Embedded newform 6336.2.a.j.1.1

• Label: 6336.2.a.j.1.1
• Level: $6336$
• Weight: $2$
• Character: 6336.1
• Self dual: yes
• Analytic conductor: $50.593$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(50.5932147207\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	6336.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{5} +2.00000 q^{7} +O(q^{10})\)

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
\(5\)	−3.00000	−1.34164				−0.670820	−	0.741620i	\(-0.734058\pi\)
			−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	2.00000	0.755929	0.377964	−	0.925820i	\(-0.376624\pi\)
			0.377964	+	0.925820i	\(0.376624\pi\)
\(11\)	−1.00000	−0.301511
			\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
0.554700	+	0.832050i				\(0.312833\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	−8.00000	−1.83533	−0.917663	−	0.397360i	\(-0.869927\pi\)
			−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	3.00000	0.625543	0.312772	−	0.949828i	\(-0.398743\pi\)
			0.312772	+	0.949828i	\(0.398743\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	5.00000	0.898027	0.449013	−	0.893525i	\(-0.351776\pi\)
			0.449013	+	0.893525i	\(0.351776\pi\)
\(37\)	1.00000	0.164399	0.0821995	−	0.996616i	\(-0.473806\pi\)
			0.0821995	+	0.996616i	\(0.473806\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	10.0000	1.52499	0.762493	−	0.646997i	\(-0.223975\pi\)
			0.762493	+	0.646997i	\(0.223975\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.a.j.1.1		1
3.2	odd	2		704.2.a.f.1.1		1
4.3	odd	2		6336.2.a.i.1.1		1
8.3	odd	2		1584.2.a.p.1.1		1
8.5	even	2		396.2.a.c.1.1		1
12.11	even	2		704.2.a.i.1.1		1
24.5	odd	2		44.2.a.a.1.1	✓	1
24.11	even	2		176.2.a.a.1.1		1
33.32	even	2		7744.2.a.m.1.1		1
40.13	odd	4		9900.2.c.g.5149.1		2
40.29	even	2		9900.2.a.h.1.1		1
40.37	odd	4		9900.2.c.g.5149.2		2
48.5	odd	4		2816.2.c.e.1409.2		2
48.11	even	4		2816.2.c.k.1409.1		2
48.29	odd	4		2816.2.c.e.1409.1		2
48.35	even	4		2816.2.c.k.1409.2		2
72.5	odd	6		3564.2.i.j.1189.1		2
72.13	even	6		3564.2.i.a.1189.1		2
72.29	odd	6		3564.2.i.j.2377.1		2
72.61	even	6		3564.2.i.a.2377.1		2
88.21	odd	2		4356.2.a.j.1.1		1
120.29	odd	2		1100.2.a.b.1.1		1
120.53	even	4		1100.2.b.c.749.2		2
120.59	even	2		4400.2.a.v.1.1		1
120.77	even	4		1100.2.b.c.749.1		2
120.83	odd	4		4400.2.b.k.4049.1		2
120.107	odd	4		4400.2.b.k.4049.2		2
132.131	odd	2		7744.2.a.bc.1.1		1
168.5	even	6		2156.2.i.c.1145.1		2
168.53	odd	6		2156.2.i.b.177.1		2
168.83	odd	2		8624.2.a.w.1.1		1
168.101	even	6		2156.2.i.c.177.1		2
168.125	even	2		2156.2.a.a.1.1		1
168.149	odd	6		2156.2.i.b.1145.1		2
264.5	odd	10		484.2.e.a.245.1		4
264.29	even	10		484.2.e.b.269.1		4
264.53	odd	10		484.2.e.a.81.1		4
264.101	even	10		484.2.e.b.81.1		4
264.125	odd	10		484.2.e.a.269.1		4
264.131	odd	2		1936.2.a.c.1.1		1
264.149	even	10		484.2.e.b.245.1		4
264.173	even	10		484.2.e.b.9.1		4
264.197	even	2		484.2.a.a.1.1		1
264.245	odd	10		484.2.e.a.9.1		4
312.77	odd	2		7436.2.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.a.a.1.1	✓	1	24.5	odd	2
176.2.a.a.1.1		1	24.11	even	2
396.2.a.c.1.1		1	8.5	even	2
484.2.a.a.1.1		1	264.197	even	2
484.2.e.a.9.1		4	264.245	odd	10
484.2.e.a.81.1		4	264.53	odd	10
484.2.e.a.245.1		4	264.5	odd	10
484.2.e.a.269.1		4	264.125	odd	10
484.2.e.b.9.1		4	264.173	even	10
484.2.e.b.81.1		4	264.101	even	10
484.2.e.b.245.1		4	264.149	even	10
484.2.e.b.269.1		4	264.29	even	10
704.2.a.f.1.1		1	3.2	odd	2
704.2.a.i.1.1		1	12.11	even	2
1100.2.a.b.1.1		1	120.29	odd	2
1100.2.b.c.749.1		2	120.77	even	4
1100.2.b.c.749.2		2	120.53	even	4
1584.2.a.p.1.1		1	8.3	odd	2
1936.2.a.c.1.1		1	264.131	odd	2
2156.2.a.a.1.1		1	168.125	even	2
2156.2.i.b.177.1		2	168.53	odd	6
2156.2.i.b.1145.1		2	168.149	odd	6
2156.2.i.c.177.1		2	168.101	even	6
2156.2.i.c.1145.1		2	168.5	even	6
2816.2.c.e.1409.1		2	48.29	odd	4
2816.2.c.e.1409.2		2	48.5	odd	4
2816.2.c.k.1409.1		2	48.11	even	4
2816.2.c.k.1409.2		2	48.35	even	4
3564.2.i.a.1189.1		2	72.13	even	6
3564.2.i.a.2377.1		2	72.61	even	6
3564.2.i.j.1189.1		2	72.5	odd	6
3564.2.i.j.2377.1		2	72.29	odd	6
4356.2.a.j.1.1		1	88.21	odd	2
4400.2.a.v.1.1		1	120.59	even	2
4400.2.b.k.4049.1		2	120.83	odd	4
4400.2.b.k.4049.2		2	120.107	odd	4
6336.2.a.i.1.1		1	4.3	odd	2
6336.2.a.j.1.1		1	1.1	even	1	trivial
7436.2.a.d.1.1		1	312.77	odd	2
7744.2.a.m.1.1		1	33.32	even	2
7744.2.a.bc.1.1		1	132.131	odd	2
8624.2.a.w.1.1		1	168.83	odd	2
9900.2.a.h.1.1		1	40.29	even	2
9900.2.c.g.5149.1		2	40.13	odd	4
9900.2.c.g.5149.2		2	40.37	odd	4