Embedded newform 7581.2.a.d.1.1

• Label: 7581.2.a.d.1.1
• Level: $7581$
• Weight: $2$
• Character: 7581.1
• Self dual: yes
• Analytic conductor: $60.535$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7581 = 3 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7581.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(60.5345897723\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	7581.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)

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
			\(5\)	−2.00000	−0.894427	−0.447214	−	0.894427i	\(-0.647584\pi\)
−0.447214	+	0.894427i				\(0.647584\pi\)
\(7\)	−1.00000	−0.377964
			\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
0.603023	+	0.797724i				\(0.293963\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	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	\(0.5\pi\)
\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−2.00000	−0.312348	−0.156174	−	0.987730i	\(-0.549916\pi\)
			−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)	−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	7581.2.a.d.1.1		1
19.18	odd	2		21.2.a.a.1.1	✓	1
57.56	even	2		63.2.a.a.1.1		1
76.75	even	2		336.2.a.a.1.1		1
95.18	even	4		525.2.d.a.274.2		2
95.37	even	4		525.2.d.a.274.1		2
95.94	odd	2		525.2.a.d.1.1		1
133.18	odd	6		147.2.e.b.79.1		2
133.37	odd	6		147.2.e.b.67.1		2
133.75	even	6		147.2.e.c.67.1		2
133.94	even	6		147.2.e.c.79.1		2
133.132	even	2		147.2.a.a.1.1		1
152.37	odd	2		1344.2.a.g.1.1		1
152.75	even	2		1344.2.a.s.1.1		1
171.56	even	6		567.2.f.b.190.1		2
171.94	odd	6		567.2.f.g.379.1		2
171.113	even	6		567.2.f.b.379.1		2
171.151	odd	6		567.2.f.g.190.1		2
209.208	even	2		2541.2.a.j.1.1		1
228.227	odd	2		1008.2.a.l.1.1		1
247.246	odd	2		3549.2.a.c.1.1		1
285.113	odd	4		1575.2.d.a.1324.1		2
285.227	odd	4		1575.2.d.a.1324.2		2
285.284	even	2		1575.2.a.c.1.1		1
304.37	odd	4		5376.2.c.r.2689.1		2
304.75	even	4		5376.2.c.l.2689.2		2
304.189	odd	4		5376.2.c.r.2689.2		2
304.227	even	4		5376.2.c.l.2689.1		2
323.322	odd	2		6069.2.a.b.1.1		1
380.379	even	2		8400.2.a.bn.1.1		1
399.170	even	6		441.2.e.a.361.1		2
399.227	odd	6		441.2.e.b.226.1		2
399.284	even	6		441.2.e.a.226.1		2
399.341	odd	6		441.2.e.b.361.1		2
399.398	odd	2		441.2.a.f.1.1		1
456.227	odd	2		4032.2.a.k.1.1		1
456.341	even	2		4032.2.a.h.1.1		1
532.75	odd	6		2352.2.q.e.1537.1		2
532.151	even	6		2352.2.q.x.961.1		2
532.227	odd	6		2352.2.q.e.961.1		2
532.303	even	6		2352.2.q.x.1537.1		2
532.531	odd	2		2352.2.a.v.1.1		1
627.626	odd	2		7623.2.a.g.1.1		1
665.664	even	2		3675.2.a.n.1.1		1
1064.531	odd	2		9408.2.a.m.1.1		1
1064.797	even	2		9408.2.a.bv.1.1		1
1596.1595	even	2		7056.2.a.p.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.a.a.1.1	✓	1	19.18	odd	2
63.2.a.a.1.1		1	57.56	even	2
147.2.a.a.1.1		1	133.132	even	2
147.2.e.b.67.1		2	133.37	odd	6
147.2.e.b.79.1		2	133.18	odd	6
147.2.e.c.67.1		2	133.75	even	6
147.2.e.c.79.1		2	133.94	even	6
336.2.a.a.1.1		1	76.75	even	2
441.2.a.f.1.1		1	399.398	odd	2
441.2.e.a.226.1		2	399.284	even	6
441.2.e.a.361.1		2	399.170	even	6
441.2.e.b.226.1		2	399.227	odd	6
441.2.e.b.361.1		2	399.341	odd	6
525.2.a.d.1.1		1	95.94	odd	2
525.2.d.a.274.1		2	95.37	even	4
525.2.d.a.274.2		2	95.18	even	4
567.2.f.b.190.1		2	171.56	even	6
567.2.f.b.379.1		2	171.113	even	6
567.2.f.g.190.1		2	171.151	odd	6
567.2.f.g.379.1		2	171.94	odd	6
1008.2.a.l.1.1		1	228.227	odd	2
1344.2.a.g.1.1		1	152.37	odd	2
1344.2.a.s.1.1		1	152.75	even	2
1575.2.a.c.1.1		1	285.284	even	2
1575.2.d.a.1324.1		2	285.113	odd	4
1575.2.d.a.1324.2		2	285.227	odd	4
2352.2.a.v.1.1		1	532.531	odd	2
2352.2.q.e.961.1		2	532.227	odd	6
2352.2.q.e.1537.1		2	532.75	odd	6
2352.2.q.x.961.1		2	532.151	even	6
2352.2.q.x.1537.1		2	532.303	even	6
2541.2.a.j.1.1		1	209.208	even	2
3549.2.a.c.1.1		1	247.246	odd	2
3675.2.a.n.1.1		1	665.664	even	2
4032.2.a.h.1.1		1	456.341	even	2
4032.2.a.k.1.1		1	456.227	odd	2
5376.2.c.l.2689.1		2	304.227	even	4
5376.2.c.l.2689.2		2	304.75	even	4
5376.2.c.r.2689.1		2	304.37	odd	4
5376.2.c.r.2689.2		2	304.189	odd	4
6069.2.a.b.1.1		1	323.322	odd	2
7056.2.a.p.1.1		1	1596.1595	even	2
7581.2.a.d.1.1		1	1.1	even	1	trivial
7623.2.a.g.1.1		1	627.626	odd	2
8400.2.a.bn.1.1		1	380.379	even	2
9408.2.a.m.1.1		1	1064.531	odd	2
9408.2.a.bv.1.1		1	1064.797	even	2