Embedded newform 720.2.a.i.1.1

• Label: 720.2.a.i.1.1
• Level: $720$
• Weight: $2$
• Character: 720.1
• Self dual: yes
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{5} -2.00000 q^{7} +2.00000 q^{11} +4.00000 q^{13} +2.00000 q^{17} -4.00000 q^{19} +8.00000 q^{23} +1.00000 q^{25} +10.0000 q^{29} -4.00000 q^{31} -2.00000 q^{35} +8.00000 q^{43} +8.00000 q^{47} -3.00000 q^{49} -6.00000 q^{53} +2.00000 q^{55} -14.0000 q^{59} -14.0000 q^{61} +4.00000 q^{65} +4.00000 q^{67} +12.0000 q^{71} +6.00000 q^{73} -4.00000 q^{77} +12.0000 q^{79} +4.00000 q^{83} +2.00000 q^{85} +12.0000 q^{89} -8.00000 q^{91} -4.00000 q^{95} -14.0000 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	1.00000	0.447214
			\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
−0.377964	+	0.925820i				\(0.623376\pi\)
\(11\)	2.00000	0.603023	0.301511	−	0.953463i	\(-0.402509\pi\)
			0.301511	+	0.953463i	\(0.402509\pi\)
\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
			0.554700	+	0.832050i	\(0.312833\pi\)
\(17\)	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	\(0.651732\pi\)
\(23\)	8.00000	1.66812	0.834058	−	0.551677i	\(-0.186012\pi\)
			0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	10.0000	1.85695	0.928477	−	0.371391i	\(-0.121119\pi\)
			0.928477	+	0.371391i	\(0.121119\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(41\)	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	\(0.291172\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)
\(53\)	−6.00000	−0.824163	−0.412082	−	0.911147i	\(-0.635198\pi\)
			−0.412082	+	0.911147i	\(0.635198\pi\)
\(59\)	−14.0000	−1.82264	−0.911322	−	0.411693i	\(-0.864937\pi\)
			−0.911322	+	0.411693i	\(0.864937\pi\)
\(61\)	−14.0000	−1.79252	−0.896258	−	0.443533i	\(-0.853725\pi\)
			−0.896258	+	0.443533i	\(0.853725\pi\)
\(67\)	4.00000	0.488678	0.244339	−	0.969690i	\(-0.421429\pi\)
			0.244339	+	0.969690i	\(0.421429\pi\)
\(71\)	12.0000	1.42414	0.712069	−	0.702109i	\(-0.247758\pi\)
			0.712069	+	0.702109i	\(0.247758\pi\)
\(73\)	6.00000	0.702247	0.351123	−	0.936329i	\(-0.385800\pi\)
			0.351123	+	0.936329i	\(0.385800\pi\)
\(79\)	12.0000	1.35011	0.675053	−	0.737769i	\(-0.264121\pi\)
			0.675053	+	0.737769i	\(0.264121\pi\)
\(83\)	4.00000	0.439057	0.219529	−	0.975606i	\(-0.429548\pi\)
			0.219529	+	0.975606i	\(0.429548\pi\)
\(89\)	12.0000	1.27200	0.635999	−	0.771690i	\(-0.280588\pi\)
			0.635999	+	0.771690i	\(0.280588\pi\)
\(97\)	−14.0000	−1.42148	−0.710742	−	0.703452i	\(-0.751641\pi\)
			−0.710742	+	0.703452i	\(0.751641\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.a.i.1.1		1
3.2	odd	2		720.2.a.a.1.1		1
4.3	odd	2		360.2.a.d.1.1	yes	1
5.2	odd	4		3600.2.f.q.2449.1		2
5.3	odd	4		3600.2.f.q.2449.2		2
5.4	even	2		3600.2.a.bh.1.1		1
8.3	odd	2		2880.2.a.n.1.1		1
8.5	even	2		2880.2.a.e.1.1		1
12.11	even	2		360.2.a.c.1.1	✓	1
15.2	even	4		3600.2.f.g.2449.1		2
15.8	even	4		3600.2.f.g.2449.2		2
15.14	odd	2		3600.2.a.bd.1.1		1
20.3	even	4		1800.2.f.d.649.1		2
20.7	even	4		1800.2.f.d.649.2		2
20.19	odd	2		1800.2.a.f.1.1		1
24.5	odd	2		2880.2.a.w.1.1		1
24.11	even	2		2880.2.a.bd.1.1		1
36.7	odd	6		3240.2.q.d.1081.1		2
36.11	even	6		3240.2.q.n.1081.1		2
36.23	even	6		3240.2.q.n.2161.1		2
36.31	odd	6		3240.2.q.d.2161.1		2
60.23	odd	4		1800.2.f.h.649.1		2
60.47	odd	4		1800.2.f.h.649.2		2
60.59	even	2		1800.2.a.i.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.a.c.1.1	✓	1	12.11	even	2
360.2.a.d.1.1	yes	1	4.3	odd	2
720.2.a.a.1.1		1	3.2	odd	2
720.2.a.i.1.1		1	1.1	even	1	trivial
1800.2.a.f.1.1		1	20.19	odd	2
1800.2.a.i.1.1		1	60.59	even	2
1800.2.f.d.649.1		2	20.3	even	4
1800.2.f.d.649.2		2	20.7	even	4
1800.2.f.h.649.1		2	60.23	odd	4
1800.2.f.h.649.2		2	60.47	odd	4
2880.2.a.e.1.1		1	8.5	even	2
2880.2.a.n.1.1		1	8.3	odd	2
2880.2.a.w.1.1		1	24.5	odd	2
2880.2.a.bd.1.1		1	24.11	even	2
3240.2.q.d.1081.1		2	36.7	odd	6
3240.2.q.d.2161.1		2	36.31	odd	6
3240.2.q.n.1081.1		2	36.11	even	6
3240.2.q.n.2161.1		2	36.23	even	6
3600.2.a.bd.1.1		1	15.14	odd	2
3600.2.a.bh.1.1		1	5.4	even	2
3600.2.f.g.2449.1		2	15.2	even	4
3600.2.f.g.2449.2		2	15.8	even	4
3600.2.f.q.2449.1		2	5.2	odd	4
3600.2.f.q.2449.2		2	5.3	odd	4