Properties

Label 2012.2.a.b
Level $2012$
Weight $2$
Character orbit 2012.a
Self dual yes
Analytic conductor $16.066$
Analytic rank $0$
Dimension $21$
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] = [2012,2,Mod(1,2012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2012, 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("2012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2012 = 2^{2} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2012.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: \(16.0659008867\)
Analytic rank: \(0\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 10 q^{3} + 3 q^{5} + 13 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 10 q^{3} + 3 q^{5} + 13 q^{7} + 21 q^{9} + 7 q^{11} + 12 q^{13} + 14 q^{15} + q^{17} + 14 q^{19} + 14 q^{21} + 26 q^{23} + 18 q^{25} + 37 q^{27} + 9 q^{29} + 28 q^{31} + 3 q^{33} + 20 q^{35} + 31 q^{37} + 29 q^{39} + 4 q^{41} + 38 q^{43} + 24 q^{45} + 9 q^{47} + 16 q^{49} + 15 q^{51} + 22 q^{53} + 35 q^{55} - q^{57} + 10 q^{59} + 22 q^{61} + 35 q^{63} - 14 q^{65} + 58 q^{67} + 15 q^{69} + 27 q^{71} - 6 q^{73} + 48 q^{75} + 16 q^{77} + 47 q^{79} + 29 q^{81} + 22 q^{83} + 14 q^{85} + 29 q^{87} + q^{89} + 51 q^{91} + 34 q^{93} + 23 q^{95} - 2 q^{97} + 22 q^{99}+O(q^{100}) \) 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 −2.88123 0 0.398394 0 3.33507 0 5.30148 0
1.2 0 −2.41348 0 −0.469286 0 −1.19654 0 2.82487 0
1.3 0 −2.22301 0 −1.09600 0 −1.95566 0 1.94176 0
1.4 0 −1.91007 0 1.16052 0 1.88100 0 0.648366 0
1.5 0 −1.30141 0 −2.99143 0 4.27894 0 −1.30633 0
1.6 0 −1.27408 0 3.64848 0 −0.903884 0 −1.37671 0
1.7 0 −0.388717 0 −4.11083 0 −1.27214 0 −2.84890 0
1.8 0 −0.273229 0 −1.56489 0 −3.17063 0 −2.92535 0
1.9 0 −0.175511 0 3.29961 0 3.77232 0 −2.96920 0
1.10 0 −0.112957 0 0.539075 0 −1.01688 0 −2.98724 0
1.11 0 0.207528 0 −1.91018 0 −4.59256 0 −2.95693 0
1.12 0 1.10653 0 2.64866 0 3.45308 0 −1.77559 0
1.13 0 1.32334 0 2.33991 0 2.12211 0 −1.24878 0
1.14 0 1.38206 0 −2.65136 0 3.15617 0 −1.08990 0
1.15 0 1.93158 0 −3.25684 0 2.49607 0 0.730999 0
1.16 0 2.30121 0 1.38461 0 −2.51996 0 2.29558 0
1.17 0 2.44429 0 3.11768 0 −1.05875 0 2.97454 0
1.18 0 2.69810 0 1.28348 0 4.83591 0 4.27974 0
1.19 0 3.07762 0 3.88947 0 −0.0828849 0 6.47174 0
1.20 0 3.16551 0 −0.445451 0 3.10818 0 7.02047 0
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(1\)

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 2012.2.a.b 21
4.b odd 2 1 8048.2.a.s 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2012.2.a.b 21 1.a even 1 1 trivial
8048.2.a.s 21 4.b odd 2 1