Embedded newform 2352.2.a.i.1.1

• Label: 2352.2.a.i.1.1
• Level: $2352$
• Weight: $2$
• Character: 2352.1
• Self dual: yes
• Analytic conductor: $18.781$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.7808145554\)
Analytic rank:	\(0\)
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	\(\chi\)	\(=\)	2352.1

$q$-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	\(-0.352416\pi\)
			0.447214	+	0.894427i	\(0.352416\pi\)
\(7\)	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	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−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	\(-0.186012\pi\)
			0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	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	\(-0.244865\pi\)
			0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	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	2352.2.a.i.1.1		1
3.2	odd	2		7056.2.a.q.1.1		1
4.3	odd	2		1176.2.a.i.1.1		1
7.2	even	3		2352.2.q.r.1537.1		2
7.3	odd	6		2352.2.q.l.961.1		2
7.4	even	3		2352.2.q.r.961.1		2
7.5	odd	6		2352.2.q.l.1537.1		2
7.6	odd	2		48.2.a.a.1.1		1
8.3	odd	2		9408.2.a.h.1.1		1
8.5	even	2		9408.2.a.cc.1.1		1
12.11	even	2		3528.2.a.d.1.1		1
21.20	even	2		144.2.a.b.1.1		1
28.3	even	6		1176.2.q.i.961.1		2
28.11	odd	6		1176.2.q.a.961.1		2
28.19	even	6		1176.2.q.i.361.1		2
28.23	odd	6		1176.2.q.a.361.1		2
28.27	even	2		24.2.a.a.1.1	✓	1
35.13	even	4		1200.2.f.b.49.2		2
35.27	even	4		1200.2.f.b.49.1		2
35.34	odd	2		1200.2.a.d.1.1		1
56.13	odd	2		192.2.a.b.1.1		1
56.27	even	2		192.2.a.d.1.1		1
63.13	odd	6		1296.2.i.m.865.1		2
63.20	even	6		1296.2.i.e.433.1		2
63.34	odd	6		1296.2.i.m.433.1		2
63.41	even	6		1296.2.i.e.865.1		2
77.76	even	2		5808.2.a.s.1.1		1
84.11	even	6		3528.2.s.y.3313.1		2
84.23	even	6		3528.2.s.y.361.1		2
84.47	odd	6		3528.2.s.j.361.1		2
84.59	odd	6		3528.2.s.j.3313.1		2
84.83	odd	2		72.2.a.a.1.1		1
91.90	odd	2		8112.2.a.be.1.1		1
105.62	odd	4		3600.2.f.r.2449.2		2
105.83	odd	4		3600.2.f.r.2449.1		2
105.104	even	2		3600.2.a.v.1.1		1
112.13	odd	4		768.2.d.d.385.2		2
112.27	even	4		768.2.d.e.385.2		2
112.69	odd	4		768.2.d.d.385.1		2
112.83	even	4		768.2.d.e.385.1		2
140.27	odd	4		600.2.f.e.49.2		2
140.83	odd	4		600.2.f.e.49.1		2
140.139	even	2		600.2.a.h.1.1		1
168.83	odd	2		576.2.a.d.1.1		1
168.125	even	2		576.2.a.b.1.1		1
252.83	odd	6		648.2.i.b.433.1		2
252.139	even	6		648.2.i.g.217.1		2
252.167	odd	6		648.2.i.b.217.1		2
252.223	even	6		648.2.i.g.433.1		2
280.13	even	4		4800.2.f.bg.3649.1		2
280.27	odd	4		4800.2.f.d.3649.1		2
280.69	odd	2		4800.2.a.cc.1.1		1
280.83	odd	4		4800.2.f.d.3649.2		2
280.139	even	2		4800.2.a.q.1.1		1
280.237	even	4		4800.2.f.bg.3649.2		2
308.307	odd	2		2904.2.a.c.1.1		1
336.83	odd	4		2304.2.d.i.1153.1		2
336.125	even	4		2304.2.d.k.1153.1		2
336.251	odd	4		2304.2.d.i.1153.2		2
336.293	even	4		2304.2.d.k.1153.2		2
364.83	odd	4		4056.2.c.e.337.2		2
364.307	odd	4		4056.2.c.e.337.1		2
364.363	even	2		4056.2.a.i.1.1		1
420.83	even	4		1800.2.f.c.649.1		2
420.167	even	4		1800.2.f.c.649.2		2
420.419	odd	2		1800.2.a.m.1.1		1
476.475	even	2		6936.2.a.p.1.1		1
532.531	odd	2		8664.2.a.j.1.1		1
924.923	even	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	28.27	even	2
48.2.a.a.1.1		1	7.6	odd	2
72.2.a.a.1.1		1	84.83	odd	2
144.2.a.b.1.1		1	21.20	even	2
192.2.a.b.1.1		1	56.13	odd	2
192.2.a.d.1.1		1	56.27	even	2
576.2.a.b.1.1		1	168.125	even	2
576.2.a.d.1.1		1	168.83	odd	2
600.2.a.h.1.1		1	140.139	even	2
600.2.f.e.49.1		2	140.83	odd	4
600.2.f.e.49.2		2	140.27	odd	4
648.2.i.b.217.1		2	252.167	odd	6
648.2.i.b.433.1		2	252.83	odd	6
648.2.i.g.217.1		2	252.139	even	6
648.2.i.g.433.1		2	252.223	even	6
768.2.d.d.385.1		2	112.69	odd	4
768.2.d.d.385.2		2	112.13	odd	4
768.2.d.e.385.1		2	112.83	even	4
768.2.d.e.385.2		2	112.27	even	4
1176.2.a.i.1.1		1	4.3	odd	2
1176.2.q.a.361.1		2	28.23	odd	6
1176.2.q.a.961.1		2	28.11	odd	6
1176.2.q.i.361.1		2	28.19	even	6
1176.2.q.i.961.1		2	28.3	even	6
1200.2.a.d.1.1		1	35.34	odd	2
1200.2.f.b.49.1		2	35.27	even	4
1200.2.f.b.49.2		2	35.13	even	4
1296.2.i.e.433.1		2	63.20	even	6
1296.2.i.e.865.1		2	63.41	even	6
1296.2.i.m.433.1		2	63.34	odd	6
1296.2.i.m.865.1		2	63.13	odd	6
1800.2.a.m.1.1		1	420.419	odd	2
1800.2.f.c.649.1		2	420.83	even	4
1800.2.f.c.649.2		2	420.167	even	4
2304.2.d.i.1153.1		2	336.83	odd	4
2304.2.d.i.1153.2		2	336.251	odd	4
2304.2.d.k.1153.1		2	336.125	even	4
2304.2.d.k.1153.2		2	336.293	even	4
2352.2.a.i.1.1		1	1.1	even	1	trivial
2352.2.q.l.961.1		2	7.3	odd	6
2352.2.q.l.1537.1		2	7.5	odd	6
2352.2.q.r.961.1		2	7.4	even	3
2352.2.q.r.1537.1		2	7.2	even	3
2904.2.a.c.1.1		1	308.307	odd	2
3528.2.a.d.1.1		1	12.11	even	2
3528.2.s.j.361.1		2	84.47	odd	6
3528.2.s.j.3313.1		2	84.59	odd	6
3528.2.s.y.361.1		2	84.23	even	6
3528.2.s.y.3313.1		2	84.11	even	6
3600.2.a.v.1.1		1	105.104	even	2
3600.2.f.r.2449.1		2	105.83	odd	4
3600.2.f.r.2449.2		2	105.62	odd	4
4056.2.a.i.1.1		1	364.363	even	2
4056.2.c.e.337.1		2	364.307	odd	4
4056.2.c.e.337.2		2	364.83	odd	4
4800.2.a.q.1.1		1	280.139	even	2
4800.2.a.cc.1.1		1	280.69	odd	2
4800.2.f.d.3649.1		2	280.27	odd	4
4800.2.f.d.3649.2		2	280.83	odd	4
4800.2.f.bg.3649.1		2	280.13	even	4
4800.2.f.bg.3649.2		2	280.237	even	4
5808.2.a.s.1.1		1	77.76	even	2
6936.2.a.p.1.1		1	476.475	even	2
7056.2.a.q.1.1		1	3.2	odd	2
8112.2.a.be.1.1		1	91.90	odd	2
8664.2.a.j.1.1		1	532.531	odd	2
8712.2.a.u.1.1		1	924.923	even	2
9408.2.a.h.1.1		1	8.3	odd	2
9408.2.a.cc.1.1		1	8.5	even	2