Properties

Label 6013.2.a.d
Level $6013$
Weight $2$
Character orbit 6013.a
Self dual yes
Analytic conductor $48.014$
Analytic rank $1$
Dimension $104$
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: \(1\)
Dimension: \(104\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 17 q^{2} - 34 q^{3} + 99 q^{4} - 46 q^{5} - 18 q^{6} + 104 q^{7} - 48 q^{8} + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 17 q^{2} - 34 q^{3} + 99 q^{4} - 46 q^{5} - 18 q^{6} + 104 q^{7} - 48 q^{8} + 98 q^{9} - 17 q^{10} - 46 q^{11} - 62 q^{12} - 37 q^{13} - 17 q^{14} - 17 q^{15} + 93 q^{16} - 75 q^{17} - 41 q^{18} - 40 q^{19} - 106 q^{20} - 34 q^{21} - 14 q^{22} - 57 q^{23} - 38 q^{24} + 102 q^{25} - 45 q^{26} - 121 q^{27} + 99 q^{28} - 29 q^{29} + 26 q^{30} - 42 q^{31} - 113 q^{32} - 42 q^{33} - 7 q^{34} - 46 q^{35} + 99 q^{36} - 31 q^{37} - 62 q^{38} - 9 q^{39} - 36 q^{40} - 106 q^{41} - 18 q^{42} - 29 q^{43} - 60 q^{44} - 121 q^{45} + 21 q^{46} - 141 q^{47} - 88 q^{48} + 104 q^{49} - 70 q^{50} - 2 q^{51} - 58 q^{52} - 70 q^{53} - 49 q^{54} - 14 q^{55} - 48 q^{56} - 11 q^{57} - 28 q^{58} - 202 q^{59} + 17 q^{60} - 79 q^{61} - 41 q^{62} + 98 q^{63} + 110 q^{64} - 34 q^{65} - 21 q^{66} - 57 q^{67} - 180 q^{68} - 99 q^{69} - 17 q^{70} - 109 q^{71} - 73 q^{72} - 46 q^{73} - 7 q^{74} - 146 q^{75} - 54 q^{76} - 46 q^{77} - 42 q^{78} - 12 q^{79} - 187 q^{80} + 100 q^{81} - 45 q^{82} - 156 q^{83} - 62 q^{84} - 13 q^{85} - 8 q^{86} - 76 q^{87} - 6 q^{88} - 140 q^{89} - 5 q^{90} - 37 q^{91} - 98 q^{92} - q^{93} - 17 q^{94} - 48 q^{95} - 12 q^{96} - 63 q^{97} - 17 q^{98} - 78 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.81212 2.27146 5.90799 −1.69764 −6.38761 1.00000 −10.9897 2.15954 4.77395
1.2 −2.80884 0.402205 5.88959 2.13491 −1.12973 1.00000 −10.9252 −2.83823 −5.99663
1.3 −2.74380 −2.93976 5.52843 −4.29158 8.06610 1.00000 −9.68130 5.64218 11.7752
1.4 −2.73548 −2.83530 5.48288 2.55962 7.75591 1.00000 −9.52736 5.03891 −7.00181
1.5 −2.73004 −0.655778 5.45311 −0.775046 1.79030 1.00000 −9.42711 −2.56996 2.11590
1.6 −2.67480 1.18584 5.15457 −2.80512 −3.17188 1.00000 −8.43786 −1.59379 7.50315
1.7 −2.63784 3.06244 4.95819 −3.52332 −8.07822 1.00000 −7.80322 6.37853 9.29393
1.8 −2.57571 −1.85963 4.63428 3.27062 4.78986 1.00000 −6.78514 0.458214 −8.42416
1.9 −2.50413 −2.14527 4.27066 −3.15595 5.37203 1.00000 −5.68604 1.60217 7.90290
1.10 −2.49952 2.02668 4.24760 1.29364 −5.06572 1.00000 −5.61792 1.10742 −3.23348
1.11 −2.42761 −1.54319 3.89329 −3.03492 3.74626 1.00000 −4.59617 −0.618564 7.36759
1.12 −2.40887 −1.48249 3.80264 1.73279 3.57112 1.00000 −4.34233 −0.802232 −4.17407
1.13 −2.35495 −3.13271 3.54579 −0.919184 7.37736 1.00000 −3.64026 6.81384 2.16463
1.14 −2.31949 1.94621 3.38001 1.57325 −4.51421 1.00000 −3.20092 0.787746 −3.64913
1.15 −2.30883 −0.594888 3.33072 −3.14300 1.37350 1.00000 −3.07241 −2.64611 7.25667
1.16 −2.23888 0.907516 3.01259 −2.38757 −2.03182 1.00000 −2.26707 −2.17641 5.34549
1.17 −2.13035 −2.00060 2.53838 1.16499 4.26197 1.00000 −1.14694 1.00240 −2.48184
1.18 −2.12186 −3.29762 2.50229 4.04257 6.99709 1.00000 −1.06579 7.87430 −8.57776
1.19 −2.10979 2.86366 2.45121 −1.33833 −6.04172 1.00000 −0.951963 5.20055 2.82359
1.20 −2.08457 2.65051 2.34543 3.08275 −5.52518 1.00000 −0.720080 4.02522 −6.42620
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.104
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.d 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6013.2.a.d 104 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{104} + 17 T_{2}^{103} - 9 T_{2}^{102} - 1752 T_{2}^{101} - 6642 T_{2}^{100} + 79784 T_{2}^{99} + 535221 T_{2}^{98} - 1950640 T_{2}^{97} - 22433400 T_{2}^{96} + 18580870 T_{2}^{95} + 627877140 T_{2}^{94} + \cdots + 2948952 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\). Copy content Toggle raw display