Embedded newform 592.2.s.a.339.20

• Label: 592.2.s.a.339.20
• Level: $592$
• Weight: $2$
• Character: 592.339
• Analytic conductor: $4.727$
• Analytic rank: $0$
• Dimension: $148$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.72714379966\)
Analytic rank:	\(0\)
Dimension:	\(148\)
Relative dimension:	\(74\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			339.20
Character	\(\chi\)	\(=\)	592.339
Dual form			592.2.s.a.475.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03953 + 0.958838i) q^{2} +(0.444118 + 0.444118i) q^{3} +(0.161261 - 1.99349i) q^{4} +3.03578 q^{5} +(-0.887513 - 0.0358386i) q^{6} -2.92719 q^{7} +(1.74380 + 2.22692i) q^{8} -2.60552i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 148 q - 4 q^{2} - 4 q^{5} + 2 q^{6} - 8 q^{7} - 4 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\)	\(113\)	\(149\)	\(223\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03953	+	0.958838i	−0.735061	+	0.678001i
							\(3\)	0.444118	+	0.444118i	0.256412	+	0.256412i	0.823593	−	0.567181i	\(-0.191966\pi\)
−0.567181	+	0.823593i	\(0.691966\pi\)
\(5\)	3.03578			1.35764			0.678822	−	0.734303i	\(-0.262491\pi\)
							0.678822	+	0.734303i	\(0.262491\pi\)
\(7\)	−2.92719			−1.10637			−0.553187	−	0.833057i	\(-0.686588\pi\)
							−0.553187	+	0.833057i	\(0.686588\pi\)
\(11\)	3.50412	−	3.50412i	1.05653	−	1.05653i	0.0582274	−	0.998303i	\(-0.481455\pi\)
							0.998303	−	0.0582274i	\(-0.0185449\pi\)
\(13\)			2.51352i			0.697126i	0.937285	+	0.348563i	\(0.113330\pi\)
							−0.937285	+	0.348563i	\(0.886670\pi\)
\(17\)	0.465045	+	0.465045i	0.112790	+	0.112790i	0.761249	−	0.648459i	\(-0.224586\pi\)
							−0.648459	+	0.761249i	\(0.724586\pi\)
\(19\)	4.21035			0.965921			0.482961	−	0.875642i	\(-0.339561\pi\)
							0.482961	+	0.875642i	\(0.339561\pi\)
\(23\)	−4.93838	−	4.93838i	−1.02972	−	1.02972i	−0.999545	−	0.0301789i	\(-0.990392\pi\)
							−0.0301789	−	0.999545i	\(-0.509608\pi\)
\(29\)	9.21241			1.71070			0.855351	−	0.518049i	\(-0.173341\pi\)
							0.855351	+	0.518049i	\(0.173341\pi\)
\(31\)	−4.88461	−	4.88461i	−0.877303	−	0.877303i	0.115952	−	0.993255i	\(-0.463008\pi\)
							−0.993255	+	0.115952i	\(0.963008\pi\)
\(37\)	4.57394	+	4.00987i	0.751952	+	0.659218i
							\(41\)	2.24229			0.350187			0.175094	−	0.984552i	\(-0.443977\pi\)
0.175094	+	0.984552i	\(0.443977\pi\)
\(43\)		−	6.53066i		−	0.995916i	−0.867201	−	0.497958i	\(-0.834083\pi\)
							0.867201	−	0.497958i	\(-0.165917\pi\)
\(47\)			7.63343i			1.11345i	0.830697	+	0.556725i	\(0.187942\pi\)
							−0.830697	+	0.556725i	\(0.812058\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.2.s.a.339.20	yes	148
16.11	odd	4		592.2.m.a.43.19	✓	148
37.31	odd	4		592.2.m.a.179.19	yes	148
592.475	even	4	inner	592.2.s.a.475.20	yes	148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
592.2.m.a.43.19	✓	148	16.11	odd	4
592.2.m.a.179.19	yes	148	37.31	odd	4
592.2.s.a.339.20	yes	148	1.1	even	1	trivial
592.2.s.a.475.20	yes	148	592.475	even	4	inner