Embedded newform 574.2.s.b.387.10

• Label: 574.2.s.b.387.10
• Level: $574$
• Weight: $2$
• Character: 574.387
• Analytic conductor: $4.583$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 574 = 2 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	574.s (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.58341307602\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			387.10
Character	\(\chi\)	\(=\)	574.387
Dual form			574.2.s.b.221.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978148 - 0.207912i) q^{2} +(0.501366 + 0.868392i) q^{3} +(0.913545 + 0.406737i) q^{4} +(0.162909 + 1.54998i) q^{5} +(-0.309861 - 0.953655i) q^{6} +(1.93357 - 1.80591i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(0.997264 - 1.72731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{2} - 2 q^{3} + 14 q^{4} + 4 q^{5} + 4 q^{6} - 2 q^{7} - 28 q^{8} - 58 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(493\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.978148	−	0.207912i	−0.691655	−	0.147016i
							\(3\)	0.501366	+	0.868392i	0.289464	+	0.501366i	0.973682	−	0.227911i	\(-0.0731896\pi\)
−0.684218	+	0.729278i	\(0.739856\pi\)
\(5\)	0.162909	+	1.54998i	0.0728552	+	0.693171i	0.968604	+	0.248610i	\(0.0799737\pi\)
							−0.895748	+	0.444561i	\(0.853360\pi\)
\(7\)	1.93357	−	1.80591i	0.730820	−	0.682570i
							\(11\)	0.390615	−	3.71645i	0.117775	−	1.12055i	−0.762797	−	0.646638i	\(-0.776174\pi\)
0.880571	−	0.473913i	\(-0.157159\pi\)
\(13\)	−1.97589	−	6.08117i	−0.548014	−	1.68661i	−0.713713	−	0.700438i	\(-0.752988\pi\)
							0.165699	−	0.986176i	\(-0.447012\pi\)
\(17\)	−0.100996	+	0.960914i	−0.0244951	+	0.233056i	0.975424	+	0.220336i	\(0.0707155\pi\)
							−0.999919	+	0.0127194i	\(0.995951\pi\)
\(19\)	−0.758998	−	0.842953i	−0.174126	−	0.193387i	0.649764	−	0.760136i	\(-0.274868\pi\)
							−0.823890	+	0.566749i	\(0.808201\pi\)
\(23\)	1.29382	+	0.275010i	0.269780	+	0.0573436i	0.340815	−	0.940130i	\(-0.389297\pi\)
							−0.0710350	+	0.997474i	\(0.522630\pi\)
\(29\)	−0.291911	+	0.212085i	−0.0542064	+	0.0393833i	−0.614559	−	0.788871i	\(-0.710666\pi\)
							0.560352	+	0.828255i	\(0.310666\pi\)
\(31\)	−0.414131	+	3.94019i	−0.0743801	+	0.707679i	0.892257	+	0.451528i	\(0.149121\pi\)
							−0.966637	+	0.256151i	\(0.917546\pi\)
\(37\)	−0.0402742	−	0.383184i	−0.00662105	−	0.0629950i	0.990712	−	0.135974i	\(-0.0434162\pi\)
							−0.997334	+	0.0729785i	\(0.976750\pi\)
\(41\)	−1.27576	+	6.27475i	−0.199240	+	0.979951i
							\(43\)	−0.523038	−	1.60975i	−0.0797625	−	0.245484i	0.903222	−	0.429175i	\(-0.141196\pi\)
−0.982984	+	0.183691i	\(0.941196\pi\)
\(47\)	−5.93462	−	1.26144i	−0.865654	−	0.184000i	−0.246378	−	0.969174i	\(-0.579240\pi\)
							−0.619276	+	0.785174i	\(0.712574\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	574.2.s.b.387.10	yes	112
7.4	even	3	inner	574.2.s.b.305.10	yes	112
41.16	even	5	inner	574.2.s.b.303.10	yes	112
287.221	even	15	inner	574.2.s.b.221.10	✓	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.s.b.221.10	✓	112	287.221	even	15	inner
574.2.s.b.303.10	yes	112	41.16	even	5	inner
574.2.s.b.305.10	yes	112	7.4	even	3	inner
574.2.s.b.387.10	yes	112	1.1	even	1	trivial