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Properties

Label 8006.2.a.d
Level $8006$
Weight $2$
Character orbit 8006.a
Self dual yes
Analytic conductor $63.928$
Analytic rank $0$
Dimension $98$
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: \(0\)
Dimension: \(98\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9} + 4 q^{10} + 51 q^{11} + 16 q^{12} + 31 q^{13} + 29 q^{14} + 57 q^{15} + 98 q^{16} + 35 q^{17} + 130 q^{18} + 77 q^{19} + 4 q^{20} + 46 q^{21} + 51 q^{22} + 73 q^{23} + 16 q^{24} + 150 q^{25} + 31 q^{26} + 52 q^{27} + 29 q^{28} + 20 q^{29} + 57 q^{30} + 59 q^{31} + 98 q^{32} + 27 q^{33} + 35 q^{34} + 48 q^{35} + 130 q^{36} + 41 q^{37} + 77 q^{38} + 64 q^{39} + 4 q^{40} + 29 q^{41} + 46 q^{42} + 94 q^{43} + 51 q^{44} - 3 q^{45} + 73 q^{46} + 58 q^{47} + 16 q^{48} + 149 q^{49} + 150 q^{50} + 58 q^{51} + 31 q^{52} - 11 q^{53} + 52 q^{54} + 56 q^{55} + 29 q^{56} + 64 q^{57} + 20 q^{58} + 45 q^{59} + 57 q^{60} + 73 q^{61} + 59 q^{62} + 53 q^{63} + 98 q^{64} + 39 q^{65} + 27 q^{66} + 133 q^{67} + 35 q^{68} + 13 q^{69} + 48 q^{70} + 67 q^{71} + 130 q^{72} + 42 q^{73} + 41 q^{74} + 36 q^{75} + 77 q^{76} - 25 q^{77} + 64 q^{78} + 154 q^{79} + 4 q^{80} + 198 q^{81} + 29 q^{82} + 69 q^{83} + 46 q^{84} + 81 q^{85} + 94 q^{86} + 25 q^{87} + 51 q^{88} + 32 q^{89} - 3 q^{90} + 95 q^{91} + 73 q^{92} - 23 q^{93} + 58 q^{94} + 50 q^{95} + 16 q^{96} + 76 q^{97} + 149 q^{98} + 149 q^{99}+O(q^{100}) \) 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 1.00000 −3.34117 1.00000 −2.63189 −3.34117 −3.23365 1.00000 8.16344 −2.63189
1.2 1.00000 −3.30357 1.00000 −2.71504 −3.30357 0.802195 1.00000 7.91355 −2.71504
1.3 1.00000 −3.23335 1.00000 1.72220 −3.23335 4.52664 1.00000 7.45455 1.72220
1.4 1.00000 −3.15680 1.00000 −2.35136 −3.15680 −0.0333758 1.00000 6.96536 −2.35136
1.5 1.00000 −3.07582 1.00000 1.83927 −3.07582 −1.83961 1.00000 6.46070 1.83927
1.6 1.00000 −3.05574 1.00000 2.27846 −3.05574 −0.374830 1.00000 6.33757 2.27846
1.7 1.00000 −3.04444 1.00000 −4.16552 −3.04444 −3.08322 1.00000 6.26861 −4.16552
1.8 1.00000 −2.80568 1.00000 1.82812 −2.80568 4.55893 1.00000 4.87183 1.82812
1.9 1.00000 −2.77826 1.00000 −3.83157 −2.77826 3.52049 1.00000 4.71872 −3.83157
1.10 1.00000 −2.73443 1.00000 −0.0905939 −2.73443 −4.21224 1.00000 4.47709 −0.0905939
1.11 1.00000 −2.69452 1.00000 −4.09763 −2.69452 4.00132 1.00000 4.26044 −4.09763
1.12 1.00000 −2.67018 1.00000 0.0387933 −2.67018 −0.245209 1.00000 4.12985 0.0387933
1.13 1.00000 −2.57565 1.00000 −1.28503 −2.57565 −4.37657 1.00000 3.63399 −1.28503
1.14 1.00000 −2.52562 1.00000 3.79609 −2.52562 −3.83950 1.00000 3.37877 3.79609
1.15 1.00000 −2.49687 1.00000 3.81192 −2.49687 2.89844 1.00000 3.23436 3.81192
1.16 1.00000 −2.29904 1.00000 −0.379667 −2.29904 −2.21994 1.00000 2.28559 −0.379667
1.17 1.00000 −2.28004 1.00000 0.593845 −2.28004 −0.744586 1.00000 2.19860 0.593845
1.18 1.00000 −2.25105 1.00000 −0.876874 −2.25105 −0.763073 1.00000 2.06722 −0.876874
1.19 1.00000 −2.20795 1.00000 −3.75545 −2.20795 −1.87926 1.00000 1.87502 −3.75545
1.20 1.00000 −2.20682 1.00000 0.892162 −2.20682 −0.549139 1.00000 1.87006 0.892162
See all 98 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.98
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.d ✓ 98
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8006.2.a.d ✓ 98 1.a even 1 1 trivial