↑

Properties

Label 6036.2.a.h
Level $6036$
Weight $2$
Character orbit 6036.a
Self dual yes
Analytic conductor $48.198$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6036,2,Mod(1,6036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6036, 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("6036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6036.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: \(48.1977026600\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 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 0 1.00000 0 −3.64052 0 2.86228 0 1.00000 0
1.2 0 1.00000 0 −3.54344 0 −1.09625 0 1.00000 0
1.3 0 1.00000 0 −2.75869 0 −0.925607 0 1.00000 0
1.4 0 1.00000 0 −2.71451 0 −1.53717 0 1.00000 0
1.5 0 1.00000 0 −1.85882 0 5.16371 0 1.00000 0
1.6 0 1.00000 0 −1.66295 0 −0.368117 0 1.00000 0
1.7 0 1.00000 0 −1.40769 0 3.41112 0 1.00000 0
1.8 0 1.00000 0 −0.766960 0 −1.64718 0 1.00000 0
1.9 0 1.00000 0 −0.497428 0 −3.93416 0 1.00000 0
1.10 0 1.00000 0 −0.0279923 0 3.66780 0 1.00000 0
1.11 0 1.00000 0 0.357952 0 −4.62593 0 1.00000 0
1.12 0 1.00000 0 1.21459 0 5.16615 0 1.00000 0
1.13 0 1.00000 0 1.56884 0 1.63329 0 1.00000 0
1.14 0 1.00000 0 1.63100 0 0.968844 0 1.00000 0
1.15 0 1.00000 0 1.65069 0 1.03307 0 1.00000 0
1.16 0 1.00000 0 1.97211 0 2.26181 0 1.00000 0
1.17 0 1.00000 0 2.44705 0 −3.28775 0 1.00000 0
1.18 0 1.00000 0 2.55581 0 −4.67011 0 1.00000 0
1.19 0 1.00000 0 3.52243 0 1.80430 0 1.00000 0
1.20 0 1.00000 0 3.68154 0 2.26895 0 1.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(503\) \(-1\)

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 6036.2.a.h ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6036.2.a.h ✓ 24 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6036))\):

\( T_{5}^{24} - 18 T_{5}^{23} + 78 T_{5}^{22} + 450 T_{5}^{21} - 4594 T_{5}^{20} + 3626 T_{5}^{19} + \cdots + 360242 \) Copy content Toggle raw display
\( T_{7}^{24} - 9 T_{7}^{23} - 75 T_{7}^{22} + 867 T_{7}^{21} + 1766 T_{7}^{20} - 34844 T_{7}^{19} + \cdots - 550408192 \) Copy content Toggle raw display