Embedded newform 1872.2.a.f.1.1

• Label: 1872.2.a.f.1.1
• Level: $1872$
• Weight: $2$
• Character: 1872.1
• Self dual: yes
• Analytic conductor: $14.948$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.9479952584\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1872.1

$q$-expansion

\(f(q)\)

\(=\)

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

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
\(5\)	−2.00000	−0.894427				−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	2.00000	0.755929	0.377964	−	0.925820i	\(-0.376624\pi\)
			0.377964	+	0.925820i	\(0.376624\pi\)
\(11\)	−2.00000	−0.603023	−0.301511	−	0.953463i	\(-0.597491\pi\)
			−0.301511	+	0.953463i	\(0.597491\pi\)
\(13\)	−1.00000	−0.277350
			\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
−0.727607	+	0.685994i				\(0.759367\pi\)
\(19\)	6.00000	1.37649	0.688247	−	0.725476i	\(-0.258380\pi\)
			0.688247	+	0.725476i	\(0.258380\pi\)
\(23\)	8.00000	1.66812	0.834058	−	0.551677i	\(-0.186012\pi\)
			0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	−10.0000	−1.79605	−0.898027	−	0.439941i	\(-0.854999\pi\)
			−0.898027	+	0.439941i	\(0.854999\pi\)
\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	−2.00000	−0.291730	−0.145865	−	0.989305i	\(-0.546597\pi\)
			−0.145865	+	0.989305i	\(0.546597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.a.f.1.1		1
3.2	odd	2		208.2.a.c.1.1		1
4.3	odd	2		468.2.a.b.1.1		1
8.3	odd	2		7488.2.a.bn.1.1		1
8.5	even	2		7488.2.a.bw.1.1		1
12.11	even	2		52.2.a.a.1.1	✓	1
15.14	odd	2		5200.2.a.q.1.1		1
24.5	odd	2		832.2.a.f.1.1		1
24.11	even	2		832.2.a.e.1.1		1
36.7	odd	6		4212.2.i.i.1405.1		2
36.11	even	6		4212.2.i.d.1405.1		2
36.23	even	6		4212.2.i.d.2809.1		2
36.31	odd	6		4212.2.i.i.2809.1		2
39.5	even	4		2704.2.f.f.337.1		2
39.8	even	4		2704.2.f.f.337.2		2
39.38	odd	2		2704.2.a.g.1.1		1
48.5	odd	4		3328.2.b.e.1665.2		2
48.11	even	4		3328.2.b.q.1665.2		2
48.29	odd	4		3328.2.b.e.1665.1		2
48.35	even	4		3328.2.b.q.1665.1		2
52.31	even	4		6084.2.b.m.4393.2		2
52.47	even	4		6084.2.b.m.4393.1		2
52.51	odd	2		6084.2.a.m.1.1		1
60.23	odd	4		1300.2.c.c.1249.2		2
60.47	odd	4		1300.2.c.c.1249.1		2
60.59	even	2		1300.2.a.d.1.1		1
84.11	even	6		2548.2.j.e.1353.1		2
84.23	even	6		2548.2.j.e.1145.1		2
84.47	odd	6		2548.2.j.f.1145.1		2
84.59	odd	6		2548.2.j.f.1353.1		2
84.83	odd	2		2548.2.a.e.1.1		1
132.131	odd	2		6292.2.a.g.1.1		1
156.11	odd	12		676.2.h.c.485.2		4
156.23	even	6		676.2.e.b.529.1		2
156.35	even	6		676.2.e.c.653.1		2
156.47	odd	4		676.2.d.c.337.2		2
156.59	odd	12		676.2.h.c.361.2		4
156.71	odd	12		676.2.h.c.361.1		4
156.83	odd	4		676.2.d.c.337.1		2
156.95	even	6		676.2.e.b.653.1		2
156.107	even	6		676.2.e.c.529.1		2
156.119	odd	12		676.2.h.c.485.1		4
156.155	even	2		676.2.a.c.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.a.a.1.1	✓	1	12.11	even	2
208.2.a.c.1.1		1	3.2	odd	2
468.2.a.b.1.1		1	4.3	odd	2
676.2.a.c.1.1		1	156.155	even	2
676.2.d.c.337.1		2	156.83	odd	4
676.2.d.c.337.2		2	156.47	odd	4
676.2.e.b.529.1		2	156.23	even	6
676.2.e.b.653.1		2	156.95	even	6
676.2.e.c.529.1		2	156.107	even	6
676.2.e.c.653.1		2	156.35	even	6
676.2.h.c.361.1		4	156.71	odd	12
676.2.h.c.361.2		4	156.59	odd	12
676.2.h.c.485.1		4	156.119	odd	12
676.2.h.c.485.2		4	156.11	odd	12
832.2.a.e.1.1		1	24.11	even	2
832.2.a.f.1.1		1	24.5	odd	2
1300.2.a.d.1.1		1	60.59	even	2
1300.2.c.c.1249.1		2	60.47	odd	4
1300.2.c.c.1249.2		2	60.23	odd	4
1872.2.a.f.1.1		1	1.1	even	1	trivial
2548.2.a.e.1.1		1	84.83	odd	2
2548.2.j.e.1145.1		2	84.23	even	6
2548.2.j.e.1353.1		2	84.11	even	6
2548.2.j.f.1145.1		2	84.47	odd	6
2548.2.j.f.1353.1		2	84.59	odd	6
2704.2.a.g.1.1		1	39.38	odd	2
2704.2.f.f.337.1		2	39.5	even	4
2704.2.f.f.337.2		2	39.8	even	4
3328.2.b.e.1665.1		2	48.29	odd	4
3328.2.b.e.1665.2		2	48.5	odd	4
3328.2.b.q.1665.1		2	48.35	even	4
3328.2.b.q.1665.2		2	48.11	even	4
4212.2.i.d.1405.1		2	36.11	even	6
4212.2.i.d.2809.1		2	36.23	even	6
4212.2.i.i.1405.1		2	36.7	odd	6
4212.2.i.i.2809.1		2	36.31	odd	6
5200.2.a.q.1.1		1	15.14	odd	2
6084.2.a.m.1.1		1	52.51	odd	2
6084.2.b.m.4393.1		2	52.47	even	4
6084.2.b.m.4393.2		2	52.31	even	4
6292.2.a.g.1.1		1	132.131	odd	2
7488.2.a.bn.1.1		1	8.3	odd	2
7488.2.a.bw.1.1		1	8.5	even	2