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Properties

Label 800.2.y.e
Level $800$
Weight $2$
Character orbit 800.y
Analytic conductor $6.388$
Analytic rank $0$
Dimension $64$
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(101,800)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.y (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: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 16 q^{14} + 20 q^{16} + 20 q^{18} - 4 q^{22} + 8 q^{23} - 28 q^{24} + 24 q^{27} - 20 q^{28} - 20 q^{32} - 20 q^{34} + 12 q^{36} - 20 q^{38} + 24 q^{39} - 100 q^{42} + 8 q^{43} + 40 q^{44} + 32 q^{46} + 16 q^{51} - 88 q^{52} - 32 q^{53} + 76 q^{54} + 48 q^{56} + 72 q^{58} + 32 q^{59} - 32 q^{61} - 48 q^{62} + 80 q^{63} + 48 q^{64} + 16 q^{66} - 40 q^{67} + 48 q^{68} - 32 q^{69} + 32 q^{71} - 36 q^{72} + 8 q^{74} + 16 q^{77} + 36 q^{78} + 40 q^{83} + 56 q^{84} - 84 q^{86} - 40 q^{88} + 48 q^{91} + 4 q^{92} + 32 q^{94} - 100 q^{96} - 40 q^{98} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.41064 + 0.100408i 0.972675 + 2.34825i 1.97984 − 0.283281i 0 −1.60788 − 3.21487i −3.40884 − 3.40884i −2.76440 + 0.598400i −2.44684 + 2.44684i 0
101.2 −1.40253 + 0.181376i −0.260005 − 0.627708i 1.93421 − 0.508773i 0 0.478517 + 0.833223i 2.12789 + 2.12789i −2.62051 + 1.06439i 1.79491 − 1.79491i 0
101.3 −1.26838 + 0.625467i −0.986257 − 2.38104i 1.21758 − 1.58666i 0 2.74021 + 2.40319i −2.66071 − 2.66071i −0.551955 + 2.77405i −2.57531 + 2.57531i 0
101.4 −1.16634 − 0.799784i −0.300116 − 0.724544i 0.720691 + 1.86564i 0 −0.229442 + 1.08509i −0.0909775 − 0.0909775i 0.651537 − 2.75236i 1.68643 − 1.68643i 0
101.5 −1.04201 + 0.956142i 0.744939 + 1.79844i 0.171586 − 1.99263i 0 −2.49580 − 1.16173i 2.51047 + 2.51047i 1.72644 + 2.24040i −0.558139 + 0.558139i 0
101.6 −0.684143 − 1.23772i 0.630387 + 1.52189i −1.06390 + 1.69355i 0 1.45240 − 1.82143i −0.129367 − 0.129367i 2.82400 + 0.158175i 0.202562 − 0.202562i 0
101.7 −0.374324 + 1.36377i −0.768823 − 1.85610i −1.71976 − 1.02099i 0 2.81910 − 0.353717i 1.12268 + 1.12268i 2.03615 − 1.96319i −0.732710 + 0.732710i 0
101.8 −0.349813 − 1.37027i −1.06125 − 2.56208i −1.75526 + 0.958675i 0 −3.13949 + 2.35044i 2.94098 + 2.94098i 1.92765 + 2.06982i −3.31668 + 3.31668i 0
101.9 0.0407443 + 1.41363i 0.245289 + 0.592181i −1.99668 + 0.115194i 0 −0.827128 + 0.370875i −1.85675 − 1.85675i −0.244195 − 2.81787i 1.83081 − 1.83081i 0
101.10 0.387097 − 1.36020i −0.450051 − 1.08652i −1.70031 − 1.05306i 0 −1.65210 + 0.191573i −2.29539 − 2.29539i −2.09056 + 1.90514i 1.14334 − 1.14334i 0
101.11 0.546111 − 1.30452i 1.11470 + 2.69113i −1.40353 − 1.42482i 0 4.11938 + 0.0155081i −0.557515 − 0.557515i −2.62518 + 1.05281i −3.87832 + 3.87832i 0
101.12 0.768699 + 1.18706i 0.591596 + 1.42824i −0.818203 + 1.82498i 0 −1.24064 + 1.80014i 1.35199 + 1.35199i −2.79530 + 0.431607i 0.431441 − 0.431441i 0
101.13 1.15969 − 0.809388i −0.0847906 − 0.204703i 0.689782 − 1.87729i 0 −0.264015 − 0.168764i 1.29676 + 1.29676i −0.719515 − 2.73538i 2.08661 − 2.08661i 0
101.14 1.29975 + 0.557360i −0.127778 − 0.308484i 1.37870 + 1.44886i 0 0.00585698 − 0.472171i −3.15349 − 3.15349i 0.984428 + 2.65158i 2.04249 − 2.04249i 0
101.15 1.37866 − 0.315126i −1.29511 − 3.12667i 1.80139 − 0.868900i 0 −2.77080 − 3.90248i −1.02642 − 1.02642i 2.20969 − 1.76558i −5.97742 + 5.97742i 0
101.16 1.41033 + 0.104687i 1.03459 + 2.49771i 1.97808 + 0.295287i 0 1.19763 + 3.63091i 1.00028 + 1.00028i 2.75884 + 0.623533i −3.04688 + 3.04688i 0
301.1 −1.41064 − 0.100408i 0.972675 − 2.34825i 1.97984 + 0.283281i 0 −1.60788 + 3.21487i −3.40884 + 3.40884i −2.76440 − 0.598400i −2.44684 − 2.44684i 0
301.2 −1.40253 − 0.181376i −0.260005 + 0.627708i 1.93421 + 0.508773i 0 0.478517 − 0.833223i 2.12789 − 2.12789i −2.62051 − 1.06439i 1.79491 + 1.79491i 0
301.3 −1.26838 − 0.625467i −0.986257 + 2.38104i 1.21758 + 1.58666i 0 2.74021 − 2.40319i −2.66071 + 2.66071i −0.551955 − 2.77405i −2.57531 − 2.57531i 0
301.4 −1.16634 + 0.799784i −0.300116 + 0.724544i 0.720691 − 1.86564i 0 −0.229442 − 1.08509i −0.0909775 + 0.0909775i 0.651537 + 2.75236i 1.68643 + 1.68643i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g 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.y.e yes 64
5.b even 2 1 800.2.y.d ✓ 64
5.c odd 4 1 800.2.ba.f 64
5.c odd 4 1 800.2.ba.h 64
32.g even 8 1 inner 800.2.y.e yes 64
160.v odd 8 1 800.2.ba.f 64
160.z even 8 1 800.2.y.d ✓ 64
160.bb odd 8 1 800.2.ba.h 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.y.d ✓ 64 5.b even 2 1
800.2.y.d ✓ 64 160.z even 8 1
800.2.y.e yes 64 1.a even 1 1 trivial
800.2.y.e yes 64 32.g even 8 1 inner
800.2.ba.f 64 5.c odd 4 1
800.2.ba.f 64 160.v odd 8 1
800.2.ba.h 64 5.c odd 4 1
800.2.ba.h 64 160.bb odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 8 T_{3}^{61} + 104 T_{3}^{59} + 48 T_{3}^{58} - 48 T_{3}^{57} + 15936 T_{3}^{56} + \cdots + 1963464721 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display