Embedded newform 605.2.m.e.118.2

• Label: 605.2.m.e.118.2
• 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			118.2
Character	\(\chi\)	\(=\)	605.118
Dual form			605.2.m.e.282.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.482327 - 0.0763931i) q^{2} +(0.517260 - 1.01518i) q^{3} +(-1.67531 - 0.544341i) q^{4} +(2.05331 + 0.885381i) q^{5} +(-0.327041 + 0.450133i) q^{6} +(2.59854 - 1.32402i) q^{7} +(1.63669 + 0.833936i) q^{8} +(1.00032 + 1.37683i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} - 4 q^{3} - 2 q^{5} + 20 q^{6} + 10 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.482327	−	0.0763931i	−0.341057	−	0.0540181i	−0.0164432	−	0.999865i	\(-0.505234\pi\)
							−0.324614	+	0.945847i	\(0.605234\pi\)
\(3\)	0.517260	−	1.01518i	0.298640	−	0.586114i	−0.692114	−	0.721789i	\(-0.743320\pi\)
							0.990753	+	0.135675i	\(0.0433202\pi\)
\(5\)	2.05331	+	0.885381i	0.918270	+	0.395954i
							\(7\)	2.59854	−	1.32402i	0.982157	−	0.500434i	0.112266	−	0.993678i	\(-0.464189\pi\)
0.869891	+	0.493244i	\(0.164189\pi\)
\(11\)		0			0
							\(13\)	−0.457808	+	2.89049i	−0.126973	+	0.801677i	0.839208	+	0.543811i	\(0.183019\pi\)
−0.966181	+	0.257866i	\(0.916981\pi\)
\(17\)	0.824113	+	5.20325i	0.199877	+	1.26197i	0.859797	+	0.510636i	\(0.170590\pi\)
							−0.659920	+	0.751336i	\(0.729410\pi\)
\(19\)	−1.26677	−	3.89873i	−0.290618	−	0.894429i	−0.984658	−	0.174494i	\(-0.944171\pi\)
							0.694041	−	0.719936i	\(-0.255829\pi\)
\(23\)	−2.12046	−	2.12046i	−0.442147	−	0.442147i	0.450586	−	0.892733i	\(-0.351215\pi\)
							−0.892733	+	0.450586i	\(0.851215\pi\)
\(29\)	−0.817241	+	2.51521i	−0.151758	+	0.467063i	−0.997818	−	0.0660248i	\(-0.978968\pi\)
							0.846060	+	0.533088i	\(0.178968\pi\)
\(31\)	5.45328	−	3.96204i	0.979438	−	0.711603i	0.0218548	−	0.999761i	\(-0.493043\pi\)
							0.957583	+	0.288158i	\(0.0930429\pi\)
\(37\)	−0.528116	−	1.03649i	−0.0868218	−	0.170397i	0.843513	−	0.537109i	\(-0.180484\pi\)
							−0.930335	+	0.366712i	\(0.880484\pi\)
\(41\)	3.38136	−	1.09867i	0.528080	−	0.171584i	−0.0328289	−	0.999461i	\(-0.510452\pi\)
							0.560909	+	0.827877i	\(0.310452\pi\)
\(43\)	5.07292	−	5.07292i	0.773613	−	0.773613i	−0.205123	−	0.978736i	\(-0.565759\pi\)
							0.978736	+	0.205123i	\(0.0657595\pi\)
\(47\)	−3.28041	−	1.67145i	−0.478497	−	0.243806i	0.198070	−	0.980188i	\(-0.436533\pi\)
							−0.676566	+	0.736382i	\(0.736533\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.e.118.2		32
5.2	odd	4	inner	605.2.m.e.602.3		32
11.2	odd	10		605.2.e.b.483.8		32
11.3	even	5		605.2.m.d.578.2		32
11.4	even	5		55.2.l.a.18.2	yes	32
11.5	even	5		605.2.m.c.233.3		32
11.6	odd	10		605.2.m.d.233.2		32
11.7	odd	10	inner	605.2.m.e.403.3		32
11.8	odd	10		605.2.m.c.578.3		32
11.9	even	5		605.2.e.b.483.9		32
11.10	odd	2		55.2.l.a.8.3	yes	32
33.26	odd	10		495.2.bj.a.73.3		32
33.32	even	2		495.2.bj.a.118.2		32
44.15	odd	10		880.2.cm.a.513.3		32
44.43	even	2		880.2.cm.a.833.2		32
55.2	even	20		605.2.e.b.362.9		32
55.4	even	10		275.2.bm.b.18.3		32
55.7	even	20	inner	605.2.m.e.282.2		32
55.17	even	20		605.2.m.d.112.2		32
55.27	odd	20		605.2.m.c.112.3		32
55.32	even	4		55.2.l.a.52.2	yes	32
55.37	odd	20		55.2.l.a.7.3	✓	32
55.42	odd	20		605.2.e.b.362.8		32
55.43	even	4		275.2.bm.b.107.3		32
55.47	odd	20		605.2.m.d.457.2		32
55.48	odd	20		275.2.bm.b.7.2		32
55.52	even	20		605.2.m.c.457.3		32
55.54	odd	2		275.2.bm.b.118.2		32
165.32	odd	4		495.2.bj.a.217.3		32
165.92	even	20		495.2.bj.a.172.2		32
220.87	odd	4		880.2.cm.a.657.3		32
220.147	even	20		880.2.cm.a.337.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.7.3	✓	32	55.37	odd	20
55.2.l.a.8.3	yes	32	11.10	odd	2
55.2.l.a.18.2	yes	32	11.4	even	5
55.2.l.a.52.2	yes	32	55.32	even	4
275.2.bm.b.7.2		32	55.48	odd	20
275.2.bm.b.18.3		32	55.4	even	10
275.2.bm.b.107.3		32	55.43	even	4
275.2.bm.b.118.2		32	55.54	odd	2
495.2.bj.a.73.3		32	33.26	odd	10
495.2.bj.a.118.2		32	33.32	even	2
495.2.bj.a.172.2		32	165.92	even	20
495.2.bj.a.217.3		32	165.32	odd	4
605.2.e.b.362.8		32	55.42	odd	20
605.2.e.b.362.9		32	55.2	even	20
605.2.e.b.483.8		32	11.2	odd	10
605.2.e.b.483.9		32	11.9	even	5
605.2.m.c.112.3		32	55.27	odd	20
605.2.m.c.233.3		32	11.5	even	5
605.2.m.c.457.3		32	55.52	even	20
605.2.m.c.578.3		32	11.8	odd	10
605.2.m.d.112.2		32	55.17	even	20
605.2.m.d.233.2		32	11.6	odd	10
605.2.m.d.457.2		32	55.47	odd	20
605.2.m.d.578.2		32	11.3	even	5
605.2.m.e.118.2		32	1.1	even	1	trivial
605.2.m.e.282.2		32	55.7	even	20	inner
605.2.m.e.403.3		32	11.7	odd	10	inner
605.2.m.e.602.3		32	5.2	odd	4	inner
880.2.cm.a.337.2		32	220.147	even	20
880.2.cm.a.513.3		32	44.15	odd	10
880.2.cm.a.657.3		32	220.87	odd	4
880.2.cm.a.833.2		32	44.43	even	2