Base field \(\Q(\sqrt{58}) \)
Generator \(w\), with minimal polynomial \(x^{2} - 58\); narrow class number \(2\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[8, 4, 2w]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $48$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} + 5x^{9} - 5x^{8} - 60x^{7} - 58x^{6} + 158x^{5} + 314x^{4} + 116x^{3} - 59x^{2} - 23x - 1\) |
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| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}0$ |
| 3 | $[3, 3, w + 1]$ | $-\frac{3}{4}e^{9} - 3e^{8} + \frac{25}{4}e^{7} + \frac{149}{4}e^{6} + \frac{47}{4}e^{5} - \frac{445}{4}e^{4} - \frac{535}{4}e^{3} - \frac{59}{4}e^{2} + \frac{55}{2}e + \frac{15}{4}$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -2w + 15]$ | $\phantom{-}\frac{1}{2}e^{9} + \frac{9}{4}e^{8} - \frac{7}{2}e^{7} - 28e^{6} - \frac{65}{4}e^{5} + \frac{335}{4}e^{4} + \frac{233}{2}e^{3} + 12e^{2} - \frac{107}{4}e - \frac{5}{2}$ |
| 7 | $[7, 7, -2w - 15]$ | $-\frac{1}{4}e^{8} - e^{7} + \frac{5}{2}e^{6} + \frac{51}{4}e^{5} - \frac{5}{4}e^{4} - 41e^{3} - \frac{55}{2}e^{2} + \frac{27}{4}e + 1$ |
| 11 | $[11, 11, w + 5]$ | $-e^{3} + 6e$ |
| 11 | $[11, 11, w + 6]$ | $\phantom{-}\frac{1}{4}e^{9} + e^{8} - \frac{9}{4}e^{7} - \frac{51}{4}e^{6} - \frac{9}{4}e^{5} + \frac{165}{4}e^{4} + \frac{171}{4}e^{3} - \frac{35}{4}e^{2} - \frac{41}{2}e + \frac{1}{4}$ |
| 19 | $[19, 19, w + 1]$ | $-\frac{1}{2}e^{8} - \frac{3}{2}e^{7} + 6e^{6} + \frac{39}{2}e^{5} - 15e^{4} - \frac{133}{2}e^{3} - 18e^{2} + \frac{53}{2}e + \frac{11}{2}$ |
| 19 | $[19, 19, w + 18]$ | $-\frac{1}{2}e^{9} - \frac{3}{2}e^{8} + \frac{13}{2}e^{7} + \frac{41}{2}e^{6} - 22e^{5} - 80e^{4} + \frac{15}{2}e^{3} + \frac{145}{2}e^{2} - \frac{1}{2}e - \frac{13}{2}$ |
| 23 | $[23, 23, w + 9]$ | $\phantom{-}\frac{3}{2}e^{9} + \frac{23}{4}e^{8} - \frac{27}{2}e^{7} - 72e^{6} - \frac{43}{4}e^{5} + \frac{881}{4}e^{4} + \frac{449}{2}e^{3} + 8e^{2} - \frac{145}{4}e - \frac{11}{2}$ |
| 23 | $[23, 23, -w + 9]$ | $-\frac{3}{4}e^{8} - 2e^{7} + \frac{17}{2}e^{6} + \frac{101}{4}e^{5} - \frac{67}{4}e^{4} - 80e^{3} - \frac{83}{2}e^{2} + \frac{37}{4}e + 6$ |
| 25 | $[25, 5, 5]$ | $-\frac{1}{2}e^{9} - e^{8} + 7e^{7} + \frac{27}{2}e^{6} - \frac{57}{2}e^{5} - 51e^{4} + 31e^{3} + \frac{79}{2}e^{2} - 16e - 3$ |
| 29 | $[29, 29, w]$ | $\phantom{-}e^{9} + 3e^{8} - \frac{23}{2}e^{7} - 39e^{6} + 25e^{5} + \frac{267}{2}e^{4} + \frac{89}{2}e^{3} - 55e^{2} - 4e - \frac{3}{2}$ |
| 37 | $[37, 37, w + 13]$ | $-\frac{3}{4}e^{9} - 3e^{8} + \frac{27}{4}e^{7} + \frac{153}{4}e^{6} + \frac{23}{4}e^{5} - \frac{491}{4}e^{4} - \frac{469}{4}e^{3} + \frac{65}{4}e^{2} + \frac{69}{2}e + \frac{25}{4}$ |
| 37 | $[37, 37, w + 24]$ | $-\frac{3}{4}e^{9} - 2e^{8} + \frac{39}{4}e^{7} + \frac{109}{4}e^{6} - \frac{133}{4}e^{5} - \frac{423}{4}e^{4} + \frac{55}{4}e^{3} + \frac{373}{4}e^{2} - \frac{9}{2}e - \frac{39}{4}$ |
| 43 | $[43, 43, w + 12]$ | $\phantom{-}\frac{1}{4}e^{9} + \frac{3}{2}e^{8} - \frac{1}{4}e^{7} - \frac{71}{4}e^{6} - \frac{107}{4}e^{5} + \frac{179}{4}e^{4} + \frac{467}{4}e^{3} + \frac{169}{4}e^{2} - 25e - \frac{35}{4}$ |
| 43 | $[43, 43, w + 31]$ | $\phantom{-}\frac{1}{2}e^{8} + e^{7} - 6e^{6} - \frac{25}{2}e^{5} + \frac{31}{2}e^{4} + 40e^{3} + 14e^{2} - \frac{21}{2}e - 9$ |
| 61 | $[61, 61, w + 27]$ | $\phantom{-}\frac{3}{4}e^{9} + 3e^{8} - \frac{13}{2}e^{7} - \frac{149}{4}e^{6} - \frac{33}{4}e^{5} + 111e^{4} + \frac{235}{2}e^{3} + \frac{59}{4}e^{2} - e - 2$ |
| 61 | $[61, 61, w + 34]$ | $-\frac{3}{4}e^{9} - 3e^{8} + 7e^{7} + \frac{153}{4}e^{6} + \frac{5}{4}e^{5} - \frac{247}{2}e^{4} - 93e^{3} + \frac{89}{4}e^{2} - 2e - \frac{5}{2}$ |
| 71 | $[71, 71, 12w - 91]$ | $\phantom{-}\frac{7}{4}e^{9} + \frac{15}{2}e^{8} - \frac{51}{4}e^{7} - \frac{367}{4}e^{6} - \frac{203}{4}e^{5} + \frac{1043}{4}e^{4} + \frac{1549}{4}e^{3} + \frac{361}{4}e^{2} - \frac{153}{2}e - \frac{35}{4}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, w]$ | $-1$ |