[N,k,chi] = [603,2,Mod(64,603)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(603, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("603.64");
S:= CuspForms(chi, 2);
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 \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).
\(n\)
\(136\)
\(470\)
\(\chi(n)\)
\(-\beta_{9}\)
\(1\)
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
\( T_{2}^{10} - 2T_{2}^{9} + 4T_{2}^{8} + 3T_{2}^{7} - 6T_{2}^{6} + 12T_{2}^{5} + 9T_{2}^{4} - 7T_{2}^{3} + 14T_{2}^{2} - 6T_{2} + 1 \)
T2^10 - 2*T2^9 + 4*T2^8 + 3*T2^7 - 6*T2^6 + 12*T2^5 + 9*T2^4 - 7*T2^3 + 14*T2^2 - 6*T2 + 1
acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} - 6 T^{6} + \cdots + 1)^{2} \)
(T^10 - 2*T^9 + 4*T^8 + 3*T^7 - 6*T^6 + 12*T^5 + 9*T^4 - 7*T^3 + 14*T^2 - 6*T + 1)^2
$3$
\( T^{20} \)
T^20
$5$
\( T^{20} + T^{19} + 7 T^{18} + \cdots + 12243001 \)
T^20 + T^19 + 7*T^18 + 57*T^17 + 154*T^16 + 672*T^15 + 3587*T^14 + 13273*T^13 + 46917*T^12 + 166705*T^11 + 530058*T^10 + 1457269*T^9 + 3770223*T^8 + 9393478*T^7 + 21256728*T^6 + 40426860*T^5 + 61743495*T^4 + 73103531*T^3 + 64417909*T^2 + 38751425*T + 12243001
$7$
\( T^{20} + 8 T^{19} + 18 T^{18} + \cdots + 1739761 \)
T^20 + 8*T^19 + 18*T^18 - 81*T^17 - 178*T^16 + 1290*T^15 + 1238*T^14 - 22574*T^13 + 11093*T^12 + 389598*T^11 + 953470*T^10 + 3029675*T^9 + 17843461*T^8 + 66827206*T^7 + 159617730*T^6 + 204122487*T^5 + 139995442*T^4 + 48115435*T^3 + 30421885*T^2 + 3592956*T + 1739761
$11$
\( T^{20} + 5 T^{18} + 22 T^{17} + \cdots + 4190209 \)
T^20 + 5*T^18 + 22*T^17 + 267*T^16 + 3476*T^15 + 20717*T^14 + 81136*T^13 + 264295*T^12 + 545171*T^11 + 321520*T^10 - 1173843*T^9 - 2017901*T^8 - 3362755*T^7 - 9051105*T^6 - 6491969*T^5 + 89582871*T^4 + 172911585*T^3 + 165359346*T^2 - 19071899*T + 4190209
$13$
\( T^{20} - 12 T^{19} + 73 T^{18} + \cdots + 4489 \)
T^20 - 12*T^19 + 73*T^18 - 552*T^17 + 4037*T^16 - 23299*T^15 + 134817*T^14 - 633607*T^13 + 2614856*T^12 - 10109858*T^11 + 32454225*T^10 - 75044530*T^9 + 119253591*T^8 - 131334264*T^7 + 109056297*T^6 - 91675024*T^5 + 100286186*T^4 - 93628046*T^3 + 43983286*T^2 + 117317*T + 4489
$17$
\( T^{20} - 4 T^{19} + 57 T^{18} + \cdots + 876811321 \)
T^20 - 4*T^19 + 57*T^18 - 480*T^17 + 3014*T^16 - 16688*T^15 + 80898*T^14 - 373507*T^13 + 1716524*T^12 - 7173254*T^11 + 21229891*T^10 - 34988327*T^9 + 88695890*T^8 - 218224049*T^7 + 21924787*T^6 - 379204145*T^5 + 1717318889*T^4 + 1968690111*T^3 + 4780842296*T^2 - 3586365876*T + 876811321
$19$
\( T^{20} - 4 T^{19} - 61 T^{18} + \cdots + 23319241 \)
T^20 - 4*T^19 - 61*T^18 + 431*T^17 + 465*T^16 - 11760*T^15 + 72802*T^14 - 417708*T^13 + 1363998*T^12 - 1119033*T^11 - 197174*T^10 - 12746349*T^9 + 111519870*T^8 - 204772403*T^7 + 422169748*T^6 - 471067014*T^5 + 1328554953*T^4 - 379730186*T^3 + 936759857*T^2 + 112134209*T + 23319241
$23$
\( T^{20} + 2 T^{19} + \cdots + 1098458449 \)
T^20 + 2*T^19 + 15*T^18 + 140*T^17 + 1248*T^16 - 6722*T^15 - 48611*T^14 + 15638*T^13 + 1123972*T^12 + 1520547*T^11 + 5136605*T^10 + 54274693*T^9 + 307830402*T^8 + 1229454809*T^7 + 3139859668*T^6 + 6215090607*T^5 + 12119529279*T^4 + 6205229569*T^3 + 12456296600*T^2 - 6410651632*T + 1098458449
$29$
\( (T^{10} - 3 T^{9} - 91 T^{8} + 70 T^{7} + \cdots + 737)^{2} \)
(T^10 - 3*T^9 - 91*T^8 + 70*T^7 + 2069*T^6 + 989*T^5 - 11726*T^4 - 19063*T^3 - 7469*T^2 + 1067*T + 737)^2
$31$
\( T^{20} - 21 T^{18} + \cdots + 867288026089 \)
T^20 - 21*T^18 - 242*T^17 + 408*T^16 + 22693*T^15 - 87889*T^14 - 670032*T^13 + 7389768*T^12 + 8277346*T^11 + 88040459*T^10 + 590016570*T^9 + 2979052847*T^8 + 11393629962*T^7 + 64235476779*T^6 + 212623989391*T^5 + 549693049632*T^4 + 902797631428*T^3 + 1126793392847*T^2 + 1044971234791*T + 867288026089
$37$
\( (T^{10} - 12 T^{9} - 54 T^{8} + \cdots - 279553)^{2} \)
(T^10 - 12*T^9 - 54*T^8 + 912*T^7 + 1046*T^6 - 25994*T^5 - 12341*T^4 + 318196*T^3 + 119670*T^2 - 1318714*T - 279553)^2
$41$
\( T^{20} + \cdots + 132185354901721 \)
T^20 - 6*T^19 + 135*T^18 - 161*T^17 + 1593*T^16 - 22373*T^15 - 215947*T^14 - 2794118*T^13 + 6112187*T^12 + 299849966*T^11 + 3742248555*T^10 + 34659723318*T^9 + 243380687143*T^8 + 1160878548654*T^7 + 3562751449026*T^6 + 6318749198498*T^5 + 3639325109178*T^4 - 6535665495304*T^3 + 3816898736255*T^2 + 64739521540100*T + 132185354901721
$43$
\( T^{20} + \cdots + 638971773407881 \)
T^20 + 22*T^19 + 290*T^18 + 957*T^17 - 6056*T^16 - 74063*T^15 + 1007180*T^14 + 10376300*T^13 + 26195326*T^12 - 187012254*T^11 + 1682399665*T^10 - 1202467497*T^9 - 87046272452*T^8 - 306805613400*T^7 + 2024782408214*T^6 + 20915806382009*T^5 + 91405475037655*T^4 + 271061066245684*T^3 + 535869419330391*T^2 + 562079588722256*T + 638971773407881
$47$
\( T^{20} + 16 T^{19} + \cdots + 2571253941169 \)
T^20 + 16*T^19 + 153*T^18 + 976*T^17 + 1925*T^16 - 11637*T^15 + 21279*T^14 + 666709*T^13 + 132118*T^12 - 17082769*T^11 - 12470941*T^10 + 294483937*T^9 + 1076913790*T^8 + 3067634022*T^7 + 25633130362*T^6 + 92137025894*T^5 + 184601905664*T^4 + 187982809186*T^3 + 1378653652444*T^2 - 442119000847*T + 2571253941169
$53$
\( T^{20} - T^{19} + 102 T^{18} + \cdots + 51457746649 \)
T^20 - T^19 + 102*T^18 + 292*T^17 - 1837*T^16 - 59190*T^15 + 1234782*T^14 + 13657614*T^13 - 36351672*T^12 - 398530209*T^11 + 2509548273*T^10 + 25130098818*T^9 + 148926003404*T^8 + 364455852011*T^7 + 701017970164*T^6 + 811515176396*T^5 + 702180170648*T^4 + 359472235201*T^3 + 22522336967*T^2 + 11639314330*T + 51457746649
$59$
\( T^{20} + \cdots + 279020843142889 \)
T^20 - 26*T^19 + 380*T^18 - 6980*T^17 + 117290*T^16 - 1274014*T^15 + 12331215*T^14 - 123899816*T^13 + 1072907694*T^12 - 9243716108*T^11 + 73167848865*T^10 - 144383952042*T^9 - 270988231359*T^8 + 2940534696304*T^7 - 4341127135889*T^6 - 5038406363183*T^5 + 50241828715669*T^4 - 123003168930119*T^3 + 222915563949810*T^2 - 254336880354894*T + 279020843142889
$61$
\( T^{20} + 26 T^{19} + \cdots + 76356191669209 \)
T^20 + 26*T^19 + 314*T^18 + 1436*T^17 - 5008*T^16 - 341534*T^15 - 4550540*T^14 - 11928866*T^13 + 458617952*T^12 + 4307694600*T^11 + 7370921680*T^10 + 1969879208*T^9 + 238773705831*T^8 + 17789585620*T^7 + 2807020092509*T^6 - 3613007470176*T^5 + 12279348643955*T^4 - 21219949944232*T^3 + 44514199075176*T^2 - 43596327832292*T + 76356191669209
$67$
\( T^{20} + 22 T^{19} + \cdots + 18\!\cdots\!49 \)
T^20 + 22*T^19 + 331*T^18 + 2222*T^17 - 2496*T^16 - 327800*T^15 - 4137748*T^14 - 27737666*T^13 + 14210054*T^12 + 2219845276*T^11 + 26753077319*T^10 + 148729633492*T^9 + 63788932406*T^8 - 8342463639158*T^7 - 83380260615508*T^6 - 442571010074600*T^5 - 225784121893824*T^4 + 13466901187027706*T^3 + 134408401271248171*T^2 + 598543756718488834*T + 1822837804551761449
$71$
\( T^{20} + 20 T^{19} + \cdots + 1590305111329 \)
T^20 + 20*T^19 + 279*T^18 + 1103*T^17 - 13415*T^16 + 264870*T^15 + 7596488*T^14 + 107866345*T^13 + 1181820445*T^12 - 2082803564*T^11 - 2767877254*T^10 + 108959197149*T^9 + 747758070169*T^8 + 3628051769705*T^7 + 15184624283906*T^6 + 42370947376153*T^5 + 53829422659650*T^4 + 13295246183109*T^3 + 192739190609768*T^2 - 22878612098067*T + 1590305111329
$73$
\( T^{20} + 55 T^{19} + \cdots + 24\!\cdots\!01 \)
T^20 + 55*T^19 + 1666*T^18 + 34034*T^17 + 514385*T^16 + 5889114*T^15 + 51412398*T^14 + 321490070*T^13 + 1157592709*T^12 - 1649597136*T^11 - 54892966952*T^10 - 359329983507*T^9 - 742934709831*T^8 + 6991792011039*T^7 + 79879396404503*T^6 + 430564267933635*T^5 + 1498295474107953*T^4 + 3525180717523982*T^3 + 5531875691713050*T^2 + 5282234106002194*T + 2420034779434801
$79$
\( T^{20} + 34 T^{19} + \cdots + 45\!\cdots\!09 \)
T^20 + 34*T^19 + 601*T^18 + 8472*T^17 + 124509*T^16 + 1366751*T^15 + 10488545*T^14 + 63741113*T^13 + 304035525*T^12 - 1006326145*T^11 - 15073141863*T^10 - 29782996265*T^9 + 123378245711*T^8 + 1554323775340*T^7 + 2642128297961*T^6 - 28504320835495*T^5 + 204575012393048*T^4 - 629924613705810*T^3 + 2080752308937561*T^2 - 3614541611092700*T + 4559598209891209
$83$
\( T^{20} + 56 T^{19} + \cdots + 25849243729 \)
T^20 + 56*T^19 + 1517*T^18 + 24901*T^17 + 267057*T^16 + 1964462*T^15 + 11782932*T^14 + 102580442*T^13 + 1418728712*T^12 + 17055751911*T^11 + 150542199380*T^10 + 947740364725*T^9 + 4215057380334*T^8 + 12290053064265*T^7 + 19876050389670*T^6 + 3974864631546*T^5 + 1827640733019*T^4 - 12574534145700*T^3 + 4816007661847*T^2 - 169042063239*T + 25849243729
$89$
\( T^{20} - 3 T^{19} + \cdots + 41\!\cdots\!49 \)
T^20 - 3*T^19 + 15*T^18 - 2615*T^17 + 35765*T^16 + 209897*T^15 + 8096622*T^14 - 833386*T^13 - 309984990*T^12 - 10523665408*T^11 - 66681515131*T^10 + 420317619014*T^9 + 18698410439578*T^8 + 284983628846087*T^7 + 2988347030093149*T^6 + 23704821826542566*T^5 + 144595954488604672*T^4 + 658092132985919927*T^3 + 2109455022029745626*T^2 + 4260900774121910166*T + 4126921988840047849
$97$
\( (T^{10} + 7 T^{9} - 405 T^{8} + \cdots - 294045257)^{2} \)
(T^10 + 7*T^9 - 405*T^8 - 2470*T^7 + 50873*T^6 + 245563*T^5 - 2630430*T^4 - 8875999*T^3 + 54143463*T^2 + 90831807*T - 294045257)^2
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