Base field \(\Q(\sqrt{101}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 25\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[13,13,-w + 4]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 5x^{7} - 5x^{6} - 53x^{5} - 30x^{4} + 132x^{3} + 101x^{2} - 94x - 61\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -w + 5]$ | $-\frac{23}{148}e^{7} - \frac{18}{37}e^{6} + \frac{211}{148}e^{5} + \frac{182}{37}e^{4} - \frac{165}{74}e^{3} - \frac{817}{74}e^{2} + \frac{127}{148}e + \frac{895}{148}$ |
| 5 | $[5, 5, -w - 4]$ | $-\frac{33}{74}e^{7} - \frac{42}{37}e^{6} + \frac{351}{74}e^{5} + \frac{400}{37}e^{4} - \frac{433}{37}e^{3} - \frac{725}{37}e^{2} + \frac{623}{74}e + \frac{557}{74}$ |
| 9 | $[9, 3, 3]$ | $-\frac{117}{148}e^{7} - \frac{61}{37}e^{6} + \frac{1453}{148}e^{5} + \frac{625}{37}e^{4} - \frac{2477}{74}e^{3} - \frac{2853}{74}e^{2} + \frac{4745}{148}e + \frac{2957}{148}$ |
| 13 | $[13, 13, w + 3]$ | $-\frac{5}{37}e^{7} - \frac{6}{37}e^{6} + \frac{70}{37}e^{5} + \frac{73}{37}e^{4} - \frac{268}{37}e^{3} - \frac{241}{37}e^{2} + \frac{211}{37}e + \frac{164}{37}$ |
| 13 | $[13, 13, w - 4]$ | $\phantom{-}1$ |
| 17 | $[17, 17, w + 6]$ | $\phantom{-}\frac{143}{148}e^{7} + \frac{91}{37}e^{6} - \frac{1595}{148}e^{5} - \frac{916}{37}e^{4} + \frac{2271}{74}e^{3} + \frac{4005}{74}e^{2} - \frac{4303}{148}e - \frac{4535}{148}$ |
| 17 | $[17, 17, -w + 7]$ | $-\frac{81}{148}e^{7} - \frac{65}{37}e^{6} + \frac{801}{148}e^{5} + \frac{649}{37}e^{4} - \frac{861}{74}e^{3} - \frac{2681}{74}e^{2} + \frac{1509}{148}e + \frac{2309}{148}$ |
| 19 | $[19, 19, w + 2]$ | $\phantom{-}\frac{35}{148}e^{7} + \frac{29}{37}e^{6} - \frac{231}{148}e^{5} - \frac{248}{37}e^{4} - \frac{209}{74}e^{3} + \frac{529}{74}e^{2} + \frac{1409}{148}e - \frac{223}{148}$ |
| 19 | $[19, 19, w - 3]$ | $\phantom{-}\frac{77}{74}e^{7} + \frac{98}{37}e^{6} - \frac{893}{74}e^{5} - \frac{1032}{37}e^{4} + \frac{1368}{37}e^{3} + \frac{2518}{37}e^{2} - \frac{2687}{74}e - \frac{3051}{74}$ |
| 23 | $[23, 23, w + 1]$ | $\phantom{-}\frac{157}{148}e^{7} + \frac{110}{37}e^{6} - \frac{1569}{148}e^{5} - \frac{1067}{37}e^{4} + \frac{1625}{74}e^{3} + \frac{4039}{74}e^{2} - \frac{1845}{148}e - \frac{2937}{148}$ |
| 23 | $[23, 23, -w + 2]$ | $-\frac{1}{37}e^{7} - \frac{16}{37}e^{6} - \frac{23}{37}e^{5} + \frac{170}{37}e^{4} + \frac{272}{37}e^{3} - \frac{433}{37}e^{2} - \frac{424}{37}e + \frac{203}{37}$ |
| 31 | $[31, 31, -w - 7]$ | $-\frac{141}{148}e^{7} - \frac{83}{37}e^{6} + \frac{1641}{148}e^{5} + \frac{831}{37}e^{4} - \frac{2469}{74}e^{3} - \frac{3461}{74}e^{2} + \frac{4337}{148}e + \frac{3093}{148}$ |
| 31 | $[31, 31, w - 8]$ | $\phantom{-}\frac{42}{37}e^{7} + \frac{117}{37}e^{6} - \frac{440}{37}e^{5} - \frac{1146}{37}e^{4} + \frac{1082}{37}e^{3} + \frac{2202}{37}e^{2} - \frac{988}{37}e - \frac{719}{37}$ |
| 37 | $[37, 37, 2w - 9]$ | $-\frac{171}{148}e^{7} - \frac{129}{37}e^{6} + \frac{1691}{148}e^{5} + \frac{1292}{37}e^{4} - \frac{1719}{74}e^{3} - \frac{5479}{74}e^{2} + \frac{1903}{148}e + \frac{4743}{148}$ |
| 37 | $[37, 37, 2w + 7]$ | $\phantom{-}\frac{62}{37}e^{7} + \frac{141}{37}e^{6} - \frac{720}{37}e^{5} - \frac{1401}{37}e^{4} + \frac{2154}{37}e^{3} + \frac{2870}{37}e^{2} - \frac{1758}{37}e - \frac{1190}{37}$ |
| 43 | $[43, 43, 4w + 17]$ | $\phantom{-}\frac{89}{148}e^{7} + \frac{60}{37}e^{6} - \frac{1061}{148}e^{5} - \frac{656}{37}e^{4} + \frac{1697}{74}e^{3} + \frac{3451}{74}e^{2} - \frac{3297}{148}e - \frac{4377}{148}$ |
| 43 | $[43, 43, 4w - 21]$ | $\phantom{-}\frac{1}{4}e^{7} + e^{6} - \frac{5}{4}e^{5} - 9e^{4} - \frac{15}{2}e^{3} + \frac{25}{2}e^{2} + \frac{83}{4}e - \frac{17}{4}$ |
| 47 | $[47, 47, -w - 8]$ | $-\frac{3}{37}e^{7} - \frac{11}{37}e^{6} + \frac{5}{37}e^{5} + \frac{29}{37}e^{4} + \frac{150}{37}e^{3} + \frac{477}{37}e^{2} - \frac{162}{37}e - \frac{649}{37}$ |
| 47 | $[47, 47, w - 9]$ | $-\frac{265}{148}e^{7} - \frac{172}{37}e^{6} + \frac{2933}{148}e^{5} + \frac{1735}{37}e^{4} - \frac{4031}{74}e^{3} - \frac{7515}{74}e^{2} + \frac{6965}{148}e + \frac{7545}{148}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13,13,-w + 4]$ | $-1$ |