Base field 5.5.138136.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 6x^{3} + 3x^{2} + 4x - 2\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[19, 19, -w^{3} + w^{2} + 5w - 1]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $29$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - 4x^{15} - 22x^{14} + 101x^{13} + 177x^{12} - 1035x^{11} - 551x^{10} + 5544x^{9} - 306x^{8} - 16618x^{7} + 6443x^{6} + 27546x^{5} - 16204x^{4} - 23056x^{3} + 16288x^{2} + 7488x - 5760\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -2w^{4} + w^{3} + 12w^{2} + w - 5]$ | $\phantom{-}\frac{27}{32}e^{15} - \frac{35}{16}e^{14} - \frac{347}{16}e^{13} + \frac{1755}{32}e^{12} + \frac{7281}{32}e^{11} - \frac{17807}{32}e^{10} - \frac{40195}{32}e^{9} + \frac{47053}{16}e^{8} + \frac{62531}{16}e^{7} - \frac{138553}{16}e^{6} - \frac{216795}{32}e^{5} + \frac{56105}{4}e^{4} + \frac{47913}{8}e^{3} - \frac{45503}{4}e^{2} - 2089e + 3521$ |
| 8 | $[8, 2, w^{4} - 7w^{2} - 3w + 5]$ | $\phantom{-}\frac{213}{16}e^{15} - \frac{531}{16}e^{14} - \frac{2743}{8}e^{13} + \frac{13243}{16}e^{12} + \frac{28829}{8}e^{11} - \frac{66757}{8}e^{10} - \frac{159309}{8}e^{9} + \frac{700333}{16}e^{8} + \frac{495361}{8}e^{7} - \frac{511307}{4}e^{6} - \frac{1711779}{16}e^{5} + \frac{3285149}{16}e^{4} + \frac{751449}{8}e^{3} - \frac{660989}{4}e^{2} - 32378e + 50847$ |
| 19 | $[19, 19, -2w^{4} + w^{3} + 12w^{2} - 5]$ | $\phantom{-}17e^{15} - \frac{339}{8}e^{14} - 438e^{13} + \frac{4229}{4}e^{12} + \frac{36841}{8}e^{11} - \frac{85311}{8}e^{10} - \frac{203667}{8}e^{9} + \frac{447705}{8}e^{8} + \frac{158395}{2}e^{7} - \frac{654041}{4}e^{6} - \frac{547649}{4}e^{5} + \frac{2101991}{8}e^{4} + \frac{481143}{4}e^{3} - \frac{423075}{2}e^{2} - 41496e + 65108$ |
| 19 | $[19, 19, -w^{3} + w^{2} + 5w - 1]$ | $\phantom{-}1$ |
| 29 | $[29, 29, 2w^{4} - 2w^{3} - 11w^{2} + 4w + 3]$ | $-\frac{63}{32}e^{15} + \frac{19}{4}e^{14} + \frac{815}{16}e^{13} - \frac{3787}{32}e^{12} - \frac{17211}{32}e^{11} + \frac{38133}{32}e^{10} + \frac{95505}{32}e^{9} - 6242e^{8} - \frac{148899}{16}e^{7} + \frac{291251}{16}e^{6} + \frac{514667}{32}e^{5} - \frac{467221}{16}e^{4} - \frac{56295}{4}e^{3} + 23468e^{2} + 4819e - 7209$ |
| 31 | $[31, 31, -2w^{4} + w^{3} + 12w^{2} + 2w - 5]$ | $-\frac{29}{32}e^{15} + \frac{37}{16}e^{14} + \frac{373}{16}e^{13} - \frac{1853}{32}e^{12} - \frac{7831}{32}e^{11} + \frac{18777}{32}e^{10} + \frac{43237}{32}e^{9} - \frac{49555}{16}e^{8} - \frac{67229}{16}e^{7} + \frac{145775}{16}e^{6} + \frac{232797}{32}e^{5} - \frac{58987}{4}e^{4} - \frac{51363}{8}e^{3} + \frac{47789}{4}e^{2} + 2235e - 3685$ |
| 37 | $[37, 37, -2w^{4} + 13w^{2} + 5w - 7]$ | $\phantom{-}\frac{31}{4}e^{15} - \frac{77}{4}e^{14} - \frac{799}{4}e^{13} + \frac{1921}{4}e^{12} + 2101e^{11} - \frac{19375}{4}e^{10} - \frac{46479}{4}e^{9} + \frac{50839}{2}e^{8} + \frac{144653}{4}e^{7} - 74277e^{6} - \frac{250179}{4}e^{5} + \frac{477577}{4}e^{4} + \frac{219875}{4}e^{3} - \frac{192373}{2}e^{2} - 18962e + 29636$ |
| 53 | $[53, 53, 3w^{4} - 2w^{3} - 18w^{2} + 3w + 9]$ | $\phantom{-}\frac{149}{16}e^{15} - \frac{93}{4}e^{14} - \frac{1919}{8}e^{13} + \frac{9281}{16}e^{12} + \frac{40341}{16}e^{11} - \frac{93607}{16}e^{10} - \frac{222939}{16}e^{9} + 30700e^{8} + \frac{346615}{8}e^{7} - \frac{717477}{8}e^{6} - \frac{1197673}{16}e^{5} + \frac{1152617}{8}e^{4} + \frac{262797}{4}e^{3} - \frac{231875}{2}e^{2} - 22632e + 35658$ |
| 59 | $[59, 59, 2w^{4} - 13w^{2} - 6w + 5]$ | $-\frac{75}{16}e^{15} + \frac{93}{8}e^{14} + \frac{967}{8}e^{13} - \frac{4643}{16}e^{12} - \frac{20349}{16}e^{11} + \frac{46859}{16}e^{10} + \frac{112543}{16}e^{9} - \frac{123039}{8}e^{8} - \frac{175023}{8}e^{7} + \frac{359749}{8}e^{6} + \frac{604419}{16}e^{5} - \frac{289239}{4}e^{4} - \frac{132401}{4}e^{3} + \frac{116477}{2}e^{2} + 11376e - 17922$ |
| 61 | $[61, 61, -3w^{4} + 2w^{3} + 18w^{2} - 2w - 11]$ | $-\frac{123}{16}e^{15} + \frac{153}{8}e^{14} + \frac{1585}{8}e^{13} - \frac{7635}{16}e^{12} - \frac{33341}{16}e^{11} + \frac{77015}{16}e^{10} + \frac{184395}{16}e^{9} - \frac{202105}{8}e^{8} - \frac{286957}{8}e^{7} + \frac{590597}{8}e^{6} + \frac{992715}{16}e^{5} - \frac{474647}{4}e^{4} - \frac{109075}{2}e^{3} + 95576e^{2} + 18816e - 29440$ |
| 61 | $[61, 61, w^{2} - w - 3]$ | $-\frac{91}{32}e^{15} + \frac{57}{8}e^{14} + \frac{1171}{16}e^{13} - \frac{5695}{32}e^{12} - \frac{24587}{32}e^{11} + \frac{57525}{32}e^{10} + \frac{135665}{32}e^{9} - \frac{75605}{8}e^{8} - \frac{210543}{16}e^{7} + \frac{442691}{16}e^{6} + \frac{726175}{32}e^{5} - \frac{712823}{16}e^{4} - 19888e^{3} + \frac{71857}{2}e^{2} + 6847e - 11065$ |
| 61 | $[61, 61, 6w^{4} - 2w^{3} - 38w^{2} - 6w + 23]$ | $-17e^{15} + \frac{169}{4}e^{14} + 438e^{13} - \frac{2107}{2}e^{12} - \frac{18419}{4}e^{11} + \frac{42473}{4}e^{10} + \frac{101801}{4}e^{9} - \frac{222703}{4}e^{8} - 79132e^{7} + \frac{325037}{2}e^{6} + \frac{273347}{2}e^{5} - \frac{1043673}{4}e^{4} - \frac{239819}{2}e^{3} + 209896e^{2} + 41298e - 64558$ |
| 67 | $[67, 67, -w^{4} + 7w^{2} + 4w - 5]$ | $-\frac{103}{4}e^{15} + \frac{257}{4}e^{14} + 663e^{13} - \frac{6409}{4}e^{12} - \frac{13931}{2}e^{11} + 16152e^{10} + 38475e^{9} - \frac{338849}{4}e^{8} - 119583e^{7} + 247347e^{6} + \frac{826133}{4}e^{5} - \frac{1588831}{4}e^{4} - 181276e^{3} + 319580e^{2} + 62478e - 98284$ |
| 67 | $[67, 67, -w^{4} + 6w^{2} + 2w - 1]$ | $\phantom{-}\frac{1065}{32}e^{15} - \frac{1325}{16}e^{14} - \frac{13721}{16}e^{13} + \frac{66105}{32}e^{12} + \frac{288547}{32}e^{11} - \frac{666621}{32}e^{10} - \frac{1595225}{32}e^{9} + \frac{1748771}{16}e^{8} + \frac{2481169}{16}e^{7} - \frac{5108203}{16}e^{6} - \frac{8577209}{32}e^{5} + \frac{1025809}{2}e^{4} + \frac{1883231}{8}e^{3} - \frac{1651333}{4}e^{2} - 81161e + 127037$ |
| 71 | $[71, 71, -w^{2} + 3]$ | $\phantom{-}\frac{229}{32}e^{15} - \frac{283}{16}e^{14} - \frac{2949}{16}e^{13} + \frac{14093}{32}e^{12} + \frac{61987}{32}e^{11} - \frac{141821}{32}e^{10} - \frac{342521}{32}e^{9} + \frac{371213}{16}e^{8} + \frac{532437}{16}e^{7} - \frac{1081979}{16}e^{6} - \frac{1839245}{32}e^{5} + \frac{867617}{8}e^{4} + \frac{403477}{8}e^{3} - \frac{348843}{4}e^{2} - 17369e + 26835$ |
| 73 | $[73, 73, 3w^{4} - w^{3} - 19w^{2} - 3w + 9]$ | $-\frac{63}{8}e^{15} + \frac{39}{2}e^{14} + 203e^{13} - \frac{3891}{8}e^{12} - \frac{17083}{8}e^{11} + \frac{39235}{8}e^{10} + \frac{94479}{8}e^{9} - \frac{102929}{4}e^{8} - \frac{73501}{2}e^{7} + \frac{300719}{4}e^{6} + \frac{508367}{8}e^{5} - \frac{483353}{4}e^{4} - \frac{223353}{4}e^{3} + \frac{194701}{2}e^{2} + 19262e - 29992$ |
| 79 | $[79, 79, 3w^{4} - w^{3} - 19w^{2} - 4w + 11]$ | $\phantom{-}\frac{77}{16}e^{15} - \frac{95}{8}e^{14} - \frac{993}{8}e^{13} + \frac{4741}{16}e^{12} + \frac{20899}{16}e^{11} - \frac{47829}{16}e^{10} - \frac{115585}{16}e^{9} + \frac{125541}{8}e^{8} + \frac{179721}{8}e^{7} - \frac{366971}{8}e^{6} - \frac{620437}{16}e^{5} + \frac{295011}{4}e^{4} + \frac{135883}{4}e^{3} - \frac{118793}{2}e^{2} - 11682e + 18278$ |
| 83 | $[83, 83, -w^{4} - w^{3} + 7w^{2} + 8w - 3]$ | $\phantom{-}\frac{785}{32}e^{15} - \frac{245}{4}e^{14} - \frac{10109}{16}e^{13} + \frac{48901}{32}e^{12} + \frac{212485}{32}e^{11} - \frac{493235}{32}e^{10} - \frac{1174167}{32}e^{9} + \frac{323561}{4}e^{8} + \frac{1825585}{16}e^{7} - \frac{3781549}{16}e^{6} - \frac{6309877}{32}e^{5} + \frac{6076819}{16}e^{4} + 173204e^{3} - \frac{611565}{2}e^{2} - 59747e + 94113$ |
| 97 | $[97, 97, -2w^{4} + w^{3} + 12w^{2} + 2w - 3]$ | $-\frac{1375}{32}e^{15} + \frac{1717}{16}e^{14} + \frac{17699}{16}e^{13} - \frac{85639}{32}e^{12} - \frac{371841}{32}e^{11} + \frac{863351}{32}e^{10} + \frac{2053675}{32}e^{9} - \frac{2264127}{16}e^{8} - \frac{3191259}{16}e^{7} + \frac{6611225}{16}e^{6} + \frac{11023543}{32}e^{5} - \frac{5308489}{8}e^{4} - \frac{2419237}{8}e^{3} + \frac{2135519}{4}e^{2} + 104265e - 164215$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, -w^{3} + w^{2} + 5w - 1]$ | $-1$ |