LMFDB - Newform orbit 4002.2.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4002,2,Mod(1,4002)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4002, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4002.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4002 = 2 \cdot 3 \cdot 23 \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4002.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$31.9561308889$
Analytic rank:	$0$
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^{8} + q^{9} + q^{10} + 5 q^{11} + q^{12} + 3 q^{13} - q^{15} + q^{16} + 4 q^{17} - q^{18} + 4 q^{19} - q^{20} - 5 q^{22} - q^{23} - q^{24} - 4 q^{25}+ \cdots + 5 q^{99}+O(q^{100})

q - q^2 + q^3 + q^4 - q^5 - q^6 - q^8 + q^9 + q^10 + 5 * q^11 + q^12 + 3 * q^13 - q^15 + q^16 + 4 * q^17 - q^18 + 4 * q^19 - q^20 - 5 * q^22 - q^23 - q^24 - 4 * q^25 - 3 * q^26 + q^27 - q^29 + q^30 + 5 * q^31 - q^32 + 5 * q^33 - 4 * q^34 + q^36 - 3 * q^37 - 4 * q^38 + 3 * q^39 + q^40 + 9 * q^41 - 6 * q^43 + 5 * q^44 - q^45 + q^46 + 10 * q^47 + q^48 - 7 * q^49 + 4 * q^50 + 4 * q^51 + 3 * q^52 + 6 * q^53 - q^54 - 5 * q^55 + 4 * q^57 + q^58 - 7 * q^59 - q^60 + q^61 - 5 * q^62 + q^64 - 3 * q^65 - 5 * q^66 - 13 * q^67 + 4 * q^68 - q^69 - 5 * q^71 - q^72 - 4 * q^73 + 3 * q^74 - 4 * q^75 + 4 * q^76 - 3 * q^78 - 2 * q^79 - q^80 + q^81 - 9 * q^82 - 6 * q^83 - 4 * q^85 + 6 * q^86 - q^87 - 5 * q^88 - 10 * q^89 + q^90 - q^92 + 5 * q^93 - 10 * q^94 - 4 * q^95 - q^96 + 18 * q^97 + 7 * q^98 + 5 * 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

1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$23$	$+1$
$29$	$+1$

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	4002.2.a.e	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4002.2.a.e	✓	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 $S_{2}^{\mathrm{new}}(\Gamma_0(4002))$ :

T_{5} + 1

T5 + 1

T_{7}

T7

T_{11} - 5

T11 - 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 1$ T + 1
$3$	$T - 1$ T - 1
$5$	$T + 1$ T + 1
$7$	$T$ T
$11$	$T - 5$ T - 5
$13$	$T - 3$ T - 3
$17$	$T - 4$ T - 4
$19$	$T - 4$ T - 4
$23$	$T + 1$ T + 1
$29$	$T + 1$ T + 1
$31$	$T - 5$ T - 5
$37$	$T + 3$ T + 3
$41$	$T - 9$ T - 9
$43$	$T + 6$ T + 6
$47$	$T - 10$ T - 10
$53$	$T - 6$ T - 6
$59$	$T + 7$ T + 7
$61$	$T - 1$ T - 1
$67$	$T + 13$ T + 13
$71$	$T + 5$ T + 5
$73$	$T + 4$ T + 4
$79$	$T + 2$ T + 2
$83$	$T + 6$ T + 6
$89$	$T + 10$ T + 10
$97$	$T - 18$ T - 18
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