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Properties

Label 819.2.dx.a
Level $819$
Weight $2$
Character orbit 819.dx
Analytic conductor $6.540$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(503,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.503");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.dx (of order \(6\), degree \(2\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 32 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 32 q^{4} - 2 q^{7} - 24 q^{16} + 24 q^{22} + 56 q^{25} - 8 q^{28} + 16 q^{37} - 4 q^{43} + 32 q^{46} + 26 q^{49} - 40 q^{58} - 64 q^{64} + 60 q^{67} - 40 q^{70} - 72 q^{79} - 80 q^{85} - 24 q^{88} + 2 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
503.1 −2.27480 − 1.31336i 0 2.44981 + 4.24319i −1.43978 0 −2.25700 + 1.38056i − 7.61644i 0 3.27522 + 1.89095i
503.2 −2.27480 − 1.31336i 0 2.44981 + 4.24319i 1.43978 0 2.32410 − 1.26434i − 7.61644i 0 −3.27522 − 1.89095i
503.3 −2.20353 − 1.27221i 0 2.23704 + 3.87466i −1.86278 0 2.64289 + 0.122982i − 6.29508i 0 4.10470 + 2.36985i
503.4 −2.20353 − 1.27221i 0 2.23704 + 3.87466i 1.86278 0 −1.21494 + 2.35030i − 6.29508i 0 −4.10470 − 2.36985i
503.5 −1.66245 − 0.959816i 0 0.842493 + 1.45924i −2.54989 0 0.0378984 − 2.64548i 0.604709i 0 4.23906 + 2.44742i
503.6 −1.66245 − 0.959816i 0 0.842493 + 1.45924i 2.54989 0 −2.31000 − 1.28992i 0.604709i 0 −4.23906 − 2.44742i
503.7 −1.54546 − 0.892271i 0 0.592297 + 1.02589i −3.88703 0 −1.88286 + 1.85872i 1.45513i 0 6.00725 + 3.46829i
503.8 −1.54546 − 0.892271i 0 0.592297 + 1.02589i 3.88703 0 2.55113 − 0.701238i 1.45513i 0 −6.00725 − 3.46829i
503.9 −1.51215 − 0.873037i 0 0.524388 + 0.908267i −1.66387 0 0.365442 − 2.62039i 1.66091i 0 2.51601 + 1.45262i
503.10 −1.51215 − 0.873037i 0 0.524388 + 0.908267i 1.66387 0 −2.45205 − 0.993713i 1.66091i 0 −2.51601 − 1.45262i
503.11 −1.14474 − 0.660916i 0 −0.126381 − 0.218898i −1.30928 0 2.62751 + 0.310174i 2.97777i 0 1.49878 + 0.865323i
503.12 −1.14474 − 0.660916i 0 −0.126381 − 0.218898i 1.30928 0 −1.04513 + 2.43057i 2.97777i 0 −1.49878 − 0.865323i
503.13 −0.595297 − 0.343695i 0 −0.763748 − 1.32285i −2.01783 0 1.48283 + 2.19117i 2.42476i 0 1.20121 + 0.693518i
503.14 −0.595297 − 0.343695i 0 −0.763748 − 1.32285i 2.01783 0 1.15619 + 2.37975i 2.42476i 0 −1.20121 − 0.693518i
503.15 −0.569266 − 0.328666i 0 −0.783957 − 1.35785i −1.53259 0 −2.25245 + 1.38797i 2.34530i 0 0.872452 + 0.503710i
503.16 −0.569266 − 0.328666i 0 −0.783957 − 1.35785i 1.53259 0 2.32824 − 1.25669i 2.34530i 0 −0.872452 − 0.503710i
503.17 −0.205179 − 0.118460i 0 −0.971934 − 1.68344i −3.73403 0 −1.71677 − 2.01313i 0.934382i 0 0.766145 + 0.442334i
503.18 −0.205179 − 0.118460i 0 −0.971934 − 1.68344i 3.73403 0 −0.885038 − 2.49333i 0.934382i 0 −0.766145 − 0.442334i
503.19 0.205179 + 0.118460i 0 −0.971934 − 1.68344i −3.73403 0 −0.885038 − 2.49333i − 0.934382i 0 −0.766145 − 0.442334i
503.20 0.205179 + 0.118460i 0 −0.971934 − 1.68344i 3.73403 0 −1.71677 − 2.01313i − 0.934382i 0 0.766145 + 0.442334i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 503.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
13.c even 3 1 inner
21.c even 2 1 inner
39.i odd 6 1 inner
91.n odd 6 1 inner
273.bn even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.dx.a ✓ 72
3.b odd 2 1 inner 819.2.dx.a ✓ 72
7.b odd 2 1 inner 819.2.dx.a ✓ 72
13.c even 3 1 inner 819.2.dx.a ✓ 72
21.c even 2 1 inner 819.2.dx.a ✓ 72
39.i odd 6 1 inner 819.2.dx.a ✓ 72
91.n odd 6 1 inner 819.2.dx.a ✓ 72
273.bn even 6 1 inner 819.2.dx.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.dx.a ✓ 72 1.a even 1 1 trivial
819.2.dx.a ✓ 72 3.b odd 2 1 inner
819.2.dx.a ✓ 72 7.b odd 2 1 inner
819.2.dx.a ✓ 72 13.c even 3 1 inner
819.2.dx.a ✓ 72 21.c even 2 1 inner
819.2.dx.a ✓ 72 39.i odd 6 1 inner
819.2.dx.a ✓ 72 91.n odd 6 1 inner
819.2.dx.a ✓ 72 273.bn even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).