Base field 3.3.785.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 6x + 5\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[23, 23, -w^{2} + 8]$ |
| Dimension: | $13$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $23$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{13} - 5x^{12} - 17x^{11} + 115x^{10} + 42x^{9} - 886x^{8} + 418x^{7} + 2974x^{6} - 2443x^{5} - 4212x^{4} + 4208x^{3} + 1616x^{2} - 1936x + 160\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w - 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w]$ | $\phantom{-}\frac{33299}{192376}e^{12} - \frac{119591}{96188}e^{11} - \frac{44091}{96188}e^{10} + \frac{2131659}{96188}e^{9} - \frac{7362121}{192376}e^{8} - \frac{9088785}{96188}e^{7} + \frac{13738265}{48094}e^{6} + \frac{748942}{24047}e^{5} - \frac{119697615}{192376}e^{4} + \frac{67407477}{192376}e^{3} + \frac{28776279}{96188}e^{2} - \frac{5879586}{24047}e + \frac{421373}{24047}$ |
| 5 | $[5, 5, -w + 3]$ | $\phantom{-}\frac{23973}{384752}e^{12} - \frac{156321}{384752}e^{11} - \frac{159109}{384752}e^{10} + \frac{2910863}{384752}e^{9} - \frac{1728575}{192376}e^{8} - \frac{7135651}{192376}e^{7} + \frac{14752357}{192376}e^{6} + \frac{8450075}{192376}e^{5} - \frac{67264199}{384752}e^{4} + \frac{4899509}{96188}e^{3} + \frac{4230401}{48094}e^{2} - \frac{1220303}{24047}e + \frac{116416}{24047}$ |
| 8 | $[8, 2, 2]$ | $-\frac{14223}{384752}e^{12} + \frac{152189}{384752}e^{11} - \frac{200395}{384752}e^{10} - \frac{2504279}{384752}e^{9} + \frac{969389}{48094}e^{8} + \frac{3790643}{192376}e^{7} - \frac{24349513}{192376}e^{6} + \frac{8962729}{192376}e^{5} + \frac{97458737}{384752}e^{4} - \frac{42314613}{192376}e^{3} - \frac{4842427}{48094}e^{2} + \frac{6250529}{48094}e - \frac{374292}{24047}$ |
| 9 | $[9, 3, w^{2} + w - 4]$ | $\phantom{-}\frac{149759}{384752}e^{12} - \frac{1027187}{384752}e^{11} - \frac{679155}{384752}e^{10} + \frac{18614377}{384752}e^{9} - \frac{13770087}{192376}e^{8} - \frac{42022043}{192376}e^{7} + \frac{107653303}{192376}e^{6} + \frac{29763197}{192376}e^{5} - \frac{473895661}{384752}e^{4} + \frac{55438593}{96188}e^{3} + \frac{56816005}{96188}e^{2} - \frac{20826641}{48094}e + \frac{840613}{24047}$ |
| 13 | $[13, 13, w + 3]$ | $\phantom{-}\frac{43483}{96188}e^{12} - \frac{576135}{192376}e^{11} - \frac{467337}{192376}e^{10} + \frac{10454919}{192376}e^{9} - \frac{14551523}{192376}e^{8} - \frac{23676185}{96188}e^{7} + \frac{58384127}{96188}e^{6} + \frac{17023701}{96188}e^{5} - \frac{32243200}{24047}e^{4} + \frac{124351859}{192376}e^{3} + \frac{30417163}{48094}e^{2} - \frac{23625147}{48094}e + \frac{1052490}{24047}$ |
| 17 | $[17, 17, -w^{2} + w + 3]$ | $-\frac{115189}{384752}e^{12} + \frac{778035}{384752}e^{11} + \frac{546999}{384752}e^{10} - \frac{14055397}{384752}e^{9} + \frac{1293199}{24047}e^{8} + \frac{31356747}{192376}e^{7} - \frac{81355811}{192376}e^{6} - \frac{19514413}{192376}e^{5} + \frac{357563499}{384752}e^{4} - \frac{90908961}{192376}e^{3} - \frac{41787441}{96188}e^{2} + \frac{16844755}{48094}e - \frac{825600}{24047}$ |
| 23 | $[23, 23, w^{2} - 2]$ | $\phantom{-}\frac{15453}{48094}e^{12} - \frac{53381}{24047}e^{11} - \frac{33946}{24047}e^{10} + \frac{969725}{24047}e^{9} - \frac{2888961}{48094}e^{8} - \frac{4418027}{24047}e^{7} + \frac{11266458}{24047}e^{6} + \frac{3423454}{24047}e^{5} - \frac{49666803}{48094}e^{4} + \frac{21471977}{48094}e^{3} + \frac{11905001}{24047}e^{2} - \frac{8220448}{24047}e + \frac{823194}{24047}$ |
| 23 | $[23, 23, w^{2} - 3]$ | $-\frac{56429}{192376}e^{12} + \frac{95061}{48094}e^{11} + \frac{142295}{96188}e^{10} - \frac{3456981}{96188}e^{9} + \frac{9812239}{192376}e^{8} + \frac{7891711}{48094}e^{7} - \frac{19480021}{48094}e^{6} - \frac{3078190}{24047}e^{5} + \frac{172015033}{192376}e^{4} - \frac{77324181}{192376}e^{3} - \frac{20377891}{48094}e^{2} + \frac{7593171}{24047}e - \frac{765419}{24047}$ |
| 23 | $[23, 23, -w^{2} + 8]$ | $\phantom{-}1$ |
| 29 | $[29, 29, w - 4]$ | $-\frac{17387}{24047}e^{12} + \frac{948993}{192376}e^{11} + \frac{647841}{192376}e^{10} - \frac{17222727}{192376}e^{9} + \frac{25268333}{192376}e^{8} + \frac{19539749}{48094}e^{7} - \frac{99282921}{96188}e^{6} - \frac{29091541}{96188}e^{5} + \frac{219451985}{96188}e^{4} - \frac{197785271}{192376}e^{3} - \frac{106694587}{96188}e^{2} + \frac{18806549}{24047}e - \frac{1305941}{24047}$ |
| 37 | $[37, 37, w^{2} + w - 8]$ | $\phantom{-}\frac{263265}{384752}e^{12} - \frac{1853831}{384752}e^{11} - \frac{954719}{384752}e^{10} + \frac{33378285}{384752}e^{9} - \frac{13285277}{96188}e^{8} - \frac{73705217}{192376}e^{7} + \frac{202741927}{192376}e^{6} + \frac{41478045}{192376}e^{5} - \frac{886281687}{384752}e^{4} + \frac{224239249}{192376}e^{3} + \frac{26310438}{24047}e^{2} - \frac{20137321}{24047}e + \frac{1720064}{24047}$ |
| 41 | $[41, 41, w^{2} + 2w - 4]$ | $-\frac{925}{96188}e^{12} - \frac{12411}{192376}e^{11} + \frac{110265}{192376}e^{10} + \frac{197729}{192376}e^{9} - \frac{1711169}{192376}e^{8} - \frac{137573}{48094}e^{7} + \frac{4785439}{96188}e^{6} - \frac{650289}{96188}e^{5} - \frac{5271871}{48094}e^{4} + \frac{4995051}{192376}e^{3} + \frac{7387705}{96188}e^{2} - \frac{248381}{24047}e - \frac{263091}{24047}$ |
| 47 | $[47, 47, 2w^{2} + w - 8]$ | $\phantom{-}\frac{72005}{96188}e^{12} - \frac{1041877}{192376}e^{11} - \frac{393107}{192376}e^{10} + \frac{18661545}{192376}e^{9} - \frac{31585557}{192376}e^{8} - \frac{40494709}{96188}e^{7} + \frac{117891587}{96188}e^{6} + \frac{18251783}{96188}e^{5} - \frac{63917804}{24047}e^{4} + \frac{272213001}{192376}e^{3} + \frac{60044603}{48094}e^{2} - \frac{24113337}{24047}e + \frac{2105609}{24047}$ |
| 59 | $[59, 59, -2w^{2} - 3w + 6]$ | $\phantom{-}\frac{5667}{48094}e^{12} - \frac{42845}{48094}e^{11} - \frac{2481}{96188}e^{10} + \frac{1489523}{96188}e^{9} - \frac{3040943}{96188}e^{8} - \frac{5754467}{96188}e^{7} + \frac{5414809}{24047}e^{6} - \frac{1147877}{48094}e^{5} - \frac{11552139}{24047}e^{4} + \frac{16679647}{48094}e^{3} + \frac{21339705}{96188}e^{2} - \frac{10598921}{48094}e + \frac{385399}{24047}$ |
| 61 | $[61, 61, -2w - 1]$ | $\phantom{-}\frac{235993}{384752}e^{12} - \frac{1654169}{384752}e^{11} - \frac{909057}{384752}e^{10} + \frac{29913475}{384752}e^{9} - \frac{23311951}{192376}e^{8} - \frac{67070201}{192376}e^{7} + \frac{179031085}{192376}e^{6} + \frac{44777855}{192376}e^{5} - \frac{783617795}{384752}e^{4} + \frac{45824033}{48094}e^{3} + \frac{92450747}{96188}e^{2} - \frac{33577695}{48094}e + \frac{1488311}{24047}$ |
| 67 | $[67, 67, -2w - 3]$ | $-\frac{204545}{192376}e^{12} + \frac{1444983}{192376}e^{11} + \frac{735639}{192376}e^{10} - \frac{26070485}{192376}e^{9} + \frac{10361413}{48094}e^{8} + \frac{57996869}{96188}e^{7} - \frac{158223131}{96188}e^{6} - \frac{35583401}{96188}e^{5} + \frac{693996199}{192376}e^{4} - \frac{168603449}{96188}e^{3} - \frac{41935960}{24047}e^{2} + \frac{31046544}{24047}e - \frac{2154876}{24047}$ |
| 79 | $[79, 79, 2w^{2} - 9]$ | $\phantom{-}\frac{191355}{384752}e^{12} - \frac{1285591}{384752}e^{11} - \frac{930207}{384752}e^{10} + \frac{23186485}{384752}e^{9} - \frac{16933551}{192376}e^{8} - \frac{51450887}{192376}e^{7} + \frac{133256199}{192376}e^{6} + \frac{30379377}{192376}e^{5} - \frac{583799825}{384752}e^{4} + \frac{76499647}{96188}e^{3} + \frac{67853199}{96188}e^{2} - \frac{13761142}{24047}e + \frac{1294915}{24047}$ |
| 109 | $[109, 109, w^{2} + 2w - 6]$ | $-\frac{107225}{192376}e^{12} + \frac{716661}{192376}e^{11} + \frac{569901}{192376}e^{10} - \frac{13067999}{192376}e^{9} + \frac{9048257}{96188}e^{8} + \frac{30107241}{96188}e^{7} - \frac{72668949}{96188}e^{6} - \frac{25241527}{96188}e^{5} + \frac{323882691}{192376}e^{4} - \frac{34353127}{48094}e^{3} - \frac{40165473}{48094}e^{2} + \frac{13775032}{24047}e - \frac{813588}{24047}$ |
| 109 | $[109, 109, 2w^{2} + w - 14]$ | $\phantom{-}\frac{193131}{384752}e^{12} - \frac{1189031}{384752}e^{11} - \frac{1536239}{384752}e^{10} + \frac{22196381}{384752}e^{9} - \frac{11325875}{192376}e^{8} - \frac{54968859}{192376}e^{7} + \frac{103257003}{192376}e^{6} + \frac{70163713}{192376}e^{5} - \frac{479033521}{384752}e^{4} + \frac{28647731}{96188}e^{3} + \frac{63229795}{96188}e^{2} - \frac{15334403}{48094}e + \frac{232315}{24047}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $23$ | $[23, 23, -w^{2} + 8]$ | $-1$ |