Base field \(\Q(\sqrt{197}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 49\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[9, 3, 3]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - 17x^{15} + 93x^{14} - 17x^{13} - 1683x^{12} + 5415x^{11} + 1749x^{10} - 36793x^{9} + 46489x^{8} + 64707x^{7} - 171535x^{6} + 27692x^{5} + 173164x^{4} - 99132x^{3} - 47112x^{2} + 40132x - 3200\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w - 7]$ | $\phantom{-}\frac{324654260849071}{71119928360410861}e^{15} - \frac{3681803091012116}{71119928360410861}e^{14} + \frac{7142973262924599}{71119928360410861}e^{13} + \frac{59709964451882331}{71119928360410861}e^{12} - \frac{253175584063741919}{71119928360410861}e^{11} - \frac{96845694826496926}{71119928360410861}e^{10} + \frac{1720121486519871010}{71119928360410861}e^{9} - \frac{1238551503065855066}{71119928360410861}e^{8} - \frac{4255907950524529675}{71119928360410861}e^{7} + \frac{4680785160490571678}{71119928360410861}e^{6} + \frac{4057844437023237206}{71119928360410861}e^{5} - \frac{4564771288091387020}{71119928360410861}e^{4} - \frac{2343671895727450530}{71119928360410861}e^{3} + \frac{1436004027180097757}{71119928360410861}e^{2} + \frac{926202657180049825}{71119928360410861}e - \frac{83495983864953357}{71119928360410861}$ |
| 7 | $[7, 7, w + 6]$ | $\phantom{-}\frac{324654260849071}{71119928360410861}e^{15} - \frac{3681803091012116}{71119928360410861}e^{14} + \frac{7142973262924599}{71119928360410861}e^{13} + \frac{59709964451882331}{71119928360410861}e^{12} - \frac{253175584063741919}{71119928360410861}e^{11} - \frac{96845694826496926}{71119928360410861}e^{10} + \frac{1720121486519871010}{71119928360410861}e^{9} - \frac{1238551503065855066}{71119928360410861}e^{8} - \frac{4255907950524529675}{71119928360410861}e^{7} + \frac{4680785160490571678}{71119928360410861}e^{6} + \frac{4057844437023237206}{71119928360410861}e^{5} - \frac{4564771288091387020}{71119928360410861}e^{4} - \frac{2343671895727450530}{71119928360410861}e^{3} + \frac{1436004027180097757}{71119928360410861}e^{2} + \frac{926202657180049825}{71119928360410861}e - \frac{83495983864953357}{71119928360410861}$ |
| 9 | $[9, 3, 3]$ | $-1$ |
| 19 | $[19, 19, w + 5]$ | $-\frac{454236509922863}{71119928360410861}e^{15} + \frac{7092378243900001}{71119928360410861}e^{14} - \frac{32862015782163556}{71119928360410861}e^{13} - \frac{32573161883145417}{71119928360410861}e^{12} + \frac{707453215171897066}{71119928360410861}e^{11} - \frac{1558332335808346263}{71119928360410861}e^{10} - \frac{2562213524481001452}{71119928360410861}e^{9} + \frac{13147370791669985417}{71119928360410861}e^{8} - \frac{5513743655529348793}{71119928360410861}e^{7} - \frac{33954052701578631063}{71119928360410861}e^{6} + \frac{36978601015298889535}{71119928360410861}e^{5} + \frac{27048414908860540962}{71119928360410861}e^{4} - \frac{45066816784863669804}{71119928360410861}e^{3} - \frac{3102085387317646450}{71119928360410861}e^{2} + \frac{15761054623713113454}{71119928360410861}e - \frac{1838168813031281225}{71119928360410861}$ |
| 19 | $[19, 19, w - 6]$ | $-\frac{454236509922863}{71119928360410861}e^{15} + \frac{7092378243900001}{71119928360410861}e^{14} - \frac{32862015782163556}{71119928360410861}e^{13} - \frac{32573161883145417}{71119928360410861}e^{12} + \frac{707453215171897066}{71119928360410861}e^{11} - \frac{1558332335808346263}{71119928360410861}e^{10} - \frac{2562213524481001452}{71119928360410861}e^{9} + \frac{13147370791669985417}{71119928360410861}e^{8} - \frac{5513743655529348793}{71119928360410861}e^{7} - \frac{33954052701578631063}{71119928360410861}e^{6} + \frac{36978601015298889535}{71119928360410861}e^{5} + \frac{27048414908860540962}{71119928360410861}e^{4} - \frac{45066816784863669804}{71119928360410861}e^{3} - \frac{3102085387317646450}{71119928360410861}e^{2} + \frac{15761054623713113454}{71119928360410861}e - \frac{1838168813031281225}{71119928360410861}$ |
| 23 | $[23, 23, w + 8]$ | $\phantom{-}\frac{832268894939982}{71119928360410861}e^{15} - \frac{23194192976783297}{142239856720821722}e^{14} + \frac{43323337539849473}{71119928360410861}e^{13} + \frac{101404738760728724}{71119928360410861}e^{12} - \frac{1052976236821276625}{71119928360410861}e^{11} + \frac{1556554962606259280}{71119928360410861}e^{10} + \frac{4946974934537650110}{71119928360410861}e^{9} - \frac{15784402105546015532}{71119928360410861}e^{8} - \frac{566463122085744864}{71119928360410861}e^{7} + \frac{44379207322538254980}{71119928360410861}e^{6} - \frac{31334955633167434716}{71119928360410861}e^{5} - \frac{41188697126914354058}{71119928360410861}e^{4} + \frac{86229213107491883739}{142239856720821722}e^{3} + \frac{10267265613409348169}{71119928360410861}e^{2} - \frac{14988325733263897862}{71119928360410861}e + \frac{1191656520585744679}{71119928360410861}$ |
| 23 | $[23, 23, -w + 9]$ | $\phantom{-}\frac{832268894939982}{71119928360410861}e^{15} - \frac{23194192976783297}{142239856720821722}e^{14} + \frac{43323337539849473}{71119928360410861}e^{13} + \frac{101404738760728724}{71119928360410861}e^{12} - \frac{1052976236821276625}{71119928360410861}e^{11} + \frac{1556554962606259280}{71119928360410861}e^{10} + \frac{4946974934537650110}{71119928360410861}e^{9} - \frac{15784402105546015532}{71119928360410861}e^{8} - \frac{566463122085744864}{71119928360410861}e^{7} + \frac{44379207322538254980}{71119928360410861}e^{6} - \frac{31334955633167434716}{71119928360410861}e^{5} - \frac{41188697126914354058}{71119928360410861}e^{4} + \frac{86229213107491883739}{142239856720821722}e^{3} + \frac{10267265613409348169}{71119928360410861}e^{2} - \frac{14988325733263897862}{71119928360410861}e + \frac{1191656520585744679}{71119928360410861}$ |
| 25 | $[25, 5, 5]$ | $-\frac{121525081414308}{71119928360410861}e^{15} + \frac{1695657238598819}{71119928360410861}e^{14} - \frac{6432463522278442}{71119928360410861}e^{13} - \frac{14056449796884394}{71119928360410861}e^{12} + \frac{155151843390446708}{71119928360410861}e^{11} - \frac{249033717773246146}{71119928360410861}e^{10} - \frac{713709282530793082}{71119928360410861}e^{9} + \frac{2529558504960524482}{71119928360410861}e^{8} - \frac{92149995327183442}{71119928360410861}e^{7} - \frac{7612299945735933175}{71119928360410861}e^{6} + \frac{5393323532567147303}{71119928360410861}e^{5} + \frac{8627509891559759426}{71119928360410861}e^{4} - \frac{7579995897970710271}{71119928360410861}e^{3} - \frac{3491310142354895152}{71119928360410861}e^{2} + \frac{2480674540292516152}{71119928360410861}e + \frac{225797170848637506}{71119928360410861}$ |
| 29 | $[29, 29, -w - 4]$ | $-\frac{415343520203248}{71119928360410861}e^{15} + \frac{5117388082195351}{71119928360410861}e^{14} - \frac{25367472773685853}{142239856720821722}e^{13} - \frac{158153645744427801}{142239856720821722}e^{12} + \frac{836121020724020837}{142239856720821722}e^{11} + \frac{2497879521512693}{142239856720821722}e^{10} - \frac{6113119041889518457}{142239856720821722}e^{9} + \frac{6558472031641598033}{142239856720821722}e^{8} + \frac{17556854387076633031}{142239856720821722}e^{7} - \frac{27700135179606415151}{142239856720821722}e^{6} - \frac{20911392140560448581}{142239856720821722}e^{5} + \frac{40600280859986890849}{142239856720821722}e^{4} + \frac{9571754597536076101}{142239856720821722}e^{3} - \frac{10067587241821636360}{71119928360410861}e^{2} - \frac{494524834956752347}{71119928360410861}e + \frac{665772187538904005}{71119928360410861}$ |
| 29 | $[29, 29, w - 5]$ | $-\frac{415343520203248}{71119928360410861}e^{15} + \frac{5117388082195351}{71119928360410861}e^{14} - \frac{25367472773685853}{142239856720821722}e^{13} - \frac{158153645744427801}{142239856720821722}e^{12} + \frac{836121020724020837}{142239856720821722}e^{11} + \frac{2497879521512693}{142239856720821722}e^{10} - \frac{6113119041889518457}{142239856720821722}e^{9} + \frac{6558472031641598033}{142239856720821722}e^{8} + \frac{17556854387076633031}{142239856720821722}e^{7} - \frac{27700135179606415151}{142239856720821722}e^{6} - \frac{20911392140560448581}{142239856720821722}e^{5} + \frac{40600280859986890849}{142239856720821722}e^{4} + \frac{9571754597536076101}{142239856720821722}e^{3} - \frac{10067587241821636360}{71119928360410861}e^{2} - \frac{494524834956752347}{71119928360410861}e + \frac{665772187538904005}{71119928360410861}$ |
| 37 | $[37, 37, -w - 3]$ | $-\frac{348395177121975}{71119928360410861}e^{15} + \frac{5412402593030784}{71119928360410861}e^{14} - \frac{23658424513963797}{71119928360410861}e^{13} - \frac{37325144879534673}{71119928360410861}e^{12} + \frac{543610606240285901}{71119928360410861}e^{11} - \frac{936216088590521125}{71119928360410861}e^{10} - \frac{2508448284771733946}{71119928360410861}e^{9} + \frac{8622867124612024690}{71119928360410861}e^{8} + \frac{344645966099315202}{71119928360410861}e^{7} - \frac{23319852507076698054}{71119928360410861}e^{6} + \frac{14836695207575701830}{71119928360410861}e^{5} + \frac{20127486634555050571}{71119928360410861}e^{4} - \frac{19319186215905544830}{71119928360410861}e^{3} - \frac{3746976878788393503}{71119928360410861}e^{2} + \frac{6342789922056739236}{71119928360410861}e - \frac{747894165366267957}{71119928360410861}$ |
| 37 | $[37, 37, w - 4]$ | $-\frac{348395177121975}{71119928360410861}e^{15} + \frac{5412402593030784}{71119928360410861}e^{14} - \frac{23658424513963797}{71119928360410861}e^{13} - \frac{37325144879534673}{71119928360410861}e^{12} + \frac{543610606240285901}{71119928360410861}e^{11} - \frac{936216088590521125}{71119928360410861}e^{10} - \frac{2508448284771733946}{71119928360410861}e^{9} + \frac{8622867124612024690}{71119928360410861}e^{8} + \frac{344645966099315202}{71119928360410861}e^{7} - \frac{23319852507076698054}{71119928360410861}e^{6} + \frac{14836695207575701830}{71119928360410861}e^{5} + \frac{20127486634555050571}{71119928360410861}e^{4} - \frac{19319186215905544830}{71119928360410861}e^{3} - \frac{3746976878788393503}{71119928360410861}e^{2} + \frac{6342789922056739236}{71119928360410861}e - \frac{747894165366267957}{71119928360410861}$ |
| 41 | $[41, 41, -w - 9]$ | $\phantom{-}\frac{617864265750627}{142239856720821722}e^{15} - \frac{3926299144259683}{71119928360410861}e^{14} + \frac{22855738693361567}{142239856720821722}e^{13} + \frac{97856896698927289}{142239856720821722}e^{12} - \frac{634244171751945843}{142239856720821722}e^{11} + \frac{398608126687064397}{142239856720821722}e^{10} + \frac{3584999305670057007}{142239856720821722}e^{9} - \frac{6081945369539477683}{142239856720821722}e^{8} - \frac{5698008121339474585}{142239856720821722}e^{7} + \frac{16316741811303834405}{142239856720821722}e^{6} - \frac{977096083838785825}{142239856720821722}e^{5} - \frac{5128364857113836002}{71119928360410861}e^{4} + \frac{1684210426700487845}{142239856720821722}e^{3} - \frac{587950704682654736}{71119928360410861}e^{2} + \frac{633491713704161400}{71119928360410861}e + \frac{783209710409448037}{71119928360410861}$ |
| 41 | $[41, 41, w - 10]$ | $\phantom{-}\frac{617864265750627}{142239856720821722}e^{15} - \frac{3926299144259683}{71119928360410861}e^{14} + \frac{22855738693361567}{142239856720821722}e^{13} + \frac{97856896698927289}{142239856720821722}e^{12} - \frac{634244171751945843}{142239856720821722}e^{11} + \frac{398608126687064397}{142239856720821722}e^{10} + \frac{3584999305670057007}{142239856720821722}e^{9} - \frac{6081945369539477683}{142239856720821722}e^{8} - \frac{5698008121339474585}{142239856720821722}e^{7} + \frac{16316741811303834405}{142239856720821722}e^{6} - \frac{977096083838785825}{142239856720821722}e^{5} - \frac{5128364857113836002}{71119928360410861}e^{4} + \frac{1684210426700487845}{142239856720821722}e^{3} - \frac{587950704682654736}{71119928360410861}e^{2} + \frac{633491713704161400}{71119928360410861}e + \frac{783209710409448037}{71119928360410861}$ |
| 43 | $[43, 43, -w - 2]$ | $-\frac{331010881119850}{71119928360410861}e^{15} + \frac{4640642037831556}{71119928360410861}e^{14} - \frac{18303301097318877}{71119928360410861}e^{13} - \frac{31477056205510105}{71119928360410861}e^{12} + \frac{410907518781988663}{71119928360410861}e^{11} - \frac{785759658351444712}{71119928360410861}e^{10} - \frac{1449812718412138236}{71119928360410861}e^{9} + \frac{6878767605621668364}{71119928360410861}e^{8} - \frac{3653911722361824276}{71119928360410861}e^{7} - \frac{16722713467605173575}{71119928360410861}e^{6} + \frac{22678917645484124677}{71119928360410861}e^{5} + \frac{10229003674137980694}{71119928360410861}e^{4} - \frac{26538486673488143624}{71119928360410861}e^{3} + \frac{737363506538031459}{71119928360410861}e^{2} + \frac{8573721571252548655}{71119928360410861}e - \frac{597095704784357041}{71119928360410861}$ |
| 43 | $[43, 43, w - 3]$ | $-\frac{331010881119850}{71119928360410861}e^{15} + \frac{4640642037831556}{71119928360410861}e^{14} - \frac{18303301097318877}{71119928360410861}e^{13} - \frac{31477056205510105}{71119928360410861}e^{12} + \frac{410907518781988663}{71119928360410861}e^{11} - \frac{785759658351444712}{71119928360410861}e^{10} - \frac{1449812718412138236}{71119928360410861}e^{9} + \frac{6878767605621668364}{71119928360410861}e^{8} - \frac{3653911722361824276}{71119928360410861}e^{7} - \frac{16722713467605173575}{71119928360410861}e^{6} + \frac{22678917645484124677}{71119928360410861}e^{5} + \frac{10229003674137980694}{71119928360410861}e^{4} - \frac{26538486673488143624}{71119928360410861}e^{3} + \frac{737363506538031459}{71119928360410861}e^{2} + \frac{8573721571252548655}{71119928360410861}e - \frac{597095704784357041}{71119928360410861}$ |
| 47 | $[47, 47, -w - 1]$ | $\phantom{-}\frac{2363877228669316}{71119928360410861}e^{15} - \frac{33670995167987351}{71119928360410861}e^{14} + \frac{260880663442679429}{142239856720821722}e^{13} + \frac{557759574293260845}{142239856720821722}e^{12} - \frac{6230496243503766441}{142239856720821722}e^{11} + \frac{9638049207569065911}{142239856720821722}e^{10} + \frac{28729934729256536817}{142239856720821722}e^{9} - \frac{94447379017684034437}{142239856720821722}e^{8} - \frac{272270269643540347}{142239856720821722}e^{7} + \frac{258907021489645032867}{142239856720821722}e^{6} - \frac{190033589398670492739}{142239856720821722}e^{5} - \frac{226109382580808703129}{142239856720821722}e^{4} + \frac{255022313725480137519}{142239856720821722}e^{3} + \frac{22618414914659596934}{71119928360410861}e^{2} - \frac{44519229207570677141}{71119928360410861}e + \frac{4226014268485738693}{71119928360410861}$ |
| 47 | $[47, 47, w - 2]$ | $\phantom{-}\frac{2363877228669316}{71119928360410861}e^{15} - \frac{33670995167987351}{71119928360410861}e^{14} + \frac{260880663442679429}{142239856720821722}e^{13} + \frac{557759574293260845}{142239856720821722}e^{12} - \frac{6230496243503766441}{142239856720821722}e^{11} + \frac{9638049207569065911}{142239856720821722}e^{10} + \frac{28729934729256536817}{142239856720821722}e^{9} - \frac{94447379017684034437}{142239856720821722}e^{8} - \frac{272270269643540347}{142239856720821722}e^{7} + \frac{258907021489645032867}{142239856720821722}e^{6} - \frac{190033589398670492739}{142239856720821722}e^{5} - \frac{226109382580808703129}{142239856720821722}e^{4} + \frac{255022313725480137519}{142239856720821722}e^{3} + \frac{22618414914659596934}{71119928360410861}e^{2} - \frac{44519229207570677141}{71119928360410861}e + \frac{4226014268485738693}{71119928360410861}$ |
| 53 | $[53, 53, 2w - 13]$ | $-\frac{1900625390196219}{142239856720821722}e^{15} + \frac{26586092519117077}{142239856720821722}e^{14} - \frac{98859206774478809}{142239856720821722}e^{13} - \frac{242348669789734981}{142239856720821722}e^{12} + \frac{2442202985252451885}{142239856720821722}e^{11} - \frac{3414418305588054597}{142239856720821722}e^{10} - \frac{12077738091865493455}{142239856720821722}e^{9} + \frac{35989941541383703327}{142239856720821722}e^{8} + \frac{5622883198537580249}{142239856720821722}e^{7} - \frac{104573197853122874917}{142239856720821722}e^{6} + \frac{63687224171876150145}{142239856720821722}e^{5} + \frac{51239199393462935442}{71119928360410861}e^{4} - \frac{48204288477456303849}{71119928360410861}e^{3} - \frac{13303514341611107949}{71119928360410861}e^{2} + \frac{17966056185341666610}{71119928360410861}e - \frac{2005023591107049126}{71119928360410861}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9, 3, 3]$ | $1$ |