Embedded newform 55.2.l.a.13.2

• Label: 55.2.l.a.13.2
• Level: $55$
• Weight: $2$
• Character: 55.13
• Analytic conductor: $0.439$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	55.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.439177211117\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	55.13
Dual form			55.2.l.a.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.930933 + 0.474334i) q^{2} +(-2.78152 - 0.440550i) q^{3} +(-0.533928 + 0.734888i) q^{4} +(-1.77672 + 1.35766i) q^{5} +(2.79838 - 0.909249i) q^{6} +(0.0860119 + 0.543058i) q^{7} +(0.475357 - 3.00128i) q^{8} +(4.68963 + 1.52375i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 10 q^{2} - 4 q^{3} - 2 q^{5} - 20 q^{6} - 10 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\)	\(12\)	\(46\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.930933	+	0.474334i	−0.658269	+	0.335405i	−0.751015	−	0.660285i	\(-0.770435\pi\)
							0.0927463	+	0.995690i	\(0.470435\pi\)
\(3\)	−2.78152	−	0.440550i	−1.60591	−	0.254352i	−0.711863	−	0.702318i	\(-0.752149\pi\)
							−0.894051	+	0.447966i	\(0.852149\pi\)
\(5\)	−1.77672	+	1.35766i	−0.794575	+	0.607166i
							\(7\)	0.0860119	+	0.543058i	0.0325095	+	0.205257i	0.998597	−	0.0529594i	\(-0.0168654\pi\)
−0.966087	+	0.258216i	\(0.916865\pi\)
\(11\)	−3.29961	−	0.335528i	−0.994870	−	0.101166i
							\(13\)	1.47301	+	2.89095i	0.408540	+	0.801805i	0.999990	−	0.00453687i	\(-0.00144414\pi\)
−0.591450	+	0.806342i	\(0.701444\pi\)
\(17\)	−2.57079	+	5.04545i	−0.623507	+	1.22370i	0.335959	+	0.941877i	\(0.390940\pi\)
							−0.959466	+	0.281825i	\(0.909060\pi\)
\(19\)	−1.25925	+	0.914902i	−0.288893	+	0.209893i	−0.722787	−	0.691071i	\(-0.757139\pi\)
							0.433894	+	0.900964i	\(0.357139\pi\)
\(23\)	−0.803543	−	0.803543i	−0.167550	−	0.167550i	0.618351	−	0.785902i	\(-0.287801\pi\)
							−0.785902	+	0.618351i	\(0.787801\pi\)
\(29\)	−3.44380	−	2.50207i	−0.639498	−	0.464623i	0.220180	−	0.975459i	\(-0.429336\pi\)
							−0.859678	+	0.510837i	\(0.829336\pi\)
\(31\)	−0.509209	+	1.56718i	−0.0914566	+	0.281475i	−0.986314	−	0.164878i	\(-0.947277\pi\)
							0.894857	+	0.446352i	\(0.147277\pi\)
\(37\)	−0.945121	+	0.149692i	−0.155377	+	0.0246093i	−0.233638	−	0.972324i	\(-0.575063\pi\)
							0.0782613	+	0.996933i	\(0.475063\pi\)
\(41\)	5.25869	+	7.23797i	0.821270	+	1.13038i	0.989486	+	0.144631i	\(0.0461994\pi\)
							−0.168216	+	0.985750i	\(0.553801\pi\)
\(43\)	−2.55312	+	2.55312i	−0.389348	+	0.389348i	−0.874455	−	0.485107i	\(-0.838781\pi\)
							0.485107	+	0.874455i	\(0.338781\pi\)
\(47\)	0.636959	−	4.02160i	0.0929101	−	0.586611i	−0.896678	−	0.442683i	\(-0.854027\pi\)
							0.989588	−	0.143928i	\(-0.0459734\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.2.l.a.13.2	yes	32
3.2	odd	2		495.2.bj.a.343.3		32
4.3	odd	2		880.2.cm.a.673.4		32
5.2	odd	4	inner	55.2.l.a.2.2	✓	32
5.3	odd	4		275.2.bm.b.57.3		32
5.4	even	2		275.2.bm.b.68.3		32
11.2	odd	10		605.2.m.d.118.2		32
11.3	even	5		605.2.m.d.403.3		32
11.4	even	5		605.2.e.b.483.6		32
11.5	even	5		605.2.m.e.578.3		32
11.6	odd	10	inner	55.2.l.a.28.2	yes	32
11.7	odd	10		605.2.e.b.483.11		32
11.8	odd	10		605.2.m.c.403.2		32
11.9	even	5		605.2.m.c.118.3		32
11.10	odd	2		605.2.m.e.233.3		32
15.2	even	4		495.2.bj.a.442.3		32
20.7	even	4		880.2.cm.a.497.4		32
33.17	even	10		495.2.bj.a.28.3		32
44.39	even	10		880.2.cm.a.193.4		32
55.2	even	20		605.2.m.d.602.3		32
55.7	even	20		605.2.e.b.362.6		32
55.17	even	20	inner	55.2.l.a.17.2	yes	32
55.27	odd	20		605.2.m.e.457.3		32
55.28	even	20		275.2.bm.b.182.3		32
55.32	even	4		605.2.m.e.112.3		32
55.37	odd	20		605.2.e.b.362.11		32
55.39	odd	10		275.2.bm.b.193.3		32
55.42	odd	20		605.2.m.c.602.2		32
55.47	odd	20		605.2.m.d.282.2		32
55.52	even	20		605.2.m.c.282.3		32
165.17	odd	20		495.2.bj.a.127.3		32
220.127	odd	20		880.2.cm.a.17.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.2.2	✓	32	5.2	odd	4	inner
55.2.l.a.13.2	yes	32	1.1	even	1	trivial
55.2.l.a.17.2	yes	32	55.17	even	20	inner
55.2.l.a.28.2	yes	32	11.6	odd	10	inner
275.2.bm.b.57.3		32	5.3	odd	4
275.2.bm.b.68.3		32	5.4	even	2
275.2.bm.b.182.3		32	55.28	even	20
275.2.bm.b.193.3		32	55.39	odd	10
495.2.bj.a.28.3		32	33.17	even	10
495.2.bj.a.127.3		32	165.17	odd	20
495.2.bj.a.343.3		32	3.2	odd	2
495.2.bj.a.442.3		32	15.2	even	4
605.2.e.b.362.6		32	55.7	even	20
605.2.e.b.362.11		32	55.37	odd	20
605.2.e.b.483.6		32	11.4	even	5
605.2.e.b.483.11		32	11.7	odd	10
605.2.m.c.118.3		32	11.9	even	5
605.2.m.c.282.3		32	55.52	even	20
605.2.m.c.403.2		32	11.8	odd	10
605.2.m.c.602.2		32	55.42	odd	20
605.2.m.d.118.2		32	11.2	odd	10
605.2.m.d.282.2		32	55.47	odd	20
605.2.m.d.403.3		32	11.3	even	5
605.2.m.d.602.3		32	55.2	even	20
605.2.m.e.112.3		32	55.32	even	4
605.2.m.e.233.3		32	11.10	odd	2
605.2.m.e.457.3		32	55.27	odd	20
605.2.m.e.578.3		32	11.5	even	5
880.2.cm.a.17.4		32	220.127	odd	20
880.2.cm.a.193.4		32	44.39	even	10
880.2.cm.a.497.4		32	20.7	even	4
880.2.cm.a.673.4		32	4.3	odd	2