# Properties

 Label 6037.2.a.b Level 6037 Weight 2 Character orbit 6037.a Self dual yes Analytic conductor 48.206 Analytic rank 0 Dimension 259 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: $$0$$ Dimension: $$259$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$259q + 47q^{2} + 29q^{3} + 273q^{4} + 38q^{5} + 24q^{6} + 42q^{7} + 141q^{8} + 286q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$259q + 47q^{2} + 29q^{3} + 273q^{4} + 38q^{5} + 24q^{6} + 42q^{7} + 141q^{8} + 286q^{9} + 18q^{10} + 108q^{11} + 46q^{12} + 33q^{13} + 35q^{14} + 40q^{15} + 301q^{16} + 67q^{17} + 117q^{18} + 69q^{19} + 103q^{20} + 24q^{21} + 42q^{22} + 162q^{23} + 45q^{24} + 291q^{25} + 41q^{26} + 101q^{27} + 87q^{28} + 78q^{29} + 48q^{30} + 25q^{31} + 314q^{32} + 67q^{33} + 9q^{34} + 252q^{35} + 337q^{36} + 49q^{37} + 59q^{38} + 93q^{39} + 44q^{40} + 60q^{41} + 38q^{42} + 178q^{43} + 171q^{44} + 67q^{45} + 43q^{46} + 185q^{47} + 67q^{48} + 273q^{49} + 204q^{50} + 145q^{51} + 83q^{52} + 112q^{53} + 60q^{54} + 57q^{55} + 93q^{56} + 109q^{57} + 63q^{58} + 228q^{59} + 53q^{60} + 20q^{61} + 126q^{62} + 153q^{63} + 345q^{64} + 113q^{65} + 5q^{66} + 208q^{67} + 166q^{68} + 10q^{69} + 69q^{70} + 150q^{71} + 331q^{72} + 75q^{73} + 84q^{74} + 72q^{75} + 102q^{76} + 166q^{77} + 69q^{78} + 52q^{79} + 180q^{80} + 327q^{81} + 43q^{82} + 434q^{83} + 75q^{85} + 133q^{86} + 144q^{87} + 111q^{88} + 78q^{89} - 8q^{90} + 35q^{91} + 372q^{92} + 160q^{93} + 36q^{94} + 154q^{95} + 60q^{96} + 35q^{97} + 254q^{98} + 234q^{99} + O(q^{100})$$

## 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 −2.75717 −1.09794 5.60201 2.84974 3.02722 3.79647 −9.93137 −1.79452 −7.85722
1.2 −2.66503 2.65532 5.10237 −0.476820 −7.07650 −1.66384 −8.26789 4.05073 1.27074
1.3 −2.65823 −1.47107 5.06620 −0.123247 3.91044 −2.04180 −8.15068 −0.835960 0.327620
1.4 −2.65707 2.92115 5.06000 0.928477 −7.76170 3.41739 −8.13063 5.53314 −2.46702
1.5 −2.64523 1.84208 4.99727 3.41959 −4.87275 2.32041 −7.92847 0.393274 −9.04562
1.6 −2.61164 1.28286 4.82065 −1.11903 −3.35038 −4.20141 −7.36653 −1.35426 2.92250
1.7 −2.60200 −2.50808 4.77039 0.421972 6.52603 −1.06326 −7.20855 3.29049 −1.09797
1.8 −2.60071 −0.749599 4.76372 −3.60076 1.94949 −2.40322 −7.18764 −2.43810 9.36455
1.9 −2.58809 −2.19108 4.69820 0.221659 5.67071 0.704833 −6.98319 1.80083 −0.573673
1.10 −2.58711 −3.26786 4.69312 2.66521 8.45431 2.69955 −6.96738 7.67894 −6.89517
1.11 −2.58398 0.626068 4.67694 −1.39938 −1.61774 0.699914 −6.91715 −2.60804 3.61596
1.12 −2.56682 2.69132 4.58858 1.90137 −6.90815 −1.37100 −6.64441 4.24322 −4.88048
1.13 −2.47379 −1.46896 4.11964 2.18568 3.63389 −2.37075 −5.24354 −0.842162 −5.40693
1.14 −2.46629 1.74062 4.08257 −1.65001 −4.29287 1.73775 −5.13620 0.0297629 4.06940
1.15 −2.46237 −0.199276 4.06325 −0.688498 0.490692 0.363231 −5.08049 −2.96029 1.69534
1.16 −2.44806 0.638947 3.99301 2.71697 −1.56418 −2.58742 −4.87901 −2.59175 −6.65132
1.17 −2.42988 0.381455 3.90430 −2.17078 −0.926887 −2.70764 −4.62721 −2.85449 5.27472
1.18 −2.42952 −2.16225 3.90258 −1.13309 5.25323 1.20579 −4.62237 1.67531 2.75288
1.19 −2.38315 −2.49497 3.67942 −2.67299 5.94590 −2.23204 −4.00231 3.22488 6.37013
1.20 −2.38263 −0.344809 3.67692 0.273793 0.821552 0.727572 −3.99549 −2.88111 −0.652346
See next 80 embeddings (of 259 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.259 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 259

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6037.2.a.b 259 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