Base field 4.4.17989.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} - 3x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $36$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} + 2x^{13} - 22x^{12} - 46x^{11} + 177x^{10} + 391x^{9} - 644x^{8} - 1570x^{7} + 996x^{6} + 3089x^{5} - 270x^{4} - 2709x^{3} - 594x^{2} + 729x + 243\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w^{3} + 2w^{2} + 4w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^{3} - 2w^{2} - 6w + 2]$ | $\phantom{-}\frac{869}{4779}e^{13} + \frac{376}{4779}e^{12} - \frac{19502}{4779}e^{11} - \frac{9245}{4779}e^{10} + \frac{54599}{1593}e^{9} + \frac{78815}{4779}e^{8} - \frac{647110}{4779}e^{7} - \frac{312116}{4779}e^{6} + \frac{408860}{1593}e^{5} + \frac{616861}{4779}e^{4} - \frac{336176}{1593}e^{3} - \frac{60961}{531}e^{2} + \frac{3107}{59}e + \frac{1605}{59}$ |
| 11 | $[11, 11, w^{3} - 2w^{2} - 4w]$ | $\phantom{-}\frac{193}{531}e^{13} + \frac{406}{1593}e^{12} - \frac{13297}{1593}e^{11} - \frac{9730}{1593}e^{10} + \frac{115601}{1593}e^{9} + \frac{9266}{177}e^{8} - \frac{481859}{1593}e^{7} - \frac{331916}{1593}e^{6} + \frac{992234}{1593}e^{5} + \frac{213505}{531}e^{4} - \frac{917209}{1593}e^{3} - \frac{19987}{59}e^{2} + \frac{29686}{177}e + \frac{4730}{59}$ |
| 13 | $[13, 13, w^{2} - 3w - 2]$ | $-\frac{575}{4779}e^{13} - \frac{913}{4779}e^{12} + \frac{13187}{4779}e^{11} + \frac{20552}{4779}e^{10} - \frac{38048}{1593}e^{9} - \frac{168278}{4779}e^{8} + \frac{476224}{4779}e^{7} + \frac{635030}{4779}e^{6} - \frac{331991}{1593}e^{5} - \frac{1126384}{4779}e^{4} + \frac{316994}{1593}e^{3} + \frac{91972}{531}e^{2} - \frac{10490}{177}e - \frac{2032}{59}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}1$ |
| 17 | $[17, 17, -w^{3} + 3w^{2} + 2w - 2]$ | $-\frac{5977}{4779}e^{13} - \frac{2999}{4779}e^{12} + \frac{135310}{4779}e^{11} + \frac{71425}{4779}e^{10} - \frac{383591}{1593}e^{9} - \frac{595534}{4779}e^{8} + \frac{4635824}{4779}e^{7} + \frac{2307262}{4779}e^{6} - \frac{3023717}{1593}e^{5} - \frac{4454246}{4779}e^{4} + \frac{873394}{531}e^{3} + \frac{432823}{531}e^{2} - \frac{79924}{177}e - \frac{11934}{59}$ |
| 17 | $[17, 17, -w^{2} + 2w + 3]$ | $\phantom{-}\frac{464}{531}e^{13} + \frac{446}{1593}e^{12} - \frac{31613}{1593}e^{11} - \frac{11360}{1593}e^{10} + \frac{270505}{1593}e^{9} + \frac{32770}{531}e^{8} - \frac{1099462}{1593}e^{7} - \frac{395722}{1593}e^{6} + \frac{2177350}{1593}e^{5} + \frac{89899}{177}e^{4} - \frac{1912163}{1593}e^{3} - \frac{254294}{531}e^{2} + \frac{58376}{177}e + \frac{7547}{59}$ |
| 23 | $[23, 23, -w^{3} + w^{2} + 6w + 1]$ | $-\frac{4106}{4779}e^{13} - \frac{1567}{4779}e^{12} + \frac{93740}{4779}e^{11} + \frac{39149}{4779}e^{10} - \frac{268958}{1593}e^{9} - \frac{338912}{4779}e^{8} + \frac{3300457}{4779}e^{7} + \frac{1372349}{4779}e^{6} - \frac{2191304}{1593}e^{5} - \frac{2809840}{4779}e^{4} + \frac{1928819}{1593}e^{3} + \frac{292130}{531}e^{2} - \frac{19558}{59}e - \frac{8573}{59}$ |
| 27 | $[27, 3, -2w^{3} + 3w^{2} + 11w - 1]$ | $\phantom{-}\frac{2138}{4779}e^{13} + \frac{1795}{4779}e^{12} - \frac{49001}{4779}e^{11} - \frac{40121}{4779}e^{10} + \frac{46898}{531}e^{9} + \frac{321167}{4779}e^{8} - \frac{1727212}{4779}e^{7} - \frac{1195379}{4779}e^{6} + \frac{42624}{59}e^{5} + \frac{2191069}{4779}e^{4} - \frac{1028095}{1593}e^{3} - \frac{199865}{531}e^{2} + \frac{10903}{59}e + \frac{5152}{59}$ |
| 29 | $[29, 29, w^{3} - 2w^{2} - 5w + 4]$ | $\phantom{-}\frac{85}{177}e^{13} + \frac{106}{1593}e^{12} - \frac{17173}{1593}e^{11} - \frac{2992}{1593}e^{10} + \frac{144323}{1593}e^{9} + \frac{8503}{531}e^{8} - \frac{569276}{1593}e^{7} - \frac{101525}{1593}e^{6} + \frac{1069976}{1593}e^{5} + \frac{74911}{531}e^{4} - \frac{857902}{1593}e^{3} - \frac{79375}{531}e^{2} + \frac{7515}{59}e + \frac{2276}{59}$ |
| 31 | $[31, 31, -w^{3} + 2w^{2} + 6w - 3]$ | $\phantom{-}\frac{355}{1593}e^{13} - \frac{53}{1593}e^{12} - \frac{2674}{531}e^{11} + \frac{302}{531}e^{10} + \frac{67639}{1593}e^{9} - \frac{6884}{1593}e^{8} - \frac{88237}{531}e^{7} + \frac{21115}{1593}e^{6} + \frac{482290}{1593}e^{5} - \frac{1240}{1593}e^{4} - \frac{362534}{1593}e^{3} - \frac{2313}{59}e^{2} + \frac{2998}{59}e + \frac{1399}{59}$ |
| 31 | $[31, 31, -w^{3} + 2w^{2} + 5w + 1]$ | $\phantom{-}\frac{875}{1593}e^{13} + \frac{707}{1593}e^{12} - \frac{2218}{177}e^{11} - \frac{5557}{531}e^{10} + \frac{172148}{1593}e^{9} + \frac{141056}{1593}e^{8} - \frac{237509}{531}e^{7} - \frac{554353}{1593}e^{6} + \frac{1466087}{1593}e^{5} + \frac{1056523}{1593}e^{4} - \frac{1377034}{1593}e^{3} - \frac{294569}{531}e^{2} + \frac{15450}{59}e + \frac{7781}{59}$ |
| 31 | $[31, 31, w^{3} - 2w^{2} - 6w]$ | $\phantom{-}\frac{10646}{4779}e^{13} + \frac{6115}{4779}e^{12} - \frac{242615}{4779}e^{11} - \frac{143591}{4779}e^{10} + \frac{693860}{1593}e^{9} + \frac{1191704}{4779}e^{8} - \frac{8489941}{4779}e^{7} - \frac{4602005}{4779}e^{6} + \frac{5635652}{1593}e^{5} + \frac{8803069}{4779}e^{4} - \frac{4994345}{1593}e^{3} - \frac{838382}{531}e^{2} + \frac{155474}{177}e + \frac{22371}{59}$ |
| 31 | $[31, 31, -w^{2} + 2w + 1]$ | $-\frac{2275}{4779}e^{13} - \frac{965}{4779}e^{12} + \frac{52090}{4779}e^{11} + \frac{23674}{4779}e^{10} - \frac{50045}{531}e^{9} - \frac{203410}{4779}e^{8} + \frac{1856063}{4779}e^{7} + \frac{821452}{4779}e^{6} - \frac{415330}{531}e^{5} - \frac{1670198}{4779}e^{4} + \frac{1109795}{1593}e^{3} + \frac{167863}{531}e^{2} - \frac{33326}{177}e - \frac{4415}{59}$ |
| 37 | $[37, 37, w^{3} - w^{2} - 7w - 1]$ | $-\frac{926}{531}e^{13} - \frac{364}{531}e^{12} + \frac{7033}{177}e^{11} + \frac{8992}{531}e^{10} - \frac{181093}{531}e^{9} - \frac{8576}{59}e^{8} + \frac{738073}{531}e^{7} + \frac{103621}{177}e^{6} - \frac{1464479}{531}e^{5} - \frac{636038}{531}e^{4} + \frac{143058}{59}e^{3} + \frac{594965}{531}e^{2} - \frac{118720}{177}e - \frac{17143}{59}$ |
| 43 | $[43, 43, w^{2} - w - 4]$ | $\phantom{-}\frac{397}{4779}e^{13} - \frac{1099}{4779}e^{12} - \frac{8056}{4779}e^{11} + \frac{23618}{4779}e^{10} + \frac{6439}{531}e^{9} - \frac{192290}{4779}e^{8} - \frac{165257}{4779}e^{7} + \frac{731594}{4779}e^{6} + \frac{10892}{531}e^{5} - \frac{1297276}{4779}e^{4} + \frac{74995}{1593}e^{3} + \frac{105478}{531}e^{2} - \frac{7022}{177}e - \frac{2407}{59}$ |
| 71 | $[71, 71, w^{3} - 2w^{2} - 6w - 1]$ | $-\frac{8431}{4779}e^{13} - \frac{3563}{4779}e^{12} + \frac{190759}{4779}e^{11} + \frac{85735}{4779}e^{10} - \frac{540862}{1593}e^{9} - \frac{712783}{4779}e^{8} + \frac{6542996}{4779}e^{7} + \frac{2745700}{4779}e^{6} - \frac{4276036}{1593}e^{5} - \frac{5309357}{4779}e^{4} + \frac{3710611}{1593}e^{3} + \frac{174509}{177}e^{2} - \frac{112579}{177}e - \frac{15079}{59}$ |
| 71 | $[71, 71, 5w^{3} - 8w^{2} - 29w + 3]$ | $-\frac{928}{531}e^{13} - \frac{1187}{1593}e^{12} + \frac{63167}{1593}e^{11} + \frac{28679}{1593}e^{10} - \frac{539122}{1593}e^{9} - \frac{79936}{531}e^{8} + \frac{2182345}{1593}e^{7} + \frac{927793}{1593}e^{6} - \frac{4297234}{1593}e^{5} - \frac{597155}{531}e^{4} + \frac{3750989}{1593}e^{3} + \frac{525403}{531}e^{2} - \frac{38406}{59}e - \frac{14740}{59}$ |
| 89 | $[89, 89, -2w^{3} + 4w^{2} + 10w - 7]$ | $-\frac{682}{1593}e^{13} + \frac{392}{1593}e^{12} + \frac{4934}{531}e^{11} - \frac{301}{59}e^{10} - \frac{118648}{1593}e^{9} + \frac{67223}{1593}e^{8} + \frac{144190}{531}e^{7} - \frac{258661}{1593}e^{6} - \frac{696697}{1593}e^{5} + \frac{429199}{1593}e^{4} + \frac{386642}{1593}e^{3} - \frac{79567}{531}e^{2} - \frac{1684}{177}e + \frac{856}{59}$ |
| 97 | $[97, 97, -w^{3} + 2w^{2} + 4w - 4]$ | $\phantom{-}\frac{2497}{4779}e^{13} + \frac{704}{4779}e^{12} - \frac{55540}{4779}e^{11} - \frac{16711}{4779}e^{10} + \frac{153637}{1593}e^{9} + \frac{131695}{4779}e^{8} - \frac{1792061}{4779}e^{7} - \frac{491485}{4779}e^{6} + \frac{1106815}{1593}e^{5} + \frac{1023749}{4779}e^{4} - \frac{876607}{1593}e^{3} - \frac{120572}{531}e^{2} + \frac{21857}{177}e + \frac{3629}{59}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $16$ | $[16, 2, 2]$ | $-1$ |