Properties

Label 2.2.229.1-4.1-k
Base field \(\Q(\sqrt{229}) \)
Weight $[2, 2]$
Level norm $4$
Level $[4, 2, 2]$
Dimension $8$
CM no
Base change no

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Base field \(\Q(\sqrt{229}) \)

Generator \(w\), with minimal polynomial \(x^{2} - x - 57\); narrow class number \(3\) and class number \(3\).

Form

Weight: $[2, 2]$
Level: $[4, 2, 2]$
Dimension: $8$
CM: no
Base change: no
Newspace dimension: $60$

Hecke eigenvalues ($q$-expansion)

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

\(x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, w]$ $\phantom{-}e$
3 $[3, 3, w + 2]$ $\phantom{-}\frac{134}{2055}e^{7} - \frac{232}{2055}e^{6} + \frac{979}{2055}e^{5} - \frac{97}{2055}e^{4} + \frac{3424}{2055}e^{3} - \frac{1471}{2055}e^{2} + \frac{5759}{2055}e + \frac{787}{685}$
4 $[4, 2, 2]$ $\phantom{-}1$
5 $[5, 5, w + 1]$ $\phantom{-}\frac{1327}{6165}e^{7} - \frac{2686}{6165}e^{6} + \frac{10927}{6165}e^{5} - \frac{2806}{6165}e^{4} + \frac{30217}{6165}e^{3} - \frac{5923}{6165}e^{2} + \frac{16129}{2055}e + \frac{2177}{685}$
5 $[5, 5, w + 3]$ $\phantom{-}\frac{263}{6165}e^{7} - \frac{1079}{6165}e^{6} + \frac{3818}{6165}e^{5} - \frac{4694}{6165}e^{4} + \frac{10043}{6165}e^{3} - \frac{9047}{6165}e^{2} + \frac{9931}{2055}e - \frac{332}{685}$
11 $[11, 11, w + 1]$ $\phantom{-}\frac{794}{6165}e^{7} - \frac{1937}{6165}e^{6} + \frac{6854}{6165}e^{5} - \frac{4787}{6165}e^{4} + \frac{18029}{6165}e^{3} - \frac{16241}{6165}e^{2} + \frac{11828}{2055}e - \frac{596}{685}$
11 $[11, 11, w + 9]$ $-\frac{98}{6165}e^{7} + \frac{824}{6165}e^{6} - \frac{1493}{6165}e^{5} + \frac{3179}{6165}e^{4} + \frac{3712}{6165}e^{3} + \frac{5012}{6165}e^{2} + \frac{3449}{2055}e + \frac{517}{685}$
17 $[17, 17, w + 2]$ $\phantom{-}\frac{857}{6165}e^{7} - \frac{1586}{6165}e^{6} + \frac{5612}{6165}e^{5} - \frac{1106}{6165}e^{4} + \frac{14762}{6165}e^{3} - \frac{13298}{6165}e^{2} + \frac{3299}{2055}e - \frac{488}{685}$
17 $[17, 17, w + 14]$ $-\frac{2153}{6165}e^{7} + \frac{4934}{6165}e^{6} - \frac{17933}{6165}e^{5} + \frac{7289}{6165}e^{4} - \frac{43553}{6165}e^{3} + \frac{21452}{6165}e^{2} - \frac{21896}{2055}e - \frac{2908}{685}$
19 $[19, 19, w]$ $\phantom{-}\frac{401}{6165}e^{7} - \frac{1778}{6165}e^{6} + \frac{3446}{6165}e^{5} - \frac{5438}{6165}e^{4} - \frac{49}{6165}e^{3} - \frac{23444}{6165}e^{2} - \frac{2293}{2055}e - \frac{389}{685}$
19 $[19, 19, w + 18]$ $-\frac{2177}{6165}e^{7} + \frac{5681}{6165}e^{6} - \frac{20102}{6165}e^{5} + \frac{15281}{6165}e^{4} - \frac{52877}{6165}e^{3} + \frac{47633}{6165}e^{2} - \frac{10293}{685}e + \frac{1748}{685}$
37 $[37, 37, -w - 4]$ $\phantom{-}\frac{148}{2055}e^{7} - \frac{154}{2055}e^{6} + \frac{703}{2055}e^{5} + \frac{1406}{2055}e^{4} + \frac{643}{2055}e^{3} + \frac{3293}{2055}e^{2} + \frac{481}{685}e + \frac{2914}{685}$
37 $[37, 37, w - 5]$ $-\frac{112}{2055}e^{7} + \frac{61}{2055}e^{6} - \frac{532}{2055}e^{5} - \frac{1064}{2055}e^{4} - \frac{1042}{2055}e^{3} - \frac{2492}{2055}e^{2} - \frac{364}{685}e + \frac{794}{685}$
43 $[43, 43, w + 16]$ $\phantom{-}\frac{16}{6165}e^{7} - \frac{1183}{6165}e^{6} + \frac{4186}{6165}e^{5} - \frac{10123}{6165}e^{4} + \frac{11011}{6165}e^{3} - \frac{9919}{6165}e^{2} + \frac{3899}{685}e - \frac{364}{685}$
43 $[43, 43, w + 26]$ $\phantom{-}\frac{2654}{6165}e^{7} - \frac{5372}{6165}e^{6} + \frac{21854}{6165}e^{5} - \frac{5612}{6165}e^{4} + \frac{60434}{6165}e^{3} - \frac{18011}{6165}e^{2} + \frac{32258}{2055}e + \frac{4354}{685}$
49 $[49, 7, -7]$ $-\frac{24}{685}e^{7} + \frac{62}{685}e^{6} - \frac{114}{685}e^{5} - \frac{228}{685}e^{4} + \frac{266}{685}e^{3} - \frac{534}{685}e^{2} - \frac{234}{685}e - \frac{6731}{685}$
53 $[53, 53, -w - 10]$ $\phantom{-}\frac{116}{2055}e^{7} + \frac{157}{2055}e^{6} + \frac{551}{2055}e^{5} + \frac{1102}{2055}e^{4} + \frac{5336}{2055}e^{3} + \frac{2581}{2055}e^{2} + \frac{377}{685}e - \frac{2437}{685}$
53 $[53, 53, w - 11]$ $-\frac{296}{2055}e^{7} + \frac{308}{2055}e^{6} - \frac{1406}{2055}e^{5} - \frac{2812}{2055}e^{4} - \frac{3341}{2055}e^{3} - \frac{6586}{2055}e^{2} - \frac{962}{685}e - \frac{8568}{685}$
61 $[61, 61, w + 15]$ $-\frac{2881}{6165}e^{7} + \frac{6358}{6165}e^{6} - \frac{23446}{6165}e^{5} + \frac{8593}{6165}e^{4} - \frac{60601}{6165}e^{3} + \frac{25804}{6165}e^{2} - \frac{31112}{2055}e - \frac{4156}{685}$
61 $[61, 61, w + 45]$ $\phantom{-}\frac{673}{6165}e^{7} - \frac{1339}{6165}e^{6} + \frac{4738}{6165}e^{5} - \frac{2854}{6165}e^{4} + \frac{12463}{6165}e^{3} - \frac{11227}{6165}e^{2} - \frac{3464}{2055}e - \frac{412}{685}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$4$ $[4, 2, 2]$ $-1$