Base field \(\Q(\sqrt{197}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 49\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[37, 37, -w - 3]$ |
| Dimension: | $77$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $148$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{77} - 5x^{76} - 181x^{75} + 946x^{74} + 15479x^{73} - 85137x^{72} - 831292x^{71} + 4852696x^{70} + 31415386x^{69} - 196782082x^{68} - 886537805x^{67} + 6045763304x^{66} + 19322641616x^{65} - 146334704775x^{64} - 331081494397x^{63} + 2864712350527x^{62} + 4475529433711x^{61} - 46206500110833x^{60} - 46957413397231x^{59} + 622386446232296x^{58} + 358832254381988x^{57} - 7070497482501271x^{56} - 1501617758807686x^{55} + 68236345934314888x^{54} - 6635313419295758x^{53} - 562327563167276094x^{52} + 206639541574338803x^{51} + 3970444532454149878x^{50} - 2421961947288521276x^{49} - 24062976033491359812x^{48} + 20082700681454769035x^{47} + 125210880944412587347x^{46} - 131515436432997269507x^{45} - 558647531505329567154x^{44} + 707801900658166444364x^{43} + 2130264189176263230353x^{42} - 3188055681068822041095x^{41} - 6903030679827127497928x^{40} + 12124839391653480584274x^{39} + 18829635942109548462719x^{38} - 39088006648439775003141x^{37} - 42550792488189925703213x^{36} + 106882368447744091571389x^{35} + 77350789431959806337888x^{34} - 247441176476935541395969x^{33} - 105958880689839218653730x^{32} + 483057810379560897578320x^{31} + 87932478813158335139888x^{30} - 790262460017046572463178x^{29} + 23271680635953236428141x^{28} + 1073828430516544038109184x^{27} - 234212552421492058261723x^{26} - 1197093378395056685975680x^{25} + 473477647954317887671619x^{24} + 1075733290762560447170096x^{23} - 619849287973305189756601x^{22} - 758526238044593677022621x^{21} + 595333774820947877705431x^{20} + 400334581498567186063358x^{19} - 432206438210420530035297x^{18} - 141986578757973022097780x^{17} + 237100259728442343462905x^{16} + 21079045606720489592323x^{15} - 96475128155344256997149x^{14} + 9167055772644127677762x^{13} + 27994074533038278548020x^{12} - 7149446675873075083745x^{11} - 5348457810982884317057x^{10} + 2316468524853945848138x^{9} + 541695298028815924653x^{8} - 421905723652936034243x^{7} + 2349642777762362066x^{6} + 40580163941261266035x^{5} - 6525013568226387306x^{4} - 1332640750803767438x^{3} + 472050642166433495x^{2} - 37919121332580855x + 28877033988883\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w - 7]$ | $...$ |
| 7 | $[7, 7, w + 6]$ | $...$ |
| 9 | $[9, 3, 3]$ | $...$ |
| 19 | $[19, 19, w + 5]$ | $...$ |
| 19 | $[19, 19, w - 6]$ | $...$ |
| 23 | $[23, 23, w + 8]$ | $...$ |
| 23 | $[23, 23, -w + 9]$ | $...$ |
| 25 | $[25, 5, 5]$ | $...$ |
| 29 | $[29, 29, -w - 4]$ | $...$ |
| 29 | $[29, 29, w - 5]$ | $...$ |
| 37 | $[37, 37, -w - 3]$ | $-1$ |
| 37 | $[37, 37, w - 4]$ | $...$ |
| 41 | $[41, 41, -w - 9]$ | $...$ |
| 41 | $[41, 41, w - 10]$ | $...$ |
| 43 | $[43, 43, -w - 2]$ | $...$ |
| 43 | $[43, 43, w - 3]$ | $...$ |
| 47 | $[47, 47, -w - 1]$ | $...$ |
| 47 | $[47, 47, w - 2]$ | $...$ |
| 53 | $[53, 53, 2w - 13]$ | $...$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $37$ | $[37, 37, -w - 3]$ | $1$ |