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

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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: \(95\)
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.

\(\operatorname{Tr}(f)(q) = \) \(95q \) \(\mathstrut +\mathstrut 95q^{2} \) \(\mathstrut +\mathstrut 24q^{3} \) \(\mathstrut +\mathstrut 95q^{4} \) \(\mathstrut +\mathstrut 36q^{5} \) \(\mathstrut +\mathstrut 24q^{6} \) \(\mathstrut +\mathstrut 21q^{7} \) \(\mathstrut +\mathstrut 95q^{8} \) \(\mathstrut +\mathstrut 121q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(95q \) \(\mathstrut +\mathstrut 95q^{2} \) \(\mathstrut +\mathstrut 24q^{3} \) \(\mathstrut +\mathstrut 95q^{4} \) \(\mathstrut +\mathstrut 36q^{5} \) \(\mathstrut +\mathstrut 24q^{6} \) \(\mathstrut +\mathstrut 21q^{7} \) \(\mathstrut +\mathstrut 95q^{8} \) \(\mathstrut +\mathstrut 121q^{9} \) \(\mathstrut +\mathstrut 36q^{10} \) \(\mathstrut +\mathstrut 40q^{11} \) \(\mathstrut +\mathstrut 24q^{12} \) \(\mathstrut +\mathstrut 52q^{13} \) \(\mathstrut +\mathstrut 21q^{14} \) \(\mathstrut +\mathstrut 15q^{15} \) \(\mathstrut +\mathstrut 95q^{16} \) \(\mathstrut +\mathstrut 84q^{17} \) \(\mathstrut +\mathstrut 121q^{18} \) \(\mathstrut +\mathstrut 37q^{19} \) \(\mathstrut +\mathstrut 36q^{20} \) \(\mathstrut +\mathstrut 36q^{21} \) \(\mathstrut +\mathstrut 40q^{22} \) \(\mathstrut +\mathstrut 37q^{23} \) \(\mathstrut +\mathstrut 24q^{24} \) \(\mathstrut +\mathstrut 133q^{25} \) \(\mathstrut +\mathstrut 52q^{26} \) \(\mathstrut +\mathstrut 93q^{27} \) \(\mathstrut +\mathstrut 21q^{28} \) \(\mathstrut +\mathstrut 66q^{29} \) \(\mathstrut +\mathstrut 15q^{30} \) \(\mathstrut +\mathstrut 10q^{31} \) \(\mathstrut +\mathstrut 95q^{32} \) \(\mathstrut +\mathstrut 63q^{33} \) \(\mathstrut +\mathstrut 84q^{34} \) \(\mathstrut +\mathstrut 55q^{35} \) \(\mathstrut +\mathstrut 121q^{36} \) \(\mathstrut +\mathstrut 49q^{37} \) \(\mathstrut +\mathstrut 37q^{38} \) \(\mathstrut +\mathstrut 14q^{39} \) \(\mathstrut +\mathstrut 36q^{40} \) \(\mathstrut +\mathstrut 98q^{41} \) \(\mathstrut +\mathstrut 36q^{42} \) \(\mathstrut +\mathstrut 37q^{43} \) \(\mathstrut +\mathstrut 40q^{44} \) \(\mathstrut +\mathstrut 97q^{45} \) \(\mathstrut +\mathstrut 37q^{46} \) \(\mathstrut +\mathstrut 91q^{47} \) \(\mathstrut +\mathstrut 24q^{48} \) \(\mathstrut +\mathstrut 170q^{49} \) \(\mathstrut +\mathstrut 133q^{50} \) \(\mathstrut +\mathstrut 22q^{51} \) \(\mathstrut +\mathstrut 52q^{52} \) \(\mathstrut +\mathstrut 70q^{53} \) \(\mathstrut +\mathstrut 93q^{54} \) \(\mathstrut -\mathstrut q^{55} \) \(\mathstrut +\mathstrut 21q^{56} \) \(\mathstrut +\mathstrut 50q^{57} \) \(\mathstrut +\mathstrut 66q^{58} \) \(\mathstrut +\mathstrut 72q^{59} \) \(\mathstrut +\mathstrut 15q^{60} \) \(\mathstrut +\mathstrut 97q^{61} \) \(\mathstrut +\mathstrut 10q^{62} \) \(\mathstrut +\mathstrut 75q^{63} \) \(\mathstrut +\mathstrut 95q^{64} \) \(\mathstrut +\mathstrut 75q^{65} \) \(\mathstrut +\mathstrut 63q^{66} \) \(\mathstrut +\mathstrut 39q^{67} \) \(\mathstrut +\mathstrut 84q^{68} \) \(\mathstrut +\mathstrut 65q^{69} \) \(\mathstrut +\mathstrut 55q^{70} \) \(\mathstrut +\mathstrut 28q^{71} \) \(\mathstrut +\mathstrut 121q^{72} \) \(\mathstrut +\mathstrut 117q^{73} \) \(\mathstrut +\mathstrut 49q^{74} \) \(\mathstrut +\mathstrut 62q^{75} \) \(\mathstrut +\mathstrut 37q^{76} \) \(\mathstrut +\mathstrut 92q^{77} \) \(\mathstrut +\mathstrut 14q^{78} \) \(\mathstrut +\mathstrut q^{79} \) \(\mathstrut +\mathstrut 36q^{80} \) \(\mathstrut +\mathstrut 155q^{81} \) \(\mathstrut +\mathstrut 98q^{82} \) \(\mathstrut +\mathstrut 117q^{83} \) \(\mathstrut +\mathstrut 36q^{84} \) \(\mathstrut +\mathstrut 81q^{85} \) \(\mathstrut +\mathstrut 37q^{86} \) \(\mathstrut +\mathstrut 46q^{87} \) \(\mathstrut +\mathstrut 40q^{88} \) \(\mathstrut +\mathstrut 90q^{89} \) \(\mathstrut +\mathstrut 97q^{90} \) \(\mathstrut +\mathstrut 65q^{91} \) \(\mathstrut +\mathstrut 37q^{92} \) \(\mathstrut +\mathstrut 36q^{93} \) \(\mathstrut +\mathstrut 91q^{94} \) \(\mathstrut +\mathstrut 38q^{95} \) \(\mathstrut +\mathstrut 24q^{96} \) \(\mathstrut +\mathstrut 111q^{97} \) \(\mathstrut +\mathstrut 170q^{98} \) \(\mathstrut +\mathstrut 97q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.41089 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:

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 \(T_{3}^{95} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\).