# Properties

 Label 4.4.19429.1-21.1-h Base field 4.4.19429.1 Weight $[2, 2, 2, 2]$ Level norm $21$ Level $[21, 21, w^{2} - w - 6]$ Dimension $9$ CM no Base change no

# Related objects

• L-function not available

## 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:

 $$x^{9} + x^{8} - 39x^{7} - 29x^{6} + 510x^{5} + 212x^{4} - 2635x^{3} - 414x^{2} + 4376x + 208$$
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 $[79, 79, w^{2} - 2w - 1]$ $\phantom{-}\frac{182795}{9729646}e^{8} + \frac{278806}{4864823}e^{7} - \frac{2786469}{4864823}e^{6} - \frac{8236680}{4864823}e^{5} + \frac{47294789}{9729646}e^{4} + \frac{64879062}{4864823}e^{3} - \frac{119975363}{9729646}e^{2} - \frac{319476509}{9729646}e + \frac{2288849}{4864823}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$3$ $[3, 3, -w^{3} + 2w^{2} + 5w - 3]$ $1$
$7$ $[7, 7, -w^{3} + 2w^{2} + 4w - 3]$ $1$