Properties

Label 720.6.o.d
Level $720$
Weight $6$
Character orbit 720.o
Analytic conductor $115.476$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,6,Mod(719,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.719");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.o (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: \(115.476350265\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10648 q^{25} + 36248 q^{49} - 133136 q^{61} - 336816 q^{85}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1 0 0 0 −55.8906 1.11466i 0 −188.577 0 0 0
719.2 0 0 0 −55.8906 1.11466i 0 188.577 0 0 0
719.3 0 0 0 −55.8906 + 1.11466i 0 −188.577 0 0 0
719.4 0 0 0 −55.8906 + 1.11466i 0 188.577 0 0 0
719.5 0 0 0 −48.8519 27.1752i 0 −21.9879 0 0 0
719.6 0 0 0 −48.8519 27.1752i 0 21.9879 0 0 0
719.7 0 0 0 −48.8519 + 27.1752i 0 −21.9879 0 0 0
719.8 0 0 0 −48.8519 + 27.1752i 0 21.9879 0 0 0
719.9 0 0 0 −42.7952 35.9662i 0 −7.21390 0 0 0
719.10 0 0 0 −42.7952 35.9662i 0 7.21390 0 0 0
719.11 0 0 0 −42.7952 + 35.9662i 0 −7.21390 0 0 0
719.12 0 0 0 −42.7952 + 35.9662i 0 7.21390 0 0 0
719.13 0 0 0 −25.8892 49.5454i 0 −154.886 0 0 0
719.14 0 0 0 −25.8892 49.5454i 0 154.886 0 0 0
719.15 0 0 0 −25.8892 + 49.5454i 0 −154.886 0 0 0
719.16 0 0 0 −25.8892 + 49.5454i 0 154.886 0 0 0
719.17 0 0 0 −21.5882 51.5650i 0 −168.759 0 0 0
719.18 0 0 0 −21.5882 51.5650i 0 168.759 0 0 0
719.19 0 0 0 −21.5882 + 51.5650i 0 −168.759 0 0 0
719.20 0 0 0 −21.5882 + 51.5650i 0 168.759 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 719.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.6.o.d 40
3.b odd 2 1 inner 720.6.o.d 40
4.b odd 2 1 inner 720.6.o.d 40
5.b even 2 1 inner 720.6.o.d 40
12.b even 2 1 inner 720.6.o.d 40
15.d odd 2 1 inner 720.6.o.d 40
20.d odd 2 1 inner 720.6.o.d 40
60.h even 2 1 inner 720.6.o.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.6.o.d 40 1.a even 1 1 trivial
720.6.o.d 40 3.b odd 2 1 inner
720.6.o.d 40 4.b odd 2 1 inner
720.6.o.d 40 5.b even 2 1 inner
720.6.o.d 40 12.b even 2 1 inner
720.6.o.d 40 15.d odd 2 1 inner
720.6.o.d 40 20.d odd 2 1 inner
720.6.o.d 40 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - 88566 T_{7}^{8} + 2596254156 T_{7}^{6} - 25663288906440 T_{7}^{4} + \cdots - 61\!\cdots\!52 \) acting on \(S_{6}^{\mathrm{new}}(720, [\chi])\). Copy content Toggle raw display