Embedded newform 2160.4.a.p.1.1

• Label: 2160.4.a.p.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

\(f(q)\)

\(=\)

\(q+5.00000 q^{5} +6.00000 q^{7} +47.0000 q^{11} -5.00000 q^{13} -131.000 q^{17} +56.0000 q^{19} -3.00000 q^{23} +25.0000 q^{25} -157.000 q^{29} -225.000 q^{31} +30.0000 q^{35} -70.0000 q^{37} +140.000 q^{41} -397.000 q^{43} +347.000 q^{47} -307.000 q^{49} +4.00000 q^{53} +235.000 q^{55} -748.000 q^{59} -338.000 q^{61} -25.0000 q^{65} -492.000 q^{67} -32.0000 q^{71} +970.000 q^{73} +282.000 q^{77} +1257.00 q^{79} +102.000 q^{83} -655.000 q^{85} -1488.00 q^{89} -30.0000 q^{91} +280.000 q^{95} +974.000 q^{97} +O(q^{100})\)

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
			\(7\)	6.00000	0.323970	0.161985	−	0.986793i	\(-0.448210\pi\)
0.161985	+	0.986793i				\(0.448210\pi\)
\(11\)	47.0000	1.28828	0.644138	−	0.764909i	\(-0.277216\pi\)
			0.644138	+	0.764909i	\(0.277216\pi\)
\(13\)	−5.00000	−0.106673	−0.0533366	−	0.998577i	\(-0.516986\pi\)
			−0.0533366	+	0.998577i	\(0.516986\pi\)
\(17\)	−131.000	−1.86895	−0.934475	−	0.356027i	\(-0.884131\pi\)
			−0.934475	+	0.356027i	\(0.884131\pi\)
\(19\)	56.0000	0.676173	0.338086	−	0.941115i	\(-0.390220\pi\)
			0.338086	+	0.941115i	\(0.390220\pi\)
\(23\)	−3.00000	−0.0271975	−0.0135988	−	0.999908i	\(-0.504329\pi\)
			−0.0135988	+	0.999908i	\(0.504329\pi\)
\(29\)	−157.000	−1.00532	−0.502658	−	0.864485i	\(-0.667645\pi\)
			−0.502658	+	0.864485i	\(0.667645\pi\)
\(31\)	−225.000	−1.30359	−0.651793	−	0.758397i	\(-0.725983\pi\)
			−0.651793	+	0.758397i	\(0.725983\pi\)
\(37\)	−70.0000	−0.311025	−0.155513	−	0.987834i	\(-0.549703\pi\)
			−0.155513	+	0.987834i	\(0.549703\pi\)
\(41\)	140.000	0.533276	0.266638	−	0.963797i	\(-0.414087\pi\)
			0.266638	+	0.963797i	\(0.414087\pi\)
\(43\)	−397.000	−1.40795	−0.703976	−	0.710224i	\(-0.748594\pi\)
			−0.703976	+	0.710224i	\(0.748594\pi\)
\(47\)	347.000	1.07692	0.538459	−	0.842652i	\(-0.319007\pi\)
			0.538459	+	0.842652i	\(0.319007\pi\)

(See \(a_n\) instead)

Twists

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

    

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