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Properties

Label 800.2.bg.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{9} - 30 q^{13} - 20 q^{17} + 40 q^{21} - 6 q^{29} + 60 q^{33} - 40 q^{37} + 40 q^{41} - 90 q^{45} - 76 q^{49} - 40 q^{53} + 22 q^{61} + 30 q^{65} - 94 q^{69} - 50 q^{73} + 60 q^{77} - 44 q^{81} + 90 q^{85} + 74 q^{89} + 80 q^{97}+O(q^{100}) \) 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 −1.88228 + 2.59074i 0 0.955724 − 2.02153i 0 2.63648i 0 −2.24190 − 6.89984i 0
129.2 0 −1.76164 + 2.42469i 0 −1.88260 − 1.20658i 0 − 2.84052i 0 −1.84870 − 5.68973i 0
129.3 0 −1.44235 + 1.98523i 0 1.44312 + 1.70804i 0 − 4.75882i 0 −0.933699 − 2.87363i 0
129.4 0 −1.02796 + 1.41486i 0 −2.19937 + 0.403458i 0 − 0.0261722i 0 −0.0180863 − 0.0556640i 0
129.5 0 −1.00913 + 1.38895i 0 2.23016 − 0.162406i 0 1.68744i 0 0.0162113 + 0.0498933i 0
129.6 0 −0.267488 + 0.368166i 0 0.570991 − 2.16194i 0 − 3.14132i 0 0.863055 + 2.65621i 0
129.7 0 0.267488 − 0.368166i 0 0.570991 − 2.16194i 0 3.14132i 0 0.863055 + 2.65621i 0
129.8 0 1.00913 − 1.38895i 0 2.23016 − 0.162406i 0 − 1.68744i 0 0.0162113 + 0.0498933i 0
129.9 0 1.02796 − 1.41486i 0 −2.19937 + 0.403458i 0 0.0261722i 0 −0.0180863 − 0.0556640i 0
129.10 0 1.44235 − 1.98523i 0 1.44312 + 1.70804i 0 4.75882i 0 −0.933699 − 2.87363i 0
129.11 0 1.76164 − 2.42469i 0 −1.88260 − 1.20658i 0 2.84052i 0 −1.84870 − 5.68973i 0
129.12 0 1.88228 − 2.59074i 0 0.955724 − 2.02153i 0 − 2.63648i 0 −2.24190 − 6.89984i 0
289.1 0 −2.77655 − 0.902154i 0 −1.30499 + 1.81577i 0 − 1.64108i 0 4.46827 + 3.24639i 0
289.2 0 −2.54226 − 0.826029i 0 0.388966 + 2.20198i 0 4.83597i 0 3.35369 + 2.43660i 0
289.3 0 −1.76381 − 0.573095i 0 2.12569 − 0.693864i 0 − 4.24179i 0 0.355523 + 0.258303i 0
289.4 0 −1.45812 − 0.473772i 0 −1.46578 − 1.68864i 0 2.73876i 0 −0.525394 − 0.381721i 0
289.5 0 −0.868926 − 0.282331i 0 1.05735 − 1.97028i 0 1.01266i 0 −1.75173 − 1.27271i 0
289.6 0 −0.460663 − 0.149678i 0 −1.91928 + 1.14734i 0 − 0.138643i 0 −2.23724 − 1.62545i 0
289.7 0 0.460663 + 0.149678i 0 −1.91928 + 1.14734i 0 0.138643i 0 −2.23724 − 1.62545i 0
289.8 0 0.868926 + 0.282331i 0 1.05735 − 1.97028i 0 − 1.01266i 0 −1.75173 − 1.27271i 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.d ✓ 48
4.b odd 2 1 inner 800.2.bg.d ✓ 48
25.e even 10 1 inner 800.2.bg.d ✓ 48
100.h odd 10 1 inner 800.2.bg.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.bg.d ✓ 48 1.a even 1 1 trivial
800.2.bg.d ✓ 48 4.b odd 2 1 inner
800.2.bg.d ✓ 48 25.e even 10 1 inner
800.2.bg.d ✓ 48 100.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 17 T_{3}^{46} + 295 T_{3}^{44} - 4283 T_{3}^{42} + 52264 T_{3}^{40} - 502343 T_{3}^{38} + \cdots + 9971220736 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display