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Properties

Label 6013.2.a.f
Level $6013$
Weight $2$
Character orbit 6013.a
Self dual yes
Analytic conductor $48.014$
Analytic rank $0$
Dimension $110$
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: \(110\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 110 q + 16 q^{2} + 29 q^{3} + 118 q^{4} + 12 q^{6} - 110 q^{7} + 57 q^{8} + 127 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 110 q + 16 q^{2} + 29 q^{3} + 118 q^{4} + 12 q^{6} - 110 q^{7} + 57 q^{8} + 127 q^{9} + 3 q^{10} + 52 q^{11} + 62 q^{12} - 9 q^{13} - 16 q^{14} + 39 q^{15} + 146 q^{16} + 11 q^{17} + 60 q^{18} + 14 q^{19} + 18 q^{20} - 29 q^{21} + 32 q^{22} + 73 q^{23} + 24 q^{24} + 132 q^{25} - 7 q^{26} + 116 q^{27} - 118 q^{28} + 35 q^{29} + 18 q^{30} + 36 q^{31} + 140 q^{32} + 42 q^{33} - 7 q^{34} + 180 q^{36} + 49 q^{37} + 45 q^{39} + 6 q^{40} - 14 q^{41} - 12 q^{42} + 58 q^{43} + 92 q^{44} + 17 q^{45} + 27 q^{46} + 87 q^{47} + 98 q^{48} + 110 q^{49} + 91 q^{50} + 42 q^{51} + 16 q^{52} + 95 q^{53} + 41 q^{54} + 8 q^{55} - 57 q^{56} + 61 q^{57} + 46 q^{58} + 114 q^{59} + 81 q^{60} - 47 q^{61} + 31 q^{62} - 127 q^{63} + 199 q^{64} + 62 q^{65} + 21 q^{66} + 95 q^{67} + 60 q^{68} - 39 q^{69} - 3 q^{70} + 131 q^{71} + 186 q^{72} + 31 q^{73} + 23 q^{74} + 121 q^{75} + 14 q^{76} - 52 q^{77} + 110 q^{78} + 9 q^{79} + 61 q^{80} + 194 q^{81} + 45 q^{82} + 73 q^{83} - 62 q^{84} + 59 q^{85} + 72 q^{86} + 64 q^{87} + 100 q^{88} - 17 q^{89} + 11 q^{90} + 9 q^{91} + 192 q^{92} + 85 q^{93} - 11 q^{94} + 108 q^{95} + 68 q^{96} + 32 q^{97} + 16 q^{98} + 160 q^{99}+O(q^{100}) \) 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 −2.71420 2.40545 5.36690 2.49981 −6.52887 −1.00000 −9.13847 2.78617 −6.78498
1.2 −2.67470 1.47568 5.15405 −1.35988 −3.94701 −1.00000 −8.43614 −0.822364 3.63728
1.3 −2.64365 1.43905 4.98889 0.402261 −3.80434 −1.00000 −7.90158 −0.929138 −1.06344
1.4 −2.63934 −1.02733 4.96612 −0.539298 2.71147 −1.00000 −7.82859 −1.94460 1.42339
1.5 −2.59895 −1.56127 4.75456 1.42920 4.05766 −1.00000 −7.15899 −0.562451 −3.71443
1.6 −2.57438 3.21457 4.62742 −1.73819 −8.27551 −1.00000 −6.76398 7.33344 4.47476
1.7 −2.55830 −1.85077 4.54491 3.83295 4.73481 −1.00000 −6.51064 0.425331 −9.80585
1.8 −2.38030 −0.0622570 3.66584 −0.894729 0.148190 −1.00000 −3.96519 −2.99612 2.12972
1.9 −2.32328 −2.34605 3.39764 −4.09939 5.45055 −1.00000 −3.24712 2.50397 9.52403
1.10 −2.31912 −2.74143 3.37834 −1.26450 6.35772 −1.00000 −3.19654 4.51544 2.93254
1.11 −2.30986 1.63849 3.33543 −0.463511 −3.78466 −1.00000 −3.08465 −0.315366 1.07064
1.12 −2.24570 −1.05382 3.04316 1.30809 2.36656 −1.00000 −2.34263 −1.88947 −2.93757
1.13 −2.23147 2.27145 2.97947 −3.38307 −5.06867 −1.00000 −2.18565 2.15947 7.54923
1.14 −2.10326 2.95040 2.42369 4.24716 −6.20544 −1.00000 −0.891125 5.70483 −8.93287
1.15 −2.09677 −1.04747 2.39646 −2.55395 2.19632 −1.00000 −0.831292 −1.90280 5.35506
1.16 −2.05159 3.13328 2.20904 2.94582 −6.42821 −1.00000 −0.428856 6.81742 −6.04363
1.17 −1.99923 0.235231 1.99692 2.61420 −0.470281 −1.00000 0.00615784 −2.94467 −5.22639
1.18 −1.94077 −1.00614 1.76657 −2.69594 1.95267 −1.00000 0.453027 −1.98769 5.23218
1.19 −1.93774 1.87483 1.75485 1.38259 −3.63294 −1.00000 0.475036 0.514992 −2.67910
1.20 −1.87802 0.475349 1.52695 −2.58742 −0.892713 −1.00000 0.888402 −2.77404 4.85921
See next 80 embeddings (of 110 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.110
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.f ✓ 110
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6013.2.a.f ✓ 110 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{110} - 16 T_{2}^{109} - 41 T_{2}^{108} + 1981 T_{2}^{107} - 4409 T_{2}^{106} - 113449 T_{2}^{105} + \cdots + 82552988 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\). Copy content Toggle raw display