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Properties

Label 819.2.w.b
Level $819$
Weight $2$
Character orbit 819.w
Analytic conductor $6.540$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(8,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("819.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.w (of order \(4\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{5} + 4 q^{13} - 44 q^{16} - 8 q^{19} - 72 q^{20} - 16 q^{22} + 16 q^{23} + 44 q^{26} + 16 q^{31} - 12 q^{34} + 12 q^{37} - 32 q^{38} - 16 q^{40} - 48 q^{41} + 56 q^{44} + 24 q^{46} + 12 q^{47} + 96 q^{55} - 12 q^{56} + 60 q^{58} - 64 q^{59} + 24 q^{61} + 160 q^{62} + 24 q^{65} - 40 q^{67} - 24 q^{70} - 16 q^{71} - 4 q^{73} - 104 q^{76} - 8 q^{77} + 16 q^{79} - 152 q^{80} + 20 q^{83} - 16 q^{85} + 8 q^{86} - 4 q^{89} - 8 q^{91} - 48 q^{95} + 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −1.96872 + 1.96872i 0 − 5.75174i 2.26345 − 2.26345i 0 −0.707107 + 0.707107i 7.38614 + 7.38614i 0 8.91222i
8.2 −1.60647 + 1.60647i 0 − 3.16152i 0.0202562 − 0.0202562i 0 0.707107 − 0.707107i 1.86595 + 1.86595i 0 0.0650821i
8.3 −1.29386 + 1.29386i 0 − 1.34812i 1.01108 − 1.01108i 0 −0.707107 + 0.707107i −0.843433 − 0.843433i 0 2.61637i
8.4 −1.24107 + 1.24107i 0 − 1.08050i −2.48707 + 2.48707i 0 0.707107 − 0.707107i −1.14116 − 1.14116i 0 − 6.17323i
8.5 −0.760832 + 0.760832i 0 0.842269i 0.510590 − 0.510590i 0 0.707107 − 0.707107i −2.16249 − 2.16249i 0 0.776947i
8.6 −0.372095 + 0.372095i 0 1.72309i −3.11188 + 3.11188i 0 −0.707107 + 0.707107i −1.38534 − 1.38534i 0 − 2.31583i
8.7 −0.344835 + 0.344835i 0 1.76218i 1.64485 − 1.64485i 0 −0.707107 + 0.707107i −1.29733 − 1.29733i 0 1.13441i
8.8 0.299083 − 0.299083i 0 1.82110i −0.982214 + 0.982214i 0 −0.707107 + 0.707107i 1.14283 + 1.14283i 0 0.587527i
8.9 0.450371 − 0.450371i 0 1.59433i 0.700601 − 0.700601i 0 0.707107 − 0.707107i 1.61878 + 1.61878i 0 − 0.631060i
8.10 0.781312 − 0.781312i 0 0.779103i −2.54738 + 2.54738i 0 0.707107 − 0.707107i 2.17135 + 2.17135i 0 3.98060i
8.11 1.18422 − 1.18422i 0 − 0.804750i −0.631453 + 0.631453i 0 −0.707107 + 0.707107i 1.41544 + 1.41544i 0 1.49556i
8.12 1.20136 − 1.20136i 0 − 0.886509i 1.65527 − 1.65527i 0 0.707107 − 0.707107i 1.33770 + 1.33770i 0 − 3.97713i
8.13 1.78910 − 1.78910i 0 − 4.40175i 2.22038 − 2.22038i 0 −0.707107 + 0.707107i −4.29697 − 4.29697i 0 − 7.94497i
8.14 1.88244 − 1.88244i 0 − 5.08718i 1.73351 − 1.73351i 0 0.707107 − 0.707107i −5.81144 − 5.81144i 0 − 6.52648i
512.1 −1.96872 − 1.96872i 0 5.75174i 2.26345 + 2.26345i 0 −0.707107 − 0.707107i 7.38614 − 7.38614i 0 − 8.91222i
512.2 −1.60647 − 1.60647i 0 3.16152i 0.0202562 + 0.0202562i 0 0.707107 + 0.707107i 1.86595 − 1.86595i 0 − 0.0650821i
512.3 −1.29386 − 1.29386i 0 1.34812i 1.01108 + 1.01108i 0 −0.707107 − 0.707107i −0.843433 + 0.843433i 0 − 2.61637i
512.4 −1.24107 − 1.24107i 0 1.08050i −2.48707 − 2.48707i 0 0.707107 + 0.707107i −1.14116 + 1.14116i 0 6.17323i
512.5 −0.760832 − 0.760832i 0 − 0.842269i 0.510590 + 0.510590i 0 0.707107 + 0.707107i −2.16249 + 2.16249i 0 − 0.776947i
512.6 −0.372095 − 0.372095i 0 − 1.72309i −3.11188 − 3.11188i 0 −0.707107 − 0.707107i −1.38534 + 1.38534i 0 2.31583i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.w.b yes 28
3.b odd 2 1 819.2.w.a ✓ 28
13.d odd 4 1 819.2.w.a ✓ 28
39.f even 4 1 inner 819.2.w.b yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.w.a ✓ 28 3.b odd 2 1
819.2.w.a ✓ 28 13.d odd 4 1
819.2.w.b yes 28 1.a even 1 1 trivial
819.2.w.b yes 28 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 109 T_{2}^{24} + 8 T_{2}^{21} + 3722 T_{2}^{20} + 136 T_{2}^{19} - 680 T_{2}^{17} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display