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Properties

Label 880.2.ch.a
Level $880$
Weight $2$
Character orbit 880.ch
Analytic conductor $7.027$
Analytic rank $0$
Dimension $1120$
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(3,880)]
 
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([10, 15, 15, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.ch (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.

\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 6 q^{2} - 12 q^{3} - 6 q^{5} - 12 q^{6} - 12 q^{7} - 6 q^{8} - 264 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 6 q^{2} - 12 q^{3} - 6 q^{5} - 12 q^{6} - 12 q^{7} - 6 q^{8} - 264 q^{9} - 24 q^{10} - 16 q^{11} - 20 q^{12} + 24 q^{15} - 12 q^{16} - 12 q^{17} + 14 q^{18} - 6 q^{20} - 32 q^{21} - 22 q^{22} - 32 q^{23} + 24 q^{24} + 20 q^{26} + 12 q^{27} - 22 q^{28} - 36 q^{32} - 16 q^{33} + 32 q^{34} - 26 q^{35} + 20 q^{36} - 42 q^{38} + 36 q^{40} + 42 q^{42} - 80 q^{44} + 16 q^{45} - 12 q^{46} + 6 q^{48} - 68 q^{50} - 12 q^{51} + 2 q^{52} - 12 q^{53} + 168 q^{54} - 16 q^{55} - 80 q^{56} + 24 q^{57} - 50 q^{58} - 16 q^{59} + 52 q^{60} - 12 q^{61} - 124 q^{62} - 36 q^{63} - 32 q^{65} - 24 q^{66} + 50 q^{68} - 36 q^{69} + 30 q^{70} - 56 q^{71} - 52 q^{72} - 100 q^{74} - 42 q^{75} - 32 q^{76} + 12 q^{77} - 132 q^{78} + 46 q^{80} - 240 q^{81} + 30 q^{82} + 108 q^{83} - 24 q^{84} - 26 q^{85} - 4 q^{86} - 32 q^{87} - 130 q^{88} - 4 q^{90} - 28 q^{91} - 68 q^{92} + 48 q^{94} + 120 q^{95} + 20 q^{96} - 12 q^{97} - 172 q^{98} - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41419 − 0.00742043i −0.420156 − 1.29311i 1.99989 + 0.0209879i −1.45621 + 1.69690i 0.584587 + 1.83182i −2.03662 − 3.99710i −2.82808 − 0.0445209i 0.931452 − 0.676740i 2.07195 − 2.38894i
3.2 −1.41413 + 0.0154771i 0.615081 + 1.89302i 1.99952 − 0.0437732i 2.17850 + 0.504120i −0.899102 − 2.66746i 1.39271 + 2.73334i −2.82690 + 0.0928478i −0.778167 + 0.565372i −3.08848 − 0.679174i
3.3 −1.41368 + 0.0389587i −0.937940 − 2.88668i 1.99696 − 0.110150i 1.44999 + 1.70221i 1.43841 + 4.04430i 0.678210 + 1.33106i −2.81877 + 0.233516i −5.02616 + 3.65172i −2.11613 − 2.34989i
3.4 −1.41234 − 0.0728289i 0.576842 + 1.77534i 1.98939 + 0.205718i −2.23607 + 0.00245174i −0.685400 − 2.54939i −0.104289 − 0.204678i −2.79471 − 0.435428i −0.392025 + 0.284823i 3.15826 + 0.159388i
3.5 −1.41156 − 0.0865730i 0.181125 + 0.557445i 1.98501 + 0.244406i 1.50243 − 1.65611i −0.207409 − 0.802548i −0.999464 − 1.96156i −2.78080 − 0.516843i 2.14911 − 1.56142i −2.26414 + 2.20763i
3.6 −1.40992 + 0.110113i −0.245249 − 0.754798i 1.97575 − 0.310501i −0.273486 + 2.21928i 0.428894 + 1.03720i 1.26650 + 2.48564i −2.75146 + 0.655336i 1.91748 − 1.39313i 0.141222 − 3.15912i
3.7 −1.40476 − 0.163251i −0.00637238 − 0.0196122i 1.94670 + 0.458656i 2.20539 − 0.369125i 0.00574996 + 0.0285907i 0.841532 + 1.65160i −2.65977 − 0.962102i 2.42671 − 1.76311i −3.15830 + 0.158501i
3.8 −1.39587 + 0.227068i −0.278371 − 0.856738i 1.89688 − 0.633912i −1.64351 − 1.51620i 0.583106 + 1.13268i 0.635022 + 1.24630i −2.50385 + 1.31558i 1.77054 − 1.28637i 2.63840 + 1.74322i
3.9 −1.38155 + 0.302196i 0.944887 + 2.90806i 1.81736 − 0.834996i −0.886842 − 2.05268i −2.18421 − 3.73209i 2.37324 + 4.65774i −2.25843 + 1.70279i −5.13697 + 3.73223i 1.84553 + 2.56788i
3.10 −1.38081 − 0.305559i 0.672650 + 2.07020i 1.81327 + 0.843837i −1.89509 − 1.18685i −0.296232 − 3.06409i −1.34718 − 2.64400i −2.24593 − 1.71924i −1.40623 + 1.02169i 2.25411 + 2.21788i
3.11 −1.37207 + 0.342675i 0.979144 + 3.01349i 1.76515 − 0.940347i 1.59160 − 1.57061i −2.37610 − 3.79919i −1.58669 − 3.11406i −2.09967 + 1.89509i −5.69537 + 4.13793i −1.64557 + 2.70039i
3.12 −1.36438 − 0.372123i −0.640082 − 1.96997i 1.72305 + 1.01543i −2.10934 − 0.742086i 0.140241 + 2.92597i 1.78514 + 3.50354i −1.97302 − 2.02662i −1.04403 + 0.758530i 2.60178 + 1.79742i
3.13 −1.36287 − 0.377605i −0.495583 − 1.52525i 1.71483 + 1.02925i −0.149487 − 2.23107i 0.0994734 + 2.26585i −1.50066 − 2.94521i −1.94844 − 2.05027i 0.346271 − 0.251580i −0.638731 + 3.09710i
3.14 −1.35013 + 0.420899i −0.200873 − 0.618224i 1.64569 − 1.13653i 0.939878 + 2.02895i 0.531414 + 0.750134i −1.18591 − 2.32748i −1.74352 + 2.22713i 2.08520 − 1.51499i −2.12294 − 2.34374i
3.15 −1.33212 + 0.474831i −0.880415 − 2.70964i 1.54907 − 1.26506i 0.453949 − 2.18950i 2.45944 + 3.19151i −1.13236 − 2.22239i −1.46286 + 2.42075i −4.13996 + 3.00786i 0.434930 + 3.13223i
3.16 −1.32110 − 0.504675i −0.718182 − 2.21034i 1.49061 + 1.33345i 2.20754 − 0.356039i −0.166712 + 3.28252i −0.492736 − 0.967049i −1.29628 − 2.51389i −1.94275 + 1.41149i −3.09606 − 0.643729i
3.17 −1.29404 + 0.570493i 0.316697 + 0.974695i 1.34907 − 1.47648i −2.08879 + 0.798093i −0.965876 − 1.08062i 0.205045 + 0.402424i −0.903433 + 2.68026i 1.57732 − 1.14599i 2.24767 − 2.22441i
3.18 −1.27747 − 0.606694i 0.250972 + 0.772414i 1.26385 + 1.55006i −1.43109 + 1.71813i 0.148009 − 1.13900i 1.73454 + 3.40422i −0.674108 − 2.74692i 1.89342 − 1.37565i 2.87055 − 1.32662i
3.19 −1.27435 + 0.613223i −0.0348820 − 0.107356i 1.24791 − 1.56292i −0.0740804 − 2.23484i 0.110285 + 0.115418i 0.817935 + 1.60529i −0.631859 + 2.75695i 2.41674 − 1.75587i 1.46486 + 2.80253i
3.20 −1.27188 − 0.618313i 0.560562 + 1.72523i 1.23538 + 1.57285i 1.60613 + 1.55575i 0.353764 − 2.54090i −2.02820 − 3.98057i −0.598746 − 2.76433i −0.235147 + 0.170845i −1.08087 − 2.97182i
See next 80 embeddings (of 1120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
80.s even 4 1 inner
880.ch 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.ch.a ✓ 1120
5.c odd 4 1 880.2.cz.a yes 1120
11.c even 5 1 inner 880.2.ch.a ✓ 1120
16.f odd 4 1 880.2.cz.a yes 1120
55.k odd 20 1 880.2.cz.a yes 1120
80.s even 4 1 inner 880.2.ch.a ✓ 1120
176.v odd 20 1 880.2.cz.a yes 1120
880.ch even 20 1 inner 880.2.ch.a ✓ 1120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.ch.a ✓ 1120 1.a even 1 1 trivial
880.2.ch.a ✓ 1120 11.c even 5 1 inner
880.2.ch.a ✓ 1120 80.s even 4 1 inner
880.2.ch.a ✓ 1120 880.ch even 20 1 inner
880.2.cz.a yes 1120 5.c odd 4 1
880.2.cz.a yes 1120 16.f odd 4 1
880.2.cz.a yes 1120 55.k odd 20 1
880.2.cz.a yes 1120 176.v odd 20 1

Hecke kernels

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