Properties

Label 8045.2.a.e
Level 8045
Weight 2
Character orbit 8045.a
Self dual Yes
Analytic conductor 64.240
Analytic rank 0
Dimension 142
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: \(142\)
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.

\(\operatorname{Tr}(f)(q) = \) \(142q \) \(\mathstrut +\mathstrut 21q^{2} \) \(\mathstrut +\mathstrut 33q^{3} \) \(\mathstrut +\mathstrut 157q^{4} \) \(\mathstrut +\mathstrut 142q^{5} \) \(\mathstrut +\mathstrut 15q^{6} \) \(\mathstrut +\mathstrut 63q^{7} \) \(\mathstrut +\mathstrut 60q^{8} \) \(\mathstrut +\mathstrut 157q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(142q \) \(\mathstrut +\mathstrut 21q^{2} \) \(\mathstrut +\mathstrut 33q^{3} \) \(\mathstrut +\mathstrut 157q^{4} \) \(\mathstrut +\mathstrut 142q^{5} \) \(\mathstrut +\mathstrut 15q^{6} \) \(\mathstrut +\mathstrut 63q^{7} \) \(\mathstrut +\mathstrut 60q^{8} \) \(\mathstrut +\mathstrut 157q^{9} \) \(\mathstrut +\mathstrut 21q^{10} \) \(\mathstrut +\mathstrut 36q^{11} \) \(\mathstrut +\mathstrut 55q^{12} \) \(\mathstrut +\mathstrut 57q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 33q^{15} \) \(\mathstrut +\mathstrut 179q^{16} \) \(\mathstrut +\mathstrut 55q^{17} \) \(\mathstrut +\mathstrut 65q^{18} \) \(\mathstrut +\mathstrut 130q^{19} \) \(\mathstrut +\mathstrut 157q^{20} \) \(\mathstrut +\mathstrut 28q^{21} \) \(\mathstrut +\mathstrut 30q^{22} \) \(\mathstrut +\mathstrut 117q^{23} \) \(\mathstrut +\mathstrut 21q^{24} \) \(\mathstrut +\mathstrut 142q^{25} \) \(\mathstrut +\mathstrut 21q^{26} \) \(\mathstrut +\mathstrut 120q^{27} \) \(\mathstrut +\mathstrut 135q^{28} \) \(\mathstrut +\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 15q^{30} \) \(\mathstrut +\mathstrut 74q^{31} \) \(\mathstrut +\mathstrut 126q^{32} \) \(\mathstrut +\mathstrut 55q^{33} \) \(\mathstrut +\mathstrut 35q^{34} \) \(\mathstrut +\mathstrut 63q^{35} \) \(\mathstrut +\mathstrut 186q^{36} \) \(\mathstrut +\mathstrut 75q^{37} \) \(\mathstrut +\mathstrut 65q^{38} \) \(\mathstrut +\mathstrut 23q^{39} \) \(\mathstrut +\mathstrut 60q^{40} \) \(\mathstrut +\mathstrut 22q^{41} \) \(\mathstrut +\mathstrut 10q^{42} \) \(\mathstrut +\mathstrut 190q^{43} \) \(\mathstrut +\mathstrut 22q^{44} \) \(\mathstrut +\mathstrut 157q^{45} \) \(\mathstrut +\mathstrut 56q^{46} \) \(\mathstrut +\mathstrut 102q^{47} \) \(\mathstrut +\mathstrut 78q^{48} \) \(\mathstrut +\mathstrut 197q^{49} \) \(\mathstrut +\mathstrut 21q^{50} \) \(\mathstrut +\mathstrut 30q^{51} \) \(\mathstrut +\mathstrut 120q^{52} \) \(\mathstrut +\mathstrut 56q^{53} \) \(\mathstrut -\mathstrut 6q^{54} \) \(\mathstrut +\mathstrut 36q^{55} \) \(\mathstrut +\mathstrut 3q^{56} \) \(\mathstrut +\mathstrut 68q^{57} \) \(\mathstrut +\mathstrut 31q^{58} \) \(\mathstrut +\mathstrut 55q^{59} \) \(\mathstrut +\mathstrut 55q^{60} \) \(\mathstrut +\mathstrut 90q^{61} \) \(\mathstrut +\mathstrut 68q^{62} \) \(\mathstrut +\mathstrut 167q^{63} \) \(\mathstrut +\mathstrut 180q^{64} \) \(\mathstrut +\mathstrut 57q^{65} \) \(\mathstrut +\mathstrut 17q^{66} \) \(\mathstrut +\mathstrut 151q^{67} \) \(\mathstrut +\mathstrut 119q^{68} \) \(\mathstrut +\mathstrut 21q^{69} \) \(\mathstrut +\mathstrut 2q^{70} \) \(\mathstrut +\mathstrut 4q^{71} \) \(\mathstrut +\mathstrut 130q^{72} \) \(\mathstrut +\mathstrut 143q^{73} \) \(\mathstrut -\mathstrut 46q^{74} \) \(\mathstrut +\mathstrut 33q^{75} \) \(\mathstrut +\mathstrut 213q^{76} \) \(\mathstrut +\mathstrut 75q^{77} \) \(\mathstrut -\mathstrut 24q^{78} \) \(\mathstrut +\mathstrut 47q^{79} \) \(\mathstrut +\mathstrut 179q^{80} \) \(\mathstrut +\mathstrut 150q^{81} \) \(\mathstrut +\mathstrut 69q^{82} \) \(\mathstrut +\mathstrut 201q^{83} \) \(\mathstrut -\mathstrut 31q^{84} \) \(\mathstrut +\mathstrut 55q^{85} \) \(\mathstrut -\mathstrut 4q^{86} \) \(\mathstrut +\mathstrut 153q^{87} \) \(\mathstrut +\mathstrut 37q^{88} \) \(\mathstrut +\mathstrut 25q^{89} \) \(\mathstrut +\mathstrut 65q^{90} \) \(\mathstrut +\mathstrut 132q^{91} \) \(\mathstrut +\mathstrut 194q^{92} \) \(\mathstrut +\mathstrut 52q^{93} \) \(\mathstrut +\mathstrut 18q^{94} \) \(\mathstrut +\mathstrut 130q^{95} \) \(\mathstrut +\mathstrut 13q^{96} \) \(\mathstrut +\mathstrut 80q^{97} \) \(\mathstrut +\mathstrut 58q^{98} \) \(\mathstrut +\mathstrut 103q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.77343 2.65510 5.69191 1.00000 −7.36373 0.492807 −10.2393 4.04956 −2.77343
1.2 −2.68924 −0.0494506 5.23199 1.00000 0.132984 2.47489 −8.69158 −2.99755 −2.68924
1.3 −2.64935 2.86876 5.01907 1.00000 −7.60037 4.90549 −7.99858 5.22980 −2.64935
1.4 −2.64700 −1.25206 5.00659 1.00000 3.31419 4.95452 −7.95843 −1.43235 −2.64700
1.5 −2.63029 −2.20415 4.91841 1.00000 5.79754 3.10194 −7.67624 1.85826 −2.63029
1.6 −2.60196 1.36677 4.77022 1.00000 −3.55629 2.24153 −7.20801 −1.13194 −2.60196
1.7 −2.59468 −0.861703 4.73239 1.00000 2.23585 −3.11370 −7.08968 −2.25747 −2.59468
1.8 −2.58612 1.58438 4.68803 1.00000 −4.09739 −2.71567 −6.95157 −0.489749 −2.58612
1.9 −2.51445 −1.12018 4.32245 1.00000 2.81662 0.310792 −5.83967 −1.74521 −2.51445
1.10 −2.48735 −2.65295 4.18689 1.00000 6.59881 2.41015 −5.43957 4.03816 −2.48735
1.11 −2.46500 2.31843 4.07621 1.00000 −5.71491 −0.590235 −5.11785 2.37510 −2.46500
1.12 −2.39887 −2.11291 3.75456 1.00000 5.06858 1.17746 −4.20895 1.46438 −2.39887
1.13 −2.28247 0.951173 3.20967 1.00000 −2.17102 −3.35163 −2.76104 −2.09527 −2.28247
1.14 −2.20465 −2.96481 2.86047 1.00000 6.53637 0.166097 −1.89703 5.79013 −2.20465
1.15 −2.19407 0.184631 2.81393 1.00000 −0.405094 1.35112 −1.78582 −2.96591 −2.19407
1.16 −2.18359 0.444286 2.76808 1.00000 −0.970139 −0.684124 −1.67716 −2.80261 −2.18359
1.17 −2.16169 −1.95272 2.67292 1.00000 4.22118 −2.55513 −1.45466 0.813116 −2.16169
1.18 −2.15729 −0.777397 2.65392 1.00000 1.67707 3.51600 −1.41069 −2.39565 −2.15729
1.19 −2.15373 3.04079 2.63855 1.00000 −6.54904 −4.47994 −1.37525 6.24641 −2.15373
1.20 −2.15214 2.27833 2.63171 1.00000 −4.90329 4.73674 −1.35954 2.19079 −2.15214
See next 80 embeddings (of 142 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.142
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 \(T_{2}^{142} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\).