Properties

Label 880.2.cl.a
Level $880$
Weight $2$
Character orbit 880.cl
Analytic conductor $7.027$
Analytic rank $0$
Dimension $1120$
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] = [880,2,Mod(69,880)]
 
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, 5, 10, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.69");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.cl (of order \(20\), degree \(8\), 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: \(1120\)
Relative dimension: \(140\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 12 q^{4} - 6 q^{5} - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 12 q^{4} - 6 q^{5} - 12 q^{6} - 16 q^{10} - 16 q^{11} - 4 q^{14} - 12 q^{15} - 12 q^{16} - 12 q^{19} - 6 q^{20} - 8 q^{21} + 20 q^{24} + 16 q^{26} - 12 q^{29} + 10 q^{30} - 24 q^{31} - 48 q^{34} + 14 q^{35} - 44 q^{36} + 14 q^{40} - 44 q^{44} - 48 q^{45} - 36 q^{46} - 256 q^{49} - 18 q^{50} - 36 q^{51} - 24 q^{54} + 72 q^{56} - 28 q^{59} - 14 q^{60} - 12 q^{61} + 108 q^{64} - 32 q^{65} + 76 q^{66} - 48 q^{69} - 72 q^{70} + 28 q^{74} + 54 q^{75} - 32 q^{76} - 104 q^{79} - 112 q^{80} + 192 q^{81} - 152 q^{84} - 26 q^{85} - 20 q^{86} + 36 q^{90} + 28 q^{91} - 112 q^{94} - 12 q^{95} + 20 q^{96} - 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −1.41410 0.0182095i 0.519642 1.01985i 1.99934 + 0.0515000i 0.598527 + 2.15448i −0.753395 + 1.43271i −0.828964 + 2.55129i −2.82632 0.109233i 0.993280 + 1.36713i −0.807142 3.05754i
69.2 −1.41376 0.0359969i 1.29983 2.55107i 1.99741 + 0.101782i 0.705695 + 2.12179i −1.92948 + 3.55979i 1.21258 3.73193i −2.82018 0.215795i −3.05502 4.20487i −0.921302 3.02510i
69.3 −1.41320 + 0.0534818i 0.992902 1.94868i 1.99428 0.151161i −2.21272 + 0.322262i −1.29895 + 2.80698i 1.01028 3.10932i −2.81024 + 0.320279i −1.04814 1.44264i 3.10979 0.573762i
69.4 −1.40924 + 0.118487i 0.0780044 0.153092i 1.97192 0.333953i 1.99339 1.01311i −0.0917875 + 0.224986i 1.27618 3.92767i −2.73934 + 0.704268i 1.74600 + 2.40317i −2.68913 + 1.66391i
69.5 −1.40868 + 0.124994i −1.15920 + 2.27506i 1.96875 0.352153i −0.861575 + 2.06342i 1.34857 3.34972i −0.139307 + 0.428744i −2.72932 + 0.742152i −2.06879 2.84744i 0.955769 3.01438i
69.6 −1.40837 + 0.128402i 0.325764 0.639347i 1.96703 0.361676i −1.22240 1.87236i −0.376703 + 0.942268i −0.728483 + 2.24204i −2.72386 + 0.761945i 1.46071 + 2.01050i 1.96201 + 2.48002i
69.7 −1.40125 + 0.191080i −1.06155 + 2.08341i 1.92698 0.535499i −0.115276 2.23309i 1.08940 3.12221i 0.969208 2.98291i −2.59785 + 1.11857i −1.45035 1.99624i 0.588229 + 3.10709i
69.8 −1.39937 0.204351i 1.47694 2.89867i 1.91648 + 0.571925i −1.23044 1.86709i −2.65914 + 3.75450i −1.38653 + 4.26730i −2.56500 1.19197i −4.45755 6.13529i 1.34029 + 2.86419i
69.9 −1.39897 0.207055i −1.22930 + 2.41263i 1.91426 + 0.579330i −1.90794 1.16610i 2.21930 3.12068i 0.116717 0.359217i −2.55804 1.20683i −2.54626 3.50463i 2.42771 + 2.02639i
69.10 −1.38353 + 0.293000i −0.630219 + 1.23687i 1.82830 0.810748i 1.78995 + 1.34018i 0.509521 1.89590i 0.715623 2.20246i −2.29196 + 1.65739i 0.630674 + 0.868048i −2.86912 1.32972i
69.11 −1.37063 0.348388i −0.457643 + 0.898176i 1.75725 + 0.955021i 1.98781 1.02402i 0.940173 1.07163i −0.891410 + 2.74348i −2.07582 1.92119i 1.16607 + 1.60496i −3.08130 + 0.711028i
69.12 −1.36915 0.354156i −1.47318 + 2.89128i 1.74915 + 0.969787i 2.22551 0.217049i 3.04097 3.43686i −0.722522 + 2.22369i −2.05139 1.94726i −4.42589 6.09171i −3.12393 0.491005i
69.13 −1.36740 + 0.360857i 0.198607 0.389788i 1.73956 0.986872i −1.80717 + 1.31687i −0.130917 + 0.604664i −0.812900 + 2.50185i −2.02256 + 1.97718i 1.65087 + 2.27222i 1.99592 2.45282i
69.14 −1.36193 0.380967i 0.825023 1.61920i 1.70973 + 1.03770i 2.23309 + 0.115335i −1.74049 + 1.89093i −0.130304 + 0.401035i −1.93320 2.06464i −0.177784 0.244698i −2.99738 1.00781i
69.15 −1.34173 0.446957i −0.587275 + 1.15259i 1.60046 + 1.19939i −1.04830 + 1.97511i 1.30312 1.28398i 0.778431 2.39576i −1.61130 2.32459i 0.779780 + 1.07328i 2.28932 2.18151i
69.16 −1.32327 + 0.498959i 1.01937 2.00062i 1.50208 1.32051i 1.53011 1.63057i −0.350669 + 3.15598i 0.0518810 0.159673i −1.32877 + 2.49687i −1.20002 1.65169i −1.21116 + 2.92114i
69.17 −1.30926 + 0.534642i −0.752040 + 1.47596i 1.42832 1.39997i 0.990806 2.00457i 0.195503 2.33449i −1.27730 + 3.93112i −1.12155 + 2.59656i 0.150458 + 0.207088i −0.225493 + 3.15423i
69.18 −1.30917 0.534848i −0.250760 + 0.492144i 1.42787 + 1.40042i 0.339996 2.21007i 0.591511 0.510184i 0.413479 1.27256i −1.12032 2.59709i 1.58403 + 2.18023i −1.62717 + 2.71152i
69.19 −1.28530 0.589926i −0.507149 + 0.995335i 1.30397 + 1.51646i −2.19950 0.402737i 1.23901 0.980121i −0.969160 + 2.98277i −0.781394 2.71835i 1.02986 + 1.41749i 2.58943 + 1.81518i
69.20 −1.26834 0.625557i 0.750300 1.47255i 1.21736 + 1.58683i −1.25626 1.84981i −1.87280 + 1.39833i 1.11355 3.42715i −0.551362 2.77417i 0.157911 + 0.217346i 0.436198 + 3.13205i
See next 80 embeddings (of 1120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
16.e even 4 1 inner
55.j even 10 1 inner
80.q even 4 1 inner
176.w even 20 1 inner
880.cl 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.cl.a 1120
5.b even 2 1 inner 880.2.cl.a 1120
11.c even 5 1 inner 880.2.cl.a 1120
16.e even 4 1 inner 880.2.cl.a 1120
55.j even 10 1 inner 880.2.cl.a 1120
80.q even 4 1 inner 880.2.cl.a 1120
176.w even 20 1 inner 880.2.cl.a 1120
880.cl even 20 1 inner 880.2.cl.a 1120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.cl.a 1120 1.a even 1 1 trivial
880.2.cl.a 1120 5.b even 2 1 inner
880.2.cl.a 1120 11.c even 5 1 inner
880.2.cl.a 1120 16.e even 4 1 inner
880.2.cl.a 1120 55.j even 10 1 inner
880.2.cl.a 1120 80.q even 4 1 inner
880.2.cl.a 1120 176.w even 20 1 inner
880.2.cl.a 1120 880.cl even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(880, [\chi])\).