Embedded newform 1872.2.a.h.1.1

• Label: 1872.2.a.h.1.1
• Level: $1872$
• Weight: $2$
• Character: 1872.1
• Self dual: yes
• Analytic conductor: $14.948$
• Analytic rank: $0$
• 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:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{5} +4.00000 q^{7} +4.00000 q^{11} +1.00000 q^{13} -2.00000 q^{17} -1.00000 q^{25} +10.0000 q^{29} -4.00000 q^{31} -8.00000 q^{35} -2.00000 q^{37} -6.00000 q^{41} +12.0000 q^{43} +9.00000 q^{49} -6.00000 q^{53} -8.00000 q^{55} +12.0000 q^{59} -2.00000 q^{61} -2.00000 q^{65} +8.00000 q^{67} +2.00000 q^{73} +16.0000 q^{77} -8.00000 q^{79} +4.00000 q^{83} +4.00000 q^{85} +2.00000 q^{89} +4.00000 q^{91} +10.0000 q^{97} +O(q^{100})\)

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
\(5\)	−2.00000	−0.894427				−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
			0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)	1.00000	0.277350
			\(17\)	−2.00000	−0.485071	−0.242536	−	0.970143i	\(-0.577979\pi\)
−0.242536	+	0.970143i				\(0.577979\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	10.0000	1.85695	0.928477	−	0.371391i	\(-0.121119\pi\)
			0.928477	+	0.371391i	\(0.121119\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	12.0000	1.82998	0.914991	−	0.403473i	\(-0.132197\pi\)
			0.914991	+	0.403473i	\(0.132197\pi\)
\(47\)	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	1872.2.a.h.1.1		1
3.2	odd	2		624.2.a.i.1.1		1
4.3	odd	2		117.2.a.a.1.1		1
8.3	odd	2		7488.2.a.bl.1.1		1
8.5	even	2		7488.2.a.by.1.1		1
12.11	even	2		39.2.a.a.1.1	✓	1
20.3	even	4		2925.2.c.e.2224.2		2
20.7	even	4		2925.2.c.e.2224.1		2
20.19	odd	2		2925.2.a.p.1.1		1
24.5	odd	2		2496.2.a.e.1.1		1
24.11	even	2		2496.2.a.q.1.1		1
28.27	even	2		5733.2.a.e.1.1		1
36.7	odd	6		1053.2.e.d.352.1		2
36.11	even	6		1053.2.e.b.352.1		2
36.23	even	6		1053.2.e.b.703.1		2
36.31	odd	6		1053.2.e.d.703.1		2
39.38	odd	2		8112.2.a.s.1.1		1
52.31	even	4		1521.2.b.b.1351.2		2
52.47	even	4		1521.2.b.b.1351.1		2
52.51	odd	2		1521.2.a.e.1.1		1
60.23	odd	4		975.2.c.f.274.1		2
60.47	odd	4		975.2.c.f.274.2		2
60.59	even	2		975.2.a.f.1.1		1
84.83	odd	2		1911.2.a.f.1.1		1
132.131	odd	2		4719.2.a.c.1.1		1
156.11	odd	12		507.2.j.e.316.1		4
156.23	even	6		507.2.e.b.22.1		2
156.35	even	6		507.2.e.a.484.1		2
156.47	odd	4		507.2.b.a.337.2		2
156.59	odd	12		507.2.j.e.361.2		4
156.71	odd	12		507.2.j.e.361.1		4
156.83	odd	4		507.2.b.a.337.1		2
156.95	even	6		507.2.e.b.484.1		2
156.107	even	6		507.2.e.a.22.1		2
156.119	odd	12		507.2.j.e.316.2		4
156.155	even	2		507.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.a.1.1	✓	1	12.11	even	2
117.2.a.a.1.1		1	4.3	odd	2
507.2.a.a.1.1		1	156.155	even	2
507.2.b.a.337.1		2	156.83	odd	4
507.2.b.a.337.2		2	156.47	odd	4
507.2.e.a.22.1		2	156.107	even	6
507.2.e.a.484.1		2	156.35	even	6
507.2.e.b.22.1		2	156.23	even	6
507.2.e.b.484.1		2	156.95	even	6
507.2.j.e.316.1		4	156.11	odd	12
507.2.j.e.316.2		4	156.119	odd	12
507.2.j.e.361.1		4	156.71	odd	12
507.2.j.e.361.2		4	156.59	odd	12
624.2.a.i.1.1		1	3.2	odd	2
975.2.a.f.1.1		1	60.59	even	2
975.2.c.f.274.1		2	60.23	odd	4
975.2.c.f.274.2		2	60.47	odd	4
1053.2.e.b.352.1		2	36.11	even	6
1053.2.e.b.703.1		2	36.23	even	6
1053.2.e.d.352.1		2	36.7	odd	6
1053.2.e.d.703.1		2	36.31	odd	6
1521.2.a.e.1.1		1	52.51	odd	2
1521.2.b.b.1351.1		2	52.47	even	4
1521.2.b.b.1351.2		2	52.31	even	4
1872.2.a.h.1.1		1	1.1	even	1	trivial
1911.2.a.f.1.1		1	84.83	odd	2
2496.2.a.e.1.1		1	24.5	odd	2
2496.2.a.q.1.1		1	24.11	even	2
2925.2.a.p.1.1		1	20.19	odd	2
2925.2.c.e.2224.1		2	20.7	even	4
2925.2.c.e.2224.2		2	20.3	even	4
4719.2.a.c.1.1		1	132.131	odd	2
5733.2.a.e.1.1		1	28.27	even	2
7488.2.a.bl.1.1		1	8.3	odd	2
7488.2.a.by.1.1		1	8.5	even	2
8112.2.a.s.1.1		1	39.38	odd	2