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Properties

Label 864.3.q.b
Level $864$
Weight $3$
Character orbit 864.q
Analytic conductor $23.542$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [864,3,Mod(449,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 864.q (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: \(23.5422948407\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 60 q^{25} - 72 q^{29} + 36 q^{41} - 132 q^{49} + 96 q^{61} - 576 q^{65} + 24 q^{73} - 432 q^{77} - 96 q^{85} + 252 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1 0 0 0 −6.92504 − 3.99817i 0 6.10647 + 10.5767i 0 0 0
449.2 0 0 0 −6.92504 − 3.99817i 0 −6.10647 − 10.5767i 0 0 0
449.3 0 0 0 −3.32266 − 1.91834i 0 2.70775 + 4.68995i 0 0 0
449.4 0 0 0 −3.32266 − 1.91834i 0 −2.70775 − 4.68995i 0 0 0
449.5 0 0 0 −1.15965 − 0.669525i 0 −0.328661 − 0.569258i 0 0 0
449.6 0 0 0 −1.15965 − 0.669525i 0 0.328661 + 0.569258i 0 0 0
449.7 0 0 0 −0.439631 − 0.253821i 0 −6.44757 − 11.1675i 0 0 0
449.8 0 0 0 −0.439631 − 0.253821i 0 6.44757 + 11.1675i 0 0 0
449.9 0 0 0 4.45884 + 2.57431i 0 1.35076 + 2.33958i 0 0 0
449.10 0 0 0 4.45884 + 2.57431i 0 −1.35076 − 2.33958i 0 0 0
449.11 0 0 0 7.38813 + 4.26554i 0 1.36942 + 2.37191i 0 0 0
449.12 0 0 0 7.38813 + 4.26554i 0 −1.36942 − 2.37191i 0 0 0
737.1 0 0 0 −6.92504 + 3.99817i 0 6.10647 − 10.5767i 0 0 0
737.2 0 0 0 −6.92504 + 3.99817i 0 −6.10647 + 10.5767i 0 0 0
737.3 0 0 0 −3.32266 + 1.91834i 0 2.70775 − 4.68995i 0 0 0
737.4 0 0 0 −3.32266 + 1.91834i 0 −2.70775 + 4.68995i 0 0 0
737.5 0 0 0 −1.15965 + 0.669525i 0 −0.328661 + 0.569258i 0 0 0
737.6 0 0 0 −1.15965 + 0.669525i 0 0.328661 − 0.569258i 0 0 0
737.7 0 0 0 −0.439631 + 0.253821i 0 −6.44757 + 11.1675i 0 0 0
737.8 0 0 0 −0.439631 + 0.253821i 0 6.44757 − 11.1675i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.3.q.b 24
3.b odd 2 1 288.3.q.a ✓ 24
4.b odd 2 1 inner 864.3.q.b 24
8.b even 2 1 1728.3.q.l 24
8.d odd 2 1 1728.3.q.l 24
9.c even 3 1 288.3.q.a ✓ 24
9.c even 3 1 2592.3.e.j 24
9.d odd 6 1 inner 864.3.q.b 24
9.d odd 6 1 2592.3.e.j 24
12.b even 2 1 288.3.q.a ✓ 24
24.f even 2 1 576.3.q.k 24
24.h odd 2 1 576.3.q.k 24
36.f odd 6 1 288.3.q.a ✓ 24
36.f odd 6 1 2592.3.e.j 24
36.h even 6 1 inner 864.3.q.b 24
36.h even 6 1 2592.3.e.j 24
72.j odd 6 1 1728.3.q.l 24
72.l even 6 1 1728.3.q.l 24
72.n even 6 1 576.3.q.k 24
72.p odd 6 1 576.3.q.k 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.q.a ✓ 24 3.b odd 2 1
288.3.q.a ✓ 24 9.c even 3 1
288.3.q.a ✓ 24 12.b even 2 1
288.3.q.a ✓ 24 36.f odd 6 1
576.3.q.k 24 24.f even 2 1
576.3.q.k 24 24.h odd 2 1
576.3.q.k 24 72.n even 6 1
576.3.q.k 24 72.p odd 6 1
864.3.q.b 24 1.a even 1 1 trivial
864.3.q.b 24 4.b odd 2 1 inner
864.3.q.b 24 9.d odd 6 1 inner
864.3.q.b 24 36.h even 6 1 inner
1728.3.q.l 24 8.b even 2 1
1728.3.q.l 24 8.d odd 2 1
1728.3.q.l 24 72.j odd 6 1
1728.3.q.l 24 72.l even 6 1
2592.3.e.j 24 9.c even 3 1
2592.3.e.j 24 9.d odd 6 1
2592.3.e.j 24 36.f odd 6 1
2592.3.e.j 24 36.h even 6 1