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Properties

Label 800.2.v.b
Level $800$
Weight $2$
Character orbit 800.v
Analytic conductor $6.388$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,2,Mod(43,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.v (of order \(8\), degree \(4\), 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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{2} + 4 q^{3} - 8 q^{6} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{2} + 4 q^{3} - 8 q^{6} + 16 q^{8} - 8 q^{11} + 20 q^{12} + 4 q^{13} + 16 q^{14} - 8 q^{16} - 4 q^{18} + 16 q^{19} - 8 q^{21} + 20 q^{22} - 32 q^{24} - 8 q^{26} + 16 q^{27} - 12 q^{28} + 24 q^{32} + 8 q^{33} + 8 q^{34} - 8 q^{36} + 4 q^{37} + 60 q^{38} - 8 q^{41} - 60 q^{42} + 36 q^{43} + 16 q^{44} - 8 q^{46} + 8 q^{47} + 24 q^{48} - 40 q^{49} + 8 q^{51} + 56 q^{52} + 4 q^{53} - 56 q^{54} + 40 q^{56} + 8 q^{57} + 36 q^{58} - 40 q^{61} - 40 q^{62} - 56 q^{63} + 48 q^{64} + 72 q^{66} - 20 q^{67} - 8 q^{68} + 24 q^{69} + 24 q^{71} - 24 q^{72} + 8 q^{73} + 56 q^{76} - 24 q^{77} - 32 q^{78} - 84 q^{82} + 44 q^{83} + 56 q^{84} + 24 q^{86} + 120 q^{87} - 104 q^{88} - 8 q^{91} - 124 q^{92} - 8 q^{93} - 32 q^{94} + 24 q^{96} + 8 q^{97} + 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.40295 − 0.178125i 1.05468 − 2.54622i 1.93654 + 0.499802i 0 −1.93321 + 3.38436i − 3.70855i −2.62785 − 1.04614i −3.24959 − 3.24959i 0
43.2 −1.31460 + 0.521383i −0.923741 + 2.23011i 1.45632 − 1.37081i 0 0.0516058 − 3.41331i − 3.63945i −1.19975 + 2.56137i −1.99876 − 1.99876i 0
43.3 −1.30876 − 0.535863i −0.461737 + 1.11473i 1.42570 + 1.40263i 0 1.20165 − 1.21149i 2.85280i −1.11428 − 2.59969i 1.09189 + 1.09189i 0
43.4 −1.30208 + 0.551886i 0.577765 − 1.39485i 1.39084 − 1.43720i 0 0.0174990 + 2.13507i 1.62907i −1.01782 + 2.63895i 0.509534 + 0.509534i 0
43.5 −1.02775 − 0.971453i 1.05354 − 2.54348i 0.112559 + 1.99683i 0 −3.55365 + 1.59060i 4.43630i 1.82414 − 2.16160i −3.23800 − 3.23800i 0
43.6 −0.992527 + 1.00742i 0.278775 − 0.673021i −0.0297801 − 1.99978i 0 0.401322 + 0.948834i 0.467309i 2.04417 + 1.95483i 1.74608 + 1.74608i 0
43.7 −0.816290 − 1.15485i −1.28377 + 3.09930i −0.667342 + 1.88538i 0 4.62714 − 1.04737i 0.906290i 2.72207 − 0.768338i −5.83625 − 5.83625i 0
43.8 −0.640905 − 1.26065i −0.0983610 + 0.237464i −1.17848 + 1.61591i 0 0.362400 − 0.0281932i − 4.12414i 2.79240 + 0.450008i 2.07461 + 2.07461i 0
43.9 −0.400521 + 1.35631i −0.627770 + 1.51557i −1.67917 − 1.08646i 0 −1.80415 − 1.45847i 4.80429i 2.14613 − 1.84232i 0.218458 + 0.218458i 0
43.10 −0.375898 − 1.36334i 0.508197 − 1.22690i −1.71740 + 1.02496i 0 −1.86371 − 0.231658i 0.810621i 2.04293 + 1.95613i 0.874309 + 0.874309i 0
43.11 −0.0754956 + 1.41220i 0.698571 − 1.68650i −1.98860 − 0.213229i 0 2.32893 + 1.11384i − 2.70081i 0.451252 − 2.79220i −0.234961 − 0.234961i 0
43.12 0.201889 + 1.39973i −0.867966 + 2.09546i −1.91848 + 0.565181i 0 −3.10830 − 0.791867i − 1.82364i −1.17842 − 2.57125i −1.51625 − 1.51625i 0
43.13 0.458371 − 1.33787i −0.218753 + 0.528116i −1.57979 − 1.22648i 0 0.606280 + 0.534736i − 0.814088i −2.36500 + 1.55137i 1.89027 + 1.89027i 0
43.14 0.491080 − 1.32621i −0.896482 + 2.16430i −1.51768 − 1.30255i 0 2.43008 + 2.25177i 0.225996i −2.47277 + 1.37311i −1.75919 − 1.75919i 0
43.15 0.759267 + 1.19311i 0.252131 − 0.608697i −0.847028 + 1.81178i 0 0.917677 − 0.161344i − 1.49067i −2.80477 + 0.365026i 1.81438 + 1.81438i 0
43.16 0.998231 + 1.00177i −0.509063 + 1.22899i −0.00706901 + 1.99999i 0 −1.73932 + 0.716851i 2.73471i −2.01058 + 1.98937i 0.870059 + 0.870059i 0
43.17 1.04802 − 0.949552i 0.255424 − 0.616647i 0.196703 − 1.99030i 0 −0.317849 − 0.888798i 2.27809i −1.68375 − 2.27266i 1.80631 + 1.80631i 0
43.18 1.19101 − 0.762560i 1.11089 − 2.68192i 0.837005 − 1.81643i 0 −0.722048 − 4.04131i − 0.874514i −0.388258 − 2.80165i −3.83732 − 3.83732i 0
43.19 1.32278 + 0.500260i 1.03647 − 2.50226i 1.49948 + 1.32347i 0 2.62280 − 2.79143i − 2.65674i 1.32140 + 2.50078i −3.06572 − 3.06572i 0
43.20 1.37186 − 0.343520i −1.11476 + 2.69126i 1.76399 − 0.942521i 0 −0.604785 + 4.07496i − 0.518179i 2.09617 − 1.89897i −3.87887 − 3.87887i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
160.u even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.v.b 88
5.b even 2 1 160.2.u.a ✓ 88
5.c odd 4 1 160.2.ba.a yes 88
5.c odd 4 1 800.2.bb.b 88
20.d odd 2 1 640.2.u.a 88
20.e even 4 1 640.2.ba.a 88
32.h odd 8 1 800.2.bb.b 88
160.u even 8 1 inner 800.2.v.b 88
160.v odd 8 1 640.2.u.a 88
160.y odd 8 1 160.2.ba.a yes 88
160.z even 8 1 640.2.ba.a 88
160.ba even 8 1 160.2.u.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.u.a ✓ 88 5.b even 2 1
160.2.u.a ✓ 88 160.ba even 8 1
160.2.ba.a yes 88 5.c odd 4 1
160.2.ba.a yes 88 160.y odd 8 1
640.2.u.a 88 20.d odd 2 1
640.2.u.a 88 160.v odd 8 1
640.2.ba.a 88 20.e even 4 1
640.2.ba.a 88 160.z even 8 1
800.2.v.b 88 1.a even 1 1 trivial
800.2.v.b 88 160.u even 8 1 inner
800.2.bb.b 88 5.c odd 4 1
800.2.bb.b 88 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 4 T_{3}^{87} + 8 T_{3}^{86} - 24 T_{3}^{85} + 64 T_{3}^{84} - 8 T_{3}^{83} + \cdots + 154719502336 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display