Properties

Label 8006.2.a.a
Level $8006$
Weight $2$
Character orbit 8006.a
Self dual yes
Analytic conductor $63.928$
Analytic rank $1$
Dimension $69$
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] = [8006,2,Mod(1,8006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8006, 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("8006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8006 = 2 \cdot 4003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8006.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: \(63.9282318582\)
Analytic rank: \(1\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 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 −3.36322 1.00000 3.90848 −3.36322 −1.71915 1.00000 8.31124 3.90848
1.2 1.00000 −3.28301 1.00000 −1.18461 −3.28301 0.314789 1.00000 7.77813 −1.18461
1.3 1.00000 −3.15967 1.00000 −0.278823 −3.15967 1.77562 1.00000 6.98353 −0.278823
1.4 1.00000 −3.08249 1.00000 1.32186 −3.08249 1.26521 1.00000 6.50177 1.32186
1.5 1.00000 −2.86585 1.00000 2.47496 −2.86585 −3.03619 1.00000 5.21309 2.47496
1.6 1.00000 −2.78283 1.00000 2.27257 −2.78283 −2.57189 1.00000 4.74415 2.27257
1.7 1.00000 −2.75233 1.00000 −1.92549 −2.75233 3.54771 1.00000 4.57534 −1.92549
1.8 1.00000 −2.73399 1.00000 1.40012 −2.73399 2.82233 1.00000 4.47471 1.40012
1.9 1.00000 −2.72077 1.00000 0.316882 −2.72077 −4.75223 1.00000 4.40260 0.316882
1.10 1.00000 −2.65418 1.00000 −3.04088 −2.65418 −4.15627 1.00000 4.04466 −3.04088
1.11 1.00000 −2.51689 1.00000 −2.29473 −2.51689 0.132144 1.00000 3.33476 −2.29473
1.12 1.00000 −2.38200 1.00000 2.27463 −2.38200 −1.51830 1.00000 2.67393 2.27463
1.13 1.00000 −2.26149 1.00000 −2.88813 −2.26149 2.25047 1.00000 2.11432 −2.88813
1.14 1.00000 −2.16991 1.00000 −2.35323 −2.16991 −2.98482 1.00000 1.70849 −2.35323
1.15 1.00000 −2.10040 1.00000 −0.571471 −2.10040 3.01756 1.00000 1.41167 −0.571471
1.16 1.00000 −2.04988 1.00000 4.28371 −2.04988 3.78593 1.00000 1.20199 4.28371
1.17 1.00000 −1.97198 1.00000 2.38225 −1.97198 2.34177 1.00000 0.888724 2.38225
1.18 1.00000 −1.76933 1.00000 3.70543 −1.76933 −4.46804 1.00000 0.130514 3.70543
1.19 1.00000 −1.67001 1.00000 1.85681 −1.67001 0.410918 1.00000 −0.211058 1.85681
1.20 1.00000 −1.63641 1.00000 −3.89748 −1.63641 0.627952 1.00000 −0.322178 −3.89748
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(4003\) \(-1\)

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 8006.2.a.a 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8006.2.a.a 69 1.a even 1 1 trivial