Base field 4.4.19429.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} - x + 5\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[21, 21, w^{2} - w - 6]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $37$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + x^{8} - 39x^{7} - 29x^{6} + 510x^{5} + 212x^{4} - 2635x^{3} - 414x^{2} + 4376x + 208\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w^{3} + 2w^{2} + 5w - 3]$ | $-1$ |
| 5 | $[5, 5, w]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -w^{3} + 2w^{2} + 4w - 3]$ | $-1$ |
| 13 | $[13, 13, -w^{2} + w + 4]$ | $\phantom{-}\frac{207969}{19459292}e^{8} + \frac{1115551}{19459292}e^{7} - \frac{5533797}{19459292}e^{6} - \frac{33368695}{19459292}e^{5} + \frac{8066193}{4864823}e^{4} + \frac{66693608}{4864823}e^{3} - \frac{16629179}{19459292}e^{2} - \frac{146209830}{4864823}e - \frac{28354999}{4864823}$ |
| 13 | $[13, 13, -w^{3} + 2w^{2} + 5w - 2]$ | $\phantom{-}\frac{229337}{9729646}e^{8} + \frac{891125}{9729646}e^{7} - \frac{6764415}{9729646}e^{6} - \frac{25711093}{9729646}e^{5} + \frac{26720335}{4864823}e^{4} + \frac{95721983}{4864823}e^{3} - \frac{121583571}{9729646}e^{2} - \frac{189463940}{4864823}e - \frac{7273224}{4864823}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{141379}{19459292}e^{8} + \frac{855999}{19459292}e^{7} - \frac{4049221}{19459292}e^{6} - \frac{24359787}{19459292}e^{5} + \frac{13557339}{9729646}e^{4} + \frac{44732927}{4864823}e^{3} - \frac{11438733}{19459292}e^{2} - \frac{162275241}{9729646}e - \frac{40114725}{4864823}$ |
| 17 | $[17, 17, -w + 2]$ | $-\frac{94865}{4864823}e^{8} - \frac{298166}{4864823}e^{7} + \frac{2715470}{4864823}e^{6} + \frac{8374850}{4864823}e^{5} - \frac{20186045}{4864823}e^{4} - \frac{57776644}{4864823}e^{3} + \frac{42408100}{4864823}e^{2} + \frac{99071967}{4864823}e - \frac{14119444}{4864823}$ |
| 19 | $[19, 19, -w^{3} + 2w^{2} + 3w - 2]$ | $\phantom{-}\frac{74912}{4864823}e^{8} + \frac{306016}{4864823}e^{7} - \frac{2287143}{4864823}e^{6} - \frac{9340516}{4864823}e^{5} + \frac{19221041}{4864823}e^{4} + \frac{77607673}{4864823}e^{3} - \frac{46674406}{4864823}e^{2} - \frac{177932447}{4864823}e - \frac{12134482}{4864823}$ |
| 27 | $[27, 3, w^{3} - 3w^{2} - 4w + 7]$ | $\phantom{-}\frac{299947}{9729646}e^{8} + \frac{913571}{9729646}e^{7} - \frac{8671751}{9729646}e^{6} - \frac{26469339}{9729646}e^{5} + \frac{32872796}{4864823}e^{4} + \frac{97438978}{4864823}e^{3} - \frac{131667995}{9729646}e^{2} - \frac{186612441}{4864823}e - \frac{25270600}{4864823}$ |
| 31 | $[31, 31, w^{3} - 3w^{2} - 3w + 7]$ | $\phantom{-}\frac{295575}{9729646}e^{8} + \frac{1190801}{9729646}e^{7} - \frac{8647629}{9729646}e^{6} - \frac{34755079}{9729646}e^{5} + \frac{32825588}{4864823}e^{4} + \frac{134688664}{4864823}e^{3} - \frac{123092593}{9729646}e^{2} - \frac{299896237}{4864823}e - \frac{29628328}{4864823}$ |
| 31 | $[31, 31, -w^{3} + w^{2} + 6w + 1]$ | $-\frac{229337}{9729646}e^{8} - \frac{891125}{9729646}e^{7} + \frac{6764415}{9729646}e^{6} + \frac{25711093}{9729646}e^{5} - \frac{26720335}{4864823}e^{4} - \frac{95721983}{4864823}e^{3} + \frac{131313217}{9729646}e^{2} + \frac{189463940}{4864823}e - \frac{21915714}{4864823}$ |
| 41 | $[41, 41, w^{2} - w - 1]$ | $-\frac{308607}{4864823}e^{8} - \frac{1032071}{4864823}e^{7} + \frac{9146817}{4864823}e^{6} + \frac{30139471}{4864823}e^{5} - \frac{73147503}{4864823}e^{4} - \frac{229151701}{4864823}e^{3} + \frac{167841826}{4864823}e^{2} + \frac{471684183}{4864823}e + \frac{22720228}{4864823}$ |
| 43 | $[43, 43, 2w^{3} - 5w^{2} - 6w + 4]$ | $-\frac{427755}{19459292}e^{8} - \frac{1381637}{19459292}e^{7} + \frac{12528147}{19459292}e^{6} + \frac{39383721}{19459292}e^{5} - \frac{23891774}{4864823}e^{4} - \frac{71666650}{4864823}e^{3} + \frac{177854717}{19459292}e^{2} + \frac{150483902}{4864823}e + \frac{40757323}{4864823}$ |
| 47 | $[47, 47, -w^{3} + 2w^{2} + 5w - 1]$ | $\phantom{-}\frac{49541}{19459292}e^{8} - \frac{63621}{19459292}e^{7} - \frac{199897}{19459292}e^{6} + \frac{2476877}{19459292}e^{5} - \frac{5002387}{4864823}e^{4} - \frac{9197070}{4864823}e^{3} + \frac{149899597}{19459292}e^{2} + \frac{34574116}{4864823}e - \frac{34217893}{4864823}$ |
| 53 | $[53, 53, -w - 3]$ | $-\frac{138221}{9729646}e^{8} - \frac{221013}{4864823}e^{7} + \frac{2403126}{4864823}e^{6} + \frac{6615551}{4864823}e^{5} - \frac{50765273}{9729646}e^{4} - \frac{52671051}{4864823}e^{3} + \frac{181406299}{9729646}e^{2} + \frac{215530777}{9729646}e - \frac{52067899}{4864823}$ |
| 53 | $[53, 53, -w^{3} + 3w^{2} + 3w - 6]$ | $-\frac{120470}{4864823}e^{8} - \frac{1061131}{9729646}e^{7} + \frac{6532449}{9729646}e^{6} + \frac{31161023}{9729646}e^{5} - \frac{41117479}{9729646}e^{4} - \frac{120658605}{4864823}e^{3} + \frac{21627865}{4864823}e^{2} + \frac{519261997}{9729646}e + \frac{42286667}{4864823}$ |
| 59 | $[59, 59, 2w^{2} - 3w - 6]$ | $\phantom{-}\frac{71756}{4864823}e^{8} + \frac{447015}{9729646}e^{7} - \frac{3983099}{9729646}e^{6} - \frac{14224093}{9729646}e^{5} + \frac{27948549}{9729646}e^{4} + \frac{61623015}{4864823}e^{3} - \frac{24723509}{4864823}e^{2} - \frac{305225367}{9729646}e - \frac{22765495}{4864823}$ |
| 59 | $[59, 59, w^{3} - w^{2} - 7w - 3]$ | $-\frac{180179}{19459292}e^{8} + \frac{28707}{19459292}e^{7} + \frac{5660875}{19459292}e^{6} - \frac{2946367}{19459292}e^{5} - \frac{12558436}{4864823}e^{4} + \frac{16432039}{4864823}e^{3} + \frac{114728521}{19459292}e^{2} - \frac{77408087}{4864823}e + \frac{6570813}{4864823}$ |
| 79 | $[79, 79, 2w^{3} - 4w^{2} - 8w + 3]$ | $-\frac{115397}{4864823}e^{8} - \frac{338069}{9729646}e^{7} + \frac{7353295}{9729646}e^{6} + \frac{8898351}{9729646}e^{5} - \frac{67612613}{9729646}e^{4} - \frac{23370348}{4864823}e^{3} + \frac{99700526}{4864823}e^{2} + \frac{7701027}{9729646}e - \frac{79952125}{4864823}$ |
| 79 | $[79, 79, w^{2} - 2w - 1]$ | $\phantom{-}\frac{182795}{9729646}e^{8} + \frac{278806}{4864823}e^{7} - \frac{2786469}{4864823}e^{6} - \frac{8236680}{4864823}e^{5} + \frac{47294789}{9729646}e^{4} + \frac{64879062}{4864823}e^{3} - \frac{119975363}{9729646}e^{2} - \frac{319476509}{9729646}e + \frac{2288849}{4864823}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, -w^{3} + 2w^{2} + 5w - 3]$ | $1$ |
| $7$ | $[7, 7, -w^{3} + 2w^{2} + 4w - 3]$ | $1$ |