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Properties

Label 8038.2.a.b
Level $8038$
Weight $2$
Character orbit 8038.a
Self dual yes
Analytic conductor $64.184$
Analytic rank $0$
Dimension $83$
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] = [8038,2,Mod(1,8038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8038, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8038 = 2 \cdot 4019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8038.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: \(64.1837531447\)
Analytic rank: \(0\)
Dimension: \(83\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 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 −1.00000 −3.18441 1.00000 −1.27929 3.18441 1.89743 −1.00000 7.14045 1.27929
1.2 −1.00000 −3.15134 1.00000 4.13084 3.15134 −2.97931 −1.00000 6.93095 −4.13084
1.3 −1.00000 −3.02523 1.00000 2.41251 3.02523 3.55505 −1.00000 6.15201 −2.41251
1.4 −1.00000 −3.00434 1.00000 −0.184095 3.00434 1.88237 −1.00000 6.02606 0.184095
1.5 −1.00000 −2.95780 1.00000 0.0755052 2.95780 −2.37738 −1.00000 5.74860 −0.0755052
1.6 −1.00000 −2.90051 1.00000 2.46896 2.90051 0.644361 −1.00000 5.41299 −2.46896
1.7 −1.00000 −2.88068 1.00000 −2.36782 2.88068 0.285889 −1.00000 5.29829 2.36782
1.8 −1.00000 −2.78821 1.00000 0.878256 2.78821 −4.20087 −1.00000 4.77411 −0.878256
1.9 −1.00000 −2.34219 1.00000 −3.72898 2.34219 −1.27056 −1.00000 2.48587 3.72898
1.10 −1.00000 −2.20646 1.00000 0.820064 2.20646 3.76837 −1.00000 1.86848 −0.820064
1.11 −1.00000 −2.18666 1.00000 −3.58998 2.18666 −4.12584 −1.00000 1.78150 3.58998
1.12 −1.00000 −2.14458 1.00000 3.80241 2.14458 4.02154 −1.00000 1.59920 −3.80241
1.13 −1.00000 −2.12932 1.00000 2.23951 2.12932 −4.41338 −1.00000 1.53399 −2.23951
1.14 −1.00000 −2.12834 1.00000 −2.40910 2.12834 0.412842 −1.00000 1.52982 2.40910
1.15 −1.00000 −2.12762 1.00000 −1.48235 2.12762 0.377312 −1.00000 1.52675 1.48235
1.16 −1.00000 −2.05749 1.00000 3.25924 2.05749 −0.840215 −1.00000 1.23326 −3.25924
1.17 −1.00000 −1.99762 1.00000 2.51698 1.99762 −0.403674 −1.00000 0.990467 −2.51698
1.18 −1.00000 −1.88709 1.00000 −0.769273 1.88709 −1.40180 −1.00000 0.561125 0.769273
1.19 −1.00000 −1.79852 1.00000 4.20751 1.79852 −1.35268 −1.00000 0.234660 −4.20751
1.20 −1.00000 −1.59172 1.00000 0.670129 1.59172 −4.47737 −1.00000 −0.466431 −0.670129
See all 83 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.83
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(4019\) \(-1\)

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 8038.2.a.b ✓ 83
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8038.2.a.b ✓ 83 1.a even 1 1 trivial