↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,2,Mod(107,800)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bb (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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{2} + 4 q^{3} - 8 q^{6} + 8 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{2} + 4 q^{3} - 8 q^{6} + 8 q^{7} - 8 q^{8} - 8 q^{11} + 20 q^{12} + 4 q^{13} - 16 q^{14} - 8 q^{16} + 12 q^{18} - 16 q^{19} - 8 q^{21} + 20 q^{22} + 8 q^{23} + 32 q^{24} - 8 q^{26} - 8 q^{27} + 20 q^{28} - 16 q^{32} + 8 q^{33} - 8 q^{34} - 8 q^{36} + 4 q^{37} + 4 q^{38} - 8 q^{41} + 68 q^{42} - 28 q^{43} - 16 q^{44} - 8 q^{46} + 8 q^{47} + 88 q^{48} + 40 q^{49} + 8 q^{51} - 48 q^{52} + 4 q^{53} + 56 q^{54} + 40 q^{56} + 36 q^{58} - 40 q^{61} + 32 q^{62} + 56 q^{63} - 48 q^{64} + 72 q^{66} + 28 q^{67} - 8 q^{68} - 24 q^{69} + 24 q^{71} - 88 q^{72} + 56 q^{76} + 32 q^{77} - 112 q^{78} - 68 q^{82} - 36 q^{83} - 56 q^{84} + 24 q^{86} + 24 q^{88} - 8 q^{91} - 92 q^{92} - 8 q^{93} + 32 q^{94} + 24 q^{96} + 8 q^{97} - 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.41220 − 0.0754956i −1.68650 − 0.698571i 1.98860 + 0.213229i 0 2.32893 + 1.11384i 2.70081 −2.79220 − 0.451252i 0.234961 + 0.234961i 0
107.2 −1.39973 + 0.201889i 2.09546 + 0.867966i 1.91848 − 0.565181i 0 −3.10830 − 0.791867i 1.82364 −2.57125 + 1.17842i 1.51625 + 1.51625i 0
107.3 −1.35631 − 0.400521i 1.51557 + 0.627770i 1.67917 + 1.08646i 0 −1.80415 − 1.45847i −4.80429 −1.84232 − 2.14613i −0.218458 − 0.218458i 0
107.4 −1.19311 + 0.759267i −0.608697 − 0.252131i 0.847028 − 1.81178i 0 0.917677 − 0.161344i 1.49067 0.365026 + 2.80477i −1.81438 − 1.81438i 0
107.5 −1.00742 − 0.992527i −0.673021 − 0.278775i 0.0297801 + 1.99978i 0 0.401322 + 0.948834i −0.467309 1.95483 − 2.04417i −1.74608 − 1.74608i 0
107.6 −1.00177 + 0.998231i 1.22899 + 0.509063i 0.00706901 − 1.99999i 0 −1.73932 + 0.716851i −2.73471 1.98937 + 2.01058i −0.870059 − 0.870059i 0
107.7 −0.551886 − 1.30208i −1.39485 − 0.577765i −1.39084 + 1.43720i 0 0.0174990 + 2.13507i −1.62907 2.63895 + 1.01782i −0.509534 − 0.509534i 0
107.8 −0.521383 − 1.31460i 2.23011 + 0.923741i −1.45632 + 1.37081i 0 0.0516058 − 3.41331i 3.63945 2.56137 + 1.19975i 1.99876 + 1.99876i 0
107.9 −0.500260 + 1.32278i −2.50226 − 1.03647i −1.49948 − 1.32347i 0 2.62280 − 2.79143i 2.65674 2.50078 − 1.32140i 3.06572 + 3.06572i 0
107.10 −0.184269 + 1.40216i 1.10776 + 0.458849i −1.93209 − 0.516749i 0 −0.847505 + 1.46870i 4.27741 1.08059 − 2.61387i −1.10473 − 1.10473i 0
107.11 −0.0556644 + 1.41312i −0.532554 − 0.220591i −1.99380 − 0.157321i 0 0.341365 − 0.740282i −3.48272 0.333297 − 2.80872i −1.88637 − 1.88637i 0
107.12 0.178125 − 1.40295i −2.54622 − 1.05468i −1.93654 − 0.499802i 0 −1.93321 + 3.38436i 3.70855 −1.04614 + 2.62785i 3.24959 + 3.24959i 0
107.13 0.343520 + 1.37186i 2.69126 + 1.11476i −1.76399 + 0.942521i 0 −0.604785 + 4.07496i 0.518179 −1.89897 − 2.09617i 3.87887 + 3.87887i 0
107.14 0.535863 − 1.30876i 1.11473 + 0.461737i −1.42570 − 1.40263i 0 1.20165 − 1.21149i −2.85280 −2.59969 + 1.11428i −1.09189 − 1.09189i 0
107.15 0.762560 + 1.19101i −2.68192 − 1.11089i −0.837005 + 1.81643i 0 −0.722048 − 4.04131i 0.874514 −2.80165 + 0.388258i 3.83732 + 3.83732i 0
107.16 0.949552 + 1.04802i −0.616647 − 0.255424i −0.196703 + 1.99030i 0 −0.317849 − 0.888798i −2.27809 −2.27266 + 1.68375i −1.80631 − 1.80631i 0
107.17 0.971453 − 1.02775i −2.54348 − 1.05354i −0.112559 − 1.99683i 0 −3.55365 + 1.59060i −4.43630 −2.16160 − 1.82414i 3.23800 + 3.23800i 0
107.18 1.15485 − 0.816290i 3.09930 + 1.28377i 0.667342 − 1.88538i 0 4.62714 − 1.04737i −0.906290 −0.768338 − 2.72207i 5.83625 + 5.83625i 0
107.19 1.26065 − 0.640905i 0.237464 + 0.0983610i 1.17848 − 1.61591i 0 0.362400 − 0.0281932i 4.12414 0.450008 − 2.79240i −2.07461 − 2.07461i 0
107.20 1.32621 + 0.491080i 2.16430 + 0.896482i 1.51768 + 1.30255i 0 2.43008 + 2.25177i −0.225996 1.37311 + 2.47277i 1.75919 + 1.75919i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.22
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 4 T_{3}^{87} + 8 T_{3}^{86} - 32 T_{3}^{84} - 8 T_{3}^{83} + 288 T_{3}^{82} + \cdots + 154719502336 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display