Embedded newform 720.6.a.p.1.1

• Label: 720.6.a.p.1.1
• Level: $720$
• Weight: $6$
• Character: 720.1
• Self dual: yes
• Analytic conductor: $115.476$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(115.476350265\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 60)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q+25.0000 q^{5} -44.0000 q^{7} +216.000 q^{11} +770.000 q^{13} -534.000 q^{17} -1580.00 q^{19} +2904.00 q^{23} +625.000 q^{25} +4566.00 q^{29} -2744.00 q^{31} -1100.00 q^{35} +1442.00 q^{37} +13350.0 q^{41} -17204.0 q^{43} -10824.0 q^{47} -14871.0 q^{49} +9942.00 q^{53} +5400.00 q^{55} -15576.0 q^{59} +39302.0 q^{61} +19250.0 q^{65} -55796.0 q^{67} +57120.0 q^{71} +50402.0 q^{73} -9504.00 q^{77} +10552.0 q^{79} +108564. q^{83} -13350.0 q^{85} +116430. q^{89} -33880.0 q^{91} -39500.0 q^{95} -2782.00 q^{97} +O(q^{100})\)

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
\(5\)	25.0000	0.447214
			\(7\)	−44.0000	−0.339397	−0.169698	−	0.985496i	\(-0.554279\pi\)
−0.169698	+	0.985496i				\(0.554279\pi\)
\(11\)	216.000	0.538235	0.269118	−	0.963107i	\(-0.413268\pi\)
			0.269118	+	0.963107i	\(0.413268\pi\)
\(13\)	770.000	1.26367	0.631833	−	0.775104i	\(-0.282303\pi\)
			0.631833	+	0.775104i	\(0.282303\pi\)
\(17\)	−534.000	−0.448145	−0.224073	−	0.974572i	\(-0.571935\pi\)
			−0.224073	+	0.974572i	\(0.571935\pi\)
\(19\)	−1580.00	−1.00409	−0.502046	−	0.864841i	\(-0.667419\pi\)
			−0.502046	+	0.864841i	\(0.667419\pi\)
\(23\)	2904.00	1.14466	0.572331	−	0.820023i	\(-0.306039\pi\)
			0.572331	+	0.820023i	\(0.306039\pi\)
\(29\)	4566.00	1.00819	0.504093	−	0.863649i	\(-0.331827\pi\)
			0.504093	+	0.863649i	\(0.331827\pi\)
\(31\)	−2744.00	−0.512838	−0.256419	−	0.966566i	\(-0.582543\pi\)
			−0.256419	+	0.966566i	\(0.582543\pi\)
\(37\)	1442.00	0.173165	0.0865827	−	0.996245i	\(-0.472405\pi\)
			0.0865827	+	0.996245i	\(0.472405\pi\)
\(41\)	13350.0	1.24029	0.620143	−	0.784489i	\(-0.287075\pi\)
			0.620143	+	0.784489i	\(0.287075\pi\)
\(43\)	−17204.0	−1.41892	−0.709461	−	0.704745i	\(-0.751061\pi\)
			−0.709461	+	0.704745i	\(0.751061\pi\)
\(47\)	−10824.0	−0.714732	−0.357366	−	0.933964i	\(-0.616325\pi\)
			−0.357366	+	0.933964i	\(0.616325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.a.p.1.1		1
3.2	odd	2		240.6.a.i.1.1		1
4.3	odd	2		180.6.a.d.1.1		1
12.11	even	2		60.6.a.a.1.1	✓	1
20.3	even	4		900.6.d.b.649.1		2
20.7	even	4		900.6.d.b.649.2		2
20.19	odd	2		900.6.a.f.1.1		1
24.5	odd	2		960.6.a.i.1.1		1
24.11	even	2		960.6.a.z.1.1		1
60.23	odd	4		300.6.d.e.49.1		2
60.47	odd	4		300.6.d.e.49.2		2
60.59	even	2		300.6.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.a.a.1.1	✓	1	12.11	even	2
180.6.a.d.1.1		1	4.3	odd	2
240.6.a.i.1.1		1	3.2	odd	2
300.6.a.d.1.1		1	60.59	even	2
300.6.d.e.49.1		2	60.23	odd	4
300.6.d.e.49.2		2	60.47	odd	4
720.6.a.p.1.1		1	1.1	even	1	trivial
900.6.a.f.1.1		1	20.19	odd	2
900.6.d.b.649.1		2	20.3	even	4
900.6.d.b.649.2		2	20.7	even	4
960.6.a.i.1.1		1	24.5	odd	2
960.6.a.z.1.1		1	24.11	even	2