Properties

Label 8012.2.a.b
Level $8012$
Weight $2$
Character orbit 8012.a
Self dual yes
Analytic conductor $63.976$
Analytic rank $0$
Dimension $88$
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: \(0\)
Dimension: \(88\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) \) 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 −3.23075 0 0.167791 0 3.73844 0 7.43772 0
1.2 0 −3.15616 0 −3.31880 0 0.188587 0 6.96136 0
1.3 0 −3.04318 0 2.19188 0 3.84643 0 6.26097 0
1.4 0 −2.94029 0 −0.651594 0 3.16105 0 5.64533 0
1.5 0 −2.81005 0 4.18979 0 3.58597 0 4.89636 0
1.6 0 −2.80891 0 −0.455894 0 −2.00885 0 4.88996 0
1.7 0 −2.80739 0 2.13337 0 −1.30534 0 4.88143 0
1.8 0 −2.71734 0 −0.520933 0 −1.18624 0 4.38393 0
1.9 0 −2.71377 0 −3.80725 0 4.98615 0 4.36453 0
1.10 0 −2.59974 0 −0.601241 0 −1.05390 0 3.75864 0
1.11 0 −2.56711 0 −4.07865 0 0.492202 0 3.59005 0
1.12 0 −2.52878 0 2.30231 0 0.0873376 0 3.39472 0
1.13 0 −2.48630 0 −2.95359 0 −3.19667 0 3.18170 0
1.14 0 −2.24743 0 −3.82088 0 −1.51483 0 2.05096 0
1.15 0 −2.08516 0 1.63979 0 1.21456 0 1.34791 0
1.16 0 −2.06140 0 1.88031 0 −1.75026 0 1.24936 0
1.17 0 −2.04818 0 1.65769 0 −0.109887 0 1.19506 0
1.18 0 −1.98040 0 1.50325 0 0.615748 0 0.921997 0
1.19 0 −1.83871 0 −1.10593 0 4.51431 0 0.380865 0
1.20 0 −1.72366 0 3.80404 0 −2.57590 0 −0.0289913 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.88
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.b 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8012.2.a.b 88 1.a even 1 1 trivial