Embedded newform 1872.2.a.m.1.1

• Label: 1872.2.a.m.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 26)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{11} -1.00000 q^{13} +3.00000 q^{17} -6.00000 q^{19} -4.00000 q^{23} -4.00000 q^{25} -2.00000 q^{29} -4.00000 q^{31} -1.00000 q^{35} +3.00000 q^{37} +5.00000 q^{43} +13.0000 q^{47} -6.00000 q^{49} -12.0000 q^{53} -2.00000 q^{55} -10.0000 q^{59} -8.00000 q^{61} -1.00000 q^{65} +2.00000 q^{67} -5.00000 q^{71} -10.0000 q^{73} +2.00000 q^{77} +4.00000 q^{79} +3.00000 q^{85} -6.00000 q^{89} +1.00000 q^{91} -6.00000 q^{95} +14.0000 q^{97} +O(q^{100})\)

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
\(5\)	1.00000	0.447214				0.223607	−	0.974679i	\(-0.428217\pi\)
			0.223607	+	0.974679i	\(0.428217\pi\)
\(7\)	−1.00000	−0.377964	−0.188982	−	0.981981i	\(-0.560519\pi\)
			−0.188982	+	0.981981i	\(0.560519\pi\)
\(11\)	−2.00000	−0.603023	−0.301511	−	0.953463i	\(-0.597491\pi\)
			−0.301511	+	0.953463i	\(0.597491\pi\)
\(13\)	−1.00000	−0.277350
			\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
0.363803	+	0.931476i				\(0.381478\pi\)
\(19\)	−6.00000	−1.37649	−0.688247	−	0.725476i	\(-0.741620\pi\)
			−0.688247	+	0.725476i	\(0.741620\pi\)
\(23\)	−4.00000	−0.834058	−0.417029	−	0.908893i	\(-0.636929\pi\)
			−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	3.00000	0.493197	0.246598	−	0.969118i	\(-0.420687\pi\)
			0.246598	+	0.969118i	\(0.420687\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	5.00000	0.762493	0.381246	−	0.924473i	\(-0.375495\pi\)
			0.381246	+	0.924473i	\(0.375495\pi\)
\(47\)	13.0000	1.89624	0.948122	−	0.317905i	\(-0.102979\pi\)
			0.948122	+	0.317905i	\(0.102979\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.a.m.1.1		1
3.2	odd	2		208.2.a.d.1.1		1
4.3	odd	2		234.2.a.b.1.1		1
8.3	odd	2		7488.2.a.w.1.1		1
8.5	even	2		7488.2.a.v.1.1		1
12.11	even	2		26.2.a.b.1.1	✓	1
15.14	odd	2		5200.2.a.c.1.1		1
20.3	even	4		5850.2.e.v.5149.2		2
20.7	even	4		5850.2.e.v.5149.1		2
20.19	odd	2		5850.2.a.bn.1.1		1
24.5	odd	2		832.2.a.a.1.1		1
24.11	even	2		832.2.a.j.1.1		1
36.7	odd	6		2106.2.e.t.1405.1		2
36.11	even	6		2106.2.e.h.1405.1		2
36.23	even	6		2106.2.e.h.703.1		2
36.31	odd	6		2106.2.e.t.703.1		2
39.5	even	4		2704.2.f.j.337.2		2
39.8	even	4		2704.2.f.j.337.1		2
39.38	odd	2		2704.2.a.n.1.1		1
48.5	odd	4		3328.2.b.k.1665.1		2
48.11	even	4		3328.2.b.g.1665.2		2
48.29	odd	4		3328.2.b.k.1665.2		2
48.35	even	4		3328.2.b.g.1665.1		2
52.31	even	4		3042.2.b.f.1351.2		2
52.47	even	4		3042.2.b.f.1351.1		2
52.51	odd	2		3042.2.a.l.1.1		1
60.23	odd	4		650.2.b.a.599.1		2
60.47	odd	4		650.2.b.a.599.2		2
60.59	even	2		650.2.a.g.1.1		1
84.11	even	6		1274.2.f.l.79.1		2
84.23	even	6		1274.2.f.l.1145.1		2
84.47	odd	6		1274.2.f.a.1145.1		2
84.59	odd	6		1274.2.f.a.79.1		2
84.83	odd	2		1274.2.a.o.1.1		1
132.131	odd	2		3146.2.a.a.1.1		1
156.11	odd	12		338.2.e.d.147.1		4
156.23	even	6		338.2.c.g.191.1		2
156.35	even	6		338.2.c.c.315.1		2
156.47	odd	4		338.2.b.a.337.2		2
156.59	odd	12		338.2.e.d.23.2		4
156.71	odd	12		338.2.e.d.23.1		4
156.83	odd	4		338.2.b.a.337.1		2
156.95	even	6		338.2.c.g.315.1		2
156.107	even	6		338.2.c.c.191.1		2
156.119	odd	12		338.2.e.d.147.2		4
156.155	even	2		338.2.a.a.1.1		1
204.203	even	2		7514.2.a.i.1.1		1
228.227	odd	2		9386.2.a.f.1.1		1
780.779	even	2		8450.2.a.y.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.b.1.1	✓	1	12.11	even	2
208.2.a.d.1.1		1	3.2	odd	2
234.2.a.b.1.1		1	4.3	odd	2
338.2.a.a.1.1		1	156.155	even	2
338.2.b.a.337.1		2	156.83	odd	4
338.2.b.a.337.2		2	156.47	odd	4
338.2.c.c.191.1		2	156.107	even	6
338.2.c.c.315.1		2	156.35	even	6
338.2.c.g.191.1		2	156.23	even	6
338.2.c.g.315.1		2	156.95	even	6
338.2.e.d.23.1		4	156.71	odd	12
338.2.e.d.23.2		4	156.59	odd	12
338.2.e.d.147.1		4	156.11	odd	12
338.2.e.d.147.2		4	156.119	odd	12
650.2.a.g.1.1		1	60.59	even	2
650.2.b.a.599.1		2	60.23	odd	4
650.2.b.a.599.2		2	60.47	odd	4
832.2.a.a.1.1		1	24.5	odd	2
832.2.a.j.1.1		1	24.11	even	2
1274.2.a.o.1.1		1	84.83	odd	2
1274.2.f.a.79.1		2	84.59	odd	6
1274.2.f.a.1145.1		2	84.47	odd	6
1274.2.f.l.79.1		2	84.11	even	6
1274.2.f.l.1145.1		2	84.23	even	6
1872.2.a.m.1.1		1	1.1	even	1	trivial
2106.2.e.h.703.1		2	36.23	even	6
2106.2.e.h.1405.1		2	36.11	even	6
2106.2.e.t.703.1		2	36.31	odd	6
2106.2.e.t.1405.1		2	36.7	odd	6
2704.2.a.n.1.1		1	39.38	odd	2
2704.2.f.j.337.1		2	39.8	even	4
2704.2.f.j.337.2		2	39.5	even	4
3042.2.a.l.1.1		1	52.51	odd	2
3042.2.b.f.1351.1		2	52.47	even	4
3042.2.b.f.1351.2		2	52.31	even	4
3146.2.a.a.1.1		1	132.131	odd	2
3328.2.b.g.1665.1		2	48.35	even	4
3328.2.b.g.1665.2		2	48.11	even	4
3328.2.b.k.1665.1		2	48.5	odd	4
3328.2.b.k.1665.2		2	48.29	odd	4
5200.2.a.c.1.1		1	15.14	odd	2
5850.2.a.bn.1.1		1	20.19	odd	2
5850.2.e.v.5149.1		2	20.7	even	4
5850.2.e.v.5149.2		2	20.3	even	4
7488.2.a.v.1.1		1	8.5	even	2
7488.2.a.w.1.1		1	8.3	odd	2
7514.2.a.i.1.1		1	204.203	even	2
8450.2.a.y.1.1		1	780.779	even	2
9386.2.a.f.1.1		1	228.227	odd	2