Embedded newform 552.2.bb.a.85.33

• Label: 552.2.bb.a.85.33
• Level: $552$
• Weight: $2$
• Character: 552.85
• Analytic conductor: $4.408$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.bb (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.40774219157\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(48\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			85.33
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.764435 + 1.18981i) q^{2} +(0.540641 + 0.841254i) q^{3} +(-0.831279 + 1.81906i) q^{4} +(-2.37560 + 1.08490i) q^{5} +(-0.587644 + 1.28634i) q^{6} +(1.76843 - 0.519257i) q^{7} +(-2.79979 + 0.401492i) q^{8} +(-0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.764435	+	1.18981i	0.540537	+	0.841320i
							\(3\)	0.540641	+	0.841254i	0.312139	+	0.485698i
\(5\)	−2.37560	+	1.08490i	−1.06240	+	0.485182i								−0.868423	−	0.495824i	\(-0.834866\pi\)
							−0.193977	+	0.981006i	\(0.562139\pi\)
\(7\)	1.76843	−	0.519257i	0.668403	−	0.196261i	0.0701110	−	0.997539i	\(-0.477665\pi\)
							0.598292	+	0.801278i	\(0.295846\pi\)
\(11\)	−3.66757	+	3.17797i	−1.10581	+	0.958193i	−0.999347	−	0.0361238i	\(-0.988499\pi\)
							−0.106466	+	0.994316i	\(0.533953\pi\)
\(13\)	0.735357	−	2.50439i	0.203951	−	0.694594i	−0.792458	−	0.609926i	\(-0.791199\pi\)
							0.996409	−	0.0846677i	\(-0.0269829\pi\)
\(17\)	−0.317717	+	2.20977i	−0.0770577	+	0.535948i	0.914328	+	0.404974i	\(0.132719\pi\)
							−0.991386	+	0.130974i	\(0.958190\pi\)
\(19\)	−1.84049	+	0.264623i	−0.422238	+	0.0607087i	−0.350159	−	0.936690i	\(-0.613872\pi\)
							−0.0720790	+	0.997399i	\(0.522963\pi\)
\(23\)	4.13830	+	2.42373i	0.862895	+	0.505383i
							\(29\)	−1.40144	−	0.201497i	−0.260242	−	0.0374171i	0.0109589	−	0.999940i	\(-0.496512\pi\)
−0.271201	+	0.962523i	\(0.587421\pi\)
\(31\)	0.445741	+	0.286460i	0.0800574	+	0.0514498i	0.580057	−	0.814576i	\(-0.303030\pi\)
							−0.500000	+	0.866026i	\(0.666667\pi\)
\(37\)	6.09278	+	2.78248i	1.00165	+	0.457437i	0.847605	−	0.530627i	\(-0.178044\pi\)
							0.154042	+	0.988064i	\(0.450771\pi\)
\(41\)	−1.56265	−	3.42173i	−0.244045	−	0.534385i	0.747482	−	0.664282i	\(-0.231263\pi\)
							−0.991527	+	0.129897i	\(0.958535\pi\)
\(43\)	0.469155	+	0.730019i	0.0715454	+	0.111327i	0.875183	−	0.483793i	\(-0.160741\pi\)
							−0.803637	+	0.595119i	\(0.797105\pi\)
\(47\)	2.04867			0.298829			0.149414	−	0.988775i	\(-0.452261\pi\)
							0.149414	+	0.988775i	\(0.452261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.85.33	yes	480
8.5	even	2	inner	552.2.bb.a.85.8	yes	480
23.13	even	11	inner	552.2.bb.a.13.8	✓	480
184.13	even	22	inner	552.2.bb.a.13.33	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.8	✓	480	23.13	even	11	inner
552.2.bb.a.13.33	yes	480	184.13	even	22	inner
552.2.bb.a.85.8	yes	480	8.5	even	2	inner
552.2.bb.a.85.33	yes	480	1.1	even	1	trivial