Properties

Label 1440.2.bi.e
Level $1440$
Weight $2$
Character orbit 1440.bi
Analytic conductor $11.498$
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] = [1440,2,Mod(847,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.847");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.bi (of order \(4\), 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: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{17} - 8 q^{25} - 48 q^{35} + 32 q^{43} + 8 q^{65} - 48 q^{67} - 40 q^{73} + 80 q^{83} - 64 q^{91} - 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
847.1 0 0 0 −2.22965 + 0.169312i 0 0.645414 0.645414i 0 0 0
847.2 0 0 0 −2.11218 + 0.733965i 0 −1.93078 + 1.93078i 0 0 0
847.3 0 0 0 −1.51371 1.64581i 0 3.43671 3.43671i 0 0 0
847.4 0 0 0 −1.28903 + 1.82713i 0 1.45533 1.45533i 0 0 0
847.5 0 0 0 −0.780766 2.09533i 0 2.10796 2.10796i 0 0 0
847.6 0 0 0 −0.0696909 2.23498i 0 −1.21782 + 1.21782i 0 0 0
847.7 0 0 0 0.0696909 + 2.23498i 0 1.21782 1.21782i 0 0 0
847.8 0 0 0 0.780766 + 2.09533i 0 −2.10796 + 2.10796i 0 0 0
847.9 0 0 0 1.28903 1.82713i 0 −1.45533 + 1.45533i 0 0 0
847.10 0 0 0 1.51371 + 1.64581i 0 −3.43671 + 3.43671i 0 0 0
847.11 0 0 0 2.11218 0.733965i 0 1.93078 1.93078i 0 0 0
847.12 0 0 0 2.22965 0.169312i 0 −0.645414 + 0.645414i 0 0 0
1423.1 0 0 0 −2.22965 0.169312i 0 0.645414 + 0.645414i 0 0 0
1423.2 0 0 0 −2.11218 0.733965i 0 −1.93078 1.93078i 0 0 0
1423.3 0 0 0 −1.51371 + 1.64581i 0 3.43671 + 3.43671i 0 0 0
1423.4 0 0 0 −1.28903 1.82713i 0 1.45533 + 1.45533i 0 0 0
1423.5 0 0 0 −0.780766 + 2.09533i 0 2.10796 + 2.10796i 0 0 0
1423.6 0 0 0 −0.0696909 + 2.23498i 0 −1.21782 1.21782i 0 0 0
1423.7 0 0 0 0.0696909 2.23498i 0 1.21782 + 1.21782i 0 0 0
1423.8 0 0 0 0.780766 2.09533i 0 −2.10796 2.10796i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 847.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.bi.e 24
3.b odd 2 1 480.2.bh.a 24
4.b odd 2 1 360.2.w.e 24
5.c odd 4 1 inner 1440.2.bi.e 24
8.b even 2 1 360.2.w.e 24
8.d odd 2 1 inner 1440.2.bi.e 24
12.b even 2 1 120.2.v.a 24
15.d odd 2 1 2400.2.bh.b 24
15.e even 4 1 480.2.bh.a 24
15.e even 4 1 2400.2.bh.b 24
20.e even 4 1 360.2.w.e 24
24.f even 2 1 480.2.bh.a 24
24.h odd 2 1 120.2.v.a 24
40.i odd 4 1 360.2.w.e 24
40.k even 4 1 inner 1440.2.bi.e 24
60.h even 2 1 600.2.v.b 24
60.l odd 4 1 120.2.v.a 24
60.l odd 4 1 600.2.v.b 24
120.i odd 2 1 600.2.v.b 24
120.m even 2 1 2400.2.bh.b 24
120.q odd 4 1 480.2.bh.a 24
120.q odd 4 1 2400.2.bh.b 24
120.w even 4 1 120.2.v.a 24
120.w even 4 1 600.2.v.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.v.a 24 12.b even 2 1
120.2.v.a 24 24.h odd 2 1
120.2.v.a 24 60.l odd 4 1
120.2.v.a 24 120.w even 4 1
360.2.w.e 24 4.b odd 2 1
360.2.w.e 24 8.b even 2 1
360.2.w.e 24 20.e even 4 1
360.2.w.e 24 40.i odd 4 1
480.2.bh.a 24 3.b odd 2 1
480.2.bh.a 24 15.e even 4 1
480.2.bh.a 24 24.f even 2 1
480.2.bh.a 24 120.q odd 4 1
600.2.v.b 24 60.h even 2 1
600.2.v.b 24 60.l odd 4 1
600.2.v.b 24 120.i odd 2 1
600.2.v.b 24 120.w even 4 1
1440.2.bi.e 24 1.a even 1 1 trivial
1440.2.bi.e 24 5.c odd 4 1 inner
1440.2.bi.e 24 8.d odd 2 1 inner
1440.2.bi.e 24 40.k even 4 1 inner
2400.2.bh.b 24 15.d odd 2 1
2400.2.bh.b 24 15.e even 4 1
2400.2.bh.b 24 120.m even 2 1
2400.2.bh.b 24 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 720T_{7}^{20} + 98656T_{7}^{16} + 4752640T_{7}^{12} + 81309952T_{7}^{8} + 440926208T_{7}^{4} + 268435456 \) acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\). Copy content Toggle raw display