Properties

Label 6005.2.a.d
Level 6005
Weight 2
Character orbit 6005.a
Self dual Yes
Analytic conductor 47.950
Analytic rank 1
Dimension 83
CM No

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Newspace parameters

Level: \( N \) = \( 6005 = 5 \cdot 1201 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9501664138\)
Analytic rank: \(1\)
Dimension: \(83\)
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.

\(\operatorname{Tr}(f)(q) = \) \(83q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 61q^{4} \) \(\mathstrut -\mathstrut 83q^{5} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 61q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(83q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 61q^{4} \) \(\mathstrut -\mathstrut 83q^{5} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 61q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut -\mathstrut 26q^{11} \) \(\mathstrut -\mathstrut 12q^{12} \) \(\mathstrut -\mathstrut 15q^{13} \) \(\mathstrut -\mathstrut 21q^{14} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 5q^{16} \) \(\mathstrut +\mathstrut 8q^{17} \) \(\mathstrut -\mathstrut 12q^{18} \) \(\mathstrut -\mathstrut 79q^{19} \) \(\mathstrut -\mathstrut 61q^{20} \) \(\mathstrut -\mathstrut 34q^{21} \) \(\mathstrut -\mathstrut 25q^{22} \) \(\mathstrut +\mathstrut 31q^{23} \) \(\mathstrut -\mathstrut 42q^{24} \) \(\mathstrut +\mathstrut 83q^{25} \) \(\mathstrut -\mathstrut 13q^{26} \) \(\mathstrut -\mathstrut 25q^{27} \) \(\mathstrut -\mathstrut 16q^{28} \) \(\mathstrut -\mathstrut 16q^{29} \) \(\mathstrut +\mathstrut 6q^{30} \) \(\mathstrut -\mathstrut 40q^{31} \) \(\mathstrut +\mathstrut 15q^{32} \) \(\mathstrut -\mathstrut 33q^{33} \) \(\mathstrut -\mathstrut 54q^{34} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 11q^{36} \) \(\mathstrut -\mathstrut 45q^{37} \) \(\mathstrut +\mathstrut 10q^{38} \) \(\mathstrut -\mathstrut 54q^{39} \) \(\mathstrut +\mathstrut 3q^{40} \) \(\mathstrut -\mathstrut 27q^{41} \) \(\mathstrut -\mathstrut 28q^{42} \) \(\mathstrut -\mathstrut 101q^{43} \) \(\mathstrut -\mathstrut 51q^{44} \) \(\mathstrut -\mathstrut 61q^{45} \) \(\mathstrut -\mathstrut 46q^{46} \) \(\mathstrut +\mathstrut 71q^{47} \) \(\mathstrut -\mathstrut 14q^{48} \) \(\mathstrut +\mathstrut 23q^{49} \) \(\mathstrut +\mathstrut q^{50} \) \(\mathstrut -\mathstrut 71q^{51} \) \(\mathstrut -\mathstrut 34q^{52} \) \(\mathstrut -\mathstrut 49q^{53} \) \(\mathstrut -\mathstrut 25q^{54} \) \(\mathstrut +\mathstrut 26q^{55} \) \(\mathstrut -\mathstrut 41q^{56} \) \(\mathstrut -\mathstrut 20q^{57} \) \(\mathstrut -\mathstrut 43q^{58} \) \(\mathstrut -\mathstrut 60q^{59} \) \(\mathstrut +\mathstrut 12q^{60} \) \(\mathstrut -\mathstrut 38q^{61} \) \(\mathstrut -\mathstrut 2q^{62} \) \(\mathstrut +\mathstrut 36q^{63} \) \(\mathstrut -\mathstrut 113q^{64} \) \(\mathstrut +\mathstrut 15q^{65} \) \(\mathstrut -\mathstrut 42q^{66} \) \(\mathstrut -\mathstrut 164q^{67} \) \(\mathstrut +\mathstrut 10q^{68} \) \(\mathstrut -\mathstrut 93q^{69} \) \(\mathstrut +\mathstrut 21q^{70} \) \(\mathstrut -\mathstrut 78q^{71} \) \(\mathstrut +\mathstrut q^{72} \) \(\mathstrut -\mathstrut 18q^{73} \) \(\mathstrut -\mathstrut 23q^{74} \) \(\mathstrut -\mathstrut 4q^{75} \) \(\mathstrut -\mathstrut 112q^{76} \) \(\mathstrut -\mathstrut 35q^{77} \) \(\mathstrut -\mathstrut 44q^{78} \) \(\mathstrut -\mathstrut 124q^{79} \) \(\mathstrut -\mathstrut 5q^{80} \) \(\mathstrut -\mathstrut 45q^{81} \) \(\mathstrut -\mathstrut 34q^{82} \) \(\mathstrut +\mathstrut 5q^{83} \) \(\mathstrut -\mathstrut 60q^{84} \) \(\mathstrut -\mathstrut 8q^{85} \) \(\mathstrut -\mathstrut 25q^{86} \) \(\mathstrut +\mathstrut 12q^{87} \) \(\mathstrut -\mathstrut 149q^{88} \) \(\mathstrut -\mathstrut 44q^{89} \) \(\mathstrut +\mathstrut 12q^{90} \) \(\mathstrut -\mathstrut 192q^{91} \) \(\mathstrut +\mathstrut 35q^{92} \) \(\mathstrut -\mathstrut 13q^{93} \) \(\mathstrut -\mathstrut 32q^{94} \) \(\mathstrut +\mathstrut 79q^{95} \) \(\mathstrut -\mathstrut 59q^{96} \) \(\mathstrut -\mathstrut 31q^{97} \) \(\mathstrut +\mathstrut 25q^{98} \) \(\mathstrut -\mathstrut 134q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.64292 0.725582 4.98500 −1.00000 −1.91765 −1.58856 −7.88911 −2.47353 2.64292
1.2 −2.62595 0.137720 4.89563 −1.00000 −0.361646 0.826916 −7.60380 −2.98103 2.62595
1.3 −2.46128 −2.46204 4.05791 −1.00000 6.05978 3.81143 −5.06509 3.06165 2.46128
1.4 −2.43937 1.48645 3.95052 −1.00000 −3.62599 1.16444 −4.75805 −0.790480 2.43937
1.5 −2.43196 −2.45366 3.91445 −1.00000 5.96721 −1.89640 −4.65586 3.02044 2.43196
1.6 −2.41534 2.54592 3.83384 −1.00000 −6.14926 −0.649970 −4.42935 3.48173 2.41534
1.7 −2.36067 1.42227 3.57275 −1.00000 −3.35751 −1.80334 −3.71273 −0.977139 2.36067
1.8 −2.35753 −1.70273 3.55796 −1.00000 4.01424 1.43471 −3.67294 −0.100706 2.35753
1.9 −2.30347 2.77237 3.30599 −1.00000 −6.38609 3.31837 −3.00832 4.68605 2.30347
1.10 −2.27002 2.83748 3.15300 −1.00000 −6.44114 −2.52643 −2.61733 5.05129 2.27002
1.11 −2.26906 −1.09200 3.14865 −1.00000 2.47782 −1.23548 −2.60637 −1.80753 2.26906
1.12 −2.05565 −1.34402 2.22569 −1.00000 2.76284 1.91384 −0.463948 −1.19360 2.05565
1.13 −1.98693 −0.154242 1.94789 −1.00000 0.306469 −4.64687 0.103547 −2.97621 1.98693
1.14 −1.97272 −2.98994 1.89163 −1.00000 5.89832 0.991432 0.213787 5.93974 1.97272
1.15 −1.95246 1.27997 1.81211 −1.00000 −2.49910 5.02234 0.366847 −1.36166 1.95246
1.16 −1.93139 −1.99415 1.73027 −1.00000 3.85147 0.495869 0.520963 0.976620 1.93139
1.17 −1.68633 −0.538907 0.843700 −1.00000 0.908773 −3.51769 1.94990 −2.70958 1.68633
1.18 −1.67860 −2.56726 0.817709 −1.00000 4.30941 −1.41200 1.98460 3.59082 1.67860
1.19 −1.67033 1.77820 0.789988 −1.00000 −2.97017 2.94591 2.02111 0.161995 1.67033
1.20 −1.55566 −0.523024 0.420073 −1.00000 0.813646 2.34977 2.45783 −2.72645 1.55566
See all 83 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.83
Significant digits:
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Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(1201\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{83} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).