Properties

Label 8002.2.a.f
Level 8002
Weight 2
Character orbit 8002.a
Self dual Yes
Analytic conductor 63.896
Analytic rank 1
Dimension 89
CM No

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Newspace parameters

Level: \( N \) = \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8962916974\)
Analytic rank: \(1\)
Dimension: \(89\)
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.

\(\operatorname{Tr}(f)(q) = \) \(89q \) \(\mathstrut -\mathstrut 89q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 89q^{4} \) \(\mathstrut -\mathstrut 18q^{5} \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut -\mathstrut 27q^{7} \) \(\mathstrut -\mathstrut 89q^{8} \) \(\mathstrut +\mathstrut 95q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(89q \) \(\mathstrut -\mathstrut 89q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 89q^{4} \) \(\mathstrut -\mathstrut 18q^{5} \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut -\mathstrut 27q^{7} \) \(\mathstrut -\mathstrut 89q^{8} \) \(\mathstrut +\mathstrut 95q^{9} \) \(\mathstrut +\mathstrut 18q^{10} \) \(\mathstrut -\mathstrut 26q^{11} \) \(\mathstrut -\mathstrut 12q^{12} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 27q^{14} \) \(\mathstrut -\mathstrut 21q^{15} \) \(\mathstrut +\mathstrut 89q^{16} \) \(\mathstrut -\mathstrut 60q^{17} \) \(\mathstrut -\mathstrut 95q^{18} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut -\mathstrut 18q^{20} \) \(\mathstrut -\mathstrut 6q^{21} \) \(\mathstrut +\mathstrut 26q^{22} \) \(\mathstrut -\mathstrut 45q^{23} \) \(\mathstrut +\mathstrut 12q^{24} \) \(\mathstrut +\mathstrut 107q^{25} \) \(\mathstrut -\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 45q^{27} \) \(\mathstrut -\mathstrut 27q^{28} \) \(\mathstrut -\mathstrut 18q^{29} \) \(\mathstrut +\mathstrut 21q^{30} \) \(\mathstrut -\mathstrut 38q^{31} \) \(\mathstrut -\mathstrut 89q^{32} \) \(\mathstrut -\mathstrut 29q^{33} \) \(\mathstrut +\mathstrut 60q^{34} \) \(\mathstrut -\mathstrut 47q^{35} \) \(\mathstrut +\mathstrut 95q^{36} \) \(\mathstrut -\mathstrut 15q^{37} \) \(\mathstrut -\mathstrut q^{38} \) \(\mathstrut -\mathstrut 38q^{39} \) \(\mathstrut +\mathstrut 18q^{40} \) \(\mathstrut -\mathstrut 50q^{41} \) \(\mathstrut +\mathstrut 6q^{42} \) \(\mathstrut -\mathstrut 15q^{43} \) \(\mathstrut -\mathstrut 26q^{44} \) \(\mathstrut -\mathstrut 35q^{45} \) \(\mathstrut +\mathstrut 45q^{46} \) \(\mathstrut -\mathstrut 121q^{47} \) \(\mathstrut -\mathstrut 12q^{48} \) \(\mathstrut +\mathstrut 132q^{49} \) \(\mathstrut -\mathstrut 107q^{50} \) \(\mathstrut +\mathstrut 6q^{51} \) \(\mathstrut +\mathstrut 2q^{52} \) \(\mathstrut -\mathstrut 46q^{53} \) \(\mathstrut +\mathstrut 45q^{54} \) \(\mathstrut -\mathstrut 37q^{55} \) \(\mathstrut +\mathstrut 27q^{56} \) \(\mathstrut -\mathstrut 42q^{57} \) \(\mathstrut +\mathstrut 18q^{58} \) \(\mathstrut -\mathstrut 34q^{59} \) \(\mathstrut -\mathstrut 21q^{60} \) \(\mathstrut +\mathstrut 41q^{61} \) \(\mathstrut +\mathstrut 38q^{62} \) \(\mathstrut -\mathstrut 131q^{63} \) \(\mathstrut +\mathstrut 89q^{64} \) \(\mathstrut -\mathstrut 57q^{65} \) \(\mathstrut +\mathstrut 29q^{66} \) \(\mathstrut -\mathstrut 11q^{67} \) \(\mathstrut -\mathstrut 60q^{68} \) \(\mathstrut +\mathstrut 15q^{69} \) \(\mathstrut +\mathstrut 47q^{70} \) \(\mathstrut -\mathstrut 66q^{71} \) \(\mathstrut -\mathstrut 95q^{72} \) \(\mathstrut -\mathstrut 47q^{73} \) \(\mathstrut +\mathstrut 15q^{74} \) \(\mathstrut -\mathstrut 46q^{75} \) \(\mathstrut +\mathstrut q^{76} \) \(\mathstrut -\mathstrut 106q^{77} \) \(\mathstrut +\mathstrut 38q^{78} \) \(\mathstrut -\mathstrut 51q^{79} \) \(\mathstrut -\mathstrut 18q^{80} \) \(\mathstrut +\mathstrut 113q^{81} \) \(\mathstrut +\mathstrut 50q^{82} \) \(\mathstrut -\mathstrut 141q^{83} \) \(\mathstrut -\mathstrut 6q^{84} \) \(\mathstrut -\mathstrut 7q^{85} \) \(\mathstrut +\mathstrut 15q^{86} \) \(\mathstrut -\mathstrut 110q^{87} \) \(\mathstrut +\mathstrut 26q^{88} \) \(\mathstrut -\mathstrut 30q^{89} \) \(\mathstrut +\mathstrut 35q^{90} \) \(\mathstrut +\mathstrut 37q^{91} \) \(\mathstrut -\mathstrut 45q^{92} \) \(\mathstrut -\mathstrut 44q^{93} \) \(\mathstrut +\mathstrut 121q^{94} \) \(\mathstrut -\mathstrut 98q^{95} \) \(\mathstrut +\mathstrut 12q^{96} \) \(\mathstrut +\mathstrut 3q^{97} \) \(\mathstrut -\mathstrut 132q^{98} \) \(\mathstrut -\mathstrut 71q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −1.00000 −3.34684 1.00000 −3.62971 3.34684 −3.29144 −1.00000 8.20136 3.62971
1.2 −1.00000 −3.32456 1.00000 2.37351 3.32456 −1.54900 −1.00000 8.05269 −2.37351
1.3 −1.00000 −3.29759 1.00000 −2.40937 3.29759 1.21018 −1.00000 7.87411 2.40937
1.4 −1.00000 −3.26711 1.00000 0.417272 3.26711 −4.66212 −1.00000 7.67400 −0.417272
1.5 −1.00000 −3.23014 1.00000 4.30971 3.23014 −1.93951 −1.00000 7.43383 −4.30971
1.6 −1.00000 −3.19543 1.00000 −0.753590 3.19543 −0.614331 −1.00000 7.21079 0.753590
1.7 −1.00000 −2.99019 1.00000 2.92742 2.99019 4.68595 −1.00000 5.94125 −2.92742
1.8 −1.00000 −2.96590 1.00000 −3.55325 2.96590 2.76315 −1.00000 5.79657 3.55325
1.9 −1.00000 −2.88148 1.00000 0.529257 2.88148 −3.48565 −1.00000 5.30294 −0.529257
1.10 −1.00000 −2.87972 1.00000 −2.30779 2.87972 −0.146606 −1.00000 5.29280 2.30779
1.11 −1.00000 −2.78190 1.00000 1.30300 2.78190 4.15781 −1.00000 4.73899 −1.30300
1.12 −1.00000 −2.76801 1.00000 −2.71168 2.76801 −4.48343 −1.00000 4.66186 2.71168
1.13 −1.00000 −2.61902 1.00000 1.94954 2.61902 −2.60943 −1.00000 3.85928 −1.94954
1.14 −1.00000 −2.61462 1.00000 2.63805 2.61462 1.49750 −1.00000 3.83623 −2.63805
1.15 −1.00000 −2.39406 1.00000 3.62374 2.39406 −4.52806 −1.00000 2.73150 −3.62374
1.16 −1.00000 −2.39329 1.00000 −1.79652 2.39329 −1.61915 −1.00000 2.72784 1.79652
1.17 −1.00000 −2.23021 1.00000 −2.99176 2.23021 3.17674 −1.00000 1.97381 2.99176
1.18 −1.00000 −2.19996 1.00000 −2.22300 2.19996 1.97846 −1.00000 1.83980 2.22300
1.19 −1.00000 −2.15966 1.00000 1.78112 2.15966 1.44604 −1.00000 1.66414 −1.78112
1.20 −1.00000 −2.15321 1.00000 0.345955 2.15321 4.65391 −1.00000 1.63633 −0.345955
See all 89 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.89
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{89} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\).