Embedded newform 4275.2.a.i.1.1

• Label: 4275.2.a.i.1.1
• Level: $4275$
• Weight: $2$
• Character: 4275.1
• Self dual: yes
• Analytic conductor: $34.136$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(34.1360468641\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4275.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{4} +1.00000 q^{7} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(3\)	0	0
			\(5\)	0	0
\(7\)	1.00000	0.377964				0.188982	−	0.981981i	\(-0.439481\pi\)
			0.188982	+	0.981981i	\(0.439481\pi\)
\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
			−0.452267	+	0.891883i	\(0.649385\pi\)
\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
			0.554700	+	0.832050i	\(0.312833\pi\)
\(17\)	−3.00000	−0.727607	−0.363803	−	0.931476i	\(-0.618522\pi\)
			−0.363803	+	0.931476i	\(0.618522\pi\)
\(19\)	1.00000	0.229416
			\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−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	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	1.00000	0.152499	0.0762493	−	0.997089i	\(-0.475706\pi\)
			0.0762493	+	0.997089i	\(0.475706\pi\)
\(47\)	−3.00000	−0.437595	−0.218797	−	0.975770i	\(-0.570213\pi\)
			−0.218797	+	0.975770i	\(0.570213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.i.1.1		1
3.2	odd	2		475.2.a.b.1.1		1
5.4	even	2		171.2.a.b.1.1		1
12.11	even	2		7600.2.a.c.1.1		1
15.2	even	4		475.2.b.a.324.1		2
15.8	even	4		475.2.b.a.324.2		2
15.14	odd	2		19.2.a.a.1.1	✓	1
20.19	odd	2		2736.2.a.c.1.1		1
35.34	odd	2		8379.2.a.j.1.1		1
57.56	even	2		9025.2.a.d.1.1		1
60.59	even	2		304.2.a.f.1.1		1
95.94	odd	2		3249.2.a.d.1.1		1
105.44	odd	6		931.2.f.c.704.1		2
105.59	even	6		931.2.f.b.324.1		2
105.74	odd	6		931.2.f.c.324.1		2
105.89	even	6		931.2.f.b.704.1		2
105.104	even	2		931.2.a.a.1.1		1
120.29	odd	2		1216.2.a.o.1.1		1
120.59	even	2		1216.2.a.b.1.1		1
165.164	even	2		2299.2.a.b.1.1		1
195.194	odd	2		3211.2.a.a.1.1		1
255.254	odd	2		5491.2.a.b.1.1		1
285.14	even	18		361.2.e.e.234.1		6
285.29	even	18		361.2.e.e.62.1		6
285.44	odd	18		361.2.e.d.245.1		6
285.59	even	18		361.2.e.e.99.1		6
285.74	odd	18		361.2.e.d.99.1		6
285.89	even	18		361.2.e.e.245.1		6
285.104	odd	18		361.2.e.d.62.1		6
285.119	odd	18		361.2.e.d.234.1		6
285.149	odd	18		361.2.e.d.28.1		6
285.164	even	6		361.2.c.a.68.1		2
285.179	even	6		361.2.c.a.292.1		2
285.194	odd	18		361.2.e.d.54.1		6
285.224	even	18		361.2.e.e.54.1		6
285.239	odd	6		361.2.c.c.292.1		2
285.254	odd	6		361.2.c.c.68.1		2
285.269	even	18		361.2.e.e.28.1		6
285.284	even	2		361.2.a.b.1.1		1
1140.1139	odd	2		5776.2.a.c.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.a.a.1.1	✓	1	15.14	odd	2
171.2.a.b.1.1		1	5.4	even	2
304.2.a.f.1.1		1	60.59	even	2
361.2.a.b.1.1		1	285.284	even	2
361.2.c.a.68.1		2	285.164	even	6
361.2.c.a.292.1		2	285.179	even	6
361.2.c.c.68.1		2	285.254	odd	6
361.2.c.c.292.1		2	285.239	odd	6
361.2.e.d.28.1		6	285.149	odd	18
361.2.e.d.54.1		6	285.194	odd	18
361.2.e.d.62.1		6	285.104	odd	18
361.2.e.d.99.1		6	285.74	odd	18
361.2.e.d.234.1		6	285.119	odd	18
361.2.e.d.245.1		6	285.44	odd	18
361.2.e.e.28.1		6	285.269	even	18
361.2.e.e.54.1		6	285.224	even	18
361.2.e.e.62.1		6	285.29	even	18
361.2.e.e.99.1		6	285.59	even	18
361.2.e.e.234.1		6	285.14	even	18
361.2.e.e.245.1		6	285.89	even	18
475.2.a.b.1.1		1	3.2	odd	2
475.2.b.a.324.1		2	15.2	even	4
475.2.b.a.324.2		2	15.8	even	4
931.2.a.a.1.1		1	105.104	even	2
931.2.f.b.324.1		2	105.59	even	6
931.2.f.b.704.1		2	105.89	even	6
931.2.f.c.324.1		2	105.74	odd	6
931.2.f.c.704.1		2	105.44	odd	6
1216.2.a.b.1.1		1	120.59	even	2
1216.2.a.o.1.1		1	120.29	odd	2
2299.2.a.b.1.1		1	165.164	even	2
2736.2.a.c.1.1		1	20.19	odd	2
3211.2.a.a.1.1		1	195.194	odd	2
3249.2.a.d.1.1		1	95.94	odd	2
4275.2.a.i.1.1		1	1.1	even	1	trivial
5491.2.a.b.1.1		1	255.254	odd	2
5776.2.a.c.1.1		1	1140.1139	odd	2
7600.2.a.c.1.1		1	12.11	even	2
8379.2.a.j.1.1		1	35.34	odd	2
9025.2.a.d.1.1		1	57.56	even	2