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Properties

Label 880.2.cm.c
Level $880$
Weight $2$
Character orbit 880.cm
Analytic conductor $7.027$
Analytic rank $0$
Dimension $48$
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] = [880,2,Mod(17,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.cm (of order \(20\), degree \(8\), 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} - 8 q^{5} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} - 8 q^{5} + 20 q^{7} - 12 q^{11} + 16 q^{15} - 20 q^{17} + 8 q^{23} - 20 q^{25} - 8 q^{27} - 16 q^{31} - 104 q^{33} + 20 q^{37} - 20 q^{41} + 16 q^{45} - 40 q^{47} - 40 q^{51} - 96 q^{55} + 80 q^{61} - 100 q^{63} + 56 q^{71} - 20 q^{73} + 76 q^{75} - 96 q^{77} - 68 q^{81} - 80 q^{85} + 68 q^{91} + 76 q^{93} - 20 q^{95} + 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −2.03488 + 0.322293i 0 0.667967 + 2.13397i 0 0.0611439 − 0.386047i 0 1.18368 − 0.384601i 0
17.2 0 −1.61854 + 0.256351i 0 −2.22860 + 0.182641i 0 −0.119289 + 0.753160i 0 −0.299227 + 0.0972249i 0
17.3 0 0.216404 − 0.0342750i 0 1.60893 − 1.55285i 0 0.0165943 − 0.104772i 0 −2.80751 + 0.912216i 0
17.4 0 1.23025 − 0.194853i 0 −1.87069 + 1.22496i 0 0.466963 − 2.94829i 0 −1.37761 + 0.447613i 0
17.5 0 2.20142 − 0.348671i 0 2.18748 − 0.463629i 0 −0.620489 + 3.91761i 0 1.87153 − 0.608097i 0
17.6 0 2.79893 − 0.443308i 0 0.537022 + 2.17062i 0 0.516545 − 3.26134i 0 4.78434 − 1.55453i 0
193.1 0 −0.322293 − 2.03488i 0 −1.79471 + 1.33380i 0 −0.386047 − 0.0611439i 0 −1.18368 + 0.384601i 0
193.2 0 −0.256351 − 1.61854i 0 1.69562 + 1.45770i 0 0.753160 + 0.119289i 0 0.299227 − 0.0972249i 0
193.3 0 0.0342750 + 0.216404i 0 −0.388914 − 2.20199i 0 −0.104772 − 0.0165943i 0 2.80751 − 0.912216i 0
193.4 0 0.194853 + 1.23025i 0 0.793405 + 2.09058i 0 −2.94829 − 0.466963i 0 1.37761 − 0.447613i 0
193.5 0 0.348671 + 2.20142i 0 −1.49719 − 1.66085i 0 3.91761 + 0.620489i 0 −1.87153 + 0.608097i 0
193.6 0 0.443308 + 2.79893i 0 −1.71032 + 1.44042i 0 −3.26134 − 0.516545i 0 −4.78434 + 1.55453i 0
337.1 0 −1.18907 − 2.33368i 0 −1.05320 − 1.97251i 0 2.88379 + 1.46937i 0 −2.26883 + 3.12278i 0
337.2 0 −0.992087 − 1.94708i 0 0.244349 + 2.22268i 0 3.05681 + 1.55752i 0 −1.04353 + 1.43630i 0
337.3 0 0.176422 + 0.346247i 0 0.0767418 − 2.23475i 0 0.0136870 + 0.00697389i 0 1.67459 − 2.30488i 0
337.4 0 0.757609 + 1.48689i 0 −1.65111 − 1.50793i 0 −2.92930 − 1.49255i 0 0.126480 − 0.174085i 0
337.5 0 1.07350 + 2.10686i 0 1.91408 + 1.15599i 0 3.37294 + 1.71860i 0 −1.52312 + 2.09639i 0
337.6 0 1.45771 + 2.86091i 0 −1.70643 + 1.44502i 0 1.35828 + 0.692077i 0 −4.29653 + 5.91367i 0
497.1 0 −0.322293 + 2.03488i 0 −1.79471 − 1.33380i 0 −0.386047 + 0.0611439i 0 −1.18368 − 0.384601i 0
497.2 0 −0.256351 + 1.61854i 0 1.69562 − 1.45770i 0 0.753160 − 0.119289i 0 0.299227 + 0.0972249i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.cm.c 48
4.b odd 2 1 110.2.k.a ✓ 48
5.c odd 4 1 inner 880.2.cm.c 48
11.d odd 10 1 inner 880.2.cm.c 48
12.b even 2 1 990.2.bh.c 48
20.d odd 2 1 550.2.bh.b 48
20.e even 4 1 110.2.k.a ✓ 48
20.e even 4 1 550.2.bh.b 48
44.g even 10 1 110.2.k.a ✓ 48
55.l even 20 1 inner 880.2.cm.c 48
60.l odd 4 1 990.2.bh.c 48
132.n odd 10 1 990.2.bh.c 48
220.o even 10 1 550.2.bh.b 48
220.w odd 20 1 110.2.k.a ✓ 48
220.w odd 20 1 550.2.bh.b 48
660.bv even 20 1 990.2.bh.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.k.a ✓ 48 4.b odd 2 1
110.2.k.a ✓ 48 20.e even 4 1
110.2.k.a ✓ 48 44.g even 10 1
110.2.k.a ✓ 48 220.w odd 20 1
550.2.bh.b 48 20.d odd 2 1
550.2.bh.b 48 20.e even 4 1
550.2.bh.b 48 220.o even 10 1
550.2.bh.b 48 220.w odd 20 1
880.2.cm.c 48 1.a even 1 1 trivial
880.2.cm.c 48 5.c odd 4 1 inner
880.2.cm.c 48 11.d odd 10 1 inner
880.2.cm.c 48 55.l even 20 1 inner
990.2.bh.c 48 12.b even 2 1
990.2.bh.c 48 60.l odd 4 1
990.2.bh.c 48 132.n odd 10 1
990.2.bh.c 48 660.bv even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 4 T_{3}^{47} + 8 T_{3}^{46} - 60 T_{3}^{44} - 64 T_{3}^{43} + 736 T_{3}^{42} + \cdots + 723394816 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display