Embedded newform 720.4.a.p.1.1

• Label: 720.4.a.p.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: $1$

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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 360)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{5} -34.0000 q^{7} +18.0000 q^{11} +12.0000 q^{13} +106.000 q^{17} +44.0000 q^{19} +56.0000 q^{23} +25.0000 q^{25} -270.000 q^{29} -204.000 q^{31} -170.000 q^{35} +120.000 q^{37} -80.0000 q^{41} -536.000 q^{43} -536.000 q^{47} +813.000 q^{49} -542.000 q^{53} +90.0000 q^{55} -174.000 q^{59} +186.000 q^{61} +60.0000 q^{65} -332.000 q^{67} -132.000 q^{71} -602.000 q^{73} -612.000 q^{77} +548.000 q^{79} -492.000 q^{83} +530.000 q^{85} +1052.00 q^{89} -408.000 q^{91} +220.000 q^{95} +482.000 q^{97} +O(q^{100})\)

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
			\(7\)	−34.0000	−1.83583	−0.917914	−	0.396780i	\(-0.870128\pi\)
−0.917914	+	0.396780i				\(0.870128\pi\)
\(11\)	18.0000	0.493382	0.246691	−	0.969094i	\(-0.420657\pi\)
			0.246691	+	0.969094i	\(0.420657\pi\)
\(13\)	12.0000	0.256015	0.128008	−	0.991773i	\(-0.459142\pi\)
			0.128008	+	0.991773i	\(0.459142\pi\)
\(17\)	106.000	1.51228	0.756140	−	0.654409i	\(-0.227083\pi\)
			0.756140	+	0.654409i	\(0.227083\pi\)
\(19\)	44.0000	0.531279	0.265639	−	0.964072i	\(-0.414417\pi\)
			0.265639	+	0.964072i	\(0.414417\pi\)
\(23\)	56.0000	0.507687	0.253844	−	0.967245i	\(-0.418305\pi\)
			0.253844	+	0.967245i	\(0.418305\pi\)
\(29\)	−270.000	−1.72889	−0.864444	−	0.502729i	\(-0.832329\pi\)
			−0.864444	+	0.502729i	\(0.832329\pi\)
\(31\)	−204.000	−1.18192	−0.590959	−	0.806701i	\(-0.701251\pi\)
			−0.590959	+	0.806701i	\(0.701251\pi\)
\(37\)	120.000	0.533186	0.266593	−	0.963809i	\(-0.414102\pi\)
			0.266593	+	0.963809i	\(0.414102\pi\)
\(41\)	−80.0000	−0.304729	−0.152365	−	0.988324i	\(-0.548689\pi\)
			−0.152365	+	0.988324i	\(0.548689\pi\)
\(43\)	−536.000	−1.90091	−0.950456	−	0.310858i	\(-0.899383\pi\)
			−0.950456	+	0.310858i	\(0.899383\pi\)
\(47\)	−536.000	−1.66348	−0.831741	−	0.555164i	\(-0.812655\pi\)
			−0.831741	+	0.555164i	\(0.812655\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.a.p.1.1		1
3.2	odd	2		720.4.a.a.1.1		1
4.3	odd	2		360.4.a.o.1.1	yes	1
12.11	even	2		360.4.a.g.1.1	✓	1
20.3	even	4		1800.4.f.i.649.1		2
20.7	even	4		1800.4.f.i.649.2		2
20.19	odd	2		1800.4.a.a.1.1		1
60.23	odd	4		1800.4.f.o.649.1		2
60.47	odd	4		1800.4.f.o.649.2		2
60.59	even	2		1800.4.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.a.g.1.1	✓	1	12.11	even	2
360.4.a.o.1.1	yes	1	4.3	odd	2
720.4.a.a.1.1		1	3.2	odd	2
720.4.a.p.1.1		1	1.1	even	1	trivial
1800.4.a.a.1.1		1	20.19	odd	2
1800.4.a.b.1.1		1	60.59	even	2
1800.4.f.i.649.1		2	20.3	even	4
1800.4.f.i.649.2		2	20.7	even	4
1800.4.f.o.649.1		2	60.23	odd	4
1800.4.f.o.649.2		2	60.47	odd	4