Properties

Label 8026.2.a.a
Level $8026$
Weight $2$
Character orbit 8026.a
Self dual yes
Analytic conductor $64.088$
Analytic rank $1$
Dimension $71$
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] = [8026,2,Mod(1,8026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8026, 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("8026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8026 = 2 \cdot 4013 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8026.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: \(64.0879326623\)
Analytic rank: \(1\)
Dimension: \(71\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 71 q + 71 q^{2} - 9 q^{3} + 71 q^{4} - 34 q^{5} - 9 q^{6} - 19 q^{7} + 71 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 71 q + 71 q^{2} - 9 q^{3} + 71 q^{4} - 34 q^{5} - 9 q^{6} - 19 q^{7} + 71 q^{8} + 34 q^{9} - 34 q^{10} - 37 q^{11} - 9 q^{12} - 62 q^{13} - 19 q^{14} - 29 q^{15} + 71 q^{16} - 52 q^{17} + 34 q^{18} - 30 q^{19} - 34 q^{20} - 51 q^{21} - 37 q^{22} - 45 q^{23} - 9 q^{24} + 27 q^{25} - 62 q^{26} - 27 q^{27} - 19 q^{28} - 55 q^{29} - 29 q^{30} - 61 q^{31} + 71 q^{32} - 73 q^{33} - 52 q^{34} - 33 q^{35} + 34 q^{36} - 43 q^{37} - 30 q^{38} - 40 q^{39} - 34 q^{40} - 87 q^{41} - 51 q^{42} - 4 q^{43} - 37 q^{44} - 81 q^{45} - 45 q^{46} - 89 q^{47} - 9 q^{48} - 2 q^{49} + 27 q^{50} - 19 q^{51} - 62 q^{52} - 50 q^{53} - 27 q^{54} - 66 q^{55} - 19 q^{56} - 45 q^{57} - 55 q^{58} - 118 q^{59} - 29 q^{60} - 92 q^{61} - 61 q^{62} - 54 q^{63} + 71 q^{64} - 51 q^{65} - 73 q^{66} - 17 q^{67} - 52 q^{68} - 89 q^{69} - 33 q^{70} - 95 q^{71} + 34 q^{72} - 114 q^{73} - 43 q^{74} - 38 q^{75} - 30 q^{76} - 73 q^{77} - 40 q^{78} - 47 q^{79} - 34 q^{80} - 57 q^{81} - 87 q^{82} - 68 q^{83} - 51 q^{84} - 67 q^{85} - 4 q^{86} - 55 q^{87} - 37 q^{88} - 150 q^{89} - 81 q^{90} - 23 q^{91} - 45 q^{92} - 59 q^{93} - 89 q^{94} - 47 q^{95} - 9 q^{96} - 97 q^{97} - 2 q^{98} - 57 q^{99}+O(q^{100}) \) 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 1.00000 −3.33028 1.00000 −2.81719 −3.33028 2.39962 1.00000 8.09075 −2.81719
1.2 1.00000 −3.18837 1.00000 1.76077 −3.18837 −1.56118 1.00000 7.16572 1.76077
1.3 1.00000 −3.08324 1.00000 −2.29351 −3.08324 −1.56272 1.00000 6.50635 −2.29351
1.4 1.00000 −3.00113 1.00000 2.08729 −3.00113 1.33710 1.00000 6.00675 2.08729
1.5 1.00000 −2.91430 1.00000 −0.167675 −2.91430 0.471775 1.00000 5.49314 −0.167675
1.6 1.00000 −2.72893 1.00000 0.723438 −2.72893 −3.46721 1.00000 4.44707 0.723438
1.7 1.00000 −2.59864 1.00000 −3.30872 −2.59864 −3.09209 1.00000 3.75294 −3.30872
1.8 1.00000 −2.57686 1.00000 −0.429942 −2.57686 3.15501 1.00000 3.64023 −0.429942
1.9 1.00000 −2.52707 1.00000 −2.67563 −2.52707 −1.45299 1.00000 3.38607 −2.67563
1.10 1.00000 −2.46200 1.00000 −1.19890 −2.46200 2.88631 1.00000 3.06146 −1.19890
1.11 1.00000 −2.45931 1.00000 3.45635 −2.45931 3.30103 1.00000 3.04822 3.45635
1.12 1.00000 −2.11692 1.00000 −3.74472 −2.11692 4.88608 1.00000 1.48134 −3.74472
1.13 1.00000 −2.05160 1.00000 2.89935 −2.05160 −3.15492 1.00000 1.20907 2.89935
1.14 1.00000 −2.02406 1.00000 1.61352 −2.02406 −3.12008 1.00000 1.09682 1.61352
1.15 1.00000 −2.02372 1.00000 3.74567 −2.02372 1.02525 1.00000 1.09543 3.74567
1.16 1.00000 −1.97637 1.00000 −0.168031 −1.97637 −1.30361 1.00000 0.906032 −0.168031
1.17 1.00000 −1.95374 1.00000 0.792660 −1.95374 2.85182 1.00000 0.817104 0.792660
1.18 1.00000 −1.89918 1.00000 −0.474376 −1.89918 1.52174 1.00000 0.606869 −0.474376
1.19 1.00000 −1.72230 1.00000 −4.08988 −1.72230 −0.697853 1.00000 −0.0336847 −4.08988
1.20 1.00000 −1.70684 1.00000 −2.82175 −1.70684 3.08724 1.00000 −0.0867059 −2.82175
See all 71 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.71
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(4013\) \(-1\)

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 8026.2.a.a 71
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8026.2.a.a 71 1.a even 1 1 trivial