Embedded newform 605.2.m.e.118.4

• Label: 605.2.m.e.118.4
• Level: $605$
• Weight: $2$
• Character: 605.118
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.m (of order \(20\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{20})\)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			118.4
Character	\(\chi\)	\(=\)	605.118
Dual form			605.2.m.e.282.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.40264 + 0.380541i) q^{2} +(-0.271495 + 0.532840i) q^{3} +(3.72576 + 1.21057i) q^{4} +(1.85044 - 1.25533i) q^{5} +(-0.855073 + 1.17691i) q^{6} +(-2.30033 + 1.17208i) q^{7} +(4.15610 + 2.11764i) q^{8} +(1.55315 + 2.13772i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} - 4 q^{3} - 2 q^{5} + 20 q^{6} + 10 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.40264	+	0.380541i	1.69892	+	0.269083i	0.929276	−	0.369385i	\(-0.120432\pi\)
							0.769648	+	0.638468i	\(0.220432\pi\)
\(3\)	−0.271495	+	0.532840i	−0.156748	+	0.307635i	−0.956003	−	0.293357i	\(-0.905228\pi\)
							0.799255	+	0.600992i	\(0.205228\pi\)
\(5\)	1.85044	−	1.25533i	0.827543	−	0.561403i
							\(7\)	−2.30033	+	1.17208i	−0.869443	+	0.443003i	−0.831010	−	0.556258i	\(-0.812237\pi\)
−0.0384331	+	0.999261i	\(0.512237\pi\)
\(11\)		0			0
							\(13\)	0.439393	−	2.77422i	0.121866	−	0.769429i	−0.848750	−	0.528794i	\(-0.822644\pi\)
0.970616	−	0.240635i	\(-0.0773556\pi\)
\(17\)	−0.147723	−	0.932684i	−0.0358280	−	0.226209i	0.963277	−	0.268510i	\(-0.0865312\pi\)
							−0.999105	+	0.0423009i	\(0.986531\pi\)
\(19\)	1.28236	+	3.94669i	0.294193	+	0.905433i	0.983491	+	0.180955i	\(0.0579189\pi\)
							−0.689298	+	0.724478i	\(0.742081\pi\)
\(23\)	0.104710	+	0.104710i	0.0218334	+	0.0218334i	0.717939	−	0.696106i	\(-0.245086\pi\)
							−0.696106	+	0.717939i	\(0.745086\pi\)
\(29\)	2.14459	−	6.60038i	0.398241	−	1.22566i	−0.528168	−	0.849140i	\(-0.677121\pi\)
							0.926409	−	0.376520i	\(-0.122879\pi\)
\(31\)	−7.33171	+	5.32680i	−1.31681	+	0.956722i	−0.316849	+	0.948476i	\(0.602625\pi\)
							−0.999966	+	0.00824590i	\(0.997375\pi\)
\(37\)	−3.44950	−	6.77002i	−0.567094	−	1.11298i	−0.979398	−	0.201938i	\(-0.935276\pi\)
							0.412305	−	0.911046i	\(-0.364724\pi\)
\(41\)	−3.27880	+	1.06535i	−0.512063	+	0.166379i	−0.553640	−	0.832756i	\(-0.686762\pi\)
							0.0415772	+	0.999135i	\(0.486762\pi\)
\(43\)	3.91833	−	3.91833i	0.597540	−	0.597540i	−0.342117	−	0.939657i	\(-0.611144\pi\)
							0.939657	+	0.342117i	\(0.111144\pi\)
\(47\)	−0.942051	−	0.479999i	−0.137412	−	0.0700150i	0.383933	−	0.923361i	\(-0.374569\pi\)
							−0.521345	+	0.853346i	\(0.674569\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.e.118.4		32
5.2	odd	4	inner	605.2.m.e.602.1		32
11.2	odd	10		605.2.e.b.483.15		32
11.3	even	5		605.2.m.d.578.4		32
11.4	even	5		55.2.l.a.18.4	yes	32
11.5	even	5		605.2.m.c.233.1		32
11.6	odd	10		605.2.m.d.233.4		32
11.7	odd	10	inner	605.2.m.e.403.1		32
11.8	odd	10		605.2.m.c.578.1		32
11.9	even	5		605.2.e.b.483.2		32
11.10	odd	2		55.2.l.a.8.1	yes	32
33.26	odd	10		495.2.bj.a.73.1		32
33.32	even	2		495.2.bj.a.118.4		32
44.15	odd	10		880.2.cm.a.513.2		32
44.43	even	2		880.2.cm.a.833.3		32
55.2	even	20		605.2.e.b.362.2		32
55.4	even	10		275.2.bm.b.18.1		32
55.7	even	20	inner	605.2.m.e.282.4		32
55.17	even	20		605.2.m.d.112.4		32
55.27	odd	20		605.2.m.c.112.1		32
55.32	even	4		55.2.l.a.52.4	yes	32
55.37	odd	20		55.2.l.a.7.1	✓	32
55.42	odd	20		605.2.e.b.362.15		32
55.43	even	4		275.2.bm.b.107.1		32
55.47	odd	20		605.2.m.d.457.4		32
55.48	odd	20		275.2.bm.b.7.4		32
55.52	even	20		605.2.m.c.457.1		32
55.54	odd	2		275.2.bm.b.118.4		32
165.32	odd	4		495.2.bj.a.217.1		32
165.92	even	20		495.2.bj.a.172.4		32
220.87	odd	4		880.2.cm.a.657.2		32
220.147	even	20		880.2.cm.a.337.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.7.1	✓	32	55.37	odd	20
55.2.l.a.8.1	yes	32	11.10	odd	2
55.2.l.a.18.4	yes	32	11.4	even	5
55.2.l.a.52.4	yes	32	55.32	even	4
275.2.bm.b.7.4		32	55.48	odd	20
275.2.bm.b.18.1		32	55.4	even	10
275.2.bm.b.107.1		32	55.43	even	4
275.2.bm.b.118.4		32	55.54	odd	2
495.2.bj.a.73.1		32	33.26	odd	10
495.2.bj.a.118.4		32	33.32	even	2
495.2.bj.a.172.4		32	165.92	even	20
495.2.bj.a.217.1		32	165.32	odd	4
605.2.e.b.362.2		32	55.2	even	20
605.2.e.b.362.15		32	55.42	odd	20
605.2.e.b.483.2		32	11.9	even	5
605.2.e.b.483.15		32	11.2	odd	10
605.2.m.c.112.1		32	55.27	odd	20
605.2.m.c.233.1		32	11.5	even	5
605.2.m.c.457.1		32	55.52	even	20
605.2.m.c.578.1		32	11.8	odd	10
605.2.m.d.112.4		32	55.17	even	20
605.2.m.d.233.4		32	11.6	odd	10
605.2.m.d.457.4		32	55.47	odd	20
605.2.m.d.578.4		32	11.3	even	5
605.2.m.e.118.4		32	1.1	even	1	trivial
605.2.m.e.282.4		32	55.7	even	20	inner
605.2.m.e.403.1		32	11.7	odd	10	inner
605.2.m.e.602.1		32	5.2	odd	4	inner
880.2.cm.a.337.3		32	220.147	even	20
880.2.cm.a.513.2		32	44.15	odd	10
880.2.cm.a.657.2		32	220.87	odd	4
880.2.cm.a.833.3		32	44.43	even	2