Properties

Label 880.2.cm.b
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 220)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3} + 4 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3} + 4 q^{5} - 10 q^{7} + 16 q^{15} + 10 q^{17} - 16 q^{23} - 26 q^{25} + 10 q^{27} - 16 q^{31} + 28 q^{33} - 34 q^{37} - 20 q^{41} - 56 q^{45} + 2 q^{47} + 80 q^{51} + 6 q^{53} + 18 q^{55} - 120 q^{57} - 40 q^{61} + 50 q^{63} + 72 q^{67} - 4 q^{71} - 20 q^{73} - 20 q^{75} - 36 q^{77} + 100 q^{81} + 40 q^{85} + 8 q^{91} - 14 q^{93} - 50 q^{95} + 6 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −2.95172 + 0.467507i 0 0.0236267 2.23594i 0 0.776647 4.90356i 0 5.64095 1.83285i 0
17.2 0 −2.50020 + 0.395993i 0 −0.797801 + 2.08890i 0 −0.371734 + 2.34703i 0 3.24104 1.05308i 0
17.3 0 −0.194131 + 0.0307474i 0 1.80035 + 1.32618i 0 0.509098 3.21432i 0 −2.81643 + 0.915113i 0
17.4 0 0.464116 0.0735087i 0 −0.794013 2.09035i 0 −0.413999 + 2.61389i 0 −2.64317 + 0.858818i 0
17.5 0 0.834729 0.132208i 0 −2.08639 + 0.804354i 0 0.228981 1.44573i 0 −2.17388 + 0.706335i 0
17.6 0 2.95041 0.467300i 0 1.40317 1.74101i 0 0.372695 2.35311i 0 5.63340 1.83040i 0
193.1 0 −0.467507 2.95172i 0 1.29514 1.82280i 0 −4.90356 0.776647i 0 −5.64095 + 1.83285i 0
193.2 0 −0.395993 2.50020i 0 −0.582392 + 2.15889i 0 2.34703 + 0.371734i 0 −3.24104 + 1.05308i 0
193.3 0 −0.0307474 0.194131i 0 −2.23602 + 0.0146855i 0 −3.21432 0.509098i 0 2.81643 0.915113i 0
193.4 0 0.0735087 + 0.464116i 0 1.87104 1.22442i 0 2.61389 + 0.413999i 0 2.64317 0.858818i 0
193.5 0 0.132208 + 0.834729i 0 1.21514 + 1.87708i 0 −1.44573 0.228981i 0 2.17388 0.706335i 0
193.6 0 0.467300 + 2.95041i 0 −0.111852 2.23327i 0 −2.35311 0.372695i 0 −5.63340 + 1.83040i 0
337.1 0 −1.37132 2.69137i 0 2.10381 + 0.757611i 0 0.645435 + 0.328865i 0 −3.59960 + 4.95443i 0
337.2 0 −0.629920 1.23629i 0 −1.37813 + 1.76090i 0 −1.43256 0.729926i 0 0.631750 0.869529i 0
337.3 0 −0.574804 1.12812i 0 0.182995 2.22857i 0 −2.04212 1.04051i 0 0.821108 1.13016i 0
337.4 0 0.140599 + 0.275942i 0 −2.21755 0.287183i 0 4.33567 + 2.20914i 0 1.70698 2.34946i 0
337.5 0 0.748461 + 1.46894i 0 0.857567 + 2.06509i 0 −2.43189 1.23911i 0 0.165773 0.228167i 0
337.6 0 1.04495 + 2.05082i 0 1.53909 1.62210i 0 1.49127 + 0.759842i 0 −1.35061 + 1.85895i 0
497.1 0 −0.467507 + 2.95172i 0 1.29514 + 1.82280i 0 −4.90356 + 0.776647i 0 −5.64095 1.83285i 0
497.2 0 −0.395993 + 2.50020i 0 −0.582392 2.15889i 0 2.34703 0.371734i 0 −3.24104 1.05308i 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.b 48
4.b odd 2 1 220.2.u.a 48
5.c odd 4 1 inner 880.2.cm.b 48
11.d odd 10 1 inner 880.2.cm.b 48
20.e even 4 1 220.2.u.a 48
44.g even 10 1 220.2.u.a 48
55.l even 20 1 inner 880.2.cm.b 48
220.w odd 20 1 220.2.u.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.u.a 48 4.b odd 2 1
220.2.u.a 48 20.e even 4 1
220.2.u.a 48 44.g even 10 1
220.2.u.a 48 220.w odd 20 1
880.2.cm.b 48 1.a even 1 1 trivial
880.2.cm.b 48 5.c odd 4 1 inner
880.2.cm.b 48 11.d odd 10 1 inner
880.2.cm.b 48 55.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 2 T_{3}^{47} + 2 T_{3}^{46} - 6 T_{3}^{45} - 138 T_{3}^{44} - 100 T_{3}^{43} + 94 T_{3}^{42} + 1666 T_{3}^{41} + 10527 T_{3}^{40} - 3072 T_{3}^{39} - 19222 T_{3}^{38} - 137620 T_{3}^{37} - 488387 T_{3}^{36} + 410386 T_{3}^{35} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display