LMFDB - Newform orbit 2450.2.a.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2450,2,Mod(1,2450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$2450 = 2 \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,1,0,0,0,-1,-3,0,-2,0,0,0,0,1,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.5633484952$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 350)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

f(q)

=

q - q^{2} + q^{4} - q^{8} - 3 q^{9} - 2 q^{11} + q^{16} + 7 q^{17} + 3 q^{18} + 2 q^{22} - 3 q^{23} + 6 q^{29} - 7 q^{31} - q^{32} - 7 q^{34} - 3 q^{36} + 4 q^{37} - 7 q^{41} + 8 q^{43} - 2 q^{44}+ \cdots + 6 q^{99}+O(q^{100})

q - q^2 + q^4 - q^8 - 3 * q^9 - 2 * q^11 + q^16 + 7 * q^17 + 3 * q^18 + 2 * q^22 - 3 * q^23 + 6 * q^29 - 7 * q^31 - q^32 - 7 * q^34 - 3 * q^36 + 4 * q^37 - 7 * q^41 + 8 * q^43 - 2 * q^44 + 3 * q^46 + 7 * q^47 - 4 * q^53 - 6 * q^58 - 14 * q^59 - 14 * q^61 + 7 * q^62 + q^64 - 12 * q^67 + 7 * q^68 - q^71 + 3 * q^72 - 14 * q^73 - 4 * q^74 - 11 * q^79 + 9 * q^81 + 7 * q^82 - 14 * q^83 - 8 * q^86 + 2 * q^88 + 7 * q^89 - 3 * q^92 - 7 * q^94 + 7 * q^97 + 6 * 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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

−1.00000

1.00000

−1.00000

−3.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$
$7$	$+1$

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	2450.2.a.i		1
5.b	even	2	1		2450.2.a.y		1
5.c	odd	4	2		2450.2.c.i		2
7.b	odd	2	1		2450.2.a.h		1
7.c	even	3	2		350.2.e.i	yes	2
35.c	odd	2	1		2450.2.a.z		1
35.f	even	4	2		2450.2.c.j		2
35.j	even	6	2		350.2.e.d	✓	2
35.l	odd	12	4		350.2.j.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
350.2.e.d	✓	2	35.j	even	6	2
350.2.e.i	yes	2	7.c	even	3	2
350.2.j.c		4	35.l	odd	12	4
2450.2.a.h		1	7.b	odd	2	1
2450.2.a.i		1	1.a	even	1	1	trivial
2450.2.a.y		1	5.b	even	2	1
2450.2.a.z		1	35.c	odd	2	1
2450.2.c.i		2	5.c	odd	4	2
2450.2.c.j		2	35.f	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2450))$ :

T_{3}

T3

T_{11} + 2

T11 + 2

T_{13}

T13

T_{17} - 7

T17 - 7

T_{19}

T19

T_{23} + 3

T23 + 3

T_{37} - 4

T37 - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 1$ T + 1
$3$	$T$ T
$5$	$T$ T
$7$	$T$ T
$11$	$T + 2$ T + 2
$13$	$T$ T
$17$	$T - 7$ T - 7
$19$	$T$ T
$23$	$T + 3$ T + 3
$29$	$T - 6$ T - 6
$31$	$T + 7$ T + 7
$37$	$T - 4$ T - 4
$41$	$T + 7$ T + 7
$43$	$T - 8$ T - 8
$47$	$T - 7$ T - 7
$53$	$T + 4$ T + 4
$59$	$T + 14$ T + 14
$61$	$T + 14$ T + 14
$67$	$T + 12$ T + 12
$71$	$T + 1$ T + 1
$73$	$T + 14$ T + 14
$79$	$T + 11$ T + 11
$83$	$T + 14$ T + 14
$89$	$T - 7$ T - 7
$97$	$T - 7$ T - 7
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