Embedded newform 1800.4.a.ba.1.1

• Label: 1800.4.a.ba.1.1
• Level: $1800$
• Weight: $4$
• Character: 1800.1
• Self dual: yes
• Analytic conductor: $106.203$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1800.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$106.203438010$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1800.1

$q$-expansion

$f(q)$

$=$

$q+12.0000 q^{7} +64.0000 q^{11} -58.0000 q^{13} -32.0000 q^{17} -136.000 q^{19} +128.000 q^{23} -144.000 q^{29} +20.0000 q^{31} +18.0000 q^{37} -288.000 q^{41} +200.000 q^{43} -384.000 q^{47} -199.000 q^{49} -496.000 q^{53} -128.000 q^{59} -458.000 q^{61} +496.000 q^{67} +512.000 q^{71} +602.000 q^{73} +768.000 q^{77} +1108.00 q^{79} -704.000 q^{83} -960.000 q^{89} -696.000 q^{91} -206.000 q^{97} +O(q^{100})$

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
			$7$	12.0000	0.647939	0.323970	−	0.946068i	$-0.394982\pi$
0.323970	+	0.946068i				$0.394982\pi$
$11$	64.0000	1.75425	0.877124	−	0.480264i	$-0.159459\pi$
			0.877124	+	0.480264i	$0.159459\pi$
$13$	−58.0000	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
			−0.618704	+	0.785624i	$0.712342\pi$
$17$	−32.0000	−0.456538	−0.228269	−	0.973598i	$-0.573307\pi$
			−0.228269	+	0.973598i	$0.573307\pi$
$19$	−136.000	−1.64213	−0.821067	−	0.570832i	$-0.806621\pi$
			−0.821067	+	0.570832i	$0.806621\pi$
$23$	128.000	1.16043	0.580214	−	0.814464i	$-0.302969\pi$
			0.580214	+	0.814464i	$0.302969\pi$
$29$	−144.000	−0.922073	−0.461037	−	0.887381i	$-0.652522\pi$
			−0.461037	+	0.887381i	$0.652522\pi$
$31$	20.0000	0.115874	0.0579372	−	0.998320i	$-0.481548\pi$
			0.0579372	+	0.998320i	$0.481548\pi$
$37$	18.0000	0.0799779	0.0399889	−	0.999200i	$-0.487268\pi$
			0.0399889	+	0.999200i	$0.487268\pi$
$41$	−288.000	−1.09703	−0.548513	−	0.836142i	$-0.684806\pi$
			−0.548513	+	0.836142i	$0.684806\pi$
$43$	200.000	0.709296	0.354648	−	0.935000i	$-0.384601\pi$
			0.354648	+	0.935000i	$0.384601\pi$
$47$	−384.000	−1.19175	−0.595874	−	0.803078i	$-0.703194\pi$
			−0.595874	+	0.803078i	$0.703194\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.ba.1.1		1
3.2	odd	2		1800.4.a.z.1.1		1
5.2	odd	4		1800.4.f.x.649.2		2
5.3	odd	4		1800.4.f.x.649.1		2
5.4	even	2		72.4.a.d.1.1	yes	1
15.2	even	4		1800.4.f.b.649.2		2
15.8	even	4		1800.4.f.b.649.1		2
15.14	odd	2		72.4.a.a.1.1	✓	1
20.19	odd	2		144.4.a.f.1.1		1
40.19	odd	2		576.4.a.d.1.1		1
40.29	even	2		576.4.a.c.1.1		1
45.4	even	6		648.4.i.a.217.1		2
45.14	odd	6		648.4.i.l.217.1		2
45.29	odd	6		648.4.i.l.433.1		2
45.34	even	6		648.4.i.a.433.1		2
60.59	even	2		144.4.a.a.1.1		1
120.29	odd	2		576.4.a.w.1.1		1
120.59	even	2		576.4.a.x.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.a.a.1.1	✓	1	15.14	odd	2
72.4.a.d.1.1	yes	1	5.4	even	2
144.4.a.a.1.1		1	60.59	even	2
144.4.a.f.1.1		1	20.19	odd	2
576.4.a.c.1.1		1	40.29	even	2
576.4.a.d.1.1		1	40.19	odd	2
576.4.a.w.1.1		1	120.29	odd	2
576.4.a.x.1.1		1	120.59	even	2
648.4.i.a.217.1		2	45.4	even	6
648.4.i.a.433.1		2	45.34	even	6
648.4.i.l.217.1		2	45.14	odd	6
648.4.i.l.433.1		2	45.29	odd	6
1800.4.a.z.1.1		1	3.2	odd	2
1800.4.a.ba.1.1		1	1.1	even	1	trivial
1800.4.f.b.649.1		2	15.8	even	4
1800.4.f.b.649.2		2	15.2	even	4
1800.4.f.x.649.1		2	5.3	odd	4
1800.4.f.x.649.2		2	5.2	odd	4