Properties

Label 576.3.b.e
Level $576$
Weight $3$
Character orbit 576.b
Analytic conductor $15.695$
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] = [576,3,Mod(415,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.b (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: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 192)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} - \beta_1 q^{7} + 2 \beta_{3} q^{11} + 6 \beta_{2} q^{13} + 18 q^{17} - 3 \beta_{3} q^{19} - 18 \beta_1 q^{23} + 13 q^{25} - 9 \beta_{2} q^{29} - 11 \beta_1 q^{31} - \beta_{3} q^{35} - 12 \beta_{2} q^{37} + 54 q^{41} + 3 \beta_{3} q^{43} - 18 \beta_1 q^{47} + 45 q^{49} + 29 \beta_{2} q^{53} - 24 \beta_1 q^{55} - 9 \beta_{3} q^{59} + 72 q^{65} + 9 \beta_{3} q^{67} - 54 \beta_1 q^{71} + 10 q^{73} - 8 \beta_{2} q^{77} + 25 \beta_1 q^{79} - 2 \beta_{3} q^{83} - 18 \beta_{2} q^{85} + 18 q^{89} + 6 \beta_{3} q^{91} + 36 \beta_1 q^{95} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 72 q^{17} + 52 q^{25} + 216 q^{41} + 180 q^{49} + 288 q^{65} + 40 q^{73} + 72 q^{89} - 136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{12}^{3} + 8\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 3.46410i 0 2.00000i 0 0 0
415.2 0 0 0 3.46410i 0 2.00000i 0 0 0
415.3 0 0 0 3.46410i 0 2.00000i 0 0 0
415.4 0 0 0 3.46410i 0 2.00000i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.b.e 4
3.b odd 2 1 192.3.b.a 4
4.b odd 2 1 inner 576.3.b.e 4
8.b even 2 1 inner 576.3.b.e 4
8.d odd 2 1 inner 576.3.b.e 4
12.b even 2 1 192.3.b.a 4
16.e even 4 2 2304.3.g.v 4
16.f odd 4 2 2304.3.g.v 4
24.f even 2 1 192.3.b.a 4
24.h odd 2 1 192.3.b.a 4
48.i odd 4 2 768.3.g.d 4
48.k even 4 2 768.3.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.3.b.a 4 3.b odd 2 1
192.3.b.a 4 12.b even 2 1
192.3.b.a 4 24.f even 2 1
192.3.b.a 4 24.h odd 2 1
576.3.b.e 4 1.a even 1 1 trivial
576.3.b.e 4 4.b odd 2 1 inner
576.3.b.e 4 8.b even 2 1 inner
576.3.b.e 4 8.d odd 2 1 inner
768.3.g.d 4 48.i odd 4 2
768.3.g.d 4 48.k even 4 2
2304.3.g.v 4 16.e even 4 2
2304.3.g.v 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 432)^{2} \) Copy content Toggle raw display
$17$ \( (T - 18)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 972)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 484)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1728)^{2} \) Copy content Toggle raw display
$41$ \( (T - 54)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 1296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 10092)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3888)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3888)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 11664)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2500)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T - 18)^{4} \) Copy content Toggle raw display
$97$ \( (T + 34)^{4} \) Copy content Toggle raw display
show more
show less