Properties

Label 6009.2.a.d
Level 6009
Weight 2
Character orbit 6009.a
Self dual Yes
Analytic conductor 47.982
Analytic rank 0
Dimension 93
CM No

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Newspace parameters

Level: \( N \) = \( 6009 = 3 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6009.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9821065746\)
Analytic rank: \(0\)
Dimension: \(93\)
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.

\(\operatorname{Tr}(f)(q) = \) \(93q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 93q^{3} \) \(\mathstrut +\mathstrut 114q^{4} \) \(\mathstrut -\mathstrut 20q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 28q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 93q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(93q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 93q^{3} \) \(\mathstrut +\mathstrut 114q^{4} \) \(\mathstrut -\mathstrut 20q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 28q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 93q^{9} \) \(\mathstrut +\mathstrut 19q^{10} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut -\mathstrut 114q^{12} \) \(\mathstrut +\mathstrut 20q^{13} \) \(\mathstrut +\mathstrut 13q^{14} \) \(\mathstrut +\mathstrut 20q^{15} \) \(\mathstrut +\mathstrut 148q^{16} \) \(\mathstrut -\mathstrut 43q^{17} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 50q^{19} \) \(\mathstrut -\mathstrut 31q^{20} \) \(\mathstrut -\mathstrut 28q^{21} \) \(\mathstrut +\mathstrut 36q^{22} \) \(\mathstrut +\mathstrut 21q^{23} \) \(\mathstrut -\mathstrut 6q^{24} \) \(\mathstrut +\mathstrut 137q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 93q^{27} \) \(\mathstrut +\mathstrut 62q^{28} \) \(\mathstrut -\mathstrut q^{29} \) \(\mathstrut -\mathstrut 19q^{30} \) \(\mathstrut +\mathstrut 58q^{31} \) \(\mathstrut +\mathstrut 19q^{32} \) \(\mathstrut -\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 30q^{34} \) \(\mathstrut +\mathstrut 30q^{35} \) \(\mathstrut +\mathstrut 114q^{36} \) \(\mathstrut +\mathstrut 42q^{37} \) \(\mathstrut -\mathstrut 6q^{38} \) \(\mathstrut -\mathstrut 20q^{39} \) \(\mathstrut +\mathstrut 53q^{40} \) \(\mathstrut -\mathstrut 7q^{41} \) \(\mathstrut -\mathstrut 13q^{42} \) \(\mathstrut +\mathstrut 60q^{43} \) \(\mathstrut +\mathstrut 25q^{44} \) \(\mathstrut -\mathstrut 20q^{45} \) \(\mathstrut +\mathstrut 57q^{46} \) \(\mathstrut +\mathstrut 9q^{47} \) \(\mathstrut -\mathstrut 148q^{48} \) \(\mathstrut +\mathstrut 145q^{49} \) \(\mathstrut +\mathstrut 41q^{50} \) \(\mathstrut +\mathstrut 43q^{51} \) \(\mathstrut +\mathstrut 71q^{52} \) \(\mathstrut -\mathstrut 45q^{53} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 78q^{55} \) \(\mathstrut +\mathstrut 44q^{56} \) \(\mathstrut -\mathstrut 50q^{57} \) \(\mathstrut +\mathstrut 40q^{58} \) \(\mathstrut +\mathstrut 42q^{59} \) \(\mathstrut +\mathstrut 31q^{60} \) \(\mathstrut +\mathstrut 69q^{61} \) \(\mathstrut -\mathstrut 42q^{62} \) \(\mathstrut +\mathstrut 28q^{63} \) \(\mathstrut +\mathstrut 230q^{64} \) \(\mathstrut -\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 36q^{66} \) \(\mathstrut +\mathstrut 76q^{67} \) \(\mathstrut -\mathstrut 91q^{68} \) \(\mathstrut -\mathstrut 21q^{69} \) \(\mathstrut +\mathstrut 57q^{70} \) \(\mathstrut +\mathstrut 92q^{71} \) \(\mathstrut +\mathstrut 6q^{72} \) \(\mathstrut +\mathstrut 29q^{73} \) \(\mathstrut +\mathstrut 59q^{74} \) \(\mathstrut -\mathstrut 137q^{75} \) \(\mathstrut +\mathstrut 131q^{76} \) \(\mathstrut -\mathstrut 98q^{77} \) \(\mathstrut -\mathstrut 2q^{78} \) \(\mathstrut +\mathstrut 215q^{79} \) \(\mathstrut -\mathstrut 37q^{80} \) \(\mathstrut +\mathstrut 93q^{81} \) \(\mathstrut +\mathstrut 50q^{82} \) \(\mathstrut -\mathstrut 27q^{83} \) \(\mathstrut -\mathstrut 62q^{84} \) \(\mathstrut +\mathstrut 52q^{85} \) \(\mathstrut +\mathstrut 82q^{86} \) \(\mathstrut +\mathstrut q^{87} \) \(\mathstrut +\mathstrut 136q^{88} \) \(\mathstrut -\mathstrut 14q^{89} \) \(\mathstrut +\mathstrut 19q^{90} \) \(\mathstrut +\mathstrut 101q^{91} \) \(\mathstrut -\mathstrut 14q^{92} \) \(\mathstrut -\mathstrut 58q^{93} \) \(\mathstrut +\mathstrut 112q^{94} \) \(\mathstrut +\mathstrut 59q^{95} \) \(\mathstrut -\mathstrut 19q^{96} \) \(\mathstrut +\mathstrut 38q^{97} \) \(\mathstrut -\mathstrut 16q^{98} \) \(\mathstrut +\mathstrut 10q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81022 −1.00000 5.89735 −2.74818 2.81022 1.73585 −10.9524 1.00000 7.72298
1.2 −2.77500 −1.00000 5.70062 0.612657 2.77500 3.28614 −10.2692 1.00000 −1.70012
1.3 −2.71219 −1.00000 5.35600 0.0136473 2.71219 −2.53463 −9.10211 1.00000 −0.0370143
1.4 −2.70585 −1.00000 5.32163 −3.73335 2.70585 −4.22653 −8.98782 1.00000 10.1019
1.5 −2.66976 −1.00000 5.12761 3.59325 2.66976 3.20381 −8.34997 1.00000 −9.59311
1.6 −2.59266 −1.00000 4.72188 −2.86941 2.59266 1.94917 −7.05690 1.00000 7.43940
1.7 −2.58147 −1.00000 4.66401 −0.715328 2.58147 −2.52841 −6.87707 1.00000 1.84660
1.8 −2.57633 −1.00000 4.63748 2.33581 2.57633 −3.18098 −6.79503 1.00000 −6.01782
1.9 −2.46607 −1.00000 4.08152 −3.84806 2.46607 3.41335 −5.13317 1.00000 9.48961
1.10 −2.41534 −1.00000 3.83389 2.77205 2.41534 2.36070 −4.42946 1.00000 −6.69545
1.11 −2.40646 −1.00000 3.79107 −1.64760 2.40646 4.01941 −4.31013 1.00000 3.96488
1.12 −2.31785 −1.00000 3.37244 −1.66098 2.31785 0.0321142 −3.18111 1.00000 3.84991
1.13 −2.29477 −1.00000 3.26596 1.38948 2.29477 −4.16968 −2.90508 1.00000 −3.18854
1.14 −2.26328 −1.00000 3.12245 3.34447 2.26328 −1.47338 −2.54043 1.00000 −7.56947
1.15 −2.15308 −1.00000 2.63577 −3.48527 2.15308 4.31227 −1.36887 1.00000 7.50407
1.16 −2.07852 −1.00000 2.32023 −3.95771 2.07852 −1.61443 −0.665606 1.00000 8.22618
1.17 −1.99887 −1.00000 1.99550 2.90160 1.99887 2.75939 0.00900131 1.00000 −5.79994
1.18 −1.98067 −1.00000 1.92305 0.325740 1.98067 2.23155 0.152420 1.00000 −0.645183
1.19 −1.89273 −1.00000 1.58242 −0.0965283 1.89273 5.12594 0.790373 1.00000 0.182702
1.20 −1.84702 −1.00000 1.41149 −1.45421 1.84702 −3.40926 1.08699 1.00000 2.68596
See all 93 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.93
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(2003\) \(-1\)