Newform orbit 720.3.j.c

• Label: 720.3.j.c
• Level: $720$
• Weight: $3$
• Character orbit: 720.j
• Analytic conductor: $19.619$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$720 = 2^{4} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	720.j (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$3$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 3i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (b + 4) * q^5 - 8*b * q^13 - 10*b * q^17 + (8*b + 7) * q^25 + 40 * q^29 + 8*b * q^37 + 80 * q^41 - 49 * q^49 - 30*b * q^53 + 22 * q^61 + (-32*b + 72) * q^65 + 32*b * q^73 + (-40*b + 90) * q^85 + 160 * q^89 - 48*b * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 8 q^{5} + 14 q^{25} + 80 q^{29} + 160 q^{41} - 98 q^{49} + 44 q^{61} + 144 q^{65} + 180 q^{85} + 320 q^{89}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

Label  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

559.1

	−	1.00000i
		1.00000i

0

0

0

4.00000

−

3.00000i

0

0

0

0

0

559.2

0

0

0

4.00000

+

3.00000i

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
5.b	even	2	1	inner
20.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	720.3.j.c	yes	2
3.b	odd	2	1		720.3.j.b	✓	2
4.b	odd	2	1	CM	720.3.j.c	yes	2
5.b	even	2	1	inner	720.3.j.c	yes	2
5.c	odd	4	1		3600.3.e.b		1
5.c	odd	4	1		3600.3.e.d		1
12.b	even	2	1		720.3.j.b	✓	2
15.d	odd	2	1		720.3.j.b	✓	2
15.e	even	4	1		3600.3.e.a		1
15.e	even	4	1		3600.3.e.e		1
20.d	odd	2	1	inner	720.3.j.c	yes	2
20.e	even	4	1		3600.3.e.b		1
20.e	even	4	1		3600.3.e.d		1
60.h	even	2	1		720.3.j.b	✓	2
60.l	odd	4	1		3600.3.e.a		1
60.l	odd	4	1		3600.3.e.e		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
720.3.j.b	✓	2	3.b	odd	2	1
720.3.j.b	✓	2	12.b	even	2	1
720.3.j.b	✓	2	15.d	odd	2	1
720.3.j.b	✓	2	60.h	even	2	1
720.3.j.c	yes	2	1.a	even	1	1	trivial
720.3.j.c	yes	2	4.b	odd	2	1	CM
720.3.j.c	yes	2	5.b	even	2	1	inner
720.3.j.c	yes	2	20.d	odd	2	1	inner
3600.3.e.a		1	15.e	even	4	1
3600.3.e.a		1	60.l	odd	4	1
3600.3.e.b		1	5.c	odd	4	1
3600.3.e.b		1	20.e	even	4	1
3600.3.e.d		1	5.c	odd	4	1
3600.3.e.d		1	20.e	even	4	1
3600.3.e.e		1	15.e	even	4	1
3600.3.e.e		1	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(720, [\chi])$:

$T_{7}$

$T_{11}$

$T_{29} - 40$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} - 8T + 25$
$7$	$T^{2}$
$11$	$T^{2}$