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Properties

Label 6028.2.a.d
Level $6028$
Weight $2$
Character orbit 6028.a
Self dual yes
Analytic conductor $48.134$
Analytic rank $1$
Dimension $27$
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] = [6028,2,Mod(1,6028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6028, 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("6028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6028.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: \(48.1338223384\)
Analytic rank: \(1\)
Dimension: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) \) 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 −3.20019 0 4.35825 0 −3.09018 0 7.24120 0
1.2 0 −3.14651 0 −2.46312 0 3.25354 0 6.90051 0
1.3 0 −3.06151 0 0.299561 0 −0.131528 0 6.37287 0
1.4 0 −2.85963 0 3.21944 0 0.600905 0 5.17750 0
1.5 0 −2.82460 0 0.0657942 0 −4.70856 0 4.97835 0
1.6 0 −2.14728 0 −3.45750 0 −3.94164 0 1.61081 0
1.7 0 −1.86912 0 −2.88318 0 3.98409 0 0.493612 0
1.8 0 −1.79483 0 −0.551345 0 −0.762057 0 0.221406 0
1.9 0 −1.62290 0 2.34735 0 −4.47964 0 −0.366188 0
1.10 0 −1.47615 0 0.527876 0 4.17702 0 −0.820967 0
1.11 0 −1.09297 0 −1.45016 0 −0.643920 0 −1.80542 0
1.12 0 −0.968293 0 −3.33985 0 0.948972 0 −2.06241 0
1.13 0 −0.900414 0 4.31358 0 4.18187 0 −2.18926 0
1.14 0 −0.736924 0 −3.54572 0 −4.24213 0 −2.45694 0
1.15 0 −0.119901 0 2.02323 0 −2.56219 0 −2.98562 0
1.16 0 0.236483 0 2.48335 0 0.448018 0 −2.94408 0
1.17 0 0.362063 0 1.87197 0 −1.02731 0 −2.86891 0
1.18 0 1.09294 0 −4.37649 0 1.34275 0 −1.80549 0
1.19 0 1.56304 0 2.78499 0 −0.687006 0 −0.556912 0
1.20 0 1.68597 0 −0.919775 0 −2.37296 0 −0.157490 0
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(137\) \(-1\)

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 6028.2.a.d ✓ 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6028.2.a.d ✓ 27 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6028))\):

\( T_{3}^{27} + 6 T_{3}^{26} - 39 T_{3}^{25} - 288 T_{3}^{24} + 566 T_{3}^{23} + 6013 T_{3}^{22} + \cdots + 37578 \) Copy content Toggle raw display
\( T_{5}^{27} - T_{5}^{26} - 86 T_{5}^{25} + 74 T_{5}^{24} + 3179 T_{5}^{23} - 2326 T_{5}^{22} + \cdots + 10512 \) Copy content Toggle raw display