Properties

Label 6009.2.a.c
Level 6009
Weight 2
Character orbit 6009.a
Self dual Yes
Analytic conductor 47.982
Analytic rank 0
Dimension 92
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: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \(92q \) \(\mathstrut +\mathstrut 17q^{2} \) \(\mathstrut +\mathstrut 92q^{3} \) \(\mathstrut +\mathstrut 107q^{4} \) \(\mathstrut +\mathstrut 34q^{5} \) \(\mathstrut +\mathstrut 17q^{6} \) \(\mathstrut +\mathstrut 22q^{7} \) \(\mathstrut +\mathstrut 51q^{8} \) \(\mathstrut +\mathstrut 92q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(92q \) \(\mathstrut +\mathstrut 17q^{2} \) \(\mathstrut +\mathstrut 92q^{3} \) \(\mathstrut +\mathstrut 107q^{4} \) \(\mathstrut +\mathstrut 34q^{5} \) \(\mathstrut +\mathstrut 17q^{6} \) \(\mathstrut +\mathstrut 22q^{7} \) \(\mathstrut +\mathstrut 51q^{8} \) \(\mathstrut +\mathstrut 92q^{9} \) \(\mathstrut +\mathstrut 13q^{10} \) \(\mathstrut +\mathstrut 40q^{11} \) \(\mathstrut +\mathstrut 107q^{12} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 37q^{14} \) \(\mathstrut +\mathstrut 34q^{15} \) \(\mathstrut +\mathstrut 133q^{16} \) \(\mathstrut +\mathstrut 77q^{17} \) \(\mathstrut +\mathstrut 17q^{18} \) \(\mathstrut +\mathstrut 34q^{19} \) \(\mathstrut +\mathstrut 55q^{20} \) \(\mathstrut +\mathstrut 22q^{21} \) \(\mathstrut +\mathstrut 8q^{22} \) \(\mathstrut +\mathstrut 83q^{23} \) \(\mathstrut +\mathstrut 51q^{24} \) \(\mathstrut +\mathstrut 110q^{25} \) \(\mathstrut +\mathstrut 22q^{26} \) \(\mathstrut +\mathstrut 92q^{27} \) \(\mathstrut +\mathstrut 32q^{28} \) \(\mathstrut +\mathstrut 97q^{29} \) \(\mathstrut +\mathstrut 13q^{30} \) \(\mathstrut +\mathstrut 44q^{31} \) \(\mathstrut +\mathstrut 104q^{32} \) \(\mathstrut +\mathstrut 40q^{33} \) \(\mathstrut +\mathstrut 20q^{34} \) \(\mathstrut +\mathstrut 80q^{35} \) \(\mathstrut +\mathstrut 107q^{36} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 54q^{38} \) \(\mathstrut +\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 23q^{40} \) \(\mathstrut +\mathstrut 67q^{41} \) \(\mathstrut +\mathstrut 37q^{42} \) \(\mathstrut +\mathstrut 30q^{43} \) \(\mathstrut +\mathstrut 87q^{44} \) \(\mathstrut +\mathstrut 34q^{45} \) \(\mathstrut +\mathstrut 33q^{46} \) \(\mathstrut +\mathstrut 69q^{47} \) \(\mathstrut +\mathstrut 133q^{48} \) \(\mathstrut +\mathstrut 112q^{49} \) \(\mathstrut +\mathstrut 58q^{50} \) \(\mathstrut +\mathstrut 77q^{51} \) \(\mathstrut -\mathstrut 3q^{52} \) \(\mathstrut +\mathstrut 113q^{53} \) \(\mathstrut +\mathstrut 17q^{54} \) \(\mathstrut +\mathstrut 42q^{55} \) \(\mathstrut +\mathstrut 92q^{56} \) \(\mathstrut +\mathstrut 34q^{57} \) \(\mathstrut -\mathstrut 30q^{58} \) \(\mathstrut +\mathstrut 72q^{59} \) \(\mathstrut +\mathstrut 55q^{60} \) \(\mathstrut +\mathstrut 19q^{61} \) \(\mathstrut +\mathstrut 60q^{62} \) \(\mathstrut +\mathstrut 22q^{63} \) \(\mathstrut +\mathstrut 147q^{64} \) \(\mathstrut +\mathstrut 74q^{65} \) \(\mathstrut +\mathstrut 8q^{66} \) \(\mathstrut +\mathstrut 26q^{67} \) \(\mathstrut +\mathstrut 171q^{68} \) \(\mathstrut +\mathstrut 83q^{69} \) \(\mathstrut -\mathstrut 35q^{70} \) \(\mathstrut +\mathstrut 134q^{71} \) \(\mathstrut +\mathstrut 51q^{72} \) \(\mathstrut -\mathstrut 17q^{73} \) \(\mathstrut +\mathstrut 95q^{74} \) \(\mathstrut +\mathstrut 110q^{75} \) \(\mathstrut +\mathstrut 27q^{76} \) \(\mathstrut +\mathstrut 108q^{77} \) \(\mathstrut +\mathstrut 22q^{78} \) \(\mathstrut +\mathstrut 159q^{79} \) \(\mathstrut +\mathstrut 79q^{80} \) \(\mathstrut +\mathstrut 92q^{81} \) \(\mathstrut -\mathstrut 64q^{82} \) \(\mathstrut +\mathstrut 73q^{83} \) \(\mathstrut +\mathstrut 32q^{84} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut +\mathstrut 22q^{86} \) \(\mathstrut +\mathstrut 97q^{87} \) \(\mathstrut -\mathstrut 16q^{88} \) \(\mathstrut +\mathstrut 50q^{89} \) \(\mathstrut +\mathstrut 13q^{90} \) \(\mathstrut +\mathstrut 17q^{91} \) \(\mathstrut +\mathstrut 154q^{92} \) \(\mathstrut +\mathstrut 44q^{93} \) \(\mathstrut +\mathstrut 8q^{94} \) \(\mathstrut +\mathstrut 155q^{95} \) \(\mathstrut +\mathstrut 104q^{96} \) \(\mathstrut -\mathstrut 20q^{97} \) \(\mathstrut +\mathstrut 63q^{98} \) \(\mathstrut +\mathstrut 40q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.78023 1.00000 5.72970 −1.51058 −2.78023 −3.00947 −10.3695 1.00000 4.19978
1.2 −2.66639 1.00000 5.10965 3.96550 −2.66639 2.39934 −8.29154 1.00000 −10.5736
1.3 −2.62844 1.00000 4.90869 −1.09828 −2.62844 −4.21500 −7.64532 1.00000 2.88675
1.4 −2.60458 1.00000 4.78383 2.52259 −2.60458 −2.57820 −7.25072 1.00000 −6.57030
1.5 −2.49751 1.00000 4.23755 1.03851 −2.49751 3.59057 −5.58831 1.00000 −2.59368
1.6 −2.48563 1.00000 4.17837 2.55956 −2.48563 1.79588 −5.41463 1.00000 −6.36212
1.7 −2.40248 1.00000 3.77189 0.0504655 −2.40248 −0.217870 −4.25692 1.00000 −0.121242
1.8 −2.37626 1.00000 3.64660 −0.379121 −2.37626 2.10974 −3.91276 1.00000 0.900889
1.9 −2.31255 1.00000 3.34790 −0.928530 −2.31255 −3.24971 −3.11708 1.00000 2.14727
1.10 −2.28351 1.00000 3.21443 −3.38813 −2.28351 0.924728 −2.77317 1.00000 7.73684
1.11 −2.23454 1.00000 2.99319 2.17127 −2.23454 −3.00622 −2.21933 1.00000 −4.85180
1.12 −2.22934 1.00000 2.96995 −1.56663 −2.22934 2.33278 −2.16235 1.00000 3.49254
1.13 −2.22794 1.00000 2.96373 3.46015 −2.22794 3.74184 −2.14714 1.00000 −7.70902
1.14 −2.15336 1.00000 2.63696 −3.93365 −2.15336 −2.88832 −1.37159 1.00000 8.47056
1.15 −1.90307 1.00000 1.62167 −1.77542 −1.90307 −0.953587 0.719990 1.00000 3.37874
1.16 −1.89797 1.00000 1.60229 4.27518 −1.89797 −0.912076 0.754847 1.00000 −8.11416
1.17 −1.84272 1.00000 1.39563 −0.293700 −1.84272 0.646833 1.11368 1.00000 0.541209
1.18 −1.77532 1.00000 1.15178 0.669700 −1.77532 2.59999 1.50587 1.00000 −1.18893
1.19 −1.73440 1.00000 1.00813 1.28297 −1.73440 −0.642029 1.72030 1.00000 −2.22518
1.20 −1.66735 1.00000 0.780065 −0.964493 −1.66735 4.83846 2.03406 1.00000 1.60815
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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