Embedded newform 605.2.m.e.118.3

• Label: 605.2.m.e.118.3
• 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			118.3
Character	\(\chi\)	\(=\)	605.118
Dual form			605.2.m.e.282.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.22307 + 0.193716i) q^{2} +(1.15501 - 2.26684i) q^{3} +(-0.443732 - 0.144177i) q^{4} +(-2.23029 + 0.160637i) q^{5} +(1.85178 - 2.54876i) q^{6} +(-3.09022 + 1.57455i) q^{7} +(-2.72149 - 1.38667i) q^{8} +(-2.04114 - 2.80938i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} - 4 q^{3} - 2 q^{5} + 20 q^{6} + 10 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.22307	+	0.193716i	0.864843	+	0.136978i	0.573067	−	0.819509i	\(-0.305754\pi\)
							0.291777	+	0.956487i	\(0.405754\pi\)
\(3\)	1.15501	−	2.26684i	0.666845	−	1.30876i	−0.271293	−	0.962497i	\(-0.587451\pi\)
							0.938138	−	0.346261i	\(-0.112549\pi\)
\(5\)	−2.23029	+	0.160637i	−0.997416	+	0.0718389i
							\(7\)	−3.09022	+	1.57455i	−1.16799	+	0.595123i	−0.926874	−	0.375373i	\(-0.877515\pi\)
−0.241120	+	0.970495i	\(0.577515\pi\)
\(11\)		0			0
							\(13\)	0.115699	−	0.730492i	0.0320890	−	0.202602i	−0.966435	−	0.256910i	\(-0.917295\pi\)
0.998524	+	0.0543085i	\(0.0172954\pi\)
\(17\)	−0.0965640	−	0.609681i	−0.0234202	−	0.147869i	0.973207	−	0.229930i	\(-0.0738498\pi\)
							−0.996627	+	0.0820609i	\(0.973850\pi\)
\(19\)	−0.971971	−	2.99142i	−0.222985	−	0.686279i	−0.998490	−	0.0549364i	\(-0.982504\pi\)
							0.775504	−	0.631342i	\(-0.217496\pi\)
\(23\)	−4.30752	−	4.30752i	−0.898181	−	0.898181i	0.0970945	−	0.995275i	\(-0.469045\pi\)
							−0.995275	+	0.0970945i	\(0.969045\pi\)
\(29\)	0.896353	−	2.75869i	0.166449	−	0.512276i	−0.832691	−	0.553737i	\(-0.813201\pi\)
							0.999140	+	0.0414610i	\(0.0132012\pi\)
\(31\)	−2.45570	+	1.78417i	−0.441056	+	0.320446i	−0.786054	−	0.618157i	\(-0.787879\pi\)
							0.344998	+	0.938603i	\(0.387879\pi\)
\(37\)	−1.04453	−	2.05000i	−0.171719	−	0.337018i	0.789068	−	0.614306i	\(-0.210564\pi\)
							−0.960787	+	0.277289i	\(0.910564\pi\)
\(41\)	−0.970387	+	0.315298i	−0.151549	+	0.0492413i	−0.383809	−	0.923412i	\(-0.625388\pi\)
							0.232260	+	0.972654i	\(0.425388\pi\)
\(43\)	4.07488	−	4.07488i	0.621413	−	0.621413i	−0.324480	−	0.945893i	\(-0.605189\pi\)
							0.945893	+	0.324480i	\(0.105189\pi\)
\(47\)	0.967050	+	0.492737i	0.141059	+	0.0718730i	0.523095	−	0.852274i	\(-0.324777\pi\)
							−0.382037	+	0.924147i	\(0.624777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.e.118.3		32
5.2	odd	4	inner	605.2.m.e.602.2		32
11.2	odd	10		605.2.e.b.483.12		32
11.3	even	5		605.2.m.d.578.3		32
11.4	even	5		55.2.l.a.18.3	yes	32
11.5	even	5		605.2.m.c.233.2		32
11.6	odd	10		605.2.m.d.233.3		32
11.7	odd	10	inner	605.2.m.e.403.2		32
11.8	odd	10		605.2.m.c.578.2		32
11.9	even	5		605.2.e.b.483.5		32
11.10	odd	2		55.2.l.a.8.2	yes	32
33.26	odd	10		495.2.bj.a.73.2		32
33.32	even	2		495.2.bj.a.118.3		32
44.15	odd	10		880.2.cm.a.513.4		32
44.43	even	2		880.2.cm.a.833.1		32
55.2	even	20		605.2.e.b.362.5		32
55.4	even	10		275.2.bm.b.18.2		32
55.7	even	20	inner	605.2.m.e.282.3		32
55.17	even	20		605.2.m.d.112.3		32
55.27	odd	20		605.2.m.c.112.2		32
55.32	even	4		55.2.l.a.52.3	yes	32
55.37	odd	20		55.2.l.a.7.2	✓	32
55.42	odd	20		605.2.e.b.362.12		32
55.43	even	4		275.2.bm.b.107.2		32
55.47	odd	20		605.2.m.d.457.3		32
55.48	odd	20		275.2.bm.b.7.3		32
55.52	even	20		605.2.m.c.457.2		32
55.54	odd	2		275.2.bm.b.118.3		32
165.32	odd	4		495.2.bj.a.217.2		32
165.92	even	20		495.2.bj.a.172.3		32
220.87	odd	4		880.2.cm.a.657.4		32
220.147	even	20		880.2.cm.a.337.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.7.2	✓	32	55.37	odd	20
55.2.l.a.8.2	yes	32	11.10	odd	2
55.2.l.a.18.3	yes	32	11.4	even	5
55.2.l.a.52.3	yes	32	55.32	even	4
275.2.bm.b.7.3		32	55.48	odd	20
275.2.bm.b.18.2		32	55.4	even	10
275.2.bm.b.107.2		32	55.43	even	4
275.2.bm.b.118.3		32	55.54	odd	2
495.2.bj.a.73.2		32	33.26	odd	10
495.2.bj.a.118.3		32	33.32	even	2
495.2.bj.a.172.3		32	165.92	even	20
495.2.bj.a.217.2		32	165.32	odd	4
605.2.e.b.362.5		32	55.2	even	20
605.2.e.b.362.12		32	55.42	odd	20
605.2.e.b.483.5		32	11.9	even	5
605.2.e.b.483.12		32	11.2	odd	10
605.2.m.c.112.2		32	55.27	odd	20
605.2.m.c.233.2		32	11.5	even	5
605.2.m.c.457.2		32	55.52	even	20
605.2.m.c.578.2		32	11.8	odd	10
605.2.m.d.112.3		32	55.17	even	20
605.2.m.d.233.3		32	11.6	odd	10
605.2.m.d.457.3		32	55.47	odd	20
605.2.m.d.578.3		32	11.3	even	5
605.2.m.e.118.3		32	1.1	even	1	trivial
605.2.m.e.282.3		32	55.7	even	20	inner
605.2.m.e.403.2		32	11.7	odd	10	inner
605.2.m.e.602.2		32	5.2	odd	4	inner
880.2.cm.a.337.1		32	220.147	even	20
880.2.cm.a.513.4		32	44.15	odd	10
880.2.cm.a.657.4		32	220.87	odd	4
880.2.cm.a.833.1		32	44.43	even	2