Properties

Label 8042.2.a.b
Level 8042
Weight 2
Character orbit 8042.a
Self dual Yes
Analytic conductor 64.216
Analytic rank 1
Dimension 82
CM No

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Newspace parameters

Level: \( N \) = \( 8042 = 2 \cdot 4021 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8042.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2156933055\)
Analytic rank: \(1\)
Dimension: \(82\)
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.

\(\operatorname{Tr}(f)(q) = \) \(82q \) \(\mathstrut -\mathstrut 82q^{2} \) \(\mathstrut -\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 82q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 13q^{6} \) \(\mathstrut -\mathstrut 37q^{7} \) \(\mathstrut -\mathstrut 82q^{8} \) \(\mathstrut +\mathstrut 91q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(82q \) \(\mathstrut -\mathstrut 82q^{2} \) \(\mathstrut -\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 82q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 13q^{6} \) \(\mathstrut -\mathstrut 37q^{7} \) \(\mathstrut -\mathstrut 82q^{8} \) \(\mathstrut +\mathstrut 91q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut -\mathstrut 16q^{11} \) \(\mathstrut -\mathstrut 13q^{12} \) \(\mathstrut -\mathstrut 42q^{13} \) \(\mathstrut +\mathstrut 37q^{14} \) \(\mathstrut -\mathstrut 9q^{15} \) \(\mathstrut +\mathstrut 82q^{16} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 91q^{18} \) \(\mathstrut -\mathstrut 42q^{19} \) \(\mathstrut +\mathstrut 3q^{20} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut 16q^{22} \) \(\mathstrut -\mathstrut 6q^{23} \) \(\mathstrut +\mathstrut 13q^{24} \) \(\mathstrut +\mathstrut 53q^{25} \) \(\mathstrut +\mathstrut 42q^{26} \) \(\mathstrut -\mathstrut 49q^{27} \) \(\mathstrut -\mathstrut 37q^{28} \) \(\mathstrut +\mathstrut 15q^{29} \) \(\mathstrut +\mathstrut 9q^{30} \) \(\mathstrut -\mathstrut 40q^{31} \) \(\mathstrut -\mathstrut 82q^{32} \) \(\mathstrut -\mathstrut 37q^{33} \) \(\mathstrut -\mathstrut 3q^{34} \) \(\mathstrut -\mathstrut 42q^{35} \) \(\mathstrut +\mathstrut 91q^{36} \) \(\mathstrut -\mathstrut 72q^{37} \) \(\mathstrut +\mathstrut 42q^{38} \) \(\mathstrut -\mathstrut 14q^{39} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut +\mathstrut q^{42} \) \(\mathstrut -\mathstrut 93q^{43} \) \(\mathstrut -\mathstrut 16q^{44} \) \(\mathstrut -\mathstrut 11q^{45} \) \(\mathstrut +\mathstrut 6q^{46} \) \(\mathstrut +\mathstrut 7q^{47} \) \(\mathstrut -\mathstrut 13q^{48} \) \(\mathstrut +\mathstrut 61q^{49} \) \(\mathstrut -\mathstrut 53q^{50} \) \(\mathstrut -\mathstrut 70q^{51} \) \(\mathstrut -\mathstrut 42q^{52} \) \(\mathstrut +\mathstrut 18q^{53} \) \(\mathstrut +\mathstrut 49q^{54} \) \(\mathstrut -\mathstrut 62q^{55} \) \(\mathstrut +\mathstrut 37q^{56} \) \(\mathstrut -\mathstrut 51q^{57} \) \(\mathstrut -\mathstrut 15q^{58} \) \(\mathstrut -\mathstrut 47q^{59} \) \(\mathstrut -\mathstrut 9q^{60} \) \(\mathstrut -\mathstrut 14q^{61} \) \(\mathstrut +\mathstrut 40q^{62} \) \(\mathstrut -\mathstrut 100q^{63} \) \(\mathstrut +\mathstrut 82q^{64} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut +\mathstrut 37q^{66} \) \(\mathstrut -\mathstrut 150q^{67} \) \(\mathstrut +\mathstrut 3q^{68} \) \(\mathstrut +\mathstrut 31q^{69} \) \(\mathstrut +\mathstrut 42q^{70} \) \(\mathstrut +\mathstrut 7q^{71} \) \(\mathstrut -\mathstrut 91q^{72} \) \(\mathstrut -\mathstrut 78q^{73} \) \(\mathstrut +\mathstrut 72q^{74} \) \(\mathstrut -\mathstrut 49q^{75} \) \(\mathstrut -\mathstrut 42q^{76} \) \(\mathstrut +\mathstrut 29q^{77} \) \(\mathstrut +\mathstrut 14q^{78} \) \(\mathstrut -\mathstrut 59q^{79} \) \(\mathstrut +\mathstrut 3q^{80} \) \(\mathstrut +\mathstrut 122q^{81} \) \(\mathstrut -\mathstrut 8q^{82} \) \(\mathstrut -\mathstrut 52q^{83} \) \(\mathstrut -\mathstrut q^{84} \) \(\mathstrut -\mathstrut 108q^{85} \) \(\mathstrut +\mathstrut 93q^{86} \) \(\mathstrut -\mathstrut 49q^{87} \) \(\mathstrut +\mathstrut 16q^{88} \) \(\mathstrut +\mathstrut 38q^{89} \) \(\mathstrut +\mathstrut 11q^{90} \) \(\mathstrut -\mathstrut 69q^{91} \) \(\mathstrut -\mathstrut 6q^{92} \) \(\mathstrut -\mathstrut 63q^{93} \) \(\mathstrut -\mathstrut 7q^{94} \) \(\mathstrut +\mathstrut 5q^{95} \) \(\mathstrut +\mathstrut 13q^{96} \) \(\mathstrut -\mathstrut 74q^{97} \) \(\mathstrut -\mathstrut 61q^{98} \) \(\mathstrut -\mathstrut 89q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.40861 1.00000 −1.73046 3.40861 −5.08691 −1.00000 8.61865 1.73046
1.2 −1.00000 −3.35064 1.00000 3.93349 3.35064 2.37767 −1.00000 8.22677 −3.93349
1.3 −1.00000 −3.32799 1.00000 −3.39172 3.32799 1.47014 −1.00000 8.07554 3.39172
1.4 −1.00000 −3.26666 1.00000 1.36820 3.26666 −2.10777 −1.00000 7.67104 −1.36820
1.5 −1.00000 −3.16571 1.00000 −2.70471 3.16571 0.275737 −1.00000 7.02173 2.70471
1.6 −1.00000 −3.05101 1.00000 −2.07881 3.05101 −1.72566 −1.00000 6.30864 2.07881
1.7 −1.00000 −2.97766 1.00000 2.45317 2.97766 −2.95290 −1.00000 5.86643 −2.45317
1.8 −1.00000 −2.93784 1.00000 2.35767 2.93784 −3.95734 −1.00000 5.63089 −2.35767
1.9 −1.00000 −2.89017 1.00000 1.86034 2.89017 −1.49646 −1.00000 5.35311 −1.86034
1.10 −1.00000 −2.84903 1.00000 −1.49595 2.84903 −3.40125 −1.00000 5.11699 1.49595
1.11 −1.00000 −2.83011 1.00000 −3.27582 2.83011 1.55181 −1.00000 5.00950 3.27582
1.12 −1.00000 −2.79380 1.00000 −0.886780 2.79380 2.62201 −1.00000 4.80532 0.886780
1.13 −1.00000 −2.65806 1.00000 4.11943 2.65806 −4.31253 −1.00000 4.06529 −4.11943
1.14 −1.00000 −2.65492 1.00000 1.59716 2.65492 4.41180 −1.00000 4.04858 −1.59716
1.15 −1.00000 −2.63165 1.00000 1.97422 2.63165 0.692516 −1.00000 3.92558 −1.97422
1.16 −1.00000 −2.37311 1.00000 2.96553 2.37311 1.19055 −1.00000 2.63165 −2.96553
1.17 −1.00000 −2.27252 1.00000 1.18592 2.27252 3.11806 −1.00000 2.16434 −1.18592
1.18 −1.00000 −2.21914 1.00000 −2.28006 2.21914 5.09322 −1.00000 1.92459 2.28006
1.19 −1.00000 −1.99235 1.00000 −2.03322 1.99235 −0.500887 −1.00000 0.969445 2.03322
1.20 −1.00000 −1.88117 1.00000 0.290138 1.88117 2.48398 −1.00000 0.538819 −0.290138
See all 82 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.82
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(4021\) \(1\)