Embedded newform 552.2.t.a.19.9

• Label: 552.2.t.a.19.9
• 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			19.9
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.a.523.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.675020 + 1.24272i) q^{2} +(0.959493 - 0.281733i) q^{3} +(-1.08870 - 1.67772i) q^{4} +(-0.991910 - 0.637462i) q^{5} +(-0.297563 + 1.38255i) q^{6} +(-0.126423 - 0.879292i) q^{7} +(2.81982 - 0.220446i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{3} - 4 q^{4} + 11 q^{6} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.675020	+	1.24272i	−0.477311	+	0.878734i
							\(3\)	0.959493	−	0.281733i	0.553964	−	0.162658i
\(5\)	−0.991910	−	0.637462i	−0.443596	−	0.285082i								0.299714	−	0.954029i	\(-0.403109\pi\)
							−0.743310	+	0.668947i	\(0.766745\pi\)
\(7\)	−0.126423	−	0.879292i	−0.0477835	−	0.332341i	−0.999665	−	0.0258756i	\(-0.991763\pi\)
							0.951882	−	0.306466i	\(-0.0991465\pi\)
\(11\)	−4.36627	−	1.99401i	−1.31648	−	0.601216i	−0.371527	−	0.928422i	\(-0.621166\pi\)
							−0.944953	+	0.327206i	\(0.893893\pi\)
\(13\)	−3.32880	−	0.478610i	−0.923244	−	0.132742i	−0.335732	−	0.941958i	\(-0.608984\pi\)
							−0.587512	+	0.809215i	\(0.699893\pi\)
\(17\)	−3.33243	+	2.88756i	−0.808232	+	0.700337i	−0.957492	−	0.288461i	\(-0.906856\pi\)
							0.149260	+	0.988798i	\(0.452311\pi\)
\(19\)	−2.16222	−	1.87357i	−0.496047	−	0.429827i	0.370568	−	0.928806i	\(-0.379163\pi\)
							−0.866614	+	0.498979i	\(0.833709\pi\)
\(23\)	−1.50468	+	4.55367i	−0.313748	+	0.949506i
							\(29\)	−1.68533	+	1.46034i	−0.312957	+	0.271179i	−0.797149	−	0.603783i	\(-0.793659\pi\)
0.484191	+	0.874962i	\(0.339114\pi\)
\(31\)	1.46968	−	5.00527i	0.263962	−	0.898973i	−0.715715	−	0.698393i	\(-0.753899\pi\)
							0.979677	−	0.200581i	\(-0.0642828\pi\)
\(37\)	2.94109	−	1.89012i	0.483511	−	0.310734i	−0.276080	−	0.961135i	\(-0.589035\pi\)
							0.759591	+	0.650401i	\(0.225399\pi\)
\(41\)	−2.62214	−	1.68515i	−0.409509	−	0.263176i	0.319628	−	0.947543i	\(-0.396442\pi\)
							−0.729137	+	0.684367i	\(0.760078\pi\)
\(43\)	−1.16783	−	3.97728i	−0.178093	−	0.606529i	−0.999353	−	0.0359747i	\(-0.988546\pi\)
							0.821260	−	0.570555i	\(-0.193272\pi\)
\(47\)			0.446855i			0.0651805i	0.999469	+	0.0325902i	\(0.0103756\pi\)
							−0.999469	+	0.0325902i	\(0.989624\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.a.19.9	✓	240
8.3	odd	2	inner	552.2.t.a.19.23	yes	240
23.17	odd	22	inner	552.2.t.a.523.23	yes	240
184.155	even	22	inner	552.2.t.a.523.9	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.a.19.9	✓	240	1.1	even	1	trivial
552.2.t.a.19.23	yes	240	8.3	odd	2	inner
552.2.t.a.523.9	yes	240	184.155	even	22	inner
552.2.t.a.523.23	yes	240	23.17	odd	22	inner