Properties

Label 8045.2.a.b
Level 8045
Weight 2
Character orbit 8045.a
Self dual Yes
Analytic conductor 64.240
Analytic rank 1
Dimension 126
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: \(1\)
Dimension: \(126\)
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.

\(\operatorname{Tr}(f)(q) = \) \(126q \) \(\mathstrut +\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 109q^{4} \) \(\mathstrut -\mathstrut 126q^{5} \) \(\mathstrut -\mathstrut 21q^{6} \) \(\mathstrut -\mathstrut 23q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 109q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(126q \) \(\mathstrut +\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 109q^{4} \) \(\mathstrut -\mathstrut 126q^{5} \) \(\mathstrut -\mathstrut 21q^{6} \) \(\mathstrut -\mathstrut 23q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 109q^{9} \) \(\mathstrut -\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 44q^{11} \) \(\mathstrut -\mathstrut 11q^{12} \) \(\mathstrut -\mathstrut 35q^{13} \) \(\mathstrut -\mathstrut 14q^{14} \) \(\mathstrut +\mathstrut 9q^{15} \) \(\mathstrut +\mathstrut 75q^{16} \) \(\mathstrut +\mathstrut 11q^{17} \) \(\mathstrut -\mathstrut 15q^{18} \) \(\mathstrut -\mathstrut 130q^{19} \) \(\mathstrut -\mathstrut 109q^{20} \) \(\mathstrut -\mathstrut 44q^{21} \) \(\mathstrut -\mathstrut 14q^{22} \) \(\mathstrut +\mathstrut 75q^{23} \) \(\mathstrut -\mathstrut 63q^{24} \) \(\mathstrut +\mathstrut 126q^{25} \) \(\mathstrut -\mathstrut 43q^{26} \) \(\mathstrut -\mathstrut 42q^{27} \) \(\mathstrut -\mathstrut 77q^{28} \) \(\mathstrut -\mathstrut 24q^{29} \) \(\mathstrut +\mathstrut 21q^{30} \) \(\mathstrut -\mathstrut 78q^{31} \) \(\mathstrut +\mathstrut 24q^{32} \) \(\mathstrut -\mathstrut 29q^{33} \) \(\mathstrut -\mathstrut 57q^{34} \) \(\mathstrut +\mathstrut 23q^{35} \) \(\mathstrut +\mathstrut 50q^{36} \) \(\mathstrut -\mathstrut 31q^{37} \) \(\mathstrut -\mathstrut 3q^{38} \) \(\mathstrut -\mathstrut 57q^{39} \) \(\mathstrut -\mathstrut 12q^{40} \) \(\mathstrut -\mathstrut 38q^{41} \) \(\mathstrut -\mathstrut 10q^{42} \) \(\mathstrut -\mathstrut 100q^{43} \) \(\mathstrut -\mathstrut 90q^{44} \) \(\mathstrut -\mathstrut 109q^{45} \) \(\mathstrut -\mathstrut 96q^{46} \) \(\mathstrut +\mathstrut 12q^{47} \) \(\mathstrut -\mathstrut 22q^{48} \) \(\mathstrut +\mathstrut 65q^{49} \) \(\mathstrut +\mathstrut 5q^{50} \) \(\mathstrut -\mathstrut 74q^{51} \) \(\mathstrut -\mathstrut 112q^{52} \) \(\mathstrut +\mathstrut 20q^{53} \) \(\mathstrut -\mathstrut 90q^{54} \) \(\mathstrut +\mathstrut 44q^{55} \) \(\mathstrut -\mathstrut 57q^{56} \) \(\mathstrut +\mathstrut 6q^{57} \) \(\mathstrut -\mathstrut 35q^{58} \) \(\mathstrut -\mathstrut 97q^{59} \) \(\mathstrut +\mathstrut 11q^{60} \) \(\mathstrut -\mathstrut 102q^{61} \) \(\mathstrut -\mathstrut 16q^{62} \) \(\mathstrut -\mathstrut 15q^{63} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut +\mathstrut 35q^{65} \) \(\mathstrut -\mathstrut 83q^{66} \) \(\mathstrut -\mathstrut 121q^{67} \) \(\mathstrut +\mathstrut 41q^{68} \) \(\mathstrut -\mathstrut 71q^{69} \) \(\mathstrut +\mathstrut 14q^{70} \) \(\mathstrut -\mathstrut 32q^{71} \) \(\mathstrut -\mathstrut 32q^{72} \) \(\mathstrut -\mathstrut 85q^{73} \) \(\mathstrut -\mathstrut 42q^{74} \) \(\mathstrut -\mathstrut 9q^{75} \) \(\mathstrut -\mathstrut 275q^{76} \) \(\mathstrut +\mathstrut 13q^{77} \) \(\mathstrut +\mathstrut 10q^{78} \) \(\mathstrut -\mathstrut 97q^{79} \) \(\mathstrut -\mathstrut 75q^{80} \) \(\mathstrut +\mathstrut 86q^{81} \) \(\mathstrut -\mathstrut 55q^{82} \) \(\mathstrut -\mathstrut 73q^{83} \) \(\mathstrut -\mathstrut 111q^{84} \) \(\mathstrut -\mathstrut 11q^{85} \) \(\mathstrut -\mathstrut 56q^{86} \) \(\mathstrut -\mathstrut q^{87} \) \(\mathstrut -\mathstrut 37q^{88} \) \(\mathstrut -\mathstrut 67q^{89} \) \(\mathstrut +\mathstrut 15q^{90} \) \(\mathstrut -\mathstrut 180q^{91} \) \(\mathstrut +\mathstrut 98q^{92} \) \(\mathstrut -\mathstrut 44q^{93} \) \(\mathstrut -\mathstrut 86q^{94} \) \(\mathstrut +\mathstrut 130q^{95} \) \(\mathstrut -\mathstrut 179q^{96} \) \(\mathstrut -\mathstrut 50q^{97} \) \(\mathstrut +\mathstrut 18q^{98} \) \(\mathstrut -\mathstrut 217q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.77015 2.48053 5.67370 −1.00000 −6.87144 2.38067 −10.1767 3.15305 2.77015
1.2 −2.68479 −1.35137 5.20811 −1.00000 3.62815 −0.768058 −8.61310 −1.17380 2.68479
1.3 −2.59235 −0.476573 4.72028 −1.00000 1.23544 −4.74079 −7.05191 −2.77288 2.59235
1.4 −2.58262 2.30051 4.66995 −1.00000 −5.94134 −3.25366 −6.89548 2.29232 2.58262
1.5 −2.57758 3.01279 4.64389 −1.00000 −7.76569 −3.99315 −6.81483 6.07689 2.57758
1.6 −2.55817 −0.0316431 4.54425 −1.00000 0.0809484 0.209260 −6.50863 −2.99900 2.55817
1.7 −2.52674 −0.799891 4.38441 −1.00000 2.02112 −0.269880 −6.02477 −2.36017 2.52674
1.8 −2.51572 −0.0214177 4.32884 −1.00000 0.0538809 1.26771 −5.85870 −2.99954 2.51572
1.9 −2.44573 0.659690 3.98158 −1.00000 −1.61342 −0.125305 −4.84641 −2.56481 2.44573
1.10 −2.44115 2.21626 3.95922 −1.00000 −5.41023 1.20932 −4.78275 1.91181 2.44115
1.11 −2.40759 −3.16484 3.79648 −1.00000 7.61962 −1.46420 −4.32518 7.01619 2.40759
1.12 −2.37056 −2.51302 3.61953 −1.00000 5.95725 3.42592 −3.83919 3.31526 2.37056
1.13 −2.28169 −1.33238 3.20613 −1.00000 3.04008 −1.04082 −2.75203 −1.22477 2.28169
1.14 −2.25161 −2.78948 3.06974 −1.00000 6.28081 1.00140 −2.40864 4.78119 2.25161
1.15 −2.22019 1.33183 2.92925 −1.00000 −2.95691 2.14660 −2.06312 −1.22624 2.22019
1.16 −2.20326 −1.91505 2.85434 −1.00000 4.21934 2.75796 −1.88232 0.667415 2.20326
1.17 −2.11951 0.398701 2.49230 −1.00000 −0.845049 −4.30325 −1.04344 −2.84104 2.11951
1.18 −2.05998 0.444613 2.24351 −1.00000 −0.915893 4.66376 −0.501618 −2.80232 2.05998
1.19 −2.03743 2.30287 2.15111 −1.00000 −4.69193 −3.80447 −0.307877 2.30320 2.03743
1.20 −2.02597 −2.39891 2.10454 −1.00000 4.86012 −3.38637 −0.211794 2.75479 2.02597
See next 80 embeddings (of 126 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.126
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 \(T_{2}^{126} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\).