Embedded newform 33.2.a.a.1.1

• Label: 33.2.a.a.1.1
• Level: $33$
• Weight: $2$
• Character: 33.1
• Self dual: yes
• Analytic conductor: $0.264$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.263506326670\)
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	\(\chi\)	\(=\)	33.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107	0.353553	−	0.935414i	\(-0.384973\pi\)
			0.353553	+	0.935414i	\(0.384973\pi\)
\(3\)	−1.00000	−0.577350
			\(5\)	−2.00000	−0.894427	−0.447214	−	0.894427i	\(-0.647584\pi\)
−0.447214	+	0.894427i				\(0.647584\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	1.00000	0.301511
			\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
−0.277350	+	0.960769i				\(0.589456\pi\)
\(17\)	−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	\(0.5\pi\)
\(23\)	8.00000	1.66812	0.834058	−	0.551677i	\(-0.186012\pi\)
			0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	−2.00000	−0.312348	−0.156174	−	0.987730i	\(-0.549916\pi\)
			−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	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	33.2.a.a.1.1	✓	1
3.2	odd	2		99.2.a.b.1.1		1
4.3	odd	2		528.2.a.g.1.1		1
5.2	odd	4		825.2.c.a.199.2		2
5.3	odd	4		825.2.c.a.199.1		2
5.4	even	2		825.2.a.a.1.1		1
7.6	odd	2		1617.2.a.j.1.1		1
8.3	odd	2		2112.2.a.j.1.1		1
8.5	even	2		2112.2.a.bb.1.1		1
9.2	odd	6		891.2.e.g.595.1		2
9.4	even	3		891.2.e.e.298.1		2
9.5	odd	6		891.2.e.g.298.1		2
9.7	even	3		891.2.e.e.595.1		2
11.2	odd	10		363.2.e.g.202.1		4
11.3	even	5		363.2.e.e.130.1		4
11.4	even	5		363.2.e.e.148.1		4
11.5	even	5		363.2.e.e.124.1		4
11.6	odd	10		363.2.e.g.124.1		4
11.7	odd	10		363.2.e.g.148.1		4
11.8	odd	10		363.2.e.g.130.1		4
11.9	even	5		363.2.e.e.202.1		4
11.10	odd	2		363.2.a.b.1.1		1
12.11	even	2		1584.2.a.o.1.1		1
13.12	even	2		5577.2.a.a.1.1		1
15.2	even	4		2475.2.c.d.199.1		2
15.8	even	4		2475.2.c.d.199.2		2
15.14	odd	2		2475.2.a.g.1.1		1
17.16	even	2		9537.2.a.m.1.1		1
21.20	even	2		4851.2.a.b.1.1		1
24.5	odd	2		6336.2.a.x.1.1		1
24.11	even	2		6336.2.a.n.1.1		1
33.32	even	2		1089.2.a.j.1.1		1
44.43	even	2		5808.2.a.t.1.1		1
55.54	odd	2		9075.2.a.q.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	1.1	even	1	trivial
99.2.a.b.1.1		1	3.2	odd	2
363.2.a.b.1.1		1	11.10	odd	2
363.2.e.e.124.1		4	11.5	even	5
363.2.e.e.130.1		4	11.3	even	5
363.2.e.e.148.1		4	11.4	even	5
363.2.e.e.202.1		4	11.9	even	5
363.2.e.g.124.1		4	11.6	odd	10
363.2.e.g.130.1		4	11.8	odd	10
363.2.e.g.148.1		4	11.7	odd	10
363.2.e.g.202.1		4	11.2	odd	10
528.2.a.g.1.1		1	4.3	odd	2
825.2.a.a.1.1		1	5.4	even	2
825.2.c.a.199.1		2	5.3	odd	4
825.2.c.a.199.2		2	5.2	odd	4
891.2.e.e.298.1		2	9.4	even	3
891.2.e.e.595.1		2	9.7	even	3
891.2.e.g.298.1		2	9.5	odd	6
891.2.e.g.595.1		2	9.2	odd	6
1089.2.a.j.1.1		1	33.32	even	2
1584.2.a.o.1.1		1	12.11	even	2
1617.2.a.j.1.1		1	7.6	odd	2
2112.2.a.j.1.1		1	8.3	odd	2
2112.2.a.bb.1.1		1	8.5	even	2
2475.2.a.g.1.1		1	15.14	odd	2
2475.2.c.d.199.1		2	15.2	even	4
2475.2.c.d.199.2		2	15.8	even	4
4851.2.a.b.1.1		1	21.20	even	2
5577.2.a.a.1.1		1	13.12	even	2
5808.2.a.t.1.1		1	44.43	even	2
6336.2.a.n.1.1		1	24.11	even	2
6336.2.a.x.1.1		1	24.5	odd	2
9075.2.a.q.1.1		1	55.54	odd	2
9537.2.a.m.1.1		1	17.16	even	2