Embedded newform 4032.2.a.r.1.1

• Label: 4032.2.a.r.1.1
• Level: $4032$
• Weight: $2$
• Character: 4032.1
• Self dual: yes
• Analytic conductor: $32.196$
• Analytic rank: $1$
• 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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

$f(q)$

$=$

$q-1.00000 q^{7} +4.00000 q^{13} -6.00000 q^{17} +2.00000 q^{19} -5.00000 q^{25} -6.00000 q^{29} +4.00000 q^{31} -2.00000 q^{37} -6.00000 q^{41} +8.00000 q^{43} -12.0000 q^{47} +1.00000 q^{49} +6.00000 q^{53} +6.00000 q^{59} -8.00000 q^{61} -4.00000 q^{67} +2.00000 q^{73} -8.00000 q^{79} +6.00000 q^{83} +6.00000 q^{89} -4.00000 q^{91} -10.0000 q^{97} +O(q^{100})$

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
$5$	0	0					−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$7$	−1.00000	−0.377964
			$11$	0	0		−	1.00000i	$-0.5\pi$
		1.00000i				$0.5\pi$
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
			0.554700	+	0.832050i	$0.312833\pi$
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
			−0.727607	+	0.685994i	$0.759367\pi$
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
			0.229416	+	0.973329i	$0.426318\pi$
$23$	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	$-0.383046\pi$
			0.359211	+	0.933257i	$0.383046\pi$
$37$	−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	$-0.655213\pi$
			−0.468521	+	0.883452i	$0.655213\pi$
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
			0.609994	+	0.792406i	$0.291172\pi$
$47$	−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.r.1.1		1
3.2	odd	2		448.2.a.a.1.1		1
4.3	odd	2		4032.2.a.w.1.1		1
8.3	odd	2		126.2.a.b.1.1		1
8.5	even	2		1008.2.a.h.1.1		1
12.11	even	2		448.2.a.g.1.1		1
21.20	even	2		3136.2.a.z.1.1		1
24.5	odd	2		112.2.a.c.1.1		1
24.11	even	2		14.2.a.a.1.1	✓	1
40.3	even	4		3150.2.g.j.2899.1		2
40.19	odd	2		3150.2.a.i.1.1		1
40.27	even	4		3150.2.g.j.2899.2		2
48.5	odd	4		1792.2.b.g.897.2		2
48.11	even	4		1792.2.b.c.897.1		2
48.29	odd	4		1792.2.b.g.897.1		2
48.35	even	4		1792.2.b.c.897.2		2
56.3	even	6		882.2.g.d.667.1		2
56.11	odd	6		882.2.g.c.667.1		2
56.13	odd	2		7056.2.a.bd.1.1		1
56.19	even	6		882.2.g.d.361.1		2
56.27	even	2		882.2.a.i.1.1		1
56.51	odd	6		882.2.g.c.361.1		2
72.11	even	6		1134.2.f.l.757.1		2
72.43	odd	6		1134.2.f.f.757.1		2
72.59	even	6		1134.2.f.l.379.1		2
72.67	odd	6		1134.2.f.f.379.1		2
84.83	odd	2		3136.2.a.e.1.1		1
120.29	odd	2		2800.2.a.g.1.1		1
120.53	even	4		2800.2.g.h.449.2		2
120.59	even	2		350.2.a.f.1.1		1
120.77	even	4		2800.2.g.h.449.1		2
120.83	odd	4		350.2.c.d.99.2		2
120.107	odd	4		350.2.c.d.99.1		2
168.5	even	6		784.2.i.i.753.1		2
168.11	even	6		98.2.c.b.79.1		2
168.53	odd	6		784.2.i.c.177.1		2
168.59	odd	6		98.2.c.a.79.1		2
168.83	odd	2		98.2.a.a.1.1		1
168.101	even	6		784.2.i.i.177.1		2
168.107	even	6		98.2.c.b.67.1		2
168.125	even	2		784.2.a.b.1.1		1
168.131	odd	6		98.2.c.a.67.1		2
168.149	odd	6		784.2.i.c.753.1		2
264.131	odd	2		1694.2.a.e.1.1		1
312.83	odd	4		2366.2.d.b.337.2		2
312.155	even	2		2366.2.a.j.1.1		1
312.203	odd	4		2366.2.d.b.337.1		2
408.203	even	2		4046.2.a.f.1.1		1
456.227	odd	2		5054.2.a.c.1.1		1
552.275	odd	2		7406.2.a.a.1.1		1
840.83	even	4		2450.2.c.c.99.2		2
840.419	odd	2		2450.2.a.t.1.1		1
840.587	even	4		2450.2.c.c.99.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	24.11	even	2
98.2.a.a.1.1		1	168.83	odd	2
98.2.c.a.67.1		2	168.131	odd	6
98.2.c.a.79.1		2	168.59	odd	6
98.2.c.b.67.1		2	168.107	even	6
98.2.c.b.79.1		2	168.11	even	6
112.2.a.c.1.1		1	24.5	odd	2
126.2.a.b.1.1		1	8.3	odd	2
350.2.a.f.1.1		1	120.59	even	2
350.2.c.d.99.1		2	120.107	odd	4
350.2.c.d.99.2		2	120.83	odd	4
448.2.a.a.1.1		1	3.2	odd	2
448.2.a.g.1.1		1	12.11	even	2
784.2.a.b.1.1		1	168.125	even	2
784.2.i.c.177.1		2	168.53	odd	6
784.2.i.c.753.1		2	168.149	odd	6
784.2.i.i.177.1		2	168.101	even	6
784.2.i.i.753.1		2	168.5	even	6
882.2.a.i.1.1		1	56.27	even	2
882.2.g.c.361.1		2	56.51	odd	6
882.2.g.c.667.1		2	56.11	odd	6
882.2.g.d.361.1		2	56.19	even	6
882.2.g.d.667.1		2	56.3	even	6
1008.2.a.h.1.1		1	8.5	even	2
1134.2.f.f.379.1		2	72.67	odd	6
1134.2.f.f.757.1		2	72.43	odd	6
1134.2.f.l.379.1		2	72.59	even	6
1134.2.f.l.757.1		2	72.11	even	6
1694.2.a.e.1.1		1	264.131	odd	2
1792.2.b.c.897.1		2	48.11	even	4
1792.2.b.c.897.2		2	48.35	even	4
1792.2.b.g.897.1		2	48.29	odd	4
1792.2.b.g.897.2		2	48.5	odd	4
2366.2.a.j.1.1		1	312.155	even	2
2366.2.d.b.337.1		2	312.203	odd	4
2366.2.d.b.337.2		2	312.83	odd	4
2450.2.a.t.1.1		1	840.419	odd	2
2450.2.c.c.99.1		2	840.587	even	4
2450.2.c.c.99.2		2	840.83	even	4
2800.2.a.g.1.1		1	120.29	odd	2
2800.2.g.h.449.1		2	120.77	even	4
2800.2.g.h.449.2		2	120.53	even	4
3136.2.a.e.1.1		1	84.83	odd	2
3136.2.a.z.1.1		1	21.20	even	2
3150.2.a.i.1.1		1	40.19	odd	2
3150.2.g.j.2899.1		2	40.3	even	4
3150.2.g.j.2899.2		2	40.27	even	4
4032.2.a.r.1.1		1	1.1	even	1	trivial
4032.2.a.w.1.1		1	4.3	odd	2
4046.2.a.f.1.1		1	408.203	even	2
5054.2.a.c.1.1		1	456.227	odd	2
7056.2.a.bd.1.1		1	56.13	odd	2
7406.2.a.a.1.1		1	552.275	odd	2