[N,k,chi] = [21,5,Mod(2,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.2");
S:= CuspForms(chi, 5);
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 \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
\(n\)
\(8\)
\(10\)
\(\chi(n)\)
\(-1\)
\(-1 - \beta_{2}\)
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
\( T_{2}^{16} - 103 T_{2}^{14} + 7227 T_{2}^{12} - 270898 T_{2}^{10} + 7374256 T_{2}^{8} - 115494792 T_{2}^{6} + 1245573504 T_{2}^{4} - 2908017504 T_{2}^{2} + 5639409216 \)
T2^16 - 103*T2^14 + 7227*T2^12 - 270898*T2^10 + 7374256*T2^8 - 115494792*T2^6 + 1245573504*T2^4 - 2908017504*T2^2 + 5639409216
acting on \(S_{5}^{\mathrm{new}}(21, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} - 103 T^{14} + \cdots + 5639409216 \)
T^16 - 103*T^14 + 7227*T^12 - 270898*T^10 + 7374256*T^8 - 115494792*T^6 + 1245573504*T^4 - 2908017504*T^2 + 5639409216
$3$
\( T^{16} - 8 T^{15} + \cdots + 18\!\cdots\!41 \)
T^16 - 8*T^15 + T^14 - 180*T^13 + 8397*T^12 - 114912*T^11 + 1108566*T^10 - 4977612*T^9 + 29575530*T^8 - 403186572*T^7 + 7273301526*T^6 - 61068948192*T^5 + 361463316237*T^4 - 627621192180*T^3 + 282429536481*T^2 - 183014339639688*T + 1853020188851841
$5$
\( T^{16} - 3547 T^{14} + \cdots + 88\!\cdots\!16 \)
T^16 - 3547*T^14 + 8240634*T^12 - 11243531563*T^10 + 11177868938314*T^8 - 6896058986680899*T^6 + 3016060690773138561*T^4 - 619312418648064933624*T^2 + 88973936275855348823616
$7$
\( (T^{8} + 84 T^{7} + \cdots + 33232930569601)^{2} \)
(T^8 + 84*T^7 + 2135*T^6 - 219324*T^5 - 19275228*T^4 - 526596924*T^3 + 12307850135*T^2 + 1162668124884*T + 33232930569601)^2
$11$
\( T^{16} - 49255 T^{14} + \cdots + 23\!\cdots\!36 \)
T^16 - 49255*T^14 + 1772602830*T^12 - 29834307230299*T^10 + 368603345865319966*T^8 - 720479645547057631095*T^6 + 1065221444706741072809289*T^4 - 570629056355072383125929472*T^2 + 235550994721164638679720001536
$13$
\( (T^{4} + 123 T^{3} - 25046 T^{2} + \cdots + 114515728)^{4} \)
(T^4 + 123*T^3 - 25046*T^2 - 1759704*T + 114515728)^4
$17$
\( T^{16} - 320190 T^{14} + \cdots + 15\!\cdots\!16 \)
T^16 - 320190*T^14 + 71483950683*T^12 - 7744269460819806*T^10 + 599504684178321684513*T^8 - 25952342690981491786628424*T^6 + 810907451197230215759536734912*T^4 - 13858515102189882978431601654500352*T^2 + 159640858278800012280225478133950857216
$19$
\( (T^{8} + 663 T^{7} + \cdots + 27\!\cdots\!16)^{2} \)
(T^8 + 663*T^7 + 381680*T^6 + 75286737*T^5 + 16112841420*T^4 - 369092879181*T^3 + 371041661855081*T^2 + 9729554956268160*T + 277999631912943616)^2
$23$
\( T^{16} - 1456782 T^{14} + \cdots + 53\!\cdots\!96 \)
T^16 - 1456782*T^14 + 1519083112851*T^12 - 732964885764275862*T^10 + 255345787309981123672281*T^8 - 37168840112414575441283962380*T^6 + 3905474438842746361484727088103472*T^4 - 168951889762285740767881501966105431168*T^2 + 5381154427507742928626610777741329750762496
$29$
\( (T^{8} + 472213 T^{6} + \cdots + 43\!\cdots\!36)^{2} \)
(T^8 + 472213*T^6 + 62942341336*T^4 + 2935135874614608*T^2 + 43734935370689127936)^2
$31$
\( (T^{8} + 1252 T^{7} + \cdots + 42\!\cdots\!69)^{2} \)
(T^8 + 1252*T^7 + 1723284*T^6 - 16823792*T^5 + 129277610905*T^4 - 2522687119032*T^3 + 8960460932725596*T^2 - 583735053955138992*T + 42915473425920809169)^2
$37$
\( (T^{8} - 671 T^{7} + \cdots + 66\!\cdots\!44)^{2} \)
(T^8 - 671*T^7 + 1529010*T^6 - 377232455*T^5 + 1558866218290*T^4 - 628410655962867*T^3 + 275363602547633901*T^2 - 14154002595162093924*T + 660960785118657542544)^2
$41$
\( (T^{8} + 9934876 T^{6} + \cdots + 27\!\cdots\!76)^{2} \)
(T^8 + 9934876*T^6 + 32416802249776*T^4 + 35768790900903493440*T^2 + 2764923019239285276696576)^2
$43$
\( (T^{4} - 365 T^{3} + \cdots + 885324127312)^{4} \)
(T^4 - 365*T^3 - 3850284*T^2 + 1737835060*T + 885324127312)^4
$47$
\( T^{16} - 15501246 T^{14} + \cdots + 77\!\cdots\!76 \)
T^16 - 15501246*T^14 + 197109277981731*T^12 - 663363315573360753510*T^10 + 1818154102513573217696823801*T^8 - 128038010508394567623349296348492*T^6 + 7712151418027549089529153774908410160*T^4 - 82912181393987125343643507369243440515200*T^2 + 771416989916879817339921074483458338629477376
$53$
\( T^{16} - 30342651 T^{14} + \cdots + 68\!\cdots\!96 \)
T^16 - 30342651*T^14 + 588431159454906*T^12 - 6980101064294588111691*T^10 + 60730012719764816282056879434*T^8 - 356829404401916050333043830714639587*T^6 + 1537348140621243695653684349830459012821249*T^4 - 4045541741114386561754911501575073856159122041528*T^2 + 6807397979424663361924788902572175500165104045853932096
$59$
\( T^{16} - 44267415 T^{14} + \cdots + 21\!\cdots\!96 \)
T^16 - 44267415*T^14 + 1497278120716278*T^12 - 18812549152000702659699*T^10 + 177003187243511699378037905478*T^8 - 369314193085872281969718303197156511*T^6 + 616109373419172868058536578432115011113217*T^4 - 120420245989096289455066646158216031458350804408*T^2 + 21217326303967122635987300359010079432250829170360896
$61$
\( (T^{8} - 7506 T^{7} + \cdots + 29\!\cdots\!64)^{2} \)
(T^8 - 7506*T^7 + 55108751*T^6 - 98296807482*T^5 + 318589265015949*T^4 + 312272635573494864*T^3 + 1961572767814409522276*T^2 + 763617845720348703856512*T + 294101769517052431194776464)^2
$67$
\( (T^{8} - 4329 T^{7} + \cdots + 30\!\cdots\!84)^{2} \)
(T^8 - 4329*T^7 + 59066504*T^6 + 168357464865*T^5 + 1465187785851396*T^4 + 1385265415736239623*T^3 + 7045456565814395847797*T^2 + 542165157986362003336932*T + 30440448292643045951864621584)^2
$71$
\( (T^{8} + 155086092 T^{6} + \cdots + 10\!\cdots\!24)^{2} \)
(T^8 + 155086092*T^6 + 8128946327751648*T^4 + 164192587365745297078272*T^2 + 1007473568242599016860294733824)^2
$73$
\( (T^{8} - 6161 T^{7} + \cdots + 30\!\cdots\!84)^{2} \)
(T^8 - 6161*T^7 + 58704516*T^6 - 203558217239*T^5 + 1626437390284240*T^4 - 5596374382545810531*T^3 + 23817908943827418743379*T^2 - 29029754184034280448127626*T + 30697244395807007046038276484)^2
$79$
\( (T^{8} + 20084 T^{7} + \cdots + 45\!\cdots\!49)^{2} \)
(T^8 + 20084*T^7 + 302093344*T^6 + 2038558036160*T^5 + 10369822932436285*T^4 + 2479501502776808264*T^3 + 6841001783767795280560*T^2 + 154462389004812061836032*T + 4555969322387957607729946249)^2
$83$
\( (T^{8} + 116812773 T^{6} + \cdots + 51\!\cdots\!44)^{2} \)
(T^8 + 116812773*T^6 + 3194595703525464*T^4 + 24782493786131396915712*T^2 + 5131869800810930097339654144)^2
$89$
\( T^{16} - 202431178 T^{14} + \cdots + 32\!\cdots\!56 \)
T^16 - 202431178*T^14 + 35537146704474579*T^12 - 1050571515725155536932482*T^10 + 24454743442760922607952511535129*T^8 - 138488317325097948319363602942474577716*T^6 + 647776106812873630043063042000966020328948496*T^4 - 146158067464219196137402529816263454744983911936*T^2 + 32976131847208198889866401588906041676630687891456
$97$
\( (T^{4} + 7067 T^{3} + \cdots + 368225255164464)^{4} \)
(T^4 + 7067*T^3 - 30719906*T^2 - 161925558552*T + 368225255164464)^4
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