Properties

Label 531.1.g.a
Level $531$
Weight $1$
Character orbit 531.g
Analytic conductor $0.265$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -59
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,1,Mod(58,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.58");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 531.g (of order \(6\), degree \(2\), 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: \(0.265003521708\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.6439662447201.3

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{18}^{8} q^{3} + \zeta_{18}^{6} q^{4} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{7} - \zeta_{18}^{7} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{18}^{8} q^{3} + \zeta_{18}^{6} q^{4} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{7} - \zeta_{18}^{7} q^{9} - \zeta_{18}^{5} q^{12} + ( - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{15} - \zeta_{18}^{3} q^{16} - q^{17} + (\zeta_{18}^{8} - \zeta_{18}) q^{19} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{20} + (\zeta_{18}^{4} + 1) q^{21} + (\zeta_{18}^{8} - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{25} + \zeta_{18}^{6} q^{27} + ( - \zeta_{18}^{7} + \zeta_{18}^{2}) q^{28} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{29} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2) q^{35} + \zeta_{18}^{4} q^{36} + (\zeta_{18}^{2} - \zeta_{18}) q^{41} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{45} + \zeta_{18}^{2} q^{48} + (\zeta_{18}^{6} + \zeta_{18}^{2} - \zeta_{18}) q^{49} - \zeta_{18}^{8} q^{51} + (\zeta_{18}^{8} - \zeta_{18}) q^{53} + ( - \zeta_{18}^{7} + 1) q^{57} + \zeta_{18}^{6} q^{59} + (\zeta_{18}^{4} + 1) q^{60} + (\zeta_{18}^{8} - \zeta_{18}^{3}) q^{63} + q^{64} - \zeta_{18}^{6} q^{68} - q^{71} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} + \zeta_{18}^{2}) q^{75} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{76} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{79} + ( - \zeta_{18}^{7} + \zeta_{18}^{2}) q^{80} - \zeta_{18}^{5} q^{81} + (\zeta_{18}^{6} - \zeta_{18}) q^{84} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{85} + ( - \zeta_{18}^{3} - \zeta_{18}) q^{87} + ( - \zeta_{18}^{7} - \zeta_{18}^{5} - \zeta_{18}^{3} + 1) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{4} - 3 q^{15} - 3 q^{16} - 6 q^{17} + 6 q^{21} - 3 q^{25} - 3 q^{27} + 12 q^{35} - 3 q^{45} - 3 q^{49} + 6 q^{57} - 3 q^{59} + 6 q^{60} - 3 q^{63} + 6 q^{64} + 3 q^{68} - 6 q^{71} - 3 q^{75} - 3 q^{84} - 3 q^{87} + 3 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(\zeta_{18}^{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
0 −0.939693 + 0.342020i −0.500000 + 0.866025i −0.766044 + 1.32683i 0 −0.766044 1.32683i 0 0.766044 0.642788i 0
58.2 0 0.173648 0.984808i −0.500000 + 0.866025i 0.939693 1.62760i 0 0.939693 + 1.62760i 0 −0.939693 0.342020i 0
58.3 0 0.766044 + 0.642788i −0.500000 + 0.866025i −0.173648 + 0.300767i 0 −0.173648 0.300767i 0 0.173648 + 0.984808i 0
412.1 0 −0.939693 0.342020i −0.500000 0.866025i −0.766044 1.32683i 0 −0.766044 + 1.32683i 0 0.766044 + 0.642788i 0
412.2 0 0.173648 + 0.984808i −0.500000 0.866025i 0.939693 + 1.62760i 0 0.939693 1.62760i 0 −0.939693 + 0.342020i 0
412.3 0 0.766044 0.642788i −0.500000 0.866025i −0.173648 0.300767i 0 −0.173648 + 0.300767i 0 0.173648 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)
9.c even 3 1 inner
531.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.1.g.a 6
3.b odd 2 1 1593.1.g.a 6
9.c even 3 1 inner 531.1.g.a 6
9.d odd 6 1 1593.1.g.a 6
59.b odd 2 1 CM 531.1.g.a 6
177.d even 2 1 1593.1.g.a 6
531.f even 6 1 1593.1.g.a 6
531.g odd 6 1 inner 531.1.g.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.1.g.a 6 1.a even 1 1 trivial
531.1.g.a 6 9.c even 3 1 inner
531.1.g.a 6 59.b odd 2 1 CM
531.1.g.a 6 531.g odd 6 1 inner
1593.1.g.a 6 3.b odd 2 1
1593.1.g.a 6 9.d odd 6 1
1593.1.g.a 6 177.d even 2 1
1593.1.g.a 6 531.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(531, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T + 1)^{6} \) Copy content Toggle raw display
$19$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( (T + 1)^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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