Properties

Label 891.2.bb.a
Level $891$
Weight $2$
Character orbit 891.bb
Analytic conductor $7.115$
Analytic rank $0$
Dimension $816$
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] = [891,2,Mod(8,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([5, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.bb (of order \(90\), degree \(24\), 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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(816\)
Relative dimension: \(34\) over \(\Q(\zeta_{90})\)
Twist minimal: no (minimal twist has level 297)
Sato-Tate group: $\mathrm{SU}(2)[C_{90}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 816 q + 30 q^{2} - 18 q^{4} + 21 q^{5} - 30 q^{7} + 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 816 q + 30 q^{2} - 18 q^{4} + 21 q^{5} - 30 q^{7} + 45 q^{8} + 33 q^{11} - 30 q^{13} + 18 q^{14} - 30 q^{16} + 45 q^{17} - 15 q^{19} + 60 q^{20} - 15 q^{22} + 84 q^{23} - 27 q^{25} - 60 q^{28} - 60 q^{29} - 9 q^{31} - 42 q^{34} + 45 q^{35} - 9 q^{37} + 18 q^{38} - 90 q^{40} + 30 q^{41} + 108 q^{44} - 15 q^{46} + 6 q^{47} - 18 q^{49} + 105 q^{50} - 30 q^{52} - 48 q^{55} - 54 q^{56} - 18 q^{58} - 81 q^{59} - 30 q^{61} + 45 q^{62} + 51 q^{64} + 6 q^{67} + 225 q^{68} - 93 q^{70} + 27 q^{71} - 15 q^{73} + 30 q^{74} + 141 q^{77} - 30 q^{79} - 36 q^{82} - 15 q^{83} - 30 q^{85} - 93 q^{86} - 108 q^{88} - 54 q^{89} - 9 q^{91} - 276 q^{92} - 30 q^{94} - 90 q^{95} - 81 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −1.16150 2.38143i 0 −3.09081 + 3.95605i 0.0429102 + 0.613644i 0 −1.71414 + 0.0598591i 7.82768 + 1.66383i 0 1.41151 0.814937i
8.2 −1.11997 2.29627i 0 −2.78723 + 3.56749i 0.0405112 + 0.579337i 0 −1.72991 + 0.0604096i 6.31552 + 1.34240i 0 1.28495 0.741864i
8.3 −1.11045 2.27675i 0 −2.71918 + 3.48040i −0.237504 3.39647i 0 1.01734 0.0355264i 5.98798 + 1.27279i 0 −7.46918 + 4.31233i
8.4 −1.10307 2.26163i 0 −2.66689 + 3.41346i 0.284326 + 4.06604i 0 4.04513 0.141259i 5.73915 + 1.21989i 0 8.88227 5.12818i
8.5 −0.936912 1.92095i 0 −1.58094 + 2.02351i 0.0440697 + 0.630226i 0 1.09943 0.0383930i 1.18716 + 0.252338i 0 1.16935 0.675122i
8.6 −0.847489 1.73761i 0 −1.06973 + 1.36919i −0.164217 2.34841i 0 −0.729987 + 0.0254917i −0.496339 0.105500i 0 −3.94146 + 2.27560i
8.7 −0.831591 1.70501i 0 −0.984207 + 1.25973i −0.134621 1.92517i 0 −4.51387 + 0.157628i −0.744786 0.158309i 0 −3.17049 + 1.83049i
8.8 −0.738685 1.51453i 0 −0.516818 + 0.661497i 0.106711 + 1.52603i 0 2.22195 0.0775923i −1.91287 0.406592i 0 2.23239 1.28887i
8.9 −0.626833 1.28520i 0 −0.0274932 + 0.0351897i −0.0191541 0.273916i 0 −0.0722617 + 0.00252344i −2.73488 0.581316i 0 −0.340030 + 0.196317i
8.10 −0.626532 1.28458i 0 −0.0262823 + 0.0336398i 0.179932 + 2.57315i 0 −4.29989 + 0.150155i −2.73631 0.581621i 0 3.19268 1.84330i
8.11 −0.469140 0.961880i 0 0.526202 0.673508i 0.120802 + 1.72754i 0 3.20463 0.111908i −2.98830 0.635184i 0 1.60502 0.926656i
8.12 −0.361589 0.741367i 0 0.812444 1.03988i −0.228051 3.26129i 0 −3.64248 + 0.127198i −2.67835 0.569301i 0 −2.33535 + 1.34832i
8.13 −0.337236 0.691437i 0 0.866966 1.10967i 0.278822 + 3.98735i 0 −2.50352 + 0.0874247i −2.56460 0.545123i 0 2.66297 1.53747i
8.14 −0.294358 0.603523i 0 0.953729 1.22072i −0.136587 1.95329i 0 1.03639 0.0361914i −2.33108 0.495487i 0 −1.13865 + 0.657401i
8.15 −0.131962 0.270562i 0 1.17553 1.50461i −0.115008 1.64470i 0 4.68773 0.163699i −1.15112 0.244677i 0 −0.429816 + 0.248154i
8.16 −0.0361822 0.0741844i 0 1.22713 1.57065i 0.116205 + 1.66180i 0 −4.36097 + 0.152288i −0.322386 0.0685253i 0 0.119076 0.0687483i
8.17 −0.0350715 0.0719072i 0 1.22738 1.57098i 0.103569 + 1.48111i 0 −0.412525 + 0.0144057i −0.312522 0.0664287i 0 0.102870 0.0593920i
8.18 −0.00652561 0.0133795i 0 1.23119 1.57585i −0.107633 1.53923i 0 −0.727962 + 0.0254210i −0.0582398 0.0123792i 0 −0.0198917 + 0.0114845i
8.19 0.234840 + 0.481493i 0 1.05464 1.34987i 0.140627 + 2.01106i 0 2.77243 0.0968153i 1.94563 + 0.413557i 0 −0.935288 + 0.539989i
8.20 0.257823 + 0.528615i 0 1.01836 1.30344i −0.252670 3.61335i 0 1.32393 0.0462327i 2.10215 + 0.446826i 0 1.84493 1.06517i
See next 80 embeddings (of 816 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
27.f odd 18 1 inner
297.x even 90 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.bb.a 816
3.b odd 2 1 297.2.x.a 816
11.d odd 10 1 inner 891.2.bb.a 816
27.e even 9 1 297.2.x.a 816
27.f odd 18 1 inner 891.2.bb.a 816
33.f even 10 1 297.2.x.a 816
297.w odd 90 1 297.2.x.a 816
297.x even 90 1 inner 891.2.bb.a 816
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.x.a 816 3.b odd 2 1
297.2.x.a 816 27.e even 9 1
297.2.x.a 816 33.f even 10 1
297.2.x.a 816 297.w odd 90 1
891.2.bb.a 816 1.a even 1 1 trivial
891.2.bb.a 816 11.d odd 10 1 inner
891.2.bb.a 816 27.f odd 18 1 inner
891.2.bb.a 816 297.x even 90 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(891, [\chi])\).