# Properties

 Label 4.4.18097.1-21.1-h Base field 4.4.18097.1 Weight $[2, 2, 2, 2]$ Level norm $21$ Level $[21, 21, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{9}{2}w - 1]$ Dimension $9$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.18097.1

Generator $$w$$, with minimal polynomial $$x^{4} - x^{3} - 7x^{2} + 6x + 4$$; narrow class number $$2$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[21, 21, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{9}{2}w - 1]$ Dimension: $9$ CM: no Base change: no Newspace dimension: $34$

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:

 $$x^{9} + 3x^{8} - 14x^{7} - 41x^{6} + 57x^{5} + 158x^{4} - 78x^{3} - 171x^{2} + 34x + 19$$
Norm Prime Eigenvalue
3 $[3, 3, -w + 1]$ $-1$
4 $[4, 2, w]$ $\phantom{-}e$
4 $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ $\phantom{-}\frac{21}{554}e^{8} + \frac{29}{277}e^{7} - \frac{321}{554}e^{6} - \frac{745}{554}e^{5} + \frac{1625}{554}e^{4} + \frac{2509}{554}e^{3} - \frac{3317}{554}e^{2} - \frac{1150}{277}e + \frac{644}{277}$
7 $[7, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{5}{2}w - 2]$ $\phantom{-}1$
13 $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ $-\frac{147}{1108}e^{8} - \frac{203}{554}e^{7} + \frac{985}{554}e^{6} + \frac{5215}{1108}e^{5} - \frac{1805}{277}e^{4} - \frac{8643}{554}e^{3} + \frac{1996}{277}e^{2} + \frac{10837}{1108}e - \frac{2091}{1108}$
17 $[17, 17, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w]$ $-\frac{20}{277}e^{8} - \frac{108}{277}e^{7} + \frac{187}{277}e^{6} + \frac{1435}{277}e^{5} - \frac{189}{277}e^{4} - \frac{5054}{277}e^{3} - \frac{376}{277}e^{2} + \frac{4235}{277}e - \frac{488}{277}$
27 $[27, 3, -w^{3} + 7w + 1]$ $\phantom{-}\frac{107}{554}e^{8} + \frac{95}{277}e^{7} - \frac{798}{277}e^{6} - \frac{2345}{554}e^{5} + \frac{3421}{277}e^{4} + \frac{3866}{277}e^{3} - \frac{4368}{277}e^{2} - \frac{7353}{554}e + \frac{2223}{554}$
31 $[31, 31, w + 3]$ $-\frac{1}{1108}e^{8} + \frac{25}{554}e^{7} + \frac{67}{554}e^{6} - \frac{967}{1108}e^{5} - \frac{359}{277}e^{4} + \frac{3073}{554}e^{3} + \frac{854}{277}e^{2} - \frac{12461}{1108}e + \frac{31}{1108}$
31 $[31, 31, -w^{2} + 5]$ $\phantom{-}\frac{247}{1108}e^{8} + \frac{98}{277}e^{7} - \frac{934}{277}e^{6} - \frac{4357}{1108}e^{5} + \frac{4188}{277}e^{4} + \frac{5489}{554}e^{3} - \frac{5958}{277}e^{2} - \frac{2373}{1108}e + \frac{3977}{1108}$
37 $[37, 37, w^{2} - 3]$ $-\frac{33}{277}e^{8} - \frac{12}{277}e^{7} + \frac{544}{277}e^{6} - \frac{56}{277}e^{5} - \frac{2791}{277}e^{4} + \frac{1439}{277}e^{3} + \frac{4975}{277}e^{2} - \frac{2915}{277}e - \frac{2301}{277}$
41 $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{3}{2}w - 2]$ $-\frac{115}{1108}e^{8} - \frac{86}{277}e^{7} + \frac{390}{277}e^{6} + \frac{4581}{1108}e^{5} - \frac{1397}{277}e^{4} - \frac{8367}{554}e^{3} + \frac{983}{277}e^{2} + \frac{16249}{1108}e - \frac{1421}{1108}$
47 $[47, 47, w^{3} - 5w - 3]$ $\phantom{-}\frac{281}{1108}e^{8} + \frac{227}{277}e^{7} - \frac{965}{277}e^{6} - \frac{12475}{1108}e^{5} + \frac{3652}{277}e^{4} + \frac{23441}{554}e^{3} - \frac{3693}{277}e^{2} - \frac{41843}{1108}e + \frac{4031}{1108}$
53 $[53, 53, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w]$ $-\frac{183}{1108}e^{8} - \frac{67}{277}e^{7} + \frac{729}{277}e^{6} + \frac{3089}{1108}e^{5} - \frac{3372}{277}e^{4} - \frac{3829}{554}e^{3} + \frac{3655}{277}e^{2} - \frac{2315}{1108}e + \frac{5119}{1108}$
61 $[61, 61, w^{3} - w^{2} - 5w + 3]$ $\phantom{-}\frac{32}{277}e^{8} + \frac{62}{277}e^{7} - \frac{410}{277}e^{6} - \frac{634}{277}e^{5} + \frac{1078}{277}e^{4} + \frac{829}{277}e^{3} + \frac{1488}{277}e^{2} + \frac{2365}{277}e - \frac{3208}{277}$
83 $[83, 83, w^{3} + w^{2} - 6w - 7]$ $-\frac{393}{1108}e^{8} - \frac{212}{277}e^{7} + \frac{1393}{277}e^{6} + \frac{10539}{1108}e^{5} - \frac{5634}{277}e^{4} - \frac{17759}{554}e^{3} + \frac{6823}{277}e^{2} + \frac{33427}{1108}e - \frac{9423}{1108}$
83 $[83, 83, -w^{3} + 5w + 1]$ $-\frac{45}{554}e^{8} + \frac{17}{277}e^{7} + \frac{522}{277}e^{6} - \frac{303}{554}e^{5} - \frac{3502}{277}e^{4} - \frac{215}{277}e^{3} + \frac{6779}{277}e^{2} + \frac{5443}{554}e - \frac{4145}{554}$
83 $[83, 83, 2w - 1]$ $\phantom{-}\frac{9}{277}e^{8} + \frac{104}{277}e^{7} + \frac{179}{277}e^{6} - \frac{1269}{277}e^{5} - \frac{3142}{277}e^{4} + \frac{3964}{277}e^{3} + \frac{9421}{277}e^{2} - \frac{3083}{277}e - \frac{3603}{277}$
83 $[83, 83, -\frac{3}{2}w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 2]$ $\phantom{-}\frac{125}{1108}e^{8} + \frac{199}{554}e^{7} - \frac{619}{554}e^{6} - \frac{4329}{1108}e^{5} + \frac{1}{277}e^{4} + \frac{4229}{554}e^{3} + \frac{4050}{277}e^{2} + \frac{4209}{1108}e - \frac{12739}{1108}$
89 $[89, 89, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 2]$ $-\frac{79}{1108}e^{8} - \frac{241}{554}e^{7} + \frac{307}{554}e^{6} + \frac{6707}{1108}e^{5} + \frac{447}{277}e^{4} - \frac{13181}{554}e^{3} - \frac{4000}{277}e^{2} + \frac{27185}{1108}e + \frac{10205}{1108}$
89 $[89, 89, w^{3} - 5w + 5]$ $-\frac{33}{554}e^{8} - \frac{6}{277}e^{7} + \frac{272}{277}e^{6} + \frac{221}{554}e^{5} - \frac{1257}{277}e^{4} - \frac{1081}{277}e^{3} + \frac{1241}{277}e^{2} + \frac{8165}{554}e + \frac{1577}{554}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

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