[N,k,chi] = [570,4,Mod(121,570)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(570, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("570.121");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).
\(n\)
\(191\)
\(211\)
\(457\)
\(\chi(n)\)
\(1\)
\(-1 + \beta_{3}\)
\(1\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} - 4T_{7}^{5} - 1288T_{7}^{4} - 1560T_{7}^{3} + 392979T_{7}^{2} + 2391124T_{7} - 2276916 \)
T7^6 - 4*T7^5 - 1288*T7^4 - 1560*T7^3 + 392979*T7^2 + 2391124*T7 - 2276916
acting on \(S_{4}^{\mathrm{new}}(570, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T + 4)^{6} \)
(T^2 + 2*T + 4)^6
$3$
\( (T^{2} + 3 T + 9)^{6} \)
(T^2 + 3*T + 9)^6
$5$
\( (T^{2} - 5 T + 25)^{6} \)
(T^2 - 5*T + 25)^6
$7$
\( (T^{6} - 4 T^{5} - 1288 T^{4} + \cdots - 2276916)^{2} \)
(T^6 - 4*T^5 - 1288*T^4 - 1560*T^3 + 392979*T^2 + 2391124*T - 2276916)^2
$11$
\( (T^{6} + 43 T^{5} - 5937 T^{4} + \cdots - 3783540528)^{2} \)
(T^6 + 43*T^5 - 5937*T^4 - 143947*T^3 + 11027012*T^2 + 26579460*T - 3783540528)^2
$13$
\( T^{12} + 28 T^{11} + \cdots + 11\!\cdots\!44 \)
T^12 + 28*T^11 + 8279*T^10 + 410880*T^9 + 64238665*T^8 + 2375791886*T^7 + 97973648356*T^6 + 1065058635776*T^5 + 24112477562560*T^4 - 158776600012800*T^3 + 7382050366186496*T^2 - 28893363250454528*T + 116462952338489344
$17$
\( T^{12} + 15 T^{11} + \cdots + 44\!\cdots\!16 \)
T^12 + 15*T^11 + 12827*T^10 + 669830*T^9 + 158283274*T^8 + 5507511236*T^7 + 271886209228*T^6 + 4723710631464*T^5 + 206152933590288*T^4 + 3941759622827520*T^3 + 78215515286777856*T^2 + 645061940347797504*T + 4476219935048073216
$19$
\( T^{12} - 36 T^{11} + \cdots + 10\!\cdots\!41 \)
T^12 - 36*T^11 + 7938*T^10 - 1067472*T^9 + 77267718*T^8 - 5747519988*T^7 + 881391500422*T^6 - 39422239597692*T^5 + 3635127866169558*T^4 - 344460082123544688*T^3 + 17569293827547186018*T^2 - 546520573075492738764*T + 104127350297911241532841
$23$
\( T^{12} - 74 T^{11} + \cdots + 12\!\cdots\!84 \)
T^12 - 74*T^11 + 20583*T^10 - 639362*T^9 + 220782637*T^8 - 4984925952*T^7 + 1459742815476*T^6 - 1522488147840*T^5 + 5677462641032208*T^4 + 22736326789252224*T^3 + 14142661152896332800*T^2 + 276011326110201053184*T + 12847325900820194525184
$29$
\( T^{12} + 121 T^{11} + \cdots + 68\!\cdots\!84 \)
T^12 + 121*T^11 + 124641*T^10 + 12846724*T^9 + 10839740434*T^8 + 1050440810856*T^7 + 430941767106348*T^6 + 27769591215167472*T^5 + 11092354464263219568*T^4 + 635943234171320849088*T^3 + 113356574137211518751040*T^2 - 2477829066459960714948864*T + 68325461098748674311619584
$31$
\( (T^{6} - 88 T^{5} + \cdots - 73102333248616)^{2} \)
(T^6 - 88*T^5 - 145314*T^4 + 9473276*T^3 + 6168641169*T^2 - 195447396892*T - 73102333248616)^2
$37$
\( (T^{6} + 166 T^{5} + \cdots - 7829761512672)^{2} \)
(T^6 + 166*T^5 - 115906*T^4 - 13730044*T^3 + 1741942785*T^2 + 112753690038*T - 7829761512672)^2
$41$
\( T^{12} + 184 T^{11} + \cdots + 19\!\cdots\!44 \)
T^12 + 184*T^11 + 211423*T^10 + 32312056*T^9 + 36580604117*T^8 + 5794459179264*T^7 + 1180702244288788*T^6 + 102515211524591808*T^5 + 15217390058228228304*T^4 + 1236801884985292756608*T^3 + 121458745066375027065600*T^2 + 5056497888431387113728000*T + 190430901175614490000134144
$43$
\( T^{12} + 604 T^{11} + \cdots + 13\!\cdots\!24 \)
T^12 + 604*T^11 + 367775*T^10 + 96546904*T^9 + 38817354549*T^8 + 10578119877370*T^7 + 2714313766818800*T^6 + 437652271021191480*T^5 + 54752924139970102976*T^4 + 4102169122340246157152*T^3 + 225047254674801730310080*T^2 + 6602511675569455389051648*T + 132179151791077131114906624
$47$
\( T^{12} - 589 T^{11} + \cdots + 14\!\cdots\!24 \)
T^12 - 589*T^11 + 419339*T^10 - 87949934*T^9 + 45083288656*T^8 - 7317720991416*T^7 + 3519539119147392*T^6 - 207881590867647264*T^5 + 62367093252271344960*T^4 + 6240553638291670298112*T^3 + 884808029340765819039744*T^2 + 36373954603464027425931264*T + 1412735081985650417571201024
$53$
\( T^{12} + 767 T^{11} + \cdots + 53\!\cdots\!76 \)
T^12 + 767*T^11 + 810424*T^10 + 301160297*T^9 + 239680848548*T^8 + 83094492490491*T^7 + 45715643742816625*T^6 + 11186340283619396484*T^5 + 4410427951532870553348*T^4 + 936580350837578069230416*T^3 + 282648553356298471445107728*T^2 + 37322122944012348833740879488*T + 5343415854260919202583182181376
$59$
\( T^{12} - 527 T^{11} + \cdots + 67\!\cdots\!64 \)
T^12 - 527*T^11 + 946889*T^10 - 494119048*T^9 + 622288126720*T^8 - 292314358552752*T^7 + 196414619880081744*T^6 - 64353119010309166464*T^5 + 33346808776562665259520*T^4 - 9862753162200626604347136*T^3 + 3230540700904427464325452032*T^2 - 493319369096686999034849912832*T + 67067717037607982070282707288064
$61$
\( T^{12} - 107 T^{11} + \cdots + 10\!\cdots\!36 \)
T^12 - 107*T^11 + 500444*T^10 + 265518287*T^9 + 226729760398*T^8 + 53253992189843*T^7 + 11920469475523969*T^6 + 794714664597169078*T^5 + 93890260990040985198*T^4 - 1264237009936966575468*T^3 + 853314577854925250573404*T^2 - 9323899802704708681859112*T + 103081734554627273264433936
$67$
\( T^{12} - 266 T^{11} + \cdots + 72\!\cdots\!44 \)
T^12 - 266*T^11 + 1174961*T^10 - 118797814*T^9 + 1087352899279*T^8 - 175399821984448*T^7 + 241668636069466114*T^6 - 66021657269707580576*T^5 + 47559303218313342422292*T^4 - 9893033644776155050066680*T^3 + 1707247476016434357461222800*T^2 - 126671401101457671318713004288*T + 7258902539111605421116331986944
$71$
\( T^{12} + 545 T^{11} + \cdots + 29\!\cdots\!44 \)
T^12 + 545*T^11 + 1339951*T^10 + 544301462*T^9 + 1337732867594*T^8 + 536610661470756*T^7 + 358251998726423884*T^6 + 621220303872463368*T^5 + 11371812959070583728912*T^4 + 1368201076325441714221056*T^3 + 176618678921343359005434624*T^2 + 7806638537865532819945820160*T + 296462067001303738404133638144
$73$
\( T^{12} + 1146 T^{11} + \cdots + 32\!\cdots\!44 \)
T^12 + 1146*T^11 + 2798235*T^10 + 1132298846*T^9 + 3031500123039*T^8 + 672197279892030*T^7 + 2379974143757077870*T^6 - 160363340434083886380*T^5 + 930206966502852291727836*T^4 - 192626451428466992737594752*T^3 + 299489257230766572238809204240*T^2 - 70932749838644197979342835815808*T + 32143536846205945510130961007546944
$79$
\( T^{12} - 199 T^{11} + \cdots + 31\!\cdots\!56 \)
T^12 - 199*T^11 + 1131034*T^10 - 709409759*T^9 + 1238173367190*T^8 - 512395440350179*T^7 + 257988661652398617*T^6 - 32930674094342270364*T^5 + 12896775175128513803788*T^4 - 598720777944702007604624*T^3 + 637564030963534871954910608*T^2 + 13906334438538991278956545152*T + 315923080264190613170998445056
$83$
\( (T^{6} - 1779 T^{5} + \cdots - 25\!\cdots\!84)^{2} \)
(T^6 - 1779*T^5 - 926948*T^4 + 2064081424*T^3 + 619066674144*T^2 - 692180085248784*T - 256912492297958784)^2
$89$
\( T^{12} + 1189 T^{11} + \cdots + 18\!\cdots\!96 \)
T^12 + 1189*T^11 + 2958886*T^10 + 2371417181*T^9 + 5007552414278*T^8 + 3825788972620127*T^7 + 3887911707666405295*T^6 + 1150746942531710225902*T^5 + 625238684150529882833068*T^4 - 52924947113308431676014768*T^3 + 85744236052600981490542208832*T^2 - 1275009325651122838370574898176*T + 18846685770192684728612748804096
$97$
\( T^{12} - 1736 T^{11} + \cdots + 30\!\cdots\!84 \)
T^12 - 1736*T^11 + 4204090*T^10 - 3622175800*T^9 + 6540944789708*T^8 - 4707247392735352*T^7 + 6586337495751300544*T^6 - 2040987313810925259104*T^5 + 1889333774047729628327296*T^4 + 439573692753160330393944192*T^3 + 445567101705660677155092331776*T^2 + 37256842070155867059235024791552*T + 3045103893630766164407349906653184
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