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Properties

Label 8039.2.a.b
Level $8039$
Weight $2$
Character orbit 8039.a
Self dual yes
Analytic conductor $64.192$
Analytic rank $0$
Dimension $391$
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] = [8039,2,Mod(1,8039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8039, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8039 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8039.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: \(64.1917381849\)
Analytic rank: \(0\)
Dimension: \(391\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 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 −2.81050 0.273089 5.89891 −1.07280 −0.767516 3.48763 −10.9579 −2.92542 3.01510
1.2 −2.80167 −1.48127 5.84934 2.23536 4.15003 2.13740 −10.7846 −0.805840 −6.26274
1.3 −2.79911 −1.99039 5.83499 −0.815598 5.57131 −3.39799 −10.7345 0.961647 2.28295
1.4 −2.79609 −0.0994905 5.81814 −1.46865 0.278185 0.0832220 −10.6759 −2.99010 4.10649
1.5 −2.78936 −2.33525 5.78055 3.95215 6.51386 5.17788 −10.5453 2.45339 −11.0240
1.6 −2.78914 1.94751 5.77928 −0.505263 −5.43186 −0.334314 −10.5409 0.792785 1.40925
1.7 −2.78421 2.45333 5.75180 2.41719 −6.83056 1.41777 −10.4458 3.01881 −6.72994
1.8 −2.77323 3.30634 5.69082 −2.95004 −9.16925 3.29315 −10.2355 7.93189 8.18116
1.9 −2.73934 −2.46230 5.50400 −1.26420 6.74509 −0.402196 −9.59867 3.06293 3.46308
1.10 −2.73322 2.36794 5.47049 3.42125 −6.47211 −3.92603 −9.48561 2.60715 −9.35102
1.11 −2.72603 −3.12098 5.43123 −3.26463 8.50788 −3.62664 −9.35363 6.74052 8.89948
1.12 −2.71686 −1.39132 5.38132 −4.28068 3.78003 2.13526 −9.18657 −1.06422 11.6300
1.13 −2.70592 0.720832 5.32200 1.39019 −1.95051 −3.88734 −8.98906 −2.48040 −3.76173
1.14 −2.67673 −3.14138 5.16489 −3.87922 8.40863 4.00887 −8.47155 6.86827 10.3836
1.15 −2.65165 1.77560 5.03124 −0.216981 −4.70826 −4.11451 −8.03778 0.152748 0.575357
1.16 −2.64673 −3.15136 5.00518 2.70892 8.34080 −2.33255 −7.95389 6.93109 −7.16979
1.17 −2.62368 −0.0902052 4.88368 −3.95870 0.236669 −3.68990 −7.56585 −2.99186 10.3864
1.18 −2.62255 0.822293 4.87774 −3.51986 −2.15650 −0.523743 −7.54702 −2.32383 9.23098
1.19 −2.61852 1.44668 4.85663 3.17736 −3.78815 3.56525 −7.48013 −0.907124 −8.31997
1.20 −2.61507 3.14612 4.83861 2.05578 −8.22734 4.83721 −7.42319 6.89808 −5.37602
See next 80 embeddings (of 391 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.391
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(8039\) \(-1\)

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 8039.2.a.b ✓ 391
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8039.2.a.b ✓ 391 1.a even 1 1 trivial