Embedded newform 720.4.a.y.1.1

• Label: 720.4.a.y.1.1
• Level: $720$
• Weight: $4$
• Character: 720.1
• Self dual: yes
• Analytic conductor: $42.481$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{5} +4.00000 q^{7} +O(q^{10})\)

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	\(-0.465559\pi\)
0.107990	+	0.994152i				\(0.465559\pi\)
\(11\)	−48.0000	−1.31569	−0.657843	−	0.753155i	\(-0.728531\pi\)
			−0.657843	+	0.753155i	\(0.728531\pi\)
\(13\)	2.00000	0.0426692	0.0213346	−	0.999772i	\(-0.493208\pi\)
			0.0213346	+	0.999772i	\(0.493208\pi\)
\(17\)	114.000	1.62642	0.813208	−	0.581974i	\(-0.197719\pi\)
			0.813208	+	0.581974i	\(0.197719\pi\)
\(19\)	−140.000	−1.69043	−0.845216	−	0.534425i	\(-0.820528\pi\)
			−0.845216	+	0.534425i	\(0.820528\pi\)
\(23\)	72.0000	0.652741	0.326370	−	0.945242i	\(-0.394174\pi\)
			0.326370	+	0.945242i	\(0.394174\pi\)
\(29\)	−210.000	−1.34469	−0.672345	−	0.740238i	\(-0.734713\pi\)
			−0.672345	+	0.740238i	\(0.734713\pi\)
\(31\)	−272.000	−1.57589	−0.787946	−	0.615745i	\(-0.788855\pi\)
			−0.787946	+	0.615745i	\(0.788855\pi\)
\(37\)	−334.000	−1.48403	−0.742017	−	0.670381i	\(-0.766131\pi\)
			−0.742017	+	0.670381i	\(0.766131\pi\)
\(41\)	198.000	0.754205	0.377102	−	0.926172i	\(-0.376920\pi\)
			0.377102	+	0.926172i	\(0.376920\pi\)
\(43\)	268.000	0.950456	0.475228	−	0.879863i	\(-0.342366\pi\)
			0.475228	+	0.879863i	\(0.342366\pi\)
\(47\)	216.000	0.670358	0.335179	−	0.942154i	\(-0.391203\pi\)
			0.335179	+	0.942154i	\(0.391203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.a.y.1.1		1
3.2	odd	2		240.4.a.b.1.1		1
4.3	odd	2		90.4.a.c.1.1		1
12.11	even	2		30.4.a.b.1.1	✓	1
15.2	even	4		1200.4.f.r.49.2		2
15.8	even	4		1200.4.f.r.49.1		2
15.14	odd	2		1200.4.a.ba.1.1		1
20.3	even	4		450.4.c.j.199.2		2
20.7	even	4		450.4.c.j.199.1		2
20.19	odd	2		450.4.a.r.1.1		1
24.5	odd	2		960.4.a.bg.1.1		1
24.11	even	2		960.4.a.n.1.1		1
36.7	odd	6		810.4.e.p.271.1		2
36.11	even	6		810.4.e.i.271.1		2
36.23	even	6		810.4.e.i.541.1		2
36.31	odd	6		810.4.e.p.541.1		2
60.23	odd	4		150.4.c.c.49.1		2
60.47	odd	4		150.4.c.c.49.2		2
60.59	even	2		150.4.a.b.1.1		1
84.83	odd	2		1470.4.a.r.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.b.1.1	✓	1	12.11	even	2
90.4.a.c.1.1		1	4.3	odd	2
150.4.a.b.1.1		1	60.59	even	2
150.4.c.c.49.1		2	60.23	odd	4
150.4.c.c.49.2		2	60.47	odd	4
240.4.a.b.1.1		1	3.2	odd	2
450.4.a.r.1.1		1	20.19	odd	2
450.4.c.j.199.1		2	20.7	even	4
450.4.c.j.199.2		2	20.3	even	4
720.4.a.y.1.1		1	1.1	even	1	trivial
810.4.e.i.271.1		2	36.11	even	6
810.4.e.i.541.1		2	36.23	even	6
810.4.e.p.271.1		2	36.7	odd	6
810.4.e.p.541.1		2	36.31	odd	6
960.4.a.n.1.1		1	24.11	even	2
960.4.a.bg.1.1		1	24.5	odd	2
1200.4.a.ba.1.1		1	15.14	odd	2
1200.4.f.r.49.1		2	15.8	even	4
1200.4.f.r.49.2		2	15.2	even	4
1470.4.a.r.1.1		1	84.83	odd	2