Properties

Label 8002.2.a.d
Level $8002$
Weight $2$
Character orbit 8002.a
Self dual yes
Analytic conductor $63.896$
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] = [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: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 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 −3.43715 1.00000 −2.28285 −3.43715 −3.85963 1.00000 8.81400 −2.28285
1.2 1.00000 −3.40667 1.00000 2.18421 −3.40667 −1.24057 1.00000 8.60538 2.18421
1.3 1.00000 −3.22228 1.00000 −3.24951 −3.22228 4.15498 1.00000 7.38311 −3.24951
1.4 1.00000 −3.09800 1.00000 −2.64538 −3.09800 3.07801 1.00000 6.59762 −2.64538
1.5 1.00000 −3.04857 1.00000 −0.106661 −3.04857 −0.613952 1.00000 6.29376 −0.106661
1.6 1.00000 −3.03319 1.00000 −3.55972 −3.03319 −4.22109 1.00000 6.20023 −3.55972
1.7 1.00000 −2.87355 1.00000 2.00628 −2.87355 0.474155 1.00000 5.25731 2.00628
1.8 1.00000 −2.80104 1.00000 0.676703 −2.80104 3.76033 1.00000 4.84585 0.676703
1.9 1.00000 −2.76808 1.00000 4.07666 −2.76808 2.61903 1.00000 4.66227 4.07666
1.10 1.00000 −2.73719 1.00000 −4.28393 −2.73719 2.06732 1.00000 4.49223 −4.28393
1.11 1.00000 −2.64782 1.00000 −0.675974 −2.64782 −0.999289 1.00000 4.01094 −0.675974
1.12 1.00000 −2.62997 1.00000 −0.133179 −2.62997 −2.98290 1.00000 3.91674 −0.133179
1.13 1.00000 −2.42898 1.00000 −1.63165 −2.42898 3.93412 1.00000 2.89993 −1.63165
1.14 1.00000 −2.38948 1.00000 1.42184 −2.38948 −5.14183 1.00000 2.70964 1.42184
1.15 1.00000 −2.15867 1.00000 0.496482 −2.15867 −0.313465 1.00000 1.65985 0.496482
1.16 1.00000 −2.04234 1.00000 −4.37946 −2.04234 −3.34038 1.00000 1.17114 −4.37946
1.17 1.00000 −2.01476 1.00000 2.07868 −2.01476 −1.64070 1.00000 1.05926 2.07868
1.18 1.00000 −1.99879 1.00000 2.05838 −1.99879 2.31041 1.00000 0.995149 2.05838
1.19 1.00000 −1.96903 1.00000 3.30115 −1.96903 1.85739 1.00000 0.877064 3.30115
1.20 1.00000 −1.96349 1.00000 0.136598 −1.96349 −0.645165 1.00000 0.855285 0.136598
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
\(2\) \(-1\)
\(4001\) \(-1\)

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.d 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8002.2.a.d 69 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{69} + 25 T_{3}^{68} + 182 T_{3}^{67} - 583 T_{3}^{66} - 14542 T_{3}^{65} - 42332 T_{3}^{64} + 387895 T_{3}^{63} + 2646653 T_{3}^{62} - 2844455 T_{3}^{61} - 69153744 T_{3}^{60} - 97890854 T_{3}^{59} + \cdots + 388376961 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\). Copy content Toggle raw display