Base field \(\Q(\sqrt{393}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 98\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[1, 1, 1]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $31$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - 30x^{14} + 378x^{12} - 2602x^{10} + 10654x^{8} - 26443x^{6} + 38661x^{4} - 30305x^{2} + 9710\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -17w - 160]$ | $\phantom{-}e$ |
| 2 | $[2, 2, -17w + 177]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -842w + 8767]$ | $-\frac{1}{116}e^{14} + \frac{21}{116}e^{12} - \frac{189}{116}e^{10} + \frac{1017}{116}e^{8} - \frac{3589}{116}e^{6} + \frac{129}{2}e^{4} - \frac{7283}{116}e^{2} + \frac{1113}{58}$ |
| 7 | $[7, 7, -2w + 21]$ | $\phantom{-}\frac{7}{58}e^{14} - \frac{88}{29}e^{12} + \frac{1787}{58}e^{10} - \frac{4705}{29}e^{8} + \frac{27443}{58}e^{6} - \frac{1511}{2}e^{4} + \frac{17646}{29}e^{2} - \frac{5587}{29}$ |
| 7 | $[7, 7, 2w + 19]$ | $\phantom{-}\frac{7}{58}e^{14} - \frac{88}{29}e^{12} + \frac{1787}{58}e^{10} - \frac{4705}{29}e^{8} + \frac{27443}{58}e^{6} - \frac{1511}{2}e^{4} + \frac{17646}{29}e^{2} - \frac{5587}{29}$ |
| 13 | $[13, 13, -12w - 113]$ | $-\frac{7}{58}e^{14} + \frac{88}{29}e^{12} - \frac{1787}{58}e^{10} + \frac{4705}{29}e^{8} - \frac{27443}{58}e^{6} + \frac{1509}{2}e^{4} - \frac{17414}{29}e^{2} + \frac{5239}{29}$ |
| 13 | $[13, 13, 12w - 125]$ | $-\frac{7}{58}e^{14} + \frac{88}{29}e^{12} - \frac{1787}{58}e^{10} + \frac{4705}{29}e^{8} - \frac{27443}{58}e^{6} + \frac{1509}{2}e^{4} - \frac{17414}{29}e^{2} + \frac{5239}{29}$ |
| 17 | $[17, 17, 182w - 1895]$ | $\phantom{-}\frac{11}{116}e^{15} - \frac{289}{116}e^{13} + \frac{3123}{116}e^{11} - \frac{17973}{116}e^{9} + \frac{59431}{116}e^{7} - 971e^{5} + \frac{112419}{116}e^{3} - \frac{22393}{58}e$ |
| 17 | $[17, 17, 182w + 1713]$ | $\phantom{-}\frac{11}{116}e^{15} - \frac{289}{116}e^{13} + \frac{3123}{116}e^{11} - \frac{17973}{116}e^{9} + \frac{59431}{116}e^{7} - 971e^{5} + \frac{112419}{116}e^{3} - \frac{22393}{58}e$ |
| 23 | $[23, 23, -512w - 4819]$ | $-\frac{5}{29}e^{15} + \frac{134}{29}e^{13} - \frac{1467}{29}e^{11} + \frac{8449}{29}e^{9} - \frac{27428}{29}e^{7} + 1714e^{5} - \frac{46014}{29}e^{3} + \frac{16553}{29}e$ |
| 23 | $[23, 23, 512w - 5331]$ | $-\frac{5}{29}e^{15} + \frac{134}{29}e^{13} - \frac{1467}{29}e^{11} + \frac{8449}{29}e^{9} - \frac{27428}{29}e^{7} + 1714e^{5} - \frac{46014}{29}e^{3} + \frac{16553}{29}e$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{15}{116}e^{14} - \frac{431}{116}e^{12} + \frac{5039}{116}e^{10} - \frac{30799}{116}e^{8} + \frac{105223}{116}e^{6} - \frac{3427}{2}e^{4} + \frac{189981}{116}e^{2} - \frac{34733}{58}$ |
| 29 | $[29, 29, 22w - 229]$ | $\phantom{-}\frac{7}{116}e^{15} - \frac{205}{116}e^{13} + \frac{2483}{116}e^{11} - \frac{15993}{116}e^{9} + \frac{58531}{116}e^{7} - 1036e^{5} + \frac{126555}{116}e^{3} - \frac{26119}{58}e$ |
| 29 | $[29, 29, 22w + 207]$ | $\phantom{-}\frac{7}{116}e^{15} - \frac{205}{116}e^{13} + \frac{2483}{116}e^{11} - \frac{15993}{116}e^{9} + \frac{58531}{116}e^{7} - 1036e^{5} + \frac{126555}{116}e^{3} - \frac{26119}{58}e$ |
| 43 | $[43, 43, 114w + 1073]$ | $-\frac{13}{29}e^{14} + \frac{691}{58}e^{12} - \frac{3762}{29}e^{10} + \frac{43291}{58}e^{8} - \frac{70640}{29}e^{6} + \frac{8951}{2}e^{4} - \frac{246401}{58}e^{2} + \frac{46164}{29}$ |
| 43 | $[43, 43, 114w - 1187]$ | $-\frac{13}{29}e^{14} + \frac{691}{58}e^{12} - \frac{3762}{29}e^{10} + \frac{43291}{58}e^{8} - \frac{70640}{29}e^{6} + \frac{8951}{2}e^{4} - \frac{246401}{58}e^{2} + \frac{46164}{29}$ |
| 47 | $[47, 47, 8w - 83]$ | $-\frac{1}{58}e^{15} + \frac{21}{58}e^{13} - \frac{131}{58}e^{11} - \frac{85}{58}e^{9} + \frac{4067}{58}e^{7} - 281e^{5} + \frac{25023}{58}e^{3} - \frac{6514}{29}e$ |
| 47 | $[47, 47, -8w - 75]$ | $-\frac{1}{58}e^{15} + \frac{21}{58}e^{13} - \frac{131}{58}e^{11} - \frac{85}{58}e^{9} + \frac{4067}{58}e^{7} - 281e^{5} + \frac{25023}{58}e^{3} - \frac{6514}{29}e$ |
| 61 | $[61, 61, -1172w - 11031]$ | $-\frac{25}{116}e^{14} + \frac{641}{116}e^{12} - \frac{6697}{116}e^{10} + \frac{36793}{116}e^{8} - \frac{114433}{116}e^{6} + \frac{3477}{2}e^{4} - \frac{187875}{116}e^{2} + \frac{35829}{58}$ |
| 61 | $[61, 61, 1172w - 12203]$ | $-\frac{25}{116}e^{14} + \frac{641}{116}e^{12} - \frac{6697}{116}e^{10} + \frac{36793}{116}e^{8} - \frac{114433}{116}e^{6} + \frac{3477}{2}e^{4} - \frac{187875}{116}e^{2} + \frac{35829}{58}$ |
Atkin-Lehner eigenvalues
This form has no Atkin-Lehner eigenvalues since the level is \((1)\).