Embedded newform 36.4.a.a.1.1

• Label: 36.4.a.a.1.1
• Level: $36$
• Weight: $4$
• Character: 36.1
• Self dual: yes
• Analytic conductor: $2.124$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$36 = 2^{2} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	36.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$2.12406876021$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 12)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	36.1

$q$-expansion

$f(q)$

$=$

$q+18.0000 q^{5} +8.00000 q^{7} -36.0000 q^{11} -10.0000 q^{13} -18.0000 q^{17} -100.000 q^{19} -72.0000 q^{23} +199.000 q^{25} +234.000 q^{29} -16.0000 q^{31} +144.000 q^{35} -226.000 q^{37} -90.0000 q^{41} +452.000 q^{43} -432.000 q^{47} -279.000 q^{49} -414.000 q^{53} -648.000 q^{55} +684.000 q^{59} +422.000 q^{61} -180.000 q^{65} +332.000 q^{67} +360.000 q^{71} +26.0000 q^{73} -288.000 q^{77} +512.000 q^{79} +1188.00 q^{83} -324.000 q^{85} +630.000 q^{89} -80.0000 q^{91} -1800.00 q^{95} -1054.00 q^{97} +O(q^{100})$

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
$5$	18.0000	1.60997				0.804984	−	0.593296i	$-0.202174\pi$
			0.804984	+	0.593296i	$0.202174\pi$
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
			0.215980	+	0.976398i	$0.430705\pi$
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
			−0.493382	+	0.869813i	$0.664240\pi$
$13$	−10.0000	−0.213346	−0.106673	−	0.994294i	$-0.534020\pi$
			−0.106673	+	0.994294i	$0.534020\pi$
$17$	−18.0000	−0.256802	−0.128401	−	0.991722i	$-0.540985\pi$
			−0.128401	+	0.991722i	$0.540985\pi$
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
			−0.603726	+	0.797192i	$0.706318\pi$
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
			−0.326370	+	0.945242i	$0.605826\pi$
$29$	234.000	1.49837	0.749185	−	0.662361i	$-0.230446\pi$
			0.749185	+	0.662361i	$0.230446\pi$
$31$	−16.0000	−0.0926995	−0.0463498	−	0.998925i	$-0.514759\pi$
			−0.0463498	+	0.998925i	$0.514759\pi$
$37$	−226.000	−1.00417	−0.502083	−	0.864819i	$-0.667433\pi$
			−0.502083	+	0.864819i	$0.667433\pi$
$41$	−90.0000	−0.342820	−0.171410	−	0.985200i	$-0.554832\pi$
			−0.171410	+	0.985200i	$0.554832\pi$
$43$	452.000	1.60301	0.801504	−	0.597989i	$-0.204033\pi$
			0.801504	+	0.597989i	$0.204033\pi$
$47$	−432.000	−1.34072	−0.670358	−	0.742038i	$-0.733860\pi$
			−0.670358	+	0.742038i	$0.733860\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.4.a.a.1.1		1
3.2	odd	2		12.4.a.a.1.1	✓	1
4.3	odd	2		144.4.a.g.1.1		1
5.2	odd	4		900.4.d.c.649.2		2
5.3	odd	4		900.4.d.c.649.1		2
5.4	even	2		900.4.a.g.1.1		1
7.2	even	3		1764.4.k.b.361.1		2
7.3	odd	6		1764.4.k.o.1549.1		2
7.4	even	3		1764.4.k.b.1549.1		2
7.5	odd	6		1764.4.k.o.361.1		2
7.6	odd	2		1764.4.a.b.1.1		1
8.3	odd	2		576.4.a.a.1.1		1
8.5	even	2		576.4.a.b.1.1		1
9.2	odd	6		324.4.e.h.109.1		2
9.4	even	3		324.4.e.a.217.1		2
9.5	odd	6		324.4.e.h.217.1		2
9.7	even	3		324.4.e.a.109.1		2
12.11	even	2		48.4.a.a.1.1		1
15.2	even	4		300.4.d.e.49.1		2
15.8	even	4		300.4.d.e.49.2		2
15.14	odd	2		300.4.a.b.1.1		1
21.2	odd	6		588.4.i.d.361.1		2
21.5	even	6		588.4.i.e.361.1		2
21.11	odd	6		588.4.i.d.373.1		2
21.17	even	6		588.4.i.e.373.1		2
21.20	even	2		588.4.a.c.1.1		1
24.5	odd	2		192.4.a.f.1.1		1
24.11	even	2		192.4.a.l.1.1		1
33.32	even	2		1452.4.a.d.1.1		1
39.5	even	4		2028.4.b.c.337.2		2
39.8	even	4		2028.4.b.c.337.1		2
39.38	odd	2		2028.4.a.c.1.1		1
48.5	odd	4		768.4.d.g.385.1		2
48.11	even	4		768.4.d.j.385.2		2
48.29	odd	4		768.4.d.g.385.2		2
48.35	even	4		768.4.d.j.385.1		2
60.23	odd	4		1200.4.f.d.49.1		2
60.47	odd	4		1200.4.f.d.49.2		2
60.59	even	2		1200.4.a.be.1.1		1
84.83	odd	2		2352.4.a.bk.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.4.a.a.1.1	✓	1	3.2	odd	2
36.4.a.a.1.1		1	1.1	even	1	trivial
48.4.a.a.1.1		1	12.11	even	2
144.4.a.g.1.1		1	4.3	odd	2
192.4.a.f.1.1		1	24.5	odd	2
192.4.a.l.1.1		1	24.11	even	2
300.4.a.b.1.1		1	15.14	odd	2
300.4.d.e.49.1		2	15.2	even	4
300.4.d.e.49.2		2	15.8	even	4
324.4.e.a.109.1		2	9.7	even	3
324.4.e.a.217.1		2	9.4	even	3
324.4.e.h.109.1		2	9.2	odd	6
324.4.e.h.217.1		2	9.5	odd	6
576.4.a.a.1.1		1	8.3	odd	2
576.4.a.b.1.1		1	8.5	even	2
588.4.a.c.1.1		1	21.20	even	2
588.4.i.d.361.1		2	21.2	odd	6
588.4.i.d.373.1		2	21.11	odd	6
588.4.i.e.361.1		2	21.5	even	6
588.4.i.e.373.1		2	21.17	even	6
768.4.d.g.385.1		2	48.5	odd	4
768.4.d.g.385.2		2	48.29	odd	4
768.4.d.j.385.1		2	48.35	even	4
768.4.d.j.385.2		2	48.11	even	4
900.4.a.g.1.1		1	5.4	even	2
900.4.d.c.649.1		2	5.3	odd	4
900.4.d.c.649.2		2	5.2	odd	4
1200.4.a.be.1.1		1	60.59	even	2
1200.4.f.d.49.1		2	60.23	odd	4
1200.4.f.d.49.2		2	60.47	odd	4
1452.4.a.d.1.1		1	33.32	even	2
1764.4.a.b.1.1		1	7.6	odd	2
1764.4.k.b.361.1		2	7.2	even	3
1764.4.k.b.1549.1		2	7.4	even	3
1764.4.k.o.361.1		2	7.5	odd	6
1764.4.k.o.1549.1		2	7.3	odd	6
2028.4.a.c.1.1		1	39.38	odd	2
2028.4.b.c.337.1		2	39.8	even	4
2028.4.b.c.337.2		2	39.5	even	4
2352.4.a.bk.1.1		1	84.83	odd	2