Embedded newform 2160.4.a.t.1.1

• Label: 2160.4.a.t.1.1
• Level: $2160$
• Weight: $4$
• Character: 2160.1
• Self dual: yes
• Analytic conductor: $127.444$
• Analytic rank: $0$
• 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:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 270)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{5} +34.0000 q^{7} +48.0000 q^{11} -70.0000 q^{13} -27.0000 q^{17} -119.000 q^{19} +51.0000 q^{23} +25.0000 q^{25} -30.0000 q^{29} +133.000 q^{31} +170.000 q^{35} +218.000 q^{37} +156.000 q^{41} +88.0000 q^{43} +516.000 q^{47} +813.000 q^{49} -639.000 q^{53} +240.000 q^{55} +654.000 q^{59} +461.000 q^{61} -350.000 q^{65} -182.000 q^{67} -900.000 q^{71} +704.000 q^{73} +1632.00 q^{77} +1375.00 q^{79} -915.000 q^{83} -135.000 q^{85} -1116.00 q^{89} -2380.00 q^{91} -595.000 q^{95} -16.0000 q^{97} +O(q^{100})\)

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	\(-0.129872\pi\)
0.917914	+	0.396780i				\(0.129872\pi\)
\(11\)	48.0000	1.31569	0.657843	−	0.753155i	\(-0.271469\pi\)
			0.657843	+	0.753155i	\(0.271469\pi\)
\(13\)	−70.0000	−1.49342	−0.746712	−	0.665148i	\(-0.768369\pi\)
			−0.746712	+	0.665148i	\(0.768369\pi\)
\(17\)	−27.0000	−0.385204	−0.192602	−	0.981277i	\(-0.561693\pi\)
			−0.192602	+	0.981277i	\(0.561693\pi\)
\(19\)	−119.000	−1.43687	−0.718433	−	0.695596i	\(-0.755141\pi\)
			−0.718433	+	0.695596i	\(0.755141\pi\)
\(23\)	51.0000	0.462358	0.231179	−	0.972911i	\(-0.425742\pi\)
			0.231179	+	0.972911i	\(0.425742\pi\)
\(29\)	−30.0000	−0.192099	−0.0960493	−	0.995377i	\(-0.530621\pi\)
			−0.0960493	+	0.995377i	\(0.530621\pi\)
\(31\)	133.000	0.770565	0.385282	−	0.922799i	\(-0.374104\pi\)
			0.385282	+	0.922799i	\(0.374104\pi\)
\(37\)	218.000	0.968621	0.484311	−	0.874896i	\(-0.339070\pi\)
			0.484311	+	0.874896i	\(0.339070\pi\)
\(41\)	156.000	0.594222	0.297111	−	0.954843i	\(-0.403977\pi\)
			0.297111	+	0.954843i	\(0.403977\pi\)
\(43\)	88.0000	0.312090	0.156045	−	0.987750i	\(-0.450125\pi\)
			0.156045	+	0.987750i	\(0.450125\pi\)
\(47\)	516.000	1.60141	0.800706	−	0.599058i	\(-0.204458\pi\)
			0.800706	+	0.599058i	\(0.204458\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.t.1.1		1
3.2	odd	2		2160.4.a.j.1.1		1
4.3	odd	2		270.4.a.k.1.1	yes	1
12.11	even	2		270.4.a.a.1.1	✓	1
20.3	even	4		1350.4.c.c.649.1		2
20.7	even	4		1350.4.c.c.649.2		2
20.19	odd	2		1350.4.a.n.1.1		1
36.7	odd	6		810.4.e.d.271.1		2
36.11	even	6		810.4.e.x.271.1		2
36.23	even	6		810.4.e.x.541.1		2
36.31	odd	6		810.4.e.d.541.1		2
60.23	odd	4		1350.4.c.r.649.2		2
60.47	odd	4		1350.4.c.r.649.1		2
60.59	even	2		1350.4.a.bb.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.a.1.1	✓	1	12.11	even	2
270.4.a.k.1.1	yes	1	4.3	odd	2
810.4.e.d.271.1		2	36.7	odd	6
810.4.e.d.541.1		2	36.31	odd	6
810.4.e.x.271.1		2	36.11	even	6
810.4.e.x.541.1		2	36.23	even	6
1350.4.a.n.1.1		1	20.19	odd	2
1350.4.a.bb.1.1		1	60.59	even	2
1350.4.c.c.649.1		2	20.3	even	4
1350.4.c.c.649.2		2	20.7	even	4
1350.4.c.r.649.1		2	60.47	odd	4
1350.4.c.r.649.2		2	60.23	odd	4
2160.4.a.j.1.1		1	3.2	odd	2
2160.4.a.t.1.1		1	1.1	even	1	trivial