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Properties

Label 880.2.cx.a
Level $880$
Weight $2$
Character orbit 880.cx
Analytic conductor $7.027$
Analytic rank $0$
Dimension $768$
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(51,880)]
 
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, 0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.cx (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: \(768\)
Relative dimension: \(96\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 768 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 768 q + 8 q^{4} - 8 q^{11} + 4 q^{14} - 40 q^{16} - 100 q^{18} + 8 q^{20} + 100 q^{22} - 32 q^{23} - 32 q^{26} - 40 q^{34} + 48 q^{37} - 40 q^{42} + 48 q^{44} - 40 q^{48} + 192 q^{49} - 48 q^{53} - 152 q^{56} - 40 q^{58} - 16 q^{59} + 84 q^{60} - 64 q^{64} - 40 q^{66} + 80 q^{67} + 260 q^{68} - 8 q^{70} - 32 q^{71} - 100 q^{72} - 100 q^{74} - 16 q^{77} + 152 q^{78} - 48 q^{80} + 192 q^{81} - 120 q^{82} - 240 q^{84} - 80 q^{88} - 160 q^{91} - 88 q^{92} - 240 q^{94} - 200 q^{96} + 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −1.41381 + 0.0336718i 1.50564 + 2.95499i 1.99773 − 0.0952111i 0.156434 − 0.987688i −2.22820 − 4.12710i −2.52067 + 0.819017i −2.82121 + 0.201878i −4.70165 + 6.47126i −0.187912 + 1.40167i
51.2 −1.40610 + 0.151292i −0.812644 − 1.59490i 1.95422 − 0.425463i −0.156434 + 0.987688i 1.38395 + 2.11964i −3.18505 + 1.03488i −2.68346 + 0.893901i −0.119972 + 0.165128i 0.0705326 − 1.41245i
51.3 −1.39955 + 0.203098i 0.00126045 + 0.00247377i 1.91750 − 0.568494i −0.156434 + 0.987688i −0.00226648 − 0.00320618i 3.51826 − 1.14315i −2.56819 + 1.18508i 1.76335 − 2.42704i 0.0183406 − 1.41409i
51.4 −1.39420 + 0.237077i −0.887359 − 1.74154i 1.88759 − 0.661067i 0.156434 − 0.987688i 1.65004 + 2.21768i −0.0515426 + 0.0167472i −2.47495 + 1.36916i −0.482198 + 0.663689i 0.0160576 + 1.41412i
51.5 −1.39173 + 0.251179i 1.14770 + 2.25249i 1.87382 − 0.699145i 0.156434 − 0.987688i −2.16307 − 2.84658i 4.56056 − 1.48182i −2.43224 + 1.44368i −1.99314 + 2.74332i 0.0303718 + 1.41389i
51.6 −1.38457 − 0.288046i −1.18522 − 2.32613i 1.83406 + 0.797639i 0.156434 − 0.987688i 0.970987 + 3.56208i 2.28768 − 0.743314i −2.30962 − 1.63268i −2.24276 + 3.08689i −0.501094 + 1.32246i
51.7 −1.38424 − 0.289600i 0.974740 + 1.91304i 1.83226 + 0.801753i −0.156434 + 0.987688i −0.795264 − 2.93039i 1.55614 − 0.505620i −2.30411 − 1.64045i −0.946230 + 1.30237i 0.502578 − 1.32190i
51.8 −1.36223 − 0.379895i 0.236345 + 0.463852i 1.71136 + 1.03501i 0.156434 − 0.987688i −0.145741 − 0.721662i −1.86341 + 0.605459i −1.93807 − 2.06007i 1.60406 − 2.20779i −0.588318 + 1.28603i
51.9 −1.32786 + 0.486622i −1.46252 − 2.87037i 1.52640 − 1.29233i −0.156434 + 0.987688i 3.33880 + 3.09973i 1.58221 − 0.514090i −1.39796 + 2.45880i −4.33666 + 5.96891i −0.272908 − 1.38763i
51.10 −1.32174 − 0.502983i −0.659135 − 1.29363i 1.49402 + 1.32963i −0.156434 + 0.987688i 0.220537 + 2.04138i −1.69361 + 0.550286i −1.30593 − 2.50889i 0.524347 − 0.721701i 0.703556 − 1.22679i
51.11 −1.32131 + 0.504130i 0.640530 + 1.25711i 1.49171 − 1.33222i −0.156434 + 0.987688i −1.48008 − 1.33812i −4.06110 + 1.31953i −1.29939 + 2.51229i 0.593308 − 0.816619i −0.291225 − 1.38390i
51.12 −1.29447 + 0.569508i 0.111287 + 0.218413i 1.35132 − 1.47443i 0.156434 − 0.987688i −0.268446 − 0.219351i 3.82962 − 1.24432i −0.909553 + 2.67819i 1.72804 − 2.37844i 0.359996 + 1.36763i
51.13 −1.27150 − 0.619111i 0.941549 + 1.84789i 1.23340 + 1.57439i −0.156434 + 0.987688i −0.0531238 − 2.93251i 1.42689 − 0.463625i −0.593541 − 2.76545i −0.764839 + 1.05271i 0.810395 − 1.15899i
51.14 −1.26544 + 0.631398i −0.499740 − 0.980794i 1.20267 − 1.59799i 0.156434 − 0.987688i 1.25166 + 0.925601i −1.11934 + 0.363697i −0.512939 + 2.78153i 1.05114 − 1.44677i 0.425666 + 1.34863i
51.15 −1.22957 + 0.698691i 1.16283 + 2.28218i 1.02366 − 1.71817i −0.156434 + 0.987688i −3.02431 − 1.99363i 0.412829 − 0.134136i −0.0581881 + 2.82783i −2.09282 + 2.88052i −0.497742 − 1.32373i
51.16 −1.20651 − 0.737787i −1.56157 − 3.06476i 0.911340 + 1.78030i −0.156434 + 0.987688i −0.377085 + 4.84978i −0.633127 + 0.205715i 0.213939 − 2.82032i −5.19090 + 7.14466i 0.917444 − 1.07624i
51.17 −1.20365 − 0.742441i 1.00105 + 1.96467i 0.897564 + 1.78728i 0.156434 − 0.987688i 0.253734 − 3.10800i 1.92443 − 0.625284i 0.246596 − 2.81766i −1.09447 + 1.50640i −0.921593 + 1.07269i
51.18 −1.19869 − 0.750427i −0.434969 − 0.853674i 0.873719 + 1.79906i −0.156434 + 0.987688i −0.119227 + 1.34970i 3.95226 − 1.28417i 0.302745 − 2.81218i 1.22379 − 1.68441i 0.928705 − 1.06654i
51.19 −1.19823 − 0.751166i −0.645323 − 1.26652i 0.871500 + 1.80014i 0.156434 − 0.987688i −0.178121 + 2.00232i −4.56009 + 1.48166i 0.307944 − 2.81161i 0.575731 − 0.792426i −0.929362 + 1.06597i
51.20 −1.18918 + 0.765413i 0.820566 + 1.61045i 0.828285 − 1.82042i 0.156434 − 0.987688i −2.20846 − 1.28704i −4.15614 + 1.35041i 0.408399 + 2.79879i −0.156869 + 0.215911i 0.569961 + 1.29427i
See next 80 embeddings (of 768 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
16.f odd 4 1 inner
176.x 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.cx.a ✓ 768
11.d odd 10 1 inner 880.2.cx.a ✓ 768
16.f odd 4 1 inner 880.2.cx.a ✓ 768
176.x even 20 1 inner 880.2.cx.a ✓ 768
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.cx.a ✓ 768 1.a even 1 1 trivial
880.2.cx.a ✓ 768 11.d odd 10 1 inner
880.2.cx.a ✓ 768 16.f odd 4 1 inner
880.2.cx.a ✓ 768 176.x even 20 1 inner

Hecke kernels

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