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Properties

Label 6028.2.a.e
Level $6028$
Weight $2$
Character orbit 6028.a
Self dual yes
Analytic conductor $48.134$
Analytic rank $0$
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: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) \) 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 −2.99791 0 1.37488 0 4.31566 0 5.98748 0
1.2 0 −2.85465 0 −3.73429 0 −2.11521 0 5.14904 0
1.3 0 −2.62082 0 −2.77332 0 −0.589123 0 3.86872 0
1.4 0 −2.52741 0 2.61441 0 2.36761 0 3.38782 0
1.5 0 −2.12719 0 −0.932078 0 −0.358800 0 1.52492 0
1.6 0 −2.08259 0 1.30696 0 −1.23590 0 1.33717 0
1.7 0 −1.58174 0 −1.88933 0 3.30001 0 −0.498107 0
1.8 0 −1.06195 0 −3.38178 0 4.44479 0 −1.87225 0
1.9 0 −0.925674 0 2.67976 0 2.79766 0 −2.14313 0
1.10 0 −0.836719 0 1.09288 0 −3.78282 0 −2.29990 0
1.11 0 −0.700606 0 3.28744 0 −1.66773 0 −2.50915 0
1.12 0 −0.600596 0 −1.19632 0 −0.0561599 0 −2.63928 0
1.13 0 0.0546031 0 −0.338001 0 −3.60189 0 −2.99702 0
1.14 0 0.137892 0 −0.136126 0 0.729350 0 −2.98099 0
1.15 0 0.283778 0 −2.65499 0 −2.51132 0 −2.91947 0
1.16 0 0.470040 0 −0.875088 0 3.59259 0 −2.77906 0
1.17 0 0.863194 0 3.07549 0 2.16824 0 −2.25490 0
1.18 0 1.35334 0 3.55583 0 −1.94424 0 −1.16848 0
1.19 0 1.41110 0 −1.10550 0 −3.91271 0 −1.00881 0
1.20 0 1.62144 0 −3.05441 0 −4.42036 0 −0.370921 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.e ✓ 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6028.2.a.e ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6028))\):

\( T_{3}^{27} - 5 T_{3}^{26} - 42 T_{3}^{25} + 231 T_{3}^{24} + 741 T_{3}^{23} - 4617 T_{3}^{22} + \cdots - 500 \) Copy content Toggle raw display
\( T_{5}^{27} + 2 T_{5}^{26} - 76 T_{5}^{25} - 143 T_{5}^{24} + 2519 T_{5}^{23} + 4423 T_{5}^{22} + \cdots + 369676 \) Copy content Toggle raw display