Embedded newform 4032.2.a.bm.1.1

• Label: 4032.2.a.bm.1.1
• Level: $4032$
• Weight: $2$
• Character: 4032.1
• Self dual: yes
• Analytic conductor: $32.196$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(32.1956820950\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4032.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+4.00000 q^{5} -1.00000 q^{7} +O(q^{10})\)

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
			\(3\)	0	0
\(5\)	4.00000	1.78885				0.894427	−	0.447214i	\(-0.147584\pi\)
			0.894427	+	0.447214i	\(0.147584\pi\)
\(7\)	−1.00000	−0.377964
			\(11\)	2.00000	0.603023	0.301511	−	0.953463i	\(-0.402509\pi\)
0.301511	+	0.953463i				\(0.402509\pi\)
\(13\)	6.00000	1.66410	0.832050	−	0.554700i	\(-0.187167\pi\)
			0.832050	+	0.554700i	\(0.187167\pi\)
\(17\)	4.00000	0.970143	0.485071	−	0.874475i	\(-0.338794\pi\)
			0.485071	+	0.874475i	\(0.338794\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	−2.00000	−0.417029	−0.208514	−	0.978019i	\(-0.566863\pi\)
			−0.208514	+	0.978019i	\(0.566863\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−12.0000	−1.75038	−0.875190	−	0.483779i	\(-0.839264\pi\)
			−0.875190	+	0.483779i	\(0.839264\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.a.bm.1.1		1
3.2	odd	2		1344.2.a.k.1.1		1
4.3	odd	2		4032.2.a.bn.1.1		1
8.3	odd	2		1008.2.a.a.1.1		1
8.5	even	2		252.2.a.a.1.1		1
12.11	even	2		1344.2.a.a.1.1		1
21.20	even	2		9408.2.a.bn.1.1		1
24.5	odd	2		84.2.a.a.1.1	✓	1
24.11	even	2		336.2.a.f.1.1		1
40.13	odd	4		6300.2.k.g.6049.2		2
40.29	even	2		6300.2.a.w.1.1		1
40.37	odd	4		6300.2.k.g.6049.1		2
48.5	odd	4		5376.2.c.q.2689.1		2
48.11	even	4		5376.2.c.p.2689.2		2
48.29	odd	4		5376.2.c.q.2689.2		2
48.35	even	4		5376.2.c.p.2689.1		2
56.5	odd	6		1764.2.k.a.361.1		2
56.13	odd	2		1764.2.a.k.1.1		1
56.27	even	2		7056.2.a.cd.1.1		1
56.37	even	6		1764.2.k.k.361.1		2
56.45	odd	6		1764.2.k.a.1549.1		2
56.53	even	6		1764.2.k.k.1549.1		2
72.5	odd	6		2268.2.j.a.1513.1		2
72.13	even	6		2268.2.j.n.1513.1		2
72.29	odd	6		2268.2.j.a.757.1		2
72.61	even	6		2268.2.j.n.757.1		2
84.83	odd	2		9408.2.a.df.1.1		1
120.29	odd	2		2100.2.a.r.1.1		1
120.53	even	4		2100.2.k.i.1849.1		2
120.59	even	2		8400.2.a.e.1.1		1
120.77	even	4		2100.2.k.i.1849.2		2
168.5	even	6		588.2.i.d.361.1		2
168.11	even	6		2352.2.q.b.961.1		2
168.53	odd	6		588.2.i.e.373.1		2
168.59	odd	6		2352.2.q.z.961.1		2
168.83	odd	2		2352.2.a.a.1.1		1
168.101	even	6		588.2.i.d.373.1		2
168.107	even	6		2352.2.q.b.1537.1		2
168.125	even	2		588.2.a.d.1.1		1
168.131	odd	6		2352.2.q.z.1537.1		2
168.149	odd	6		588.2.i.e.361.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.a.1.1	✓	1	24.5	odd	2
252.2.a.a.1.1		1	8.5	even	2
336.2.a.f.1.1		1	24.11	even	2
588.2.a.d.1.1		1	168.125	even	2
588.2.i.d.361.1		2	168.5	even	6
588.2.i.d.373.1		2	168.101	even	6
588.2.i.e.361.1		2	168.149	odd	6
588.2.i.e.373.1		2	168.53	odd	6
1008.2.a.a.1.1		1	8.3	odd	2
1344.2.a.a.1.1		1	12.11	even	2
1344.2.a.k.1.1		1	3.2	odd	2
1764.2.a.k.1.1		1	56.13	odd	2
1764.2.k.a.361.1		2	56.5	odd	6
1764.2.k.a.1549.1		2	56.45	odd	6
1764.2.k.k.361.1		2	56.37	even	6
1764.2.k.k.1549.1		2	56.53	even	6
2100.2.a.r.1.1		1	120.29	odd	2
2100.2.k.i.1849.1		2	120.53	even	4
2100.2.k.i.1849.2		2	120.77	even	4
2268.2.j.a.757.1		2	72.29	odd	6
2268.2.j.a.1513.1		2	72.5	odd	6
2268.2.j.n.757.1		2	72.61	even	6
2268.2.j.n.1513.1		2	72.13	even	6
2352.2.a.a.1.1		1	168.83	odd	2
2352.2.q.b.961.1		2	168.11	even	6
2352.2.q.b.1537.1		2	168.107	even	6
2352.2.q.z.961.1		2	168.59	odd	6
2352.2.q.z.1537.1		2	168.131	odd	6
4032.2.a.bm.1.1		1	1.1	even	1	trivial
4032.2.a.bn.1.1		1	4.3	odd	2
5376.2.c.p.2689.1		2	48.35	even	4
5376.2.c.p.2689.2		2	48.11	even	4
5376.2.c.q.2689.1		2	48.5	odd	4
5376.2.c.q.2689.2		2	48.29	odd	4
6300.2.a.w.1.1		1	40.29	even	2
6300.2.k.g.6049.1		2	40.37	odd	4
6300.2.k.g.6049.2		2	40.13	odd	4
7056.2.a.cd.1.1		1	56.27	even	2
8400.2.a.e.1.1		1	120.59	even	2
9408.2.a.bn.1.1		1	21.20	even	2
9408.2.a.df.1.1		1	84.83	odd	2