Embedded newform 3528.2.a.j.1.1

• Label: 3528.2.a.j.1.1
• Level: $3528$
• Weight: $2$
• Character: 3528.1
• Self dual: yes
• Analytic conductor: $28.171$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(28.1712218331\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	3528.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{5} +O(q^{10})\)

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
			\(3\)	0	0
\(5\)	−1.00000	−0.447214				−0.223607	−	0.974679i	\(-0.571783\pi\)
			−0.223607	+	0.974679i	\(0.571783\pi\)
\(7\)	0	0
			\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
−0.452267	+	0.891883i				\(0.649385\pi\)
\(13\)	6.00000	1.66410	0.832050	−	0.554700i	\(-0.187167\pi\)
			0.832050	+	0.554700i	\(0.187167\pi\)
\(17\)	−5.00000	−1.21268	−0.606339	−	0.795206i	\(-0.707363\pi\)
			−0.606339	+	0.795206i	\(0.707363\pi\)
\(19\)	−1.00000	−0.229416	−0.114708	−	0.993399i	\(-0.536593\pi\)
			−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)	7.00000	1.45960	0.729800	−	0.683660i	\(-0.239613\pi\)
			0.729800	+	0.683660i	\(0.239613\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	5.00000	0.898027	0.449013	−	0.893525i	\(-0.351776\pi\)
			0.449013	+	0.893525i	\(0.351776\pi\)
\(37\)	3.00000	0.493197	0.246598	−	0.969118i	\(-0.420687\pi\)
			0.246598	+	0.969118i	\(0.420687\pi\)
\(41\)	−2.00000	−0.312348	−0.156174	−	0.987730i	\(-0.549916\pi\)
			−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	5.00000	0.729325	0.364662	−	0.931140i	\(-0.381184\pi\)
			0.364662	+	0.931140i	\(0.381184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.a.j.1.1		1
3.2	odd	2		392.2.a.e.1.1		1
4.3	odd	2		7056.2.a.u.1.1		1
7.2	even	3		3528.2.s.q.361.1		2
7.3	odd	6		504.2.s.c.289.1		2
7.4	even	3		3528.2.s.q.3313.1		2
7.5	odd	6		504.2.s.c.361.1		2
7.6	odd	2		3528.2.a.p.1.1		1
12.11	even	2		784.2.a.c.1.1		1
15.14	odd	2		9800.2.a.s.1.1		1
21.2	odd	6		392.2.i.b.361.1		2
21.5	even	6		56.2.i.b.25.1	yes	2
21.11	odd	6		392.2.i.b.177.1		2
21.17	even	6		56.2.i.b.9.1	✓	2
21.20	even	2		392.2.a.c.1.1		1
24.5	odd	2		3136.2.a.i.1.1		1
24.11	even	2		3136.2.a.t.1.1		1
28.3	even	6		1008.2.s.g.289.1		2
28.19	even	6		1008.2.s.g.865.1		2
28.27	even	2		7056.2.a.bj.1.1		1
84.11	even	6		784.2.i.h.177.1		2
84.23	even	6		784.2.i.h.753.1		2
84.47	odd	6		112.2.i.a.81.1		2
84.59	odd	6		112.2.i.a.65.1		2
84.83	odd	2		784.2.a.h.1.1		1
105.17	odd	12		1400.2.bh.a.849.2		4
105.38	odd	12		1400.2.bh.a.849.1		4
105.47	odd	12		1400.2.bh.a.249.1		4
105.59	even	6		1400.2.q.d.401.1		2
105.68	odd	12		1400.2.bh.a.249.2		4
105.89	even	6		1400.2.q.d.1201.1		2
105.104	even	2		9800.2.a.be.1.1		1
168.5	even	6		448.2.i.b.193.1		2
168.59	odd	6		448.2.i.d.65.1		2
168.83	odd	2		3136.2.a.j.1.1		1
168.101	even	6		448.2.i.b.65.1		2
168.125	even	2		3136.2.a.u.1.1		1
168.131	odd	6		448.2.i.d.193.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.i.b.9.1	✓	2	21.17	even	6
56.2.i.b.25.1	yes	2	21.5	even	6
112.2.i.a.65.1		2	84.59	odd	6
112.2.i.a.81.1		2	84.47	odd	6
392.2.a.c.1.1		1	21.20	even	2
392.2.a.e.1.1		1	3.2	odd	2
392.2.i.b.177.1		2	21.11	odd	6
392.2.i.b.361.1		2	21.2	odd	6
448.2.i.b.65.1		2	168.101	even	6
448.2.i.b.193.1		2	168.5	even	6
448.2.i.d.65.1		2	168.59	odd	6
448.2.i.d.193.1		2	168.131	odd	6
504.2.s.c.289.1		2	7.3	odd	6
504.2.s.c.361.1		2	7.5	odd	6
784.2.a.c.1.1		1	12.11	even	2
784.2.a.h.1.1		1	84.83	odd	2
784.2.i.h.177.1		2	84.11	even	6
784.2.i.h.753.1		2	84.23	even	6
1008.2.s.g.289.1		2	28.3	even	6
1008.2.s.g.865.1		2	28.19	even	6
1400.2.q.d.401.1		2	105.59	even	6
1400.2.q.d.1201.1		2	105.89	even	6
1400.2.bh.a.249.1		4	105.47	odd	12
1400.2.bh.a.249.2		4	105.68	odd	12
1400.2.bh.a.849.1		4	105.38	odd	12
1400.2.bh.a.849.2		4	105.17	odd	12
3136.2.a.i.1.1		1	24.5	odd	2
3136.2.a.j.1.1		1	168.83	odd	2
3136.2.a.t.1.1		1	24.11	even	2
3136.2.a.u.1.1		1	168.125	even	2
3528.2.a.j.1.1		1	1.1	even	1	trivial
3528.2.a.p.1.1		1	7.6	odd	2
3528.2.s.q.361.1		2	7.2	even	3
3528.2.s.q.3313.1		2	7.4	even	3
7056.2.a.u.1.1		1	4.3	odd	2
7056.2.a.bj.1.1		1	28.27	even	2
9800.2.a.s.1.1		1	15.14	odd	2
9800.2.a.be.1.1		1	105.104	even	2