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Properties

Label 6038.2.a.b
Level $6038$
Weight $2$
Character orbit 6038.a
Self dual yes
Analytic conductor $48.214$
Analytic rank $1$
Dimension $54$
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] = [6038,2,Mod(1,6038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6038, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6038 = 2 \cdot 3019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6038.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: \(48.2136727404\)
Analytic rank: \(1\)
Dimension: \(54\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9} - 14 q^{10} - 31 q^{11} - 21 q^{12} - 34 q^{13} - 44 q^{14} - 22 q^{15} + 54 q^{16} - 40 q^{17} + 39 q^{18} - 44 q^{19} - 14 q^{20} - 3 q^{21} - 31 q^{22} - 33 q^{23} - 21 q^{24} + 14 q^{25} - 34 q^{26} - 66 q^{27} - 44 q^{28} - 22 q^{30} - 65 q^{31} + 54 q^{32} - 43 q^{33} - 40 q^{34} - 46 q^{35} + 39 q^{36} - 58 q^{37} - 44 q^{38} - 36 q^{39} - 14 q^{40} - 49 q^{41} - 3 q^{42} - 47 q^{43} - 31 q^{44} - 45 q^{45} - 33 q^{46} - 66 q^{47} - 21 q^{48} + 16 q^{49} + 14 q^{50} - 33 q^{51} - 34 q^{52} - 16 q^{53} - 66 q^{54} - 50 q^{55} - 44 q^{56} - 33 q^{57} - 70 q^{59} - 22 q^{60} - 40 q^{61} - 65 q^{62} - 117 q^{63} + 54 q^{64} - 33 q^{65} - 43 q^{66} - 82 q^{67} - 40 q^{68} - q^{69} - 46 q^{70} - 60 q^{71} + 39 q^{72} - 92 q^{73} - 58 q^{74} - 68 q^{75} - 44 q^{76} + 13 q^{77} - 36 q^{78} - 57 q^{79} - 14 q^{80} + 26 q^{81} - 49 q^{82} - 77 q^{83} - 3 q^{84} - 24 q^{85} - 47 q^{86} - 61 q^{87} - 31 q^{88} - 54 q^{89} - 45 q^{90} - 46 q^{91} - 33 q^{92} - 24 q^{93} - 66 q^{94} - 66 q^{95} - 21 q^{96} - 137 q^{97} + 16 q^{98} - 71 q^{99}+O(q^{100}) \) 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 1.00000 −3.41141 1.00000 0.211160 −3.41141 −3.35690 1.00000 8.63772 0.211160
1.2 1.00000 −3.37741 1.00000 −2.92787 −3.37741 −0.403159 1.00000 8.40689 −2.92787
1.3 1.00000 −3.34616 1.00000 0.269513 −3.34616 1.67569 1.00000 8.19680 0.269513
1.4 1.00000 −3.17799 1.00000 −0.0583780 −3.17799 −5.13519 1.00000 7.09963 −0.0583780
1.5 1.00000 −2.98721 1.00000 2.38189 −2.98721 −0.220104 1.00000 5.92341 2.38189
1.6 1.00000 −2.82052 1.00000 −0.526511 −2.82052 2.78989 1.00000 4.95531 −0.526511
1.7 1.00000 −2.77973 1.00000 3.72096 −2.77973 −2.52302 1.00000 4.72690 3.72096
1.8 1.00000 −2.56435 1.00000 −4.01554 −2.56435 −3.36655 1.00000 3.57587 −4.01554
1.9 1.00000 −2.48349 1.00000 3.70736 −2.48349 −1.05138 1.00000 3.16770 3.70736
1.10 1.00000 −2.40402 1.00000 −1.73865 −2.40402 4.25060 1.00000 2.77932 −1.73865
1.11 1.00000 −2.32590 1.00000 3.29375 −2.32590 −5.00561 1.00000 2.40980 3.29375
1.12 1.00000 −2.27315 1.00000 −2.64091 −2.27315 0.657447 1.00000 2.16722 −2.64091
1.13 1.00000 −2.11795 1.00000 1.57843 −2.11795 1.37381 1.00000 1.48570 1.57843
1.14 1.00000 −2.02410 1.00000 −4.35432 −2.02410 −1.05145 1.00000 1.09700 −4.35432
1.15 1.00000 −1.99905 1.00000 −3.02278 −1.99905 −5.14057 1.00000 0.996217 −3.02278
1.16 1.00000 −1.93892 1.00000 0.524165 −1.93892 −1.38269 1.00000 0.759396 0.524165
1.17 1.00000 −1.77991 1.00000 −0.815447 −1.77991 −3.69164 1.00000 0.168091 −0.815447
1.18 1.00000 −1.61049 1.00000 1.72129 −1.61049 1.31284 1.00000 −0.406311 1.72129
1.19 1.00000 −1.56837 1.00000 −1.56733 −1.56837 2.76703 1.00000 −0.540224 −1.56733
1.20 1.00000 −1.51709 1.00000 2.00493 −1.51709 2.93208 1.00000 −0.698426 2.00493
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.54
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3019\) \(-1\)

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 6038.2.a.b ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6038.2.a.b ✓ 54 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{54} + 21 T_{3}^{53} + 120 T_{3}^{52} - 503 T_{3}^{51} - 8000 T_{3}^{50} - 14065 T_{3}^{49} + \cdots + 5806105 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\). Copy content Toggle raw display