Base field 5.5.70601.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 5x^{3} + 2x^{2} + 3x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[47, 47, w^{3} - w^{2} - 4w - 1]$ |
| 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:
| \(x^{9} - 6x^{8} - 29x^{7} + 220x^{6} + 38x^{5} - 1892x^{4} + 1640x^{3} + 3892x^{2} - 4624x + 616\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 7 | $[7, 7, w^{4} - 6w^{2} - 2w + 4]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -w^{4} + w^{3} + 5w^{2} - w - 4]$ | $\phantom{-}\frac{101333}{10165153}e^{8} - \frac{1220933}{40660612}e^{7} - \frac{7460837}{20330306}e^{6} + \frac{44134991}{40660612}e^{5} + \frac{31638896}{10165153}e^{4} - \frac{91948688}{10165153}e^{3} - \frac{47560815}{10165153}e^{2} + \frac{179532440}{10165153}e - \frac{80975845}{10165153}$ |
| 11 | $[11, 11, -2w^{4} + w^{3} + 10w^{2} + w - 3]$ | $-\frac{98815}{10165153}e^{8} + \frac{241297}{10165153}e^{7} + \frac{3710099}{10165153}e^{6} - \frac{16864209}{20330306}e^{5} - \frac{33566412}{10165153}e^{4} + \frac{126803867}{20330306}e^{3} + \frac{74683180}{10165153}e^{2} - \frac{96787374}{10165153}e - \frac{23021380}{10165153}$ |
| 11 | $[11, 11, w^{4} - 6w^{2} - 3w + 3]$ | $\phantom{-}\frac{42637}{40660612}e^{8} - \frac{74535}{20330306}e^{7} - \frac{1298817}{40660612}e^{6} + \frac{1265754}{10165153}e^{5} + \frac{564345}{10165153}e^{4} - \frac{18192313}{20330306}e^{3} + \frac{23085354}{10165153}e^{2} + \frac{18604695}{10165153}e - \frac{54608276}{10165153}$ |
| 17 | $[17, 17, w^{2} - 2]$ | $\phantom{-}\frac{101333}{10165153}e^{8} - \frac{1220933}{40660612}e^{7} - \frac{7460837}{20330306}e^{6} + \frac{44134991}{40660612}e^{5} + \frac{31638896}{10165153}e^{4} - \frac{91948688}{10165153}e^{3} - \frac{47560815}{10165153}e^{2} + \frac{189697593}{10165153}e - \frac{80975845}{10165153}$ |
| 23 | $[23, 23, -w^{3} + w^{2} + 3w]$ | $\phantom{-}\frac{202733}{40660612}e^{8} - \frac{347070}{10165153}e^{7} - \frac{6030263}{40660612}e^{6} + \frac{25676731}{20330306}e^{5} + \frac{8876897}{20330306}e^{4} - \frac{223895151}{20330306}e^{3} + \frac{38615150}{10165153}e^{2} + \frac{228214337}{10165153}e - \frac{80428396}{10165153}$ |
| 27 | $[27, 3, w^{4} - w^{3} - 4w^{2} + 2w - 1]$ | $\phantom{-}\frac{54769}{10165153}e^{8} - \frac{348123}{10165153}e^{7} - \frac{1848862}{10165153}e^{6} + \frac{13177509}{10165153}e^{5} + \frac{11451759}{10165153}e^{4} - \frac{121703453}{10165153}e^{3} + \frac{14614122}{10165153}e^{2} + \frac{274279656}{10165153}e - \frac{135608348}{10165153}$ |
| 29 | $[29, 29, 2w^{4} - 2w^{3} - 9w^{2} + 2w + 3]$ | $\phantom{-}\frac{113276}{10165153}e^{8} - \frac{1129995}{40660612}e^{7} - \frac{4143699}{10165153}e^{6} + \frac{38352575}{40660612}e^{5} + \frac{35132385}{10165153}e^{4} - \frac{67492470}{10165153}e^{3} - \frac{69526247}{10165153}e^{2} + \frac{109542522}{10165153}e + \frac{22246285}{10165153}$ |
| 32 | $[32, 2, -2]$ | $-\frac{440683}{40660612}e^{8} + \frac{437040}{10165153}e^{7} + \frac{15519813}{40660612}e^{6} - \frac{32254639}{20330306}e^{5} - \frac{28785492}{10165153}e^{4} + \frac{144036577}{10165153}e^{3} + \frac{21409659}{10165153}e^{2} - \frac{359503331}{10165153}e + \frac{123320431}{10165153}$ |
| 47 | $[47, 47, w^{4} - 2w^{3} - 3w^{2} + 5w]$ | $-\frac{74945}{20330306}e^{8} - \frac{286081}{20330306}e^{7} + \frac{1877829}{10165153}e^{6} + \frac{5672015}{10165153}e^{5} - \frac{57127237}{20330306}e^{4} - \frac{115283597}{20330306}e^{3} + \frac{136383684}{10165153}e^{2} + \frac{150031658}{10165153}e - \frac{158058052}{10165153}$ |
| 47 | $[47, 47, w^{3} - w^{2} - 4w - 1]$ | $\phantom{-}1$ |
| 53 | $[53, 53, -w^{4} + 7w^{2} - 3]$ | $-\frac{603249}{40660612}e^{8} + \frac{2228559}{40660612}e^{7} + \frac{22737795}{40660612}e^{6} - \frac{82293153}{40660612}e^{5} - \frac{104937529}{20330306}e^{4} + \frac{361518975}{20330306}e^{3} + \frac{125083861}{10165153}e^{2} - \frac{392678107}{10165153}e + \frac{38995157}{10165153}$ |
| 53 | $[53, 53, 2w^{4} - 2w^{3} - 9w^{2} + 2w + 2]$ | $-\frac{166367}{40660612}e^{8} + \frac{987677}{40660612}e^{7} + \frac{6015353}{40660612}e^{6} - \frac{37240447}{40660612}e^{5} - \frac{12814930}{10165153}e^{4} + \frac{84060542}{10165153}e^{3} + \frac{35593597}{10165153}e^{2} - \frac{152599589}{10165153}e - \frac{43327459}{10165153}$ |
| 53 | $[53, 53, 3w^{4} - 2w^{3} - 16w^{2} + 8]$ | $\phantom{-}\frac{299413}{40660612}e^{8} - \frac{1194879}{40660612}e^{7} - \frac{10266109}{40660612}e^{6} + \frac{41584063}{40660612}e^{5} + \frac{17437109}{10165153}e^{4} - \frac{77796637}{10165153}e^{3} - \frac{11388321}{10165153}e^{2} + \frac{129984214}{10165153}e + \frac{17002731}{10165153}$ |
| 67 | $[67, 67, -w^{4} + 6w^{2} + 4w - 3]$ | $\phantom{-}\frac{186891}{20330306}e^{8} - \frac{300246}{10165153}e^{7} - \frac{6560453}{20330306}e^{6} + \frac{10804319}{10165153}e^{5} + \frac{47247915}{20330306}e^{4} - \frac{174396861}{20330306}e^{3} - \frac{5631366}{10165153}e^{2} + \frac{139316668}{10165153}e - \frac{99959380}{10165153}$ |
| 73 | $[73, 73, 2w^{4} - 12w^{2} - 4w + 5]$ | $\phantom{-}\frac{518489}{40660612}e^{8} - \frac{2587371}{40660612}e^{7} - \frac{17661557}{40660612}e^{6} + \frac{94294099}{40660612}e^{5} + \frac{28888868}{10165153}e^{4} - \frac{199500090}{10165153}e^{3} + \frac{3225801}{10165153}e^{2} + \frac{394098717}{10165153}e - \frac{138935923}{10165153}$ |
| 83 | $[83, 83, w^{4} - 5w^{2} - 3w + 3]$ | $\phantom{-}\frac{66901}{20330306}e^{8} - \frac{273588}{10165153}e^{7} - \frac{2398907}{20330306}e^{6} + \frac{10646001}{10165153}e^{5} + \frac{10323069}{10165153}e^{4} - \frac{103511140}{10165153}e^{3} - \frac{31556586}{10165153}e^{2} + \frac{237070266}{10165153}e + \frac{14268816}{10165153}$ |
| 97 | $[97, 97, -w^{4} + w^{3} + 5w^{2} - 3w - 3]$ | $\phantom{-}\frac{152933}{10165153}e^{8} - \frac{390216}{10165153}e^{7} - \frac{5652614}{10165153}e^{6} + \frac{14093273}{10165153}e^{5} + \frac{47971001}{10165153}e^{4} - \frac{119287432}{10165153}e^{3} - \frac{65656175}{10165153}e^{2} + \frac{250275356}{10165153}e - \frac{92025650}{10165153}$ |
| 103 | $[103, 103, 2w^{3} - 3w^{2} - 7w + 3]$ | $\phantom{-}\frac{304851}{40660612}e^{8} - \frac{226764}{10165153}e^{7} - \frac{11888457}{40660612}e^{6} + \frac{17223909}{20330306}e^{5} + \frac{29508578}{10165153}e^{4} - \frac{78761995}{10165153}e^{3} - \frac{75803102}{10165153}e^{2} + \frac{168502903}{10165153}e - \frac{25592474}{10165153}$ |
| 109 | $[109, 109, -3w^{4} + 2w^{3} + 14w^{2} + w - 6]$ | $\phantom{-}\frac{57525}{40660612}e^{8} - \frac{785469}{40660612}e^{7} + \frac{405763}{40660612}e^{6} + \frac{28664889}{40660612}e^{5} - \frac{17754591}{10165153}e^{4} - \frac{60779687}{10165153}e^{3} + \frac{166345915}{10165153}e^{2} + \frac{116418431}{10165153}e - \frac{220023667}{10165153}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $47$ | $[47, 47, w^{3} - w^{2} - 4w - 1]$ | $-1$ |