# Properties

 Label 4.4.17600.1-19.2-e Base field 4.4.17600.1 Weight $[2, 2, 2, 2]$ Level norm $19$ Level $[19, 19, \frac{1}{2}w^{3} + w^{2} - 4w - 9]$ Dimension $18$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.17600.1

Generator $$w$$, with minimal polynomial $$x^{4} - 14x^{2} + 44$$; narrow class number $$2$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[19, 19, \frac{1}{2}w^{3} + w^{2} - 4w - 9]$ Dimension: $18$ 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:

 $$x^{18} - 40x^{16} + 616x^{14} - 4678x^{12} + 18495x^{10} - 36933x^{8} + 34206x^{6} - 15148x^{4} + 3066x^{2} - 225$$
Norm Prime Eigenvalue
4 $[4, 2, -\frac{1}{2}w^{3} + w^{2} + 4w - 8]$ $\phantom{-}e$
11 $[11, 11, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - 4w + 11]$ $\phantom{-}\frac{15461}{454128}e^{16} - \frac{155189}{113532}e^{14} + \frac{9621143}{454128}e^{12} - \frac{18463901}{113532}e^{10} + \frac{24758759}{37844}e^{8} - \frac{202726725}{151376}e^{6} + \frac{95258835}{75688}e^{4} - \frac{219572465}{454128}e^{2} + \frac{4435105}{75688}$
11 $[11, 11, -\frac{1}{2}w^{2} - w + 2]$ $...$
11 $[11, 11, -\frac{1}{2}w^{2} + w + 2]$ $...$
19 $[19, 19, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - 4w + 5]$ $\phantom{-}\frac{69227}{756880}e^{17} - \frac{69729}{18922}e^{15} + \frac{43368877}{756880}e^{13} - \frac{83391999}{189220}e^{11} + \frac{67082071}{37844}e^{9} - \frac{2739006161}{756880}e^{7} + \frac{1282546121}{378440}e^{5} - \frac{995037811}{756880}e^{3} + \frac{60329641}{378440}e$
19 $[19, 19, \frac{1}{2}w^{3} + w^{2} - 4w - 9]$ $-1$
19 $[19, 19, \frac{1}{2}w^{3} - w^{2} - 4w + 9]$ $-\frac{202039}{908256}e^{16} + \frac{7992901}{908256}e^{14} - \frac{120891847}{908256}e^{12} + \frac{890344615}{908256}e^{10} - \frac{554631709}{151376}e^{8} + \frac{1974315807}{302752}e^{6} - \frac{1394592907}{302752}e^{4} + \frac{1193529991}{908256}e^{2} - \frac{38537697}{302752}$
19 $[19, 19, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 4w - 5]$ $...$
25 $[25, 5, \frac{1}{2}w^{2} - 1]$ $-\frac{36227}{302752}e^{16} + \frac{1428039}{302752}e^{14} - \frac{21508323}{302752}e^{12} + \frac{157671297}{302752}e^{10} - \frac{293055893}{151376}e^{8} + \frac{1034860081}{302752}e^{6} - \frac{719902639}{302752}e^{4} + \frac{204690311}{302752}e^{2} - \frac{20773249}{302752}$
29 $[29, 29, \frac{1}{2}w^{3} - 4w - 1]$ $...$
29 $[29, 29, -\frac{1}{2}w^{3} + 4w - 1]$ $...$
31 $[31, 31, -w - 1]$ $-\frac{1088947}{6811920}e^{17} + \frac{8518583}{1362384}e^{15} - \frac{632956447}{6811920}e^{13} + \frac{4528540201}{6811920}e^{11} - \frac{534671555}{227064}e^{9} + \frac{8420521547}{2270640}e^{7} - \frac{3955347329}{2270640}e^{5} + \frac{746986051}{6811920}e^{3} + \frac{58356581}{2270640}e$
31 $[31, 31, w - 1]$ $...$
41 $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 4w - 2]$ $\phantom{-}\frac{886417}{3405960}e^{17} - \frac{870457}{85149}e^{15} + \frac{520755667}{3405960}e^{13} - \frac{470592377}{425745}e^{11} + \frac{226614553}{56766}e^{9} - \frac{7479204257}{1135320}e^{7} + \frac{2112043597}{567660}e^{5} - \frac{2285107261}{3405960}e^{3} + \frac{20413307}{567660}e$
41 $[41, 41, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 4w - 2]$ $...$
49 $[49, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 5w - 7]$ $\phantom{-}\frac{152513}{756880}e^{17} - \frac{1202477}{151376}e^{15} + \frac{90488023}{756880}e^{13} - \frac{661600349}{756880}e^{11} + \frac{244275157}{75688}e^{9} - \frac{4237673579}{756880}e^{7} + \frac{2781725873}{756880}e^{5} - \frac{665140279}{756880}e^{3} + \frac{38863093}{756880}e$
49 $[49, 7, \frac{3}{2}w^{3} - \frac{7}{2}w^{2} - 13w + 29]$ $...$
59 $[59, 59, -\frac{1}{2}w^{3} - \frac{3}{2}w^{2} + 3w + 10]$ $-\frac{50835}{302752}e^{16} + \frac{2008461}{302752}e^{14} - \frac{30354611}{302752}e^{12} + \frac{223739079}{302752}e^{10} - \frac{419914063}{151376}e^{8} + \frac{1513510481}{302752}e^{6} - \frac{1110631873}{302752}e^{4} + \frac{344604907}{302752}e^{2} - \frac{38356171}{302752}$
59 $[59, 59, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - 3w + 10]$ $-\frac{54959}{605504}e^{16} + \frac{2148295}{605504}e^{14} - \frac{31952879}{605504}e^{12} + \frac{229682057}{605504}e^{10} - \frac{412381749}{302752}e^{8} + \frac{1351763381}{605504}e^{6} - \frac{750486455}{605504}e^{4} + \frac{118500283}{605504}e^{2} + \frac{2369471}{605504}$
61 $[61, 61, \frac{1}{2}w^{2} + w - 6]$ $\phantom{-}\frac{196159}{605504}e^{16} - \frac{7744835}{605504}e^{14} + \frac{117023735}{605504}e^{12} - \frac{863366813}{605504}e^{10} + \frac{1625579793}{302752}e^{8} - \frac{5903725445}{605504}e^{6} + \frac{4396240139}{605504}e^{4} - \frac{1337353243}{605504}e^{2} + \frac{136379869}{605504}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$19$ $[19, 19, \frac{1}{2}w^{3} + w^{2} - 4w - 9]$ $1$