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Properties

Label 8043.2.a.t
Level $8043$
Weight $2$
Character orbit 8043.a
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $52$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(0\)
Dimension: \(52\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 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 −2.75477 −1.00000 5.58878 −0.0622071 2.75477 1.00000 −9.88627 1.00000 0.171367
1.2 −2.71076 −1.00000 5.34821 0.130231 2.71076 1.00000 −9.07620 1.00000 −0.353025
1.3 −2.56328 −1.00000 4.57042 −4.02293 2.56328 1.00000 −6.58873 1.00000 10.3119
1.4 −2.55848 −1.00000 4.54580 3.55238 2.55848 1.00000 −6.51338 1.00000 −9.08869
1.5 −2.45692 −1.00000 4.03647 −2.79922 2.45692 1.00000 −5.00345 1.00000 6.87748
1.6 −2.35415 −1.00000 3.54201 −1.85354 2.35415 1.00000 −3.63012 1.00000 4.36349
1.7 −2.28128 −1.00000 3.20426 2.19375 2.28128 1.00000 −2.74725 1.00000 −5.00457
1.8 −1.95494 −1.00000 1.82179 −1.46647 1.95494 1.00000 0.348396 1.00000 2.86686
1.9 −1.95089 −1.00000 1.80596 3.26605 1.95089 1.00000 0.378554 1.00000 −6.37169
1.10 −1.90192 −1.00000 1.61732 0.767188 1.90192 1.00000 0.727834 1.00000 −1.45913
1.11 −1.85290 −1.00000 1.43325 −3.21014 1.85290 1.00000 1.05014 1.00000 5.94808
1.12 −1.77779 −1.00000 1.16053 3.56820 1.77779 1.00000 1.49240 1.00000 −6.34350
1.13 −1.64173 −1.00000 0.695293 0.237832 1.64173 1.00000 2.14198 1.00000 −0.390457
1.14 −1.61741 −1.00000 0.616031 −3.81830 1.61741 1.00000 2.23845 1.00000 6.17577
1.15 −1.58348 −1.00000 0.507395 −2.03406 1.58348 1.00000 2.36350 1.00000 3.22089
1.16 −1.33568 −1.00000 −0.215968 1.69750 1.33568 1.00000 2.95982 1.00000 −2.26731
1.17 −1.10272 −1.00000 −0.784010 0.760743 1.10272 1.00000 3.06998 1.00000 −0.838886
1.18 −0.890431 −1.00000 −1.20713 −2.94587 0.890431 1.00000 2.85573 1.00000 2.62309
1.19 −0.869058 −1.00000 −1.24474 −3.05181 0.869058 1.00000 2.81987 1.00000 2.65220
1.20 −0.837626 −1.00000 −1.29838 2.34855 0.837626 1.00000 2.76281 1.00000 −1.96721
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.52
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(383\) \(1\)

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 8043.2.a.t ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8043.2.a.t ✓ 52 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\( T_{2}^{52} - 3 T_{2}^{51} - 78 T_{2}^{50} + 231 T_{2}^{49} + 2852 T_{2}^{48} - 8311 T_{2}^{47} + \cdots - 384 \) Copy content Toggle raw display
\( T_{5}^{52} + 7 T_{5}^{51} - 147 T_{5}^{50} - 1121 T_{5}^{49} + 9710 T_{5}^{48} + 82735 T_{5}^{47} + \cdots - 10636333056 \) Copy content Toggle raw display
\( T_{11}^{52} - 9 T_{11}^{51} - 300 T_{11}^{50} + 2922 T_{11}^{49} + 40661 T_{11}^{48} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display