Properties

Label 576.4.d.f
Level $576$
Weight $4$
Character orbit 576.d
Analytic conductor $33.985$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,4,Mod(289,576)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 576.d (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: \(33.9851001633\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + \beta_{2} q^{7} + 12 \beta_1 q^{11} - 4 \beta_{3} q^{13} + 42 q^{17} - 23 \beta_1 q^{19} + 2 \beta_{2} q^{23} - 271 q^{25} - \beta_{3} q^{29} + 7 \beta_{2} q^{31} + 99 \beta_1 q^{35} + 10 \beta_{3} q^{37} + 6 q^{41} + 23 \beta_1 q^{43} + 2 \beta_{2} q^{47} + 53 q^{49} + 25 \beta_{3} q^{53} - 48 \beta_{2} q^{55} + 129 \beta_1 q^{59} + 18 \beta_{3} q^{61} - 1584 q^{65} + 131 \beta_1 q^{67} - 50 \beta_{2} q^{71} - 430 q^{73} - 48 \beta_{3} q^{77} + 59 \beta_{2} q^{79} - 108 \beta_1 q^{83} - 42 \beta_{3} q^{85} - 630 q^{89} + 396 \beta_1 q^{91} + 92 \beta_{2} q^{95} + 862 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 168 q^{17} - 1084 q^{25} + 24 q^{41} + 212 q^{49} - 6336 q^{65} - 1720 q^{73} - 2520 q^{89} + 3448 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{3} + 16\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 12\nu^{2} - 30 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{2} - 3\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 30 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} - 6\beta_1 ) / 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.65831 0.500000i
1.65831 + 0.500000i
−1.65831 + 0.500000i
1.65831 0.500000i
0 0 0 19.8997i 0 −19.8997 0 0 0
289.2 0 0 0 19.8997i 0 19.8997 0 0 0
289.3 0 0 0 19.8997i 0 −19.8997 0 0 0
289.4 0 0 0 19.8997i 0 19.8997 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 576.4.d.f 4
3.b odd 2 1 192.4.d.b 4
4.b odd 2 1 inner 576.4.d.f 4
8.b even 2 1 inner 576.4.d.f 4
8.d odd 2 1 inner 576.4.d.f 4
12.b even 2 1 192.4.d.b 4
16.e even 4 1 2304.4.a.y 2
16.e even 4 1 2304.4.a.bl 2
16.f odd 4 1 2304.4.a.y 2
16.f odd 4 1 2304.4.a.bl 2
24.f even 2 1 192.4.d.b 4
24.h odd 2 1 192.4.d.b 4
48.i odd 4 1 768.4.a.i 2
48.i odd 4 1 768.4.a.l 2
48.k even 4 1 768.4.a.i 2
48.k even 4 1 768.4.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.d.b 4 3.b odd 2 1
192.4.d.b 4 12.b even 2 1
192.4.d.b 4 24.f even 2 1
192.4.d.b 4 24.h odd 2 1
576.4.d.f 4 1.a even 1 1 trivial
576.4.d.f 4 4.b odd 2 1 inner
576.4.d.f 4 8.b even 2 1 inner
576.4.d.f 4 8.d odd 2 1 inner
768.4.a.i 2 48.i odd 4 1
768.4.a.i 2 48.k even 4 1
768.4.a.l 2 48.i odd 4 1
768.4.a.l 2 48.k even 4 1
2304.4.a.y 2 16.e even 4 1
2304.4.a.y 2 16.f odd 4 1
2304.4.a.bl 2 16.e even 4 1
2304.4.a.bl 2 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} + 396 \) Copy content Toggle raw display
\( T_{7}^{2} - 396 \) Copy content Toggle raw display
\( T_{17} - 42 \) 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
$5$ \( (T^{2} + 396)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 396)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2304)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6336)^{2} \) Copy content Toggle raw display
$17$ \( (T - 42)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8464)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 1584)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 396)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 19404)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 39600)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8464)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1584)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 247500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 266256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 128304)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 274576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 990000)^{2} \) Copy content Toggle raw display
$73$ \( (T + 430)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 1378476)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 186624)^{2} \) Copy content Toggle raw display
$89$ \( (T + 630)^{4} \) Copy content Toggle raw display
$97$ \( (T - 862)^{4} \) Copy content Toggle raw display
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