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Properties

Label 800.2.be.a
Level $800$
Weight $2$
Character orbit 800.be
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(209,800)]
 
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, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.be (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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 30 q^{9} + 2 q^{15} - 10 q^{17} + 10 q^{23} - 6 q^{25} + 18 q^{31} - 10 q^{33} + 10 q^{39} - 10 q^{41} + 10 q^{47} - 80 q^{49} + 34 q^{55} - 60 q^{63} + 40 q^{65} - 22 q^{71} - 10 q^{73} - 14 q^{79} - 6 q^{81} + 10 q^{87} + 24 q^{89} + 86 q^{95} - 50 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 −0.970606 − 2.98722i 0 0.762120 − 2.10218i 0 1.71800i 0 −5.55435 + 4.03547i 0
209.2 0 −0.943803 − 2.90473i 0 1.19199 + 1.89187i 0 2.27753i 0 −5.11962 + 3.71962i 0
209.3 0 −0.880259 − 2.70916i 0 −2.09944 + 0.769644i 0 − 4.00784i 0 −4.13763 + 3.00616i 0
209.4 0 −0.853735 − 2.62753i 0 2.18267 − 0.485731i 0 − 2.67391i 0 −3.74798 + 2.72307i 0
209.5 0 −0.732250 − 2.25363i 0 −2.17427 + 0.522052i 0 3.97385i 0 −2.11562 + 1.53709i 0
209.6 0 −0.639712 − 1.96883i 0 −2.01712 − 0.964996i 0 − 2.05891i 0 −1.04001 + 0.755610i 0
209.7 0 −0.615546 − 1.89446i 0 0.987308 + 2.00630i 0 1.38088i 0 −0.783019 + 0.568896i 0
209.8 0 −0.455515 − 1.40193i 0 −0.0782993 − 2.23470i 0 2.41973i 0 0.669132 − 0.486153i 0
209.9 0 −0.423859 − 1.30450i 0 −1.11400 + 1.93881i 0 − 0.927462i 0 0.904979 − 0.657506i 0
209.10 0 −0.327557 − 1.00812i 0 1.12940 − 1.92988i 0 − 3.62100i 0 1.51804 − 1.10292i 0
209.11 0 −0.305611 − 0.940574i 0 −1.04427 − 1.97724i 0 3.68114i 0 1.63577 − 1.18846i 0
209.12 0 −0.209612 − 0.645120i 0 2.16529 + 0.558124i 0 0.544271i 0 2.05481 − 1.49291i 0
209.13 0 −0.112530 − 0.346333i 0 −1.22900 − 1.86803i 0 − 4.30063i 0 2.31977 − 1.68541i 0
209.14 0 −0.0756938 − 0.232961i 0 2.03665 − 0.923061i 0 1.59435i 0 2.37851 − 1.72809i 0
209.15 0 0.0756938 + 0.232961i 0 −2.03665 + 0.923061i 0 1.59435i 0 2.37851 − 1.72809i 0
209.16 0 0.112530 + 0.346333i 0 1.22900 + 1.86803i 0 − 4.30063i 0 2.31977 − 1.68541i 0
209.17 0 0.209612 + 0.645120i 0 −2.16529 − 0.558124i 0 0.544271i 0 2.05481 − 1.49291i 0
209.18 0 0.305611 + 0.940574i 0 1.04427 + 1.97724i 0 3.68114i 0 1.63577 − 1.18846i 0
209.19 0 0.327557 + 1.00812i 0 −1.12940 + 1.92988i 0 − 3.62100i 0 1.51804 − 1.10292i 0
209.20 0 0.423859 + 1.30450i 0 1.11400 − 1.93881i 0 − 0.927462i 0 0.904979 − 0.657506i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.28
Significant digits:
Format:

Inner twists

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

Hecke kernels

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