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Properties

Label 8002.2.a.g
Level $8002$
Weight $2$
Character orbit 8002.a
Self dual yes
Analytic conductor $63.896$
Analytic rank $0$
Dimension $95$
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] = [8002,2,Mod(1,8002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8002, 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("8002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8002.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: \(63.8962916974\)
Analytic rank: \(0\)
Dimension: \(95\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9} + 36 q^{10} + 40 q^{11} + 24 q^{12} + 52 q^{13} + 21 q^{14} + 15 q^{15} + 95 q^{16} + 84 q^{17} + 121 q^{18} + 37 q^{19} + 36 q^{20} + 36 q^{21} + 40 q^{22} + 37 q^{23} + 24 q^{24} + 133 q^{25} + 52 q^{26} + 93 q^{27} + 21 q^{28} + 66 q^{29} + 15 q^{30} + 10 q^{31} + 95 q^{32} + 63 q^{33} + 84 q^{34} + 55 q^{35} + 121 q^{36} + 49 q^{37} + 37 q^{38} + 14 q^{39} + 36 q^{40} + 98 q^{41} + 36 q^{42} + 37 q^{43} + 40 q^{44} + 97 q^{45} + 37 q^{46} + 91 q^{47} + 24 q^{48} + 170 q^{49} + 133 q^{50} + 22 q^{51} + 52 q^{52} + 70 q^{53} + 93 q^{54} - q^{55} + 21 q^{56} + 50 q^{57} + 66 q^{58} + 72 q^{59} + 15 q^{60} + 97 q^{61} + 10 q^{62} + 75 q^{63} + 95 q^{64} + 75 q^{65} + 63 q^{66} + 39 q^{67} + 84 q^{68} + 65 q^{69} + 55 q^{70} + 28 q^{71} + 121 q^{72} + 117 q^{73} + 49 q^{74} + 62 q^{75} + 37 q^{76} + 92 q^{77} + 14 q^{78} + q^{79} + 36 q^{80} + 155 q^{81} + 98 q^{82} + 117 q^{83} + 36 q^{84} + 81 q^{85} + 37 q^{86} + 46 q^{87} + 40 q^{88} + 90 q^{89} + 97 q^{90} + 65 q^{91} + 37 q^{92} + 36 q^{93} + 91 q^{94} + 38 q^{95} + 24 q^{96} + 111 q^{97} + 170 q^{98} + 97 q^{99}+O(q^{100}) \) 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 −3.41089 1.00000 2.42267 −3.41089 3.23449 1.00000 8.63416 2.42267
1.2 1.00000 −3.18851 1.00000 0.351128 −3.18851 −0.0577605 1.00000 7.16662 0.351128
1.3 1.00000 −3.08942 1.00000 −3.01572 −3.08942 −0.515913 1.00000 6.54453 −3.01572
1.4 1.00000 −2.96030 1.00000 3.24617 −2.96030 −2.38509 1.00000 5.76337 3.24617
1.5 1.00000 −2.82799 1.00000 4.14459 −2.82799 4.30276 1.00000 4.99754 4.14459
1.6 1.00000 −2.80236 1.00000 1.73987 −2.80236 −0.386891 1.00000 4.85324 1.73987
1.7 1.00000 −2.78398 1.00000 3.98298 −2.78398 −2.83528 1.00000 4.75054 3.98298
1.8 1.00000 −2.76392 1.00000 −0.367238 −2.76392 3.23477 1.00000 4.63923 −0.367238
1.9 1.00000 −2.73826 1.00000 −2.29649 −2.73826 −2.81204 1.00000 4.49807 −2.29649
1.10 1.00000 −2.72633 1.00000 0.195975 −2.72633 −4.75513 1.00000 4.43286 0.195975
1.11 1.00000 −2.68330 1.00000 −1.68032 −2.68330 0.225636 1.00000 4.20011 −1.68032
1.12 1.00000 −2.66813 1.00000 2.32067 −2.66813 −4.26871 1.00000 4.11889 2.32067
1.13 1.00000 −2.37543 1.00000 −2.94724 −2.37543 −3.40112 1.00000 2.64268 −2.94724
1.14 1.00000 −2.24343 1.00000 −3.17394 −2.24343 1.39363 1.00000 2.03297 −3.17394
1.15 1.00000 −2.22371 1.00000 3.47056 −2.22371 3.09791 1.00000 1.94489 3.47056
1.16 1.00000 −2.22303 1.00000 −1.84239 −2.22303 1.24832 1.00000 1.94184 −1.84239
1.17 1.00000 −2.18707 1.00000 −0.511085 −2.18707 4.44664 1.00000 1.78325 −0.511085
1.18 1.00000 −2.11368 1.00000 1.91949 −2.11368 3.49294 1.00000 1.46764 1.91949
1.19 1.00000 −1.94136 1.00000 −0.368072 −1.94136 −3.56861 1.00000 0.768897 −0.368072
1.20 1.00000 −1.87090 1.00000 4.22840 −1.87090 −1.96654 1.00000 0.500276 4.22840
See all 95 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.95
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(4001\) \(1\)

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 8002.2.a.g ✓ 95
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8002.2.a.g ✓ 95 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{95} - 24 T_{3}^{94} + 85 T_{3}^{93} + 2489 T_{3}^{92} - 22878 T_{3}^{91} - 82622 T_{3}^{90} + \cdots - 50539270125376 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\). Copy content Toggle raw display