Properties

Label 8039.2.a.a
Level $8039$
Weight $2$
Character orbit 8039.a
Self dual yes
Analytic conductor $64.192$
Analytic rank $1$
Dimension $279$
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: \(1\)
Dimension: \(279\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 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 −2.75826 0.543669 5.60799 4.07368 −1.49958 −1.45954 −9.95176 −2.70442 −11.2363
1.2 −2.72307 −0.226737 5.41513 0.679891 0.617421 −1.82848 −9.29967 −2.94859 −1.85139
1.3 −2.72088 2.51741 5.40317 −2.91261 −6.84957 −4.81699 −9.25961 3.33737 7.92486
1.4 −2.69202 −2.07215 5.24695 2.43355 5.57825 −0.413821 −8.74083 1.29379 −6.55116
1.5 −2.68146 1.09614 5.19020 0.161648 −2.93925 3.85315 −8.55439 −1.79847 −0.433452
1.6 −2.66452 2.42009 5.09965 1.98258 −6.44836 1.42101 −8.25906 2.85681 −5.28261
1.7 −2.65513 −1.36605 5.04972 2.66534 3.62703 −0.858733 −8.09741 −1.13392 −7.07683
1.8 −2.65123 3.28882 5.02904 −0.364681 −8.71943 −0.0824305 −8.03069 7.81633 0.966855
1.9 −2.64944 0.544800 5.01955 −2.56078 −1.44342 −0.780813 −8.00012 −2.70319 6.78465
1.10 −2.64457 −1.20149 4.99374 −2.42138 3.17741 3.07885 −7.91715 −1.55643 6.40351
1.11 −2.62410 1.81984 4.88588 −0.731341 −4.77543 −1.89048 −7.57282 0.311808 1.91911
1.12 −2.61158 −1.07466 4.82033 −1.60450 2.80656 −4.85582 −7.36550 −1.84510 4.19028
1.13 −2.60916 −2.67838 4.80773 −2.56224 6.98832 −1.44552 −7.32584 4.17369 6.68531
1.14 −2.59800 −2.63260 4.74962 0.476137 6.83949 1.24122 −7.14351 3.93057 −1.23700
1.15 −2.55932 −0.558527 4.55013 0.514137 1.42945 2.79738 −6.52662 −2.68805 −1.31584
1.16 −2.55424 2.06857 4.52414 −3.35787 −5.28363 0.600266 −6.44725 1.27899 8.57679
1.17 −2.54048 −0.00862555 4.45405 1.93048 0.0219131 2.12668 −6.23447 −2.99993 −4.90434
1.18 −2.53338 3.09967 4.41800 3.04096 −7.85264 −3.65810 −6.12571 6.60796 −7.70391
1.19 −2.51220 −0.224247 4.31117 3.23846 0.563355 −3.25585 −5.80612 −2.94971 −8.13568
1.20 −2.47449 2.58655 4.12308 0.927265 −6.40039 −0.828816 −5.25355 3.69026 −2.29451
See next 80 embeddings (of 279 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.279
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.a 279
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8039.2.a.a 279 1.a even 1 1 trivial