Embedded newform 1323.2.a.i.1.1

• Label: 1323.2.a.i.1.1
• Level: $1323$
• Weight: $2$
• Character: 1323.1
• Self dual: yes
• Analytic conductor: $10.564$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$1323 = 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1323.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1323.1

$q$-expansion

$f(q)$

$=$

$q-2.00000 q^{4} -5.00000 q^{13} +4.00000 q^{16} +7.00000 q^{19} -5.00000 q^{25} +4.00000 q^{31} +11.0000 q^{37} +8.00000 q^{43} +10.0000 q^{52} +1.00000 q^{61} -8.00000 q^{64} +5.00000 q^{67} +7.00000 q^{73} -14.0000 q^{76} +17.0000 q^{79} +19.0000 q^{97} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$3$	0	0
			$5$	0	0		−	1.00000i	$-0.5\pi$
		1.00000i				$0.5\pi$
$7$	0	0
			$11$	0	0		−	1.00000i	$-0.5\pi$
		1.00000i				$0.5\pi$
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
			−0.693375	+	0.720577i	$0.743877\pi$
$17$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
			0.802955	+	0.596040i	$0.203260\pi$
$23$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$29$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
			0.359211	+	0.933257i	$0.383046\pi$
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
			0.904194	+	0.427121i	$0.140472\pi$
$41$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
			0.609994	+	0.792406i	$0.291172\pi$
$47$	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	1323.2.a.i.1.1		1
3.2	odd	2	CM	1323.2.a.i.1.1		1
7.6	odd	2		27.2.a.a.1.1	✓	1
21.20	even	2		27.2.a.a.1.1	✓	1
28.27	even	2		432.2.a.e.1.1		1
35.13	even	4		675.2.b.f.649.2		2
35.27	even	4		675.2.b.f.649.1		2
35.34	odd	2		675.2.a.e.1.1		1
56.13	odd	2		1728.2.a.n.1.1		1
56.27	even	2		1728.2.a.o.1.1		1
63.13	odd	6		81.2.c.a.55.1		2
63.20	even	6		81.2.c.a.28.1		2
63.34	odd	6		81.2.c.a.28.1		2
63.41	even	6		81.2.c.a.55.1		2
77.76	even	2		3267.2.a.f.1.1		1
84.83	odd	2		432.2.a.e.1.1		1
91.90	odd	2		4563.2.a.e.1.1		1
105.62	odd	4		675.2.b.f.649.1		2
105.83	odd	4		675.2.b.f.649.2		2
105.104	even	2		675.2.a.e.1.1		1
119.118	odd	2		7803.2.a.k.1.1		1
133.132	even	2		9747.2.a.f.1.1		1
168.83	odd	2		1728.2.a.o.1.1		1
168.125	even	2		1728.2.a.n.1.1		1
189.13	odd	18		729.2.e.f.163.1		6
189.20	even	18		729.2.e.f.325.1		6
189.34	odd	18		729.2.e.f.325.1		6
189.41	even	18		729.2.e.f.163.1		6
189.76	odd	18		729.2.e.f.649.1		6
189.83	even	18		729.2.e.f.568.1		6
189.97	odd	18		729.2.e.f.82.1		6
189.104	even	18		729.2.e.f.406.1		6
189.139	odd	18		729.2.e.f.406.1		6
189.146	even	18		729.2.e.f.82.1		6
189.160	odd	18		729.2.e.f.568.1		6
189.167	even	18		729.2.e.f.649.1		6
231.230	odd	2		3267.2.a.f.1.1		1
252.83	odd	6		1296.2.i.i.433.1		2
252.139	even	6		1296.2.i.i.865.1		2
252.167	odd	6		1296.2.i.i.865.1		2
252.223	even	6		1296.2.i.i.433.1		2
273.272	even	2		4563.2.a.e.1.1		1
357.356	even	2		7803.2.a.k.1.1		1
399.398	odd	2		9747.2.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.a.a.1.1	✓	1	7.6	odd	2
27.2.a.a.1.1	✓	1	21.20	even	2
81.2.c.a.28.1		2	63.20	even	6
81.2.c.a.28.1		2	63.34	odd	6
81.2.c.a.55.1		2	63.13	odd	6
81.2.c.a.55.1		2	63.41	even	6
432.2.a.e.1.1		1	28.27	even	2
432.2.a.e.1.1		1	84.83	odd	2
675.2.a.e.1.1		1	35.34	odd	2
675.2.a.e.1.1		1	105.104	even	2
675.2.b.f.649.1		2	35.27	even	4
675.2.b.f.649.1		2	105.62	odd	4
675.2.b.f.649.2		2	35.13	even	4
675.2.b.f.649.2		2	105.83	odd	4
729.2.e.f.82.1		6	189.97	odd	18
729.2.e.f.82.1		6	189.146	even	18
729.2.e.f.163.1		6	189.13	odd	18
729.2.e.f.163.1		6	189.41	even	18
729.2.e.f.325.1		6	189.20	even	18
729.2.e.f.325.1		6	189.34	odd	18
729.2.e.f.406.1		6	189.104	even	18
729.2.e.f.406.1		6	189.139	odd	18
729.2.e.f.568.1		6	189.83	even	18
729.2.e.f.568.1		6	189.160	odd	18
729.2.e.f.649.1		6	189.76	odd	18
729.2.e.f.649.1		6	189.167	even	18
1296.2.i.i.433.1		2	252.83	odd	6
1296.2.i.i.433.1		2	252.223	even	6
1296.2.i.i.865.1		2	252.139	even	6
1296.2.i.i.865.1		2	252.167	odd	6
1323.2.a.i.1.1		1	1.1	even	1	trivial
1323.2.a.i.1.1		1	3.2	odd	2	CM
1728.2.a.n.1.1		1	56.13	odd	2
1728.2.a.n.1.1		1	168.125	even	2
1728.2.a.o.1.1		1	56.27	even	2
1728.2.a.o.1.1		1	168.83	odd	2
3267.2.a.f.1.1		1	77.76	even	2
3267.2.a.f.1.1		1	231.230	odd	2
4563.2.a.e.1.1		1	91.90	odd	2
4563.2.a.e.1.1		1	273.272	even	2
7803.2.a.k.1.1		1	119.118	odd	2
7803.2.a.k.1.1		1	357.356	even	2
9747.2.a.f.1.1		1	133.132	even	2
9747.2.a.f.1.1		1	399.398	odd	2