Properties

Label 432.2.k.a
Level $432$
Weight $2$
Character orbit 432.k
Analytic conductor $3.450$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,2,Mod(109,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.k (of order \(4\), 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: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{3} + \zeta_{8}) q^{2} - 2 q^{4} - \zeta_{8} q^{5} + 3 \zeta_{8}^{2} q^{7} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{8}^{3} + \zeta_{8}) q^{2} - 2 q^{4} - \zeta_{8} q^{5} + 3 \zeta_{8}^{2} q^{7} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{8} + ( - \zeta_{8}^{2} + 1) q^{10} - \zeta_{8} q^{11} + (5 \zeta_{8}^{2} - 5) q^{13} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{14} + 4 q^{16} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{17} + (2 \zeta_{8}^{2} - 2) q^{19} + 2 \zeta_{8} q^{20} + ( - \zeta_{8}^{2} + 1) q^{22} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{23} - 4 \zeta_{8}^{2} q^{25} - 10 \zeta_{8} q^{26} - 6 \zeta_{8}^{2} q^{28} + 8 \zeta_{8}^{3} q^{29} - q^{31} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{32} - 6 \zeta_{8}^{2} q^{34} - 3 \zeta_{8}^{3} q^{35} + (4 \zeta_{8}^{2} + 4) q^{37} - 4 \zeta_{8} q^{38} + (2 \zeta_{8}^{2} - 2) q^{40} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{41} + (\zeta_{8}^{2} + 1) q^{43} + 2 \zeta_{8} q^{44} + 8 q^{46} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{47} - 2 q^{49} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{50} + ( - 10 \zeta_{8}^{2} + 10) q^{52} + 11 \zeta_{8} q^{53} + \zeta_{8}^{2} q^{55} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{56} + ( - 8 \zeta_{8}^{2} - 8) q^{58} - 10 \zeta_{8} q^{59} + (2 \zeta_{8}^{2} - 2) q^{61} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{62} - 8 q^{64} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{65} + (5 \zeta_{8}^{2} - 5) q^{67} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{68} + (3 \zeta_{8}^{2} + 3) q^{70} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{71} + 3 \zeta_{8}^{2} q^{73} + 8 \zeta_{8}^{3} q^{74} + ( - 4 \zeta_{8}^{2} + 4) q^{76} - 3 \zeta_{8}^{3} q^{77} + 14 q^{79} - 4 \zeta_{8} q^{80} - 4 q^{82} - 7 \zeta_{8}^{3} q^{83} + (3 \zeta_{8}^{2} + 3) q^{85} + 2 \zeta_{8}^{3} q^{86} + (2 \zeta_{8}^{2} - 2) q^{88} + (11 \zeta_{8}^{3} + 11 \zeta_{8}) q^{89} + ( - 15 \zeta_{8}^{2} - 15) q^{91} + (8 \zeta_{8}^{3} + 8 \zeta_{8}) q^{92} + 6 \zeta_{8}^{2} q^{94} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{95} + 5 q^{97} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 4 q^{10} - 20 q^{13} + 16 q^{16} - 8 q^{19} + 4 q^{22} - 4 q^{31} + 16 q^{37} - 8 q^{40} + 4 q^{43} + 32 q^{46} - 8 q^{49} + 40 q^{52} - 32 q^{58} - 8 q^{61} - 32 q^{64} - 20 q^{67} + 12 q^{70} + 16 q^{76} + 56 q^{79} - 16 q^{82} + 12 q^{85} - 8 q^{88} - 60 q^{91} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(\zeta_{8}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.41421i 0 −2.00000 −0.707107 + 0.707107i 0 3.00000i 2.82843i 0 1.00000 + 1.00000i
109.2 1.41421i 0 −2.00000 0.707107 0.707107i 0 3.00000i 2.82843i 0 1.00000 + 1.00000i
325.1 1.41421i 0 −2.00000 0.707107 + 0.707107i 0 3.00000i 2.82843i 0 1.00000 1.00000i
325.2 1.41421i 0 −2.00000 −0.707107 0.707107i 0 3.00000i 2.82843i 0 1.00000 1.00000i
\(n\): 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 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.k.a 4
3.b odd 2 1 inner 432.2.k.a 4
4.b odd 2 1 1728.2.k.a 4
12.b even 2 1 1728.2.k.a 4
16.e even 4 1 inner 432.2.k.a 4
16.f odd 4 1 1728.2.k.a 4
48.i odd 4 1 inner 432.2.k.a 4
48.k even 4 1 1728.2.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.k.a 4 1.a even 1 1 trivial
432.2.k.a 4 3.b odd 2 1 inner
432.2.k.a 4 16.e even 4 1 inner
432.2.k.a 4 48.i odd 4 1 inner
1728.2.k.a 4 4.b odd 2 1
1728.2.k.a 4 12.b even 2 1
1728.2.k.a 4 16.f odd 4 1
1728.2.k.a 4 48.k even 4 1

Hecke kernels

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

\( T_{5}^{4} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 4096 \) Copy content Toggle raw display
$31$ \( (T + 1)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 32)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 14641 \) Copy content Toggle raw display
$59$ \( T^{4} + 10000 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$79$ \( (T - 14)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 2401 \) Copy content Toggle raw display
$89$ \( (T^{2} + 242)^{2} \) Copy content Toggle raw display
$97$ \( (T - 5)^{4} \) Copy content Toggle raw display
show more
show less