Properties

Label 3240.1.z.a
Level $3240$
Weight $1$
Character orbit 3240.z
Analytic conductor $1.617$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3240,1,Mod(379,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.379");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3240.z (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: \(1.61697064093\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1080)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.27993600.1

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \zeta_{6}^{2} q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \zeta_{6}^{2} q^{5} - q^{8} - \zeta_{6}^{2} q^{10} + q^{16} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{17} + q^{19} + \zeta_{6}^{2} q^{20} - \zeta_{6}^{2} q^{23} - \zeta_{6} q^{25} + ( - \zeta_{6} - 1) q^{31} - q^{32} + (\zeta_{6}^{2} + \zeta_{6}) q^{34} - q^{38} - \zeta_{6}^{2} q^{40} + \zeta_{6}^{2} q^{46} + \zeta_{6} q^{47} - \zeta_{6}^{2} q^{49} + \zeta_{6} q^{50} + q^{53} + ( - \zeta_{6}^{2} + 1) q^{61} + (\zeta_{6} + 1) q^{62} + q^{64} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{68} + q^{76} + (\zeta_{6}^{2} - 1) q^{79} + \zeta_{6}^{2} q^{80} + ( - \zeta_{6}^{2} + 1) q^{83} + (\zeta_{6} + 1) q^{85} - \zeta_{6}^{2} q^{92} - 2 \zeta_{6} q^{94} + \zeta_{6}^{2} q^{95} + \zeta_{6}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{8} + q^{10} + 2 q^{16} + 2 q^{19} - q^{20} + q^{23} - q^{25} - 3 q^{31} - 2 q^{32} - 2 q^{38} + q^{40} - q^{46} + 2 q^{47} + q^{49} + q^{50} + 2 q^{53} + 3 q^{61} + 3 q^{62} + 2 q^{64} + 2 q^{76} - 3 q^{79} - q^{80} + 3 q^{83} + 3 q^{85} + q^{92} - 2 q^{94} - q^{95} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 0 1.00000 −0.500000 0.866025i 0 0 −1.00000 0 0.500000 + 0.866025i
2539.1 −1.00000 0 1.00000 −0.500000 + 0.866025i 0 0 −1.00000 0 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
72.l even 6 1 inner
360.z odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.1.z.a 2
3.b odd 2 1 3240.1.z.j 2
5.b even 2 1 3240.1.z.j 2
8.d odd 2 1 3240.1.z.e 2
9.c even 3 1 1080.1.p.b yes 2
9.c even 3 1 3240.1.z.g 2
9.d odd 6 1 1080.1.p.a 2
9.d odd 6 1 3240.1.z.e 2
15.d odd 2 1 CM 3240.1.z.a 2
24.f even 2 1 3240.1.z.g 2
40.e odd 2 1 3240.1.z.g 2
45.h odd 6 1 1080.1.p.b yes 2
45.h odd 6 1 3240.1.z.g 2
45.j even 6 1 1080.1.p.a 2
45.j even 6 1 3240.1.z.e 2
72.l even 6 1 1080.1.p.b yes 2
72.l even 6 1 inner 3240.1.z.a 2
72.p odd 6 1 1080.1.p.a 2
72.p odd 6 1 3240.1.z.j 2
120.m even 2 1 3240.1.z.e 2
360.z odd 6 1 1080.1.p.b yes 2
360.z odd 6 1 inner 3240.1.z.a 2
360.bd even 6 1 1080.1.p.a 2
360.bd even 6 1 3240.1.z.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.1.p.a 2 9.d odd 6 1
1080.1.p.a 2 45.j even 6 1
1080.1.p.a 2 72.p odd 6 1
1080.1.p.a 2 360.bd even 6 1
1080.1.p.b yes 2 9.c even 3 1
1080.1.p.b yes 2 45.h odd 6 1
1080.1.p.b yes 2 72.l even 6 1
1080.1.p.b yes 2 360.z odd 6 1
3240.1.z.a 2 1.a even 1 1 trivial
3240.1.z.a 2 15.d odd 2 1 CM
3240.1.z.a 2 72.l even 6 1 inner
3240.1.z.a 2 360.z odd 6 1 inner
3240.1.z.e 2 8.d odd 2 1
3240.1.z.e 2 9.d odd 6 1
3240.1.z.e 2 45.j even 6 1
3240.1.z.e 2 120.m even 2 1
3240.1.z.g 2 9.c even 3 1
3240.1.z.g 2 24.f even 2 1
3240.1.z.g 2 40.e odd 2 1
3240.1.z.g 2 45.h odd 6 1
3240.1.z.j 2 3.b odd 2 1
3240.1.z.j 2 5.b even 2 1
3240.1.z.j 2 72.p odd 6 1
3240.1.z.j 2 360.bd even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17}^{2} + 3 \) Copy content Toggle raw display
\( T_{23}^{2} - T_{23} + 1 \) Copy content Toggle raw display
\( T_{31}^{2} + 3T_{31} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

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