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Properties

Label 1080.2.s.a
Level $1080$
Weight $2$
Character orbit 1080.s
Analytic conductor $8.624$
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] = [1080,2,Mod(377,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.377");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.s (of order \(4\), degree \(2\), 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: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{7} - 8 q^{13} - 16 q^{25} - 16 q^{31} + 8 q^{37} - 28 q^{55} + 8 q^{67} - 20 q^{73} + 48 q^{85} - 64 q^{91} + 20 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1 0 0 0 −2.16799 − 0.547562i 0 0.431925 + 0.431925i 0 0 0
377.2 0 0 0 −2.03577 + 0.925011i 0 −3.35220 − 3.35220i 0 0 0
377.3 0 0 0 −1.82892 + 1.28648i 0 0.197131 + 0.197131i 0 0 0
377.4 0 0 0 −0.787352 − 2.09286i 0 −1.99508 − 1.99508i 0 0 0
377.5 0 0 0 −0.431161 + 2.19411i 0 0.952374 + 0.952374i 0 0 0
377.6 0 0 0 −0.0684245 + 2.23502i 0 2.76585 + 2.76585i 0 0 0
377.7 0 0 0 0.0684245 − 2.23502i 0 2.76585 + 2.76585i 0 0 0
377.8 0 0 0 0.431161 − 2.19411i 0 0.952374 + 0.952374i 0 0 0
377.9 0 0 0 0.787352 + 2.09286i 0 −1.99508 − 1.99508i 0 0 0
377.10 0 0 0 1.82892 − 1.28648i 0 0.197131 + 0.197131i 0 0 0
377.11 0 0 0 2.03577 − 0.925011i 0 −3.35220 − 3.35220i 0 0 0
377.12 0 0 0 2.16799 + 0.547562i 0 0.431925 + 0.431925i 0 0 0
593.1 0 0 0 −2.16799 + 0.547562i 0 0.431925 − 0.431925i 0 0 0
593.2 0 0 0 −2.03577 − 0.925011i 0 −3.35220 + 3.35220i 0 0 0
593.3 0 0 0 −1.82892 − 1.28648i 0 0.197131 − 0.197131i 0 0 0
593.4 0 0 0 −0.787352 + 2.09286i 0 −1.99508 + 1.99508i 0 0 0
593.5 0 0 0 −0.431161 − 2.19411i 0 0.952374 − 0.952374i 0 0 0
593.6 0 0 0 −0.0684245 − 2.23502i 0 2.76585 − 2.76585i 0 0 0
593.7 0 0 0 0.0684245 + 2.23502i 0 2.76585 − 2.76585i 0 0 0
593.8 0 0 0 0.431161 + 2.19411i 0 0.952374 − 0.952374i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.s.a ✓ 24
3.b odd 2 1 inner 1080.2.s.a ✓ 24
4.b odd 2 1 2160.2.w.h 24
5.c odd 4 1 inner 1080.2.s.a ✓ 24
12.b even 2 1 2160.2.w.h 24
15.e even 4 1 inner 1080.2.s.a ✓ 24
20.e even 4 1 2160.2.w.h 24
60.l odd 4 1 2160.2.w.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.s.a ✓ 24 1.a even 1 1 trivial
1080.2.s.a ✓ 24 3.b odd 2 1 inner
1080.2.s.a ✓ 24 5.c odd 4 1 inner
1080.2.s.a ✓ 24 15.e even 4 1 inner
2160.2.w.h 24 4.b odd 2 1
2160.2.w.h 24 12.b even 2 1
2160.2.w.h 24 20.e even 4 1
2160.2.w.h 24 60.l odd 4 1