Properties

Label 2527.1.bd.a
Level $2527$
Weight $1$
Character orbit 2527.bd
Analytic conductor $1.261$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -19
Inner twists $12$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2527,1,Mod(116,2527)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2527, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([12, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2527.116");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2527 = 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2527.bd (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.26113728692\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.931.1
Artin image: $C_3^2:C_{18}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{54} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{18} q^{4} - \zeta_{18}^{8} q^{5} + \zeta_{18}^{6} q^{7} - \zeta_{18} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18} q^{4} - \zeta_{18}^{8} q^{5} + \zeta_{18}^{6} q^{7} - \zeta_{18} q^{9} + \zeta_{18}^{3} q^{11} + \zeta_{18}^{2} q^{16} + \zeta_{18}^{8} q^{17} - q^{20} + \zeta_{18}^{7} q^{23} - \zeta_{18}^{7} q^{28} + \zeta_{18}^{5} q^{35} + \zeta_{18}^{2} q^{36} + \zeta_{18}^{5} q^{43} - \zeta_{18}^{4} q^{44} - q^{45} + \zeta_{18} q^{47} - \zeta_{18}^{3} q^{49} + \zeta_{18}^{2} q^{55} + \zeta_{18}^{7} q^{61} - \zeta_{18}^{7} q^{63} - \zeta_{18}^{3} q^{64} + 2 q^{68} + \zeta_{18}^{5} q^{73} - q^{77} + \zeta_{18} q^{80} + \zeta_{18}^{2} q^{81} + \zeta_{18}^{3} q^{83} + 2 \zeta_{18}^{7} q^{85} - \zeta_{18}^{8} q^{92} - \zeta_{18}^{4} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{7} + 3 q^{11} - 6 q^{20} - 6 q^{45} - 3 q^{49} - 3 q^{64} + 12 q^{68} - 6 q^{77} + 3 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2527\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(1807\)
\(\chi(n)\) \(-\zeta_{18}^{3}\) \(-\zeta_{18}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.766044 + 0.642788i
0.939693 0.342020i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.173648 + 0.984808i
−0.173648 0.984808i
0 0 0.766044 0.642788i −0.766044 0.642788i 0 −0.500000 + 0.866025i 0 0.766044 0.642788i 0
849.1 0 0 −0.939693 + 0.342020i 0.939693 + 0.342020i 0 −0.500000 0.866025i 0 −0.939693 + 0.342020i 0
1390.1 0 0 −0.939693 0.342020i 0.939693 0.342020i 0 −0.500000 + 0.866025i 0 −0.939693 0.342020i 0
1416.1 0 0 0.766044 + 0.642788i −0.766044 + 0.642788i 0 −0.500000 0.866025i 0 0.766044 + 0.642788i 0
2067.1 0 0 0.173648 0.984808i −0.173648 0.984808i 0 −0.500000 0.866025i 0 0.173648 0.984808i 0
2104.1 0 0 0.173648 + 0.984808i −0.173648 + 0.984808i 0 −0.500000 + 0.866025i 0 0.173648 + 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
19.c even 3 2 inner
19.d odd 6 2 inner
133.u even 9 3 inner
133.bd odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2527.1.bd.a 6
7.c even 3 1 2527.1.be.a 6
19.b odd 2 1 CM 2527.1.bd.a 6
19.c even 3 2 inner 2527.1.bd.a 6
19.d odd 6 2 inner 2527.1.bd.a 6
19.e even 9 1 133.1.r.a 2
19.e even 9 1 2527.1.j.a 2
19.e even 9 1 2527.1.n.a 2
19.e even 9 3 2527.1.be.a 6
19.f odd 18 1 133.1.r.a 2
19.f odd 18 1 2527.1.j.a 2
19.f odd 18 1 2527.1.n.a 2
19.f odd 18 3 2527.1.be.a 6
57.j even 18 1 1197.1.cz.a 2
57.l odd 18 1 1197.1.cz.a 2
76.k even 18 1 2128.1.cl.c 2
76.l odd 18 1 2128.1.cl.c 2
95.o odd 18 1 3325.1.bm.a 2
95.p even 18 1 3325.1.bm.a 2
95.q odd 36 2 3325.1.y.a 4
95.r even 36 2 3325.1.y.a 4
133.g even 3 1 2527.1.be.a 6
133.h even 3 1 2527.1.be.a 6
133.j odd 6 1 2527.1.be.a 6
133.n odd 6 1 2527.1.be.a 6
133.r odd 6 1 2527.1.be.a 6
133.u even 9 1 931.1.b.a 1
133.u even 9 3 inner 2527.1.bd.a 6
133.w even 9 1 133.1.r.a 2
133.w even 9 1 2527.1.j.a 2
133.w even 9 1 2527.1.n.a 2
133.x odd 18 1 931.1.b.b 1
133.y odd 18 1 931.1.r.a 2
133.z odd 18 1 931.1.r.a 2
133.ba even 18 1 931.1.r.a 2
133.bb even 18 1 931.1.b.b 1
133.bd odd 18 1 931.1.b.a 1
133.bd odd 18 3 inner 2527.1.bd.a 6
133.be odd 18 1 133.1.r.a 2
133.be odd 18 1 2527.1.j.a 2
133.be odd 18 1 2527.1.n.a 2
133.bf even 18 1 931.1.r.a 2
399.br even 18 1 1197.1.cz.a 2
399.ca odd 18 1 1197.1.cz.a 2
532.bs even 18 1 2128.1.cl.c 2
532.bt odd 18 1 2128.1.cl.c 2
665.ct odd 18 1 3325.1.bm.a 2
665.dc even 18 1 3325.1.bm.a 2
665.dq odd 36 2 3325.1.y.a 4
665.dr even 36 2 3325.1.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.1.r.a 2 19.e even 9 1
133.1.r.a 2 19.f odd 18 1
133.1.r.a 2 133.w even 9 1
133.1.r.a 2 133.be odd 18 1
931.1.b.a 1 133.u even 9 1
931.1.b.a 1 133.bd odd 18 1
931.1.b.b 1 133.x odd 18 1
931.1.b.b 1 133.bb even 18 1
931.1.r.a 2 133.y odd 18 1
931.1.r.a 2 133.z odd 18 1
931.1.r.a 2 133.ba even 18 1
931.1.r.a 2 133.bf even 18 1
1197.1.cz.a 2 57.j even 18 1
1197.1.cz.a 2 57.l odd 18 1
1197.1.cz.a 2 399.br even 18 1
1197.1.cz.a 2 399.ca odd 18 1
2128.1.cl.c 2 76.k even 18 1
2128.1.cl.c 2 76.l odd 18 1
2128.1.cl.c 2 532.bs even 18 1
2128.1.cl.c 2 532.bt odd 18 1
2527.1.j.a 2 19.e even 9 1
2527.1.j.a 2 19.f odd 18 1
2527.1.j.a 2 133.w even 9 1
2527.1.j.a 2 133.be odd 18 1
2527.1.n.a 2 19.e even 9 1
2527.1.n.a 2 19.f odd 18 1
2527.1.n.a 2 133.w even 9 1
2527.1.n.a 2 133.be odd 18 1
2527.1.bd.a 6 1.a even 1 1 trivial
2527.1.bd.a 6 19.b odd 2 1 CM
2527.1.bd.a 6 19.c even 3 2 inner
2527.1.bd.a 6 19.d odd 6 2 inner
2527.1.bd.a 6 133.u even 9 3 inner
2527.1.bd.a 6 133.bd odd 18 3 inner
2527.1.be.a 6 7.c even 3 1
2527.1.be.a 6 19.e even 9 3
2527.1.be.a 6 19.f odd 18 3
2527.1.be.a 6 133.g even 3 1
2527.1.be.a 6 133.h even 3 1
2527.1.be.a 6 133.j odd 6 1
2527.1.be.a 6 133.n odd 6 1
2527.1.be.a 6 133.r odd 6 1
3325.1.y.a 4 95.q odd 36 2
3325.1.y.a 4 95.r even 36 2
3325.1.y.a 4 665.dq odd 36 2
3325.1.y.a 4 665.dr even 36 2
3325.1.bm.a 2 95.o odd 18 1
3325.1.bm.a 2 95.p even 18 1
3325.1.bm.a 2 665.ct odd 18 1
3325.1.bm.a 2 665.dc even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2527, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 8T^{3} + 64 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
show more
show less