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Properties

Label 6013.2.a.e
Level $6013$
Weight $2$
Character orbit 6013.a
Self dual yes
Analytic conductor $48.014$
Analytic rank $0$
Dimension $109$
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] = [6013,2,Mod(1,6013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6013 = 7 \cdot 859 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6013.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: \(48.0140467354\)
Analytic rank: \(0\)
Dimension: \(109\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 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 −2.71096 3.20670 5.34933 3.06653 −8.69325 1.00000 −9.07991 7.28293 −8.31326
1.2 −2.68013 −1.39322 5.18311 −1.95866 3.73403 1.00000 −8.53117 −1.05892 5.24948
1.3 −2.65451 −2.36774 5.04644 −0.160322 6.28521 1.00000 −8.08683 2.60621 0.425577
1.4 −2.64232 0.320634 4.98185 3.17576 −0.847217 1.00000 −7.87900 −2.89719 −8.39136
1.5 −2.53130 1.72523 4.40746 −1.30306 −4.36707 1.00000 −6.09399 −0.0235740 3.29843
1.6 −2.53104 0.289125 4.40615 −2.52674 −0.731786 1.00000 −6.09005 −2.91641 6.39527
1.7 −2.44055 1.60583 3.95629 4.05418 −3.91911 1.00000 −4.77442 −0.421314 −9.89444
1.8 −2.42826 1.22129 3.89645 0.497994 −2.96562 1.00000 −4.60506 −1.50844 −1.20926
1.9 −2.40830 −1.24102 3.79989 3.14441 2.98874 1.00000 −4.33466 −1.45987 −7.57267
1.10 −2.31924 3.03113 3.37885 1.64684 −7.02991 1.00000 −3.19788 6.18776 −3.81941
1.11 −2.28251 −0.277049 3.20984 0.585497 0.632366 1.00000 −2.76147 −2.92324 −1.33640
1.12 −2.23377 −0.900406 2.98971 4.46215 2.01130 1.00000 −2.21077 −2.18927 −9.96739
1.13 −2.18834 3.06167 2.78881 −2.53450 −6.69995 1.00000 −1.72618 6.37380 5.54634
1.14 −2.11721 0.0595888 2.48258 −0.217642 −0.126162 1.00000 −1.02172 −2.99645 0.460795
1.15 −2.09226 2.05769 2.37754 −3.21058 −4.30522 1.00000 −0.789909 1.23410 6.71736
1.16 −2.06498 −2.80647 2.26415 −1.02377 5.79530 1.00000 −0.545455 4.87626 2.11406
1.17 −2.02275 2.43668 2.09152 0.154664 −4.92881 1.00000 −0.185129 2.93743 −0.312846
1.18 −1.88015 −2.98311 1.53497 2.43013 5.60869 1.00000 0.874328 5.89893 −4.56900
1.19 −1.82060 2.13458 1.31457 −4.26535 −3.88622 1.00000 1.24789 1.55645 7.76548
1.20 −1.74589 −1.96009 1.04812 −3.28540 3.42209 1.00000 1.66188 0.841938 5.73592
See next 80 embeddings (of 109 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.109
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(859\) \(1\)

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 6013.2.a.e ✓ 109
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6013.2.a.e ✓ 109 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{109} - 19 T_{2}^{108} + 16 T_{2}^{107} + 1941 T_{2}^{106} - 10360 T_{2}^{105} - 82633 T_{2}^{104} + \cdots - 451224 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\). Copy content Toggle raw display