Embedded newform 8400.2.a.ct.1.1

• Label: 8400.2.a.ct.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 84)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	8400.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{11} -2.00000 q^{13} +4.00000 q^{19} +1.00000 q^{21} -6.00000 q^{23} +1.00000 q^{27} +6.00000 q^{29} -8.00000 q^{31} +6.00000 q^{33} -2.00000 q^{37} -2.00000 q^{39} +12.0000 q^{41} -4.00000 q^{43} +12.0000 q^{47} +1.00000 q^{49} +6.00000 q^{53} +4.00000 q^{57} -10.0000 q^{61} +1.00000 q^{63} +8.00000 q^{67} -6.00000 q^{69} -6.00000 q^{71} +10.0000 q^{73} +6.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +6.00000 q^{87} +12.0000 q^{89} -2.00000 q^{91} -8.00000 q^{93} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(11\)	6.00000	1.80907				0.904534	−	0.426401i	\(-0.140219\pi\)
			0.904534	+	0.426401i	\(0.140219\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	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	\(0.348268\pi\)
\(23\)	−6.00000	−1.25109	−0.625543	−	0.780189i	\(-0.715123\pi\)
			−0.625543	+	0.780189i	\(0.715123\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	12.0000	1.87409	0.937043	−	0.349215i	\(-0.113552\pi\)
			0.937043	+	0.349215i	\(0.113552\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	12.0000	1.75038	0.875190	−	0.483779i	\(-0.160736\pi\)
			0.875190	+	0.483779i	\(0.160736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8400.2.a.ct.1.1		1
4.3	odd	2		2100.2.a.a.1.1		1
5.4	even	2		336.2.a.b.1.1		1
12.11	even	2		6300.2.a.p.1.1		1
15.14	odd	2		1008.2.a.g.1.1		1
20.3	even	4		2100.2.k.a.1849.1		2
20.7	even	4		2100.2.k.a.1849.2		2
20.19	odd	2		84.2.a.b.1.1	✓	1
35.4	even	6		2352.2.q.s.961.1		2
35.9	even	6		2352.2.q.s.1537.1		2
35.19	odd	6		2352.2.q.g.1537.1		2
35.24	odd	6		2352.2.q.g.961.1		2
35.34	odd	2		2352.2.a.s.1.1		1
40.19	odd	2		1344.2.a.f.1.1		1
40.29	even	2		1344.2.a.o.1.1		1
60.23	odd	4		6300.2.k.r.6049.2		2
60.47	odd	4		6300.2.k.r.6049.1		2
60.59	even	2		252.2.a.b.1.1		1
80.19	odd	4		5376.2.c.i.2689.2		2
80.29	even	4		5376.2.c.x.2689.1		2
80.59	odd	4		5376.2.c.i.2689.1		2
80.69	even	4		5376.2.c.x.2689.2		2
105.104	even	2		7056.2.a.x.1.1		1
120.29	odd	2		4032.2.a.t.1.1		1
120.59	even	2		4032.2.a.u.1.1		1
140.19	even	6		588.2.i.f.361.1		2
140.39	odd	6		588.2.i.c.373.1		2
140.59	even	6		588.2.i.f.373.1		2
140.79	odd	6		588.2.i.c.361.1		2
140.139	even	2		588.2.a.c.1.1		1
180.59	even	6		2268.2.j.f.1513.1		2
180.79	odd	6		2268.2.j.i.757.1		2
180.119	even	6		2268.2.j.f.757.1		2
180.139	odd	6		2268.2.j.i.1513.1		2
280.69	odd	2		9408.2.a.r.1.1		1
280.139	even	2		9408.2.a.co.1.1		1
420.59	odd	6		1764.2.k.d.1549.1		2
420.179	even	6		1764.2.k.e.1549.1		2
420.299	odd	6		1764.2.k.d.361.1		2
420.359	even	6		1764.2.k.e.361.1		2
420.419	odd	2		1764.2.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.b.1.1	✓	1	20.19	odd	2
252.2.a.b.1.1		1	60.59	even	2
336.2.a.b.1.1		1	5.4	even	2
588.2.a.c.1.1		1	140.139	even	2
588.2.i.c.361.1		2	140.79	odd	6
588.2.i.c.373.1		2	140.39	odd	6
588.2.i.f.361.1		2	140.19	even	6
588.2.i.f.373.1		2	140.59	even	6
1008.2.a.g.1.1		1	15.14	odd	2
1344.2.a.f.1.1		1	40.19	odd	2
1344.2.a.o.1.1		1	40.29	even	2
1764.2.a.g.1.1		1	420.419	odd	2
1764.2.k.d.361.1		2	420.299	odd	6
1764.2.k.d.1549.1		2	420.59	odd	6
1764.2.k.e.361.1		2	420.359	even	6
1764.2.k.e.1549.1		2	420.179	even	6
2100.2.a.a.1.1		1	4.3	odd	2
2100.2.k.a.1849.1		2	20.3	even	4
2100.2.k.a.1849.2		2	20.7	even	4
2268.2.j.f.757.1		2	180.119	even	6
2268.2.j.f.1513.1		2	180.59	even	6
2268.2.j.i.757.1		2	180.79	odd	6
2268.2.j.i.1513.1		2	180.139	odd	6
2352.2.a.s.1.1		1	35.34	odd	2
2352.2.q.g.961.1		2	35.24	odd	6
2352.2.q.g.1537.1		2	35.19	odd	6
2352.2.q.s.961.1		2	35.4	even	6
2352.2.q.s.1537.1		2	35.9	even	6
4032.2.a.t.1.1		1	120.29	odd	2
4032.2.a.u.1.1		1	120.59	even	2
5376.2.c.i.2689.1		2	80.59	odd	4
5376.2.c.i.2689.2		2	80.19	odd	4
5376.2.c.x.2689.1		2	80.29	even	4
5376.2.c.x.2689.2		2	80.69	even	4
6300.2.a.p.1.1		1	12.11	even	2
6300.2.k.r.6049.1		2	60.47	odd	4
6300.2.k.r.6049.2		2	60.23	odd	4
7056.2.a.x.1.1		1	105.104	even	2
8400.2.a.ct.1.1		1	1.1	even	1	trivial
9408.2.a.r.1.1		1	280.69	odd	2
9408.2.a.co.1.1		1	280.139	even	2