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Properties

Label 6039.2.a.o
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $25$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(25\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) \) 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 −2.47686 0 4.13484 0.821809 0 −3.73620 −5.28771 0 −2.03551
1.2 −2.44801 0 3.99278 1.89686 0 −0.969789 −4.87834 0 −4.64354
1.3 −2.23548 0 2.99739 −2.10645 0 −1.54970 −2.22964 0 4.70893
1.4 −1.97018 0 1.88162 1.48404 0 2.73292 0.233227 0 −2.92383
1.5 −1.83299 0 1.35986 −0.470771 0 0.751426 1.17337 0 0.862920
1.6 −1.76243 0 1.10617 −2.60834 0 0.445474 1.57532 0 4.59703
1.7 −1.07776 0 −0.838436 3.04682 0 3.44893 3.05915 0 −3.28374
1.8 −1.06662 0 −0.862332 3.84706 0 −1.30714 3.05301 0 −4.10333
1.9 −1.04470 0 −0.908610 −2.79878 0 −4.32870 3.03861 0 2.92388
1.10 −0.363962 0 −1.86753 0.137887 0 3.90557 1.40764 0 −0.0501855
1.11 0.0238689 0 −1.99943 −2.56388 0 0.894211 −0.0954618 0 −0.0611970
1.12 0.0892703 0 −1.99203 0.250715 0 −4.16429 −0.356370 0 0.0223814
1.13 0.110609 0 −1.98777 −0.0761036 0 3.79648 −0.441081 0 −0.00841771
1.14 0.226426 0 −1.94873 3.14756 0 −2.84661 −0.894095 0 0.712690
1.15 0.702784 0 −1.50609 −2.41068 0 0.380608 −2.46403 0 −1.69419
1.16 0.926278 0 −1.14201 3.01280 0 2.23888 −2.91037 0 2.79069
1.17 1.56609 0 0.452630 1.36136 0 3.35990 −2.42332 0 2.13201
1.18 1.65004 0 0.722641 −2.19434 0 −0.319690 −2.10770 0 −3.62075
1.19 1.77462 0 1.14926 −1.38341 0 −3.00047 −1.50973 0 −2.45503
1.20 1.79702 0 1.22928 −4.13243 0 4.22858 −1.38500 0 −7.42606
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)
\(61\) \(1\)

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 6039.2.a.o yes 25
3.b odd 2 1 6039.2.a.n ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6039.2.a.n ✓ 25 3.b odd 2 1
6039.2.a.o yes 25 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} - 5 T_{2}^{24} - 25 T_{2}^{23} + 155 T_{2}^{22} + 226 T_{2}^{21} - 2056 T_{2}^{20} - 607 T_{2}^{19} + \cdots - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\). Copy content Toggle raw display