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Properties

Label 6028.2.a.f
Level $6028$
Weight $2$
Character orbit 6028.a
Self dual yes
Analytic conductor $48.134$
Analytic rank $0$
Dimension $29$
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: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 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 −2.97866 0 −3.46603 0 0.700745 0 5.87240 0
1.2 0 −2.90851 0 0.776599 0 −2.27589 0 5.45943 0
1.3 0 −2.65731 0 2.47175 0 0.711504 0 4.06129 0
1.4 0 −2.38462 0 −1.71860 0 3.80618 0 2.68641 0
1.5 0 −2.00219 0 0.977497 0 −3.03319 0 1.00878 0
1.6 0 −1.84910 0 −0.463178 0 −1.96260 0 0.419175 0
1.7 0 −1.36461 0 2.36459 0 1.37904 0 −1.13783 0
1.8 0 −1.28547 0 2.42568 0 5.14778 0 −1.34758 0
1.9 0 −1.14725 0 4.20688 0 −0.775275 0 −1.68381 0
1.10 0 −0.886923 0 −0.640762 0 1.27780 0 −2.21337 0
1.11 0 −0.774336 0 −2.66500 0 0.769081 0 −2.40040 0
1.12 0 −0.191141 0 −3.65958 0 2.84676 0 −2.96346 0
1.13 0 0.213524 0 2.27425 0 4.54069 0 −2.95441 0
1.14 0 0.780624 0 0.801470 0 −4.30496 0 −2.39063 0
1.15 0 0.785650 0 1.28190 0 −3.55486 0 −2.38275 0
1.16 0 1.04417 0 −2.78682 0 −0.254204 0 −1.90970 0
1.17 0 1.09615 0 3.71316 0 3.72361 0 −1.79846 0
1.18 0 1.58135 0 3.56375 0 0.0558785 0 −0.499338 0
1.19 0 1.60026 0 −1.38303 0 0.488214 0 −0.439167 0
1.20 0 1.76391 0 −2.22231 0 2.22847 0 0.111381 0
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
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.f ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6028.2.a.f ✓ 29 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6028))\):

\( T_{3}^{29} - 14 T_{3}^{28} + 33 T_{3}^{27} + 408 T_{3}^{26} - 2274 T_{3}^{25} - 2691 T_{3}^{24} + \cdots - 382200 \) Copy content Toggle raw display
\( T_{5}^{29} - 9 T_{5}^{28} - 50 T_{5}^{27} + 650 T_{5}^{26} + 599 T_{5}^{25} - 20234 T_{5}^{24} + \cdots + 188778816 \) Copy content Toggle raw display