Embedded newform 825.2.a.a.1.1

• Label: 825.2.a.a.1.1
• Level: $825$
• Weight: $2$
• Character: 825.1
• Self dual: yes
• Analytic conductor: $6.588$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	825.1

$q$-expansion

\(f(q)\)

\(=\)

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

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
			\(5\)	0	0
\(7\)	−4.00000	−1.51186				−0.755929	−	0.654654i	\(-0.772814\pi\)
			−0.755929	+	0.654654i	\(0.772814\pi\)
\(11\)	1.00000	0.301511
			\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
0.277350	+	0.960769i				\(0.410544\pi\)
\(17\)	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	\(0.5\pi\)
\(23\)	−8.00000	−1.66812	−0.834058	−	0.551677i	\(-0.813988\pi\)
			−0.834058	+	0.551677i	\(0.813988\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−2.00000	−0.312348	−0.156174	−	0.987730i	\(-0.549916\pi\)
			−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.a.a.1.1		1
3.2	odd	2		2475.2.a.g.1.1		1
5.2	odd	4		825.2.c.a.199.1		2
5.3	odd	4		825.2.c.a.199.2		2
5.4	even	2		33.2.a.a.1.1	✓	1
11.10	odd	2		9075.2.a.q.1.1		1
15.2	even	4		2475.2.c.d.199.2		2
15.8	even	4		2475.2.c.d.199.1		2
15.14	odd	2		99.2.a.b.1.1		1
20.19	odd	2		528.2.a.g.1.1		1
35.34	odd	2		1617.2.a.j.1.1		1
40.19	odd	2		2112.2.a.j.1.1		1
40.29	even	2		2112.2.a.bb.1.1		1
45.4	even	6		891.2.e.e.298.1		2
45.14	odd	6		891.2.e.g.298.1		2
45.29	odd	6		891.2.e.g.595.1		2
45.34	even	6		891.2.e.e.595.1		2
55.4	even	10		363.2.e.e.148.1		4
55.9	even	10		363.2.e.e.202.1		4
55.14	even	10		363.2.e.e.130.1		4
55.19	odd	10		363.2.e.g.130.1		4
55.24	odd	10		363.2.e.g.202.1		4
55.29	odd	10		363.2.e.g.148.1		4
55.39	odd	10		363.2.e.g.124.1		4
55.49	even	10		363.2.e.e.124.1		4
55.54	odd	2		363.2.a.b.1.1		1
60.59	even	2		1584.2.a.o.1.1		1
65.64	even	2		5577.2.a.a.1.1		1
85.84	even	2		9537.2.a.m.1.1		1
105.104	even	2		4851.2.a.b.1.1		1
120.29	odd	2		6336.2.a.x.1.1		1
120.59	even	2		6336.2.a.n.1.1		1
165.164	even	2		1089.2.a.j.1.1		1
220.219	even	2		5808.2.a.t.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	5.4	even	2
99.2.a.b.1.1		1	15.14	odd	2
363.2.a.b.1.1		1	55.54	odd	2
363.2.e.e.124.1		4	55.49	even	10
363.2.e.e.130.1		4	55.14	even	10
363.2.e.e.148.1		4	55.4	even	10
363.2.e.e.202.1		4	55.9	even	10
363.2.e.g.124.1		4	55.39	odd	10
363.2.e.g.130.1		4	55.19	odd	10
363.2.e.g.148.1		4	55.29	odd	10
363.2.e.g.202.1		4	55.24	odd	10
528.2.a.g.1.1		1	20.19	odd	2
825.2.a.a.1.1		1	1.1	even	1	trivial
825.2.c.a.199.1		2	5.2	odd	4
825.2.c.a.199.2		2	5.3	odd	4
891.2.e.e.298.1		2	45.4	even	6
891.2.e.e.595.1		2	45.34	even	6
891.2.e.g.298.1		2	45.14	odd	6
891.2.e.g.595.1		2	45.29	odd	6
1089.2.a.j.1.1		1	165.164	even	2
1584.2.a.o.1.1		1	60.59	even	2
1617.2.a.j.1.1		1	35.34	odd	2
2112.2.a.j.1.1		1	40.19	odd	2
2112.2.a.bb.1.1		1	40.29	even	2
2475.2.a.g.1.1		1	3.2	odd	2
2475.2.c.d.199.1		2	15.8	even	4
2475.2.c.d.199.2		2	15.2	even	4
4851.2.a.b.1.1		1	105.104	even	2
5577.2.a.a.1.1		1	65.64	even	2
5808.2.a.t.1.1		1	220.219	even	2
6336.2.a.n.1.1		1	120.59	even	2
6336.2.a.x.1.1		1	120.29	odd	2
9075.2.a.q.1.1		1	11.10	odd	2
9537.2.a.m.1.1		1	85.84	even	2