Embedded newform 9747.2.a.f.1.1

• Label: 9747.2.a.f.1.1
• Level: $9747$
• Weight: $2$
• Character: 9747.1
• Self dual: yes
• Analytic conductor: $77.830$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9747 = 3^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9747.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(77.8301868501\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	\(-1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	9747.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{4} -1.00000 q^{7} -5.00000 q^{13} +4.00000 q^{16} -5.00000 q^{25} +2.00000 q^{28} +4.00000 q^{31} -11.0000 q^{37} +8.00000 q^{43} -6.00000 q^{49} +10.0000 q^{52} -1.00000 q^{61} -8.00000 q^{64} -5.00000 q^{67} -7.00000 q^{73} -17.0000 q^{79} +5.00000 q^{91} +19.0000 q^{97} +O(q^{100})\)

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		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(3\)	0	0
			\(5\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(7\)	−1.00000	−0.377964	−0.188982	−	0.981981i	\(-0.560519\pi\)
			−0.188982	+	0.981981i	\(0.560519\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	−5.00000	−1.38675	−0.693375	−	0.720577i	\(-0.743877\pi\)
			−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	0	0
			\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−11.0000	−1.80839	−0.904194	−	0.427121i	\(-0.859528\pi\)
			−0.904194	+	0.427121i	\(0.859528\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	8.00000	1.21999	0.609994	−	0.792406i	\(-0.291172\pi\)
			0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	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	9747.2.a.f.1.1		1
3.2	odd	2	CM	9747.2.a.f.1.1		1
19.18	odd	2		27.2.a.a.1.1	✓	1
57.56	even	2		27.2.a.a.1.1	✓	1
76.75	even	2		432.2.a.e.1.1		1
95.18	even	4		675.2.b.f.649.2		2
95.37	even	4		675.2.b.f.649.1		2
95.94	odd	2		675.2.a.e.1.1		1
133.132	even	2		1323.2.a.i.1.1		1
152.37	odd	2		1728.2.a.n.1.1		1
152.75	even	2		1728.2.a.o.1.1		1
171.56	even	6		81.2.c.a.28.1		2
171.94	odd	6		81.2.c.a.55.1		2
171.113	even	6		81.2.c.a.55.1		2
171.151	odd	6		81.2.c.a.28.1		2
209.208	even	2		3267.2.a.f.1.1		1
228.227	odd	2		432.2.a.e.1.1		1
247.246	odd	2		4563.2.a.e.1.1		1
285.113	odd	4		675.2.b.f.649.2		2
285.227	odd	4		675.2.b.f.649.1		2
285.284	even	2		675.2.a.e.1.1		1
323.322	odd	2		7803.2.a.k.1.1		1
399.398	odd	2		1323.2.a.i.1.1		1
456.227	odd	2		1728.2.a.o.1.1		1
456.341	even	2		1728.2.a.n.1.1		1
513.56	even	18		729.2.e.f.568.1		6
513.94	odd	18		729.2.e.f.163.1		6
513.113	even	18		729.2.e.f.649.1		6
513.151	odd	18		729.2.e.f.82.1		6
513.227	even	18		729.2.e.f.82.1		6
513.265	odd	18		729.2.e.f.649.1		6
513.284	even	18		729.2.e.f.163.1		6
513.322	odd	18		729.2.e.f.568.1		6
513.398	even	18		729.2.e.f.325.1		6
513.436	odd	18		729.2.e.f.406.1		6
513.455	even	18		729.2.e.f.406.1		6
513.493	odd	18		729.2.e.f.325.1		6
627.626	odd	2		3267.2.a.f.1.1		1
684.151	even	6		1296.2.i.i.433.1		2
684.227	odd	6		1296.2.i.i.433.1		2
684.455	odd	6		1296.2.i.i.865.1		2
684.607	even	6		1296.2.i.i.865.1		2
741.740	even	2		4563.2.a.e.1.1		1
969.968	even	2		7803.2.a.k.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.a.a.1.1	✓	1	19.18	odd	2
27.2.a.a.1.1	✓	1	57.56	even	2
81.2.c.a.28.1		2	171.56	even	6
81.2.c.a.28.1		2	171.151	odd	6
81.2.c.a.55.1		2	171.94	odd	6
81.2.c.a.55.1		2	171.113	even	6
432.2.a.e.1.1		1	76.75	even	2
432.2.a.e.1.1		1	228.227	odd	2
675.2.a.e.1.1		1	95.94	odd	2
675.2.a.e.1.1		1	285.284	even	2
675.2.b.f.649.1		2	95.37	even	4
675.2.b.f.649.1		2	285.227	odd	4
675.2.b.f.649.2		2	95.18	even	4
675.2.b.f.649.2		2	285.113	odd	4
729.2.e.f.82.1		6	513.151	odd	18
729.2.e.f.82.1		6	513.227	even	18
729.2.e.f.163.1		6	513.94	odd	18
729.2.e.f.163.1		6	513.284	even	18
729.2.e.f.325.1		6	513.398	even	18
729.2.e.f.325.1		6	513.493	odd	18
729.2.e.f.406.1		6	513.436	odd	18
729.2.e.f.406.1		6	513.455	even	18
729.2.e.f.568.1		6	513.56	even	18
729.2.e.f.568.1		6	513.322	odd	18
729.2.e.f.649.1		6	513.113	even	18
729.2.e.f.649.1		6	513.265	odd	18
1296.2.i.i.433.1		2	684.151	even	6
1296.2.i.i.433.1		2	684.227	odd	6
1296.2.i.i.865.1		2	684.455	odd	6
1296.2.i.i.865.1		2	684.607	even	6
1323.2.a.i.1.1		1	133.132	even	2
1323.2.a.i.1.1		1	399.398	odd	2
1728.2.a.n.1.1		1	152.37	odd	2
1728.2.a.n.1.1		1	456.341	even	2
1728.2.a.o.1.1		1	152.75	even	2
1728.2.a.o.1.1		1	456.227	odd	2
3267.2.a.f.1.1		1	209.208	even	2
3267.2.a.f.1.1		1	627.626	odd	2
4563.2.a.e.1.1		1	247.246	odd	2
4563.2.a.e.1.1		1	741.740	even	2
7803.2.a.k.1.1		1	323.322	odd	2
7803.2.a.k.1.1		1	969.968	even	2
9747.2.a.f.1.1		1	1.1	even	1	trivial
9747.2.a.f.1.1		1	3.2	odd	2	CM