Embedded newform 648.2.a.c.1.1

• Label: 648.2.a.c.1.1
• Level: $648$
• Weight: $2$
• Character: 648.1
• Self dual: yes
• Analytic conductor: $5.174$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	648.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{5} -3.00000 q^{7} -5.00000 q^{11} -5.00000 q^{13} +2.00000 q^{17} -4.00000 q^{19} +1.00000 q^{23} -4.00000 q^{25} +9.00000 q^{29} -1.00000 q^{31} -3.00000 q^{35} -6.00000 q^{37} -3.00000 q^{41} +1.00000 q^{43} +3.00000 q^{47} +2.00000 q^{49} -2.00000 q^{53} -5.00000 q^{55} -11.0000 q^{59} +7.00000 q^{61} -5.00000 q^{65} -1.00000 q^{67} -4.00000 q^{71} -2.00000 q^{73} +15.0000 q^{77} +1.00000 q^{79} -1.00000 q^{83} +2.00000 q^{85} +18.0000 q^{89} +15.0000 q^{91} -4.00000 q^{95} -13.0000 q^{97} +O(q^{100})\)

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
			\(3\)	0	0
\(5\)	1.00000	0.447214				0.223607	−	0.974679i	\(-0.428217\pi\)
			0.223607	+	0.974679i	\(0.428217\pi\)
\(7\)	−3.00000	−1.13389	−0.566947	−	0.823754i	\(-0.691875\pi\)
			−0.566947	+	0.823754i	\(0.691875\pi\)
\(11\)	−5.00000	−1.50756	−0.753778	−	0.657129i	\(-0.771771\pi\)
			−0.753778	+	0.657129i	\(0.771771\pi\)
\(13\)	−5.00000	−1.38675	−0.693375	−	0.720577i	\(-0.743877\pi\)
			−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	1.00000	0.208514	0.104257	−	0.994550i	\(-0.466753\pi\)
			0.104257	+	0.994550i	\(0.466753\pi\)
\(29\)	9.00000	1.67126	0.835629	−	0.549294i	\(-0.185103\pi\)
			0.835629	+	0.549294i	\(0.185103\pi\)
\(31\)	−1.00000	−0.179605	−0.0898027	−	0.995960i	\(-0.528624\pi\)
			−0.0898027	+	0.995960i	\(0.528624\pi\)
\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−3.00000	−0.468521	−0.234261	−	0.972174i	\(-0.575267\pi\)
			−0.234261	+	0.972174i	\(0.575267\pi\)
\(43\)	1.00000	0.152499	0.0762493	−	0.997089i	\(-0.475706\pi\)
			0.0762493	+	0.997089i	\(0.475706\pi\)
\(47\)	3.00000	0.437595	0.218797	−	0.975770i	\(-0.429787\pi\)
			0.218797	+	0.975770i	\(0.429787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.a.c.1.1		1
3.2	odd	2		648.2.a.a.1.1		1
4.3	odd	2		1296.2.a.i.1.1		1
8.3	odd	2		5184.2.a.n.1.1		1
8.5	even	2		5184.2.a.i.1.1		1
9.2	odd	6		72.2.i.a.49.1	yes	2
9.4	even	3		216.2.i.a.73.1		2
9.5	odd	6		72.2.i.a.25.1	✓	2
9.7	even	3		216.2.i.a.145.1		2
12.11	even	2		1296.2.a.e.1.1		1
24.5	odd	2		5184.2.a.s.1.1		1
24.11	even	2		5184.2.a.x.1.1		1
36.7	odd	6		432.2.i.a.145.1		2
36.11	even	6		144.2.i.b.49.1		2
36.23	even	6		144.2.i.b.97.1		2
36.31	odd	6		432.2.i.a.289.1		2
72.5	odd	6		576.2.i.d.385.1		2
72.11	even	6		576.2.i.c.193.1		2
72.13	even	6		1728.2.i.h.1153.1		2
72.29	odd	6		576.2.i.d.193.1		2
72.43	odd	6		1728.2.i.g.577.1		2
72.59	even	6		576.2.i.c.385.1		2
72.61	even	6		1728.2.i.h.577.1		2
72.67	odd	6		1728.2.i.g.1153.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.i.a.25.1	✓	2	9.5	odd	6
72.2.i.a.49.1	yes	2	9.2	odd	6
144.2.i.b.49.1		2	36.11	even	6
144.2.i.b.97.1		2	36.23	even	6
216.2.i.a.73.1		2	9.4	even	3
216.2.i.a.145.1		2	9.7	even	3
432.2.i.a.145.1		2	36.7	odd	6
432.2.i.a.289.1		2	36.31	odd	6
576.2.i.c.193.1		2	72.11	even	6
576.2.i.c.385.1		2	72.59	even	6
576.2.i.d.193.1		2	72.29	odd	6
576.2.i.d.385.1		2	72.5	odd	6
648.2.a.a.1.1		1	3.2	odd	2
648.2.a.c.1.1		1	1.1	even	1	trivial
1296.2.a.e.1.1		1	12.11	even	2
1296.2.a.i.1.1		1	4.3	odd	2
1728.2.i.g.577.1		2	72.43	odd	6
1728.2.i.g.1153.1		2	72.67	odd	6
1728.2.i.h.577.1		2	72.61	even	6
1728.2.i.h.1153.1		2	72.13	even	6
5184.2.a.i.1.1		1	8.5	even	2
5184.2.a.n.1.1		1	8.3	odd	2
5184.2.a.s.1.1		1	24.5	odd	2
5184.2.a.x.1.1		1	24.11	even	2