Embedded newform 552.2.t.b.19.15

• Label: 552.2.t.b.19.15
• Level: $552$
• Weight: $2$
• Character: 552.19
• Analytic conductor: $4.408$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.40774219157\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			19.15
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.b.523.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.559215 - 1.29895i) q^{2} +(-0.959493 + 0.281733i) q^{3} +(-1.37456 - 1.45279i) q^{4} +(0.781202 + 0.502048i) q^{5} +(-0.170606 + 1.40389i) q^{6} +(-0.233208 - 1.62200i) q^{7} +(-2.65578 + 0.973064i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{15}{22}\right)\)	\(1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.559215	−	1.29895i	0.395425	−	0.918498i
							\(3\)	−0.959493	+	0.281733i	−0.553964	+	0.162658i
\(5\)	0.781202	+	0.502048i	0.349364	+	0.224523i								0.703539	−	0.710656i	\(-0.251602\pi\)
							−0.354175	+	0.935179i	\(0.615238\pi\)
\(7\)	−0.233208	−	1.62200i	−0.0881445	−	0.613058i	−0.985234	−	0.171210i	\(-0.945232\pi\)
							0.897090	−	0.441848i	\(-0.145677\pi\)
\(11\)	−3.93727	−	1.79809i	−1.18713	−	0.542145i	−0.278782	−	0.960354i	\(-0.589931\pi\)
							−0.908351	+	0.418209i	\(0.862658\pi\)
\(13\)	−1.76083	−	0.253170i	−0.488367	−	0.0702166i	−0.106267	−	0.994338i	\(-0.533890\pi\)
							−0.382100	+	0.924121i	\(0.624799\pi\)
\(17\)	1.58730	−	1.37541i	0.384978	−	0.333585i	−0.440775	−	0.897618i	\(-0.645296\pi\)
							0.825753	+	0.564033i	\(0.190751\pi\)
\(19\)	−1.55505	−	1.34746i	−0.356752	−	0.309128i	0.457983	−	0.888961i	\(-0.348572\pi\)
							−0.814735	+	0.579834i	\(0.803118\pi\)
\(23\)	−4.31254	−	2.09810i	−0.899226	−	0.437484i
							\(29\)	−2.26983	+	1.96682i	−0.421497	+	0.365229i	−0.839632	−	0.543156i	\(-0.817229\pi\)
0.418135	+	0.908385i	\(0.362684\pi\)
\(31\)	−0.269391	+	0.917461i	−0.0483840	+	0.164781i	−0.980141	−	0.198302i	\(-0.936457\pi\)
							0.931757	+	0.363082i	\(0.118276\pi\)
\(37\)	−3.95389	+	2.54101i	−0.650016	+	0.417740i	−0.823672	−	0.567066i	\(-0.808078\pi\)
							0.173656	+	0.984806i	\(0.444442\pi\)
\(41\)	−1.14329	−	0.734745i	−0.178551	−	0.114748i	0.448312	−	0.893877i	\(-0.352025\pi\)
							−0.626864	+	0.779129i	\(0.715662\pi\)
\(43\)	−2.72152	−	9.26863i	−0.415027	−	1.41345i	−0.856485	−	0.516171i	\(-0.827357\pi\)
							0.441458	−	0.897282i	\(-0.354461\pi\)
\(47\)			7.78128i			1.13502i	0.823368	+	0.567508i	\(0.192093\pi\)
							−0.823368	+	0.567508i	\(0.807907\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.19.15	yes	240
8.3	odd	2	inner	552.2.t.b.19.3	✓	240
23.17	odd	22	inner	552.2.t.b.523.3	yes	240
184.155	even	22	inner	552.2.t.b.523.15	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.19.3	✓	240	8.3	odd	2	inner
552.2.t.b.19.15	yes	240	1.1	even	1	trivial
552.2.t.b.523.3	yes	240	23.17	odd	22	inner
552.2.t.b.523.15	yes	240	184.155	even	22	inner