Embedded newform 144.6.a.c.1.1

• Label: 144.6.a.c.1.1
• Level: $144$
• Weight: $6$
• Character: 144.1
• Self dual: yes
• Analytic conductor: $23.095$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(23.0952700531\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 4)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	144.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-54.0000 q^{5} +88.0000 q^{7} +540.000 q^{11} -418.000 q^{13} -594.000 q^{17} -836.000 q^{19} -4104.00 q^{23} -209.000 q^{25} +594.000 q^{29} -4256.00 q^{31} -4752.00 q^{35} -298.000 q^{37} -17226.0 q^{41} +12100.0 q^{43} -1296.00 q^{47} -9063.00 q^{49} -19494.0 q^{53} -29160.0 q^{55} -7668.00 q^{59} -34738.0 q^{61} +22572.0 q^{65} -21812.0 q^{67} -46872.0 q^{71} +67562.0 q^{73} +47520.0 q^{77} +76912.0 q^{79} +67716.0 q^{83} +32076.0 q^{85} -29754.0 q^{89} -36784.0 q^{91} +45144.0 q^{95} -122398. q^{97} +O(q^{100})\)

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
\(5\)	−54.0000	−0.965981				−0.482991	−	0.875625i	\(-0.660450\pi\)
			−0.482991	+	0.875625i	\(0.660450\pi\)
\(7\)	88.0000	0.678793	0.339397	−	0.940643i	\(-0.389777\pi\)
			0.339397	+	0.940643i	\(0.389777\pi\)
\(11\)	540.000	1.34559	0.672794	−	0.739830i	\(-0.265094\pi\)
			0.672794	+	0.739830i	\(0.265094\pi\)
\(13\)	−418.000	−0.685990	−0.342995	−	0.939337i	\(-0.611441\pi\)
			−0.342995	+	0.939337i	\(0.611441\pi\)
\(17\)	−594.000	−0.498499	−0.249249	−	0.968439i	\(-0.580184\pi\)
			−0.249249	+	0.968439i	\(0.580184\pi\)
\(19\)	−836.000	−0.531279	−0.265639	−	0.964072i	\(-0.585583\pi\)
			−0.265639	+	0.964072i	\(0.585583\pi\)
\(23\)	−4104.00	−1.61766	−0.808831	−	0.588041i	\(-0.799899\pi\)
			−0.808831	+	0.588041i	\(0.799899\pi\)
\(29\)	594.000	0.131157	0.0655785	−	0.997847i	\(-0.479111\pi\)
			0.0655785	+	0.997847i	\(0.479111\pi\)
\(31\)	−4256.00	−0.795422	−0.397711	−	0.917511i	\(-0.630195\pi\)
			−0.397711	+	0.917511i	\(0.630195\pi\)
\(37\)	−298.000	−0.0357859	−0.0178930	−	0.999840i	\(-0.505696\pi\)
			−0.0178930	+	0.999840i	\(0.505696\pi\)
\(41\)	−17226.0	−1.60039	−0.800193	−	0.599742i	\(-0.795270\pi\)
			−0.800193	+	0.599742i	\(0.795270\pi\)
\(43\)	12100.0	0.997963	0.498981	−	0.866613i	\(-0.333708\pi\)
			0.498981	+	0.866613i	\(0.333708\pi\)
\(47\)	−1296.00	−0.0855777	−0.0427888	−	0.999084i	\(-0.513624\pi\)
			−0.0427888	+	0.999084i	\(0.513624\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.6.a.c.1.1		1
3.2	odd	2		16.6.a.b.1.1		1
4.3	odd	2		36.6.a.a.1.1		1
8.3	odd	2		576.6.a.bc.1.1		1
8.5	even	2		576.6.a.bd.1.1		1
12.11	even	2		4.6.a.a.1.1	✓	1
15.2	even	4		400.6.c.f.49.1		2
15.8	even	4		400.6.c.f.49.2		2
15.14	odd	2		400.6.a.d.1.1		1
20.3	even	4		900.6.d.a.649.2		2
20.7	even	4		900.6.d.a.649.1		2
20.19	odd	2		900.6.a.h.1.1		1
21.20	even	2		784.6.a.d.1.1		1
24.5	odd	2		64.6.a.b.1.1		1
24.11	even	2		64.6.a.f.1.1		1
36.7	odd	6		324.6.e.d.109.1		2
36.11	even	6		324.6.e.a.109.1		2
36.23	even	6		324.6.e.a.217.1		2
36.31	odd	6		324.6.e.d.217.1		2
48.5	odd	4		256.6.b.c.129.1		2
48.11	even	4		256.6.b.g.129.2		2
48.29	odd	4		256.6.b.c.129.2		2
48.35	even	4		256.6.b.g.129.1		2
60.23	odd	4		100.6.c.b.49.1		2
60.47	odd	4		100.6.c.b.49.2		2
60.59	even	2		100.6.a.b.1.1		1
84.11	even	6		196.6.e.g.177.1		2
84.23	even	6		196.6.e.g.165.1		2
84.47	odd	6		196.6.e.d.165.1		2
84.59	odd	6		196.6.e.d.177.1		2
84.83	odd	2		196.6.a.e.1.1		1
132.131	odd	2		484.6.a.a.1.1		1
156.47	odd	4		676.6.d.a.337.2		2
156.83	odd	4		676.6.d.a.337.1		2
156.155	even	2		676.6.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.6.a.a.1.1	✓	1	12.11	even	2
16.6.a.b.1.1		1	3.2	odd	2
36.6.a.a.1.1		1	4.3	odd	2
64.6.a.b.1.1		1	24.5	odd	2
64.6.a.f.1.1		1	24.11	even	2
100.6.a.b.1.1		1	60.59	even	2
100.6.c.b.49.1		2	60.23	odd	4
100.6.c.b.49.2		2	60.47	odd	4
144.6.a.c.1.1		1	1.1	even	1	trivial
196.6.a.e.1.1		1	84.83	odd	2
196.6.e.d.165.1		2	84.47	odd	6
196.6.e.d.177.1		2	84.59	odd	6
196.6.e.g.165.1		2	84.23	even	6
196.6.e.g.177.1		2	84.11	even	6
256.6.b.c.129.1		2	48.5	odd	4
256.6.b.c.129.2		2	48.29	odd	4
256.6.b.g.129.1		2	48.35	even	4
256.6.b.g.129.2		2	48.11	even	4
324.6.e.a.109.1		2	36.11	even	6
324.6.e.a.217.1		2	36.23	even	6
324.6.e.d.109.1		2	36.7	odd	6
324.6.e.d.217.1		2	36.31	odd	6
400.6.a.d.1.1		1	15.14	odd	2
400.6.c.f.49.1		2	15.2	even	4
400.6.c.f.49.2		2	15.8	even	4
484.6.a.a.1.1		1	132.131	odd	2
576.6.a.bc.1.1		1	8.3	odd	2
576.6.a.bd.1.1		1	8.5	even	2
676.6.a.a.1.1		1	156.155	even	2
676.6.d.a.337.1		2	156.83	odd	4
676.6.d.a.337.2		2	156.47	odd	4
784.6.a.d.1.1		1	21.20	even	2
900.6.a.h.1.1		1	20.19	odd	2
900.6.d.a.649.1		2	20.7	even	4
900.6.d.a.649.2		2	20.3	even	4