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Properties

Label 6019.2.a.d
Level $6019$
Weight $2$
Character orbit 6019.a
Self dual yes
Analytic conductor $48.062$
Analytic rank $0$
Dimension $123$
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: \(123\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 123 q + 10 q^{2} + q^{3} + 136 q^{4} + 46 q^{5} + 16 q^{6} + 12 q^{7} + 30 q^{8} + 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 123 q + 10 q^{2} + q^{3} + 136 q^{4} + 46 q^{5} + 16 q^{6} + 12 q^{7} + 30 q^{8} + 154 q^{9} + 5 q^{10} + 53 q^{11} - 6 q^{12} - 123 q^{13} + 21 q^{14} + 29 q^{15} + 166 q^{16} - 35 q^{17} + 28 q^{18} + 23 q^{19} + 93 q^{20} + 72 q^{21} + 8 q^{22} + 42 q^{23} + 55 q^{24} + 153 q^{25} - 10 q^{26} + 7 q^{27} + 39 q^{28} + 86 q^{29} + 44 q^{30} + 16 q^{31} + 70 q^{32} + 40 q^{33} + 10 q^{34} + 6 q^{35} + 222 q^{36} + 52 q^{37} + 12 q^{38} - q^{39} + 14 q^{40} + 80 q^{41} + 29 q^{42} + 2 q^{43} + 143 q^{44} + 137 q^{45} + 39 q^{46} + 45 q^{47} - 27 q^{48} + 163 q^{49} + 102 q^{50} + 48 q^{51} - 136 q^{52} + 117 q^{53} + 75 q^{54} + 20 q^{55} + 88 q^{56} + 67 q^{57} + 56 q^{58} + 88 q^{59} + 96 q^{60} + 57 q^{61} - 13 q^{62} + 48 q^{63} + 228 q^{64} - 46 q^{65} + 28 q^{66} + 43 q^{67} - 56 q^{68} + 92 q^{69} + 14 q^{70} + 90 q^{71} + 98 q^{72} + 25 q^{73} + 80 q^{74} + 21 q^{75} + 75 q^{76} + 112 q^{77} - 16 q^{78} + 36 q^{79} + 208 q^{80} + 231 q^{81} - 27 q^{82} + 93 q^{83} + 175 q^{84} + 77 q^{85} + 199 q^{86} + 15 q^{87} + 43 q^{88} + 140 q^{89} + 11 q^{90} - 12 q^{91} + 93 q^{92} + 140 q^{93} + 4 q^{94} + 23 q^{95} + 105 q^{96} + 43 q^{97} + 67 q^{98} + 140 q^{99}+O(q^{100}) \) 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 −2.75439 −0.861643 5.58668 3.86069 2.37330 2.91861 −9.87912 −2.25757 −10.6339
1.2 −2.74057 −2.54852 5.51070 −0.432732 6.98439 3.36424 −9.62130 3.49496 1.18593
1.3 −2.71192 −2.81974 5.35451 4.11778 7.64691 −3.85646 −9.09715 4.95095 −11.1671
1.4 −2.69112 1.73469 5.24211 2.53592 −4.66825 −4.16450 −8.72491 0.00913932 −6.82446
1.5 −2.68053 2.57455 5.18522 0.534092 −6.90114 −0.399106 −8.53805 3.62830 −1.43165
1.6 −2.65517 −0.380089 5.04992 1.30385 1.00920 0.922271 −8.09806 −2.85553 −3.46193
1.7 −2.60439 −3.42443 4.78284 −1.92258 8.91854 −3.26289 −7.24759 8.72671 5.00714
1.8 −2.52129 0.498790 4.35689 −0.537697 −1.25759 −2.34122 −5.94240 −2.75121 1.35569
1.9 −2.52093 1.18363 4.35509 −0.839879 −2.98385 2.35424 −5.93701 −1.59902 2.11728
1.10 −2.49040 −2.13188 4.20211 −1.49942 5.30925 −0.951507 −5.48415 1.54493 3.73417
1.11 −2.47080 2.36731 4.10488 −1.23232 −5.84916 4.18566 −5.20074 2.60415 3.04483
1.12 −2.26767 −1.90418 3.14232 −3.16055 4.31804 −3.88251 −2.59040 0.625886 7.16707
1.13 −2.26442 2.63903 3.12758 3.87429 −5.97587 0.802453 −2.55330 3.96450 −8.77301
1.14 −2.25898 −0.131578 3.10301 3.21477 0.297232 0.192917 −2.49169 −2.98269 −7.26212
1.15 −2.22237 3.41646 2.93895 1.79804 −7.59266 3.28634 −2.08670 8.67221 −3.99592
1.16 −2.19076 0.906147 2.79943 −2.56920 −1.98515 −1.27799 −1.75137 −2.17890 5.62849
1.17 −2.17075 −1.42264 2.71217 −2.44924 3.08820 1.96298 −1.54596 −0.976102 5.31670
1.18 −2.16481 1.72504 2.68638 −0.175994 −3.73438 −2.81780 −1.48588 −0.0242310 0.380992
1.19 −2.14591 −1.26575 2.60492 3.62169 2.71619 0.00950737 −1.29811 −1.39787 −7.77182
1.20 −2.11210 −0.313775 2.46095 −4.01398 0.662722 2.82593 −0.973578 −2.90155 8.47792
See next 80 embeddings (of 123 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.123
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.d ✓ 123
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6019.2.a.d ✓ 123 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{123} - 10 T_{2}^{122} - 141 T_{2}^{121} + 1720 T_{2}^{120} + 8849 T_{2}^{119} - 143302 T_{2}^{118} + \cdots + 586577920 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6019))\). Copy content Toggle raw display