Properties

Label 576.1.o.a
Level $576$
Weight $1$
Character orbit 576.o
Analytic conductor $0.287$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,1,Mod(319,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.319");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 576.o (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: \(0.287461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.5184.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{3} q^{3} - \zeta_{12}^{2} q^{5} - \zeta_{12} q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{3} q^{3} - \zeta_{12}^{2} q^{5} - \zeta_{12} q^{7} - q^{9} - \zeta_{12} q^{11} + \zeta_{12}^{2} q^{13} + \zeta_{12}^{5} q^{15} + \zeta_{12}^{4} q^{21} - \zeta_{12}^{5} q^{23} + \zeta_{12}^{3} q^{27} - \zeta_{12}^{4} q^{29} - \zeta_{12}^{5} q^{31} + \zeta_{12}^{4} q^{33} + \zeta_{12}^{3} q^{35} - \zeta_{12}^{5} q^{39} - \zeta_{12}^{2} q^{41} + \zeta_{12} q^{43} + \zeta_{12}^{2} q^{45} - \zeta_{12} q^{47} + \zeta_{12}^{3} q^{55} - \zeta_{12}^{5} q^{59} - \zeta_{12}^{4} q^{61} + \zeta_{12} q^{63} - \zeta_{12}^{4} q^{65} + \zeta_{12}^{5} q^{67} - \zeta_{12}^{2} q^{69} + \zeta_{12}^{3} q^{71} + \zeta_{12}^{2} q^{77} + \zeta_{12} q^{79} + q^{81} + \zeta_{12} q^{83} - \zeta_{12} q^{87} - \zeta_{12}^{3} q^{91} - \zeta_{12}^{2} q^{93} + \zeta_{12}^{4} q^{97} + \zeta_{12} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 4 q^{9} + 2 q^{13} - 2 q^{21} + 2 q^{29} - 2 q^{33} - 2 q^{41} + 2 q^{45} + 2 q^{61} + 2 q^{65} - 2 q^{69} + 2 q^{77} + 4 q^{81} - 2 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 1.00000i 0 −0.500000 + 0.866025i 0 0.866025 0.500000i 0 −1.00000 0
319.2 0 1.00000i 0 −0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0 −1.00000 0
511.1 0 1.00000i 0 −0.500000 0.866025i 0 −0.866025 0.500000i 0 −1.00000 0
511.2 0 1.00000i 0 −0.500000 0.866025i 0 0.866025 + 0.500000i 0 −1.00000 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
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.1.o.a 4
3.b odd 2 1 1728.1.o.a 4
4.b odd 2 1 inner 576.1.o.a 4
8.b even 2 1 288.1.o.a 4
8.d odd 2 1 288.1.o.a 4
9.c even 3 1 inner 576.1.o.a 4
9.d odd 6 1 1728.1.o.a 4
12.b even 2 1 1728.1.o.a 4
16.e even 4 1 2304.1.t.a 4
16.e even 4 1 2304.1.t.b 4
16.f odd 4 1 2304.1.t.a 4
16.f odd 4 1 2304.1.t.b 4
24.f even 2 1 864.1.o.a 4
24.h odd 2 1 864.1.o.a 4
36.f odd 6 1 inner 576.1.o.a 4
36.h even 6 1 1728.1.o.a 4
72.j odd 6 1 864.1.o.a 4
72.j odd 6 1 2592.1.g.b 2
72.l even 6 1 864.1.o.a 4
72.l even 6 1 2592.1.g.b 2
72.n even 6 1 288.1.o.a 4
72.n even 6 1 2592.1.g.a 2
72.p odd 6 1 288.1.o.a 4
72.p odd 6 1 2592.1.g.a 2
144.v odd 12 1 2304.1.t.a 4
144.v odd 12 1 2304.1.t.b 4
144.x even 12 1 2304.1.t.a 4
144.x even 12 1 2304.1.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.1.o.a 4 8.b even 2 1
288.1.o.a 4 8.d odd 2 1
288.1.o.a 4 72.n even 6 1
288.1.o.a 4 72.p odd 6 1
576.1.o.a 4 1.a even 1 1 trivial
576.1.o.a 4 4.b odd 2 1 inner
576.1.o.a 4 9.c even 3 1 inner
576.1.o.a 4 36.f odd 6 1 inner
864.1.o.a 4 24.f even 2 1
864.1.o.a 4 24.h odd 2 1
864.1.o.a 4 72.j odd 6 1
864.1.o.a 4 72.l even 6 1
1728.1.o.a 4 3.b odd 2 1
1728.1.o.a 4 9.d odd 6 1
1728.1.o.a 4 12.b even 2 1
1728.1.o.a 4 36.h even 6 1
2304.1.t.a 4 16.e even 4 1
2304.1.t.a 4 16.f odd 4 1
2304.1.t.a 4 144.v odd 12 1
2304.1.t.a 4 144.x even 12 1
2304.1.t.b 4 16.e even 4 1
2304.1.t.b 4 16.f odd 4 1
2304.1.t.b 4 144.v odd 12 1
2304.1.t.b 4 144.x even 12 1
2592.1.g.a 2 72.n even 6 1
2592.1.g.a 2 72.p odd 6 1
2592.1.g.b 2 72.j odd 6 1
2592.1.g.b 2 72.l even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(576, [\chi])\).

Hecke characteristic polynomials

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