[N,k,chi] = [3,21,Mod(2,3)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 21, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.2");
S:= CuspForms(chi, 21);
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 χ \chi χ on generators for ( Z / 3 Z ) × \left(\mathbb{Z}/3\mathbb{Z}\right)^\times ( Z / 3 Z ) × .
n n n
2 2 2
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι 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 is the entire newspace S 21 n e w ( 3 , [ χ ] ) S_{21}^{\mathrm{new}}(3, [\chi]) S 2 1 n e w ( 3 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 6 + ⋯ + 84 ⋯ 20 T^{6} + \cdots + 84\!\cdots\!20 T 6 + ⋯ + 8 4 ⋯ 2 0
T^6 + 4208328*T^4 + 3964520318976*T^2 + 846951404399493120
3 3 3
T 6 + ⋯ + 42 ⋯ 01 T^{6} + \cdots + 42\!\cdots\!01 T 6 + ⋯ + 4 2 ⋯ 0 1
T^6 + 4122*T^5 + 552237183*T^4 - 356755554641556*T^3 + 1925531995336582383*T^2 + 50113897022232660517722*T + 42391158275216203514294433201
5 5 5
T 6 + ⋯ + 27 ⋯ 00 T^{6} + \cdots + 27\!\cdots\!00 T 6 + ⋯ + 2 7 ⋯ 0 0
T^6 + 484704682430400*T^4 + 66547880393157345946780800000*T^2 + 2767033894728262397343925192708608000000000
7 7 7
( T 3 + ⋯ + 30 ⋯ 00 ) 2 (T^{3} + \cdots + 30\!\cdots\!00)^{2} ( T 3 + ⋯ + 3 0 ⋯ 0 0 ) 2
(T^3 - 279803958*T^2 - 127584561562048500*T + 30590253082420358136355000)^2
11 11 1 1
T 6 + ⋯ + 72 ⋯ 00 T^{6} + \cdots + 72\!\cdots\!00 T 6 + ⋯ + 7 2 ⋯ 0 0
T^6 + 1497987675566694774720*T^4 + 214956922770980995063560331894006654080000*T^2 + 727685929765259103593665967547902342115366038733233356800000
13 13 1 3
( T 3 + ⋯ + 19 ⋯ 00 ) 2 (T^{3} + \cdots + 19\!\cdots\!00)^{2} ( T 3 + ⋯ + 1 9 ⋯ 0 0 ) 2
(T^3 - 8923939638*T^2 - 6797158417228643929140*T + 194437243363096745461703318361400)^2
17 17 1 7
T 6 + ⋯ + 21 ⋯ 80 T^{6} + \cdots + 21\!\cdots\!80 T 6 + ⋯ + 2 1 ⋯ 8 0
T^6 + 3521657843885774582258688*T^4 + 1700839880425951819093034869972315763277600915456*T^2 + 215183071234637977708649143792255523421929902342816384896592237591265280
19 19 1 9
( T 3 + ⋯ − 14 ⋯ 68 ) 2 (T^{3} + \cdots - 14\!\cdots\!68)^{2} ( T 3 + ⋯ − 1 4 ⋯ 6 8 ) 2
(T^3 + 7479010097562*T^2 - 24765328174811423258189364*T - 148642621478588729559222947970863746568)^2
23 23 2 3
T 6 + ⋯ + 24 ⋯ 20 T^{6} + \cdots + 24\!\cdots\!20 T 6 + ⋯ + 2 4 ⋯ 2 0
T^6 + 3126796700978310751720412928*T^4 + 1654944010894051160679557390030536958077559998014898176*T^2 + 240307249008070974599450780952076568989209685134735701780946057594865375846072320
29 29 2 9
T 6 + ⋯ + 80 ⋯ 00 T^{6} + \cdots + 80\!\cdots\!00 T 6 + ⋯ + 8 0 ⋯ 0 0
T^6 + 336038115294494445271637983680*T^4 + 10186611374663268043828327321598169717442408824734375040000*T^2 + 80950317413713077897938813064060324747862129976964122992462747209567841119896371200000
31 31 3 1
( T 3 + ⋯ − 58 ⋯ 88 ) 2 (T^{3} + \cdots - 58\!\cdots\!88)^{2} ( T 3 + ⋯ − 5 8 ⋯ 8 8 ) 2
(T^3 + 1091452582741674*T^2 - 685835155449678634607976232308*T - 586636376705997333807072965761108628909830088)^2
37 37 3 7
( T 3 + ⋯ + 12 ⋯ 00 ) 2 (T^{3} + \cdots + 12\!\cdots\!00)^{2} ( T 3 + ⋯ + 1 2 ⋯ 0 0 ) 2
(T^3 + 967170731084202*T^2 - 35857296656449841082807540084660*T + 12701574404253225493746450133335009205683849400)^2
41 41 4 1
T 6 + ⋯ + 47 ⋯ 00 T^{6} + \cdots + 47\!\cdots\!00 T 6 + ⋯ + 4 7 ⋯ 0 0
T^6 + 254137568716709771878848966577920*T^4 + 19795728514587001654056210257974605831082253929500616902789120000*T^2 + 477231199793790563386980257307080389915133991675442189619751905551636312791140195759082700800000
43 43 4 3
( T 3 + ⋯ + 32 ⋯ 00 ) 2 (T^{3} + \cdots + 32\!\cdots\!00)^{2} ( T 3 + ⋯ + 3 2 ⋯ 0 0 ) 2
(T^3 - 29097493751092038*T^2 - 13585824383465180129941984197300*T + 3245255887598548183176028515797863685988886075000)^2
47 47 4 7
T 6 + ⋯ + 15 ⋯ 80 T^{6} + \cdots + 15\!\cdots\!80 T 6 + ⋯ + 1 5 ⋯ 8 0
T^6 + 17029342663699560578694166737189888*T^4 + 90000059800362465924478884965135934970682035071328433577614520877056*T^2 + 150394029167159642608605366756207835485366893934484120496351250065516753756087890198969132555677204480
53 53 5 3
T 6 + ⋯ + 96 ⋯ 80 T^{6} + \cdots + 96\!\cdots\!80 T 6 + ⋯ + 9 6 ⋯ 8 0
T^6 + 53443295728380762639698447390628288*T^4 + 669518729261761602204215852226032682020909903999716471291936367072256*T^2 + 96723543209551496608561804934782495322432626355208025199103428179697278014939753103865816730473594880
59 59 5 9
T 6 + ⋯ + 12 ⋯ 00 T^{6} + \cdots + 12\!\cdots\!00 T 6 + ⋯ + 1 2 ⋯ 0 0
T^6 + 714172911770596617896253848813315520*T^4 + 126527885857196469430234844113058753371230301218225171808656533834880000*T^2 + 1266875103936222274439094422374477914126092602302790162888034962760652302028362510694337788772621516800000
61 61 6 1
( T 3 + ⋯ − 70 ⋯ 48 ) 2 (T^{3} + \cdots - 70\!\cdots\!48)^{2} ( T 3 + ⋯ − 7 0 ⋯ 4 8 ) 2
(T^3 + 1499554772690772234*T^2 + 242889361249279831927060706993269452*T - 70826644134358662762235568995917505925724691812411848)^2
67 67 6 7
( T 3 + ⋯ + 29 ⋯ 00 ) 2 (T^{3} + \cdots + 29\!\cdots\!00)^{2} ( T 3 + ⋯ + 2 9 ⋯ 0 0 ) 2
(T^3 - 2244077984945099238*T^2 - 968004934253145817392810871478164020*T + 2954591394079489169081282495249547983083995682832836600)^2
71 71 7 1
T 6 + ⋯ + 14 ⋯ 00 T^{6} + \cdots + 14\!\cdots\!00 T 6 + ⋯ + 1 4 ⋯ 0 0
T^6 + 4032706539361342129128727166898650880*T^4 + 2043274665965504517227132194961017542983522899054984198696966288066560000*T^2 + 142936393355644084897183688222382477728527585154279273677479838450477529523298065516949447975794848563200000
73 73 7 3
( T 3 + ⋯ − 11 ⋯ 00 ) 2 (T^{3} + \cdots - 11\!\cdots\!00)^{2} ( T 3 + ⋯ − 1 1 ⋯ 0 0 ) 2
(T^3 - 7869202902131298918*T^2 + 18339724644946597395216917887089644940*T - 11946437091179885872604724061686203705616654467536478600)^2
79 79 7 9
( T 3 + ⋯ + 24 ⋯ 12 ) 2 (T^{3} + \cdots + 24\!\cdots\!12)^{2} ( T 3 + ⋯ + 2 4 ⋯ 1 2 ) 2
(T^3 - 8839970375551901718*T^2 - 54031516634123460664759495721432416884*T + 247137986724215618285187593834562573574232802230668211512)^2
83 83 8 3
T 6 + ⋯ + 26 ⋯ 80 T^{6} + \cdots + 26\!\cdots\!80 T 6 + ⋯ + 2 6 ⋯ 8 0
T^6 + 922220721386856887289572246012060119488*T^4 + 275460037373900980323226353418859782144147559742641697848637075378354697491456*T^2 + 26491106145192360888306610971011221021628138329239196534682581379312653680132625069417851534261969475482786213396480
89 89 8 9
T 6 + ⋯ + 16 ⋯ 00 T^{6} + \cdots + 16\!\cdots\!00 T 6 + ⋯ + 1 6 ⋯ 0 0
T^6 + 1152126931350699539415266130009538686720*T^4 + 96042156249490966432898297426435784617239833808340447067435363534398638080000*T^2 + 1644961542482844936949646020760666491718305206079450922403343117186073416453875888909690165288806643291704524800000
97 97 9 7
( T 3 + ⋯ + 17 ⋯ 00 ) 2 (T^{3} + \cdots + 17\!\cdots\!00)^{2} ( T 3 + ⋯ + 1 7 ⋯ 0 0 ) 2
(T^3 - 111620168122774197318*T^2 - 196870150276135146250726744756377729780*T + 174570899824955145193851403566017873050082367944514138916600)^2
show more
show less