Embedded newform 7098.2.a.f.1.1

• Label: 7098.2.a.f.1.1
• Level: $7098$
• Weight: $2$
• Character: 7098.1
• Self dual: yes
• Analytic conductor: $56.678$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(56.6778153547\)
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	\(\chi\)	\(=\)	7098.1

$q$-expansion

\(f(q)\)

\(=\)

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

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
			\(3\)	−1.00000	−0.577350
\(5\)	2.00000	0.894427				0.447214	−	0.894427i	\(-0.352416\pi\)
			0.447214	+	0.894427i	\(0.352416\pi\)
\(7\)	1.00000	0.377964
			\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
0.603023	+	0.797724i				\(0.293963\pi\)
\(13\)	0	0
			\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
0.242536	+	0.970143i				\(0.422021\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	8.00000	1.66812	0.834058	−	0.551677i	\(-0.186012\pi\)
			0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	10.0000	1.64399	0.821995	−	0.569495i	\(-0.192861\pi\)
			0.821995	+	0.569495i	\(0.192861\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.f.1.1		1
13.12	even	2		42.2.a.a.1.1	✓	1
39.38	odd	2		126.2.a.a.1.1		1
52.51	odd	2		336.2.a.d.1.1		1
65.12	odd	4		1050.2.g.a.799.2		2
65.38	odd	4		1050.2.g.a.799.1		2
65.64	even	2		1050.2.a.i.1.1		1
91.12	odd	6		294.2.e.a.67.1		2
91.25	even	6		294.2.e.c.79.1		2
91.38	odd	6		294.2.e.a.79.1		2
91.51	even	6		294.2.e.c.67.1		2
91.90	odd	2		294.2.a.g.1.1		1
104.51	odd	2		1344.2.a.i.1.1		1
104.77	even	2		1344.2.a.q.1.1		1
117.25	even	6		1134.2.f.g.757.1		2
117.38	odd	6		1134.2.f.j.757.1		2
117.77	odd	6		1134.2.f.j.379.1		2
117.103	even	6		1134.2.f.g.379.1		2
143.142	odd	2		5082.2.a.d.1.1		1
156.155	even	2		1008.2.a.j.1.1		1
195.38	even	4		3150.2.g.r.2899.2		2
195.77	even	4		3150.2.g.r.2899.1		2
195.194	odd	2		3150.2.a.bo.1.1		1
208.51	odd	4		5376.2.c.e.2689.2		2
208.77	even	4		5376.2.c.bc.2689.1		2
208.155	odd	4		5376.2.c.e.2689.1		2
208.181	even	4		5376.2.c.bc.2689.2		2
260.259	odd	2		8400.2.a.k.1.1		1
273.38	even	6		882.2.g.j.667.1		2
273.116	odd	6		882.2.g.h.667.1		2
273.194	even	6		882.2.g.j.361.1		2
273.233	odd	6		882.2.g.h.361.1		2
273.272	even	2		882.2.a.b.1.1		1
312.77	odd	2		4032.2.a.e.1.1		1
312.155	even	2		4032.2.a.m.1.1		1
364.51	odd	6		2352.2.q.i.1537.1		2
364.103	even	6		2352.2.q.n.1537.1		2
364.207	odd	6		2352.2.q.i.961.1		2
364.311	even	6		2352.2.q.n.961.1		2
364.363	even	2		2352.2.a.l.1.1		1
455.454	odd	2		7350.2.a.f.1.1		1
728.181	odd	2		9408.2.a.n.1.1		1
728.363	even	2		9408.2.a.bw.1.1		1
1092.1091	odd	2		7056.2.a.k.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.a.a.1.1	✓	1	13.12	even	2
126.2.a.a.1.1		1	39.38	odd	2
294.2.a.g.1.1		1	91.90	odd	2
294.2.e.a.67.1		2	91.12	odd	6
294.2.e.a.79.1		2	91.38	odd	6
294.2.e.c.67.1		2	91.51	even	6
294.2.e.c.79.1		2	91.25	even	6
336.2.a.d.1.1		1	52.51	odd	2
882.2.a.b.1.1		1	273.272	even	2
882.2.g.h.361.1		2	273.233	odd	6
882.2.g.h.667.1		2	273.116	odd	6
882.2.g.j.361.1		2	273.194	even	6
882.2.g.j.667.1		2	273.38	even	6
1008.2.a.j.1.1		1	156.155	even	2
1050.2.a.i.1.1		1	65.64	even	2
1050.2.g.a.799.1		2	65.38	odd	4
1050.2.g.a.799.2		2	65.12	odd	4
1134.2.f.g.379.1		2	117.103	even	6
1134.2.f.g.757.1		2	117.25	even	6
1134.2.f.j.379.1		2	117.77	odd	6
1134.2.f.j.757.1		2	117.38	odd	6
1344.2.a.i.1.1		1	104.51	odd	2
1344.2.a.q.1.1		1	104.77	even	2
2352.2.a.l.1.1		1	364.363	even	2
2352.2.q.i.961.1		2	364.207	odd	6
2352.2.q.i.1537.1		2	364.51	odd	6
2352.2.q.n.961.1		2	364.311	even	6
2352.2.q.n.1537.1		2	364.103	even	6
3150.2.a.bo.1.1		1	195.194	odd	2
3150.2.g.r.2899.1		2	195.77	even	4
3150.2.g.r.2899.2		2	195.38	even	4
4032.2.a.e.1.1		1	312.77	odd	2
4032.2.a.m.1.1		1	312.155	even	2
5082.2.a.d.1.1		1	143.142	odd	2
5376.2.c.e.2689.1		2	208.155	odd	4
5376.2.c.e.2689.2		2	208.51	odd	4
5376.2.c.bc.2689.1		2	208.77	even	4
5376.2.c.bc.2689.2		2	208.181	even	4
7056.2.a.k.1.1		1	1092.1091	odd	2
7098.2.a.f.1.1		1	1.1	even	1	trivial
7350.2.a.f.1.1		1	455.454	odd	2
8400.2.a.k.1.1		1	260.259	odd	2
9408.2.a.n.1.1		1	728.181	odd	2
9408.2.a.bw.1.1		1	728.363	even	2