Embedded newform 2160.4.a.n.1.1

• Label: 2160.4.a.n.1.1
• Level: $2160$
• Weight: $4$
• Character: 2160.1
• Self dual: yes
• Analytic conductor: $127.444$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(127.444125612\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2160.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{5} -10.0000 q^{11} -80.0000 q^{13} +7.00000 q^{17} +113.000 q^{19} +81.0000 q^{23} +25.0000 q^{25} -220.000 q^{29} +189.000 q^{31} +170.000 q^{37} -130.000 q^{41} -10.0000 q^{43} -160.000 q^{47} -343.000 q^{49} +631.000 q^{53} -50.0000 q^{55} +560.000 q^{59} +229.000 q^{61} -400.000 q^{65} -750.000 q^{67} -890.000 q^{71} -890.000 q^{73} +27.0000 q^{79} -429.000 q^{83} +35.0000 q^{85} -750.000 q^{89} +565.000 q^{95} -1480.00 q^{97} +O(q^{100})\)

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
			\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(11\)	−10.0000	−0.274101	−0.137051	−	0.990564i	\(-0.543762\pi\)
			−0.137051	+	0.990564i	\(0.543762\pi\)
\(13\)	−80.0000	−1.70677	−0.853385	−	0.521281i	\(-0.825454\pi\)
			−0.853385	+	0.521281i	\(0.825454\pi\)
\(17\)	7.00000	0.0998676	0.0499338	−	0.998753i	\(-0.484099\pi\)
			0.0499338	+	0.998753i	\(0.484099\pi\)
\(19\)	113.000	1.36442	0.682210	−	0.731156i	\(-0.261019\pi\)
			0.682210	+	0.731156i	\(0.261019\pi\)
\(23\)	81.0000	0.734333	0.367167	−	0.930155i	\(-0.380328\pi\)
			0.367167	+	0.930155i	\(0.380328\pi\)
\(29\)	−220.000	−1.40872	−0.704362	−	0.709841i	\(-0.748767\pi\)
			−0.704362	+	0.709841i	\(0.748767\pi\)
\(31\)	189.000	1.09501	0.547506	−	0.836801i	\(-0.315577\pi\)
			0.547506	+	0.836801i	\(0.315577\pi\)
\(37\)	170.000	0.755347	0.377673	−	0.925939i	\(-0.376724\pi\)
			0.377673	+	0.925939i	\(0.376724\pi\)
\(41\)	−130.000	−0.495185	−0.247593	−	0.968864i	\(-0.579639\pi\)
			−0.247593	+	0.968864i	\(0.579639\pi\)
\(43\)	−10.0000	−0.0354648	−0.0177324	−	0.999843i	\(-0.505645\pi\)
			−0.0177324	+	0.999843i	\(0.505645\pi\)
\(47\)	−160.000	−0.496562	−0.248281	−	0.968688i	\(-0.579866\pi\)
			−0.248281	+	0.968688i	\(0.579866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.n.1.1		1
3.2	odd	2		2160.4.a.d.1.1		1
4.3	odd	2		135.4.a.a.1.1	✓	1
12.11	even	2		135.4.a.d.1.1	yes	1
20.3	even	4		675.4.b.d.649.2		2
20.7	even	4		675.4.b.d.649.1		2
20.19	odd	2		675.4.a.i.1.1		1
36.7	odd	6		405.4.e.j.271.1		2
36.11	even	6		405.4.e.e.271.1		2
36.23	even	6		405.4.e.e.136.1		2
36.31	odd	6		405.4.e.j.136.1		2
60.23	odd	4		675.4.b.c.649.1		2
60.47	odd	4		675.4.b.c.649.2		2
60.59	even	2		675.4.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.a.1.1	✓	1	4.3	odd	2
135.4.a.d.1.1	yes	1	12.11	even	2
405.4.e.e.136.1		2	36.23	even	6
405.4.e.e.271.1		2	36.11	even	6
405.4.e.j.136.1		2	36.31	odd	6
405.4.e.j.271.1		2	36.7	odd	6
675.4.a.b.1.1		1	60.59	even	2
675.4.a.i.1.1		1	20.19	odd	2
675.4.b.c.649.1		2	60.23	odd	4
675.4.b.c.649.2		2	60.47	odd	4
675.4.b.d.649.1		2	20.7	even	4
675.4.b.d.649.2		2	20.3	even	4
2160.4.a.d.1.1		1	3.2	odd	2
2160.4.a.n.1.1		1	1.1	even	1	trivial