Embedded newform 507.2.a.a.1.1

• Label: 507.2.a.a.1.1
• Level: $507$
• Weight: $2$
• Character: 507.1
• Self dual: yes
• Analytic conductor: $4.048$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(1\)
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	\(\chi\)	\(=\)	507.1

$q$-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	\(-0.647584\pi\)
−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	\(-0.706037\pi\)
			−0.603023	+	0.797724i	\(0.706037\pi\)
\(13\)	0	0
			\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
0.242536	+	0.970143i				\(0.422021\pi\)
\(19\)	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	\(-0.878881\pi\)
			−0.928477	+	0.371391i	\(0.878881\pi\)
\(31\)	−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	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	−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	\(-0.867803\pi\)
			−0.914991	+	0.403473i	\(0.867803\pi\)
\(47\)	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	507.2.a.a.1.1		1
3.2	odd	2		1521.2.a.e.1.1		1
4.3	odd	2		8112.2.a.s.1.1		1
13.2	odd	12		507.2.j.e.316.1		4
13.3	even	3		507.2.e.b.22.1		2
13.4	even	6		507.2.e.a.484.1		2
13.5	odd	4		507.2.b.a.337.2		2
13.6	odd	12		507.2.j.e.361.2		4
13.7	odd	12		507.2.j.e.361.1		4
13.8	odd	4		507.2.b.a.337.1		2
13.9	even	3		507.2.e.b.484.1		2
13.10	even	6		507.2.e.a.22.1		2
13.11	odd	12		507.2.j.e.316.2		4
13.12	even	2		39.2.a.a.1.1	✓	1
39.5	even	4		1521.2.b.b.1351.1		2
39.8	even	4		1521.2.b.b.1351.2		2
39.38	odd	2		117.2.a.a.1.1		1
52.51	odd	2		624.2.a.i.1.1		1
65.12	odd	4		975.2.c.f.274.2		2
65.38	odd	4		975.2.c.f.274.1		2
65.64	even	2		975.2.a.f.1.1		1
91.90	odd	2		1911.2.a.f.1.1		1
104.51	odd	2		2496.2.a.e.1.1		1
104.77	even	2		2496.2.a.q.1.1		1
117.25	even	6		1053.2.e.b.352.1		2
117.38	odd	6		1053.2.e.d.352.1		2
117.77	odd	6		1053.2.e.d.703.1		2
117.103	even	6		1053.2.e.b.703.1		2
143.142	odd	2		4719.2.a.c.1.1		1
156.155	even	2		1872.2.a.h.1.1		1
195.38	even	4		2925.2.c.e.2224.2		2
195.77	even	4		2925.2.c.e.2224.1		2
195.194	odd	2		2925.2.a.p.1.1		1
273.272	even	2		5733.2.a.e.1.1		1
312.77	odd	2		7488.2.a.bl.1.1		1
312.155	even	2		7488.2.a.by.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.a.1.1	✓	1	13.12	even	2
117.2.a.a.1.1		1	39.38	odd	2
507.2.a.a.1.1		1	1.1	even	1	trivial
507.2.b.a.337.1		2	13.8	odd	4
507.2.b.a.337.2		2	13.5	odd	4
507.2.e.a.22.1		2	13.10	even	6
507.2.e.a.484.1		2	13.4	even	6
507.2.e.b.22.1		2	13.3	even	3
507.2.e.b.484.1		2	13.9	even	3
507.2.j.e.316.1		4	13.2	odd	12
507.2.j.e.316.2		4	13.11	odd	12
507.2.j.e.361.1		4	13.7	odd	12
507.2.j.e.361.2		4	13.6	odd	12
624.2.a.i.1.1		1	52.51	odd	2
975.2.a.f.1.1		1	65.64	even	2
975.2.c.f.274.1		2	65.38	odd	4
975.2.c.f.274.2		2	65.12	odd	4
1053.2.e.b.352.1		2	117.25	even	6
1053.2.e.b.703.1		2	117.103	even	6
1053.2.e.d.352.1		2	117.38	odd	6
1053.2.e.d.703.1		2	117.77	odd	6
1521.2.a.e.1.1		1	3.2	odd	2
1521.2.b.b.1351.1		2	39.5	even	4
1521.2.b.b.1351.2		2	39.8	even	4
1872.2.a.h.1.1		1	156.155	even	2
1911.2.a.f.1.1		1	91.90	odd	2
2496.2.a.e.1.1		1	104.51	odd	2
2496.2.a.q.1.1		1	104.77	even	2
2925.2.a.p.1.1		1	195.194	odd	2
2925.2.c.e.2224.1		2	195.77	even	4
2925.2.c.e.2224.2		2	195.38	even	4
4719.2.a.c.1.1		1	143.142	odd	2
5733.2.a.e.1.1		1	273.272	even	2
7488.2.a.bl.1.1		1	312.77	odd	2
7488.2.a.by.1.1		1	312.155	even	2
8112.2.a.s.1.1		1	4.3	odd	2