Embedded newform 7803.2.a.k.1.1

• Label: 7803.2.a.k.1.1
• Level: $7803$
• Weight: $2$
• Character: 7803.1
• Self dual: yes
• Analytic conductor: $62.307$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(62.3072686972\)
Analytic rank:	\(1\)
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	\(\chi\)	\(=\)	7803.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{4} +1.00000 q^{7} +5.00000 q^{13} +4.00000 q^{16} -7.00000 q^{19} -5.00000 q^{25} -2.00000 q^{28} +4.00000 q^{31} -11.0000 q^{37} +8.00000 q^{43} -6.00000 q^{49} -10.0000 q^{52} +1.00000 q^{61} -8.00000 q^{64} +5.00000 q^{67} +7.00000 q^{73} +14.0000 q^{76} -17.0000 q^{79} +5.00000 q^{91} +19.0000 q^{97} +O(q^{100})\)

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	\(-0.439481\pi\)
			0.188982	+	0.981981i	\(0.439481\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	5.00000	1.38675	0.693375	−	0.720577i	\(-0.256123\pi\)
			0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	0	0
			\(19\)	−7.00000	−1.60591	−0.802955	−	0.596040i	\(-0.796740\pi\)
−0.802955	+	0.596040i				\(0.796740\pi\)
\(23\)	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	7803.2.a.k.1.1		1
3.2	odd	2	CM	7803.2.a.k.1.1		1
17.16	even	2		27.2.a.a.1.1	✓	1
51.50	odd	2		27.2.a.a.1.1	✓	1
68.67	odd	2		432.2.a.e.1.1		1
85.33	odd	4		675.2.b.f.649.2		2
85.67	odd	4		675.2.b.f.649.1		2
85.84	even	2		675.2.a.e.1.1		1
119.118	odd	2		1323.2.a.i.1.1		1
136.67	odd	2		1728.2.a.o.1.1		1
136.101	even	2		1728.2.a.n.1.1		1
153.16	even	6		81.2.c.a.28.1		2
153.50	odd	6		81.2.c.a.55.1		2
153.67	even	6		81.2.c.a.55.1		2
153.101	odd	6		81.2.c.a.28.1		2
187.186	odd	2		3267.2.a.f.1.1		1
204.203	even	2		432.2.a.e.1.1		1
221.220	even	2		4563.2.a.e.1.1		1
255.152	even	4		675.2.b.f.649.1		2
255.203	even	4		675.2.b.f.649.2		2
255.254	odd	2		675.2.a.e.1.1		1
323.322	odd	2		9747.2.a.f.1.1		1
357.356	even	2		1323.2.a.i.1.1		1
408.101	odd	2		1728.2.a.n.1.1		1
408.203	even	2		1728.2.a.o.1.1		1
459.16	even	18		729.2.e.f.82.1		6
459.50	odd	18		729.2.e.f.406.1		6
459.67	even	18		729.2.e.f.163.1		6
459.101	odd	18		729.2.e.f.325.1		6
459.169	even	18		729.2.e.f.325.1		6
459.203	odd	18		729.2.e.f.163.1		6
459.220	even	18		729.2.e.f.406.1		6
459.254	odd	18		729.2.e.f.82.1		6
459.322	even	18		729.2.e.f.568.1		6
459.356	odd	18		729.2.e.f.649.1		6
459.373	even	18		729.2.e.f.649.1		6
459.407	odd	18		729.2.e.f.568.1		6
561.560	even	2		3267.2.a.f.1.1		1
612.67	odd	6		1296.2.i.i.865.1		2
612.203	even	6		1296.2.i.i.865.1		2
612.407	even	6		1296.2.i.i.433.1		2
612.475	odd	6		1296.2.i.i.433.1		2
663.662	odd	2		4563.2.a.e.1.1		1
969.968	even	2		9747.2.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.a.a.1.1	✓	1	17.16	even	2
27.2.a.a.1.1	✓	1	51.50	odd	2
81.2.c.a.28.1		2	153.16	even	6
81.2.c.a.28.1		2	153.101	odd	6
81.2.c.a.55.1		2	153.50	odd	6
81.2.c.a.55.1		2	153.67	even	6
432.2.a.e.1.1		1	68.67	odd	2
432.2.a.e.1.1		1	204.203	even	2
675.2.a.e.1.1		1	85.84	even	2
675.2.a.e.1.1		1	255.254	odd	2
675.2.b.f.649.1		2	85.67	odd	4
675.2.b.f.649.1		2	255.152	even	4
675.2.b.f.649.2		2	85.33	odd	4
675.2.b.f.649.2		2	255.203	even	4
729.2.e.f.82.1		6	459.16	even	18
729.2.e.f.82.1		6	459.254	odd	18
729.2.e.f.163.1		6	459.67	even	18
729.2.e.f.163.1		6	459.203	odd	18
729.2.e.f.325.1		6	459.101	odd	18
729.2.e.f.325.1		6	459.169	even	18
729.2.e.f.406.1		6	459.50	odd	18
729.2.e.f.406.1		6	459.220	even	18
729.2.e.f.568.1		6	459.322	even	18
729.2.e.f.568.1		6	459.407	odd	18
729.2.e.f.649.1		6	459.356	odd	18
729.2.e.f.649.1		6	459.373	even	18
1296.2.i.i.433.1		2	612.407	even	6
1296.2.i.i.433.1		2	612.475	odd	6
1296.2.i.i.865.1		2	612.67	odd	6
1296.2.i.i.865.1		2	612.203	even	6
1323.2.a.i.1.1		1	119.118	odd	2
1323.2.a.i.1.1		1	357.356	even	2
1728.2.a.n.1.1		1	136.101	even	2
1728.2.a.n.1.1		1	408.101	odd	2
1728.2.a.o.1.1		1	136.67	odd	2
1728.2.a.o.1.1		1	408.203	even	2
3267.2.a.f.1.1		1	187.186	odd	2
3267.2.a.f.1.1		1	561.560	even	2
4563.2.a.e.1.1		1	221.220	even	2
4563.2.a.e.1.1		1	663.662	odd	2
7803.2.a.k.1.1		1	1.1	even	1	trivial
7803.2.a.k.1.1		1	3.2	odd	2	CM
9747.2.a.f.1.1		1	323.322	odd	2
9747.2.a.f.1.1		1	969.968	even	2