Embedded newform 12.3.c.a.5.1

• Label: 12.3.c.a.5.1
• Level: $12$
• Weight: $3$
• Character: 12.5
• Self dual: yes
• Analytic conductor: $0.327$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.326976317232\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	12.5

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} +2.00000 q^{7} +9.00000 q^{9} -22.0000 q^{13} +26.0000 q^{19} -6.00000 q^{21} +25.0000 q^{25} -27.0000 q^{27} -46.0000 q^{31} +26.0000 q^{37} +66.0000 q^{39} -22.0000 q^{43} -45.0000 q^{49} -78.0000 q^{57} +74.0000 q^{61} +18.0000 q^{63} +122.000 q^{67} -46.0000 q^{73} -75.0000 q^{75} -142.000 q^{79} +81.0000 q^{81} -44.0000 q^{91} +138.000 q^{93} +2.00000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(1\)

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	\(-0.454371\pi\)
			0.142857	+	0.989743i	\(0.454371\pi\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	−22.0000	−1.69231	−0.846154	−	0.532939i	\(-0.821088\pi\)
			−0.846154	+	0.532939i	\(0.821088\pi\)
\(17\)	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	\(-0.766092\pi\)
			−0.741935	+	0.670471i	\(0.766092\pi\)
\(37\)	26.0000	0.702703	0.351351	−	0.936244i	\(-0.385722\pi\)
			0.351351	+	0.936244i	\(0.385722\pi\)
\(41\)	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	12.3.c.a.5.1	✓	1
3.2	odd	2	CM	12.3.c.a.5.1	✓	1
4.3	odd	2		48.3.e.a.17.1		1
5.2	odd	4		300.3.b.a.149.2		2
5.3	odd	4		300.3.b.a.149.1		2
5.4	even	2		300.3.g.b.101.1		1
7.2	even	3		588.3.p.c.557.1		2
7.3	odd	6		588.3.p.b.569.1		2
7.4	even	3		588.3.p.c.569.1		2
7.5	odd	6		588.3.p.b.557.1		2
7.6	odd	2		588.3.c.c.197.1		1
8.3	odd	2		192.3.e.a.65.1		1
8.5	even	2		192.3.e.b.65.1		1
9.2	odd	6		324.3.g.b.53.1		2
9.4	even	3		324.3.g.b.269.1		2
9.5	odd	6		324.3.g.b.269.1		2
9.7	even	3		324.3.g.b.53.1		2
11.10	odd	2		1452.3.e.b.485.1		1
12.11	even	2		48.3.e.a.17.1		1
15.2	even	4		300.3.b.a.149.2		2
15.8	even	4		300.3.b.a.149.1		2
15.14	odd	2		300.3.g.b.101.1		1
16.3	odd	4		768.3.h.b.641.2		2
16.5	even	4		768.3.h.a.641.2		2
16.11	odd	4		768.3.h.b.641.1		2
16.13	even	4		768.3.h.a.641.1		2
20.3	even	4		1200.3.c.c.449.2		2
20.7	even	4		1200.3.c.c.449.1		2
20.19	odd	2		1200.3.l.b.401.1		1
21.2	odd	6		588.3.p.c.557.1		2
21.5	even	6		588.3.p.b.557.1		2
21.11	odd	6		588.3.p.c.569.1		2
21.17	even	6		588.3.p.b.569.1		2
21.20	even	2		588.3.c.c.197.1		1
24.5	odd	2		192.3.e.b.65.1		1
24.11	even	2		192.3.e.a.65.1		1
33.32	even	2		1452.3.e.b.485.1		1
36.7	odd	6		1296.3.q.b.1025.1		2
36.11	even	6		1296.3.q.b.1025.1		2
36.23	even	6		1296.3.q.b.593.1		2
36.31	odd	6		1296.3.q.b.593.1		2
48.5	odd	4		768.3.h.a.641.2		2
48.11	even	4		768.3.h.b.641.1		2
48.29	odd	4		768.3.h.a.641.1		2
48.35	even	4		768.3.h.b.641.2		2
60.23	odd	4		1200.3.c.c.449.2		2
60.47	odd	4		1200.3.c.c.449.1		2
60.59	even	2		1200.3.l.b.401.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.3.c.a.5.1	✓	1	1.1	even	1	trivial
12.3.c.a.5.1	✓	1	3.2	odd	2	CM
48.3.e.a.17.1		1	4.3	odd	2
48.3.e.a.17.1		1	12.11	even	2
192.3.e.a.65.1		1	8.3	odd	2
192.3.e.a.65.1		1	24.11	even	2
192.3.e.b.65.1		1	8.5	even	2
192.3.e.b.65.1		1	24.5	odd	2
300.3.b.a.149.1		2	5.3	odd	4
300.3.b.a.149.1		2	15.8	even	4
300.3.b.a.149.2		2	5.2	odd	4
300.3.b.a.149.2		2	15.2	even	4
300.3.g.b.101.1		1	5.4	even	2
300.3.g.b.101.1		1	15.14	odd	2
324.3.g.b.53.1		2	9.2	odd	6
324.3.g.b.53.1		2	9.7	even	3
324.3.g.b.269.1		2	9.4	even	3
324.3.g.b.269.1		2	9.5	odd	6
588.3.c.c.197.1		1	7.6	odd	2
588.3.c.c.197.1		1	21.20	even	2
588.3.p.b.557.1		2	7.5	odd	6
588.3.p.b.557.1		2	21.5	even	6
588.3.p.b.569.1		2	7.3	odd	6
588.3.p.b.569.1		2	21.17	even	6
588.3.p.c.557.1		2	7.2	even	3
588.3.p.c.557.1		2	21.2	odd	6
588.3.p.c.569.1		2	7.4	even	3
588.3.p.c.569.1		2	21.11	odd	6
768.3.h.a.641.1		2	16.13	even	4
768.3.h.a.641.1		2	48.29	odd	4
768.3.h.a.641.2		2	16.5	even	4
768.3.h.a.641.2		2	48.5	odd	4
768.3.h.b.641.1		2	16.11	odd	4
768.3.h.b.641.1		2	48.11	even	4
768.3.h.b.641.2		2	16.3	odd	4
768.3.h.b.641.2		2	48.35	even	4
1200.3.c.c.449.1		2	20.7	even	4
1200.3.c.c.449.1		2	60.47	odd	4
1200.3.c.c.449.2		2	20.3	even	4
1200.3.c.c.449.2		2	60.23	odd	4
1200.3.l.b.401.1		1	20.19	odd	2
1200.3.l.b.401.1		1	60.59	even	2
1296.3.q.b.593.1		2	36.23	even	6
1296.3.q.b.593.1		2	36.31	odd	6
1296.3.q.b.1025.1		2	36.7	odd	6
1296.3.q.b.1025.1		2	36.11	even	6
1452.3.e.b.485.1		1	11.10	odd	2
1452.3.e.b.485.1		1	33.32	even	2