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Properties

Label 800.2.bg.e
Level $800$
Weight $2$
Character orbit 800.bg
Analytic conductor $6.388$
Analytic rank $0$
Dimension $48$
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] = [800,2,Mod(129,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bg (of order \(10\), 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 16 q^{9} - 8 q^{21} + 4 q^{29} + 20 q^{37} - 8 q^{41} + 140 q^{45} - 72 q^{49} + 120 q^{53} - 12 q^{61} - 8 q^{69} + 80 q^{73} + 44 q^{81} - 140 q^{85} - 60 q^{89} + 60 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1 0 −1.77382 + 2.44145i 0 −0.968536 + 2.01543i 0 3.66163i 0 −1.88720 − 5.80819i 0
129.2 0 −1.45232 + 1.99895i 0 −1.61749 + 1.54393i 0 − 3.19472i 0 −0.959503 − 2.95305i 0
129.3 0 −1.37836 + 1.89714i 0 0.648369 − 2.14000i 0 − 0.963787i 0 −0.772241 − 2.37671i 0
129.4 0 −0.994487 + 1.36879i 0 −1.75626 − 1.38403i 0 4.05810i 0 0.0424583 + 0.130673i 0
129.5 0 −0.871452 + 1.19945i 0 1.85901 + 1.24261i 0 − 1.52381i 0 0.247798 + 0.762643i 0
129.6 0 −0.280759 + 0.386431i 0 1.83491 − 1.27793i 0 − 3.14694i 0 0.856547 + 2.63618i 0
129.7 0 0.280759 − 0.386431i 0 1.83491 − 1.27793i 0 3.14694i 0 0.856547 + 2.63618i 0
129.8 0 0.871452 − 1.19945i 0 1.85901 + 1.24261i 0 1.52381i 0 0.247798 + 0.762643i 0
129.9 0 0.994487 − 1.36879i 0 −1.75626 − 1.38403i 0 − 4.05810i 0 0.0424583 + 0.130673i 0
129.10 0 1.37836 − 1.89714i 0 0.648369 − 2.14000i 0 0.963787i 0 −0.772241 − 2.37671i 0
129.11 0 1.45232 − 1.99895i 0 −1.61749 + 1.54393i 0 3.19472i 0 −0.959503 − 2.95305i 0
129.12 0 1.77382 − 2.44145i 0 −0.968536 + 2.01543i 0 − 3.66163i 0 −1.88720 − 5.80819i 0
289.1 0 −3.07469 − 0.999028i 0 2.10387 − 0.757437i 0 0.381240i 0 6.02862 + 4.38005i 0
289.2 0 −2.20178 − 0.715402i 0 1.25686 + 1.84941i 0 − 1.24827i 0 1.90899 + 1.38696i 0
289.3 0 −2.01797 − 0.655679i 0 −2.20919 − 0.345668i 0 1.56239i 0 1.21524 + 0.882925i 0
289.4 0 −1.60806 − 0.522489i 0 −1.99045 − 1.01888i 0 − 3.47000i 0 −0.114204 − 0.0829742i 0
289.5 0 −1.48267 − 0.481747i 0 1.09686 − 1.94856i 0 3.41774i 0 −0.460833 − 0.334815i 0
289.6 0 −0.599419 − 0.194763i 0 −0.257957 + 2.22114i 0 4.57138i 0 −2.10568 − 1.52987i 0
289.7 0 0.599419 + 0.194763i 0 −0.257957 + 2.22114i 0 − 4.57138i 0 −2.10568 − 1.52987i 0
289.8 0 1.48267 + 0.481747i 0 1.09686 − 1.94856i 0 − 3.41774i 0 −0.460833 − 0.334815i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.e even 10 1 inner
100.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.bg.e ✓ 48
4.b odd 2 1 inner 800.2.bg.e ✓ 48
25.e even 10 1 inner 800.2.bg.e ✓ 48
100.h odd 10 1 inner 800.2.bg.e ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.bg.e ✓ 48 1.a even 1 1 trivial
800.2.bg.e ✓ 48 4.b odd 2 1 inner
800.2.bg.e ✓ 48 25.e even 10 1 inner
800.2.bg.e ✓ 48 100.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 26 T_{3}^{46} + 417 T_{3}^{44} - 5328 T_{3}^{42} + 61760 T_{3}^{40} - 574542 T_{3}^{38} + \cdots + 93345525625 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display