Embedded newform 8400.2.a.k.1.1

• Label: 8400.2.a.k.1.1
• Level: $8400$
• Weight: $2$
• Character: 8400.1
• Self dual: yes
• Analytic conductor: $67.074$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8400.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(67.0743376979\)
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	\(\chi\)	\(=\)	8400.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−1.00000	−0.577350
\(5\)	0	0
			\(7\)	−1.00000	−0.377964
\(11\)	4.00000	1.20605				0.603023	−	0.797724i	\(-0.293963\pi\)
			0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)	−6.00000	−1.66410	−0.832050	−	0.554700i	\(-0.812833\pi\)
			−0.832050	+	0.554700i	\(0.812833\pi\)
\(17\)	−2.00000	−0.485071	−0.242536	−	0.970143i	\(-0.577979\pi\)
			−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	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	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	−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	8400.2.a.k.1.1		1
4.3	odd	2		1050.2.a.i.1.1		1
5.4	even	2		336.2.a.d.1.1		1
12.11	even	2		3150.2.a.bo.1.1		1
15.14	odd	2		1008.2.a.j.1.1		1
20.3	even	4		1050.2.g.a.799.2		2
20.7	even	4		1050.2.g.a.799.1		2
20.19	odd	2		42.2.a.a.1.1	✓	1
28.27	even	2		7350.2.a.f.1.1		1
35.4	even	6		2352.2.q.i.961.1		2
35.9	even	6		2352.2.q.i.1537.1		2
35.19	odd	6		2352.2.q.n.1537.1		2
35.24	odd	6		2352.2.q.n.961.1		2
35.34	odd	2		2352.2.a.l.1.1		1
40.19	odd	2		1344.2.a.q.1.1		1
40.29	even	2		1344.2.a.i.1.1		1
60.23	odd	4		3150.2.g.r.2899.1		2
60.47	odd	4		3150.2.g.r.2899.2		2
60.59	even	2		126.2.a.a.1.1		1
80.19	odd	4		5376.2.c.bc.2689.1		2
80.29	even	4		5376.2.c.e.2689.2		2
80.59	odd	4		5376.2.c.bc.2689.2		2
80.69	even	4		5376.2.c.e.2689.1		2
105.104	even	2		7056.2.a.k.1.1		1
120.29	odd	2		4032.2.a.m.1.1		1
120.59	even	2		4032.2.a.e.1.1		1
140.19	even	6		294.2.e.a.67.1		2
140.39	odd	6		294.2.e.c.79.1		2
140.59	even	6		294.2.e.a.79.1		2
140.79	odd	6		294.2.e.c.67.1		2
140.139	even	2		294.2.a.g.1.1		1
180.59	even	6		1134.2.f.j.379.1		2
180.79	odd	6		1134.2.f.g.757.1		2
180.119	even	6		1134.2.f.j.757.1		2
180.139	odd	6		1134.2.f.g.379.1		2
220.219	even	2		5082.2.a.d.1.1		1
260.259	odd	2		7098.2.a.f.1.1		1
280.69	odd	2		9408.2.a.bw.1.1		1
280.139	even	2		9408.2.a.n.1.1		1
420.59	odd	6		882.2.g.j.667.1		2
420.179	even	6		882.2.g.h.667.1		2
420.299	odd	6		882.2.g.j.361.1		2
420.359	even	6		882.2.g.h.361.1		2
420.419	odd	2		882.2.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.a.a.1.1	✓	1	20.19	odd	2
126.2.a.a.1.1		1	60.59	even	2
294.2.a.g.1.1		1	140.139	even	2
294.2.e.a.67.1		2	140.19	even	6
294.2.e.a.79.1		2	140.59	even	6
294.2.e.c.67.1		2	140.79	odd	6
294.2.e.c.79.1		2	140.39	odd	6
336.2.a.d.1.1		1	5.4	even	2
882.2.a.b.1.1		1	420.419	odd	2
882.2.g.h.361.1		2	420.359	even	6
882.2.g.h.667.1		2	420.179	even	6
882.2.g.j.361.1		2	420.299	odd	6
882.2.g.j.667.1		2	420.59	odd	6
1008.2.a.j.1.1		1	15.14	odd	2
1050.2.a.i.1.1		1	4.3	odd	2
1050.2.g.a.799.1		2	20.7	even	4
1050.2.g.a.799.2		2	20.3	even	4
1134.2.f.g.379.1		2	180.139	odd	6
1134.2.f.g.757.1		2	180.79	odd	6
1134.2.f.j.379.1		2	180.59	even	6
1134.2.f.j.757.1		2	180.119	even	6
1344.2.a.i.1.1		1	40.29	even	2
1344.2.a.q.1.1		1	40.19	odd	2
2352.2.a.l.1.1		1	35.34	odd	2
2352.2.q.i.961.1		2	35.4	even	6
2352.2.q.i.1537.1		2	35.9	even	6
2352.2.q.n.961.1		2	35.24	odd	6
2352.2.q.n.1537.1		2	35.19	odd	6
3150.2.a.bo.1.1		1	12.11	even	2
3150.2.g.r.2899.1		2	60.23	odd	4
3150.2.g.r.2899.2		2	60.47	odd	4
4032.2.a.e.1.1		1	120.59	even	2
4032.2.a.m.1.1		1	120.29	odd	2
5082.2.a.d.1.1		1	220.219	even	2
5376.2.c.e.2689.1		2	80.69	even	4
5376.2.c.e.2689.2		2	80.29	even	4
5376.2.c.bc.2689.1		2	80.19	odd	4
5376.2.c.bc.2689.2		2	80.59	odd	4
7056.2.a.k.1.1		1	105.104	even	2
7098.2.a.f.1.1		1	260.259	odd	2
7350.2.a.f.1.1		1	28.27	even	2
8400.2.a.k.1.1		1	1.1	even	1	trivial
9408.2.a.n.1.1		1	280.139	even	2
9408.2.a.bw.1.1		1	280.69	odd	2