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Properties

Label 6028.2.a.c
Level $6028$
Weight $2$
Character orbit 6028.a
Self dual yes
Analytic conductor $48.134$
Analytic rank $1$
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] = [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: \(25\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) \) 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 −3.38016 0 −2.49069 0 −4.68929 0 8.42547 0
1.2 0 −3.29075 0 0.224551 0 1.38599 0 7.82903 0
1.3 0 −3.18883 0 2.53247 0 −2.00866 0 7.16863 0
1.4 0 −2.50588 0 2.86844 0 −2.58862 0 3.27945 0
1.5 0 −2.33036 0 0.868912 0 2.53747 0 2.43060 0
1.6 0 −2.30272 0 −2.25624 0 −2.39335 0 2.30252 0
1.7 0 −2.18955 0 −1.45357 0 3.11023 0 1.79411 0
1.8 0 −1.95548 0 1.12146 0 3.14826 0 0.823884 0
1.9 0 −1.61432 0 −4.02092 0 2.29943 0 −0.393974 0
1.10 0 −1.44202 0 3.24879 0 −4.91695 0 −0.920583 0
1.11 0 −0.747609 0 −1.50988 0 −0.315514 0 −2.44108 0
1.12 0 −0.509200 0 1.39955 0 1.69458 0 −2.74071 0
1.13 0 −0.397686 0 −2.70688 0 −3.71101 0 −2.84185 0
1.14 0 −0.0987018 0 0.748865 0 1.53599 0 −2.99026 0
1.15 0 0.0141418 0 −1.21436 0 −2.15622 0 −2.99980 0
1.16 0 0.0317141 0 3.98776 0 −1.35146 0 −2.99899 0
1.17 0 0.594909 0 2.93206 0 −0.580702 0 −2.64608 0
1.18 0 0.615083 0 0.179847 0 1.85704 0 −2.62167 0
1.19 0 1.10804 0 −3.19726 0 3.64868 0 −1.77224 0
1.20 0 1.71464 0 0.0212259 0 3.28745 0 −0.0600233 0
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
\(2\) \(-1\)
\(11\) \(-1\)
\(137\) \(1\)

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.c ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6028.2.a.c ✓ 25 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6028))\):

\( T_{3}^{25} + 11 T_{3}^{24} + 14 T_{3}^{23} - 253 T_{3}^{22} - 879 T_{3}^{21} + 1899 T_{3}^{20} + \cdots - 1 \) Copy content Toggle raw display
\( T_{5}^{25} + 2 T_{5}^{24} - 60 T_{5}^{23} - 119 T_{5}^{22} + 1519 T_{5}^{21} + 2967 T_{5}^{20} + \cdots + 1809 \) Copy content Toggle raw display