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Properties

Label 8006.2.a.c
Level $8006$
Weight $2$
Character orbit 8006.a
Self dual yes
Analytic conductor $63.928$
Analytic rank $0$
Dimension $92$
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: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 92 q^{2} - 2 q^{3} + 92 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{7} - 92 q^{8} + 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q - 92 q^{2} - 2 q^{3} + 92 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{7} - 92 q^{8} + 104 q^{9} - 10 q^{10} + 4 q^{11} - 2 q^{12} + 40 q^{13} - 8 q^{14} + 15 q^{15} + 92 q^{16} - 14 q^{17} - 104 q^{18} + 64 q^{19} + 10 q^{20} + 54 q^{21} - 4 q^{22} - 49 q^{23} + 2 q^{24} + 116 q^{25} - 40 q^{26} - 8 q^{27} + 8 q^{28} + 39 q^{29} - 15 q^{30} + 53 q^{31} - 92 q^{32} + q^{33} + 14 q^{34} - 22 q^{35} + 104 q^{36} + 58 q^{37} - 64 q^{38} + 58 q^{39} - 10 q^{40} + 27 q^{41} - 54 q^{42} + 40 q^{43} + 4 q^{44} + 43 q^{45} + 49 q^{46} - 28 q^{47} - 2 q^{48} + 148 q^{49} - 116 q^{50} + 48 q^{51} + 40 q^{52} + 32 q^{53} + 8 q^{54} + 36 q^{55} - 8 q^{56} + 48 q^{57} - 39 q^{58} + 8 q^{59} + 15 q^{60} + 99 q^{61} - 53 q^{62} + 92 q^{64} + 13 q^{65} - q^{66} + 48 q^{67} - 14 q^{68} + 63 q^{69} + 22 q^{70} - 13 q^{71} - 104 q^{72} + 49 q^{73} - 58 q^{74} + 16 q^{75} + 64 q^{76} + 41 q^{77} - 58 q^{78} + 143 q^{79} + 10 q^{80} + 124 q^{81} - 27 q^{82} - 24 q^{83} + 54 q^{84} + 121 q^{85} - 40 q^{86} + 5 q^{87} - 4 q^{88} + 25 q^{89} - 43 q^{90} + 67 q^{91} - 49 q^{92} + 43 q^{93} + 28 q^{94} - 38 q^{95} + 2 q^{96} + 74 q^{97} - 148 q^{98} + 86 q^{99}+O(q^{100}) \) 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 −3.33817 1.00000 2.91459 3.33817 −3.36056 −1.00000 8.14336 −2.91459
1.2 −1.00000 −3.29526 1.00000 0.810886 3.29526 0.489532 −1.00000 7.85871 −0.810886
1.3 −1.00000 −3.23769 1.00000 1.27122 3.23769 2.87873 −1.00000 7.48264 −1.27122
1.4 −1.00000 −3.23730 1.00000 −3.23542 3.23730 2.52954 −1.00000 7.48009 3.23542
1.5 −1.00000 −3.23032 1.00000 0.193126 3.23032 0.0724171 −1.00000 7.43498 −0.193126
1.6 −1.00000 −3.11194 1.00000 0.282330 3.11194 −3.77246 −1.00000 6.68416 −0.282330
1.7 −1.00000 −3.10076 1.00000 −2.40718 3.10076 −4.08337 −1.00000 6.61471 2.40718
1.8 −1.00000 −2.93049 1.00000 4.31127 2.93049 −2.38009 −1.00000 5.58776 −4.31127
1.9 −1.00000 −2.92008 1.00000 −3.64076 2.92008 −4.95153 −1.00000 5.52686 3.64076
1.10 −1.00000 −2.77089 1.00000 −0.296987 2.77089 3.25147 −1.00000 4.67784 0.296987
1.11 −1.00000 −2.73647 1.00000 2.71633 2.73647 −4.85274 −1.00000 4.48825 −2.71633
1.12 −1.00000 −2.54869 1.00000 0.707045 2.54869 4.56576 −1.00000 3.49580 −0.707045
1.13 −1.00000 −2.53187 1.00000 4.10502 2.53187 0.894818 −1.00000 3.41035 −4.10502
1.14 −1.00000 −2.53029 1.00000 −2.88347 2.53029 1.41548 −1.00000 3.40236 2.88347
1.15 −1.00000 −2.48256 1.00000 −0.944639 2.48256 −2.01033 −1.00000 3.16310 0.944639
1.16 −1.00000 −2.37750 1.00000 −3.81814 2.37750 −1.26090 −1.00000 2.65251 3.81814
1.17 −1.00000 −2.30799 1.00000 −1.49118 2.30799 3.38849 −1.00000 2.32682 1.49118
1.18 −1.00000 −2.07631 1.00000 0.619812 2.07631 2.12229 −1.00000 1.31108 −0.619812
1.19 −1.00000 −2.03971 1.00000 2.70555 2.03971 1.13459 −1.00000 1.16043 −2.70555
1.20 −1.00000 −1.98237 1.00000 −4.41598 1.98237 −1.54286 −1.00000 0.929796 4.41598
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
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.c ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8006.2.a.c ✓ 92 1.a even 1 1 trivial