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: | $[3, 3, -2w + 19]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $90$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} + 13x^{14} + 117x^{12} + 546x^{10} + 1850x^{8} + 3146x^{6} + 3757x^{4} + 585x^{2} + 81\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 2 | $[2, 2, w + 1]$ | $\phantom{-}\frac{44837}{21953088}e^{15} + \frac{573287}{21953088}e^{13} + \frac{35659}{152452}e^{11} + \frac{3918791}{3658848}e^{9} + \frac{39545831}{10976544}e^{7} + \frac{8050705}{1372068}e^{5} + \frac{159549377}{21953088}e^{3} + \frac{2760163}{2439232}e$ |
| 3 | $[3, 3, -2w + 19]$ | $\phantom{-}1$ |
| 5 | $[5, 5, w]$ | $\phantom{-}\frac{28103}{1372068}e^{15} + \frac{2928211}{10976544}e^{13} + \frac{366281}{152452}e^{11} + \frac{2568935}{228678}e^{9} + \frac{209067499}{5488272}e^{7} + \frac{89156599}{1372068}e^{5} + \frac{106220855}{1372068}e^{3} + \frac{14701967}{1219616}e$ |
| 5 | $[5, 5, w + 4]$ | $-\frac{7163}{2439232}e^{15} - \frac{84293}{2439232}e^{13} - \frac{22591}{76226}e^{11} - \frac{1439763}{1219616}e^{9} - \frac{4241941}{1219616}e^{7} - \frac{207727}{76226}e^{5} - \frac{1036431}{2439232}e^{3} + \frac{25700767}{2439232}e$ |
| 13 | $[13, 13, w + 1]$ | $\phantom{-}\frac{50659}{10976544}e^{14} + \frac{320333}{5488272}e^{12} + \frac{79153}{152452}e^{10} + \frac{4308865}{1829424}e^{8} + \frac{21630737}{2744136}e^{6} + \frac{17729587}{1372068}e^{4} + \frac{173713735}{10976544}e^{2} + \frac{1502521}{609808}$ |
| 13 | $[13, 13, w + 11]$ | $-\frac{49957}{10976544}e^{14} - \frac{1256717}{21953088}e^{12} - \frac{78169}{152452}e^{10} - \frac{4261831}{1829424}e^{8} - \frac{86689421}{10976544}e^{6} - \frac{17648155}{1372068}e^{4} - \frac{173612161}{10976544}e^{2} - \frac{703521}{2439232}$ |
| 17 | $[17, 17, w + 3]$ | $\phantom{-}\frac{226235}{21953088}e^{15} + \frac{180635}{1372068}e^{13} + \frac{89703}{76226}e^{11} + \frac{19731305}{3658848}e^{9} + \frac{24879685}{1372068}e^{7} + \frac{20662625}{686034}e^{5} + \frac{802734335}{21953088}e^{3} + \frac{216985}{38113}e$ |
| 17 | $[17, 17, w + 13]$ | $-\frac{455}{1219616}e^{15} - \frac{4801}{1219616}e^{13} - \frac{1435}{38113}e^{11} - \frac{91455}{609808}e^{9} - \frac{349413}{609808}e^{7} - \frac{13195}{38113}e^{5} - \frac{65835}{1219616}e^{3} + \frac{6110363}{1219616}e$ |
| 19 | $[19, 19, w + 6]$ | $-\frac{55505}{21953088}e^{14} - \frac{386023}{10976544}e^{12} - \frac{24011}{76226}e^{10} - \frac{5548259}{3658848}e^{8} - \frac{26628199}{5488272}e^{6} - \frac{5420945}{686034}e^{4} - \frac{151101485}{21953088}e^{2} - \frac{216099}{1219616}$ |
| 19 | $[19, 19, w + 12]$ | $\phantom{-}\frac{1151}{1372068}e^{14} + \frac{321893}{21953088}e^{12} + \frac{5507}{38113}e^{10} + \frac{95728}{114339}e^{8} + \frac{33614285}{10976544}e^{6} + \frac{2172682}{343017}e^{4} + \frac{9108437}{1372068}e^{2} + \frac{2523145}{2439232}$ |
| 37 | $[37, 37, w + 2]$ | $-\frac{94901}{5488272}e^{14} - \frac{2447599}{10976544}e^{12} - \frac{152243}{76226}e^{10} - \frac{8416469}{914712}e^{8} - \frac{168837487}{5488272}e^{6} - \frac{34371785}{686034}e^{4} - \frac{313041629}{5488272}e^{2} - \frac{1370187}{1219616}$ |
| 37 | $[37, 37, w + 34]$ | $\phantom{-}\frac{324731}{21953088}e^{14} + \frac{2123797}{10976544}e^{12} + \frac{66507}{38113}e^{10} + \frac{29989385}{3658848}e^{8} + \frac{152810233}{5488272}e^{6} + \frac{16390234}{343017}e^{4} + \frac{1244226815}{21953088}e^{2} + \frac{10763489}{1219616}$ |
| 49 | $[49, 7, -7]$ | $-\frac{39}{609808}e^{14} - \frac{2735}{2439232}e^{12} - \frac{246}{38113}e^{10} - \frac{7839}{304904}e^{8} + \frac{18497}{1219616}e^{6} - \frac{2262}{38113}e^{4} - \frac{5643}{609808}e^{2} + \frac{16646525}{2439232}$ |
| 59 | $[59, 59, -4w - 33]$ | $-\frac{111589}{3658848}e^{15} - \frac{1440637}{3658848}e^{13} - \frac{268827}{76226}e^{11} - \frac{9915695}{609808}e^{9} - \frac{99376381}{1829424}e^{7} - \frac{20230955}{228678}e^{5} - \frac{364205713}{3658848}e^{3} - \frac{2419443}{1219616}e$ |
| 59 | $[59, 59, 4w - 37]$ | $\phantom{-}\frac{111589}{3658848}e^{15} + \frac{1440637}{3658848}e^{13} + \frac{268827}{76226}e^{11} + \frac{9915695}{609808}e^{9} + \frac{99376381}{1829424}e^{7} + \frac{20230955}{228678}e^{5} + \frac{364205713}{3658848}e^{3} + \frac{2419443}{1219616}e$ |
| 61 | $[61, 61, w + 28]$ | $\phantom{-}\frac{33965}{21953088}e^{14} + \frac{169351}{10976544}e^{12} + \frac{4811}{38113}e^{10} + \frac{1360943}{3658848}e^{8} + \frac{5373091}{5488272}e^{6} - \frac{288824}{343017}e^{4} + \frac{32168201}{21953088}e^{2} + \frac{277163}{1219616}$ |
| 61 | $[61, 61, w + 32]$ | $\phantom{-}\frac{5227}{5488272}e^{14} + \frac{154451}{10976544}e^{12} + \frac{9607}{76226}e^{10} + \frac{578887}{914712}e^{8} + \frac{10654163}{5488272}e^{6} + \frac{2168965}{686034}e^{4} - \frac{6057125}{5488272}e^{2} + \frac{86463}{1219616}$ |
| 71 | $[71, 71, w + 22]$ | $-\frac{613447}{21953088}e^{15} - \frac{8132761}{21953088}e^{13} - \frac{511701}{152452}e^{11} - \frac{58480477}{3658848}e^{9} - \frac{599186425}{10976544}e^{7} - \frac{131205443}{1372068}e^{5} - \frac{2451297979}{21953088}e^{3} - \frac{42413405}{2439232}e$ |
| 71 | $[71, 71, w + 48]$ | $\phantom{-}\frac{17381}{2439232}e^{15} + \frac{103133}{1219616}e^{13} + \frac{54817}{76226}e^{11} + \frac{3493581}{1219616}e^{9} + \frac{4993005}{609808}e^{7} + \frac{504049}{76226}e^{5} + \frac{2514897}{2439232}e^{3} - \frac{18410695}{1219616}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, -2w + 19]$ | $-1$ |