Properties

Label 576.3.g.e
Level $576$
Weight $3$
Character orbit 576.g
Analytic conductor $15.695$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,3,Mod(127,576)]
 
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, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.g (of order \(2\), degree \(1\), 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: \(15.6948632272\)
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: \( 2^{3} \)
Twist minimal: no (minimal twist has level 12)
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 \(\beta = 4\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - \beta q^{7} + \beta q^{11} - 2 q^{13} - 10 q^{17} - 3 \beta q^{19} + 4 \beta q^{23} - 21 q^{25} - 26 q^{29} + \beta q^{31} + 2 \beta q^{35} - 26 q^{37} - 58 q^{41} + 7 \beta q^{43} - 10 \beta q^{47} + q^{49} - 74 q^{53} - 2 \beta q^{55} - 13 \beta q^{59} - 26 q^{61} + 4 q^{65} - \beta q^{67} - 46 q^{73} + 48 q^{77} + 17 \beta q^{79} + 7 \beta q^{83} + 20 q^{85} - 82 q^{89} + 2 \beta q^{91} + 6 \beta q^{95} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 4 q^{13} - 20 q^{17} - 42 q^{25} - 52 q^{29} - 52 q^{37} - 116 q^{41} + 2 q^{49} - 148 q^{53} - 52 q^{61} + 8 q^{65} - 92 q^{73} + 96 q^{77} + 40 q^{85} - 164 q^{89} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −2.00000 0 6.92820i 0 0 0
127.2 0 0 0 −2.00000 0 6.92820i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.g.e 2
3.b odd 2 1 192.3.g.b 2
4.b odd 2 1 inner 576.3.g.e 2
8.b even 2 1 36.3.d.c 2
8.d odd 2 1 36.3.d.c 2
12.b even 2 1 192.3.g.b 2
16.e even 4 2 2304.3.b.l 4
16.f odd 4 2 2304.3.b.l 4
24.f even 2 1 12.3.d.a 2
24.h odd 2 1 12.3.d.a 2
40.e odd 2 1 900.3.c.e 2
40.f even 2 1 900.3.c.e 2
40.i odd 4 2 900.3.f.c 4
40.k even 4 2 900.3.f.c 4
48.i odd 4 2 768.3.b.c 4
48.k even 4 2 768.3.b.c 4
72.j odd 6 1 324.3.f.d 2
72.j odd 6 1 324.3.f.j 2
72.l even 6 1 324.3.f.d 2
72.l even 6 1 324.3.f.j 2
72.n even 6 1 324.3.f.a 2
72.n even 6 1 324.3.f.g 2
72.p odd 6 1 324.3.f.a 2
72.p odd 6 1 324.3.f.g 2
120.i odd 2 1 300.3.c.b 2
120.m even 2 1 300.3.c.b 2
120.q odd 4 2 300.3.f.a 4
120.w even 4 2 300.3.f.a 4
168.e odd 2 1 588.3.g.b 2
168.i even 2 1 588.3.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 24.f even 2 1
12.3.d.a 2 24.h odd 2 1
36.3.d.c 2 8.b even 2 1
36.3.d.c 2 8.d odd 2 1
192.3.g.b 2 3.b odd 2 1
192.3.g.b 2 12.b even 2 1
300.3.c.b 2 120.i odd 2 1
300.3.c.b 2 120.m even 2 1
300.3.f.a 4 120.q odd 4 2
300.3.f.a 4 120.w even 4 2
324.3.f.a 2 72.n even 6 1
324.3.f.a 2 72.p odd 6 1
324.3.f.d 2 72.j odd 6 1
324.3.f.d 2 72.l even 6 1
324.3.f.g 2 72.n even 6 1
324.3.f.g 2 72.p odd 6 1
324.3.f.j 2 72.j odd 6 1
324.3.f.j 2 72.l even 6 1
576.3.g.e 2 1.a even 1 1 trivial
576.3.g.e 2 4.b odd 2 1 inner
588.3.g.b 2 168.e odd 2 1
588.3.g.b 2 168.i even 2 1
768.3.b.c 4 48.i odd 4 2
768.3.b.c 4 48.k even 4 2
900.3.c.e 2 40.e odd 2 1
900.3.c.e 2 40.f even 2 1
900.3.f.c 4 40.i odd 4 2
900.3.f.c 4 40.k even 4 2
2304.3.b.l 4 16.e even 4 2
2304.3.b.l 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2 \) acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\). 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
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 48 \) Copy content Toggle raw display
$11$ \( T^{2} + 48 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T + 10)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 432 \) Copy content Toggle raw display
$23$ \( T^{2} + 768 \) Copy content Toggle raw display
$29$ \( (T + 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 48 \) Copy content Toggle raw display
$37$ \( (T + 26)^{2} \) Copy content Toggle raw display
$41$ \( (T + 58)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2352 \) Copy content Toggle raw display
$47$ \( T^{2} + 4800 \) Copy content Toggle raw display
$53$ \( (T + 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8112 \) Copy content Toggle raw display
$61$ \( (T + 26)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 48 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 46)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 13872 \) Copy content Toggle raw display
$83$ \( T^{2} + 2352 \) Copy content Toggle raw display
$89$ \( (T + 82)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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