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Properties

Label 8044.2.a.b
Level $8044$
Weight $2$
Character orbit 8044.a
Self dual yes
Analytic conductor $64.232$
Analytic rank $0$
Dimension $87$
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] = [8044,2,Mod(1,8044)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8044, 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("8044.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8044 = 2^{2} \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8044.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: \(64.2316633859\)
Analytic rank: \(0\)
Dimension: \(87\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) \) 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 0 −3.34584 0 −2.78395 0 −2.30232 0 8.19464 0
1.2 0 −3.25232 0 1.12026 0 2.54903 0 7.57755 0
1.3 0 −3.19446 0 −0.458477 0 4.07842 0 7.20455 0
1.4 0 −3.01797 0 3.94367 0 1.03981 0 6.10812 0
1.5 0 −2.93853 0 0.993611 0 0.0902794 0 5.63497 0
1.6 0 −2.85294 0 −2.90004 0 −0.912476 0 5.13929 0
1.7 0 −2.80093 0 −0.853190 0 −2.91337 0 4.84521 0
1.8 0 −2.79173 0 1.70131 0 4.83461 0 4.79373 0
1.9 0 −2.70670 0 −2.00035 0 3.54114 0 4.32622 0
1.10 0 −2.55851 0 1.25574 0 −4.90748 0 3.54596 0
1.11 0 −2.49825 0 0.886500 0 −2.30257 0 3.24125 0
1.12 0 −2.42698 0 −3.68217 0 −0.776134 0 2.89025 0
1.13 0 −2.39922 0 −4.31592 0 −4.16363 0 2.75627 0
1.14 0 −2.20292 0 0.146880 0 −2.30848 0 1.85284 0
1.15 0 −2.16825 0 −3.99117 0 4.93889 0 1.70132 0
1.16 0 −2.09889 0 0.451798 0 −0.670101 0 1.40535 0
1.17 0 −2.04177 0 −2.46672 0 0.440869 0 1.16884 0
1.18 0 −1.91101 0 1.73128 0 2.74768 0 0.651964 0
1.19 0 −1.89591 0 2.40728 0 3.56479 0 0.594467 0
1.20 0 −1.85639 0 −1.86854 0 −1.80463 0 0.446172 0
See all 87 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.87
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(2011\) \(1\)

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 8044.2.a.b ✓ 87
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8044.2.a.b ✓ 87 1.a even 1 1 trivial