Properties

Label 6019.2.a.e
Level $6019$
Weight $2$
Character orbit 6019.a
Self dual yes
Analytic conductor $48.062$
Analytic rank $0$
Dimension $130$
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] = [6019,2,Mod(1,6019)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6019, 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("6019.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6019 = 13 \cdot 463 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6019.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: \(48.0619569766\)
Analytic rank: \(0\)
Dimension: \(130\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 130 q + 10 q^{2} + 11 q^{3} + 146 q^{4} + 40 q^{5} + 4 q^{6} + 8 q^{7} + 24 q^{8} + 181 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 130 q + 10 q^{2} + 11 q^{3} + 146 q^{4} + 40 q^{5} + 4 q^{6} + 8 q^{7} + 24 q^{8} + 181 q^{9} + 5 q^{10} + 43 q^{11} + 28 q^{12} + 130 q^{13} + 47 q^{14} + 29 q^{15} + 170 q^{16} + 85 q^{17} + 20 q^{18} + 3 q^{19} + 73 q^{20} + 62 q^{21} + 12 q^{22} + 62 q^{23} - 5 q^{24} + 178 q^{25} + 10 q^{26} + 35 q^{27} - q^{28} + 134 q^{29} + 24 q^{30} + 14 q^{31} + 48 q^{32} + 4 q^{33} + 4 q^{34} + 50 q^{35} + 244 q^{36} + 32 q^{37} + 76 q^{38} + 11 q^{39} - 6 q^{40} + 48 q^{41} + 9 q^{42} + 34 q^{43} + 123 q^{44} + 115 q^{45} + 5 q^{46} + 25 q^{47} + 35 q^{48} + 210 q^{49} + 24 q^{50} + 20 q^{51} + 146 q^{52} + 193 q^{53} - 39 q^{54} + 32 q^{55} + 122 q^{56} + 7 q^{57} - 4 q^{58} + 50 q^{59} + 42 q^{60} + 57 q^{61} + 51 q^{62} + 8 q^{63} + 172 q^{64} + 40 q^{65} - 4 q^{66} + 21 q^{67} + 132 q^{68} + 92 q^{69} - 46 q^{70} + 58 q^{71} - 26 q^{72} + 15 q^{73} + 120 q^{74} + 23 q^{75} - 65 q^{76} + 192 q^{77} + 4 q^{78} + 32 q^{79} + 66 q^{80} + 326 q^{81} + 11 q^{82} + 33 q^{83} + 5 q^{84} + 43 q^{85} + 105 q^{86} + 31 q^{87} - 17 q^{88} + 84 q^{89} - 73 q^{90} + 8 q^{91} + 161 q^{92} + 52 q^{93} + 4 q^{94} + 59 q^{95} - 77 q^{96} + 9 q^{97} - 61 q^{98} + 100 q^{99}+O(q^{100}) \) 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 −2.80453 3.24972 5.86540 2.53460 −9.11395 2.92678 −10.8406 7.56069 −7.10835
1.2 −2.76072 1.85217 5.62158 −1.74367 −5.11334 −4.49860 −9.99819 0.430551 4.81380
1.3 −2.70229 −1.73872 5.30239 −0.220244 4.69853 −4.26508 −8.92402 0.0231491 0.595165
1.4 −2.65690 −1.17656 5.05912 −3.97344 3.12601 −1.98444 −8.12779 −1.61571 10.5570
1.5 −2.57909 0.0546080 4.65172 2.10708 −0.140839 −1.58025 −6.83904 −2.99702 −5.43435
1.6 −2.55222 −2.23917 4.51381 1.89858 5.71484 −3.17598 −6.41579 2.01387 −4.84558
1.7 −2.52323 3.28941 4.36670 3.66616 −8.29994 −5.12423 −5.97175 7.82019 −9.25058
1.8 −2.52101 −1.65021 4.35552 2.33988 4.16020 4.41588 −5.93829 −0.276814 −5.89887
1.9 −2.51967 −3.32656 4.34873 −0.493062 8.38182 −0.741150 −5.91802 8.06597 1.24235
1.10 −2.50771 −1.46096 4.28860 1.53494 3.66367 0.422474 −5.73914 −0.865591 −3.84918
1.11 −2.49834 2.94692 4.24169 −1.72770 −7.36239 −2.26517 −5.60049 5.68431 4.31639
1.12 −2.49278 −3.38832 4.21397 3.98474 8.44636 1.44276 −5.51896 8.48073 −9.93310
1.13 −2.48853 0.814378 4.19279 −1.02272 −2.02661 0.518159 −5.45682 −2.33679 2.54508
1.14 −2.40942 1.96845 3.80532 3.79888 −4.74282 2.98166 −4.34977 0.874791 −9.15310
1.15 −2.39514 1.83196 3.73671 3.10935 −4.38781 4.25584 −4.15967 0.356083 −7.44733
1.16 −2.34374 2.49578 3.49312 −2.30500 −5.84945 0.790647 −3.49949 3.22890 5.40232
1.17 −2.33535 −1.90450 3.45386 −1.81837 4.44767 2.51906 −3.39528 0.627108 4.24653
1.18 −2.28421 0.124993 3.21761 −2.44620 −0.285511 −2.39008 −2.78127 −2.98438 5.58762
1.19 −2.17988 −0.326610 2.75186 0.291618 0.711970 2.70764 −1.63895 −2.89333 −0.635691
1.20 −2.07170 0.913939 2.29195 −0.284252 −1.89341 −4.46498 −0.604826 −2.16472 0.588884
See next 80 embeddings (of 130 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.130
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(463\) \(1\)

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 6019.2.a.e 130
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6019.2.a.e 130 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{130} - 10 T_{2}^{129} - 153 T_{2}^{128} + 1842 T_{2}^{127} + 10563 T_{2}^{126} - 164654 T_{2}^{125} - 408116 T_{2}^{124} + 9514244 T_{2}^{123} + 7510651 T_{2}^{122} - 399380265 T_{2}^{121} + 113499173 T_{2}^{120} + \cdots - 8624128 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6019))\). Copy content Toggle raw display