Base field 3.3.1573.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 7x + 2\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, -2w + 1]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $37$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} + 3x^{13} - 13x^{12} - 41x^{11} + 59x^{10} + 204x^{9} - 103x^{8} - 448x^{7} + 27x^{6} + 411x^{5} + 79x^{4} - 107x^{3} - 35x^{2} + 2x + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w^{2} - w - 7]$ | $...$ |
| 5 | $[5, 5, w - 1]$ | $...$ |
| 7 | $[7, 7, w + 1]$ | $...$ |
| 11 | $[11, 11, -w^{2} + 5]$ | $...$ |
| 13 | $[13, 13, w + 3]$ | $\phantom{-}\frac{1}{10}e^{13} + \frac{1}{10}e^{12} - 2e^{11} - \frac{8}{5}e^{10} + \frac{78}{5}e^{9} + \frac{46}{5}e^{8} - \frac{597}{10}e^{7} - \frac{112}{5}e^{6} + \frac{229}{2}e^{5} + \frac{98}{5}e^{4} - \frac{963}{10}e^{3} - \frac{13}{5}e^{2} + \frac{106}{5}e + \frac{33}{10}$ |
| 13 | $[13, 13, -2w + 1]$ | $-1$ |
| 19 | $[19, 19, 2w^{2} - w - 15]$ | $...$ |
| 25 | $[25, 5, w^{2} - 7]$ | $-3e^{13} - \frac{33}{4}e^{12} + \frac{165}{4}e^{11} + 113e^{10} - 208e^{9} - \frac{2255}{4}e^{8} + 465e^{7} + 1243e^{6} - \frac{859}{2}e^{5} - \frac{4597}{4}e^{4} + \frac{373}{4}e^{3} + \frac{615}{2}e^{2} + 10e - \frac{41}{4}$ |
| 27 | $[27, 3, 3]$ | $...$ |
| 31 | $[31, 31, w^{2} - 2w - 1]$ | $-\frac{109}{20}e^{13} - \frac{279}{20}e^{12} + 77e^{11} + \frac{1897}{10}e^{10} - \frac{8089}{20}e^{9} - \frac{9349}{10}e^{8} + \frac{9649}{10}e^{7} + \frac{20173}{10}e^{6} - \frac{4011}{4}e^{5} - \frac{35739}{20}e^{4} + \frac{1553}{5}e^{3} + \frac{4257}{10}e^{2} + \frac{287}{20}e - \frac{141}{10}$ |
| 37 | $[37, 37, -3w^{2} + 2w + 23]$ | $...$ |
| 41 | $[41, 41, -2w - 1]$ | $-\frac{17}{5}e^{13} - \frac{89}{10}e^{12} + \frac{95}{2}e^{11} + \frac{607}{5}e^{10} - \frac{1222}{5}e^{9} - \frac{6013}{10}e^{8} + \frac{2794}{5}e^{7} + \frac{6538}{5}e^{6} - 521e^{5} - \frac{11719}{10}e^{4} + \frac{897}{10}e^{3} + \frac{1432}{5}e^{2} + \frac{176}{5}e - \frac{77}{10}$ |
| 47 | $[47, 47, w^{2} - 3]$ | $\phantom{-}\frac{67}{10}e^{13} + \frac{86}{5}e^{12} - 94e^{11} - \frac{1161}{5}e^{10} + \frac{4897}{10}e^{9} + \frac{11319}{10}e^{8} - \frac{11589}{10}e^{7} - \frac{23983}{10}e^{6} + 1202e^{5} + \frac{20467}{10}e^{4} - \frac{3831}{10}e^{3} - \frac{4227}{10}e^{2} - \frac{98}{5}e + \frac{13}{5}$ |
| 49 | $[49, 7, w^{2} - 2w - 5]$ | $...$ |
| 59 | $[59, 59, 2w - 3]$ | $\phantom{-}\frac{3}{20}e^{13} + \frac{2}{5}e^{12} - \frac{9}{4}e^{11} - \frac{32}{5}e^{10} + \frac{233}{20}e^{9} + \frac{781}{20}e^{8} - \frac{213}{10}e^{7} - \frac{568}{5}e^{6} - \frac{13}{4}e^{5} + \frac{1599}{10}e^{4} + \frac{631}{20}e^{3} - \frac{477}{5}e^{2} - \frac{39}{20}e + \frac{229}{20}$ |
| 67 | $[67, 67, w - 5]$ | $...$ |
| 71 | $[71, 71, w^{2} - 2w - 11]$ | $-\frac{63}{5}e^{13} - \frac{637}{20}e^{12} + \frac{717}{4}e^{11} + \frac{4331}{10}e^{10} - \frac{9531}{10}e^{9} - \frac{42689}{20}e^{8} + \frac{23297}{10}e^{7} + \frac{23027}{5}e^{6} - 2566e^{5} - \frac{81497}{20}e^{4} + \frac{19671}{20}e^{3} + \frac{4803}{5}e^{2} - \frac{186}{5}e - \frac{651}{20}$ |
| 73 | $[73, 73, w^{2} + 1]$ | $\phantom{-}\frac{2}{5}e^{13} + \frac{12}{5}e^{12} - \frac{5}{2}e^{11} - \frac{339}{10}e^{10} - \frac{121}{10}e^{9} + \frac{1773}{10}e^{8} + \frac{631}{5}e^{7} - \frac{4201}{10}e^{6} - 319e^{5} + \frac{4379}{10}e^{4} + \frac{2743}{10}e^{3} - \frac{762}{5}e^{2} - \frac{447}{10}e + \frac{31}{5}$ |
| 83 | $[83, 83, -4w^{2} + 3w + 27]$ | $\phantom{-}\frac{11}{4}e^{13} + 7e^{12} - \frac{155}{4}e^{11} - 95e^{10} + \frac{809}{4}e^{9} + \frac{1865}{4}e^{8} - 477e^{7} - \frac{1991}{2}e^{6} + \frac{1965}{4}e^{5} + 850e^{4} - \frac{677}{4}e^{3} - \frac{321}{2}e^{2} + \frac{63}{4}e - \frac{13}{4}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -2w + 1]$ | $1$ |