Newform orbit 9000.2.a.m

• Label: 9000.2.a.m
• Level: $9000$
• Weight: $2$
• Character orbit: 9000.a
• Self dual: yes
• Analytic conductor: $71.865$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$9000 = 2^{3} \cdot 3^{2} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9000.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$71.8653618192$
Analytic rank:	$1$
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 + 2*b * q^7 + (-2*b + 2) * q^11 - 2*b * q^13 + (-b - 1) * q^17 + (b - 2) * q^19 + (b + 1) * q^23 + (4*b - 6) * q^29 + (-3*b - 1) * q^31 + (-4*b + 6) * q^37 + 6*b * q^41 - 10 * q^43 + (b + 4) * q^47 + (4*b - 3) * q^49 + (3*b - 2) * q^53 + (-2*b - 2) * q^59 + (-3*b - 4) * q^61 + (-2*b + 4) * q^67 + (2*b - 8) * q^71 + (-6*b + 2) * q^73 - 4 * q^77 + (b - 6) * q^79 + (-7*b + 4) * q^83 + (4*b + 8) * q^89 + (-4*b - 4) * q^91 + (6*b - 6) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 2 * q^7 + 2 * q^11 - 2 * q^13 - 3 * q^17 - 3 * q^19 + 3 * q^23 - 8 * q^29 - 5 * q^31 + 8 * q^37 + 6 * q^41 - 20 * q^43 + 9 * q^47 - 2 * q^49 - q^53 - 6 * q^59 - 11 * q^61 + 6 * q^67 - 14 * q^71 - 2 * q^73 - 8 * q^77 - 11 * q^79 + q^83 + 20 * q^89 - 12 * q^91 - 6 * q^97

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

0

0

0

0

−1.23607

0

0

0

1.2

0

0

0

0

0

3.23607

0

0

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	9000.2.a.m	yes	2
3.b	odd	2	1		9000.2.a.l	yes	2
5.b	even	2	1		9000.2.a.e	yes	2
15.d	odd	2	1		9000.2.a.d	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9000.2.a.d	✓	2	15.d	odd	2	1
9000.2.a.e	yes	2	5.b	even	2	1
9000.2.a.l	yes	2	3.b	odd	2	1
9000.2.a.m	yes	2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9000))$:

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

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

$T_{13}^{2} + 2T_{13} - 4$

$T_{17}^{2} + 3T_{17} + 1$

Hecke characteristic polynomials

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