LMFDB - Newform orbit 6975.2.a.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$6975 = 3^{2} \cdot 5^{2} \cdot 31$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,-2,0,0,0,0,0,0,-5,0,0,0,0,4,5] 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:	$55.6956554098$
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 - 2 q^{4} - 5 q^{11} + 4 q^{16} + 5 q^{17} - q^{19} + 5 q^{23} - 5 q^{29} - q^{31} - 10 q^{37} + 10 q^{43} + 10 q^{44} - 7 q^{49} + 5 q^{53} + 10 q^{59} + 12 q^{61} - 8 q^{64} + 5 q^{67} - 10 q^{68}+ \cdots + 5 q^{97}+O(q^{100})

q - 2 * q^4 - 5 * q^11 + 4 * q^16 + 5 * q^17 - q^19 + 5 * q^23 - 5 * q^29 - q^31 - 10 * q^37 + 10 * q^43 + 10 * q^44 - 7 * q^49 + 5 * q^53 + 10 * q^59 + 12 * q^61 - 8 * q^64 + 5 * q^67 - 10 * q^68 + 2 * q^76 - 4 * q^79 - 15 * q^83 - 15 * q^89 - 10 * q^92 + 5 * q^97

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

−2.00000

Atkin-Lehner signs

$p$	Sign
$3$	$+1$
$5$	$+1$
$31$	$+1$

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	6975.2.a.i	yes	1
3.b	odd	2	1		6975.2.a.m	yes	1
5.b	even	2	1		6975.2.a.h	✓	1
15.d	odd	2	1		6975.2.a.n	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6975.2.a.h	✓	1	5.b	even	2	1
6975.2.a.i	yes	1	1.a	even	1	1	trivial
6975.2.a.m	yes	1	3.b	odd	2	1
6975.2.a.n	yes	1	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6975))$ :

T_{2}

T2

T_{7}

T7

T_{11} + 5

T11 + 5

T_{13}

T13

T_{17} - 5

T17 - 5

T_{29} + 5

T29 + 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T$ T
$7$	$T$ T
$11$	$T + 5$ T + 5
$13$	$T$ T
$17$	$T - 5$ T - 5
$19$	$T + 1$ T + 1
$23$	$T - 5$ T - 5
$29$	$T + 5$ T + 5
$31$	$T + 1$ T + 1
$37$	$T + 10$ T + 10
$41$	$T$ T
$43$	$T - 10$ T - 10
$47$	$T$ T
$53$	$T - 5$ T - 5
$59$	$T - 10$ T - 10
$61$	$T - 12$ T - 12
$67$	$T - 5$ T - 5
$71$	$T$ T
$73$	$T$ T
$79$	$T + 4$ T + 4
$83$	$T + 15$ T + 15
$89$	$T + 15$ T + 15
$97$	$T - 5$ T - 5
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