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Properties

Label 1080.2.m.b
Level $1080$
Weight $2$
Character orbit 1080.m
Analytic conductor $8.624$
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] = [1080,2,Mod(539,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.539");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.m (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: \(8.62384341830\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} - 2 q^{10} - 28 q^{16} + 24 q^{19} + 40 q^{25} + 56 q^{34} + 18 q^{40} - 32 q^{46} + 104 q^{49} - 100 q^{64} + 10 q^{70} - 120 q^{76} - 56 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
539.1 −1.33663 − 0.461964i 0 1.57318 + 1.23495i −1.33994 + 1.79013i 0 2.78789 −1.53226 − 2.37743i 0 2.61798 − 1.77375i
539.2 −1.33663 − 0.461964i 0 1.57318 + 1.23495i 1.33994 + 1.79013i 0 −2.78789 −1.53226 − 2.37743i 0 −0.964028 − 3.01175i
539.3 −1.33663 + 0.461964i 0 1.57318 − 1.23495i −1.33994 − 1.79013i 0 2.78789 −1.53226 + 2.37743i 0 2.61798 + 1.77375i
539.4 −1.33663 + 0.461964i 0 1.57318 − 1.23495i 1.33994 − 1.79013i 0 −2.78789 −1.53226 + 2.37743i 0 −0.964028 + 3.01175i
539.5 −1.15273 − 0.819276i 0 0.657573 + 1.88881i −2.22480 + 0.224204i 0 2.27075 0.789451 − 2.71602i 0 2.74828 + 1.56428i
539.6 −1.15273 − 0.819276i 0 0.657573 + 1.88881i 2.22480 + 0.224204i 0 −2.27075 0.789451 − 2.71602i 0 −2.38091 − 2.08117i
539.7 −1.15273 + 0.819276i 0 0.657573 − 1.88881i −2.22480 − 0.224204i 0 2.27075 0.789451 + 2.71602i 0 2.74828 − 1.56428i
539.8 −1.15273 + 0.819276i 0 0.657573 − 1.88881i 2.22480 − 0.224204i 0 −2.27075 0.789451 + 2.71602i 0 −2.38091 + 2.08117i
539.9 −1.09804 − 0.891239i 0 0.411386 + 1.95723i −1.30314 − 1.81709i 0 −4.43932 1.29264 − 2.51576i 0 −0.188558 + 3.15665i
539.10 −1.09804 − 0.891239i 0 0.411386 + 1.95723i 1.30314 − 1.81709i 0 4.43932 1.29264 − 2.51576i 0 −3.05037 + 0.833828i
539.11 −1.09804 + 0.891239i 0 0.411386 − 1.95723i −1.30314 + 1.81709i 0 −4.43932 1.29264 + 2.51576i 0 −0.188558 − 3.15665i
539.12 −1.09804 + 0.891239i 0 0.411386 − 1.95723i 1.30314 + 1.81709i 0 4.43932 1.29264 + 2.51576i 0 −3.05037 − 0.833828i
539.13 −0.621745 − 1.27021i 0 −1.22687 + 1.57949i −1.66343 + 1.49432i 0 −3.65580 2.76909 + 0.576338i 0 2.93233 + 1.18383i
539.14 −0.621745 − 1.27021i 0 −1.22687 + 1.57949i 1.66343 + 1.49432i 0 3.65580 2.76909 + 0.576338i 0 0.863867 − 3.04199i
539.15 −0.621745 + 1.27021i 0 −1.22687 − 1.57949i −1.66343 − 1.49432i 0 −3.65580 2.76909 − 0.576338i 0 2.93233 − 1.18383i
539.16 −0.621745 + 1.27021i 0 −1.22687 − 1.57949i 1.66343 − 1.49432i 0 3.65580 2.76909 − 0.576338i 0 0.863867 + 3.04199i
539.17 −0.205826 − 1.39916i 0 −1.91527 + 0.575965i −1.94670 − 1.10016i 0 1.41383 1.20008 + 2.56121i 0 −1.13862 + 2.95018i
539.18 −0.205826 − 1.39916i 0 −1.91527 + 0.575965i 1.94670 − 1.10016i 0 −1.41383 1.20008 + 2.56121i 0 −1.93998 − 2.49729i
539.19 −0.205826 + 1.39916i 0 −1.91527 − 0.575965i −1.94670 + 1.10016i 0 1.41383 1.20008 − 2.56121i 0 −1.13862 − 2.95018i
539.20 −0.205826 + 1.39916i 0 −1.91527 − 0.575965i 1.94670 + 1.10016i 0 −1.41383 1.20008 − 2.56121i 0 −1.93998 + 2.49729i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 539.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.m.b ✓ 40
3.b odd 2 1 inner 1080.2.m.b ✓ 40
4.b odd 2 1 4320.2.m.b 40
5.b even 2 1 inner 1080.2.m.b ✓ 40
8.b even 2 1 4320.2.m.b 40
8.d odd 2 1 inner 1080.2.m.b ✓ 40
12.b even 2 1 4320.2.m.b 40
15.d odd 2 1 inner 1080.2.m.b ✓ 40
20.d odd 2 1 4320.2.m.b 40
24.f even 2 1 inner 1080.2.m.b ✓ 40
24.h odd 2 1 4320.2.m.b 40
40.e odd 2 1 inner 1080.2.m.b ✓ 40
40.f even 2 1 4320.2.m.b 40
60.h even 2 1 4320.2.m.b 40
120.i odd 2 1 4320.2.m.b 40
120.m even 2 1 inner 1080.2.m.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.m.b ✓ 40 1.a even 1 1 trivial
1080.2.m.b ✓ 40 3.b odd 2 1 inner
1080.2.m.b ✓ 40 5.b even 2 1 inner
1080.2.m.b ✓ 40 8.d odd 2 1 inner
1080.2.m.b ✓ 40 15.d odd 2 1 inner
1080.2.m.b ✓ 40 24.f even 2 1 inner
1080.2.m.b ✓ 40 40.e odd 2 1 inner
1080.2.m.b ✓ 40 120.m even 2 1 inner
4320.2.m.b 40 4.b odd 2 1
4320.2.m.b 40 8.b even 2 1
4320.2.m.b 40 12.b even 2 1
4320.2.m.b 40 20.d odd 2 1
4320.2.m.b 40 24.h odd 2 1
4320.2.m.b 40 40.f even 2 1
4320.2.m.b 40 60.h even 2 1
4320.2.m.b 40 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - 48T_{7}^{8} + 823T_{7}^{6} - 6192T_{7}^{4} + 20012T_{7}^{2} - 21100 \) acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\). Copy content Toggle raw display