Properties

Label 8012.2.a.a
Level $8012$
Weight $2$
Character orbit 8012.a
Self dual yes
Analytic conductor $63.976$
Analytic rank $1$
Dimension $79$
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] = [8012,2,Mod(1,8012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8012, 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("8012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8012 = 2^{2} \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8012.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: \(63.9761420994\)
Analytic rank: \(1\)
Dimension: \(79\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9} - 14 q^{11} - 3 q^{13} - 19 q^{15} - 30 q^{17} - 49 q^{19} + 7 q^{21} - 40 q^{23} + 51 q^{25} - 70 q^{27} + 2 q^{29} - 48 q^{31} - 25 q^{33} - 34 q^{35} - 35 q^{39} - 20 q^{41} - 104 q^{43} + 12 q^{45} - 38 q^{47} + 51 q^{49} - 41 q^{51} - q^{53} - 112 q^{55} - 34 q^{57} - 24 q^{59} - 120 q^{63} - 21 q^{65} - 67 q^{67} + 15 q^{69} - 28 q^{71} - 88 q^{73} - 103 q^{75} + 4 q^{77} - 99 q^{79} + 47 q^{81} - 70 q^{83} + 7 q^{85} - 109 q^{87} - 50 q^{89} - 83 q^{91} - 7 q^{93} - 61 q^{95} - 93 q^{97} - 78 q^{99}+O(q^{100}) \) 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 0 −3.41908 0 −3.90530 0 −4.72288 0 8.69013 0
1.2 0 −3.40885 0 −1.83301 0 1.65667 0 8.62026 0
1.3 0 −3.26026 0 0.973052 0 −2.06988 0 7.62930 0
1.4 0 −3.24921 0 4.44468 0 −2.04361 0 7.55734 0
1.5 0 −3.22159 0 3.40512 0 −1.04725 0 7.37863 0
1.6 0 −3.17871 0 −1.27908 0 2.10162 0 7.10419 0
1.7 0 −3.11170 0 0.355389 0 −3.83214 0 6.68266 0
1.8 0 −2.87938 0 2.41591 0 2.96745 0 5.29083 0
1.9 0 −2.79728 0 3.00547 0 −2.16180 0 4.82477 0
1.10 0 −2.71048 0 −2.47152 0 2.42570 0 4.34670 0
1.11 0 −2.68647 0 −0.588575 0 −1.13984 0 4.21715 0
1.12 0 −2.62866 0 2.26034 0 −4.56855 0 3.90985 0
1.13 0 −2.59561 0 −1.81816 0 −1.23436 0 3.73718 0
1.14 0 −2.46774 0 −2.83803 0 1.20715 0 3.08972 0
1.15 0 −2.30440 0 3.02168 0 4.18657 0 2.31024 0
1.16 0 −2.27913 0 −0.601984 0 3.53846 0 2.19442 0
1.17 0 −2.21676 0 1.73805 0 −5.05782 0 1.91402 0
1.18 0 −2.02583 0 1.86007 0 1.58288 0 1.10398 0
1.19 0 −1.92727 0 −2.52260 0 −3.66775 0 0.714366 0
1.20 0 −1.87872 0 3.33381 0 1.62469 0 0.529574 0
See all 79 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.79
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(2003\) \(-1\)

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 8012.2.a.a 79
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8012.2.a.a 79 1.a even 1 1 trivial