Embedded newform 192.3.e.a.65.1

• Label: 192.3.e.a.65.1
• Level: $192$
• Weight: $3$
• Character: 192.65
• Self dual: yes
• Analytic conductor: $5.232$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.23162107572\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			65.1
Character	\(\chi\)	\(=\)	192.65

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} -2.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−3.00000	−1.00000
\(5\)	0	0				1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	−2.00000	−0.285714	−0.142857	−	0.989743i	\(-0.545629\pi\)
			−0.142857	+	0.989743i	\(0.545629\pi\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	22.0000	1.69231	0.846154	−	0.532939i	\(-0.178912\pi\)
			0.846154	+	0.532939i	\(0.178912\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	26.0000	1.36842	0.684211	−	0.729285i	\(-0.260147\pi\)
			0.684211	+	0.729285i	\(0.260147\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	46.0000	1.48387	0.741935	−	0.670471i	\(-0.233908\pi\)
			0.741935	+	0.670471i	\(0.233908\pi\)
\(37\)	−26.0000	−0.702703	−0.351351	−	0.936244i	\(-0.614278\pi\)
			−0.351351	+	0.936244i	\(0.614278\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	−22.0000	−0.511628	−0.255814	−	0.966726i	\(-0.582343\pi\)
			−0.255814	+	0.966726i	\(0.582343\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.e.a.65.1		1
3.2	odd	2	CM	192.3.e.a.65.1		1
4.3	odd	2		192.3.e.b.65.1		1
8.3	odd	2		12.3.c.a.5.1	✓	1
8.5	even	2		48.3.e.a.17.1		1
12.11	even	2		192.3.e.b.65.1		1
16.3	odd	4		768.3.h.a.641.2		2
16.5	even	4		768.3.h.b.641.2		2
16.11	odd	4		768.3.h.a.641.1		2
16.13	even	4		768.3.h.b.641.1		2
24.5	odd	2		48.3.e.a.17.1		1
24.11	even	2		12.3.c.a.5.1	✓	1
40.3	even	4		300.3.b.a.149.1		2
40.13	odd	4		1200.3.c.c.449.2		2
40.19	odd	2		300.3.g.b.101.1		1
40.27	even	4		300.3.b.a.149.2		2
40.29	even	2		1200.3.l.b.401.1		1
40.37	odd	4		1200.3.c.c.449.1		2
48.5	odd	4		768.3.h.b.641.2		2
48.11	even	4		768.3.h.a.641.1		2
48.29	odd	4		768.3.h.b.641.1		2
48.35	even	4		768.3.h.a.641.2		2
56.3	even	6		588.3.p.b.569.1		2
56.11	odd	6		588.3.p.c.569.1		2
56.19	even	6		588.3.p.b.557.1		2
56.27	even	2		588.3.c.c.197.1		1
56.51	odd	6		588.3.p.c.557.1		2
72.5	odd	6		1296.3.q.b.593.1		2
72.11	even	6		324.3.g.b.53.1		2
72.13	even	6		1296.3.q.b.593.1		2
72.29	odd	6		1296.3.q.b.1025.1		2
72.43	odd	6		324.3.g.b.53.1		2
72.59	even	6		324.3.g.b.269.1		2
72.61	even	6		1296.3.q.b.1025.1		2
72.67	odd	6		324.3.g.b.269.1		2
88.43	even	2		1452.3.e.b.485.1		1
120.29	odd	2		1200.3.l.b.401.1		1
120.53	even	4		1200.3.c.c.449.2		2
120.59	even	2		300.3.g.b.101.1		1
120.77	even	4		1200.3.c.c.449.1		2
120.83	odd	4		300.3.b.a.149.1		2
120.107	odd	4		300.3.b.a.149.2		2
168.11	even	6		588.3.p.c.569.1		2
168.59	odd	6		588.3.p.b.569.1		2
168.83	odd	2		588.3.c.c.197.1		1
168.107	even	6		588.3.p.c.557.1		2
168.131	odd	6		588.3.p.b.557.1		2
264.131	odd	2		1452.3.e.b.485.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.3.c.a.5.1	✓	1	8.3	odd	2
12.3.c.a.5.1	✓	1	24.11	even	2
48.3.e.a.17.1		1	8.5	even	2
48.3.e.a.17.1		1	24.5	odd	2
192.3.e.a.65.1		1	1.1	even	1	trivial
192.3.e.a.65.1		1	3.2	odd	2	CM
192.3.e.b.65.1		1	4.3	odd	2
192.3.e.b.65.1		1	12.11	even	2
300.3.b.a.149.1		2	40.3	even	4
300.3.b.a.149.1		2	120.83	odd	4
300.3.b.a.149.2		2	40.27	even	4
300.3.b.a.149.2		2	120.107	odd	4
300.3.g.b.101.1		1	40.19	odd	2
300.3.g.b.101.1		1	120.59	even	2
324.3.g.b.53.1		2	72.11	even	6
324.3.g.b.53.1		2	72.43	odd	6
324.3.g.b.269.1		2	72.59	even	6
324.3.g.b.269.1		2	72.67	odd	6
588.3.c.c.197.1		1	56.27	even	2
588.3.c.c.197.1		1	168.83	odd	2
588.3.p.b.557.1		2	56.19	even	6
588.3.p.b.557.1		2	168.131	odd	6
588.3.p.b.569.1		2	56.3	even	6
588.3.p.b.569.1		2	168.59	odd	6
588.3.p.c.557.1		2	56.51	odd	6
588.3.p.c.557.1		2	168.107	even	6
588.3.p.c.569.1		2	56.11	odd	6
588.3.p.c.569.1		2	168.11	even	6
768.3.h.a.641.1		2	16.11	odd	4
768.3.h.a.641.1		2	48.11	even	4
768.3.h.a.641.2		2	16.3	odd	4
768.3.h.a.641.2		2	48.35	even	4
768.3.h.b.641.1		2	16.13	even	4
768.3.h.b.641.1		2	48.29	odd	4
768.3.h.b.641.2		2	16.5	even	4
768.3.h.b.641.2		2	48.5	odd	4
1200.3.c.c.449.1		2	40.37	odd	4
1200.3.c.c.449.1		2	120.77	even	4
1200.3.c.c.449.2		2	40.13	odd	4
1200.3.c.c.449.2		2	120.53	even	4
1200.3.l.b.401.1		1	40.29	even	2
1200.3.l.b.401.1		1	120.29	odd	2
1296.3.q.b.593.1		2	72.5	odd	6
1296.3.q.b.593.1		2	72.13	even	6
1296.3.q.b.1025.1		2	72.29	odd	6
1296.3.q.b.1025.1		2	72.61	even	6
1452.3.e.b.485.1		1	88.43	even	2
1452.3.e.b.485.1		1	264.131	odd	2