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
Inner twists 1

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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: \(243\)
Twist minimal: yes
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.

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

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 admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6037.2.a.a 243
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6037.2.a.a 243 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database