Embedded newform 4032.2.a.k.1.1

• Label: 4032.2.a.k.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 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4032.1

$q$-expansion

$f(q)$

$=$

$q-2.00000 q^{5} +1.00000 q^{7} -4.00000 q^{11} +2.00000 q^{13} +6.00000 q^{17} +4.00000 q^{19} -1.00000 q^{25} -2.00000 q^{29} -2.00000 q^{35} -6.00000 q^{37} -2.00000 q^{41} -4.00000 q^{43} +1.00000 q^{49} +6.00000 q^{53} +8.00000 q^{55} -12.0000 q^{59} +2.00000 q^{61} -4.00000 q^{65} +4.00000 q^{67} -6.00000 q^{73} -4.00000 q^{77} +16.0000 q^{79} +12.0000 q^{83} -12.0000 q^{85} +14.0000 q^{89} +2.00000 q^{91} -8.00000 q^{95} +18.0000 q^{97} +O(q^{100})$

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
$5$	−2.00000	−0.894427				−0.447214	−	0.894427i	$-0.647584\pi$
			−0.447214	+	0.894427i	$0.647584\pi$
$7$	1.00000	0.377964
			$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
−0.603023	+	0.797724i				$0.706037\pi$
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
			0.277350	+	0.960769i	$0.410544\pi$
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
			0.727607	+	0.685994i	$0.240633\pi$
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
			0.458831	+	0.888523i	$0.348268\pi$
$23$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$29$	−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	$0.5\pi$
$37$	−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	$0.549916\pi$
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
			−0.304997	+	0.952353i	$0.598656\pi$
$47$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.a.k.1.1		1
3.2	odd	2		1344.2.a.s.1.1		1
4.3	odd	2		4032.2.a.h.1.1		1
8.3	odd	2		63.2.a.a.1.1		1
8.5	even	2		1008.2.a.l.1.1		1
12.11	even	2		1344.2.a.g.1.1		1
21.20	even	2		9408.2.a.m.1.1		1
24.5	odd	2		336.2.a.a.1.1		1
24.11	even	2		21.2.a.a.1.1	✓	1
40.3	even	4		1575.2.d.a.1324.1		2
40.19	odd	2		1575.2.a.c.1.1		1
40.27	even	4		1575.2.d.a.1324.2		2
48.5	odd	4		5376.2.c.l.2689.1		2
48.11	even	4		5376.2.c.r.2689.2		2
48.29	odd	4		5376.2.c.l.2689.2		2
48.35	even	4		5376.2.c.r.2689.1		2
56.3	even	6		441.2.e.b.226.1		2
56.11	odd	6		441.2.e.a.226.1		2
56.13	odd	2		7056.2.a.p.1.1		1
56.19	even	6		441.2.e.b.361.1		2
56.27	even	2		441.2.a.f.1.1		1
56.51	odd	6		441.2.e.a.361.1		2
72.11	even	6		567.2.f.g.190.1		2
72.43	odd	6		567.2.f.b.190.1		2
72.59	even	6		567.2.f.g.379.1		2
72.67	odd	6		567.2.f.b.379.1		2
84.83	odd	2		9408.2.a.bv.1.1		1
88.43	even	2		7623.2.a.g.1.1		1
120.29	odd	2		8400.2.a.bn.1.1		1
120.59	even	2		525.2.a.d.1.1		1
120.83	odd	4		525.2.d.a.274.2		2
120.107	odd	4		525.2.d.a.274.1		2
168.5	even	6		2352.2.q.e.1537.1		2
168.11	even	6		147.2.e.b.79.1		2
168.53	odd	6		2352.2.q.x.961.1		2
168.59	odd	6		147.2.e.c.79.1		2
168.83	odd	2		147.2.a.a.1.1		1
168.101	even	6		2352.2.q.e.961.1		2
168.107	even	6		147.2.e.b.67.1		2
168.125	even	2		2352.2.a.v.1.1		1
168.131	odd	6		147.2.e.c.67.1		2
168.149	odd	6		2352.2.q.x.1537.1		2
264.131	odd	2		2541.2.a.j.1.1		1
312.155	even	2		3549.2.a.c.1.1		1
408.203	even	2		6069.2.a.b.1.1		1
456.227	odd	2		7581.2.a.d.1.1		1
840.419	odd	2		3675.2.a.n.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.a.a.1.1	✓	1	24.11	even	2
63.2.a.a.1.1		1	8.3	odd	2
147.2.a.a.1.1		1	168.83	odd	2
147.2.e.b.67.1		2	168.107	even	6
147.2.e.b.79.1		2	168.11	even	6
147.2.e.c.67.1		2	168.131	odd	6
147.2.e.c.79.1		2	168.59	odd	6
336.2.a.a.1.1		1	24.5	odd	2
441.2.a.f.1.1		1	56.27	even	2
441.2.e.a.226.1		2	56.11	odd	6
441.2.e.a.361.1		2	56.51	odd	6
441.2.e.b.226.1		2	56.3	even	6
441.2.e.b.361.1		2	56.19	even	6
525.2.a.d.1.1		1	120.59	even	2
525.2.d.a.274.1		2	120.107	odd	4
525.2.d.a.274.2		2	120.83	odd	4
567.2.f.b.190.1		2	72.43	odd	6
567.2.f.b.379.1		2	72.67	odd	6
567.2.f.g.190.1		2	72.11	even	6
567.2.f.g.379.1		2	72.59	even	6
1008.2.a.l.1.1		1	8.5	even	2
1344.2.a.g.1.1		1	12.11	even	2
1344.2.a.s.1.1		1	3.2	odd	2
1575.2.a.c.1.1		1	40.19	odd	2
1575.2.d.a.1324.1		2	40.3	even	4
1575.2.d.a.1324.2		2	40.27	even	4
2352.2.a.v.1.1		1	168.125	even	2
2352.2.q.e.961.1		2	168.101	even	6
2352.2.q.e.1537.1		2	168.5	even	6
2352.2.q.x.961.1		2	168.53	odd	6
2352.2.q.x.1537.1		2	168.149	odd	6
2541.2.a.j.1.1		1	264.131	odd	2
3549.2.a.c.1.1		1	312.155	even	2
3675.2.a.n.1.1		1	840.419	odd	2
4032.2.a.h.1.1		1	4.3	odd	2
4032.2.a.k.1.1		1	1.1	even	1	trivial
5376.2.c.l.2689.1		2	48.5	odd	4
5376.2.c.l.2689.2		2	48.29	odd	4
5376.2.c.r.2689.1		2	48.35	even	4
5376.2.c.r.2689.2		2	48.11	even	4
6069.2.a.b.1.1		1	408.203	even	2
7056.2.a.p.1.1		1	56.13	odd	2
7581.2.a.d.1.1		1	456.227	odd	2
7623.2.a.g.1.1		1	88.43	even	2
8400.2.a.bn.1.1		1	120.29	odd	2
9408.2.a.m.1.1		1	21.20	even	2
9408.2.a.bv.1.1		1	84.83	odd	2