Properties

Label 8002.2.a.e
Level 8002
Weight 2
Character orbit 8002.a
Self dual Yes
Analytic conductor 63.896
Analytic rank 0
Dimension 77
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: \(0\)
Dimension: \(77\)
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.

\(\operatorname{Tr}(f)(q) = \) \(77q \) \(\mathstrut -\mathstrut 77q^{2} \) \(\mathstrut +\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 77q^{4} \) \(\mathstrut +\mathstrut 18q^{5} \) \(\mathstrut -\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 21q^{7} \) \(\mathstrut -\mathstrut 77q^{8} \) \(\mathstrut +\mathstrut 71q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(77q \) \(\mathstrut -\mathstrut 77q^{2} \) \(\mathstrut +\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 77q^{4} \) \(\mathstrut +\mathstrut 18q^{5} \) \(\mathstrut -\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 21q^{7} \) \(\mathstrut -\mathstrut 77q^{8} \) \(\mathstrut +\mathstrut 71q^{9} \) \(\mathstrut -\mathstrut 18q^{10} \) \(\mathstrut +\mathstrut 30q^{11} \) \(\mathstrut +\mathstrut 10q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 21q^{14} \) \(\mathstrut +\mathstrut 21q^{15} \) \(\mathstrut +\mathstrut 77q^{16} \) \(\mathstrut +\mathstrut 60q^{17} \) \(\mathstrut -\mathstrut 71q^{18} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut +\mathstrut 18q^{20} \) \(\mathstrut +\mathstrut 10q^{21} \) \(\mathstrut -\mathstrut 30q^{22} \) \(\mathstrut +\mathstrut 53q^{23} \) \(\mathstrut -\mathstrut 10q^{24} \) \(\mathstrut +\mathstrut 59q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut +\mathstrut 43q^{27} \) \(\mathstrut +\mathstrut 21q^{28} \) \(\mathstrut +\mathstrut 30q^{29} \) \(\mathstrut -\mathstrut 21q^{30} \) \(\mathstrut +\mathstrut 22q^{31} \) \(\mathstrut -\mathstrut 77q^{32} \) \(\mathstrut +\mathstrut 31q^{33} \) \(\mathstrut -\mathstrut 60q^{34} \) \(\mathstrut +\mathstrut 41q^{35} \) \(\mathstrut +\mathstrut 71q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut +\mathstrut 3q^{38} \) \(\mathstrut +\mathstrut 44q^{39} \) \(\mathstrut -\mathstrut 18q^{40} \) \(\mathstrut +\mathstrut 48q^{41} \) \(\mathstrut -\mathstrut 10q^{42} \) \(\mathstrut +\mathstrut 21q^{43} \) \(\mathstrut +\mathstrut 30q^{44} \) \(\mathstrut +\mathstrut 33q^{45} \) \(\mathstrut -\mathstrut 53q^{46} \) \(\mathstrut +\mathstrut 107q^{47} \) \(\mathstrut +\mathstrut 10q^{48} \) \(\mathstrut +\mathstrut 24q^{49} \) \(\mathstrut -\mathstrut 59q^{50} \) \(\mathstrut +\mathstrut 18q^{51} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut +\mathstrut 42q^{53} \) \(\mathstrut -\mathstrut 43q^{54} \) \(\mathstrut +\mathstrut 49q^{55} \) \(\mathstrut -\mathstrut 21q^{56} \) \(\mathstrut +\mathstrut 32q^{57} \) \(\mathstrut -\mathstrut 30q^{58} \) \(\mathstrut +\mathstrut 42q^{59} \) \(\mathstrut +\mathstrut 21q^{60} \) \(\mathstrut -\mathstrut 31q^{61} \) \(\mathstrut -\mathstrut 22q^{62} \) \(\mathstrut +\mathstrut 109q^{63} \) \(\mathstrut +\mathstrut 77q^{64} \) \(\mathstrut +\mathstrut 39q^{65} \) \(\mathstrut -\mathstrut 31q^{66} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut +\mathstrut 60q^{68} \) \(\mathstrut -\mathstrut 33q^{69} \) \(\mathstrut -\mathstrut 41q^{70} \) \(\mathstrut +\mathstrut 58q^{71} \) \(\mathstrut -\mathstrut 71q^{72} \) \(\mathstrut +\mathstrut 35q^{73} \) \(\mathstrut +\mathstrut 3q^{74} \) \(\mathstrut +\mathstrut 34q^{75} \) \(\mathstrut -\mathstrut 3q^{76} \) \(\mathstrut +\mathstrut 86q^{77} \) \(\mathstrut -\mathstrut 44q^{78} \) \(\mathstrut +\mathstrut 25q^{79} \) \(\mathstrut +\mathstrut 18q^{80} \) \(\mathstrut +\mathstrut 53q^{81} \) \(\mathstrut -\mathstrut 48q^{82} \) \(\mathstrut +\mathstrut 107q^{83} \) \(\mathstrut +\mathstrut 10q^{84} \) \(\mathstrut +\mathstrut 21q^{85} \) \(\mathstrut -\mathstrut 21q^{86} \) \(\mathstrut +\mathstrut 100q^{87} \) \(\mathstrut -\mathstrut 30q^{88} \) \(\mathstrut +\mathstrut 34q^{89} \) \(\mathstrut -\mathstrut 33q^{90} \) \(\mathstrut -\mathstrut 51q^{91} \) \(\mathstrut +\mathstrut 53q^{92} \) \(\mathstrut +\mathstrut 48q^{93} \) \(\mathstrut -\mathstrut 107q^{94} \) \(\mathstrut +\mathstrut 118q^{95} \) \(\mathstrut -\mathstrut 10q^{96} \) \(\mathstrut -\mathstrut 13q^{97} \) \(\mathstrut -\mathstrut 24q^{98} \) \(\mathstrut +\mathstrut 63q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.18344 1.00000 −0.604370 3.18344 1.50072 −1.00000 7.13428 0.604370
1.2 −1.00000 −3.14716 1.00000 −2.17339 3.14716 4.77090 −1.00000 6.90459 2.17339
1.3 −1.00000 −3.14624 1.00000 3.08369 3.14624 2.61077 −1.00000 6.89883 −3.08369
1.4 −1.00000 −2.93461 1.00000 1.73622 2.93461 −0.969734 −1.00000 5.61192 −1.73622
1.5 −1.00000 −2.91842 1.00000 3.04954 2.91842 3.39338 −1.00000 5.51716 −3.04954
1.6 −1.00000 −2.69277 1.00000 0.930791 2.69277 −2.75733 −1.00000 4.25101 −0.930791
1.7 −1.00000 −2.62668 1.00000 0.854809 2.62668 1.79286 −1.00000 3.89945 −0.854809
1.8 −1.00000 −2.61328 1.00000 −3.49569 2.61328 −0.478818 −1.00000 3.82924 3.49569
1.9 −1.00000 −2.50919 1.00000 −3.69214 2.50919 −2.32427 −1.00000 3.29603 3.69214
1.10 −1.00000 −2.45617 1.00000 0.358646 2.45617 −1.91182 −1.00000 3.03279 −0.358646
1.11 −1.00000 −2.42114 1.00000 −1.19106 2.42114 −3.77401 −1.00000 2.86190 1.19106
1.12 −1.00000 −2.37610 1.00000 −3.32970 2.37610 3.18362 −1.00000 2.64585 3.32970
1.13 −1.00000 −2.19428 1.00000 0.0495220 2.19428 1.71879 −1.00000 1.81487 −0.0495220
1.14 −1.00000 −2.17233 1.00000 −1.31184 2.17233 −2.50290 −1.00000 1.71901 1.31184
1.15 −1.00000 −2.13252 1.00000 2.05525 2.13252 −1.57763 −1.00000 1.54765 −2.05525
1.16 −1.00000 −1.85869 1.00000 −0.435894 1.85869 1.03466 −1.00000 0.454718 0.435894
1.17 −1.00000 −1.73843 1.00000 4.07186 1.73843 −2.38793 −1.00000 0.0221381 −4.07186
1.18 −1.00000 −1.70235 1.00000 4.07630 1.70235 1.03752 −1.00000 −0.101993 −4.07630
1.19 −1.00000 −1.65847 1.00000 −0.802626 1.65847 −0.0977002 −1.00000 −0.249472 0.802626
1.20 −1.00000 −1.59889 1.00000 −0.0810696 1.59889 5.11538 −1.00000 −0.443564 0.0810696
See all 77 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.77
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 \(T_{3}^{77} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\).