Embedded newform 7623.2.a.g.1.1

• Label: 7623.2.a.g.1.1
• Level: $7623$
• Weight: $2$
• Character: 7623.1
• Self dual: yes
• Analytic conductor: $60.870$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(60.8699614608\)
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	\(\chi\)	\(=\)	7623.1

$q$-expansion

\(f(q)\)

\(=\)

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

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	\(-0.615027\pi\)
			−0.353553	+	0.935414i	\(0.615027\pi\)
\(3\)	0	0
			\(5\)	2.00000	0.894427	0.447214	−	0.894427i	\(-0.352416\pi\)
0.447214	+	0.894427i				\(0.352416\pi\)
\(7\)	1.00000	0.377964
			\(11\)	0	0
\(13\)	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	\(0.759367\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	2.00000	0.312348	0.156174	−	0.987730i	\(-0.450084\pi\)
			0.156174	+	0.987730i	\(0.450084\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	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	7623.2.a.g.1.1		1
3.2	odd	2		2541.2.a.j.1.1		1
11.10	odd	2		63.2.a.a.1.1		1
33.32	even	2		21.2.a.a.1.1	✓	1
44.43	even	2		1008.2.a.l.1.1		1
55.32	even	4		1575.2.d.a.1324.2		2
55.43	even	4		1575.2.d.a.1324.1		2
55.54	odd	2		1575.2.a.c.1.1		1
77.10	even	6		441.2.e.b.226.1		2
77.32	odd	6		441.2.e.a.226.1		2
77.54	even	6		441.2.e.b.361.1		2
77.65	odd	6		441.2.e.a.361.1		2
77.76	even	2		441.2.a.f.1.1		1
88.21	odd	2		4032.2.a.h.1.1		1
88.43	even	2		4032.2.a.k.1.1		1
99.32	even	6		567.2.f.g.379.1		2
99.43	odd	6		567.2.f.b.190.1		2
99.65	even	6		567.2.f.g.190.1		2
99.76	odd	6		567.2.f.b.379.1		2
132.131	odd	2		336.2.a.a.1.1		1
165.32	odd	4		525.2.d.a.274.1		2
165.98	odd	4		525.2.d.a.274.2		2
165.164	even	2		525.2.a.d.1.1		1
231.32	even	6		147.2.e.b.79.1		2
231.65	even	6		147.2.e.b.67.1		2
231.131	odd	6		147.2.e.c.67.1		2
231.164	odd	6		147.2.e.c.79.1		2
231.230	odd	2		147.2.a.a.1.1		1
264.131	odd	2		1344.2.a.s.1.1		1
264.197	even	2		1344.2.a.g.1.1		1
308.307	odd	2		7056.2.a.p.1.1		1
429.428	even	2		3549.2.a.c.1.1		1
528.131	odd	4		5376.2.c.l.2689.1		2
528.197	even	4		5376.2.c.r.2689.1		2
528.395	odd	4		5376.2.c.l.2689.2		2
528.461	even	4		5376.2.c.r.2689.2		2
561.560	even	2		6069.2.a.b.1.1		1
627.626	odd	2		7581.2.a.d.1.1		1
660.659	odd	2		8400.2.a.bn.1.1		1
924.131	even	6		2352.2.q.e.1537.1		2
924.263	odd	6		2352.2.q.x.961.1		2
924.395	even	6		2352.2.q.e.961.1		2
924.527	odd	6		2352.2.q.x.1537.1		2
924.923	even	2		2352.2.a.v.1.1		1
1155.1154	odd	2		3675.2.a.n.1.1		1
1848.461	odd	2		9408.2.a.bv.1.1		1
1848.923	even	2		9408.2.a.m.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.a.a.1.1	✓	1	33.32	even	2
63.2.a.a.1.1		1	11.10	odd	2
147.2.a.a.1.1		1	231.230	odd	2
147.2.e.b.67.1		2	231.65	even	6
147.2.e.b.79.1		2	231.32	even	6
147.2.e.c.67.1		2	231.131	odd	6
147.2.e.c.79.1		2	231.164	odd	6
336.2.a.a.1.1		1	132.131	odd	2
441.2.a.f.1.1		1	77.76	even	2
441.2.e.a.226.1		2	77.32	odd	6
441.2.e.a.361.1		2	77.65	odd	6
441.2.e.b.226.1		2	77.10	even	6
441.2.e.b.361.1		2	77.54	even	6
525.2.a.d.1.1		1	165.164	even	2
525.2.d.a.274.1		2	165.32	odd	4
525.2.d.a.274.2		2	165.98	odd	4
567.2.f.b.190.1		2	99.43	odd	6
567.2.f.b.379.1		2	99.76	odd	6
567.2.f.g.190.1		2	99.65	even	6
567.2.f.g.379.1		2	99.32	even	6
1008.2.a.l.1.1		1	44.43	even	2
1344.2.a.g.1.1		1	264.197	even	2
1344.2.a.s.1.1		1	264.131	odd	2
1575.2.a.c.1.1		1	55.54	odd	2
1575.2.d.a.1324.1		2	55.43	even	4
1575.2.d.a.1324.2		2	55.32	even	4
2352.2.a.v.1.1		1	924.923	even	2
2352.2.q.e.961.1		2	924.395	even	6
2352.2.q.e.1537.1		2	924.131	even	6
2352.2.q.x.961.1		2	924.263	odd	6
2352.2.q.x.1537.1		2	924.527	odd	6
2541.2.a.j.1.1		1	3.2	odd	2
3549.2.a.c.1.1		1	429.428	even	2
3675.2.a.n.1.1		1	1155.1154	odd	2
4032.2.a.h.1.1		1	88.21	odd	2
4032.2.a.k.1.1		1	88.43	even	2
5376.2.c.l.2689.1		2	528.131	odd	4
5376.2.c.l.2689.2		2	528.395	odd	4
5376.2.c.r.2689.1		2	528.197	even	4
5376.2.c.r.2689.2		2	528.461	even	4
6069.2.a.b.1.1		1	561.560	even	2
7056.2.a.p.1.1		1	308.307	odd	2
7581.2.a.d.1.1		1	627.626	odd	2
7623.2.a.g.1.1		1	1.1	even	1	trivial
8400.2.a.bn.1.1		1	660.659	odd	2
9408.2.a.m.1.1		1	1848.923	even	2
9408.2.a.bv.1.1		1	1848.461	odd	2