Embedded newform 5415.2.a.j.1.1

• Label: 5415.2.a.j.1.1
• Level: $5415$
• Weight: $2$
• Character: 5415.1
• Self dual: yes
• Analytic conductor: $43.239$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5415.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(43.2389926945\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	5415.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)

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	0.353553	−	0.935414i	\(-0.384973\pi\)
			0.353553	+	0.935414i	\(0.384973\pi\)
\(3\)	1.00000	0.577350
			\(5\)	1.00000	0.447214
\(7\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(11\)	−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	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	0	0
			\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	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	\(0.192861\pi\)
\(41\)	−10.0000	−1.56174	−0.780869	−	0.624695i	\(-0.785223\pi\)
			−0.780869	+	0.624695i	\(0.785223\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5415.2.a.j.1.1		1
19.18	odd	2		15.2.a.a.1.1	✓	1
57.56	even	2		45.2.a.a.1.1		1
76.75	even	2		240.2.a.d.1.1		1
95.18	even	4		75.2.b.b.49.2		2
95.37	even	4		75.2.b.b.49.1		2
95.94	odd	2		75.2.a.b.1.1		1
133.18	odd	6		735.2.i.e.226.1		2
133.37	odd	6		735.2.i.e.361.1		2
133.75	even	6		735.2.i.d.361.1		2
133.94	even	6		735.2.i.d.226.1		2
133.132	even	2		735.2.a.c.1.1		1
152.37	odd	2		960.2.a.l.1.1		1
152.75	even	2		960.2.a.a.1.1		1
171.56	even	6		405.2.e.c.271.1		2
171.94	odd	6		405.2.e.f.136.1		2
171.113	even	6		405.2.e.c.136.1		2
171.151	odd	6		405.2.e.f.271.1		2
209.208	even	2		1815.2.a.d.1.1		1
228.227	odd	2		720.2.a.c.1.1		1
247.246	odd	2		2535.2.a.j.1.1		1
285.113	odd	4		225.2.b.b.199.1		2
285.227	odd	4		225.2.b.b.199.2		2
285.284	even	2		225.2.a.b.1.1		1
304.37	odd	4		3840.2.k.m.1921.2		2
304.75	even	4		3840.2.k.r.1921.1		2
304.189	odd	4		3840.2.k.m.1921.1		2
304.227	even	4		3840.2.k.r.1921.2		2
323.322	odd	2		4335.2.a.c.1.1		1
380.227	odd	4		1200.2.f.h.49.1		2
380.303	odd	4		1200.2.f.h.49.2		2
380.379	even	2		1200.2.a.e.1.1		1
399.398	odd	2		2205.2.a.i.1.1		1
437.436	even	2		7935.2.a.d.1.1		1
456.227	odd	2		2880.2.a.bc.1.1		1
456.341	even	2		2880.2.a.y.1.1		1
627.626	odd	2		5445.2.a.c.1.1		1
665.664	even	2		3675.2.a.j.1.1		1
741.740	even	2		7605.2.a.g.1.1		1
760.37	even	4		4800.2.f.bf.3649.1		2
760.189	odd	2		4800.2.a.t.1.1		1
760.227	odd	4		4800.2.f.c.3649.2		2
760.379	even	2		4800.2.a.bz.1.1		1
760.493	even	4		4800.2.f.bf.3649.2		2
760.683	odd	4		4800.2.f.c.3649.1		2
1045.1044	even	2		9075.2.a.g.1.1		1
1140.227	even	4		3600.2.f.e.2449.2		2
1140.683	even	4		3600.2.f.e.2449.1		2
1140.1139	odd	2		3600.2.a.u.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.2.a.a.1.1	✓	1	19.18	odd	2
45.2.a.a.1.1		1	57.56	even	2
75.2.a.b.1.1		1	95.94	odd	2
75.2.b.b.49.1		2	95.37	even	4
75.2.b.b.49.2		2	95.18	even	4
225.2.a.b.1.1		1	285.284	even	2
225.2.b.b.199.1		2	285.113	odd	4
225.2.b.b.199.2		2	285.227	odd	4
240.2.a.d.1.1		1	76.75	even	2
405.2.e.c.136.1		2	171.113	even	6
405.2.e.c.271.1		2	171.56	even	6
405.2.e.f.136.1		2	171.94	odd	6
405.2.e.f.271.1		2	171.151	odd	6
720.2.a.c.1.1		1	228.227	odd	2
735.2.a.c.1.1		1	133.132	even	2
735.2.i.d.226.1		2	133.94	even	6
735.2.i.d.361.1		2	133.75	even	6
735.2.i.e.226.1		2	133.18	odd	6
735.2.i.e.361.1		2	133.37	odd	6
960.2.a.a.1.1		1	152.75	even	2
960.2.a.l.1.1		1	152.37	odd	2
1200.2.a.e.1.1		1	380.379	even	2
1200.2.f.h.49.1		2	380.227	odd	4
1200.2.f.h.49.2		2	380.303	odd	4
1815.2.a.d.1.1		1	209.208	even	2
2205.2.a.i.1.1		1	399.398	odd	2
2535.2.a.j.1.1		1	247.246	odd	2
2880.2.a.y.1.1		1	456.341	even	2
2880.2.a.bc.1.1		1	456.227	odd	2
3600.2.a.u.1.1		1	1140.1139	odd	2
3600.2.f.e.2449.1		2	1140.683	even	4
3600.2.f.e.2449.2		2	1140.227	even	4
3675.2.a.j.1.1		1	665.664	even	2
3840.2.k.m.1921.1		2	304.189	odd	4
3840.2.k.m.1921.2		2	304.37	odd	4
3840.2.k.r.1921.1		2	304.75	even	4
3840.2.k.r.1921.2		2	304.227	even	4
4335.2.a.c.1.1		1	323.322	odd	2
4800.2.a.t.1.1		1	760.189	odd	2
4800.2.a.bz.1.1		1	760.379	even	2
4800.2.f.c.3649.1		2	760.683	odd	4
4800.2.f.c.3649.2		2	760.227	odd	4
4800.2.f.bf.3649.1		2	760.37	even	4
4800.2.f.bf.3649.2		2	760.493	even	4
5415.2.a.j.1.1		1	1.1	even	1	trivial
5445.2.a.c.1.1		1	627.626	odd	2
7605.2.a.g.1.1		1	741.740	even	2
7935.2.a.d.1.1		1	437.436	even	2
9075.2.a.g.1.1		1	1045.1044	even	2