[N,k,chi] = [66,6,Mod(65,66)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(66, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("66.65");
S:= CuspForms(chi, 6);
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 \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).
\(n\)
\(13\)
\(23\)
\(\chi(n)\)
\(-1\)
\(-1\)
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
\( T_{17}^{5} + 1038T_{17}^{4} - 4533096T_{17}^{3} - 5102614656T_{17}^{2} + 2497107681792T_{17} + 2183764681359360 \)
T17^5 + 1038*T17^4 - 4533096*T17^3 - 5102614656*T17^2 + 2497107681792*T17 + 2183764681359360
acting on \(S_{6}^{\mathrm{new}}(66, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T + 4)^{10} \)
(T + 4)^10
$3$
\( T^{10} - 10 T^{9} + \cdots + 847288609443 \)
T^10 - 10*T^9 + 150*T^8 - 366*T^7 - 25407*T^6 + 241056*T^5 - 6173901*T^4 - 21611934*T^3 + 2152336050*T^2 - 34867844010*T + 847288609443
$5$
\( T^{10} + \cdots + 580964238766592 \)
T^10 + 17416*T^8 + 81910909*T^6 + 82775942666*T^4 + 13422717203264*T^2 + 580964238766592
$7$
\( T^{10} + 105078 T^{8} + \cdots + 68\!\cdots\!08 \)
T^10 + 105078*T^8 + 3665300472*T^6 + 45660812476032*T^4 + 127812842198360064*T^2 + 68081540044274270208
$11$
\( T^{10} + 30 T^{9} + \cdots + 10\!\cdots\!51 \)
T^10 + 30*T^9 + 99967*T^8 + 12084072*T^7 + 26177136370*T^6 - 6213198768876*T^5 + 4215853989524870*T^4 + 313429706373055272*T^3 + 417586967751974383517*T^2 + 20182499847976800276030*T + 108347059433883722041830251
$13$
\( T^{10} + 858042 T^{8} + \cdots + 20\!\cdots\!52 \)
T^10 + 858042*T^8 + 223183802304*T^6 + 24621122960663040*T^4 + 1190969993073553145856*T^2 + 20199250208542412035325952
$17$
\( (T^{5} + 1038 T^{4} + \cdots + 21\!\cdots\!60)^{2} \)
(T^5 + 1038*T^4 - 4533096*T^3 - 5102614656*T^2 + 2497107681792*T + 2183764681359360)^2
$19$
\( T^{10} + 16308588 T^{8} + \cdots + 74\!\cdots\!00 \)
T^10 + 16308588*T^8 + 80867135241792*T^6 + 131754966184706273280*T^4 + 60641247593624137308831744*T^2 + 7461337085091667675490077900800
$23$
\( T^{10} + 42452896 T^{8} + \cdots + 13\!\cdots\!00 \)
T^10 + 42452896*T^8 + 608243468761261*T^6 + 3935177656098926740802*T^4 + 11814990956260652944167937952*T^2 + 13375444469691423536102389693536800
$29$
\( (T^{5} + 888 T^{4} + \cdots + 12\!\cdots\!92)^{2} \)
(T^5 + 888*T^4 - 42835476*T^3 + 85992872832*T^2 + 23995237588992*T + 1210223891054592)^2
$31$
\( (T^{5} - 1084 T^{4} + \cdots + 17\!\cdots\!00)^{2} \)
(T^5 - 1084*T^4 - 60654557*T^3 + 67729931920*T^2 + 303532303938800*T + 176808810424336000)^2
$37$
\( (T^{5} - 16 T^{4} + \cdots - 44\!\cdots\!44)^{2} \)
(T^5 - 16*T^4 - 90795083*T^3 + 266406083254*T^2 + 252775388694908*T - 448183063936776344)^2
$41$
\( (T^{5} + 1332 T^{4} + \cdots + 36\!\cdots\!36)^{2} \)
(T^5 + 1332*T^4 - 416352900*T^3 + 342676829232*T^2 + 34252434295058304*T + 3645666639171035136)^2
$43$
\( T^{10} + 433628292 T^{8} + \cdots + 15\!\cdots\!48 \)
T^10 + 433628292*T^8 + 56111546290029408*T^6 + 2546716243043692806119424*T^4 + 41020540174445606460160884080640*T^2 + 151766944639122481205087389106200117248
$47$
\( T^{10} + 1150695994 T^{8} + \cdots + 14\!\cdots\!92 \)
T^10 + 1150695994*T^8 + 327732921509794336*T^6 + 29269377946095657222547520*T^4 + 410562520961515184699811395025152*T^2 + 1485981218490285124901801243956965396992
$53$
\( T^{10} + 1572253978 T^{8} + \cdots + 80\!\cdots\!00 \)
T^10 + 1572253978*T^8 + 747283516508113408*T^6 + 129297623062013532802810880*T^4 + 5975456526281300398669100310462464*T^2 + 8092119872848914751481269145374559436800
$59$
\( T^{10} + 1197786766 T^{8} + \cdots + 12\!\cdots\!00 \)
T^10 + 1197786766*T^8 + 234995102431705609*T^6 + 8797133918278594312780904*T^4 + 61926198737048729036844785509712*T^2 + 1213377756557134140821925018601932800
$61$
\( T^{10} + 8294004138 T^{8} + \cdots + 43\!\cdots\!68 \)
T^10 + 8294004138*T^8 + 24011635954862809440*T^6 + 27309819897246336804882874368*T^4 + 8929590378946859055950253738279567360*T^2 + 433277486732561877701562182748517543201210368
$67$
\( (T^{5} - 41080 T^{4} + \cdots - 21\!\cdots\!24)^{2} \)
(T^5 - 41080*T^4 - 1476692417*T^3 + 25034485160080*T^2 + 264622543061378000*T - 2176071729837979764224)^2
$71$
\( T^{10} + 12332773360 T^{8} + \cdots + 26\!\cdots\!00 \)
T^10 + 12332773360*T^8 + 55882198636434057997*T^6 + 113729666520407734967467656338*T^4 + 99803573397861142833843656236191299744*T^2 + 26426099977025720247188843338536471762521127200
$73$
\( T^{10} + 9701788080 T^{8} + \cdots + 11\!\cdots\!00 \)
T^10 + 9701788080*T^8 + 26080995082693681152*T^6 + 28364569214875852910635450368*T^4 + 12138240814769105550130969786422657024*T^2 + 1148909201586484649628809798118795237339955200
$79$
\( T^{10} + 6232034694 T^{8} + \cdots + 91\!\cdots\!88 \)
T^10 + 6232034694*T^8 + 11606420924418403224*T^6 + 6166786588617140938074069504*T^4 + 944510996216298864312621539300081664*T^2 + 9128732897965499092673741002521266815500288
$83$
\( (T^{5} + 264 T^{4} + \cdots - 92\!\cdots\!32)^{2} \)
(T^5 + 264*T^4 - 10932450960*T^3 - 145327969977984*T^2 + 16843997426561427456*T - 92909654848565242232832)^2
$89$
\( T^{10} + 37766069974 T^{8} + \cdots + 20\!\cdots\!48 \)
T^10 + 37766069974*T^8 + 385368911520939063769*T^6 + 543887064359751694865806580768*T^4 + 207588466325701730503212356494331371520*T^2 + 20994007433986994653195671607560452622281474048
$97$
\( (T^{5} - 64834 T^{4} + \cdots + 32\!\cdots\!16)^{2} \)
(T^5 - 64834*T^4 - 24522134963*T^3 + 669522620427256*T^2 + 152401462159319222204*T + 3200290045729268504708416)^2
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