Embedded newform 9075.2.a.q.1.1

• Label: 9075.2.a.q.1.1
• Level: $9075$
• Weight: $2$
• Character: 9075.1
• Self dual: yes
• Analytic conductor: $72.464$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(72.4642398343\)
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	\(\chi\)	\(=\)	9075.1

$q$-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	\(-0.384973\pi\)
			0.353553	+	0.935414i	\(0.384973\pi\)
\(3\)	1.00000	0.577350
			\(5\)	0	0
\(7\)	4.00000	1.51186				0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	0	0
			\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
−0.277350	+	0.960769i				\(0.589456\pi\)
\(17\)	−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	\(0.5\pi\)
\(23\)	−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	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−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	\(-0.450084\pi\)
			0.156174	+	0.987730i	\(0.450084\pi\)
\(43\)	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	9075.2.a.q.1.1		1
5.4	even	2		363.2.a.b.1.1		1
11.10	odd	2		825.2.a.a.1.1		1
15.14	odd	2		1089.2.a.j.1.1		1
20.19	odd	2		5808.2.a.t.1.1		1
33.32	even	2		2475.2.a.g.1.1		1
55.4	even	10		363.2.e.g.148.1		4
55.9	even	10		363.2.e.g.202.1		4
55.14	even	10		363.2.e.g.130.1		4
55.19	odd	10		363.2.e.e.130.1		4
55.24	odd	10		363.2.e.e.202.1		4
55.29	odd	10		363.2.e.e.148.1		4
55.32	even	4		825.2.c.a.199.1		2
55.39	odd	10		363.2.e.e.124.1		4
55.43	even	4		825.2.c.a.199.2		2
55.49	even	10		363.2.e.g.124.1		4
55.54	odd	2		33.2.a.a.1.1	✓	1
165.32	odd	4		2475.2.c.d.199.2		2
165.98	odd	4		2475.2.c.d.199.1		2
165.164	even	2		99.2.a.b.1.1		1
220.219	even	2		528.2.a.g.1.1		1
385.384	even	2		1617.2.a.j.1.1		1
440.109	odd	2		2112.2.a.bb.1.1		1
440.219	even	2		2112.2.a.j.1.1		1
495.164	even	6		891.2.e.g.595.1		2
495.274	odd	6		891.2.e.e.298.1		2
495.329	even	6		891.2.e.g.298.1		2
495.439	odd	6		891.2.e.e.595.1		2
660.659	odd	2		1584.2.a.o.1.1		1
715.714	odd	2		5577.2.a.a.1.1		1
935.934	odd	2		9537.2.a.m.1.1		1
1155.1154	odd	2		4851.2.a.b.1.1		1
1320.659	odd	2		6336.2.a.n.1.1		1
1320.989	even	2		6336.2.a.x.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	55.54	odd	2
99.2.a.b.1.1		1	165.164	even	2
363.2.a.b.1.1		1	5.4	even	2
363.2.e.e.124.1		4	55.39	odd	10
363.2.e.e.130.1		4	55.19	odd	10
363.2.e.e.148.1		4	55.29	odd	10
363.2.e.e.202.1		4	55.24	odd	10
363.2.e.g.124.1		4	55.49	even	10
363.2.e.g.130.1		4	55.14	even	10
363.2.e.g.148.1		4	55.4	even	10
363.2.e.g.202.1		4	55.9	even	10
528.2.a.g.1.1		1	220.219	even	2
825.2.a.a.1.1		1	11.10	odd	2
825.2.c.a.199.1		2	55.32	even	4
825.2.c.a.199.2		2	55.43	even	4
891.2.e.e.298.1		2	495.274	odd	6
891.2.e.e.595.1		2	495.439	odd	6
891.2.e.g.298.1		2	495.329	even	6
891.2.e.g.595.1		2	495.164	even	6
1089.2.a.j.1.1		1	15.14	odd	2
1584.2.a.o.1.1		1	660.659	odd	2
1617.2.a.j.1.1		1	385.384	even	2
2112.2.a.j.1.1		1	440.219	even	2
2112.2.a.bb.1.1		1	440.109	odd	2
2475.2.a.g.1.1		1	33.32	even	2
2475.2.c.d.199.1		2	165.98	odd	4
2475.2.c.d.199.2		2	165.32	odd	4
4851.2.a.b.1.1		1	1155.1154	odd	2
5577.2.a.a.1.1		1	715.714	odd	2
5808.2.a.t.1.1		1	20.19	odd	2
6336.2.a.n.1.1		1	1320.659	odd	2
6336.2.a.x.1.1		1	1320.989	even	2
9075.2.a.q.1.1		1	1.1	even	1	trivial
9537.2.a.m.1.1		1	935.934	odd	2