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Properties

Label 8019.2.a.g
Level $8019$
Weight $2$
Character orbit 8019.a
Self dual yes
Analytic conductor $64.032$
Analytic rank $1$
Dimension $36$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8019,2,Mod(1,8019)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8019, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8019.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8019 = 3^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8019.a (trivial)

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: yes
Analytic conductor: \(64.0320373809\)
Analytic rank: \(1\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 297)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 30 q^{4} - 9 q^{5} - 9 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 30 q^{4} - 9 q^{5} - 9 q^{7} - 3 q^{8} - 18 q^{10} + 36 q^{11} - 21 q^{13} - 15 q^{14} + 18 q^{16} + 12 q^{17} - 42 q^{19} - 27 q^{20} - 30 q^{23} + 9 q^{25} + 3 q^{26} - 42 q^{28} + 6 q^{29} - 45 q^{31} + 24 q^{32} - 36 q^{34} + 3 q^{35} - 48 q^{37} + 18 q^{38} - 57 q^{40} + 3 q^{41} - 15 q^{43} + 30 q^{44} - 36 q^{46} - 15 q^{47} + 3 q^{49} + 39 q^{50} - 63 q^{52} - 39 q^{53} - 9 q^{55} - 15 q^{56} - 60 q^{58} - 15 q^{59} - 75 q^{61} - 18 q^{62} - 21 q^{64} + 6 q^{65} - 18 q^{67} + 42 q^{68} - 51 q^{70} - 9 q^{71} - 54 q^{73} - 78 q^{76} - 9 q^{77} - 27 q^{79} - 9 q^{80} - 27 q^{82} + 24 q^{83} - 72 q^{85} - 48 q^{86} - 3 q^{88} - 24 q^{89} - 63 q^{91} - 30 q^{92} - 81 q^{94} + 9 q^{95} - 99 q^{97} + 60 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.57896 0 4.65103 1.81453 0 0.597625 −6.83689 0 −4.67961
1.2 −2.57869 0 4.64963 0.549951 0 −2.53498 −6.83256 0 −1.41815
1.3 −2.51998 0 4.35029 −1.98987 0 −2.89486 −5.92267 0 5.01444
1.4 −2.21310 0 2.89780 1.25911 0 0.887911 −1.98693 0 −2.78652
1.5 −2.16415 0 2.68354 3.54207 0 1.71775 −1.47929 0 −7.66558
1.6 −2.10597 0 2.43509 −0.0342445 0 2.90025 −0.916291 0 0.0721177
1.7 −2.04863 0 2.19689 −0.556265 0 2.53749 −0.403346 0 1.13958
1.8 −2.01026 0 2.04114 −1.99399 0 −3.74353 −0.0827043 0 4.00844
1.9 −1.56100 0 0.436708 −1.98729 0 −2.82745 2.44029 0 3.10215
1.10 −1.54922 0 0.400087 −4.30670 0 0.529623 2.47862 0 6.67204
1.11 −1.11300 0 −0.761231 −0.296230 0 3.44297 3.07325 0 0.329705
1.12 −1.09746 0 −0.795590 −0.844390 0 −0.470010 3.06804 0 0.926681
1.13 −1.04649 0 −0.904867 3.42777 0 −2.37542 3.03990 0 −3.58711
1.14 −0.834408 0 −1.30376 −0.396805 0 1.16343 2.75669 0 0.331097
1.15 −0.759333 0 −1.42341 −2.62463 0 1.90497 2.59951 0 1.99297
1.16 −0.272778 0 −1.92559 2.10562 0 −2.25775 1.07081 0 −0.574367
1.17 −0.271934 0 −1.92605 1.50806 0 −2.46200 1.06763 0 −0.410092
1.18 0.0197513 0 −1.99961 3.39148 0 3.53205 −0.0789976 0 0.0669862
1.19 0.403238 0 −1.83740 1.10652 0 3.55888 −1.54739 0 0.446190
1.20 0.431171 0 −1.81409 −1.32759 0 2.32692 −1.64453 0 −0.572419
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8019.2.a.g 36
3.b odd 2 1 8019.2.a.h 36
27.e even 9 2 891.2.j.b 72
27.f odd 18 2 297.2.j.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.j.b ✓ 72 27.f odd 18 2
891.2.j.b 72 27.e even 9 2
8019.2.a.g 36 1.a even 1 1 trivial
8019.2.a.h 36 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 51 T_{2}^{34} + T_{2}^{33} + 1179 T_{2}^{32} - 51 T_{2}^{31} - 16358 T_{2}^{30} + \cdots - 111 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8019))\). Copy content Toggle raw display