Base field \(\Q(\sqrt{2}, \sqrt{13})\)
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 9x^{2} + 10x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[23,23,-\frac{1}{5}w^{3} - \frac{1}{5}w^{2} + \frac{11}{5}w + \frac{3}{5}]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $16$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 4x^{8} - 11x^{7} + 57x^{6} + 17x^{5} - 251x^{4} + 119x^{3} + 319x^{2} - 266x + 43\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{2}{5}w^{3} + \frac{3}{5}w^{2} + \frac{17}{5}w - \frac{9}{5}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -\frac{2}{5}w^{3} + \frac{3}{5}w^{2} + \frac{22}{5}w - \frac{9}{5}]$ | $\phantom{-}e^{2} - 3$ |
| 9 | $[9, 3, \frac{2}{5}w^{3} - \frac{3}{5}w^{2} - \frac{22}{5}w + \frac{14}{5}]$ | $-\frac{5}{2}e^{8} + 4e^{7} + 37e^{6} - 54e^{5} - \frac{339}{2}e^{4} + \frac{447}{2}e^{3} + \frac{441}{2}e^{2} - \frac{543}{2}e + \frac{99}{2}$ |
| 17 | $[17, 17, w + 1]$ | $-e^{3} + 5e + 2$ |
| 17 | $[17, 17, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{13}{5}]$ | $\phantom{-}\frac{9}{4}e^{8} - \frac{13}{4}e^{7} - \frac{67}{2}e^{6} + \frac{175}{4}e^{5} + \frac{309}{2}e^{4} - \frac{721}{4}e^{3} - 204e^{2} + \frac{871}{4}e - \frac{145}{4}$ |
| 17 | $[17, 17, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{28}{5}]$ | $-2e^{8} + 3e^{7} + \frac{59}{2}e^{6} - 40e^{5} - \frac{269}{2}e^{4} + \frac{327}{2}e^{3} + 175e^{2} - \frac{395}{2}e + 33$ |
| 17 | $[17, 17, -w + 2]$ | $\phantom{-}\frac{9}{4}e^{8} - \frac{13}{4}e^{7} - \frac{67}{2}e^{6} + \frac{171}{4}e^{5} + \frac{311}{2}e^{4} - \frac{685}{4}e^{3} - 212e^{2} + \frac{799}{4}e - \frac{101}{4}$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{6}{5}w - \frac{22}{5}]$ | $-\frac{5}{4}e^{8} + \frac{7}{4}e^{7} + \frac{37}{2}e^{6} - \frac{93}{4}e^{5} - 85e^{4} + \frac{377}{4}e^{3} + \frac{225}{2}e^{2} - \frac{447}{4}e + \frac{85}{4}$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} - \frac{1}{5}w^{2} + \frac{11}{5}w + \frac{3}{5}]$ | $-1$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{6}{5}w - \frac{12}{5}]$ | $-2e^{8} + 3e^{7} + 30e^{6} - \frac{81}{2}e^{5} - 140e^{4} + \frac{333}{2}e^{3} + \frac{381}{2}e^{2} - 197e + \frac{57}{2}$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} - \frac{1}{5}w^{2} + \frac{11}{5}w + \frac{13}{5}]$ | $\phantom{-}\frac{15}{4}e^{8} - \frac{23}{4}e^{7} - \frac{111}{2}e^{6} + \frac{309}{4}e^{5} + \frac{509}{2}e^{4} - \frac{1267}{4}e^{3} - 334e^{2} + \frac{1509}{4}e - \frac{255}{4}$ |
| 25 | $[25, 5, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{33}{5}]$ | $\phantom{-}4e^{8} - \frac{13}{2}e^{7} - \frac{117}{2}e^{6} + 87e^{5} + 264e^{4} - 356e^{3} - 336e^{2} + \frac{853}{2}e - \frac{155}{2}$ |
| 25 | $[25, 5, -w^{3} + w^{2} + 10w - 2]$ | $-\frac{1}{2}e^{8} + e^{7} + 7e^{6} - \frac{27}{2}e^{5} - \frac{59}{2}e^{4} + 56e^{3} + 31e^{2} - \frac{139}{2}e + 22$ |
| 49 | $[49, 7, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{34}{5}w - \frac{23}{5}]$ | $\phantom{-}\frac{1}{4}e^{8} - \frac{1}{4}e^{7} - \frac{7}{2}e^{6} + \frac{17}{4}e^{5} + \frac{27}{2}e^{4} - \frac{95}{4}e^{3} - \frac{13}{2}e^{2} + \frac{167}{4}e - \frac{71}{4}$ |
| 49 | $[49, 7, \frac{4}{5}w^{3} - \frac{6}{5}w^{2} - \frac{34}{5}w + \frac{13}{5}]$ | $\phantom{-}\frac{5}{4}e^{8} - \frac{7}{4}e^{7} - 19e^{6} + \frac{91}{4}e^{5} + \frac{181}{2}e^{4} - \frac{353}{4}e^{3} - 128e^{2} + \frac{377}{4}e - \frac{47}{4}$ |
| 79 | $[79, 79, \frac{3}{5}w^{3} - \frac{7}{5}w^{2} - \frac{28}{5}w + \frac{21}{5}]$ | $-\frac{31}{4}e^{8} + \frac{49}{4}e^{7} + 114e^{6} - \frac{657}{4}e^{5} - \frac{1037}{2}e^{4} + \frac{2691}{4}e^{3} + 668e^{2} - \frac{3215}{4}e + \frac{617}{4}$ |
| 79 | $[79, 79, \frac{1}{5}w^{3} + \frac{1}{5}w^{2} - \frac{16}{5}w - \frac{13}{5}]$ | $-2e^{8} + 3e^{7} + 29e^{6} - 41e^{5} - 128e^{4} + 174e^{3} + 151e^{2} - 224e + 54$ |
| 79 | $[79, 79, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{11}{5}w - \frac{27}{5}]$ | $\phantom{-}\frac{1}{4}e^{8} - \frac{3}{4}e^{7} - \frac{7}{2}e^{6} + \frac{43}{4}e^{5} + 14e^{4} - \frac{199}{4}e^{3} - 7e^{2} + \frac{303}{4}e - \frac{107}{4}$ |
| 79 | $[79, 79, \frac{3}{5}w^{3} - \frac{2}{5}w^{2} - \frac{33}{5}w + \frac{11}{5}]$ | $-\frac{9}{2}e^{8} + 7e^{7} + \frac{133}{2}e^{6} - \frac{185}{2}e^{5} - 305e^{4} + \frac{743}{2}e^{3} + 402e^{2} - 433e + 76$ |
| 103 | $[103, 103, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{1}{5}w - \frac{17}{5}]$ | $\phantom{-}\frac{7}{2}e^{8} - 5e^{7} - 52e^{6} + \frac{133}{2}e^{5} + \frac{479}{2}e^{4} - 269e^{3} - 319e^{2} + \frac{627}{2}e - 48$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $23$ | $[23,23,-\frac{1}{5}w^{3} - \frac{1}{5}w^{2} + \frac{11}{5}w + \frac{3}{5}]$ | $1$ |