LMFDB - Newform orbit 6090.2.a.g

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6090,2,Mod(1,6090)] 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("6090.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6090, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$6090 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6090.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,1,1,-1,-1,-1,-1,1,1,0,1,0,1,-1,1,-2] 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:	$48.6288948310$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + q^{14} - q^{15} + q^{16} - 2 q^{17} - q^{18} + 6 q^{19} - q^{20} - q^{21} - 2 q^{23} - q^{24} + q^{25}+ \cdots - q^{98}+O(q^{100})

q - q^2 + q^3 + q^4 - q^5 - q^6 - q^7 - q^8 + q^9 + q^10 + q^12 + q^14 - q^15 + q^16 - 2 * q^17 - q^18 + 6 * q^19 - q^20 - q^21 - 2 * q^23 - q^24 + q^25 + q^27 - q^28 + q^29 + q^30 - 4 * q^31 - q^32 + 2 * q^34 + q^35 + q^36 + 2 * q^37 - 6 * q^38 + q^40 - 12 * q^41 + q^42 - 4 * q^43 - q^45 + 2 * q^46 - 12 * q^47 + q^48 + q^49 - q^50 - 2 * q^51 + 6 * q^53 - q^54 + q^56 + 6 * q^57 - q^58 + 4 * q^59 - q^60 + 4 * q^62 - q^63 + q^64 + 2 * q^67 - 2 * q^68 - 2 * q^69 - q^70 + 12 * q^71 - q^72 + 6 * q^73 - 2 * q^74 + q^75 + 6 * q^76 - 4 * q^79 - q^80 + q^81 + 12 * q^82 - 8 * q^83 - q^84 + 2 * q^85 + 4 * q^86 + q^87 + 4 * q^89 + q^90 - 2 * q^92 - 4 * q^93 + 12 * q^94 - 6 * q^95 - q^96 - 14 * q^97 - 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.

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

1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$5$	$+1$
$7$	$+1$
$29$	$-1$

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	6090.2.a.g	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6090.2.a.g	✓	1	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 $S_{2}^{\mathrm{new}}(\Gamma_0(6090))$ :

T_{11}

T11

T_{13}

T13

T_{17} + 2

T17 + 2

T_{19} - 6

T19 - 6

T_{23} + 2

T23 + 2

Hecke characteristic polynomials

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