Embedded newform 9408.2.a.h.1.1

• Label: 9408.2.a.h.1.1
• Level: $9408$
• Weight: $2$
• Character: 9408.1
• Self dual: yes
• Analytic conductor: $75.123$
• Analytic rank: $2$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(75.1232582216\)
Analytic rank:	\(2\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	9408.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{9} -4.00000 q^{11} -2.00000 q^{13} +2.00000 q^{15} -2.00000 q^{17} -4.00000 q^{19} -8.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} +4.00000 q^{33} -6.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} -4.00000 q^{43} -2.00000 q^{45} +2.00000 q^{51} +2.00000 q^{53} +8.00000 q^{55} +4.00000 q^{57} +4.00000 q^{59} -2.00000 q^{61} +4.00000 q^{65} +4.00000 q^{67} +8.00000 q^{69} +8.00000 q^{71} -10.0000 q^{73} +1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +4.00000 q^{85} +6.00000 q^{87} +6.00000 q^{89} +8.00000 q^{93} +8.00000 q^{95} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(5\)	−2.00000	−0.894427				−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	0	0
			\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
−0.603023	+	0.797724i				\(0.706037\pi\)
\(13\)	−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	\(0.577979\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	−8.00000	−1.66812	−0.834058	−	0.551677i	\(-0.813988\pi\)
			−0.834058	+	0.551677i	\(0.813988\pi\)
\(29\)	−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	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	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	9408.2.a.h.1.1		1
4.3	odd	2		9408.2.a.cc.1.1		1
7.6	odd	2		192.2.a.d.1.1		1
8.3	odd	2		2352.2.a.i.1.1		1
8.5	even	2		1176.2.a.i.1.1		1
21.20	even	2		576.2.a.d.1.1		1
24.5	odd	2		3528.2.a.d.1.1		1
24.11	even	2		7056.2.a.q.1.1		1
28.27	even	2		192.2.a.b.1.1		1
35.13	even	4		4800.2.f.d.3649.2		2
35.27	even	4		4800.2.f.d.3649.1		2
35.34	odd	2		4800.2.a.q.1.1		1
56.3	even	6		2352.2.q.l.961.1		2
56.5	odd	6		1176.2.q.i.361.1		2
56.11	odd	6		2352.2.q.r.961.1		2
56.13	odd	2		24.2.a.a.1.1	✓	1
56.19	even	6		2352.2.q.l.1537.1		2
56.27	even	2		48.2.a.a.1.1		1
56.37	even	6		1176.2.q.a.361.1		2
56.45	odd	6		1176.2.q.i.961.1		2
56.51	odd	6		2352.2.q.r.1537.1		2
56.53	even	6		1176.2.q.a.961.1		2
84.83	odd	2		576.2.a.b.1.1		1
112.13	odd	4		768.2.d.e.385.2		2
112.27	even	4		768.2.d.d.385.2		2
112.69	odd	4		768.2.d.e.385.1		2
112.83	even	4		768.2.d.d.385.1		2
140.27	odd	4		4800.2.f.bg.3649.2		2
140.83	odd	4		4800.2.f.bg.3649.1		2
140.139	even	2		4800.2.a.cc.1.1		1
168.5	even	6		3528.2.s.j.361.1		2
168.53	odd	6		3528.2.s.y.3313.1		2
168.83	odd	2		144.2.a.b.1.1		1
168.101	even	6		3528.2.s.j.3313.1		2
168.125	even	2		72.2.a.a.1.1		1
168.149	odd	6		3528.2.s.y.361.1		2
280.13	even	4		600.2.f.e.49.1		2
280.27	odd	4		1200.2.f.b.49.1		2
280.69	odd	2		600.2.a.h.1.1		1
280.83	odd	4		1200.2.f.b.49.2		2
280.139	even	2		1200.2.a.d.1.1		1
280.237	even	4		600.2.f.e.49.2		2
336.83	odd	4		2304.2.d.k.1153.2		2
336.125	even	4		2304.2.d.i.1153.2		2
336.251	odd	4		2304.2.d.k.1153.1		2
336.293	even	4		2304.2.d.i.1153.1		2
504.13	odd	6		648.2.i.g.217.1		2
504.83	odd	6		1296.2.i.e.433.1		2
504.139	even	6		1296.2.i.m.865.1		2
504.293	even	6		648.2.i.b.217.1		2
504.349	odd	6		648.2.i.g.433.1		2
504.419	odd	6		1296.2.i.e.865.1		2
504.461	even	6		648.2.i.b.433.1		2
504.475	even	6		1296.2.i.m.433.1		2
616.307	odd	2		5808.2.a.s.1.1		1
616.461	even	2		2904.2.a.c.1.1		1
728.125	even	4		4056.2.c.e.337.1		2
728.181	odd	2		4056.2.a.i.1.1		1
728.363	even	2		8112.2.a.be.1.1		1
728.629	even	4		4056.2.c.e.337.2		2
840.83	even	4		3600.2.f.r.2449.1		2
840.293	odd	4		1800.2.f.c.649.1		2
840.419	odd	2		3600.2.a.v.1.1		1
840.587	even	4		3600.2.f.r.2449.2		2
840.629	even	2		1800.2.a.m.1.1		1
840.797	odd	4		1800.2.f.c.649.2		2
952.237	odd	2		6936.2.a.p.1.1		1
1064.797	even	2		8664.2.a.j.1.1		1
1848.461	odd	2		8712.2.a.u.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.a.a.1.1	✓	1	56.13	odd	2
48.2.a.a.1.1		1	56.27	even	2
72.2.a.a.1.1		1	168.125	even	2
144.2.a.b.1.1		1	168.83	odd	2
192.2.a.b.1.1		1	28.27	even	2
192.2.a.d.1.1		1	7.6	odd	2
576.2.a.b.1.1		1	84.83	odd	2
576.2.a.d.1.1		1	21.20	even	2
600.2.a.h.1.1		1	280.69	odd	2
600.2.f.e.49.1		2	280.13	even	4
600.2.f.e.49.2		2	280.237	even	4
648.2.i.b.217.1		2	504.293	even	6
648.2.i.b.433.1		2	504.461	even	6
648.2.i.g.217.1		2	504.13	odd	6
648.2.i.g.433.1		2	504.349	odd	6
768.2.d.d.385.1		2	112.83	even	4
768.2.d.d.385.2		2	112.27	even	4
768.2.d.e.385.1		2	112.69	odd	4
768.2.d.e.385.2		2	112.13	odd	4
1176.2.a.i.1.1		1	8.5	even	2
1176.2.q.a.361.1		2	56.37	even	6
1176.2.q.a.961.1		2	56.53	even	6
1176.2.q.i.361.1		2	56.5	odd	6
1176.2.q.i.961.1		2	56.45	odd	6
1200.2.a.d.1.1		1	280.139	even	2
1200.2.f.b.49.1		2	280.27	odd	4
1200.2.f.b.49.2		2	280.83	odd	4
1296.2.i.e.433.1		2	504.83	odd	6
1296.2.i.e.865.1		2	504.419	odd	6
1296.2.i.m.433.1		2	504.475	even	6
1296.2.i.m.865.1		2	504.139	even	6
1800.2.a.m.1.1		1	840.629	even	2
1800.2.f.c.649.1		2	840.293	odd	4
1800.2.f.c.649.2		2	840.797	odd	4
2304.2.d.i.1153.1		2	336.293	even	4
2304.2.d.i.1153.2		2	336.125	even	4
2304.2.d.k.1153.1		2	336.251	odd	4
2304.2.d.k.1153.2		2	336.83	odd	4
2352.2.a.i.1.1		1	8.3	odd	2
2352.2.q.l.961.1		2	56.3	even	6
2352.2.q.l.1537.1		2	56.19	even	6
2352.2.q.r.961.1		2	56.11	odd	6
2352.2.q.r.1537.1		2	56.51	odd	6
2904.2.a.c.1.1		1	616.461	even	2
3528.2.a.d.1.1		1	24.5	odd	2
3528.2.s.j.361.1		2	168.5	even	6
3528.2.s.j.3313.1		2	168.101	even	6
3528.2.s.y.361.1		2	168.149	odd	6
3528.2.s.y.3313.1		2	168.53	odd	6
3600.2.a.v.1.1		1	840.419	odd	2
3600.2.f.r.2449.1		2	840.83	even	4
3600.2.f.r.2449.2		2	840.587	even	4
4056.2.a.i.1.1		1	728.181	odd	2
4056.2.c.e.337.1		2	728.125	even	4
4056.2.c.e.337.2		2	728.629	even	4
4800.2.a.q.1.1		1	35.34	odd	2
4800.2.a.cc.1.1		1	140.139	even	2
4800.2.f.d.3649.1		2	35.27	even	4
4800.2.f.d.3649.2		2	35.13	even	4
4800.2.f.bg.3649.1		2	140.83	odd	4
4800.2.f.bg.3649.2		2	140.27	odd	4
5808.2.a.s.1.1		1	616.307	odd	2
6936.2.a.p.1.1		1	952.237	odd	2
7056.2.a.q.1.1		1	24.11	even	2
8112.2.a.be.1.1		1	728.363	even	2
8664.2.a.j.1.1		1	1064.797	even	2
8712.2.a.u.1.1		1	1848.461	odd	2
9408.2.a.h.1.1		1	1.1	even	1	trivial
9408.2.a.cc.1.1		1	4.3	odd	2