Embedded newform 552.2.t.a.19.3

• Label: 552.2.t.a.19.3
• 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			19.3
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.a.523.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36604 + 0.365971i) q^{2} +(0.959493 - 0.281733i) q^{3} +(1.73213 - 0.999862i) q^{4} +(-0.824449 - 0.529841i) q^{5} +(-1.20760 + 0.736005i) q^{6} +(-0.536086 - 3.72856i) q^{7} +(-2.00024 + 1.99976i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{3} - 4 q^{4} + 11 q^{6} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36604	+	0.365971i	−0.965936	+	0.258781i
							\(3\)	0.959493	−	0.281733i	0.553964	−	0.162658i
\(5\)	−0.824449	−	0.529841i	−0.368705	−	0.236952i								0.343147	−	0.939282i	\(-0.388507\pi\)
							−0.711852	+	0.702329i	\(0.752143\pi\)
\(7\)	−0.536086	−	3.72856i	−0.202622	−	1.40926i	−0.796466	−	0.604684i	\(-0.793299\pi\)
							0.593844	−	0.804580i	\(-0.297610\pi\)
\(11\)	2.79329	+	1.27565i	0.842209	+	0.384624i	0.789324	−	0.613977i	\(-0.210431\pi\)
							0.0528846	+	0.998601i	\(0.483158\pi\)
\(13\)	−1.96450	−	0.282452i	−0.544854	−	0.0783382i	−0.135609	−	0.990762i	\(-0.543299\pi\)
							−0.409246	+	0.912424i	\(0.634208\pi\)
\(17\)	−0.726462	+	0.629483i	−0.176193	+	0.152672i	−0.738493	−	0.674261i	\(-0.764462\pi\)
							0.562300	+	0.826933i	\(0.309917\pi\)
\(19\)	−5.54449	−	4.80433i	−1.27199	−	1.10219i	−0.989745	−	0.142848i	\(-0.954374\pi\)
							−0.282249	−	0.959341i	\(-0.591081\pi\)
\(23\)	3.39143	−	3.39090i	0.707162	−	0.707052i
							\(29\)	−3.85883	+	3.34370i	−0.716567	+	0.620909i	−0.934877	−	0.354971i	\(-0.884491\pi\)
0.218311	+	0.975879i	\(0.429945\pi\)
\(31\)	−1.57929	+	5.37857i	−0.283649	+	0.966020i	0.687228	+	0.726442i	\(0.258827\pi\)
							−0.970877	+	0.239578i	\(0.922991\pi\)
\(37\)	9.08706	−	5.83990i	1.49390	−	0.960074i	0.498241	−	0.867038i	\(-0.333979\pi\)
							0.995663	−	0.0930354i	\(-0.0296570\pi\)
\(41\)	−9.00665	−	5.78823i	−1.40660	−	0.903969i	−0.406649	−	0.913584i	\(-0.633303\pi\)
							−0.999954	+	0.00961533i	\(0.996939\pi\)
\(43\)	1.59145	+	5.41997i	0.242693	+	0.826537i	0.987277	+	0.159009i	\(0.0508297\pi\)
							−0.744584	+	0.667529i	\(0.767352\pi\)
\(47\)		−	10.4088i		−	1.51828i	−0.650925	−	0.759142i	\(-0.725619\pi\)
							0.650925	−	0.759142i	\(-0.274381\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.a.19.3	✓	240
8.3	odd	2	inner	552.2.t.a.19.19	yes	240
23.17	odd	22	inner	552.2.t.a.523.19	yes	240
184.155	even	22	inner	552.2.t.a.523.3	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.a.19.3	✓	240	1.1	even	1	trivial
552.2.t.a.19.19	yes	240	8.3	odd	2	inner
552.2.t.a.523.3	yes	240	184.155	even	22	inner
552.2.t.a.523.19	yes	240	23.17	odd	22	inner