Embedded newform 11.2.a.a.1.1

• Label: 11.2.a.a.1.1
• Level: $11$
• Weight: $2$
• Character: 11.1
• Self dual: yes
• Analytic conductor: $0.088$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	11.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.0878354422234\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	11.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -2.00000 q^{7} -2.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00000	−1.41421	−0.707107	−	0.707107i	\(-0.750000\pi\)
			−0.707107	+	0.707107i	\(0.750000\pi\)
\(3\)	−1.00000	−0.577350	−0.288675	−	0.957427i	\(-0.593215\pi\)
			−0.288675	+	0.957427i	\(0.593215\pi\)
\(5\)	1.00000	0.447214	0.223607	−	0.974679i	\(-0.428217\pi\)
			0.223607	+	0.974679i	\(0.428217\pi\)
\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
			−0.377964	+	0.925820i	\(0.623376\pi\)
\(11\)	1.00000	0.301511
			\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
0.554700	+	0.832050i				\(0.312833\pi\)
\(17\)	−2.00000	−0.485071	−0.242536	−	0.970143i	\(-0.577979\pi\)
			−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	−1.00000	−0.208514	−0.104257	−	0.994550i	\(-0.533247\pi\)
			−0.104257	+	0.994550i	\(0.533247\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	7.00000	1.25724	0.628619	−	0.777714i	\(-0.283621\pi\)
			0.628619	+	0.777714i	\(0.283621\pi\)
\(37\)	3.00000	0.493197	0.246598	−	0.969118i	\(-0.420687\pi\)
			0.246598	+	0.969118i	\(0.420687\pi\)
\(41\)	−8.00000	−1.24939	−0.624695	−	0.780869i	\(-0.714777\pi\)
			−0.624695	+	0.780869i	\(0.714777\pi\)
\(43\)	−6.00000	−0.914991	−0.457496	−	0.889212i	\(-0.651253\pi\)
			−0.457496	+	0.889212i	\(0.651253\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.2.a.a.1.1	✓	1
3.2	odd	2		99.2.a.d.1.1		1
4.3	odd	2		176.2.a.b.1.1		1
5.2	odd	4		275.2.b.a.199.1		2
5.3	odd	4		275.2.b.a.199.2		2
5.4	even	2		275.2.a.b.1.1		1
7.2	even	3		539.2.e.h.67.1		2
7.3	odd	6		539.2.e.g.177.1		2
7.4	even	3		539.2.e.h.177.1		2
7.5	odd	6		539.2.e.g.67.1		2
7.6	odd	2		539.2.a.a.1.1		1
8.3	odd	2		704.2.a.c.1.1		1
8.5	even	2		704.2.a.h.1.1		1
9.2	odd	6		891.2.e.b.595.1		2
9.4	even	3		891.2.e.k.298.1		2
9.5	odd	6		891.2.e.b.298.1		2
9.7	even	3		891.2.e.k.595.1		2
11.2	odd	10		121.2.c.a.81.1		4
11.3	even	5		121.2.c.e.9.1		4
11.4	even	5		121.2.c.e.27.1		4
11.5	even	5		121.2.c.e.3.1		4
11.6	odd	10		121.2.c.a.3.1		4
11.7	odd	10		121.2.c.a.27.1		4
11.8	odd	10		121.2.c.a.9.1		4
11.9	even	5		121.2.c.e.81.1		4
11.10	odd	2		121.2.a.d.1.1		1
12.11	even	2		1584.2.a.g.1.1		1
13.12	even	2		1859.2.a.b.1.1		1
15.2	even	4		2475.2.c.a.199.2		2
15.8	even	4		2475.2.c.a.199.1		2
15.14	odd	2		2475.2.a.a.1.1		1
16.3	odd	4		2816.2.c.f.1409.2		2
16.5	even	4		2816.2.c.j.1409.2		2
16.11	odd	4		2816.2.c.f.1409.1		2
16.13	even	4		2816.2.c.j.1409.1		2
17.16	even	2		3179.2.a.a.1.1		1
19.18	odd	2		3971.2.a.b.1.1		1
20.3	even	4		4400.2.b.h.4049.2		2
20.7	even	4		4400.2.b.h.4049.1		2
20.19	odd	2		4400.2.a.i.1.1		1
21.20	even	2		4851.2.a.t.1.1		1
23.22	odd	2		5819.2.a.a.1.1		1
24.5	odd	2		6336.2.a.br.1.1		1
24.11	even	2		6336.2.a.bu.1.1		1
28.27	even	2		8624.2.a.j.1.1		1
29.28	even	2		9251.2.a.d.1.1		1
33.32	even	2		1089.2.a.b.1.1		1
44.43	even	2		1936.2.a.i.1.1		1
55.54	odd	2		3025.2.a.a.1.1		1
77.76	even	2		5929.2.a.h.1.1		1
88.21	odd	2		7744.2.a.x.1.1		1
88.43	even	2		7744.2.a.k.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.2.a.a.1.1	✓	1	1.1	even	1	trivial
99.2.a.d.1.1		1	3.2	odd	2
121.2.a.d.1.1		1	11.10	odd	2
121.2.c.a.3.1		4	11.6	odd	10
121.2.c.a.9.1		4	11.8	odd	10
121.2.c.a.27.1		4	11.7	odd	10
121.2.c.a.81.1		4	11.2	odd	10
121.2.c.e.3.1		4	11.5	even	5
121.2.c.e.9.1		4	11.3	even	5
121.2.c.e.27.1		4	11.4	even	5
121.2.c.e.81.1		4	11.9	even	5
176.2.a.b.1.1		1	4.3	odd	2
275.2.a.b.1.1		1	5.4	even	2
275.2.b.a.199.1		2	5.2	odd	4
275.2.b.a.199.2		2	5.3	odd	4
539.2.a.a.1.1		1	7.6	odd	2
539.2.e.g.67.1		2	7.5	odd	6
539.2.e.g.177.1		2	7.3	odd	6
539.2.e.h.67.1		2	7.2	even	3
539.2.e.h.177.1		2	7.4	even	3
704.2.a.c.1.1		1	8.3	odd	2
704.2.a.h.1.1		1	8.5	even	2
891.2.e.b.298.1		2	9.5	odd	6
891.2.e.b.595.1		2	9.2	odd	6
891.2.e.k.298.1		2	9.4	even	3
891.2.e.k.595.1		2	9.7	even	3
1089.2.a.b.1.1		1	33.32	even	2
1584.2.a.g.1.1		1	12.11	even	2
1859.2.a.b.1.1		1	13.12	even	2
1936.2.a.i.1.1		1	44.43	even	2
2475.2.a.a.1.1		1	15.14	odd	2
2475.2.c.a.199.1		2	15.8	even	4
2475.2.c.a.199.2		2	15.2	even	4
2816.2.c.f.1409.1		2	16.11	odd	4
2816.2.c.f.1409.2		2	16.3	odd	4
2816.2.c.j.1409.1		2	16.13	even	4
2816.2.c.j.1409.2		2	16.5	even	4
3025.2.a.a.1.1		1	55.54	odd	2
3179.2.a.a.1.1		1	17.16	even	2
3971.2.a.b.1.1		1	19.18	odd	2
4400.2.a.i.1.1		1	20.19	odd	2
4400.2.b.h.4049.1		2	20.7	even	4
4400.2.b.h.4049.2		2	20.3	even	4
4851.2.a.t.1.1		1	21.20	even	2
5819.2.a.a.1.1		1	23.22	odd	2
5929.2.a.h.1.1		1	77.76	even	2
6336.2.a.br.1.1		1	24.5	odd	2
6336.2.a.bu.1.1		1	24.11	even	2
7744.2.a.k.1.1		1	88.43	even	2
7744.2.a.x.1.1		1	88.21	odd	2
8624.2.a.j.1.1		1	28.27	even	2
9251.2.a.d.1.1		1	29.28	even	2