Base field 6.6.1397493.1
Generator \(w\), with minimal polynomial \(x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1\); narrow class number \(2\) and class number \(1\).
Form
Weight: | $[2, 2, 2, 2, 2, 2]$ |
Level: | $[57, 57, -w^{5} + 4w^{4} - 10w^{2} + w + 3]$ |
Dimension: | $14$ |
CM: | no |
Base change: | no |
Newspace dimension: | $38$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{14} - 152x^{12} + 9244x^{10} - 283408x^{8} + 4477408x^{6} - 31992576x^{4} + 60136960x^{2} - 28901376\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
3 | $[3, 3, w - 1]$ | $\phantom{-}1$ |
17 | $[17, 17, -w^{2} + 2w + 1]$ | $\phantom{-}e$ |
17 | $[17, 17, -w^{3} + w^{2} + 4w]$ | $-\frac{2129}{485173248}e^{13} + \frac{31757}{60646656}e^{11} - \frac{931285}{40431104}e^{9} + \frac{13447745}{30323328}e^{7} - \frac{48547775}{15161664}e^{5} + \frac{227129}{270744}e^{3} + \frac{2333615}{315868}e$ |
19 | $[19, 19, w^{5} - 3w^{4} - 2w^{3} + 8w^{2} + w - 4]$ | $\phantom{-}1$ |
19 | $[19, 19, w^{2} - w - 1]$ | $-\frac{3653}{4331904}e^{12} + \frac{230617}{2165952}e^{10} - \frac{912475}{180496}e^{8} + \frac{14746937}{135372}e^{6} - \frac{10991267}{11281}e^{4} + \frac{21935946}{11281}e^{2} - \frac{10791388}{11281}$ |
37 | $[37, 37, w^{4} - 2w^{3} - 3w^{2} + 3w + 2]$ | $-\frac{6949}{17327616}e^{12} + \frac{218695}{4331904}e^{10} - \frac{10346743}{4331904}e^{8} + \frac{578049}{11281}e^{6} - \frac{82200329}{180496}e^{4} + \frac{120875965}{135372}e^{2} - \frac{4779382}{11281}$ |
37 | $[37, 37, w^{4} - 4w^{3} + w^{2} + 7w - 3]$ | $\phantom{-}2$ |
53 | $[53, 53, 2w^{5} - 6w^{4} - 6w^{3} + 18w^{2} + 9w - 6]$ | $-\frac{24203}{161724416}e^{13} + \frac{143455}{7580832}e^{11} - \frac{36391989}{40431104}e^{9} + \frac{589665119}{30323328}e^{7} - \frac{2650368127}{15161664}e^{5} + \frac{48359717}{135372}e^{3} - \frac{57328083}{315868}e$ |
53 | $[53, 53, -w^{5} + 3w^{4} + 3w^{3} - 9w^{2} - 3w + 2]$ | $-\frac{17119}{242586624}e^{13} + \frac{67835}{7580832}e^{11} - \frac{25910329}{60646656}e^{9} + \frac{46899339}{5053888}e^{7} - \frac{212771479}{2526944}e^{5} + \frac{6024752}{33843}e^{3} - \frac{14572997}{157934}e$ |
64 | $[64, 2, -2]$ | $-\frac{25}{597504}e^{12} + \frac{727}{149376}e^{10} - \frac{31151}{149376}e^{8} + \frac{73781}{18672}e^{6} - \frac{188221}{6224}e^{4} + \frac{252731}{4668}e^{2} - \frac{10883}{389}$ |
71 | $[71, 71, -w^{5} + 4w^{4} - w^{3} - 8w^{2} + 3w - 1]$ | $-\frac{18721}{69310464}e^{13} + \frac{295909}{8663808}e^{11} - \frac{28131047}{17327616}e^{9} + \frac{50510719}{1443968}e^{7} - \frac{674518519}{2165952}e^{5} + \frac{161422661}{270744}e^{3} - \frac{12248721}{45124}e$ |
71 | $[71, 71, 2w^{5} - 5w^{4} - 8w^{3} + 15w^{2} + 12w - 6]$ | $\phantom{-}\frac{36367}{485173248}e^{13} - \frac{191479}{20215552}e^{11} + \frac{54614513}{121293312}e^{9} - \frac{294843779}{30323328}e^{7} + \frac{1325176649}{15161664}e^{5} - \frac{16141715}{90248}e^{3} + \frac{26812379}{315868}e$ |
71 | $[71, 71, 2w^{5} - 6w^{4} - 6w^{3} + 18w^{2} + 9w - 7]$ | $\phantom{-}\frac{6087}{40431104}e^{13} - \frac{385031}{20215552}e^{11} + \frac{27479135}{30323328}e^{9} - \frac{296681527}{15161664}e^{7} + \frac{221424061}{1263472}e^{5} - \frac{94021999}{270744}e^{3} + \frac{25299969}{157934}e$ |
73 | $[73, 73, -2w^{5} + 6w^{4} + 5w^{3} - 16w^{2} - 7w + 3]$ | $\phantom{-}\frac{905}{2887936}e^{12} - \frac{85643}{2165952}e^{10} + \frac{4062037}{2165952}e^{8} - \frac{10922959}{270744}e^{6} + \frac{97376789}{270744}e^{4} - \frac{47855705}{67686}e^{2} + \frac{3694790}{11281}$ |
73 | $[73, 73, -2w^{5} + 6w^{4} + 5w^{3} - 17w^{2} - 6w + 6]$ | $\phantom{-}\frac{1519}{17327616}e^{12} - \frac{15803}{1443968}e^{10} + \frac{2222669}{4331904}e^{8} - \frac{2950217}{270744}e^{6} + \frac{51847409}{541488}e^{4} - \frac{8388185}{45124}e^{2} + \frac{1152278}{11281}$ |
89 | $[89, 89, w^{5} - 2w^{4} - 5w^{3} + 6w^{2} + 8w - 4]$ | $-\frac{125}{485173248}e^{13} + \frac{617}{30323328}e^{11} - \frac{53059}{121293312}e^{9} + \frac{22439}{30323328}e^{7} + \frac{4943}{5053888}e^{5} + \frac{16357}{33843}e^{3} + \frac{3703325}{315868}e$ |
89 | $[89, 89, w^{5} - w^{4} - 9w^{3} + 7w^{2} + 16w - 6]$ | $\phantom{-}\frac{18721}{69310464}e^{13} - \frac{295909}{8663808}e^{11} + \frac{28131047}{17327616}e^{9} - \frac{50510719}{1443968}e^{7} + \frac{674518519}{2165952}e^{5} - \frac{161422661}{270744}e^{3} + \frac{12293845}{45124}e$ |
89 | $[89, 89, 2w^{5} - 6w^{4} - 4w^{3} + 15w^{2} + 3w - 3]$ | $-\frac{5}{5576704}e^{13} + \frac{257}{2091264}e^{11} - \frac{25537}{4182528}e^{9} + \frac{42505}{348544}e^{7} - \frac{231745}{522816}e^{5} - \frac{31005}{3112}e^{3} + \frac{232005}{10892}e$ |
89 | $[89, 89, w^{5} - 2w^{4} - 6w^{3} + 8w^{2} + 10w - 6]$ | $-\frac{68351}{485173248}e^{13} + \frac{542063}{30323328}e^{11} - \frac{34529419}{40431104}e^{9} + \frac{562769629}{30323328}e^{7} - \frac{851090859}{5053888}e^{5} + \frac{4011049}{11281}e^{3} - \frac{61995313}{315868}e$ |
107 | $[107, 107, w^{5} - 2w^{4} - 7w^{3} + 10w^{2} + 12w - 6]$ | $\phantom{-}\frac{203471}{485173248}e^{13} - \frac{1607009}{30323328}e^{11} + \frac{305458841}{121293312}e^{9} - \frac{548912387}{10107776}e^{7} + \frac{2455082499}{5053888}e^{5} - \frac{130685993}{135372}e^{3} + \frac{151866665}{315868}e$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$3$ | $[3, 3, w - 1]$ | $-1$ |
$19$ | $[19, 19, w^{5} - 3w^{4} - 2w^{3} + 8w^{2} + w - 4]$ | $-1$ |