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Properties

Label 800.2.y.d
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 −1.41033 − 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
101.2 −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
101.3 −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
101.4 −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
101.5 −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
101.6 −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
101.7 −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
101.8 −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
101.9 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
101.10 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
101.11 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
101.12 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
101.13 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
101.14 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
101.15 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
101.16 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
301.1 −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
301.2 −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
301.3 −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
301.4 −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
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.d ✓ 64
5.b even 2 1 800.2.y.e yes 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.d ✓ 64
160.v odd 8 1 800.2.ba.h 64
160.z even 8 1 800.2.y.e yes 64
160.bb odd 8 1 800.2.ba.f 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.y.d ✓ 64 1.a even 1 1 trivial
800.2.y.d ✓ 64 32.g even 8 1 inner
800.2.y.e yes 64 5.b even 2 1
800.2.y.e yes 64 160.z even 8 1
800.2.ba.f 64 5.c odd 4 1
800.2.ba.f 64 160.bb odd 8 1
800.2.ba.h 64 5.c odd 4 1
800.2.ba.h 64 160.v 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