Properties

Label 800.2.bj.a
Level $800$
Weight $2$
Character orbit 800.bj
Analytic conductor $6.388$
Analytic rank $0$
Dimension $112$
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(81,800)]
 
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, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bj (of order \(10\), degree \(4\), not 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: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 200)
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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 16 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 16 q^{7} + 18 q^{9} + 14 q^{15} - 10 q^{17} + 6 q^{23} - 6 q^{25} - 6 q^{31} - 18 q^{33} + 34 q^{39} - 2 q^{41} + 30 q^{47} + 48 q^{49} + 2 q^{55} - 28 q^{57} + 60 q^{63} - 60 q^{65} + 34 q^{71} - 26 q^{73} - 14 q^{79} - 30 q^{81} - 38 q^{87} + 24 q^{89} - 74 q^{95} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −1.91233 2.63210i 0 0.977596 2.01105i 0 1.36581 0 −2.34388 + 7.21373i 0
81.2 0 −1.75086 2.40986i 0 −2.21918 + 0.274336i 0 3.41885 0 −1.81484 + 5.58549i 0
81.3 0 −1.68097 2.31366i 0 −1.82826 1.28743i 0 −4.93157 0 −1.60031 + 4.92524i 0
81.4 0 −1.52910 2.10462i 0 0.881108 + 2.05515i 0 1.53499 0 −1.16425 + 3.58319i 0
81.5 0 −1.35783 1.86889i 0 1.91517 1.15417i 0 −0.146324 0 −0.722003 + 2.22210i 0
81.6 0 −1.13822 1.56662i 0 0.658250 + 2.13699i 0 −3.40673 0 −0.231711 + 0.713133i 0
81.7 0 −1.07287 1.47668i 0 −1.36252 + 1.77300i 0 2.31401 0 −0.102477 + 0.315391i 0
81.8 0 −1.00642 1.38522i 0 2.06230 + 0.864242i 0 −0.719899 0 0.0211043 0.0649523i 0
81.9 0 −0.910283 1.25290i 0 −0.0584731 2.23530i 0 1.01594 0 0.185915 0.572186i 0
81.10 0 −0.801761 1.10353i 0 −2.06049 + 0.868546i 0 −1.98026 0 0.352095 1.08364i 0
81.11 0 −0.289798 0.398873i 0 0.143005 2.23149i 0 4.76281 0 0.851934 2.62198i 0
81.12 0 −0.260015 0.357881i 0 −1.65958 1.49860i 0 −1.14468 0 0.866581 2.66706i 0
81.13 0 −0.239455 0.329581i 0 2.14380 + 0.635713i 0 2.90112 0 0.875766 2.69533i 0
81.14 0 −0.0135478 0.0186469i 0 −1.58615 + 1.57611i 0 −2.98406 0 0.926887 2.85266i 0
81.15 0 0.0135478 + 0.0186469i 0 1.58615 1.57611i 0 −2.98406 0 0.926887 2.85266i 0
81.16 0 0.239455 + 0.329581i 0 −2.14380 0.635713i 0 2.90112 0 0.875766 2.69533i 0
81.17 0 0.260015 + 0.357881i 0 1.65958 + 1.49860i 0 −1.14468 0 0.866581 2.66706i 0
81.18 0 0.289798 + 0.398873i 0 −0.143005 + 2.23149i 0 4.76281 0 0.851934 2.62198i 0
81.19 0 0.801761 + 1.10353i 0 2.06049 0.868546i 0 −1.98026 0 0.352095 1.08364i 0
81.20 0 0.910283 + 1.25290i 0 0.0584731 + 2.23530i 0 1.01594 0 0.185915 0.572186i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
25.d even 5 1 inner
200.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.bj.a 112
4.b odd 2 1 200.2.t.a 112
8.b even 2 1 inner 800.2.bj.a 112
8.d odd 2 1 200.2.t.a 112
20.d odd 2 1 1000.2.t.a 112
20.e even 4 2 1000.2.o.b 224
25.d even 5 1 inner 800.2.bj.a 112
40.e odd 2 1 1000.2.t.a 112
40.k even 4 2 1000.2.o.b 224
100.h odd 10 1 1000.2.t.a 112
100.j odd 10 1 200.2.t.a 112
100.l even 20 2 1000.2.o.b 224
200.n odd 10 1 200.2.t.a 112
200.s odd 10 1 1000.2.t.a 112
200.t even 10 1 inner 800.2.bj.a 112
200.v even 20 2 1000.2.o.b 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.t.a 112 4.b odd 2 1
200.2.t.a 112 8.d odd 2 1
200.2.t.a 112 100.j odd 10 1
200.2.t.a 112 200.n odd 10 1
800.2.bj.a 112 1.a even 1 1 trivial
800.2.bj.a 112 8.b even 2 1 inner
800.2.bj.a 112 25.d even 5 1 inner
800.2.bj.a 112 200.t even 10 1 inner
1000.2.o.b 224 20.e even 4 2
1000.2.o.b 224 40.k even 4 2
1000.2.o.b 224 100.l even 20 2
1000.2.o.b 224 200.v even 20 2
1000.2.t.a 112 20.d odd 2 1
1000.2.t.a 112 40.e odd 2 1
1000.2.t.a 112 100.h odd 10 1
1000.2.t.a 112 200.s odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(800, [\chi])\).