Properties

Label 432.2.c.a
Level $432$
Weight $2$
Character orbit 432.c
Analytic conductor $3.450$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,2,Mod(431,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.c (of order \(2\), degree \(1\), 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: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{7} - 5 q^{13} - \beta q^{19} + 5 q^{25} - 2 \beta q^{31} + 11 q^{37} + 2 \beta q^{43} - 20 q^{49} - q^{61} + 3 \beta q^{67} + 7 q^{73} - \beta q^{79} + 5 \beta q^{91} + 19 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{13} + 10 q^{25} + 22 q^{37} - 40 q^{49} - 2 q^{61} + 14 q^{73} + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 5.19615i 0 0 0
431.2 0 0 0 0 0 5.19615i 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 \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.c.a 2
3.b odd 2 1 CM 432.2.c.a 2
4.b odd 2 1 inner 432.2.c.a 2
8.b even 2 1 1728.2.c.b 2
8.d odd 2 1 1728.2.c.b 2
9.c even 3 1 1296.2.s.a 2
9.c even 3 1 1296.2.s.f 2
9.d odd 6 1 1296.2.s.a 2
9.d odd 6 1 1296.2.s.f 2
12.b even 2 1 inner 432.2.c.a 2
24.f even 2 1 1728.2.c.b 2
24.h odd 2 1 1728.2.c.b 2
36.f odd 6 1 1296.2.s.a 2
36.f odd 6 1 1296.2.s.f 2
36.h even 6 1 1296.2.s.a 2
36.h even 6 1 1296.2.s.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.a 2 1.a even 1 1 trivial
432.2.c.a 2 3.b odd 2 1 CM
432.2.c.a 2 4.b odd 2 1 inner
432.2.c.a 2 12.b even 2 1 inner
1296.2.s.a 2 9.c even 3 1
1296.2.s.a 2 9.d odd 6 1
1296.2.s.a 2 36.f odd 6 1
1296.2.s.a 2 36.h even 6 1
1296.2.s.f 2 9.c even 3 1
1296.2.s.f 2 9.d odd 6 1
1296.2.s.f 2 36.f odd 6 1
1296.2.s.f 2 36.h even 6 1
1728.2.c.b 2 8.b even 2 1
1728.2.c.b 2 8.d odd 2 1
1728.2.c.b 2 24.f even 2 1
1728.2.c.b 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(432, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} + 27 \) Copy content Toggle raw display

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
$7$ \( T^{2} + 27 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 27 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 108 \) Copy content Toggle raw display
$37$ \( (T - 11)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 108 \) 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
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 243 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 7)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 27 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 19)^{2} \) Copy content Toggle raw display
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