Embedded newform 576.1.g.a.127.1

• Label: 576.1.g.a.127.1
• Level: $576$
• Weight: $1$
• Character: 576.127
• Self dual: yes
• Analytic conductor: $0.287$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -3, -4, 12
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	576.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.287461447277\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 144)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\zeta_{12})\)
Artin image:	$D_4$
Artin field:	Galois closure of \(\Q(\sqrt{3 + \sqrt{-3}})\)
Stark unit:	Root of $x^{4} - 52x^{3} - 90x^{2} - 52x + 1$

Embedding invariants

Embedding label			127.1
Character	\(\chi\)	\(=\)	576.127

$q$-expansion

\(f(q)\)

\(=\)

\(q+2.00000 q^{13} -1.00000 q^{25} -2.00000 q^{37} +1.00000 q^{49} -2.00000 q^{61} -2.00000 q^{73} -2.00000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	−2.00000	−2.00000	−1.00000			\(\pi\)
			−1.00000			\(\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(59\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(61\)	−2.00000	−2.00000	−1.00000			\(\pi\)
			−1.00000			\(\pi\)
\(67\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(71\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(73\)	−2.00000	−2.00000	−1.00000			\(\pi\)
			−1.00000			\(\pi\)
\(79\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(83\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	−2.00000	−2.00000	−1.00000			\(\pi\)
			−1.00000			\(\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.1.g.a.127.1		1
3.2	odd	2	CM	576.1.g.a.127.1		1
4.3	odd	2	CM	576.1.g.a.127.1		1
8.3	odd	2		144.1.g.a.127.1	✓	1
8.5	even	2		144.1.g.a.127.1	✓	1
12.11	even	2	RM	576.1.g.a.127.1		1
16.3	odd	4		2304.1.b.a.127.2		2
16.5	even	4		2304.1.b.a.127.1		2
16.11	odd	4		2304.1.b.a.127.1		2
16.13	even	4		2304.1.b.a.127.2		2
24.5	odd	2		144.1.g.a.127.1	✓	1
24.11	even	2		144.1.g.a.127.1	✓	1
40.3	even	4		3600.1.j.a.1999.1		2
40.13	odd	4		3600.1.j.a.1999.1		2
40.19	odd	2		3600.1.e.b.3151.1		1
40.27	even	4		3600.1.j.a.1999.2		2
40.29	even	2		3600.1.e.b.3151.1		1
40.37	odd	4		3600.1.j.a.1999.2		2
48.5	odd	4		2304.1.b.a.127.1		2
48.11	even	4		2304.1.b.a.127.1		2
48.29	odd	4		2304.1.b.a.127.2		2
48.35	even	4		2304.1.b.a.127.2		2
72.5	odd	6		1296.1.o.b.703.1		2
72.11	even	6		1296.1.o.b.271.1		2
72.13	even	6		1296.1.o.b.703.1		2
72.29	odd	6		1296.1.o.b.271.1		2
72.43	odd	6		1296.1.o.b.271.1		2
72.59	even	6		1296.1.o.b.703.1		2
72.61	even	6		1296.1.o.b.271.1		2
72.67	odd	6		1296.1.o.b.703.1		2
120.29	odd	2		3600.1.e.b.3151.1		1
120.53	even	4		3600.1.j.a.1999.1		2
120.59	even	2		3600.1.e.b.3151.1		1
120.77	even	4		3600.1.j.a.1999.2		2
120.83	odd	4		3600.1.j.a.1999.1		2
120.107	odd	4		3600.1.j.a.1999.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.1.g.a.127.1	✓	1	8.3	odd	2
144.1.g.a.127.1	✓	1	8.5	even	2
144.1.g.a.127.1	✓	1	24.5	odd	2
144.1.g.a.127.1	✓	1	24.11	even	2
576.1.g.a.127.1		1	1.1	even	1	trivial
576.1.g.a.127.1		1	3.2	odd	2	CM
576.1.g.a.127.1		1	4.3	odd	2	CM
576.1.g.a.127.1		1	12.11	even	2	RM
1296.1.o.b.271.1		2	72.11	even	6
1296.1.o.b.271.1		2	72.29	odd	6
1296.1.o.b.271.1		2	72.43	odd	6
1296.1.o.b.271.1		2	72.61	even	6
1296.1.o.b.703.1		2	72.5	odd	6
1296.1.o.b.703.1		2	72.13	even	6
1296.1.o.b.703.1		2	72.59	even	6
1296.1.o.b.703.1		2	72.67	odd	6
2304.1.b.a.127.1		2	16.5	even	4
2304.1.b.a.127.1		2	16.11	odd	4
2304.1.b.a.127.1		2	48.5	odd	4
2304.1.b.a.127.1		2	48.11	even	4
2304.1.b.a.127.2		2	16.3	odd	4
2304.1.b.a.127.2		2	16.13	even	4
2304.1.b.a.127.2		2	48.29	odd	4
2304.1.b.a.127.2		2	48.35	even	4
3600.1.e.b.3151.1		1	40.19	odd	2
3600.1.e.b.3151.1		1	40.29	even	2
3600.1.e.b.3151.1		1	120.29	odd	2
3600.1.e.b.3151.1		1	120.59	even	2
3600.1.j.a.1999.1		2	40.3	even	4
3600.1.j.a.1999.1		2	40.13	odd	4
3600.1.j.a.1999.1		2	120.53	even	4
3600.1.j.a.1999.1		2	120.83	odd	4
3600.1.j.a.1999.2		2	40.27	even	4
3600.1.j.a.1999.2		2	40.37	odd	4
3600.1.j.a.1999.2		2	120.77	even	4
3600.1.j.a.1999.2		2	120.107	odd	4