Base field 4.4.13525.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 12x^{2} + 8x + 29\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $13$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $21$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{13} - x^{12} - 37x^{11} + 36x^{10} + 464x^{9} - 335x^{8} - 2385x^{7} + 462x^{6} + 5041x^{5} + 1710x^{4} - 2028x^{3} - 736x^{2} + 80x + 32\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, -w + 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -\frac{3}{5}w^{3} - w^{2} + \frac{26}{5}w + \frac{42}{5}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, \frac{2}{5}w^{3} - \frac{19}{5}w - \frac{3}{5}]$ | $\phantom{-}\frac{3137}{66048}e^{12} - \frac{1171}{16512}e^{11} - \frac{37995}{22016}e^{10} + \frac{169903}{66048}e^{9} + \frac{459961}{22016}e^{8} - \frac{219553}{8256}e^{7} - \frac{6709657}{66048}e^{6} + \frac{1663115}{22016}e^{5} + \frac{2298833}{11008}e^{4} - \frac{159495}{5504}e^{3} - \frac{134483}{1376}e^{2} + \frac{38653}{4128}e + \frac{3281}{688}$ |
| 9 | $[9, 3, -\frac{1}{5}w^{3} + \frac{2}{5}w + \frac{4}{5}]$ | $\phantom{-}\frac{3137}{66048}e^{12} - \frac{1171}{16512}e^{11} - \frac{37995}{22016}e^{10} + \frac{169903}{66048}e^{9} + \frac{459961}{22016}e^{8} - \frac{219553}{8256}e^{7} - \frac{6709657}{66048}e^{6} + \frac{1663115}{22016}e^{5} + \frac{2298833}{11008}e^{4} - \frac{159495}{5504}e^{3} - \frac{134483}{1376}e^{2} + \frac{38653}{4128}e + \frac{3281}{688}$ |
| 11 | $[11, 11, -\frac{2}{5}w^{3} - w^{2} + \frac{14}{5}w + \frac{28}{5}]$ | $\phantom{-}\frac{41}{132096}e^{12} - \frac{333}{11008}e^{11} + \frac{5383}{132096}e^{10} + \frac{135847}{132096}e^{9} - \frac{227629}{132096}e^{8} - \frac{181369}{16512}e^{7} + \frac{2368159}{132096}e^{6} + \frac{5535961}{132096}e^{5} - \frac{1028679}{22016}e^{4} - \frac{729503}{11008}e^{3} + \frac{45085}{2752}e^{2} + \frac{111925}{8256}e - \frac{821}{4128}$ |
| 11 | $[11, 11, \frac{1}{5}w^{3} - w^{2} - \frac{7}{5}w + \frac{36}{5}]$ | $\phantom{-}\frac{41}{132096}e^{12} - \frac{333}{11008}e^{11} + \frac{5383}{132096}e^{10} + \frac{135847}{132096}e^{9} - \frac{227629}{132096}e^{8} - \frac{181369}{16512}e^{7} + \frac{2368159}{132096}e^{6} + \frac{5535961}{132096}e^{5} - \frac{1028679}{22016}e^{4} - \frac{729503}{11008}e^{3} + \frac{45085}{2752}e^{2} + \frac{111925}{8256}e - \frac{821}{4128}$ |
| 16 | $[16, 2, 2]$ | $-1$ |
| 29 | $[29, 29, -w]$ | $\phantom{-}\frac{3923}{44032}e^{12} - \frac{1541}{11008}e^{11} - \frac{139747}{44032}e^{10} + \frac{219293}{44032}e^{9} + \frac{1633505}{44032}e^{8} - \frac{272043}{5504}e^{7} - \frac{7488507}{44032}e^{6} + \frac{5360707}{44032}e^{5} + \frac{7273257}{22016}e^{4} + \frac{21233}{11008}e^{3} - \frac{314387}{2752}e^{2} - \frac{20233}{2752}e - \frac{2775}{1376}$ |
| 29 | $[29, 29, \frac{1}{5}w^{3} - \frac{12}{5}w + \frac{1}{5}]$ | $\phantom{-}\frac{3923}{44032}e^{12} - \frac{1541}{11008}e^{11} - \frac{139747}{44032}e^{10} + \frac{219293}{44032}e^{9} + \frac{1633505}{44032}e^{8} - \frac{272043}{5504}e^{7} - \frac{7488507}{44032}e^{6} + \frac{5360707}{44032}e^{5} + \frac{7273257}{22016}e^{4} + \frac{21233}{11008}e^{3} - \frac{314387}{2752}e^{2} - \frac{20233}{2752}e - \frac{2775}{1376}$ |
| 41 | $[41, 41, -\frac{2}{5}w^{3} - w^{2} + \frac{14}{5}w + \frac{38}{5}]$ | $-\frac{1087}{3072}e^{12} + \frac{353}{768}e^{11} + \frac{13285}{1024}e^{10} - \frac{50801}{3072}e^{9} - \frac{163575}{1024}e^{8} + \frac{62975}{384}e^{7} + \frac{2453063}{3072}e^{6} - \frac{390341}{1024}e^{5} - \frac{855535}{512}e^{4} - \frac{42951}{256}e^{3} + \frac{46773}{64}e^{2} + \frac{11917}{192}e - \frac{1199}{32}$ |
| 41 | $[41, 41, -w^{2} + 10]$ | $\phantom{-}\frac{3283}{132096}e^{12} - \frac{1301}{33024}e^{11} - \frac{40225}{44032}e^{10} + \frac{192221}{132096}e^{9} + \frac{498187}{44032}e^{8} - \frac{261155}{16512}e^{7} - \frac{7647227}{132096}e^{6} + \frac{2343169}{44032}e^{5} + \frac{2932611}{22016}e^{4} - \frac{528453}{11008}e^{3} - \frac{272369}{2752}e^{2} + \frac{111959}{8256}e + \frac{15619}{1376}$ |
| 41 | $[41, 41, \frac{11}{5}w^{3} + 3w^{2} - \frac{102}{5}w - \frac{149}{5}]$ | $\phantom{-}\frac{3283}{132096}e^{12} - \frac{1301}{33024}e^{11} - \frac{40225}{44032}e^{10} + \frac{192221}{132096}e^{9} + \frac{498187}{44032}e^{8} - \frac{261155}{16512}e^{7} - \frac{7647227}{132096}e^{6} + \frac{2343169}{44032}e^{5} + \frac{2932611}{22016}e^{4} - \frac{528453}{11008}e^{3} - \frac{272369}{2752}e^{2} + \frac{111959}{8256}e + \frac{15619}{1376}$ |
| 41 | $[41, 41, -\frac{1}{5}w^{3} + w^{2} + \frac{7}{5}w - \frac{26}{5}]$ | $-\frac{1087}{3072}e^{12} + \frac{353}{768}e^{11} + \frac{13285}{1024}e^{10} - \frac{50801}{3072}e^{9} - \frac{163575}{1024}e^{8} + \frac{62975}{384}e^{7} + \frac{2453063}{3072}e^{6} - \frac{390341}{1024}e^{5} - \frac{855535}{512}e^{4} - \frac{42951}{256}e^{3} + \frac{46773}{64}e^{2} + \frac{11917}{192}e - \frac{1199}{32}$ |
| 49 | $[49, 7, \frac{1}{5}w^{3} + w^{2} - \frac{12}{5}w - \frac{19}{5}]$ | $-\frac{1533}{22016}e^{12} + \frac{247}{5504}e^{11} + \frac{58205}{22016}e^{10} - \frac{39635}{22016}e^{9} - \frac{763455}{22016}e^{8} + \frac{51369}{2752}e^{7} + \frac{4173301}{22016}e^{6} - \frac{842589}{22016}e^{5} - \frac{4575511}{11008}e^{4} - \frac{401311}{5504}e^{3} + \frac{285413}{1376}e^{2} + \frac{40407}{1376}e - \frac{8183}{688}$ |
| 49 | $[49, 7, \frac{1}{5}w^{3} + w^{2} - \frac{12}{5}w - \frac{44}{5}]$ | $-\frac{1533}{22016}e^{12} + \frac{247}{5504}e^{11} + \frac{58205}{22016}e^{10} - \frac{39635}{22016}e^{9} - \frac{763455}{22016}e^{8} + \frac{51369}{2752}e^{7} + \frac{4173301}{22016}e^{6} - \frac{842589}{22016}e^{5} - \frac{4575511}{11008}e^{4} - \frac{401311}{5504}e^{3} + \frac{285413}{1376}e^{2} + \frac{40407}{1376}e - \frac{8183}{688}$ |
| 59 | $[59, 59, -\frac{4}{5}w^{3} - w^{2} + \frac{33}{5}w + \frac{36}{5}]$ | $\phantom{-}\frac{48481}{132096}e^{12} - \frac{16799}{33024}e^{11} - \frac{588539}{44032}e^{10} + \frac{2421359}{132096}e^{9} + \frac{7164201}{44032}e^{8} - \frac{3042257}{16512}e^{7} - \frac{105679193}{132096}e^{6} + \frac{20373403}{44032}e^{5} + \frac{36689521}{22016}e^{4} + \frac{364505}{11008}e^{3} - \frac{2114907}{2752}e^{2} - \frac{248371}{8256}e + \frac{64241}{1376}$ |
| 59 | $[59, 59, -2w^{3} - 3w^{2} + 17w + 26]$ | $\phantom{-}\frac{48481}{132096}e^{12} - \frac{16799}{33024}e^{11} - \frac{588539}{44032}e^{10} + \frac{2421359}{132096}e^{9} + \frac{7164201}{44032}e^{8} - \frac{3042257}{16512}e^{7} - \frac{105679193}{132096}e^{6} + \frac{20373403}{44032}e^{5} + \frac{36689521}{22016}e^{4} + \frac{364505}{11008}e^{3} - \frac{2114907}{2752}e^{2} - \frac{248371}{8256}e + \frac{64241}{1376}$ |
| 61 | $[61, 61, -\frac{1}{5}w^{3} + w^{2} + \frac{12}{5}w - \frac{51}{5}]$ | $\phantom{-}\frac{31387}{132096}e^{12} - \frac{10421}{33024}e^{11} - \frac{380761}{44032}e^{10} + \frac{1500629}{132096}e^{9} + \frac{4634611}{44032}e^{8} - \frac{1863419}{16512}e^{7} - \frac{68427203}{132096}e^{6} + \frac{11763129}{44032}e^{5} + \frac{23793291}{22016}e^{4} + \frac{812387}{11008}e^{3} - \frac{1355049}{2752}e^{2} - \frac{150721}{8256}e + \frac{45707}{1376}$ |
| 61 | $[61, 61, \frac{4}{5}w^{3} + w^{2} - \frac{28}{5}w - \frac{41}{5}]$ | $\phantom{-}\frac{31387}{132096}e^{12} - \frac{10421}{33024}e^{11} - \frac{380761}{44032}e^{10} + \frac{1500629}{132096}e^{9} + \frac{4634611}{44032}e^{8} - \frac{1863419}{16512}e^{7} - \frac{68427203}{132096}e^{6} + \frac{11763129}{44032}e^{5} + \frac{23793291}{22016}e^{4} + \frac{812387}{11008}e^{3} - \frac{1355049}{2752}e^{2} - \frac{150721}{8256}e + \frac{45707}{1376}$ |
| 71 | $[71, 71, \frac{1}{5}w^{3} + w^{2} - \frac{7}{5}w - \frac{49}{5}]$ | $-\frac{419}{132096}e^{12} + \frac{47}{11008}e^{11} + \frac{19091}{132096}e^{10} - \frac{19405}{132096}e^{9} - \frac{313649}{132096}e^{8} + \frac{26611}{16512}e^{7} + \frac{2305163}{132096}e^{6} - \frac{851251}{132096}e^{5} - \frac{1236563}{22016}e^{4} + \frac{107157}{11008}e^{3} + \frac{159393}{2752}e^{2} - \frac{13351}{8256}e - \frac{7417}{4128}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $16$ | $[16, 2, 2]$ | $1$ |