Newform orbit 1500.2.a.c

• Label: 1500.2.a.c
• Level: $1500$
• Weight: $2$
• Character orbit: 1500.a
• Self dual: yes
• Analytic conductor: $11.978$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$1500 = 2^{2} \cdot 3 \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1500.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$11.9775603032$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5})$
Defining polynomial:	$x^{2} - x - 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{5})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - q^3 + 2*b * q^7 + q^9 + (-2*b + 2) * q^11 + (b - 2) * q^17 + (3*b + 1) * q^19 - 2*b * q^21 - 3*b * q^23 - q^27 + (4*b + 2) * q^29 + (b + 6) * q^31 + (2*b - 2) * q^33 - 2 * q^37 + (-4*b + 2) * q^41 + (-6*b + 4) * q^43 + (-5*b - 1) * q^47 + (4*b - 3) * q^49 + (-b + 2) * q^51 + (5*b - 3) * q^53 + (-3*b - 1) * q^57 + (2*b + 8) * q^59 + (-9*b + 9) * q^61 + 2*b * q^63 + (6*b + 6) * q^67 + 3*b * q^69 + (-4*b + 12) * q^71 + (-4*b + 10) * q^73 - 4 * q^77 + (3*b + 9) * q^79 + q^81 + (7*b - 5) * q^83 + (-4*b - 2) * q^87 + (2*b - 6) * q^89 + (-b - 6) * q^93 + 12 * q^97 + (-2*b + 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 3 * q^17 + 5 * q^19 - 2 * q^21 - 3 * q^23 - 2 * q^27 + 8 * q^29 + 13 * q^31 - 2 * q^33 - 4 * q^37 + 2 * q^43 - 7 * q^47 - 2 * q^49 + 3 * q^51 - q^53 - 5 * q^57 + 18 * q^59 + 9 * q^61 + 2 * q^63 + 18 * q^67 + 3 * q^69 + 20 * q^71 + 16 * q^73 - 8 * q^77 + 21 * q^79 + 2 * q^81 - 3 * q^83 - 8 * q^87 - 10 * q^89 - 13 * q^93 + 24 * q^97 + 2 * q^99

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

−0.618034
1.61803

0

−1.00000

0

0

0

−1.23607

0

1.00000

0

1.2

0

−1.00000

0

0

0

3.23607

0

1.00000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$5$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1500.2.a.c	✓	2
3.b	odd	2	1		4500.2.a.h		2
4.b	odd	2	1		6000.2.a.s		2
5.b	even	2	1		1500.2.a.g	yes	2
5.c	odd	4	2		1500.2.d.b		4
15.d	odd	2	1		4500.2.a.d		2
15.e	even	4	2		4500.2.d.a		4
20.d	odd	2	1		6000.2.a.i		2
20.e	even	4	2		6000.2.f.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1500.2.a.c	✓	2	1.a	even	1	1	trivial
1500.2.a.g	yes	2	5.b	even	2	1
1500.2.d.b		4	5.c	odd	4	2
4500.2.a.d		2	15.d	odd	2	1
4500.2.a.h		2	3.b	odd	2	1
4500.2.d.a		4	15.e	even	4	2
6000.2.a.i		2	20.d	odd	2	1
6000.2.a.s		2	4.b	odd	2	1
6000.2.f.f		4	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1500))$:

$T_{7}^{2} - 2T_{7} - 4$

$T_{11}^{2} - 2T_{11} - 4$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$(T + 1)^{2}$
$5$	$T^{2}$
$7$	$T^{2} - 2T - 4$
$11$	$T^{2} - 2T - 4$