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Properties

Label 880.2.cy.a
Level $880$
Weight $2$
Character orbit 880.cy
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(13,880)]
 
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, 15, 15, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.cy (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.

\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 10 q^{2} - 6 q^{5} - 20 q^{6} - 10 q^{8} + 264 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 10 q^{2} - 6 q^{5} - 20 q^{6} - 10 q^{8} + 264 q^{9} - 16 q^{11} - 12 q^{12} - 12 q^{15} - 12 q^{16} - 20 q^{17} - 10 q^{18} - 6 q^{20} - 6 q^{22} - 44 q^{26} - 10 q^{28} - 80 q^{30} - 24 q^{31} - 16 q^{33} + 32 q^{34} - 10 q^{35} + 20 q^{36} + 14 q^{38} + 60 q^{40} - 6 q^{42} - 80 q^{44} - 8 q^{45} - 20 q^{46} - 12 q^{47} - 18 q^{48} - 80 q^{50} - 20 q^{51} - 50 q^{52} - 12 q^{53} + 16 q^{56} - 66 q^{58} + 16 q^{59} - 4 q^{60} - 20 q^{61} + 140 q^{62} - 20 q^{63} - 24 q^{66} - 32 q^{67} - 10 q^{68} + 36 q^{69} - 58 q^{70} + 140 q^{72} + 140 q^{74} - 66 q^{75} + 12 q^{77} - 108 q^{78} - 114 q^{80} - 240 q^{81} - 42 q^{82} - 10 q^{85} - 20 q^{86} - 130 q^{88} + 60 q^{90} + 4 q^{91} - 96 q^{92} - 20 q^{95} - 20 q^{96} - 12 q^{97} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.41410 + 0.0176605i −1.08506 − 1.49346i 1.99938 − 0.0499476i −0.179378 − 2.22886i 1.56076 + 2.09274i −0.726530 + 0.115071i −2.82644 + 0.105941i −0.126008 + 0.387812i 0.293022 + 3.14867i
13.2 −1.41218 − 0.0757753i −1.14992 − 1.58273i 1.98852 + 0.214017i 1.10709 + 1.94277i 1.50397 + 2.32224i 3.49475 − 0.553514i −2.79193 − 0.452911i −0.255666 + 0.786860i −1.41621 − 2.82743i
13.3 −1.41164 − 0.0852557i −0.527626 − 0.726215i 1.98546 + 0.240701i 1.11256 + 1.93964i 0.682904 + 1.07014i −0.558691 + 0.0884880i −2.78224 − 0.509055i 0.678052 − 2.08683i −1.40518 − 2.83293i
13.4 −1.40983 + 0.111265i 0.797546 + 1.09773i 1.97524 − 0.313730i −0.931138 + 2.03297i −1.24654 − 1.45887i 3.98688 − 0.631460i −2.74985 + 0.662080i 0.358123 − 1.10219i 1.08655 − 2.96975i
13.5 −1.40112 + 0.191990i −1.98579 − 2.73321i 1.92628 − 0.538003i −1.27719 + 1.83542i 3.30708 + 3.44830i −2.82542 + 0.447503i −2.59566 + 1.12363i −2.60000 + 8.00199i 1.43712 − 2.81686i
13.6 −1.39515 + 0.231446i 0.592721 + 0.815811i 1.89287 − 0.645801i 2.18392 + 0.480104i −1.01575 − 1.00099i 0.794029 − 0.125762i −2.49136 + 1.33908i 0.612822 − 1.88607i −3.15800 − 0.164357i
13.7 −1.39510 + 0.231732i −0.388471 − 0.534685i 1.89260 − 0.646577i −2.23498 + 0.0697491i 0.665859 + 0.655917i −4.79109 + 0.758834i −2.49053 + 1.34061i 0.792073 − 2.43775i 3.10185 − 0.615223i
13.8 −1.39396 + 0.238513i 1.32243 + 1.82017i 1.88622 − 0.664954i 1.72535 + 1.42238i −2.27755 − 2.22182i −2.96961 + 0.470341i −2.47071 + 1.37681i −0.637151 + 1.96095i −2.74432 − 1.57121i
13.9 −1.38582 − 0.281955i 1.39718 + 1.92305i 1.84100 + 0.781479i 1.71711 − 1.43232i −1.39403 − 3.05895i 4.25021 − 0.673168i −2.33096 − 1.60207i −0.818974 + 2.52054i −2.78345 + 1.50080i
13.10 −1.37997 − 0.309305i −1.04350 − 1.43626i 1.80866 + 0.853666i −1.80347 − 1.32193i 0.995768 + 2.30477i 0.583681 − 0.0924460i −2.23186 − 1.73746i −0.0468932 + 0.144322i 2.07986 + 2.38206i
13.11 −1.37755 − 0.319939i 1.36263 + 1.87551i 1.79528 + 0.881464i 0.882898 − 2.05438i −1.27705 − 3.01956i −3.93835 + 0.623773i −2.19107 − 1.78864i −0.733698 + 2.25809i −1.87351 + 2.54754i
13.12 −1.37116 − 0.346304i 1.94749 + 2.68049i 1.76015 + 0.949676i −2.16721 + 0.550642i −1.74205 − 4.34980i −2.19745 + 0.348042i −2.08456 − 1.91170i −2.46525 + 7.58727i 3.16227 − 0.00450300i
13.13 −1.36669 − 0.363534i 0.452102 + 0.622265i 1.73569 + 0.993677i −2.21402 + 0.313255i −0.391669 − 1.01480i 1.70227 − 0.269613i −2.01091 − 1.98903i 0.744234 − 2.29052i 3.13975 + 0.376748i
13.14 −1.36252 + 0.378866i 1.80348 + 2.48227i 1.71292 − 1.03243i −1.54890 − 1.61274i −3.39772 − 2.69887i 2.92813 − 0.463770i −1.94274 + 2.05567i −1.98209 + 6.10026i 2.72141 + 1.61056i
13.15 −1.34685 − 0.431274i 0.0353882 + 0.0487077i 1.62801 + 1.16172i 2.17412 − 0.522692i −0.0266562 − 0.0808639i −3.34947 + 0.530504i −1.69166 − 2.26678i 0.925931 − 2.84972i −3.15363 − 0.233654i
13.16 −1.33772 − 0.458798i −1.62803 − 2.24079i 1.57901 + 1.22749i 1.87348 − 1.22068i 1.14978 + 3.74450i −3.37227 + 0.534116i −1.54911 − 2.36649i −1.44361 + 4.44298i −3.06625 + 0.773379i
13.17 −1.32298 + 0.499721i 0.481982 + 0.663391i 1.50056 − 1.32224i −2.20680 − 0.360606i −0.969163 − 0.636797i 0.865683 − 0.137111i −1.32446 + 2.49916i 0.719270 − 2.21369i 3.09976 − 0.625709i
13.18 −1.31485 + 0.520739i −1.67605 − 2.30688i 1.45766 − 1.36939i −2.05255 − 0.887145i 3.40503 + 2.16042i 4.49054 − 0.711232i −1.20352 + 2.55960i −1.58552 + 4.87971i 3.16077 + 0.0976194i
13.19 −1.31360 + 0.523887i −0.0572868 − 0.0788486i 1.45108 − 1.37636i 0.640694 − 2.14231i 0.116560 + 0.0735636i −2.09917 + 0.332476i −1.18509 + 2.56818i 0.924116 − 2.84414i 0.280717 + 3.14979i
13.20 −1.30093 + 0.554606i 1.07965 + 1.48601i 1.38482 − 1.44301i −0.867537 − 2.06092i −2.22869 − 1.33441i −0.368209 + 0.0583185i −1.00125 + 2.64528i −0.115526 + 0.355552i 2.27160 + 2.19996i
See next 80 embeddings (of 1120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
80.i odd 4 1 inner
880.cy 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.cy.a yes 1120
5.c odd 4 1 880.2.cg.a ✓ 1120
11.d odd 10 1 inner 880.2.cy.a yes 1120
16.e even 4 1 880.2.cg.a ✓ 1120
55.l even 20 1 880.2.cg.a ✓ 1120
80.i odd 4 1 inner 880.2.cy.a yes 1120
176.u odd 20 1 880.2.cg.a ✓ 1120
880.cy even 20 1 inner 880.2.cy.a yes 1120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.cg.a ✓ 1120 5.c odd 4 1
880.2.cg.a ✓ 1120 16.e even 4 1
880.2.cg.a ✓ 1120 55.l even 20 1
880.2.cg.a ✓ 1120 176.u odd 20 1
880.2.cy.a yes 1120 1.a even 1 1 trivial
880.2.cy.a yes 1120 11.d odd 10 1 inner
880.2.cy.a yes 1120 80.i odd 4 1 inner
880.2.cy.a yes 1120 880.cy even 20 1 inner

Hecke kernels

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