Base field \(\Q(\sqrt{357}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 89\); narrow class number \(4\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[9, 3, 3]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $114$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 262x^{10} + 32623x^{8} - 2344788x^{6} + 104702351x^{4} - 2946292166x^{2} + 49596626209\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}0$ |
| 4 | $[4, 2, 2]$ | $\phantom{-}\frac{3693}{5568984344800}e^{10} + \frac{150887}{2227593737920}e^{8} - \frac{21358837}{696123043100}e^{6} + \frac{15527143149}{5568984344800}e^{4} - \frac{540879166409}{5568984344800}e^{2} + \frac{676424474639}{301026180800}$ |
| 7 | $[7, 7, w + 3]$ | $\phantom{-}0$ |
| 11 | $[11, 11, w + 3]$ | $\phantom{-}\frac{1128726139}{1240229520539994400}e^{11} - \frac{113365778847}{496091808215997760}e^{9} + \frac{4121339287019}{155028690067499300}e^{7} - \frac{2132663582456413}{1240229520539994400}e^{5} + \frac{6429007848304181}{95402270810768800}e^{3} - \frac{182997249495321}{144793592964800}e$ |
| 11 | $[11, 11, w + 7]$ | $-\frac{1128726139}{1240229520539994400}e^{11} + \frac{113365778847}{496091808215997760}e^{9} - \frac{4121339287019}{155028690067499300}e^{7} + \frac{2132663582456413}{1240229520539994400}e^{5} - \frac{6429007848304181}{95402270810768800}e^{3} + \frac{182997249495321}{144793592964800}e$ |
| 17 | $[17, 17, w + 8]$ | $-\frac{1128726139}{1240229520539994400}e^{11} + \frac{113365778847}{496091808215997760}e^{9} - \frac{4121339287019}{155028690067499300}e^{7} + \frac{2132663582456413}{1240229520539994400}e^{5} - \frac{6429007848304181}{95402270810768800}e^{3} + \frac{327790842460121}{144793592964800}e$ |
| 23 | $[23, 23, w + 4]$ | $\phantom{-}\frac{63252753}{381609083243075200}e^{11} - \frac{16415373081}{76321816648615040}e^{9} + \frac{6628092638717}{190804541621537600}e^{7} - \frac{558318029704773}{190804541621537600}e^{5} + \frac{48006194621152261}{381609083243075200}e^{3} - \frac{57311599738971}{22275937379200}e$ |
| 23 | $[23, 23, w + 18]$ | $-\frac{63252753}{381609083243075200}e^{11} + \frac{16415373081}{76321816648615040}e^{9} - \frac{6628092638717}{190804541621537600}e^{7} + \frac{558318029704773}{190804541621537600}e^{5} - \frac{48006194621152261}{381609083243075200}e^{3} + \frac{57311599738971}{22275937379200}e$ |
| 25 | $[25, 5, 5]$ | $-\frac{3693}{5568984344800}e^{10} - \frac{150887}{2227593737920}e^{8} + \frac{21358837}{696123043100}e^{6} - \frac{15527143149}{5568984344800}e^{4} + \frac{540879166409}{5568984344800}e^{2} + \frac{2032811152561}{301026180800}$ |
| 29 | $[29, 29, w + 1]$ | $...$ |
| 29 | $[29, 29, w + 27]$ | $-\frac{12722427879}{4960918082159977600}e^{11} + \frac{466794823029}{992183616431995520}e^{9} - \frac{111659081473591}{2480459041079988800}e^{7} + \frac{5537847108576429}{2480459041079988800}e^{5} - \frac{29141899558497911}{381609083243075200}e^{3} + \frac{352932700365303}{289587185929600}e$ |
| 31 | $[31, 31, w + 13]$ | $-\frac{464064943}{20627518013139200}e^{10} + \frac{3400131569}{825100720525568}e^{8} - \frac{3822523935967}{10313759006569600}e^{6} + \frac{161787554015373}{10313759006569600}e^{4} - \frac{7907884156182771}{20627518013139200}e^{2} + \frac{5565281941751}{1204104723200}$ |
| 31 | $[31, 31, w + 17]$ | $\phantom{-}\frac{464064943}{20627518013139200}e^{10} - \frac{3400131569}{825100720525568}e^{8} + \frac{3822523935967}{10313759006569600}e^{6} - \frac{161787554015373}{10313759006569600}e^{4} + \frac{7907884156182771}{20627518013139200}e^{2} - \frac{5565281941751}{1204104723200}$ |
| 43 | $[43, 43, -w - 11]$ | $\phantom{-}\frac{7763}{293104439200}e^{10} - \frac{6163691}{1113796868960}e^{8} + \frac{1453473193}{2784492172400}e^{6} - \frac{77742426267}{2784492172400}e^{4} + \frac{4810911528709}{5568984344800}e^{2} - \frac{2980854548887}{150513090400}$ |
| 43 | $[43, 43, w - 12]$ | $\phantom{-}\frac{7763}{293104439200}e^{10} - \frac{6163691}{1113796868960}e^{8} + \frac{1453473193}{2784492172400}e^{6} - \frac{77742426267}{2784492172400}e^{4} + \frac{4810911528709}{5568984344800}e^{2} - \frac{2980854548887}{150513090400}$ |
| 47 | $[47, 47, -w - 6]$ | $-\frac{1128726139}{2480459041079988800}e^{11} + \frac{113365778847}{992183616431995520}e^{9} - \frac{4121339287019}{310057380134998600}e^{7} + \frac{2132663582456413}{2480459041079988800}e^{5} - \frac{6429007848304181}{190804541621537600}e^{3} + \frac{327790842460121}{289587185929600}e$ |
| 47 | $[47, 47, w - 7]$ | $-\frac{1128726139}{2480459041079988800}e^{11} + \frac{113365778847}{992183616431995520}e^{9} - \frac{4121339287019}{310057380134998600}e^{7} + \frac{2132663582456413}{2480459041079988800}e^{5} - \frac{6429007848304181}{190804541621537600}e^{3} + \frac{327790842460121}{289587185929600}e$ |
| 59 | $[59, 59, -w - 5]$ | $\phantom{-}\frac{1128726139}{2480459041079988800}e^{11} - \frac{113365778847}{992183616431995520}e^{9} + \frac{4121339287019}{310057380134998600}e^{7} - \frac{2132663582456413}{2480459041079988800}e^{5} + \frac{6429007848304181}{190804541621537600}e^{3} - \frac{327790842460121}{289587185929600}e$ |
| 59 | $[59, 59, w - 6]$ | $\phantom{-}\frac{1128726139}{2480459041079988800}e^{11} - \frac{113365778847}{992183616431995520}e^{9} + \frac{4121339287019}{310057380134998600}e^{7} - \frac{2132663582456413}{2480459041079988800}e^{5} + \frac{6429007848304181}{190804541621537600}e^{3} - \frac{327790842460121}{289587185929600}e$ |
| 61 | $[61, 61, w + 16]$ | $-\frac{173681091}{20627518013139200}e^{10} + \frac{4651122197}{4125503602627840}e^{8} - \frac{451264493939}{10313759006569600}e^{6} - \frac{42424044434259}{10313759006569600}e^{4} + \frac{8942199214459753}{20627518013139200}e^{2} - \frac{16113276565193}{1204104723200}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w + 1]$ | $1$ |