Embedded newform 4719.2.a.c.1.1

• Label: 4719.2.a.c.1.1
• Level: $4719$
• Weight: $2$
• Character: 4719.1
• Self dual: yes
• Analytic conductor: $37.681$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4719 = 3 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4719.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(37.6814047138\)
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	\(\chi\)	\(=\)	4719.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107	−0.353553	−	0.935414i	\(-0.615027\pi\)
			−0.353553	+	0.935414i	\(0.615027\pi\)
\(3\)	−1.00000	−0.577350
			\(5\)	2.00000	0.894427	0.447214	−	0.894427i	\(-0.352416\pi\)
0.447214	+	0.894427i				\(0.352416\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	0	0
			\(13\)	−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	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−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	4719.2.a.c.1.1		1
11.10	odd	2		39.2.a.a.1.1	✓	1
33.32	even	2		117.2.a.a.1.1		1
44.43	even	2		624.2.a.i.1.1		1
55.32	even	4		975.2.c.f.274.2		2
55.43	even	4		975.2.c.f.274.1		2
55.54	odd	2		975.2.a.f.1.1		1
77.76	even	2		1911.2.a.f.1.1		1
88.21	odd	2		2496.2.a.q.1.1		1
88.43	even	2		2496.2.a.e.1.1		1
99.32	even	6		1053.2.e.d.703.1		2
99.43	odd	6		1053.2.e.b.352.1		2
99.65	even	6		1053.2.e.d.352.1		2
99.76	odd	6		1053.2.e.b.703.1		2
132.131	odd	2		1872.2.a.h.1.1		1
143.10	odd	6		507.2.e.b.22.1		2
143.21	even	4		507.2.b.a.337.2		2
143.32	even	12		507.2.j.e.361.1		4
143.43	odd	6		507.2.e.b.484.1		2
143.54	even	12		507.2.j.e.316.2		4
143.76	even	12		507.2.j.e.316.1		4
143.87	odd	6		507.2.e.a.484.1		2
143.98	even	12		507.2.j.e.361.2		4
143.109	even	4		507.2.b.a.337.1		2
143.120	odd	6		507.2.e.a.22.1		2
143.142	odd	2		507.2.a.a.1.1		1
165.32	odd	4		2925.2.c.e.2224.1		2
165.98	odd	4		2925.2.c.e.2224.2		2
165.164	even	2		2925.2.a.p.1.1		1
231.230	odd	2		5733.2.a.e.1.1		1
264.131	odd	2		7488.2.a.by.1.1		1
264.197	even	2		7488.2.a.bl.1.1		1
429.164	odd	4		1521.2.b.b.1351.1		2
429.395	odd	4		1521.2.b.b.1351.2		2
429.428	even	2		1521.2.a.e.1.1		1
572.571	even	2		8112.2.a.s.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.a.1.1	✓	1	11.10	odd	2
117.2.a.a.1.1		1	33.32	even	2
507.2.a.a.1.1		1	143.142	odd	2
507.2.b.a.337.1		2	143.109	even	4
507.2.b.a.337.2		2	143.21	even	4
507.2.e.a.22.1		2	143.120	odd	6
507.2.e.a.484.1		2	143.87	odd	6
507.2.e.b.22.1		2	143.10	odd	6
507.2.e.b.484.1		2	143.43	odd	6
507.2.j.e.316.1		4	143.76	even	12
507.2.j.e.316.2		4	143.54	even	12
507.2.j.e.361.1		4	143.32	even	12
507.2.j.e.361.2		4	143.98	even	12
624.2.a.i.1.1		1	44.43	even	2
975.2.a.f.1.1		1	55.54	odd	2
975.2.c.f.274.1		2	55.43	even	4
975.2.c.f.274.2		2	55.32	even	4
1053.2.e.b.352.1		2	99.43	odd	6
1053.2.e.b.703.1		2	99.76	odd	6
1053.2.e.d.352.1		2	99.65	even	6
1053.2.e.d.703.1		2	99.32	even	6
1521.2.a.e.1.1		1	429.428	even	2
1521.2.b.b.1351.1		2	429.164	odd	4
1521.2.b.b.1351.2		2	429.395	odd	4
1872.2.a.h.1.1		1	132.131	odd	2
1911.2.a.f.1.1		1	77.76	even	2
2496.2.a.e.1.1		1	88.43	even	2
2496.2.a.q.1.1		1	88.21	odd	2
2925.2.a.p.1.1		1	165.164	even	2
2925.2.c.e.2224.1		2	165.32	odd	4
2925.2.c.e.2224.2		2	165.98	odd	4
4719.2.a.c.1.1		1	1.1	even	1	trivial
5733.2.a.e.1.1		1	231.230	odd	2
7488.2.a.bl.1.1		1	264.197	even	2
7488.2.a.by.1.1		1	264.131	odd	2
8112.2.a.s.1.1		1	572.571	even	2