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Properties

Label 8038.2.a.d
Level $8038$
Weight $2$
Character orbit 8038.a
Self dual yes
Analytic conductor $64.184$
Analytic rank $0$
Dimension $92$
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: \(92\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18}+ \cdots + 94 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −3.24576 1.00000 −1.41185 −3.24576 −1.89864 1.00000 7.53496 −1.41185
1.2 1.00000 −3.23664 1.00000 3.20472 −3.23664 2.34255 1.00000 7.47583 3.20472
1.3 1.00000 −3.14562 1.00000 −0.303255 −3.14562 2.95430 1.00000 6.89495 −0.303255
1.4 1.00000 −2.82708 1.00000 −2.48763 −2.82708 −1.73566 1.00000 4.99240 −2.48763
1.5 1.00000 −2.74439 1.00000 2.72118 −2.74439 4.78598 1.00000 4.53169 2.72118
1.6 1.00000 −2.70157 1.00000 0.143003 −2.70157 −2.92792 1.00000 4.29850 0.143003
1.7 1.00000 −2.67211 1.00000 1.54258 −2.67211 1.53058 1.00000 4.14019 1.54258
1.8 1.00000 −2.65852 1.00000 2.22462 −2.65852 0.870094 1.00000 4.06774 2.22462
1.9 1.00000 −2.58627 1.00000 −0.666816 −2.58627 −1.51739 1.00000 3.68880 −0.666816
1.10 1.00000 −2.54065 1.00000 −2.68248 −2.54065 5.07030 1.00000 3.45490 −2.68248
1.11 1.00000 −2.45077 1.00000 −1.39013 −2.45077 4.27652 1.00000 3.00630 −1.39013
1.12 1.00000 −2.41917 1.00000 2.79360 −2.41917 0.842715 1.00000 2.85238 2.79360
1.13 1.00000 −2.40015 1.00000 −4.19507 −2.40015 1.33387 1.00000 2.76072 −4.19507
1.14 1.00000 −2.39141 1.00000 3.31334 −2.39141 −3.74056 1.00000 2.71882 3.31334
1.15 1.00000 −2.38521 1.00000 −2.77606 −2.38521 −1.22030 1.00000 2.68923 −2.77606
1.16 1.00000 −2.15753 1.00000 −0.384990 −2.15753 −2.87024 1.00000 1.65493 −0.384990
1.17 1.00000 −1.94937 1.00000 2.21034 −1.94937 3.83725 1.00000 0.800030 2.21034
1.18 1.00000 −1.90955 1.00000 4.18412 −1.90955 −2.40151 1.00000 0.646387 4.18412
1.19 1.00000 −1.78655 1.00000 0.381834 −1.78655 1.78538 1.00000 0.191778 0.381834
1.20 1.00000 −1.75958 1.00000 2.55886 −1.75958 −0.837744 1.00000 0.0961261 2.55886
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
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.d ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8038.2.a.d ✓ 92 1.a even 1 1 trivial