Newform orbit 4050.2.a.bl

• Label: 4050.2.a.bl
• Level: $4050$
• Weight: $2$
• Character orbit: 4050.a
• Self dual: yes
• Analytic conductor: $32.339$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$4050 = 2 \cdot 3^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4050.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6})$
Defining polynomial:	$x^{2} - 6$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 450)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - q^2 + q^4 + (b - 2) * q^7 - q^8 + 2*b * q^11 + (b - 2) * q^13 + (-b + 2) * q^14 + q^16 + 2*b * q^17 + (b + 5) * q^19 - 2*b * q^22 - b * q^23 + (-b + 2) * q^26 + (b - 2) * q^28 + b * q^29 + (b + 2) * q^31 - q^32 - 2*b * q^34 + (3*b + 4) * q^37 + (-b - 5) * q^38 - 9 * q^41 + (b - 5) * q^43 + 2*b * q^44 + b * q^46 + (-2*b - 6) * q^47 + (-4*b + 3) * q^49 + (b - 2) * q^52 + (b - 6) * q^53 + (-b + 2) * q^56 - b * q^58 + (b + 3) * q^59 + 8 * q^61 + (-b - 2) * q^62 + q^64 + (-3*b + 7) * q^67 + 2*b * q^68 + (-3*b - 6) * q^71 + q^73 + (-3*b - 4) * q^74 + (b + 5) * q^76 + (-4*b + 12) * q^77 + (6*b + 2) * q^79 + 9 * q^82 + (-b - 3) * q^83 + (-b + 5) * q^86 - 2*b * q^88 + 9 * q^89 + (-4*b + 10) * q^91 - b * q^92 + (2*b + 6) * q^94 + (-4*b + 1) * q^97 + (4*b - 3) * q^98
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 2 * q^2 + 2 * q^4 - 4 * q^7 - 2 * q^8 - 4 * q^13 + 4 * q^14 + 2 * q^16 + 10 * q^19 + 4 * q^26 - 4 * q^28 + 4 * q^31 - 2 * q^32 + 8 * q^37 - 10 * q^38 - 18 * q^41 - 10 * q^43 - 12 * q^47 + 6 * q^49 - 4 * q^52 - 12 * q^53 + 4 * q^56 + 6 * q^59 + 16 * q^61 - 4 * q^62 + 2 * q^64 + 14 * q^67 - 12 * q^71 + 2 * q^73 - 8 * q^74 + 10 * q^76 + 24 * q^77 + 4 * q^79 + 18 * q^82 - 6 * q^83 + 10 * q^86 + 18 * q^89 + 20 * q^91 + 12 * q^94 + 2 * q^97 - 6 * q^98

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

−2.44949
2.44949

−1.00000

0

1.00000

0

0

−4.44949

−1.00000

0

0

1.2

−1.00000

0

1.00000

0

0

0.449490

−1.00000

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	4050.2.a.bl		2
3.b	odd	2	1		4050.2.a.bu		2
5.b	even	2	1		4050.2.a.by		2
5.c	odd	4	2		4050.2.c.w		4
9.c	even	3	2		1350.2.e.n		4
9.d	odd	6	2		450.2.e.l	✓	4
15.d	odd	2	1		4050.2.a.br		2
15.e	even	4	2		4050.2.c.y		4
45.h	odd	6	2		450.2.e.m	yes	4
45.j	even	6	2		1350.2.e.k		4
45.k	odd	12	4		1350.2.j.g		8
45.l	even	12	4		450.2.j.f		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
450.2.e.l	✓	4	9.d	odd	6	2
450.2.e.m	yes	4	45.h	odd	6	2
450.2.j.f		8	45.l	even	12	4
1350.2.e.k		4	45.j	even	6	2
1350.2.e.n		4	9.c	even	3	2
1350.2.j.g		8	45.k	odd	12	4
4050.2.a.bl		2	1.a	even	1	1	trivial
4050.2.a.br		2	15.d	odd	2	1
4050.2.a.bu		2	3.b	odd	2	1
4050.2.a.by		2	5.b	even	2	1
4050.2.c.w		4	5.c	odd	4	2
4050.2.c.y		4	15.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 $S_{2}^{\mathrm{new}}(\Gamma_0(4050))$:

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

$T_{11}^{2} - 24$

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

$T_{17}^{2} - 24$

$T_{23}^{2} - 6$

$T_{41} + 9$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T + 1)^{2}$
$3$	$T^{2}$
$5$	$T^{2}$
$7$	$T^{2} + 4T - 2$
$11$	$T^{2} - 24$