Embedded newform 633.2.k.a.23.9

• Label: 633.2.k.a.23.9
• Level: $633$
• Weight: $2$
• Character: 633.23
• Analytic conductor: $5.055$
• Analytic rank: $0$
• Dimension: $272$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 633 = 3 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	633.k (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.05453044795\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(68\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			23.9
Character	\(\chi\)	\(=\)	633.23
Dual form			633.2.k.a.578.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.706109 + 2.17318i) q^{2} +(-0.517600 + 1.65290i) q^{3} +(-2.60608 - 1.89343i) q^{4} +(1.39731 - 1.92323i) q^{5} +(-3.22657 - 2.29197i) q^{6} +(3.70895 - 1.20511i) q^{7} +(2.25771 - 1.64032i) q^{8} +(-2.46418 - 1.71108i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 10 q^{3} - 74 q^{4} + 9 q^{6} - 30 q^{7} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/633\mathbb{Z}\right)^\times\).

\(n\)	\(212\)	\(424\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.706109	+	2.17318i	−0.499294	+	1.53667i	0.310862	+	0.950455i	\(0.399382\pi\)
							−0.810156	+	0.586214i	\(0.800618\pi\)
\(3\)	−0.517600	+	1.65290i	−0.298836	+	0.954304i
							\(5\)	1.39731	−	1.92323i	0.624894	−	0.860093i	−0.372803	−	0.927910i	\(-0.621603\pi\)
0.997698	+	0.0678169i	\(0.0216034\pi\)
\(7\)	3.70895	−	1.20511i	1.40185	−	0.455489i	0.492062	−	0.870560i	\(-0.336243\pi\)
							0.909789	+	0.415071i	\(0.136243\pi\)
\(11\)	0.345470	−	0.475498i	0.104163	−	0.143368i	−0.753753	−	0.657157i	\(-0.771759\pi\)
							0.857916	+	0.513789i	\(0.171759\pi\)
\(13\)	−0.542580	−	1.66989i	−0.150485	−	0.463144i	0.847191	−	0.531289i	\(-0.178292\pi\)
							−0.997675	+	0.0681447i	\(0.978292\pi\)
\(17\)	1.57725	+	4.85428i	0.382539	+	1.17734i	0.938250	+	0.345959i	\(0.112446\pi\)
							−0.555710	+	0.831376i	\(0.687554\pi\)
\(19\)	2.24976	+	6.92406i	0.516131	+	1.58849i	0.781214	+	0.624263i	\(0.214601\pi\)
							−0.265083	+	0.964226i	\(0.585399\pi\)
\(23\)	−0.866475	+	2.66674i	−0.180673	+	0.556053i	−0.999847	−	0.0174914i	\(-0.994432\pi\)
							0.819174	+	0.573544i	\(0.194432\pi\)
\(29\)	0.742493	+	2.28516i	0.137877	+	0.424343i	0.996026	−	0.0890575i	\(-0.0283855\pi\)
							−0.858149	+	0.513401i	\(0.828385\pi\)
\(31\)			1.20911i			0.217162i	0.994088	+	0.108581i	\(0.0346307\pi\)
							−0.994088	+	0.108581i	\(0.965369\pi\)
\(37\)	−2.53375	+	1.84088i	−0.416546	+	0.302638i	−0.776247	−	0.630429i	\(-0.782879\pi\)
							0.359701	+	0.933068i	\(0.382879\pi\)
\(41\)	8.18159	+	5.94427i	1.27775	+	0.928339i	0.999483	−	0.0321647i	\(-0.0102401\pi\)
							0.278267	+	0.960504i	\(0.410240\pi\)
\(43\)	1.38621			0.211396			0.105698	−	0.994398i	\(-0.466292\pi\)
							0.105698	+	0.994398i	\(0.466292\pi\)
\(47\)	9.00010	−	2.92431i	1.31280	−	0.426554i	0.432783	−	0.901498i	\(-0.357532\pi\)
							0.880016	+	0.474944i	\(0.157532\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	633.2.k.a.23.9	✓	272
3.2	odd	2	inner	633.2.k.a.23.60	yes	272
211.156	odd	10	inner	633.2.k.a.578.60	yes	272
633.578	even	10	inner	633.2.k.a.578.9	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
633.2.k.a.23.9	✓	272	1.1	even	1	trivial
633.2.k.a.23.60	yes	272	3.2	odd	2	inner
633.2.k.a.578.9	yes	272	633.578	even	10	inner
633.2.k.a.578.60	yes	272	211.156	odd	10	inner