Properties

Label 8038.2.a.b
Level 8038
Weight 2
Character orbit 8038.a
Self dual Yes
Analytic conductor 64.184
Analytic rank 0
Dimension 83
CM No

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Newspace parameters

Level: \( N \) = \( 8038 = 2 \cdot 4019 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8038.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1837531447\)
Analytic rank: \(0\)
Dimension: \(83\)
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.

\(\operatorname{Tr}(f)(q) = \) \(83q \) \(\mathstrut -\mathstrut 83q^{2} \) \(\mathstrut +\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 83q^{4} \) \(\mathstrut +\mathstrut 31q^{5} \) \(\mathstrut -\mathstrut 20q^{6} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 83q^{8} \) \(\mathstrut +\mathstrut 91q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(83q \) \(\mathstrut -\mathstrut 83q^{2} \) \(\mathstrut +\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 83q^{4} \) \(\mathstrut +\mathstrut 31q^{5} \) \(\mathstrut -\mathstrut 20q^{6} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 83q^{8} \) \(\mathstrut +\mathstrut 91q^{9} \) \(\mathstrut -\mathstrut 31q^{10} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut +\mathstrut 20q^{12} \) \(\mathstrut +\mathstrut 28q^{13} \) \(\mathstrut +\mathstrut 3q^{14} \) \(\mathstrut +\mathstrut 12q^{15} \) \(\mathstrut +\mathstrut 83q^{16} \) \(\mathstrut +\mathstrut 36q^{17} \) \(\mathstrut -\mathstrut 91q^{18} \) \(\mathstrut -\mathstrut 38q^{19} \) \(\mathstrut +\mathstrut 31q^{20} \) \(\mathstrut +\mathstrut 21q^{21} \) \(\mathstrut -\mathstrut 3q^{22} \) \(\mathstrut +\mathstrut 50q^{23} \) \(\mathstrut -\mathstrut 20q^{24} \) \(\mathstrut +\mathstrut 94q^{25} \) \(\mathstrut -\mathstrut 28q^{26} \) \(\mathstrut +\mathstrut 74q^{27} \) \(\mathstrut -\mathstrut 3q^{28} \) \(\mathstrut +\mathstrut 48q^{29} \) \(\mathstrut -\mathstrut 12q^{30} \) \(\mathstrut -\mathstrut 41q^{31} \) \(\mathstrut -\mathstrut 83q^{32} \) \(\mathstrut +\mathstrut 40q^{33} \) \(\mathstrut -\mathstrut 36q^{34} \) \(\mathstrut +\mathstrut 40q^{35} \) \(\mathstrut +\mathstrut 91q^{36} \) \(\mathstrut +\mathstrut 37q^{37} \) \(\mathstrut +\mathstrut 38q^{38} \) \(\mathstrut +\mathstrut q^{39} \) \(\mathstrut -\mathstrut 31q^{40} \) \(\mathstrut +\mathstrut 44q^{41} \) \(\mathstrut -\mathstrut 21q^{42} \) \(\mathstrut -\mathstrut 21q^{43} \) \(\mathstrut +\mathstrut 3q^{44} \) \(\mathstrut +\mathstrut 98q^{45} \) \(\mathstrut -\mathstrut 50q^{46} \) \(\mathstrut +\mathstrut 62q^{47} \) \(\mathstrut +\mathstrut 20q^{48} \) \(\mathstrut +\mathstrut 74q^{49} \) \(\mathstrut -\mathstrut 94q^{50} \) \(\mathstrut +\mathstrut 11q^{51} \) \(\mathstrut +\mathstrut 28q^{52} \) \(\mathstrut +\mathstrut 99q^{53} \) \(\mathstrut -\mathstrut 74q^{54} \) \(\mathstrut -\mathstrut 20q^{55} \) \(\mathstrut +\mathstrut 3q^{56} \) \(\mathstrut +\mathstrut 24q^{57} \) \(\mathstrut -\mathstrut 48q^{58} \) \(\mathstrut +\mathstrut 33q^{59} \) \(\mathstrut +\mathstrut 12q^{60} \) \(\mathstrut +\mathstrut 38q^{61} \) \(\mathstrut +\mathstrut 41q^{62} \) \(\mathstrut +\mathstrut 43q^{63} \) \(\mathstrut +\mathstrut 83q^{64} \) \(\mathstrut +\mathstrut 85q^{65} \) \(\mathstrut -\mathstrut 40q^{66} \) \(\mathstrut +\mathstrut q^{67} \) \(\mathstrut +\mathstrut 36q^{68} \) \(\mathstrut +\mathstrut 73q^{69} \) \(\mathstrut -\mathstrut 40q^{70} \) \(\mathstrut +\mathstrut 46q^{71} \) \(\mathstrut -\mathstrut 91q^{72} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 37q^{74} \) \(\mathstrut +\mathstrut 89q^{75} \) \(\mathstrut -\mathstrut 38q^{76} \) \(\mathstrut +\mathstrut 118q^{77} \) \(\mathstrut -\mathstrut q^{78} \) \(\mathstrut -\mathstrut 29q^{79} \) \(\mathstrut +\mathstrut 31q^{80} \) \(\mathstrut +\mathstrut 115q^{81} \) \(\mathstrut -\mathstrut 44q^{82} \) \(\mathstrut +\mathstrut 69q^{83} \) \(\mathstrut +\mathstrut 21q^{84} \) \(\mathstrut +\mathstrut 20q^{85} \) \(\mathstrut +\mathstrut 21q^{86} \) \(\mathstrut +\mathstrut 57q^{87} \) \(\mathstrut -\mathstrut 3q^{88} \) \(\mathstrut +\mathstrut 78q^{89} \) \(\mathstrut -\mathstrut 98q^{90} \) \(\mathstrut -\mathstrut 37q^{91} \) \(\mathstrut +\mathstrut 50q^{92} \) \(\mathstrut +\mathstrut 61q^{93} \) \(\mathstrut -\mathstrut 62q^{94} \) \(\mathstrut +\mathstrut 49q^{95} \) \(\mathstrut -\mathstrut 20q^{96} \) \(\mathstrut +\mathstrut 21q^{97} \) \(\mathstrut -\mathstrut 74q^{98} \) \(\mathstrut -\mathstrut 20q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.18441 1.00000 −1.27929 3.18441 1.89743 −1.00000 7.14045 1.27929
1.2 −1.00000 −3.15134 1.00000 4.13084 3.15134 −2.97931 −1.00000 6.93095 −4.13084
1.3 −1.00000 −3.02523 1.00000 2.41251 3.02523 3.55505 −1.00000 6.15201 −2.41251
1.4 −1.00000 −3.00434 1.00000 −0.184095 3.00434 1.88237 −1.00000 6.02606 0.184095
1.5 −1.00000 −2.95780 1.00000 0.0755052 2.95780 −2.37738 −1.00000 5.74860 −0.0755052
1.6 −1.00000 −2.90051 1.00000 2.46896 2.90051 0.644361 −1.00000 5.41299 −2.46896
1.7 −1.00000 −2.88068 1.00000 −2.36782 2.88068 0.285889 −1.00000 5.29829 2.36782
1.8 −1.00000 −2.78821 1.00000 0.878256 2.78821 −4.20087 −1.00000 4.77411 −0.878256
1.9 −1.00000 −2.34219 1.00000 −3.72898 2.34219 −1.27056 −1.00000 2.48587 3.72898
1.10 −1.00000 −2.20646 1.00000 0.820064 2.20646 3.76837 −1.00000 1.86848 −0.820064
1.11 −1.00000 −2.18666 1.00000 −3.58998 2.18666 −4.12584 −1.00000 1.78150 3.58998
1.12 −1.00000 −2.14458 1.00000 3.80241 2.14458 4.02154 −1.00000 1.59920 −3.80241
1.13 −1.00000 −2.12932 1.00000 2.23951 2.12932 −4.41338 −1.00000 1.53399 −2.23951
1.14 −1.00000 −2.12834 1.00000 −2.40910 2.12834 0.412842 −1.00000 1.52982 2.40910
1.15 −1.00000 −2.12762 1.00000 −1.48235 2.12762 0.377312 −1.00000 1.52675 1.48235
1.16 −1.00000 −2.05749 1.00000 3.25924 2.05749 −0.840215 −1.00000 1.23326 −3.25924
1.17 −1.00000 −1.99762 1.00000 2.51698 1.99762 −0.403674 −1.00000 0.990467 −2.51698
1.18 −1.00000 −1.88709 1.00000 −0.769273 1.88709 −1.40180 −1.00000 0.561125 0.769273
1.19 −1.00000 −1.79852 1.00000 4.20751 1.79852 −1.35268 −1.00000 0.234660 −4.20751
1.20 −1.00000 −1.59172 1.00000 0.670129 1.59172 −4.47737 −1.00000 −0.466431 −0.670129
See all 83 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.83
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(4019\) \(-1\)