Properties

Label 2.2.321.1-5.2-k
Base field \(\Q(\sqrt{321}) \)
Weight $[2, 2]$
Level norm $5$
Level $[5,5,-w + 1]$
Dimension $8$
CM no
Base change no

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Base field \(\Q(\sqrt{321}) \)

Generator \(w\), with minimal polynomial \(x^{2} - x - 80\); narrow class number \(6\) and class number \(3\).

Form

Weight: $[2, 2]$
Level: $[5,5,-w + 1]$
Dimension: $8$
CM: no
Base change: no
Newspace dimension: $168$

Hecke eigenvalues ($q$-expansion)

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

\(x^{8} - 2x^{7} + 9x^{6} - 4x^{5} + 36x^{4} - 23x^{3} + 64x^{2} + 21x + 9\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, w]$ $-\frac{2013}{17813}e^{7} + \frac{4791}{17813}e^{6} - \frac{18398}{17813}e^{5} + \frac{12336}{17813}e^{4} - \frac{64971}{17813}e^{3} + \frac{61141}{17813}e^{2} - \frac{106035}{17813}e - \frac{34470}{17813}$
2 $[2, 2, w + 1]$ $\phantom{-}e$
3 $[3, 3, -2w + 19]$ $\phantom{-}1$
5 $[5, 5, w]$ $\phantom{-}0$
5 $[5, 5, w + 4]$ $-\frac{2153}{53439}e^{7} + \frac{4726}{53439}e^{6} - \frac{6394}{17813}e^{5} + \frac{10964}{53439}e^{4} - \frac{24464}{17813}e^{3} + \frac{74782}{53439}e^{2} - \frac{125276}{53439}e - \frac{13643}{17813}$
13 $[13, 13, w + 1]$ $\phantom{-}\frac{8612}{53439}e^{7} - \frac{18904}{53439}e^{6} + \frac{25576}{17813}e^{5} - \frac{43856}{53439}e^{4} + \frac{97856}{17813}e^{3} - \frac{245689}{53439}e^{2} + \frac{501104}{53439}e + \frac{54572}{17813}$
13 $[13, 13, w + 11]$ $-\frac{1511}{53439}e^{7} + \frac{1207}{53439}e^{6} - \frac{1633}{17813}e^{5} - \frac{4120}{53439}e^{4} - \frac{6248}{17813}e^{3} + \frac{19099}{53439}e^{2} + \frac{116404}{53439}e + \frac{1065}{17813}$
17 $[17, 17, w + 3]$ $\phantom{-}\frac{131}{17813}e^{7} + \frac{697}{17813}e^{6} - \frac{2829}{17813}e^{5} + \frac{8409}{17813}e^{4} - \frac{10824}{17813}e^{3} + \frac{11029}{17813}e^{2} - \frac{59228}{17813}e + \frac{1845}{17813}$
17 $[17, 17, w + 13]$ $\phantom{-}\frac{3347}{17813}e^{7} - \frac{5444}{17813}e^{6} + \frac{29431}{17813}e^{5} - \frac{6388}{17813}e^{4} + \frac{132742}{17813}e^{3} - \frac{46326}{17813}e^{2} + \frac{251458}{17813}e + \frac{83037}{17813}$
19 $[19, 19, w + 6]$ $-\frac{1118}{53439}e^{7} + \frac{3298}{53439}e^{6} - \frac{4462}{17813}e^{5} + \frac{21107}{53439}e^{4} - \frac{17072}{17813}e^{3} + \frac{52186}{53439}e^{2} - \frac{61280}{53439}e + \frac{2910}{17813}$
19 $[19, 19, w + 12]$ $\phantom{-}\frac{9032}{53439}e^{7} - \frac{18709}{53439}e^{6} + \frac{26360}{17813}e^{5} - \frac{39740}{53439}e^{4} + \frac{106277}{17813}e^{3} - \frac{286612}{53439}e^{2} + \frac{558827}{53439}e + \frac{61031}{17813}$
37 $[37, 37, w + 2]$ $-\frac{11966}{53439}e^{7} + \frac{28798}{53439}e^{6} - \frac{38962}{17813}e^{5} + \frac{107177}{53439}e^{4} - \frac{149072}{17813}e^{3} + \frac{455686}{53439}e^{2} - \frac{684944}{53439}e + \frac{25410}{17813}$
37 $[37, 37, w + 34]$ $\phantom{-}\frac{12836}{53439}e^{7} - \frac{24577}{53439}e^{6} + \frac{40586}{17813}e^{5} - \frac{45212}{53439}e^{4} + \frac{157609}{17813}e^{3} - \frac{216004}{53439}e^{2} + \frac{854135}{53439}e + \frac{93575}{17813}$
49 $[49, 7, -7]$ $\phantom{-}\frac{1670}{17813}e^{7} - \frac{497}{17813}e^{6} + \frac{9352}{17813}e^{5} + \frac{16366}{17813}e^{4} + \frac{55918}{17813}e^{3} + \frac{49766}{17813}e^{2} + \frac{17034}{17813}e + \frac{78319}{17813}$
59 $[59, 59, -4w - 33]$ $\phantom{-}\frac{830}{17813}e^{7} - \frac{887}{17813}e^{6} + \frac{4648}{17813}e^{5} + \frac{8134}{17813}e^{4} + \frac{5392}{17813}e^{3} + \frac{24734}{17813}e^{2} + \frac{8466}{17813}e + \frac{3939}{17813}$
59 $[59, 59, 4w - 37]$ $\phantom{-}\frac{1670}{17813}e^{7} - \frac{497}{17813}e^{6} + \frac{9352}{17813}e^{5} + \frac{16366}{17813}e^{4} + \frac{55918}{17813}e^{3} + \frac{49766}{17813}e^{2} + \frac{17034}{17813}e + \frac{42693}{17813}$
61 $[61, 61, w + 28]$ $\phantom{-}\frac{3413}{53439}e^{7} - \frac{4141}{53439}e^{6} + \frac{8746}{17813}e^{5} + \frac{1384}{53439}e^{4} + \frac{49727}{17813}e^{3} - \frac{90673}{53439}e^{2} + \frac{298445}{53439}e + \frac{33020}{17813}$
61 $[61, 61, w + 32]$ $-\frac{1760}{53439}e^{7} + \frac{6817}{53439}e^{6} - \frac{9223}{17813}e^{5} + \frac{36191}{53439}e^{4} - \frac{35288}{17813}e^{3} + \frac{107869}{53439}e^{2} - \frac{409838}{53439}e + \frac{6015}{17813}$
71 $[71, 71, w + 22]$ $\phantom{-}\frac{9599}{17813}e^{7} - \frac{22899}{17813}e^{6} + \frac{92943}{17813}e^{5} - \frac{73372}{17813}e^{4} + \frac{355608}{17813}e^{3} - \frac{362343}{17813}e^{2} + \frac{710677}{17813}e - \frac{60615}{17813}$
71 $[71, 71, w + 48]$ $-\frac{1099}{17813}e^{7} + \frac{3943}{17813}e^{6} - \frac{9717}{17813}e^{5} + \frac{14168}{17813}e^{4} - \frac{22463}{17813}e^{3} + \frac{81253}{17813}e^{2} - \frac{18335}{17813}e - \frac{5280}{17813}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$5$ $[5,5,-w + 1]$ $\frac{2153}{53439}e^{7} - \frac{4726}{53439}e^{6} + \frac{6394}{17813}e^{5} - \frac{10964}{53439}e^{4} + \frac{24464}{17813}e^{3} - \frac{74782}{53439}e^{2} + \frac{125276}{53439}e + \frac{13643}{17813}$