Base field 4.4.13676.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 6x^{2} + 7x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{2} + 4]$ |
| Dimension: | $17$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $25$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{17} + 3x^{16} - 24x^{15} - 76x^{14} + 213x^{13} + 748x^{12} - 808x^{11} - 3581x^{10} + 833x^{9} + 8474x^{8} + 2163x^{7} - 8726x^{6} - 4788x^{5} + 2572x^{4} + 1996x^{3} + 32x^{2} - 128x - 16\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w + 1]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -w^{2} - w + 3]$ | $-\frac{16605}{29144}e^{16} - \frac{42915}{29144}e^{15} + \frac{103871}{7286}e^{14} + \frac{135987}{3643}e^{13} - \frac{3964245}{29144}e^{12} - \frac{2682995}{7286}e^{11} + \frac{2201076}{3643}e^{10} + \frac{51698949}{29144}e^{9} - \frac{33917721}{29144}e^{8} - \frac{62163177}{14572}e^{7} + \frac{12143953}{29144}e^{6} + \frac{67074477}{14572}e^{5} + \frac{6984749}{7286}e^{4} - \frac{12046155}{7286}e^{3} - \frac{1857938}{3643}e^{2} + \frac{456056}{3643}e + \frac{112470}{3643}$ |
| 5 | $[5, 5, w^{3} - 5w + 1]$ | $\phantom{-}\frac{47957}{29144}e^{16} + \frac{130821}{29144}e^{15} - \frac{296159}{7286}e^{14} - \frac{1659327}{14572}e^{13} + \frac{11067813}{29144}e^{12} + \frac{16385853}{14572}e^{11} - \frac{23577595}{14572}e^{10} - \frac{158085065}{29144}e^{9} + \frac{80378219}{29144}e^{8} + \frac{190406441}{14572}e^{7} + \frac{6742417}{29144}e^{6} - \frac{51479560}{3643}e^{5} - \frac{62714039}{14572}e^{4} + \frac{37142305}{7286}e^{3} + \frac{14675143}{7286}e^{2} - \frac{1407090}{3643}e - \frac{431295}{3643}$ |
| 11 | $[11, 11, -w^{3} + 6w - 2]$ | $-\frac{31787}{14572}e^{16} - \frac{93157}{14572}e^{15} + \frac{773221}{14572}e^{14} + \frac{2369209}{14572}e^{13} - \frac{7036899}{14572}e^{12} - \frac{5865410}{3643}e^{11} + \frac{28360497}{14572}e^{10} + \frac{113516925}{14572}e^{9} - \frac{39643017}{14572}e^{8} - \frac{274319653}{14572}e^{7} - \frac{9100246}{3643}e^{6} + \frac{148884725}{7286}e^{5} + \frac{116457201}{14572}e^{4} - \frac{54169387}{7286}e^{3} - \frac{12516747}{3643}e^{2} + \frac{2157864}{3643}e + \frac{713747}{3643}$ |
| 13 | $[13, 13, -w^{2} + 4]$ | $-1$ |
| 17 | $[17, 17, 2w^{3} + w^{2} - 10w - 2]$ | $-\frac{31303}{29144}e^{16} - \frac{87549}{29144}e^{15} + \frac{384753}{14572}e^{14} + \frac{1111153}{14572}e^{13} - \frac{7135599}{29144}e^{12} - \frac{5490851}{7286}e^{11} + \frac{14983657}{14572}e^{10} + \frac{106054891}{29144}e^{9} - \frac{48871747}{29144}e^{8} - \frac{63942437}{7286}e^{7} - \frac{12165391}{29144}e^{6} + \frac{138495463}{14572}e^{5} + \frac{44451191}{14572}e^{4} - \frac{25060379}{7286}e^{3} - \frac{9937897}{7286}e^{2} + \frac{972295}{3643}e + \frac{274889}{3643}$ |
| 37 | $[37, 37, w^{3} + w^{2} - 6w - 3]$ | $\phantom{-}\frac{33637}{7286}e^{16} + \frac{46292}{3643}e^{15} - \frac{414462}{3643}e^{14} - \frac{2348847}{7286}e^{13} + \frac{7712855}{7286}e^{12} + \frac{23194369}{7286}e^{11} - \frac{32582847}{7286}e^{10} - \frac{111860159}{7286}e^{9} + \frac{27017354}{3643}e^{8} + \frac{134639708}{3643}e^{7} + \frac{5113989}{3643}e^{6} - \frac{290702971}{7286}e^{5} - \frac{93551033}{7286}e^{4} + \frac{52173669}{3643}e^{3} + \frac{21378510}{3643}e^{2} - \frac{3929472}{3643}e - \frac{1232176}{3643}$ |
| 37 | $[37, 37, -w^{3} + 6w - 4]$ | $-\frac{37981}{14572}e^{16} - \frac{99773}{14572}e^{15} + \frac{473043}{7286}e^{14} + \frac{631994}{3643}e^{13} - \frac{8965413}{14572}e^{12} - \frac{12464039}{7286}e^{11} + \frac{9830639}{3643}e^{10} + \frac{120039109}{14572}e^{9} - \frac{73221143}{14572}e^{8} - \frac{72124182}{3643}e^{7} + \frac{17855649}{14572}e^{6} + \frac{77698723}{3643}e^{5} + \frac{18475780}{3643}e^{4} - \frac{27733522}{3643}e^{3} - \frac{9306768}{3643}e^{2} + \frac{2013638}{3643}e + \frac{556660}{3643}$ |
| 41 | $[41, 41, -w^{3} + 5w - 5]$ | $\phantom{-}\frac{25601}{14572}e^{16} + \frac{74109}{14572}e^{15} - \frac{156171}{3643}e^{14} - \frac{942199}{7286}e^{13} + \frac{5716369}{14572}e^{12} + \frac{9329687}{7286}e^{11} - \frac{11660163}{7286}e^{10} - \frac{90304401}{14572}e^{9} + \frac{34210331}{14572}e^{8} + \frac{109229985}{7286}e^{7} + \frac{23929397}{14572}e^{6} - \frac{59487692}{3643}e^{5} - \frac{44109373}{7286}e^{4} + \frac{21912564}{3643}e^{3} + \frac{9655103}{3643}e^{2} - \frac{1827208}{3643}e - \frac{558500}{3643}$ |
| 43 | $[43, 43, w^{3} - 4w + 4]$ | $-\frac{5273}{3643}e^{16} - \frac{64677}{14572}e^{15} + \frac{126212}{3643}e^{14} + \frac{1644839}{14572}e^{13} - \frac{1114926}{3643}e^{12} - \frac{16286073}{14572}e^{11} + \frac{16737187}{14572}e^{10} + \frac{39384345}{7286}e^{9} - \frac{16220221}{14572}e^{8} - \frac{95092833}{7286}e^{7} - \frac{48447131}{14572}e^{6} + \frac{206095329}{14572}e^{5} + \frac{103159773}{14572}e^{4} - \frac{37438749}{7286}e^{3} - \frac{10449711}{3643}e^{2} + \frac{1529907}{3643}e + \frac{563437}{3643}$ |
| 43 | $[43, 43, -2w^{3} + 11w - 2]$ | $\phantom{-}\frac{1955}{7286}e^{16} + \frac{3739}{3643}e^{15} - \frac{43495}{7286}e^{14} - \frac{95154}{3643}e^{13} + \frac{331491}{7286}e^{12} + \frac{1886395}{7286}e^{11} - \frac{379616}{3643}e^{10} - \frac{9135785}{7286}e^{9} - \frac{1125881}{3643}e^{8} + \frac{22085193}{7286}e^{7} + \frac{13437611}{7286}e^{6} - \frac{23973047}{7286}e^{5} - \frac{9628336}{3643}e^{4} + \frac{4407413}{3643}e^{3} + \frac{3724148}{3643}e^{2} - \frac{399504}{3643}e - \frac{219788}{3643}$ |
| 47 | $[47, 47, -2w^{3} - w^{2} + 12w]$ | $\phantom{-}\frac{19533}{3643}e^{16} + \frac{53839}{3643}e^{15} - \frac{481614}{3643}e^{14} - \frac{1366411}{3643}e^{13} + \frac{4485612}{3643}e^{12} + \frac{13498989}{3643}e^{11} - \frac{18990177}{3643}e^{10} - \frac{65136082}{3643}e^{9} + \frac{31724721}{3643}e^{8} + \frac{156902348}{3643}e^{7} + \frac{5088840}{3643}e^{6} - \frac{169533453}{3643}e^{5} - \frac{53444290}{3643}e^{4} + \frac{60933065}{3643}e^{3} + \frac{24599597}{3643}e^{2} - \frac{4572999}{3643}e - \frac{1414878}{3643}$ |
| 47 | $[47, 47, 2w - 1]$ | $-\frac{41873}{14572}e^{16} - \frac{113557}{14572}e^{15} + \frac{518313}{7286}e^{14} + \frac{720201}{3643}e^{13} - \frac{9724917}{14572}e^{12} - \frac{14225267}{7286}e^{11} + \frac{10452976}{3643}e^{10} + \frac{137259073}{14572}e^{9} - \frac{73427047}{14572}e^{8} - \frac{82673271}{3643}e^{7} + \frac{2281477}{14572}e^{6} + \frac{89416187}{3643}e^{5} + \frac{25114203}{3643}e^{4} - \frac{32242735}{3643}e^{3} - \frac{12031725}{3643}e^{2} + \frac{2412406}{3643}e + \frac{712348}{3643}$ |
| 61 | $[61, 61, -w^{3} + w^{2} + 6w - 5]$ | $-\frac{42821}{7286}e^{16} - \frac{119423}{7286}e^{15} + \frac{1052453}{7286}e^{14} + \frac{3031611}{7286}e^{13} - \frac{9748911}{7286}e^{12} - \frac{14978881}{3643}e^{11} + \frac{40802717}{7286}e^{10} + \frac{144597343}{7286}e^{9} - \frac{65597163}{7286}e^{8} - \frac{348432497}{7286}e^{7} - \frac{10266177}{3643}e^{6} + \frac{188338998}{3643}e^{5} + \frac{127071661}{7286}e^{4} - \frac{67800230}{3643}e^{3} - \frac{28649407}{3643}e^{2} + \frac{5130632}{3643}e + \frac{1645878}{3643}$ |
| 71 | $[71, 71, w^{3} + w^{2} - 4w - 3]$ | $-\frac{5567}{3643}e^{16} - \frac{27515}{7286}e^{15} + \frac{280963}{7286}e^{14} + \frac{348078}{3643}e^{13} - \frac{1359900}{3643}e^{12} - \frac{6856707}{7286}e^{11} + \frac{6212744}{3643}e^{10} + \frac{16500998}{3643}e^{9} - \frac{25738127}{7286}e^{8} - \frac{79409955}{7286}e^{7} + \frac{7639967}{3643}e^{6} + \frac{86027905}{7286}e^{5} + \frac{5660530}{3643}e^{4} - \frac{15678047}{3643}e^{3} - \frac{3822227}{3643}e^{2} + \frac{1241622}{3643}e + \frac{257056}{3643}$ |
| 71 | $[71, 71, 2w^{3} + 2w^{2} - 9w - 2]$ | $\phantom{-}\frac{193947}{29144}e^{16} + \frac{549131}{29144}e^{15} - \frac{594066}{3643}e^{14} - \frac{6975223}{14572}e^{13} + \frac{43791323}{29144}e^{12} + \frac{68984847}{14572}e^{11} - \frac{90605879}{14572}e^{10} - \frac{666553047}{29144}e^{9} + \frac{279821533}{29144}e^{8} + \frac{803949645}{14572}e^{7} + \frac{134451099}{29144}e^{6} - \frac{217619210}{3643}e^{5} - \frac{310116899}{14572}e^{4} + \frac{157371651}{7286}e^{3} + \frac{68909047}{7286}e^{2} - \frac{6104794}{3643}e - \frac{1983087}{3643}$ |
| 79 | $[79, 79, w^{3} + w^{2} - 6w - 5]$ | $\phantom{-}\frac{18103}{7286}e^{16} + \frac{89443}{14572}e^{15} - \frac{457225}{7286}e^{14} - \frac{2262177}{14572}e^{13} + \frac{4431803}{7286}e^{12} + \frac{22259037}{14572}e^{11} - \frac{40595873}{14572}e^{10} - \frac{26726418}{3643}e^{9} + \frac{84692119}{14572}e^{8} + \frac{64014253}{3643}e^{7} - \frac{52574669}{14572}e^{6} - \frac{274269025}{14572}e^{5} - \frac{32563055}{14572}e^{4} + \frac{48091023}{7286}e^{3} + \frac{5644213}{3643}e^{2} - \frac{1584451}{3643}e - \frac{384667}{3643}$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{25983}{14572}e^{16} + \frac{16751}{3643}e^{15} - \frac{324865}{7286}e^{14} - \frac{1696543}{14572}e^{13} + \frac{6193771}{14572}e^{12} + \frac{16707471}{14572}e^{11} - \frac{27470195}{14572}e^{10} - \frac{80261453}{14572}e^{9} + \frac{26359379}{7286}e^{8} + \frac{96006511}{7286}e^{7} - \frac{4572097}{3643}e^{6} - \frac{204648657}{14572}e^{5} - \frac{44276629}{14572}e^{4} + \frac{35193805}{7286}e^{3} + \frac{5855414}{3643}e^{2} - \frac{1058773}{3643}e - \frac{356807}{3643}$ |
| 83 | $[83, 83, -3w^{3} - w^{2} + 16w + 3]$ | $\phantom{-}\frac{86285}{14572}e^{16} + \frac{234163}{14572}e^{15} - \frac{1066561}{7286}e^{14} - \frac{2969747}{7286}e^{13} + \frac{19952709}{14572}e^{12} + \frac{14658837}{3643}e^{11} - \frac{42598329}{7286}e^{10} - \frac{282684137}{14572}e^{9} + \frac{146195873}{14572}e^{8} + \frac{170050882}{3643}e^{7} + \frac{8522173}{14572}e^{6} - \frac{366929845}{7286}e^{5} - \frac{110862441}{7286}e^{4} + \frac{65758937}{3643}e^{3} + \frac{26006429}{3643}e^{2} - \frac{4908390}{3643}e - \frac{1532102}{3643}$ |
| 97 | $[97, 97, w^{3} + w^{2} - 6w + 1]$ | $\phantom{-}\frac{6041}{3643}e^{16} + \frac{33381}{7286}e^{15} - \frac{148395}{3643}e^{14} - \frac{846963}{7286}e^{13} + \frac{1371897}{3643}e^{12} + \frac{8363935}{7286}e^{11} - \frac{11416301}{7286}e^{10} - \frac{20170133}{3643}e^{9} + \frac{17908243}{7286}e^{8} + \frac{48578327}{3643}e^{7} + \frac{7541183}{7286}e^{6} - \frac{105121299}{7286}e^{5} - \frac{38016055}{7286}e^{4} + \frac{19091168}{3643}e^{3} + \frac{8517474}{3643}e^{2} - \frac{1529227}{3643}e - \frac{500096}{3643}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{2} + 4]$ | $1$ |