# Properties

 Base field 4.4.8768.1 Weight [2, 2, 2, 2] Level norm 23 Level $[23,23,w^{3} - 2w^{2} - 2w + 2]$ Label 4.4.8768.1-23.2-b Dimension 8 CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.8768.1

Generator $$w$$, with minimal polynomial $$x^{4} - 2x^{3} - 5x^{2} + 6x + 7$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight [2, 2, 2, 2] Level $[23,23,w^{3} - 2w^{2} - 2w + 2]$ Label 4.4.8768.1-23.2-b Dimension 8 Is CM no Is base change no Parent newspace dimension 12

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{8}$$ $$\mathstrut -\mathstrut 2x^{7}$$ $$\mathstrut -\mathstrut 16x^{6}$$ $$\mathstrut +\mathstrut 23x^{5}$$ $$\mathstrut +\mathstrut 78x^{4}$$ $$\mathstrut -\mathstrut 48x^{3}$$ $$\mathstrut -\mathstrut 126x^{2}$$ $$\mathstrut -\mathstrut 25x$$ $$\mathstrut -\mathstrut 1$$
Norm Prime Eigenvalue
4 $[4, 2, -w^{2} + w + 3]$ $\phantom{-}e$
7 $[7, 7, w]$ $-\frac{247}{397}e^{7} + \frac{1099}{794}e^{6} + \frac{3800}{397}e^{5} - \frac{12917}{794}e^{4} - \frac{35145}{794}e^{3} + \frac{29943}{794}e^{2} + \frac{55843}{794}e + \frac{4019}{794}$
7 $[7, 7, -w^{2} + 2w + 1]$ $\phantom{-}\frac{258}{397}e^{7} - \frac{492}{397}e^{6} - \frac{4183}{397}e^{5} + \frac{5748}{397}e^{4} + \frac{20631}{397}e^{3} - \frac{12893}{397}e^{2} - \frac{33098}{397}e - \frac{3777}{397}$
7 $[7, 7, w^{2} - 2]$ $-\frac{48}{397}e^{7} + \frac{110}{397}e^{6} + \frac{769}{397}e^{5} - \frac{1411}{397}e^{4} - \frac{3746}{397}e^{3} + \frac{3959}{397}e^{2} + \frac{6167}{397}e + \frac{278}{397}$
7 $[7, 7, w - 1]$ $\phantom{-}\frac{263}{794}e^{7} - \frac{260}{397}e^{6} - \frac{4321}{794}e^{5} + \frac{6201}{794}e^{4} + \frac{21931}{794}e^{3} - \frac{14637}{794}e^{2} - \frac{36263}{794}e - \frac{1812}{397}$
23 $[23, 23, -w^{3} + w^{2} + 3w - 1]$ $-\frac{657}{794}e^{7} + \frac{728}{397}e^{6} + \frac{9955}{794}e^{5} - \frac{16807}{794}e^{4} - \frac{44971}{794}e^{3} + \frac{37093}{794}e^{2} + \frac{70491}{794}e + \frac{3565}{397}$
23 $[23, 23, w^{3} - 2w^{2} - 2w + 2]$ $-1$
31 $[31, 31, w^{2} - 5]$ $\phantom{-}\frac{5}{397}e^{7} - \frac{28}{397}e^{6} - \frac{138}{397}e^{5} + \frac{453}{397}e^{4} + \frac{1300}{397}e^{3} - \frac{2141}{397}e^{2} - \frac{3165}{397}e + \frac{1344}{397}$
31 $[31, 31, -w^{2} + 2w + 4]$ $\phantom{-}\frac{271}{794}e^{7} - \frac{203}{397}e^{6} - \frac{4383}{794}e^{5} + \frac{4385}{794}e^{4} + \frac{20835}{794}e^{3} - \frac{7423}{794}e^{2} - \frac{30211}{794}e - \frac{2960}{397}$
41 $[41, 41, -w^{3} + 2w^{2} + 2w - 6]$ $\phantom{-}\frac{618}{397}e^{7} - \frac{1317}{397}e^{6} - \frac{9752}{397}e^{5} + \frac{15338}{397}e^{4} + \frac{46741}{397}e^{3} - \frac{34447}{397}e^{2} - \frac{74785}{397}e - \frac{6656}{397}$
41 $[41, 41, w^{3} - w^{2} - 3w - 3]$ $-\frac{707}{397}e^{7} + \frac{3075}{794}e^{6} + \frac{10938}{397}e^{5} - \frac{35925}{794}e^{4} - \frac{101253}{794}e^{3} + \frac{81673}{794}e^{2} + \frac{153863}{794}e + \frac{12391}{794}$
47 $[47, 47, w^{2} - 2w - 5]$ $\phantom{-}\frac{304}{397}e^{7} - \frac{1261}{794}e^{6} - \frac{4738}{397}e^{5} + \frac{14829}{794}e^{4} + \frac{44141}{794}e^{3} - \frac{34929}{794}e^{2} - \frac{67661}{794}e - \frac{3389}{794}$
47 $[47, 47, w^{2} - 6]$ $\phantom{-}\frac{297}{397}e^{7} - \frac{631}{397}e^{6} - \frac{4783}{397}e^{5} + \frac{7614}{397}e^{4} + \frac{23625}{397}e^{3} - \frac{18715}{397}e^{2} - \frac{38729}{397}e - \frac{3060}{397}$
71 $[71, 71, -w^{3} + 2w^{2} + 3w - 1]$ $\phantom{-}\frac{1}{397}e^{7} - \frac{85}{397}e^{6} + \frac{290}{397}e^{5} + \frac{964}{397}e^{4} - \frac{3313}{397}e^{3} - \frac{1778}{397}e^{2} + \frac{6116}{397}e + \frac{507}{397}$
71 $[71, 71, -w^{3} + w^{2} + 4w - 3]$ $-\frac{37}{397}e^{7} - \frac{31}{397}e^{6} + \frac{783}{397}e^{5} + \frac{459}{397}e^{4} - \frac{4459}{397}e^{3} - \frac{2498}{397}e^{2} + \frac{5159}{397}e + \frac{5458}{397}$
79 $[79, 79, -w - 3]$ $\phantom{-}\frac{143}{397}e^{7} - \frac{887}{794}e^{6} - \frac{1803}{397}e^{5} + \frac{10111}{794}e^{4} + \frac{11237}{794}e^{3} - \frac{20595}{794}e^{2} - \frac{13901}{794}e - \frac{4667}{794}$
79 $[79, 79, w - 4]$ $\phantom{-}\frac{531}{397}e^{7} - \frac{1068}{397}e^{6} - \frac{8780}{397}e^{5} + \frac{12855}{397}e^{4} + \frac{44765}{397}e^{3} - \frac{31018}{397}e^{2} - \frac{74500}{397}e - \frac{5507}{397}$
81 $[81, 3, -3]$ $\phantom{-}\frac{138}{397}e^{7} - \frac{217}{397}e^{6} - \frac{2459}{397}e^{5} + \frac{2816}{397}e^{4} + \frac{13251}{397}e^{3} - \frac{8752}{397}e^{2} - \frac{20658}{397}e + \frac{2079}{397}$
89 $[89, 89, w^{2} - 3w - 2]$ $-\frac{1109}{794}e^{7} + \frac{1279}{397}e^{6} + \frac{17031}{794}e^{5} - \frac{30127}{794}e^{4} - \frac{79121}{794}e^{3} + \frac{68981}{794}e^{2} + \frac{127935}{794}e + \frac{9274}{397}$
89 $[89, 89, w^{2} + w - 4]$ $-\frac{74}{397}e^{7} + \frac{273}{794}e^{6} + \frac{1169}{397}e^{5} - \frac{3325}{794}e^{4} - \frac{11087}{794}e^{3} + \frac{8667}{794}e^{2} + \frac{17857}{794}e - \frac{1591}{794}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
23 $[23,23,w^{3} - 2w^{2} - 2w + 2]$ $1$