Properties

Label 6037.2.a.a
Level 6037
Weight 2
Character orbit 6037.a
Self dual Yes
Analytic conductor 48.206
Analytic rank 1
Dimension 243
CM No

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Newspace parameters

Level: \( N \) = \( 6037 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6037.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2056877002\)
Analytic rank: \(1\)
Dimension: \(243\)
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.

\(\operatorname{Tr}(f)(q) = \) \(243q \) \(\mathstrut -\mathstrut 47q^{2} \) \(\mathstrut -\mathstrut 31q^{3} \) \(\mathstrut +\mathstrut 229q^{4} \) \(\mathstrut -\mathstrut 40q^{5} \) \(\mathstrut -\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 42q^{7} \) \(\mathstrut -\mathstrut 135q^{8} \) \(\mathstrut +\mathstrut 214q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(243q \) \(\mathstrut -\mathstrut 47q^{2} \) \(\mathstrut -\mathstrut 31q^{3} \) \(\mathstrut +\mathstrut 229q^{4} \) \(\mathstrut -\mathstrut 40q^{5} \) \(\mathstrut -\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 42q^{7} \) \(\mathstrut -\mathstrut 135q^{8} \) \(\mathstrut +\mathstrut 214q^{9} \) \(\mathstrut -\mathstrut 14q^{10} \) \(\mathstrut -\mathstrut 112q^{11} \) \(\mathstrut -\mathstrut 54q^{12} \) \(\mathstrut -\mathstrut 45q^{13} \) \(\mathstrut -\mathstrut 35q^{14} \) \(\mathstrut -\mathstrut 56q^{15} \) \(\mathstrut +\mathstrut 213q^{16} \) \(\mathstrut -\mathstrut 71q^{17} \) \(\mathstrut -\mathstrut 135q^{18} \) \(\mathstrut -\mathstrut 69q^{19} \) \(\mathstrut -\mathstrut 107q^{20} \) \(\mathstrut -\mathstrut 36q^{21} \) \(\mathstrut -\mathstrut 24q^{22} \) \(\mathstrut -\mathstrut 162q^{23} \) \(\mathstrut -\mathstrut 57q^{24} \) \(\mathstrut +\mathstrut 203q^{25} \) \(\mathstrut -\mathstrut 55q^{26} \) \(\mathstrut -\mathstrut 115q^{27} \) \(\mathstrut -\mathstrut 87q^{28} \) \(\mathstrut -\mathstrut 76q^{29} \) \(\mathstrut -\mathstrut 64q^{30} \) \(\mathstrut -\mathstrut 35q^{31} \) \(\mathstrut -\mathstrut 302q^{32} \) \(\mathstrut -\mathstrut 77q^{33} \) \(\mathstrut -\mathstrut 9q^{34} \) \(\mathstrut -\mathstrut 264q^{35} \) \(\mathstrut +\mathstrut 173q^{36} \) \(\mathstrut -\mathstrut 61q^{37} \) \(\mathstrut -\mathstrut 71q^{38} \) \(\mathstrut -\mathstrut 123q^{39} \) \(\mathstrut -\mathstrut 16q^{40} \) \(\mathstrut -\mathstrut 74q^{41} \) \(\mathstrut -\mathstrut 70q^{42} \) \(\mathstrut -\mathstrut 178q^{43} \) \(\mathstrut -\mathstrut 209q^{44} \) \(\mathstrut -\mathstrut 107q^{45} \) \(\mathstrut -\mathstrut 11q^{46} \) \(\mathstrut -\mathstrut 191q^{47} \) \(\mathstrut -\mathstrut 65q^{48} \) \(\mathstrut +\mathstrut 211q^{49} \) \(\mathstrut -\mathstrut 188q^{50} \) \(\mathstrut -\mathstrut 175q^{51} \) \(\mathstrut -\mathstrut 95q^{52} \) \(\mathstrut -\mathstrut 122q^{53} \) \(\mathstrut -\mathstrut 36q^{54} \) \(\mathstrut -\mathstrut 47q^{55} \) \(\mathstrut -\mathstrut 69q^{56} \) \(\mathstrut -\mathstrut 103q^{57} \) \(\mathstrut -\mathstrut 37q^{58} \) \(\mathstrut -\mathstrut 212q^{59} \) \(\mathstrut -\mathstrut 79q^{60} \) \(\mathstrut -\mathstrut 14q^{61} \) \(\mathstrut -\mathstrut 152q^{62} \) \(\mathstrut -\mathstrut 203q^{63} \) \(\mathstrut +\mathstrut 217q^{64} \) \(\mathstrut -\mathstrut 159q^{65} \) \(\mathstrut +\mathstrut 5q^{66} \) \(\mathstrut -\mathstrut 202q^{67} \) \(\mathstrut -\mathstrut 176q^{68} \) \(\mathstrut -\mathstrut 34q^{69} \) \(\mathstrut +\mathstrut 45q^{70} \) \(\mathstrut -\mathstrut 170q^{71} \) \(\mathstrut -\mathstrut 347q^{72} \) \(\mathstrut -\mathstrut 57q^{73} \) \(\mathstrut -\mathstrut 68q^{74} \) \(\mathstrut -\mathstrut 124q^{75} \) \(\mathstrut -\mathstrut 74q^{76} \) \(\mathstrut -\mathstrut 166q^{77} \) \(\mathstrut -\mathstrut 63q^{78} \) \(\mathstrut -\mathstrut 48q^{79} \) \(\mathstrut -\mathstrut 222q^{80} \) \(\mathstrut +\mathstrut 159q^{81} \) \(\mathstrut -\mathstrut 27q^{82} \) \(\mathstrut -\mathstrut 434q^{83} \) \(\mathstrut -\mathstrut 52q^{84} \) \(\mathstrut -\mathstrut 57q^{85} \) \(\mathstrut -\mathstrut 77q^{86} \) \(\mathstrut -\mathstrut 184q^{87} \) \(\mathstrut -\mathstrut 15q^{88} \) \(\mathstrut -\mathstrut 62q^{89} \) \(\mathstrut -\mathstrut 24q^{90} \) \(\mathstrut -\mathstrut 81q^{91} \) \(\mathstrut -\mathstrut 330q^{92} \) \(\mathstrut -\mathstrut 164q^{93} \) \(\mathstrut +\mathstrut 40q^{94} \) \(\mathstrut -\mathstrut 182q^{95} \) \(\mathstrut -\mathstrut 66q^{96} \) \(\mathstrut -\mathstrut 21q^{97} \) \(\mathstrut -\mathstrut 254q^{98} \) \(\mathstrut -\mathstrut 306q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81439 2.72661 5.92080 4.37726 −7.67375 −1.67878 −11.0347 4.43441 −12.3193
1.2 −2.80926 −0.794746 5.89194 −3.46470 2.23265 −2.15218 −10.9335 −2.36838 9.73325
1.3 −2.79753 −2.64112 5.82619 −2.23668 7.38863 −3.27355 −10.7039 3.97554 6.25720
1.4 −2.79471 3.05349 5.81042 −2.55663 −8.53363 4.45243 −10.6490 6.32380 7.14505
1.5 −2.76337 −0.302967 5.63621 1.94141 0.837209 −2.44102 −10.0482 −2.90821 −5.36484
1.6 −2.75691 −3.25303 5.60054 1.79806 8.96830 −3.80594 −9.92635 7.58219 −4.95708
1.7 −2.74252 −0.770641 5.52143 3.66315 2.11350 −4.05510 −9.65761 −2.40611 −10.0463
1.8 −2.73652 −0.570094 5.48855 −2.26504 1.56007 4.66290 −9.54649 −2.67499 6.19832
1.9 −2.72852 −2.79225 5.44481 1.20025 7.61871 1.91493 −9.39922 4.79667 −3.27490
1.10 −2.72738 0.874188 5.43862 −1.32205 −2.38425 −0.0787567 −9.37845 −2.23580 3.60574
1.11 −2.71870 0.776703 5.39133 1.67497 −2.11162 0.286183 −9.22001 −2.39673 −4.55375
1.12 −2.70464 1.55203 5.31508 0.654263 −4.19768 0.835894 −8.96611 −0.591204 −1.76955
1.13 −2.70152 −2.60067 5.29822 −1.66030 7.02577 3.17610 −8.91022 3.76349 4.48535
1.14 −2.63978 −0.462407 4.96844 1.16438 1.22065 1.07998 −7.83603 −2.78618 −3.07372
1.15 −2.63616 2.26078 4.94932 −3.62229 −5.95976 1.87932 −7.77485 2.11112 9.54892
1.16 −2.63608 3.35251 4.94891 0.139412 −8.83747 −3.57933 −7.77355 8.23930 −0.367500
1.17 −2.63128 2.77436 4.92361 −3.29020 −7.30010 −0.686595 −7.69283 4.69706 8.65743
1.18 −2.62743 0.461713 4.90341 −4.13985 −1.21312 1.40616 −7.62851 −2.78682 10.8772
1.19 −2.62641 −3.35140 4.89803 −4.35294 8.80215 2.67168 −7.61143 8.23188 11.4326
1.20 −2.59845 1.70782 4.75194 1.33764 −4.43768 4.30824 −7.15077 −0.0833526 −3.47580
See next 80 embeddings (of 243 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.243
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(6037\) \(1\)