Properties

Label 8043.2.a.n
Level 8043
Weight 2
Character orbit 8043.a
Self dual Yes
Analytic conductor 64.224
Analytic rank 1
Dimension 40
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: \(1\)
Dimension: \(40\)
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.

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

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 −2.75032 1.00000 5.56428 0.854023 −2.75032 1.00000 −9.80291 1.00000 −2.34884
1.2 −2.74138 1.00000 5.51515 −1.10050 −2.74138 1.00000 −9.63635 1.00000 3.01688
1.3 −2.64192 1.00000 4.97976 −3.51496 −2.64192 1.00000 −7.87228 1.00000 9.28626
1.4 −2.58104 1.00000 4.66178 2.25631 −2.58104 1.00000 −6.87016 1.00000 −5.82363
1.5 −2.32186 1.00000 3.39104 0.609544 −2.32186 1.00000 −3.22980 1.00000 −1.41528
1.6 −2.16260 1.00000 2.67684 −4.31110 −2.16260 1.00000 −1.46374 1.00000 9.32319
1.7 −2.08373 1.00000 2.34192 −2.93682 −2.08373 1.00000 −0.712460 1.00000 6.11952
1.8 −1.97104 1.00000 1.88501 1.28397 −1.97104 1.00000 0.226654 1.00000 −2.53076
1.9 −1.85383 1.00000 1.43669 −2.52734 −1.85383 1.00000 1.04428 1.00000 4.68526
1.10 −1.83203 1.00000 1.35635 1.36968 −1.83203 1.00000 1.17919 1.00000 −2.50930
1.11 −1.65595 1.00000 0.742162 −4.41540 −1.65595 1.00000 2.08291 1.00000 7.31167
1.12 −1.59287 1.00000 0.537230 2.23699 −1.59287 1.00000 2.33000 1.00000 −3.56323
1.13 −1.43635 1.00000 0.0630943 3.31438 −1.43635 1.00000 2.78207 1.00000 −4.76060
1.14 −1.36925 1.00000 −0.125168 0.412992 −1.36925 1.00000 2.90988 1.00000 −0.565488
1.15 −1.12599 1.00000 −0.732150 2.85700 −1.12599 1.00000 3.07637 1.00000 −3.21695
1.16 −1.09091 1.00000 −0.809919 −1.12988 −1.09091 1.00000 3.06536 1.00000 1.23260
1.17 −0.765993 1.00000 −1.41325 −3.34403 −0.765993 1.00000 2.61453 1.00000 2.56151
1.18 −0.717010 1.00000 −1.48590 1.86418 −0.717010 1.00000 2.49942 1.00000 −1.33664
1.19 −0.619181 1.00000 −1.61661 −0.937229 −0.619181 1.00000 2.23934 1.00000 0.580315
1.20 −0.506041 1.00000 −1.74392 −0.0764488 −0.506041 1.00000 1.89458 1.00000 0.0386862
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\(T_{2}^{40} + \cdots\)
\(T_{5}^{40} + \cdots\)
\(T_{11}^{40} + \cdots\)