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Properties

Label 1080.2.m.c
Level $1080$
Weight $2$
Character orbit 1080.m
Analytic conductor $8.624$
Analytic rank $0$
Dimension $48$
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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{4} - 4 q^{10} + 4 q^{16} - 16 q^{19} - 4 q^{34} + 16 q^{40} + 36 q^{46} + 48 q^{49} + 52 q^{64} + 28 q^{70} - 64 q^{76} + 92 q^{94}+O(q^{100}) \) 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 −1.41165 − 0.0850504i 0 1.98553 + 0.240123i −1.09556 − 1.94930i 0 4.08834 −2.78246 − 0.507841i 0 1.38076 + 2.84491i
539.2 −1.41165 − 0.0850504i 0 1.98553 + 0.240123i 1.09556 − 1.94930i 0 −4.08834 −2.78246 − 0.507841i 0 −1.71233 + 2.65855i
539.3 −1.41165 + 0.0850504i 0 1.98553 − 0.240123i −1.09556 + 1.94930i 0 4.08834 −2.78246 + 0.507841i 0 1.38076 − 2.84491i
539.4 −1.41165 + 0.0850504i 0 1.98553 − 0.240123i 1.09556 + 1.94930i 0 −4.08834 −2.78246 + 0.507841i 0 −1.71233 − 2.65855i
539.5 −1.34874 − 0.425328i 0 1.63819 + 1.14731i −2.17257 + 0.529077i 0 −1.59486 −1.72151 − 2.24419i 0 3.15527 + 0.210470i
539.6 −1.34874 − 0.425328i 0 1.63819 + 1.14731i 2.17257 + 0.529077i 0 1.59486 −1.72151 − 2.24419i 0 −2.70520 − 1.63764i
539.7 −1.34874 + 0.425328i 0 1.63819 − 1.14731i −2.17257 − 0.529077i 0 −1.59486 −1.72151 + 2.24419i 0 3.15527 − 0.210470i
539.8 −1.34874 + 0.425328i 0 1.63819 − 1.14731i 2.17257 − 0.529077i 0 1.59486 −1.72151 + 2.24419i 0 −2.70520 + 1.63764i
539.9 −1.08245 − 0.910116i 0 0.343378 + 1.97030i −1.75989 − 1.37942i 0 −1.14666 1.42152 − 2.44526i 0 0.649552 + 3.09485i
539.10 −1.08245 − 0.910116i 0 0.343378 + 1.97030i 1.75989 − 1.37942i 0 1.14666 1.42152 − 2.44526i 0 −3.16041 − 0.108555i
539.11 −1.08245 + 0.910116i 0 0.343378 − 1.97030i −1.75989 + 1.37942i 0 −1.14666 1.42152 + 2.44526i 0 0.649552 − 3.09485i
539.12 −1.08245 + 0.910116i 0 0.343378 − 1.97030i 1.75989 + 1.37942i 0 1.14666 1.42152 + 2.44526i 0 −3.16041 + 0.108555i
539.13 −0.957850 − 1.04044i 0 −0.165046 + 1.99318i −0.214370 + 2.22577i 0 2.80642 2.23188 − 1.73744i 0 2.52112 − 1.90891i
539.14 −0.957850 − 1.04044i 0 −0.165046 + 1.99318i 0.214370 + 2.22577i 0 −2.80642 2.23188 − 1.73744i 0 2.11045 − 2.35499i
539.15 −0.957850 + 1.04044i 0 −0.165046 − 1.99318i −0.214370 − 2.22577i 0 2.80642 2.23188 + 1.73744i 0 2.52112 + 1.90891i
539.16 −0.957850 + 1.04044i 0 −0.165046 − 1.99318i 0.214370 − 2.22577i 0 −2.80642 2.23188 + 1.73744i 0 2.11045 + 2.35499i
539.17 −0.482315 − 1.32943i 0 −1.53474 + 1.28240i −1.17345 − 1.90342i 0 0.450397 2.44509 + 1.42181i 0 −1.96449 + 2.47806i
539.18 −0.482315 − 1.32943i 0 −1.53474 + 1.28240i 1.17345 − 1.90342i 0 −0.450397 2.44509 + 1.42181i 0 −3.09643 − 0.641961i
539.19 −0.482315 + 1.32943i 0 −1.53474 − 1.28240i −1.17345 + 1.90342i 0 0.450397 2.44509 − 1.42181i 0 −1.96449 − 2.47806i
539.20 −0.482315 + 1.32943i 0 −1.53474 − 1.28240i 1.17345 + 1.90342i 0 −0.450397 2.44509 − 1.42181i 0 −3.09643 + 0.641961i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 539.48
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.c ✓ 48
3.b odd 2 1 inner 1080.2.m.c ✓ 48
4.b odd 2 1 4320.2.m.c 48
5.b even 2 1 inner 1080.2.m.c ✓ 48
8.b even 2 1 4320.2.m.c 48
8.d odd 2 1 inner 1080.2.m.c ✓ 48
12.b even 2 1 4320.2.m.c 48
15.d odd 2 1 inner 1080.2.m.c ✓ 48
20.d odd 2 1 4320.2.m.c 48
24.f even 2 1 inner 1080.2.m.c ✓ 48
24.h odd 2 1 4320.2.m.c 48
40.e odd 2 1 inner 1080.2.m.c ✓ 48
40.f even 2 1 4320.2.m.c 48
60.h even 2 1 4320.2.m.c 48
120.i odd 2 1 4320.2.m.c 48
120.m even 2 1 inner 1080.2.m.c ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.m.c ✓ 48 1.a even 1 1 trivial
1080.2.m.c ✓ 48 3.b odd 2 1 inner
1080.2.m.c ✓ 48 5.b even 2 1 inner
1080.2.m.c ✓ 48 8.d odd 2 1 inner
1080.2.m.c ✓ 48 15.d odd 2 1 inner
1080.2.m.c ✓ 48 24.f even 2 1 inner
1080.2.m.c ✓ 48 40.e odd 2 1 inner
1080.2.m.c ✓ 48 120.m even 2 1 inner
4320.2.m.c 48 4.b odd 2 1
4320.2.m.c 48 8.b even 2 1
4320.2.m.c 48 12.b even 2 1
4320.2.m.c 48 20.d odd 2 1
4320.2.m.c 48 24.h odd 2 1
4320.2.m.c 48 40.f even 2 1
4320.2.m.c 48 60.h even 2 1
4320.2.m.c 48 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 48T_{7}^{10} + 790T_{7}^{8} - 5196T_{7}^{6} + 12881T_{7}^{4} - 10924T_{7}^{2} + 1728 \) acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\). Copy content Toggle raw display