Newform orbit 5850.2.e.h

• Label: 5850.2.e.h
• Level: $5850$
• Weight: $2$
• Character orbit: 5850.e
• Analytic conductor: $46.712$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5850.e (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

5149.1

		1.00000i
	−	1.00000i

−

1.00000i

0

−1.00000

0

0

2.00000i

1.00000i

0

0

5149.2

1.00000i

0

−1.00000

0

0

−

2.00000i

−

1.00000i

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5850.2.e.h		2
3.b	odd	2	1		1950.2.e.m		2
5.b	even	2	1	inner	5850.2.e.h		2
5.c	odd	4	1		1170.2.a.j		1
5.c	odd	4	1		5850.2.a.s		1
15.d	odd	2	1		1950.2.e.m		2
15.e	even	4	1		390.2.a.b	✓	1
15.e	even	4	1		1950.2.a.ba		1
20.e	even	4	1		9360.2.a.v		1
60.l	odd	4	1		3120.2.a.y		1
195.j	odd	4	1		5070.2.b.f		2
195.s	even	4	1		5070.2.a.n		1
195.u	odd	4	1		5070.2.b.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.a.b	✓	1	15.e	even	4	1
1170.2.a.j		1	5.c	odd	4	1
1950.2.a.ba		1	15.e	even	4	1
1950.2.e.m		2	3.b	odd	2	1
1950.2.e.m		2	15.d	odd	2	1
3120.2.a.y		1	60.l	odd	4	1
5070.2.a.n		1	195.s	even	4	1
5070.2.b.f		2	195.j	odd	4	1
5070.2.b.f		2	195.u	odd	4	1
5850.2.a.s		1	5.c	odd	4	1
5850.2.e.h		2	1.a	even	1	1	trivial
5850.2.e.h		2	5.b	even	2	1	inner
9360.2.a.v		1	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(5850, [\chi])$:

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

$T_{11} + 4$

$T_{17}^{2} + 16$

$T_{19} - 2$

$T_{29} - 8$

$T_{31} - 4$

Hecke characteristic polynomials

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