Properties

Label 8045.2.a.d
Level 8045
Weight 2
Character orbit 8045.a
Self dual Yes
Analytic conductor 64.240
Analytic rank 0
Dimension 141
CM No

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Newspace parameters

Level: \( N \) = \( 8045 = 5 \cdot 1609 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8045.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2396484261\)
Analytic rank: \(0\)
Dimension: \(141\)
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.

\(\operatorname{Tr}(f)(q) = \) \(141q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 158q^{4} \) \(\mathstrut -\mathstrut 141q^{5} \) \(\mathstrut +\mathstrut 23q^{6} \) \(\mathstrut +\mathstrut 29q^{7} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 160q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(141q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 158q^{4} \) \(\mathstrut -\mathstrut 141q^{5} \) \(\mathstrut +\mathstrut 23q^{6} \) \(\mathstrut +\mathstrut 29q^{7} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 160q^{9} \) \(\mathstrut +\mathstrut 8q^{10} \) \(\mathstrut +\mathstrut 32q^{11} \) \(\mathstrut +\mathstrut 17q^{12} \) \(\mathstrut +\mathstrut 35q^{13} \) \(\mathstrut +\mathstrut 18q^{14} \) \(\mathstrut -\mathstrut 11q^{15} \) \(\mathstrut +\mathstrut 188q^{16} \) \(\mathstrut -\mathstrut 13q^{17} \) \(\mathstrut -\mathstrut 16q^{18} \) \(\mathstrut +\mathstrut 152q^{19} \) \(\mathstrut -\mathstrut 158q^{20} \) \(\mathstrut +\mathstrut 40q^{21} \) \(\mathstrut +\mathstrut 14q^{22} \) \(\mathstrut -\mathstrut 77q^{23} \) \(\mathstrut +\mathstrut 69q^{24} \) \(\mathstrut +\mathstrut 141q^{25} \) \(\mathstrut +\mathstrut 27q^{26} \) \(\mathstrut +\mathstrut 38q^{27} \) \(\mathstrut +\mathstrut 67q^{28} \) \(\mathstrut +\mathstrut 22q^{29} \) \(\mathstrut -\mathstrut 23q^{30} \) \(\mathstrut +\mathstrut 86q^{31} \) \(\mathstrut -\mathstrut 65q^{32} \) \(\mathstrut +\mathstrut 51q^{33} \) \(\mathstrut +\mathstrut 79q^{34} \) \(\mathstrut -\mathstrut 29q^{35} \) \(\mathstrut +\mathstrut 191q^{36} \) \(\mathstrut +\mathstrut 45q^{37} \) \(\mathstrut -\mathstrut 9q^{38} \) \(\mathstrut +\mathstrut 55q^{39} \) \(\mathstrut +\mathstrut 21q^{40} \) \(\mathstrut +\mathstrut 36q^{41} \) \(\mathstrut +\mathstrut 6q^{42} \) \(\mathstrut +\mathstrut 132q^{43} \) \(\mathstrut +\mathstrut 74q^{44} \) \(\mathstrut -\mathstrut 160q^{45} \) \(\mathstrut +\mathstrut 72q^{46} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 22q^{48} \) \(\mathstrut +\mathstrut 212q^{49} \) \(\mathstrut -\mathstrut 8q^{50} \) \(\mathstrut +\mathstrut 82q^{51} \) \(\mathstrut +\mathstrut 106q^{52} \) \(\mathstrut -\mathstrut 28q^{53} \) \(\mathstrut +\mathstrut 86q^{54} \) \(\mathstrut -\mathstrut 32q^{55} \) \(\mathstrut +\mathstrut 27q^{56} \) \(\mathstrut +\mathstrut 10q^{57} \) \(\mathstrut +\mathstrut 23q^{58} \) \(\mathstrut +\mathstrut 71q^{59} \) \(\mathstrut -\mathstrut 17q^{60} \) \(\mathstrut +\mathstrut 116q^{61} \) \(\mathstrut -\mathstrut 36q^{62} \) \(\mathstrut +\mathstrut 45q^{63} \) \(\mathstrut +\mathstrut 237q^{64} \) \(\mathstrut -\mathstrut 35q^{65} \) \(\mathstrut +\mathstrut 69q^{66} \) \(\mathstrut +\mathstrut 99q^{67} \) \(\mathstrut -\mathstrut 7q^{68} \) \(\mathstrut +\mathstrut 45q^{69} \) \(\mathstrut -\mathstrut 18q^{70} \) \(\mathstrut +\mathstrut 34q^{71} \) \(\mathstrut -\mathstrut 53q^{72} \) \(\mathstrut +\mathstrut 125q^{73} \) \(\mathstrut +\mathstrut 50q^{74} \) \(\mathstrut +\mathstrut 11q^{75} \) \(\mathstrut +\mathstrut 271q^{76} \) \(\mathstrut -\mathstrut 31q^{77} \) \(\mathstrut +\mathstrut 2q^{78} \) \(\mathstrut +\mathstrut 101q^{79} \) \(\mathstrut -\mathstrut 188q^{80} \) \(\mathstrut +\mathstrut 221q^{81} \) \(\mathstrut +\mathstrut 67q^{82} \) \(\mathstrut +\mathstrut 67q^{83} \) \(\mathstrut +\mathstrut 141q^{84} \) \(\mathstrut +\mathstrut 13q^{85} \) \(\mathstrut +\mathstrut 48q^{86} \) \(\mathstrut -\mathstrut 21q^{87} \) \(\mathstrut +\mathstrut 71q^{88} \) \(\mathstrut +\mathstrut 79q^{89} \) \(\mathstrut +\mathstrut 16q^{90} \) \(\mathstrut +\mathstrut 228q^{91} \) \(\mathstrut -\mathstrut 198q^{92} \) \(\mathstrut -\mathstrut 12q^{93} \) \(\mathstrut +\mathstrut 114q^{94} \) \(\mathstrut -\mathstrut 152q^{95} \) \(\mathstrut +\mathstrut 129q^{96} \) \(\mathstrut +\mathstrut 98q^{97} \) \(\mathstrut -\mathstrut 31q^{98} \) \(\mathstrut +\mathstrut 195q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.79563 0.863485 5.81558 −1.00000 −2.41399 −1.88296 −10.6670 −2.25439 2.79563
1.2 −2.76152 0.640687 5.62598 −1.00000 −1.76927 4.78584 −10.0132 −2.58952 2.76152
1.3 −2.76023 −1.69328 5.61886 −1.00000 4.67383 −3.35455 −9.98888 −0.132818 2.76023
1.4 −2.75850 −2.99511 5.60931 −1.00000 8.26199 0.810789 −9.95629 5.97065 2.75850
1.5 −2.73622 3.29432 5.48693 −1.00000 −9.01399 1.99589 −9.54102 7.85252 2.73622
1.6 −2.73567 −0.941263 5.48388 −1.00000 2.57498 4.04862 −9.53074 −2.11402 2.73567
1.7 −2.69245 1.61209 5.24928 −1.00000 −4.34048 −1.07253 −8.74853 −0.401159 2.69245
1.8 −2.68753 −2.90926 5.22280 −1.00000 7.81872 −1.56079 −8.66137 5.46381 2.68753
1.9 −2.61030 −1.72585 4.81367 −1.00000 4.50499 −4.19514 −7.34453 −0.0214352 2.61030
1.10 −2.59840 0.965890 4.75167 −1.00000 −2.50977 −4.65256 −7.14995 −2.06706 2.59840
1.11 −2.53833 1.49333 4.44314 −1.00000 −3.79058 3.83094 −6.20150 −0.769952 2.53833
1.12 −2.51610 −3.34886 4.33077 −1.00000 8.42608 3.53380 −5.86445 8.21488 2.51610
1.13 −2.50554 −0.175735 4.27775 −1.00000 0.440311 3.32352 −5.70700 −2.96912 2.50554
1.14 −2.49958 −1.75258 4.24791 −1.00000 4.38073 1.57689 −5.61884 0.0715494 2.49958
1.15 −2.46599 −2.26246 4.08113 −1.00000 5.57921 −2.99607 −5.13205 2.11871 2.46599
1.16 −2.41054 1.33575 3.81069 −1.00000 −3.21988 −0.579711 −4.36473 −1.21576 2.41054
1.17 −2.37811 2.67843 3.65539 −1.00000 −6.36960 −2.27861 −3.93671 4.17400 2.37811
1.18 −2.33944 3.10966 3.47296 −1.00000 −7.27485 −2.57808 −3.44589 6.66998 2.33944
1.19 −2.32435 1.69451 3.40258 −1.00000 −3.93862 3.87404 −3.26009 −0.128648 2.32435
1.20 −2.29844 3.20612 3.28283 −1.00000 −7.36907 2.27898 −2.94851 7.27918 2.29844
See next 80 embeddings (of 141 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.141
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(1609\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{141} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\).