Embedded newform 882.2.a.k.1.1

• Label: 882.2.a.k.1.1
• Level: $882$
• Weight: $2$
• Character: 882.1
• Self dual: yes
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	882.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)

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
			\(3\)	0	0
\(5\)	3.00000	1.34164				0.670820	−	0.741620i	\(-0.265942\pi\)
			0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)	0	0
			\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
−0.452267	+	0.891883i				\(0.649385\pi\)
\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
			0.554700	+	0.832050i	\(0.312833\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−9.00000	−1.67126	−0.835629	−	0.549294i	\(-0.814897\pi\)
			−0.835629	+	0.549294i	\(0.814897\pi\)
\(31\)	1.00000	0.179605	0.0898027	−	0.995960i	\(-0.471376\pi\)
			0.0898027	+	0.995960i	\(0.471376\pi\)
\(37\)	8.00000	1.31519	0.657596	−	0.753371i	\(-0.271573\pi\)
			0.657596	+	0.753371i	\(0.271573\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	−10.0000	−1.52499	−0.762493	−	0.646997i	\(-0.776025\pi\)
			−0.762493	+	0.646997i	\(0.776025\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.a.k.1.1		1
3.2	odd	2		294.2.a.a.1.1		1
4.3	odd	2		7056.2.a.bz.1.1		1
7.2	even	3		882.2.g.b.361.1		2
7.3	odd	6		126.2.g.b.37.1		2
7.4	even	3		882.2.g.b.667.1		2
7.5	odd	6		126.2.g.b.109.1		2
7.6	odd	2		882.2.a.g.1.1		1
12.11	even	2		2352.2.a.n.1.1		1
15.14	odd	2		7350.2.a.cw.1.1		1
21.2	odd	6		294.2.e.f.67.1		2
21.5	even	6		42.2.e.b.25.1	✓	2
21.11	odd	6		294.2.e.f.79.1		2
21.17	even	6		42.2.e.b.37.1	yes	2
21.20	even	2		294.2.a.d.1.1		1
24.5	odd	2		9408.2.a.db.1.1		1
24.11	even	2		9408.2.a.bm.1.1		1
28.3	even	6		1008.2.s.n.289.1		2
28.19	even	6		1008.2.s.n.865.1		2
28.27	even	2		7056.2.a.g.1.1		1
63.5	even	6		1134.2.e.a.865.1		2
63.31	odd	6		1134.2.h.a.541.1		2
63.38	even	6		1134.2.e.a.919.1		2
63.40	odd	6		1134.2.e.p.865.1		2
63.47	even	6		1134.2.h.p.109.1		2
63.52	odd	6		1134.2.e.p.919.1		2
63.59	even	6		1134.2.h.p.541.1		2
63.61	odd	6		1134.2.h.a.109.1		2
84.11	even	6		2352.2.q.m.961.1		2
84.23	even	6		2352.2.q.m.1537.1		2
84.47	odd	6		336.2.q.d.193.1		2
84.59	odd	6		336.2.q.d.289.1		2
84.83	odd	2		2352.2.a.m.1.1		1
105.17	odd	12		1050.2.o.b.499.2		4
105.38	odd	12		1050.2.o.b.499.1		4
105.47	odd	12		1050.2.o.b.949.1		4
105.59	even	6		1050.2.i.e.751.1		2
105.68	odd	12		1050.2.o.b.949.2		4
105.89	even	6		1050.2.i.e.151.1		2
105.104	even	2		7350.2.a.ce.1.1		1
168.5	even	6		1344.2.q.v.193.1		2
168.59	odd	6		1344.2.q.j.961.1		2
168.83	odd	2		9408.2.a.bu.1.1		1
168.101	even	6		1344.2.q.v.961.1		2
168.125	even	2		9408.2.a.d.1.1		1
168.131	odd	6		1344.2.q.j.193.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.e.b.25.1	✓	2	21.5	even	6
42.2.e.b.37.1	yes	2	21.17	even	6
126.2.g.b.37.1		2	7.3	odd	6
126.2.g.b.109.1		2	7.5	odd	6
294.2.a.a.1.1		1	3.2	odd	2
294.2.a.d.1.1		1	21.20	even	2
294.2.e.f.67.1		2	21.2	odd	6
294.2.e.f.79.1		2	21.11	odd	6
336.2.q.d.193.1		2	84.47	odd	6
336.2.q.d.289.1		2	84.59	odd	6
882.2.a.g.1.1		1	7.6	odd	2
882.2.a.k.1.1		1	1.1	even	1	trivial
882.2.g.b.361.1		2	7.2	even	3
882.2.g.b.667.1		2	7.4	even	3
1008.2.s.n.289.1		2	28.3	even	6
1008.2.s.n.865.1		2	28.19	even	6
1050.2.i.e.151.1		2	105.89	even	6
1050.2.i.e.751.1		2	105.59	even	6
1050.2.o.b.499.1		4	105.38	odd	12
1050.2.o.b.499.2		4	105.17	odd	12
1050.2.o.b.949.1		4	105.47	odd	12
1050.2.o.b.949.2		4	105.68	odd	12
1134.2.e.a.865.1		2	63.5	even	6
1134.2.e.a.919.1		2	63.38	even	6
1134.2.e.p.865.1		2	63.40	odd	6
1134.2.e.p.919.1		2	63.52	odd	6
1134.2.h.a.109.1		2	63.61	odd	6
1134.2.h.a.541.1		2	63.31	odd	6
1134.2.h.p.109.1		2	63.47	even	6
1134.2.h.p.541.1		2	63.59	even	6
1344.2.q.j.193.1		2	168.131	odd	6
1344.2.q.j.961.1		2	168.59	odd	6
1344.2.q.v.193.1		2	168.5	even	6
1344.2.q.v.961.1		2	168.101	even	6
2352.2.a.m.1.1		1	84.83	odd	2
2352.2.a.n.1.1		1	12.11	even	2
2352.2.q.m.961.1		2	84.11	even	6
2352.2.q.m.1537.1		2	84.23	even	6
7056.2.a.g.1.1		1	28.27	even	2
7056.2.a.bz.1.1		1	4.3	odd	2
7350.2.a.ce.1.1		1	105.104	even	2
7350.2.a.cw.1.1		1	15.14	odd	2
9408.2.a.d.1.1		1	168.125	even	2
9408.2.a.bm.1.1		1	24.11	even	2
9408.2.a.bu.1.1		1	168.83	odd	2
9408.2.a.db.1.1		1	24.5	odd	2