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: | $[9,9,-w + 4]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $99$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} + 9x^{10} + 57x^{8} + 178x^{6} + 405x^{4} + 456x^{2} + 361\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}0$ |
| 4 | $[4, 2, 2]$ | $-\frac{56}{2109}e^{11} - \frac{478}{2109}e^{9} - \frac{49}{37}e^{7} - \frac{7441}{2109}e^{5} - \frac{12371}{2109}e^{3} - \frac{133}{37}e$ |
| 5 | $[5, 5, w + 1]$ | $-\frac{16}{703}e^{10} - \frac{415}{2109}e^{8} - \frac{42}{37}e^{6} - \frac{2126}{703}e^{4} - \frac{9536}{2109}e^{2} - \frac{114}{37}$ |
| 5 | $[5, 5, w + 3]$ | $\phantom{-}\frac{3}{703}e^{11} + \frac{1}{37}e^{9} + \frac{19}{111}e^{7} + \frac{135}{703}e^{5} + \frac{8}{37}e^{3} - \frac{292}{111}e$ |
| 11 | $[11, 11, w + 1]$ | $-\frac{46}{2109}e^{10} - \frac{130}{703}e^{8} - \frac{130}{111}e^{6} - \frac{7694}{2109}e^{4} - \frac{5850}{703}e^{2} - \frac{1040}{111}$ |
| 11 | $[11, 11, w + 9]$ | $\phantom{-}\frac{3}{703}e^{11} + \frac{56}{703}e^{9} + \frac{56}{111}e^{7} + \frac{1541}{703}e^{5} + \frac{2520}{703}e^{3} + \frac{448}{111}e$ |
| 17 | $[17, 17, w + 2]$ | $\phantom{-}\frac{3}{703}e^{11} + \frac{1}{37}e^{9} + \frac{19}{111}e^{7} + \frac{135}{703}e^{5} + \frac{8}{37}e^{3} - \frac{292}{111}e$ |
| 17 | $[17, 17, w + 14]$ | $-\frac{40}{703}e^{10} - \frac{982}{2109}e^{8} - \frac{105}{37}e^{6} - \frac{5315}{703}e^{4} - \frac{28724}{2109}e^{2} - \frac{285}{37}$ |
| 19 | $[19, 19, w]$ | $-\frac{24}{703}e^{10} - \frac{530}{2109}e^{8} - \frac{63}{37}e^{6} - \frac{3189}{703}e^{4} - \frac{22444}{2109}e^{2} - \frac{171}{37}$ |
| 19 | $[19, 19, w + 18]$ | $-\frac{7}{703}e^{10} + \frac{5}{703}e^{8} + \frac{5}{111}e^{6} + \frac{1091}{703}e^{4} + \frac{225}{703}e^{2} + \frac{40}{111}$ |
| 37 | $[37, 37, -w - 4]$ | $\phantom{-}\frac{62}{703}e^{10} + \frac{62}{111}e^{8} + \frac{331}{111}e^{6} + \frac{2790}{703}e^{4} + \frac{496}{111}e^{2} - \frac{1015}{111}$ |
| 37 | $[37, 37, w - 5]$ | $-\frac{55}{703}e^{10} - \frac{55}{111}e^{8} - \frac{299}{111}e^{6} - \frac{2475}{703}e^{4} - \frac{440}{111}e^{2} + \frac{938}{111}$ |
| 43 | $[43, 43, w + 16]$ | $\phantom{-}\frac{4}{703}e^{11} + \frac{4}{111}e^{9} + \frac{13}{111}e^{7} + \frac{180}{703}e^{5} + \frac{32}{111}e^{3} + \frac{326}{111}e$ |
| 43 | $[43, 43, w + 26]$ | $\phantom{-}\frac{86}{2109}e^{11} + \frac{272}{703}e^{9} + \frac{272}{111}e^{7} + \frac{17227}{2109}e^{5} + \frac{12240}{703}e^{3} + \frac{2176}{111}e$ |
| 49 | $[49, 7, -7]$ | $-\frac{39}{703}e^{10} - \frac{13}{37}e^{8} - \frac{70}{37}e^{6} - \frac{1755}{703}e^{4} - \frac{104}{37}e^{2} + \frac{291}{37}$ |
| 53 | $[53, 53, -w - 10]$ | $-\frac{8}{2109}e^{11} + \frac{11}{2109}e^{9} - \frac{7}{37}e^{7} - \frac{1063}{2109}e^{5} - \frac{9347}{2109}e^{3} - \frac{19}{37}e$ |
| 53 | $[53, 53, w - 11]$ | $-\frac{2}{703}e^{10} - \frac{2}{111}e^{8} - \frac{25}{111}e^{6} - \frac{90}{703}e^{4} - \frac{16}{111}e^{2} + \frac{1021}{111}$ |
| 61 | $[61, 61, w + 15]$ | $\phantom{-}\frac{22}{2109}e^{11} + \frac{30}{703}e^{9} + \frac{10}{37}e^{7} + \frac{287}{2109}e^{5} + \frac{1350}{703}e^{3} + \frac{80}{37}e$ |
| 61 | $[61, 61, w + 45]$ | $\phantom{-}\frac{68}{703}e^{11} + \frac{68}{111}e^{9} + \frac{123}{37}e^{7} + \frac{3060}{703}e^{5} + \frac{544}{111}e^{3} - \frac{422}{37}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3,3,-w + 1]$ | $-\frac{8}{2109}e^{10} - \frac{21}{703}e^{8} - \frac{7}{37}e^{6} - \frac{1063}{2109}e^{4} - \frac{945}{703}e^{2} - \frac{56}{37}$ |