Properties

Label 8043.2.a.s
Level 8043
Weight 2
Character orbit 8043.a
Self dual Yes
Analytic conductor 64.224
Analytic rank 0
Dimension 50
CM No

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Newspace parameters

Level: \( N \) = \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8043.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2236783457\)
Analytic rank: \(0\)
Dimension: \(50\)
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.

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

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 −2.82109 −1.00000 5.95855 −4.23433 2.82109 −1.00000 −11.1674 1.00000 11.9454
1.2 −2.63224 −1.00000 4.92870 3.86633 2.63224 −1.00000 −7.70903 1.00000 −10.1771
1.3 −2.63191 −1.00000 4.92693 0.271396 2.63191 −1.00000 −7.70340 1.00000 −0.714288
1.4 −2.62429 −1.00000 4.88687 −1.77661 2.62429 −1.00000 −7.57598 1.00000 4.66233
1.5 −2.39697 −1.00000 3.74548 1.84555 2.39697 −1.00000 −4.18388 1.00000 −4.42374
1.6 −2.33599 −1.00000 3.45686 4.00725 2.33599 −1.00000 −3.40322 1.00000 −9.36091
1.7 −2.33165 −1.00000 3.43661 −1.59780 2.33165 −1.00000 −3.34969 1.00000 3.72551
1.8 −2.30033 −1.00000 3.29153 −3.56175 2.30033 −1.00000 −2.97095 1.00000 8.19322
1.9 −2.16078 −1.00000 2.66897 −0.263100 2.16078 −1.00000 −1.44551 1.00000 0.568501
1.10 −1.95593 −1.00000 1.82566 0.355087 1.95593 −1.00000 0.340990 1.00000 −0.694525
1.11 −1.84012 −1.00000 1.38605 −2.07033 1.84012 −1.00000 1.12975 1.00000 3.80966
1.12 −1.76145 −1.00000 1.10269 2.11729 1.76145 −1.00000 1.58056 1.00000 −3.72950
1.13 −1.46962 −1.00000 0.159775 3.40283 1.46962 −1.00000 2.70443 1.00000 −5.00085
1.14 −1.46567 −1.00000 0.148201 0.335135 1.46567 −1.00000 2.71413 1.00000 −0.491199
1.15 −1.19941 −1.00000 −0.561407 −0.150113 1.19941 −1.00000 3.07219 1.00000 0.180048
1.16 −1.18848 −1.00000 −0.587510 −4.24260 1.18848 −1.00000 3.07521 1.00000 5.04226
1.17 −1.18460 −1.00000 −0.596731 3.79711 1.18460 −1.00000 3.07608 1.00000 −4.49805
1.18 −1.15299 −1.00000 −0.670610 0.489993 1.15299 −1.00000 3.07919 1.00000 −0.564958
1.19 −1.12728 −1.00000 −0.729231 −0.145556 1.12728 −1.00000 3.07662 1.00000 0.164083
1.20 −0.719015 −1.00000 −1.48302 −1.74243 0.719015 −1.00000 2.50434 1.00000 1.25284
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.50
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(383\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\(T_{2}^{50} + \cdots\)
\(T_{5}^{50} - \cdots\)
\(T_{11}^{50} + \cdots\)