Base field 4.4.8725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 10x^{2} + 2x + 19\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $21$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + 3x^{8} - 41x^{7} - 75x^{6} + 567x^{5} + 348x^{4} - 2504x^{3} + 240x^{2} + 1344x + 128\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{10}{3}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w + 1]$ | $...$ |
| 11 | $[11, 11, w^{2} - w - 6]$ | $-\frac{989353}{180360352}e^{8} - \frac{3858187}{180360352}e^{7} + \frac{35726433}{180360352}e^{6} + \frac{99585891}{180360352}e^{5} - \frac{433221103}{180360352}e^{4} - \frac{137737399}{45090088}e^{3} + \frac{214743959}{22545044}e^{2} + \frac{727058}{5636261}e - \frac{26996172}{5636261}$ |
| 11 | $[11, 11, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{7}{3}w + \frac{23}{3}]$ | $\phantom{-}\frac{102961}{22545044}e^{8} + \frac{371871}{22545044}e^{7} - \frac{3631513}{22545044}e^{6} - \frac{8661659}{22545044}e^{5} + \frac{41769907}{22545044}e^{4} + \frac{8103901}{5636261}e^{3} - \frac{35300282}{5636261}e^{2} + \frac{16474297}{5636261}e + \frac{7191876}{5636261}$ |
| 16 | $[16, 2, 2]$ | $-\frac{295861}{45090088}e^{8} - \frac{702101}{22545044}e^{7} + \frac{4487723}{22545044}e^{6} + \frac{4277055}{5636261}e^{5} - \frac{22737689}{11272522}e^{4} - \frac{173479791}{45090088}e^{3} + \frac{167422681}{22545044}e^{2} - \frac{777463}{5636261}e - \frac{39072031}{5636261}$ |
| 19 | $[19, 19, w]$ | $\phantom{-}\frac{368103}{45090088}e^{8} + \frac{1139767}{45090088}e^{7} - \frac{14812413}{45090088}e^{6} - \frac{26823059}{45090088}e^{5} + \frac{202235231}{45090088}e^{4} + \frac{42087455}{22545044}e^{3} - \frac{220857877}{11272522}e^{2} + \frac{44047746}{5636261}e + \frac{50901574}{5636261}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{7}{3}]$ | $-\frac{101077}{22545044}e^{8} - \frac{22251}{5636261}e^{7} + \frac{2560329}{11272522}e^{6} + \frac{388105}{11272522}e^{5} - \frac{40970037}{11272522}e^{4} + \frac{32426955}{22545044}e^{3} + \frac{190217971}{11272522}e^{2} - \frac{99024215}{11272522}e - \frac{37680490}{5636261}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{2}{3}w + \frac{10}{3}]$ | $\phantom{-}\frac{1402779}{180360352}e^{8} + \frac{3877957}{180360352}e^{7} - \frac{59335191}{180360352}e^{6} - \frac{97684509}{180360352}e^{5} + \frac{840371153}{180360352}e^{4} + \frac{15240788}{5636261}e^{3} - \frac{461375577}{22545044}e^{2} - \frac{17410859}{11272522}e + \frac{42638862}{5636261}$ |
| 19 | $[19, 19, -\frac{2}{3}w^{3} + \frac{4}{3}w^{2} + \frac{13}{3}w - \frac{17}{3}]$ | $-\frac{303225}{90180176}e^{8} + \frac{290335}{90180176}e^{7} + \frac{18184847}{90180176}e^{6} - \frac{14156519}{90180176}e^{5} - \frac{311483149}{90180176}e^{4} + \frac{137909693}{45090088}e^{3} + \frac{91723298}{5636261}e^{2} - \frac{81435997}{5636261}e - \frac{46620440}{5636261}$ |
| 25 | $[25, 5, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{10}{3}w + \frac{11}{3}]$ | $-\frac{245611}{90180176}e^{8} - \frac{447185}{90180176}e^{7} + \frac{12058187}{90180176}e^{6} + \frac{14397977}{90180176}e^{5} - \frac{189431309}{90180176}e^{4} - \frac{30807385}{22545044}e^{3} + \frac{113503489}{11272522}e^{2} + \frac{36200625}{11272522}e - \frac{33579384}{5636261}$ |
| 31 | $[31, 31, w + 3]$ | $-\frac{844825}{90180176}e^{8} - \frac{4057353}{90180176}e^{7} + \frac{25966031}{90180176}e^{6} + \frac{102449273}{90180176}e^{5} - \frac{260066941}{90180176}e^{4} - \frac{286915811}{45090088}e^{3} + \frac{108011845}{11272522}e^{2} + \frac{20913954}{5636261}e - \frac{22745532}{5636261}$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{8}{3}w - \frac{16}{3}]$ | $-\frac{944821}{90180176}e^{8} - \frac{4244327}{90180176}e^{7} + \frac{31471441}{90180176}e^{6} + \frac{110036323}{90180176}e^{5} - \frac{348341495}{90180176}e^{4} - \frac{80875697}{11272522}e^{3} + \frac{299296047}{22545044}e^{2} + \frac{24779108}{5636261}e - \frac{9478154}{5636261}$ |
| 31 | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ | $\phantom{-}1$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{5}{3}w - \frac{25}{3}]$ | $-\frac{52815}{45090088}e^{8} + \frac{318233}{45090088}e^{7} + \frac{4753297}{45090088}e^{6} - \frac{9106025}{45090088}e^{5} - \frac{93648939}{45090088}e^{4} + \frac{44433439}{22545044}e^{3} + \frac{58272217}{5636261}e^{2} - \frac{35218040}{5636261}e - \frac{35855162}{5636261}$ |
| 59 | $[59, 59, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + \frac{10}{3}w + \frac{1}{3}]$ | $-\frac{15933}{45090088}e^{8} - \frac{258223}{22545044}e^{7} - \frac{682563}{11272522}e^{6} + \frac{3964187}{22545044}e^{5} + \frac{34674677}{22545044}e^{4} + \frac{38595979}{45090088}e^{3} - \frac{86399251}{11272522}e^{2} - \frac{138790745}{11272522}e + \frac{28392922}{5636261}$ |
| 59 | $[59, 59, \frac{5}{3}w^{3} - \frac{13}{3}w^{2} - \frac{25}{3}w + \frac{47}{3}]$ | $...$ |
| 61 | $[61, 61, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - \frac{8}{3}]$ | $-\frac{389509}{22545044}e^{8} - \frac{1814649}{45090088}e^{7} + \frac{33360581}{45090088}e^{6} + \frac{38760937}{45090088}e^{5} - \frac{474459461}{45090088}e^{4} - \frac{36562221}{45090088}e^{3} + \frac{514897723}{11272522}e^{2} - \frac{250613799}{11272522}e - \frac{91042356}{5636261}$ |
| 61 | $[61, 61, -\frac{2}{3}w^{3} + \frac{7}{3}w^{2} + \frac{4}{3}w - \frac{26}{3}]$ | $-\frac{1362299}{90180176}e^{8} - \frac{4908371}{90180176}e^{7} + \frac{49998677}{90180176}e^{6} + \frac{118883859}{90180176}e^{5} - \frac{614444447}{90180176}e^{4} - \frac{262880141}{45090088}e^{3} + \frac{295156843}{11272522}e^{2} - \frac{25142122}{5636261}e - \frac{65792570}{5636261}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{5}{3}]$ | $-\frac{35462}{5636261}e^{8} - \frac{726745}{22545044}e^{7} + \frac{1090926}{5636261}e^{6} + \frac{10514741}{11272522}e^{5} - \frac{9994972}{5636261}e^{4} - \frac{42720111}{5636261}e^{3} + \frac{78441325}{22545044}e^{2} + \frac{107291981}{5636261}e + \frac{13750010}{5636261}$ |
| 71 | $[71, 71, -w^{3} + 3w^{2} + 5w - 10]$ | $\phantom{-}\frac{88969}{22545044}e^{8} - \frac{406345}{45090088}e^{7} - \frac{12473263}{45090088}e^{6} + \frac{7720361}{45090088}e^{5} + \frac{224460171}{45090088}e^{4} - \frac{33447033}{45090088}e^{3} - \frac{278992137}{11272522}e^{2} + \frac{2821201}{5636261}e + \frac{62884020}{5636261}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ | $-1$ |