Properties

Label 324.1.g.a
Level $324$
Weight $1$
Character orbit 324.g
Analytic conductor $0.162$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,1,Mod(53,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.53");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 324.g (of order \(6\), degree \(2\), 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: \(0.161697064093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.108.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.314928.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{7} + \zeta_{6} q^{13} - q^{19} + \zeta_{6}^{2} q^{25} - \zeta_{6} q^{31} - q^{37} + \zeta_{6}^{2} q^{43} - \zeta_{6}^{2} q^{61} + \zeta_{6} q^{67} - q^{73} - \zeta_{6}^{2} q^{79} + q^{91} - \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{7} + q^{13} - 2 q^{19} - q^{25} - 2 q^{31} - 2 q^{37} - 2 q^{43} + q^{61} + q^{67} - 2 q^{73} + q^{79} + 2 q^{91} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 0.500000 + 0.866025i 0 0 0
269.1 0 0 0 0 0 0.500000 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.1.g.a 2
3.b odd 2 1 CM 324.1.g.a 2
4.b odd 2 1 1296.1.q.a 2
9.c even 3 1 108.1.c.a 1
9.c even 3 1 inner 324.1.g.a 2
9.d odd 6 1 108.1.c.a 1
9.d odd 6 1 inner 324.1.g.a 2
12.b even 2 1 1296.1.q.a 2
27.e even 9 6 2916.1.k.c 6
27.f odd 18 6 2916.1.k.c 6
36.f odd 6 1 432.1.e.a 1
36.f odd 6 1 1296.1.q.a 2
36.h even 6 1 432.1.e.a 1
36.h even 6 1 1296.1.q.a 2
45.h odd 6 1 2700.1.g.b 1
45.j even 6 1 2700.1.g.b 1
45.k odd 12 2 2700.1.b.b 2
45.l even 12 2 2700.1.b.b 2
72.j odd 6 1 1728.1.e.a 1
72.l even 6 1 1728.1.e.b 1
72.n even 6 1 1728.1.e.a 1
72.p odd 6 1 1728.1.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 9.c even 3 1
108.1.c.a 1 9.d odd 6 1
324.1.g.a 2 1.a even 1 1 trivial
324.1.g.a 2 3.b odd 2 1 CM
324.1.g.a 2 9.c even 3 1 inner
324.1.g.a 2 9.d odd 6 1 inner
432.1.e.a 1 36.f odd 6 1
432.1.e.a 1 36.h even 6 1
1296.1.q.a 2 4.b odd 2 1
1296.1.q.a 2 12.b even 2 1
1296.1.q.a 2 36.f odd 6 1
1296.1.q.a 2 36.h even 6 1
1728.1.e.a 1 72.j odd 6 1
1728.1.e.a 1 72.n even 6 1
1728.1.e.b 1 72.l even 6 1
1728.1.e.b 1 72.p odd 6 1
2700.1.b.b 2 45.k odd 12 2
2700.1.b.b 2 45.l even 12 2
2700.1.g.b 1 45.h odd 6 1
2700.1.g.b 1 45.j even 6 1
2916.1.k.c 6 27.e even 9 6
2916.1.k.c 6 27.f odd 18 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(324, [\chi])\).

Hecke characteristic polynomials

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