Properties

Label 4012.2.a.j
Level $4012$
Weight $2$
Character orbit 4012.a
Self dual yes
Analytic conductor $32.036$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 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 −2.75827 0 0.760103 0 0.427165 0 4.60804 0
1.2 0 −2.52360 0 −3.46979 0 1.96229 0 3.36854 0
1.3 0 −2.50436 0 2.93974 0 −0.515336 0 3.27181 0
1.4 0 −1.96950 0 −1.93705 0 −2.26754 0 0.878948 0
1.5 0 −1.89300 0 −2.37849 0 4.40428 0 0.583446 0
1.6 0 −1.58435 0 2.56834 0 3.40444 0 −0.489844 0
1.7 0 −1.03698 0 0.719057 0 −1.71880 0 −1.92468 0
1.8 0 0.0134244 0 3.34756 0 −3.13615 0 −2.99982 0
1.9 0 0.188519 0 −3.21916 0 −0.179540 0 −2.96446 0
1.10 0 0.210829 0 0.593138 0 −1.60715 0 −2.95555 0
1.11 0 0.461334 0 1.69592 0 4.42760 0 −2.78717 0
1.12 0 0.765447 0 −3.22340 0 −4.55619 0 −2.41409 0
1.13 0 0.967697 0 −3.19923 0 3.75414 0 −2.06356 0
1.14 0 1.90340 0 1.03137 0 0.216287 0 0.622916 0
1.15 0 1.97354 0 3.25764 0 2.36478 0 0.894844 0
1.16 0 2.22137 0 −1.27900 0 0.396716 0 1.93446 0
1.17 0 2.29806 0 4.01491 0 4.07651 0 2.28106 0
1.18 0 2.45518 0 −1.72896 0 −4.96392 0 3.02793 0
1.19 0 3.11094 0 3.12050 0 −1.63406 0 6.67797 0
1.20 0 3.31779 0 −0.252330 0 4.50552 0 8.00775 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
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(-1\)

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 4012.2.a.j 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.j 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\( T_{3}^{21} - 9 T_{3}^{20} - 4 T_{3}^{19} + 250 T_{3}^{18} - 437 T_{3}^{17} - 2642 T_{3}^{16} + 7806 T_{3}^{15} + 12170 T_{3}^{14} - 58757 T_{3}^{13} - 10873 T_{3}^{12} + 229457 T_{3}^{11} - 116876 T_{3}^{10} - 459309 T_{3}^{9} + \cdots + 32 \) Copy content Toggle raw display
\( T_{5}^{21} - T_{5}^{20} - 65 T_{5}^{19} + 54 T_{5}^{18} + 1805 T_{5}^{17} - 1228 T_{5}^{16} - 27934 T_{5}^{15} + 15419 T_{5}^{14} + 263489 T_{5}^{13} - 119039 T_{5}^{12} - 1556390 T_{5}^{11} + 606335 T_{5}^{10} + \cdots - 408928 \) Copy content Toggle raw display