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: | $[9, 3, 3]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $99$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} - 10x^{9} + 8x^{8} + 205x^{7} - 555x^{6} - 1071x^{5} + 4792x^{4} - 278x^{3} - 10837x^{2} + 5662x + 1699\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $-1$ |
| 3 | $[3, 3, w + 2]$ | $-1$ |
| 4 | $[4, 2, 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 1]$ | $-\frac{1013}{752}e^{9} + \frac{6657}{752}e^{8} + \frac{14725}{752}e^{7} - 209e^{6} + \frac{23031}{752}e^{5} + \frac{582011}{376}e^{4} - \frac{429621}{376}e^{3} - \frac{667009}{188}e^{2} + \frac{1815957}{752}e + \frac{500459}{752}$ |
| 5 | $[5, 5, w + 3]$ | $-\frac{1013}{752}e^{9} + \frac{6657}{752}e^{8} + \frac{14725}{752}e^{7} - 209e^{6} + \frac{23031}{752}e^{5} + \frac{582011}{376}e^{4} - \frac{429621}{376}e^{3} - \frac{667009}{188}e^{2} + \frac{1815957}{752}e + \frac{500459}{752}$ |
| 11 | $[11, 11, w + 1]$ | $-\frac{541}{376}e^{9} + \frac{3625}{376}e^{8} + \frac{7645}{376}e^{7} - 228e^{6} + \frac{17127}{376}e^{5} + \frac{318427}{188}e^{4} - \frac{244521}{188}e^{3} - \frac{367255}{94}e^{2} + \frac{1009197}{376}e + \frac{278203}{376}$ |
| 11 | $[11, 11, w + 9]$ | $-\frac{541}{376}e^{9} + \frac{3625}{376}e^{8} + \frac{7645}{376}e^{7} - 228e^{6} + \frac{17127}{376}e^{5} + \frac{318427}{188}e^{4} - \frac{244521}{188}e^{3} - \frac{367255}{94}e^{2} + \frac{1009197}{376}e + \frac{278203}{376}$ |
| 17 | $[17, 17, w + 2]$ | $\phantom{-}\frac{1725}{376}e^{9} - \frac{11347}{376}e^{8} - \frac{25029}{376}e^{7} + \frac{2851}{4}e^{6} - \frac{40405}{376}e^{5} - \frac{993007}{188}e^{4} + \frac{735875}{188}e^{3} + \frac{1139317}{94}e^{2} - \frac{3104597}{376}e - \frac{855825}{376}$ |
| 17 | $[17, 17, w + 14]$ | $\phantom{-}\frac{1725}{376}e^{9} - \frac{11347}{376}e^{8} - \frac{25029}{376}e^{7} + \frac{2851}{4}e^{6} - \frac{40405}{376}e^{5} - \frac{993007}{188}e^{4} + \frac{735875}{188}e^{3} + \frac{1139317}{94}e^{2} - \frac{3104597}{376}e - \frac{855825}{376}$ |
| 19 | $[19, 19, w]$ | $-\frac{945}{376}e^{9} + \frac{6217}{376}e^{8} + \frac{13705}{376}e^{7} - \frac{781}{2}e^{6} + \frac{22327}{376}e^{5} + \frac{543951}{188}e^{4} - \frac{403953}{188}e^{3} - \frac{623801}{94}e^{2} + \frac{1704961}{376}e + \frac{469019}{376}$ |
| 19 | $[19, 19, w + 18]$ | $-\frac{945}{376}e^{9} + \frac{6217}{376}e^{8} + \frac{13705}{376}e^{7} - \frac{781}{2}e^{6} + \frac{22327}{376}e^{5} + \frac{543951}{188}e^{4} - \frac{403953}{188}e^{3} - \frac{623801}{94}e^{2} + \frac{1704961}{376}e + \frac{469019}{376}$ |
| 37 | $[37, 37, -w - 4]$ | $\phantom{-}\frac{148}{47}e^{9} - \frac{3861}{188}e^{8} - \frac{2173}{47}e^{7} + \frac{1939}{4}e^{6} - \frac{11639}{188}e^{5} - \frac{168645}{47}e^{4} + \frac{245583}{94}e^{3} + \frac{386078}{47}e^{2} - \frac{261502}{47}e - \frac{288999}{188}$ |
| 37 | $[37, 37, w - 5]$ | $\phantom{-}\frac{148}{47}e^{9} - \frac{3861}{188}e^{8} - \frac{2173}{47}e^{7} + \frac{1939}{4}e^{6} - \frac{11639}{188}e^{5} - \frac{168645}{47}e^{4} + \frac{245583}{94}e^{3} + \frac{386078}{47}e^{2} - \frac{261502}{47}e - \frac{288999}{188}$ |
| 43 | $[43, 43, w + 16]$ | $-\frac{2593}{376}e^{9} + \frac{17157}{376}e^{8} + \frac{37297}{376}e^{7} - 1078e^{6} + \frac{67843}{376}e^{5} + \frac{1502763}{188}e^{4} - \frac{1128197}{188}e^{3} - \frac{1726591}{94}e^{2} + \frac{4725729}{376}e + \frac{1298535}{376}$ |
| 43 | $[43, 43, w + 26]$ | $-\frac{2593}{376}e^{9} + \frac{17157}{376}e^{8} + \frac{37297}{376}e^{7} - 1078e^{6} + \frac{67843}{376}e^{5} + \frac{1502763}{188}e^{4} - \frac{1128197}{188}e^{3} - \frac{1726591}{94}e^{2} + \frac{4725729}{376}e + \frac{1298535}{376}$ |
| 49 | $[49, 7, -7]$ | $-\frac{2903}{752}e^{9} + \frac{19091}{752}e^{8} + \frac{42135}{752}e^{7} - \frac{1199}{2}e^{6} + \frac{67685}{752}e^{5} + \frac{1669913}{376}e^{4} - \frac{1237527}{376}e^{3} - \frac{1914423}{188}e^{2} + \frac{5225127}{752}e + \frac{1438497}{752}$ |
| 53 | $[53, 53, -w - 10]$ | $-\frac{2853}{752}e^{9} + \frac{18961}{752}e^{8} + \frac{40821}{752}e^{7} - 596e^{6} + \frac{79415}{752}e^{5} + \frac{1663227}{376}e^{4} - \frac{1256541}{376}e^{3} - \frac{1914629}{188}e^{2} + \frac{5244037}{752}e + \frac{1452443}{752}$ |
| 53 | $[53, 53, w - 11]$ | $-\frac{2853}{752}e^{9} + \frac{18961}{752}e^{8} + \frac{40821}{752}e^{7} - 596e^{6} + \frac{79415}{752}e^{5} + \frac{1663227}{376}e^{4} - \frac{1256541}{376}e^{3} - \frac{1914629}{188}e^{2} + \frac{5244037}{752}e + \frac{1452443}{752}$ |
| 61 | $[61, 61, w + 15]$ | $\phantom{-}\frac{1369}{188}e^{9} - \frac{4501}{94}e^{8} - \frac{19877}{188}e^{7} + \frac{4523}{4}e^{6} - \frac{15859}{94}e^{5} - \frac{787509}{94}e^{4} + \frac{291374}{47}e^{3} + \frac{903210}{47}e^{2} - \frac{2460465}{188}e - \frac{339823}{94}$ |
| 61 | $[61, 61, w + 45]$ | $\phantom{-}\frac{1369}{188}e^{9} - \frac{4501}{94}e^{8} - \frac{19877}{188}e^{7} + \frac{4523}{4}e^{6} - \frac{15859}{94}e^{5} - \frac{787509}{94}e^{4} + \frac{291374}{47}e^{3} + \frac{903210}{47}e^{2} - \frac{2460465}{188}e - \frac{339823}{94}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $1$ |
| $3$ | $[3, 3, w + 2]$ | $1$ |