	−3.64031	+	7.80668i	−0.675989	+	1.44966i	0.206076	+	0.978536i	\(0.433931\pi\)
							−0.882065	+	0.471127i	\(0.843847\pi\)
\(31\)	0.0429288	−	0.0743548i	0.00771023	−	0.0133545i	−0.862145	−	0.506662i	\(-0.830879\pi\)
							0.869855	+	0.493308i	\(0.164212\pi\)
\(37\)	−0.726728	−	0.726728i	−0.119473	−	0.119473i	0.644842	−	0.764316i	\(-0.276923\pi\)
							−0.764316	+	0.644842i	\(0.776923\pi\)
\(41\)	−7.68637	+	1.35532i	−1.20041	+	0.211665i	−0.737876	−	0.674937i	\(-0.764171\pi\)
							−0.462534	+	0.886601i	\(0.653060\pi\)
\(43\)	5.14784	+	0.450377i	0.785038	+	0.0686819i	0.472624	−	0.881264i	\(-0.343307\pi\)
							0.312414	+	0.949946i	\(0.398862\pi\)
\(47\)	−1.50406	+	0.547434i	−0.219390	+	0.0798514i	−0.449377	−	0.893342i	\(-0.648354\pi\)
							0.229987	+	0.973194i	\(0.426132\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.213.2	yes	456
16.13	even	4	inner	304.2.bi.a.61.11	yes	456
19.5	even	9	inner	304.2.bi.a.5.11	✓	456
304.157	even	36	inner	304.2.bi.a.157.2	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.5.11	✓	456	19.5	even	9	inner
304.2.bi.a.61.11	yes	456	16.13	even	4	inner
304.2.bi.a.157.2	yes	456	304.157	even	36	inner
304.2.bi.a.213.2	yes	456	1.1	even	1	trivial