Embedded newform 720.4.a.v.1.1

• Label: 720.4.a.v.1.1
• Level: $720$
• Weight: $4$
• Character: 720.1
• Self dual: yes
• Analytic conductor: $42.481$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(42.4813752041\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	720.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{5} -4.00000 q^{7} +72.0000 q^{11} -6.00000 q^{13} -38.0000 q^{17} -52.0000 q^{19} +152.000 q^{23} +25.0000 q^{25} +78.0000 q^{29} -120.000 q^{31} -20.0000 q^{35} -150.000 q^{37} -362.000 q^{41} +484.000 q^{43} +280.000 q^{47} -327.000 q^{49} +670.000 q^{53} +360.000 q^{55} +696.000 q^{59} +222.000 q^{61} -30.0000 q^{65} +4.00000 q^{67} +96.0000 q^{71} +178.000 q^{73} -288.000 q^{77} +632.000 q^{79} -612.000 q^{83} -190.000 q^{85} -994.000 q^{89} +24.0000 q^{91} -260.000 q^{95} +1634.00 q^{97} +O(q^{100})\)

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
\(5\)	5.00000	0.447214
			\(7\)	−4.00000	−0.215980	−0.107990	−	0.994152i	\(-0.534441\pi\)
−0.107990	+	0.994152i				\(0.534441\pi\)
\(11\)	72.0000	1.97353	0.986764	−	0.162160i	\(-0.0518462\pi\)
			0.986764	+	0.162160i	\(0.0518462\pi\)
\(13\)	−6.00000	−0.128008	−0.0640039	−	0.997950i	\(-0.520387\pi\)
			−0.0640039	+	0.997950i	\(0.520387\pi\)
\(17\)	−38.0000	−0.542138	−0.271069	−	0.962560i	\(-0.587377\pi\)
			−0.271069	+	0.962560i	\(0.587377\pi\)
\(19\)	−52.0000	−0.627875	−0.313937	−	0.949444i	\(-0.601648\pi\)
			−0.313937	+	0.949444i	\(0.601648\pi\)
\(23\)	152.000	1.37801	0.689004	−	0.724757i	\(-0.258048\pi\)
			0.689004	+	0.724757i	\(0.258048\pi\)
\(29\)	78.0000	0.499456	0.249728	−	0.968316i	\(-0.419659\pi\)
			0.249728	+	0.968316i	\(0.419659\pi\)
\(31\)	−120.000	−0.695246	−0.347623	−	0.937634i	\(-0.613011\pi\)
			−0.347623	+	0.937634i	\(0.613011\pi\)
\(37\)	−150.000	−0.666482	−0.333241	−	0.942842i	\(-0.608142\pi\)
			−0.333241	+	0.942842i	\(0.608142\pi\)
\(41\)	−362.000	−1.37890	−0.689450	−	0.724333i	\(-0.742148\pi\)
			−0.689450	+	0.724333i	\(0.742148\pi\)
\(43\)	484.000	1.71650	0.858248	−	0.513236i	\(-0.171553\pi\)
			0.858248	+	0.513236i	\(0.171553\pi\)
\(47\)	280.000	0.868983	0.434491	−	0.900676i	\(-0.356928\pi\)
			0.434491	+	0.900676i	\(0.356928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.a.v.1.1		1
3.2	odd	2		240.4.a.h.1.1		1
4.3	odd	2		360.4.a.l.1.1		1
12.11	even	2		120.4.a.a.1.1	✓	1
15.2	even	4		1200.4.f.a.49.1		2
15.8	even	4		1200.4.f.a.49.2		2
15.14	odd	2		1200.4.a.k.1.1		1
20.3	even	4		1800.4.f.a.649.1		2
20.7	even	4		1800.4.f.a.649.2		2
20.19	odd	2		1800.4.a.n.1.1		1
24.5	odd	2		960.4.a.o.1.1		1
24.11	even	2		960.4.a.bf.1.1		1
60.23	odd	4		600.4.f.i.49.1		2
60.47	odd	4		600.4.f.i.49.2		2
60.59	even	2		600.4.a.l.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.a.1.1	✓	1	12.11	even	2
240.4.a.h.1.1		1	3.2	odd	2
360.4.a.l.1.1		1	4.3	odd	2
600.4.a.l.1.1		1	60.59	even	2
600.4.f.i.49.1		2	60.23	odd	4
600.4.f.i.49.2		2	60.47	odd	4
720.4.a.v.1.1		1	1.1	even	1	trivial
960.4.a.o.1.1		1	24.5	odd	2
960.4.a.bf.1.1		1	24.11	even	2
1200.4.a.k.1.1		1	15.14	odd	2
1200.4.f.a.49.1		2	15.2	even	4
1200.4.f.a.49.2		2	15.8	even	4
1800.4.a.n.1.1		1	20.19	odd	2
1800.4.f.a.649.1		2	20.3	even	4
1800.4.f.a.649.2		2	20.7	even	4