Embedded newform 121.2.a.d.1.1

• Label: 121.2.a.d.1.1
• Level: $121$
• Weight: $2$
• Character: 121.1
• Self dual: yes
• Analytic conductor: $0.966$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.966189864457\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	121.1

$q$-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	\(-0.250000\pi\)
			0.707107	+	0.707107i	\(0.250000\pi\)
\(3\)	−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	\(-0.376624\pi\)
			0.377964	+	0.925820i	\(0.376624\pi\)
\(11\)	0	0
			\(13\)	−4.00000	−1.10940	−0.554700	−	0.832050i	\(-0.687167\pi\)
−0.554700	+	0.832050i				\(0.687167\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	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	\(-0.285223\pi\)
			0.624695	+	0.780869i	\(0.285223\pi\)
\(43\)	6.00000	0.914991	0.457496	−	0.889212i	\(-0.348747\pi\)
			0.457496	+	0.889212i	\(0.348747\pi\)
\(47\)	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	121.2.a.d.1.1		1
3.2	odd	2		1089.2.a.b.1.1		1
4.3	odd	2		1936.2.a.i.1.1		1
5.4	even	2		3025.2.a.a.1.1		1
7.6	odd	2		5929.2.a.h.1.1		1
8.3	odd	2		7744.2.a.k.1.1		1
8.5	even	2		7744.2.a.x.1.1		1
11.2	odd	10		121.2.c.e.81.1		4
11.3	even	5		121.2.c.a.9.1		4
11.4	even	5		121.2.c.a.27.1		4
11.5	even	5		121.2.c.a.3.1		4
11.6	odd	10		121.2.c.e.3.1		4
11.7	odd	10		121.2.c.e.27.1		4
11.8	odd	10		121.2.c.e.9.1		4
11.9	even	5		121.2.c.a.81.1		4
11.10	odd	2		11.2.a.a.1.1	✓	1
33.32	even	2		99.2.a.d.1.1		1
44.43	even	2		176.2.a.b.1.1		1
55.32	even	4		275.2.b.a.199.1		2
55.43	even	4		275.2.b.a.199.2		2
55.54	odd	2		275.2.a.b.1.1		1
77.10	even	6		539.2.e.g.177.1		2
77.32	odd	6		539.2.e.h.177.1		2
77.54	even	6		539.2.e.g.67.1		2
77.65	odd	6		539.2.e.h.67.1		2
77.76	even	2		539.2.a.a.1.1		1
88.21	odd	2		704.2.a.h.1.1		1
88.43	even	2		704.2.a.c.1.1		1
99.32	even	6		891.2.e.b.298.1		2
99.43	odd	6		891.2.e.k.595.1		2
99.65	even	6		891.2.e.b.595.1		2
99.76	odd	6		891.2.e.k.298.1		2
132.131	odd	2		1584.2.a.g.1.1		1
143.142	odd	2		1859.2.a.b.1.1		1
165.32	odd	4		2475.2.c.a.199.2		2
165.98	odd	4		2475.2.c.a.199.1		2
165.164	even	2		2475.2.a.a.1.1		1
176.21	odd	4		2816.2.c.j.1409.2		2
176.43	even	4		2816.2.c.f.1409.1		2
176.109	odd	4		2816.2.c.j.1409.1		2
176.131	even	4		2816.2.c.f.1409.2		2
187.186	odd	2		3179.2.a.a.1.1		1
209.208	even	2		3971.2.a.b.1.1		1
220.43	odd	4		4400.2.b.h.4049.2		2
220.87	odd	4		4400.2.b.h.4049.1		2
220.219	even	2		4400.2.a.i.1.1		1
231.230	odd	2		4851.2.a.t.1.1		1
253.252	even	2		5819.2.a.a.1.1		1
264.131	odd	2		6336.2.a.bu.1.1		1
264.197	even	2		6336.2.a.br.1.1		1
308.307	odd	2		8624.2.a.j.1.1		1
319.318	odd	2		9251.2.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.2.a.a.1.1	✓	1	11.10	odd	2
99.2.a.d.1.1		1	33.32	even	2
121.2.a.d.1.1		1	1.1	even	1	trivial
121.2.c.a.3.1		4	11.5	even	5
121.2.c.a.9.1		4	11.3	even	5
121.2.c.a.27.1		4	11.4	even	5
121.2.c.a.81.1		4	11.9	even	5
121.2.c.e.3.1		4	11.6	odd	10
121.2.c.e.9.1		4	11.8	odd	10
121.2.c.e.27.1		4	11.7	odd	10
121.2.c.e.81.1		4	11.2	odd	10
176.2.a.b.1.1		1	44.43	even	2
275.2.a.b.1.1		1	55.54	odd	2
275.2.b.a.199.1		2	55.32	even	4
275.2.b.a.199.2		2	55.43	even	4
539.2.a.a.1.1		1	77.76	even	2
539.2.e.g.67.1		2	77.54	even	6
539.2.e.g.177.1		2	77.10	even	6
539.2.e.h.67.1		2	77.65	odd	6
539.2.e.h.177.1		2	77.32	odd	6
704.2.a.c.1.1		1	88.43	even	2
704.2.a.h.1.1		1	88.21	odd	2
891.2.e.b.298.1		2	99.32	even	6
891.2.e.b.595.1		2	99.65	even	6
891.2.e.k.298.1		2	99.76	odd	6
891.2.e.k.595.1		2	99.43	odd	6
1089.2.a.b.1.1		1	3.2	odd	2
1584.2.a.g.1.1		1	132.131	odd	2
1859.2.a.b.1.1		1	143.142	odd	2
1936.2.a.i.1.1		1	4.3	odd	2
2475.2.a.a.1.1		1	165.164	even	2
2475.2.c.a.199.1		2	165.98	odd	4
2475.2.c.a.199.2		2	165.32	odd	4
2816.2.c.f.1409.1		2	176.43	even	4
2816.2.c.f.1409.2		2	176.131	even	4
2816.2.c.j.1409.1		2	176.109	odd	4
2816.2.c.j.1409.2		2	176.21	odd	4
3025.2.a.a.1.1		1	5.4	even	2
3179.2.a.a.1.1		1	187.186	odd	2
3971.2.a.b.1.1		1	209.208	even	2
4400.2.a.i.1.1		1	220.219	even	2
4400.2.b.h.4049.1		2	220.87	odd	4
4400.2.b.h.4049.2		2	220.43	odd	4
4851.2.a.t.1.1		1	231.230	odd	2
5819.2.a.a.1.1		1	253.252	even	2
5929.2.a.h.1.1		1	7.6	odd	2
6336.2.a.br.1.1		1	264.197	even	2
6336.2.a.bu.1.1		1	264.131	odd	2
7744.2.a.k.1.1		1	8.3	odd	2
7744.2.a.x.1.1		1	8.5	even	2
8624.2.a.j.1.1		1	308.307	odd	2
9251.2.a.d.1.1		1	319.318	odd	2