Embedded newform 552.2.bb.a.469.25

• Label: 552.2.bb.a.469.25
• Level: $552$
• Weight: $2$
• Character: 552.469
• 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			469.25
Character	\(\chi\)	\(=\)	552.469
Dual form			552.2.bb.a.133.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.258184 + 1.39045i) q^{2} +(-0.755750 + 0.654861i) q^{3} +(-1.86668 + 0.717982i) q^{4} +(3.10367 - 0.446240i) q^{5} +(-1.10567 - 0.881755i) q^{6} +(1.08048 - 2.36592i) q^{7} +(-1.48026 - 2.41015i) q^{8} +(0.142315 - 0.989821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{5}{11}\right)\)	\(1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.258184	+	1.39045i	0.182564	+	0.983194i
							\(3\)	−0.755750	+	0.654861i	−0.436332	+	0.378084i
\(5\)	3.10367	−	0.446240i	1.38800	−	0.199565i								0.592528	−	0.805550i	\(-0.298130\pi\)
							0.795476	+	0.605985i	\(0.207221\pi\)
\(7\)	1.08048	−	2.36592i	0.408383	−	0.894235i	−0.587968	−	0.808884i	\(-0.700072\pi\)
							0.996351	−	0.0853502i	\(-0.0272009\pi\)
\(11\)	1.28714	−	4.38360i	0.388087	−	1.32170i	−0.501586	−	0.865108i	\(-0.667250\pi\)
							0.889674	−	0.456597i	\(-0.150932\pi\)
\(13\)	4.52507	−	2.06653i	1.25503	−	0.573152i	0.326775	−	0.945102i	\(-0.394038\pi\)
							0.928253	+	0.371950i	\(0.121311\pi\)
\(17\)	−1.93566	−	1.24397i	−0.469465	−	0.301707i	0.284436	−	0.958695i	\(-0.408194\pi\)
							−0.753901	+	0.656988i	\(0.771830\pi\)
\(19\)	−2.03158	−	3.16120i	−0.466077	−	0.725230i	0.526052	−	0.850453i	\(-0.323672\pi\)
							−0.992129	+	0.125223i	\(0.960035\pi\)
\(23\)	−2.38351	+	4.16160i	−0.496995	+	0.867753i
							\(29\)	0.665998	−	1.03631i	0.123673	−	0.192438i	−0.773897	−	0.633312i	\(-0.781695\pi\)
0.897569	+	0.440874i	\(0.145331\pi\)
\(31\)	−1.18129	+	1.36328i	−0.212166	+	0.244853i	−0.851851	−	0.523785i	\(-0.824520\pi\)
							0.639684	+	0.768638i	\(0.279065\pi\)
\(37\)	−7.40779	−	1.06508i	−1.21783	−	0.175098i	−0.496718	−	0.867912i	\(-0.665462\pi\)
							−0.721116	+	0.692814i	\(0.756371\pi\)
\(41\)	1.55118	+	10.7887i	0.242254	+	1.68491i	0.640756	+	0.767745i	\(0.278621\pi\)
							−0.398502	+	0.917167i	\(0.630470\pi\)
\(43\)	−2.79319	+	2.42031i	−0.425957	+	0.369094i	−0.841298	−	0.540572i	\(-0.818208\pi\)
							0.415340	+	0.909666i	\(0.363662\pi\)
\(47\)	−5.19881			−0.758324			−0.379162	−	0.925330i	\(-0.623788\pi\)
							−0.379162	+	0.925330i	\(0.623788\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.469.25	yes	480
8.5	even	2	inner	552.2.bb.a.469.18	yes	480
23.18	even	11	inner	552.2.bb.a.133.18	✓	480
184.133	even	22	inner	552.2.bb.a.133.25	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.133.18	✓	480	23.18	even	11	inner
552.2.bb.a.133.25	yes	480	184.133	even	22	inner
552.2.bb.a.469.18	yes	480	8.5	even	2	inner
552.2.bb.a.469.25	yes	480	1.1	even	1	trivial