Embedded newform 605.2.m.e.118.1

• Label: 605.2.m.e.118.1
• 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			118.1
Character	\(\chi\)	\(=\)	605.118
Dual form			605.2.m.e.282.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50135 - 0.237790i) q^{2} +(-0.361933 + 0.710333i) q^{3} +(0.295389 + 0.0959778i) q^{4} +(-1.89470 - 1.18749i) q^{5} +(0.712297 - 0.980393i) q^{6} +(-0.170602 + 0.0869260i) q^{7} +(2.28811 + 1.16585i) q^{8} +(1.38978 + 1.91287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} - 4 q^{3} - 2 q^{5} + 20 q^{6} + 10 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50135	−	0.237790i	−1.06161	−	0.168143i	−0.398884	−	0.917002i	\(-0.630602\pi\)
							−0.662730	+	0.748859i	\(0.730602\pi\)
\(3\)	−0.361933	+	0.710333i	−0.208962	+	0.410111i	−0.971571	−	0.236749i	\(-0.923918\pi\)
							0.762609	+	0.646860i	\(0.223918\pi\)
\(5\)	−1.89470	−	1.18749i	−0.847335	−	0.531060i
							\(7\)	−0.170602	+	0.0869260i	−0.0644815	+	0.0328550i	−0.485934	−	0.873996i	\(-0.661520\pi\)
0.421452	+	0.906851i	\(0.361520\pi\)
\(11\)		0			0
							\(13\)	0.484259	−	3.05749i	0.134309	−	0.847995i	−0.824896	−	0.565285i	\(-0.808766\pi\)
0.959205	−	0.282711i	\(-0.0912336\pi\)
\(17\)	−0.579827	−	3.66088i	−0.140629	−	0.887894i	−0.952606	−	0.304206i	\(-0.901609\pi\)
							0.811978	−	0.583688i	\(-0.198391\pi\)
\(19\)	0.229844	+	0.707388i	0.0527299	+	0.162286i	0.973954	−	0.226747i	\(-0.0728091\pi\)
							−0.921224	+	0.389033i	\(0.872809\pi\)
\(23\)	−1.14886	−	1.14886i	−0.239553	−	0.239553i	0.577112	−	0.816665i	\(-0.304180\pi\)
							−0.816665	+	0.577112i	\(0.804180\pi\)
\(29\)	−2.95025	+	9.07993i	−0.547847	+	1.68610i	0.166275	+	0.986079i	\(0.446826\pi\)
							−0.714122	+	0.700021i	\(0.753174\pi\)
\(31\)	−0.283900	+	0.206266i	−0.0509900	+	0.0370464i	−0.612988	−	0.790092i	\(-0.710033\pi\)
							0.561998	+	0.827138i	\(0.310033\pi\)
\(37\)	2.45398	+	4.81621i	0.403432	+	0.791780i	0.999941	−	0.0108264i	\(-0.00344622\pi\)
							−0.596510	+	0.802606i	\(0.703446\pi\)
\(41\)	−6.36824	+	2.06917i	−0.994552	+	0.323150i	−0.760687	−	0.649119i	\(-0.775138\pi\)
							−0.233866	+	0.972269i	\(0.575138\pi\)
\(43\)	−3.72708	+	3.72708i	−0.568374	+	0.568374i	−0.931673	−	0.363299i	\(-0.881650\pi\)
							0.363299	+	0.931673i	\(0.381650\pi\)
\(47\)	−11.0165	−	5.61318i	−1.60692	−	0.818766i	−0.999706	−	0.0242272i	\(-0.992287\pi\)
							−0.607213	−	0.794539i	\(-0.707713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.e.118.1		32
5.2	odd	4	inner	605.2.m.e.602.4		32
11.2	odd	10		605.2.e.b.483.3		32
11.3	even	5		605.2.m.d.578.1		32
11.4	even	5		55.2.l.a.18.1	yes	32
11.5	even	5		605.2.m.c.233.4		32
11.6	odd	10		605.2.m.d.233.1		32
11.7	odd	10	inner	605.2.m.e.403.4		32
11.8	odd	10		605.2.m.c.578.4		32
11.9	even	5		605.2.e.b.483.14		32
11.10	odd	2		55.2.l.a.8.4	yes	32
33.26	odd	10		495.2.bj.a.73.4		32
33.32	even	2		495.2.bj.a.118.1		32
44.15	odd	10		880.2.cm.a.513.1		32
44.43	even	2		880.2.cm.a.833.4		32
55.2	even	20		605.2.e.b.362.14		32
55.4	even	10		275.2.bm.b.18.4		32
55.7	even	20	inner	605.2.m.e.282.1		32
55.17	even	20		605.2.m.d.112.1		32
55.27	odd	20		605.2.m.c.112.4		32
55.32	even	4		55.2.l.a.52.1	yes	32
55.37	odd	20		55.2.l.a.7.4	✓	32
55.42	odd	20		605.2.e.b.362.3		32
55.43	even	4		275.2.bm.b.107.4		32
55.47	odd	20		605.2.m.d.457.1		32
55.48	odd	20		275.2.bm.b.7.1		32
55.52	even	20		605.2.m.c.457.4		32
55.54	odd	2		275.2.bm.b.118.1		32
165.32	odd	4		495.2.bj.a.217.4		32
165.92	even	20		495.2.bj.a.172.1		32
220.87	odd	4		880.2.cm.a.657.1		32
220.147	even	20		880.2.cm.a.337.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.7.4	✓	32	55.37	odd	20
55.2.l.a.8.4	yes	32	11.10	odd	2
55.2.l.a.18.1	yes	32	11.4	even	5
55.2.l.a.52.1	yes	32	55.32	even	4
275.2.bm.b.7.1		32	55.48	odd	20
275.2.bm.b.18.4		32	55.4	even	10
275.2.bm.b.107.4		32	55.43	even	4
275.2.bm.b.118.1		32	55.54	odd	2
495.2.bj.a.73.4		32	33.26	odd	10
495.2.bj.a.118.1		32	33.32	even	2
495.2.bj.a.172.1		32	165.92	even	20
495.2.bj.a.217.4		32	165.32	odd	4
605.2.e.b.362.3		32	55.42	odd	20
605.2.e.b.362.14		32	55.2	even	20
605.2.e.b.483.3		32	11.2	odd	10
605.2.e.b.483.14		32	11.9	even	5
605.2.m.c.112.4		32	55.27	odd	20
605.2.m.c.233.4		32	11.5	even	5
605.2.m.c.457.4		32	55.52	even	20
605.2.m.c.578.4		32	11.8	odd	10
605.2.m.d.112.1		32	55.17	even	20
605.2.m.d.233.1		32	11.6	odd	10
605.2.m.d.457.1		32	55.47	odd	20
605.2.m.d.578.1		32	11.3	even	5
605.2.m.e.118.1		32	1.1	even	1	trivial
605.2.m.e.282.1		32	55.7	even	20	inner
605.2.m.e.403.4		32	11.7	odd	10	inner
605.2.m.e.602.4		32	5.2	odd	4	inner
880.2.cm.a.337.4		32	220.147	even	20
880.2.cm.a.513.1		32	44.15	odd	10
880.2.cm.a.657.1		32	220.87	odd	4
880.2.cm.a.833.4		32	44.43	even	2