[N,k,chi] = [105,4,Mod(41,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.41");
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 \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
\(n\)
\(22\)
\(31\)
\(71\)
\(\chi(n)\)
\(1\)
\(-1\)
\(-1\)
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
\( T_{17}^{8} - 36 T_{17}^{7} - 17439 T_{17}^{6} + 673446 T_{17}^{5} + 80245500 T_{17}^{4} - 2696813856 T_{17}^{3} - 99699071472 T_{17}^{2} + 2613586192416 T_{17} - 14231625894720 \)
T17^8 - 36*T17^7 - 17439*T17^6 + 673446*T17^5 + 80245500*T17^4 - 2696813856*T17^3 - 99699071472*T17^2 + 2613586192416*T17 - 14231625894720
acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} + 96 T^{14} + 3618 T^{12} + \cdots + 5760000 \)
T^16 + 96*T^14 + 3618*T^12 + 68560*T^10 + 697017*T^8 + 3791184*T^6 + 10461796*T^4 + 13209792*T^2 + 5760000
$3$
\( T^{16} - 2 T^{15} + \cdots + 282429536481 \)
T^16 - 2*T^15 + 13*T^14 - 66*T^13 + 906*T^12 - 6354*T^11 + 22347*T^10 - 169506*T^9 + 544266*T^8 - 4576662*T^7 + 16290963*T^6 - 125065782*T^5 + 481485546*T^4 - 947027862*T^3 + 5036466357*T^2 - 20920706406*T + 282429536481
$5$
\( (T - 5)^{16} \)
(T - 5)^16
$7$
\( T^{16} + 4 T^{15} + \cdots + 19\!\cdots\!01 \)
T^16 + 4*T^15 - 156*T^14 + 6132*T^13 - 105180*T^12 - 971292*T^11 + 27927452*T^10 - 282573004*T^9 + 781972086*T^8 - 96922540372*T^7 + 3285636800348*T^6 - 39195135650244*T^5 - 1455826587801180*T^4 + 29112047178970476*T^3 - 254032521274030044*T^2 + 2234183456333136028*T + 191581231380566414401
$11$
\( T^{16} + 12690 T^{14} + \cdots + 15\!\cdots\!00 \)
T^16 + 12690*T^14 + 58703601*T^12 + 124231198492*T^10 + 131700393212400*T^8 + 73239720004993152*T^6 + 21312676928409491200*T^4 + 3008093170654412160000*T^2 + 158422945366440000000000
$13$
\( T^{16} + 14238 T^{14} + \cdots + 35\!\cdots\!00 \)
T^16 + 14238*T^14 + 73621641*T^12 + 166589510904*T^10 + 157504313285424*T^8 + 55293969691628928*T^6 + 3665783202760233216*T^4 + 76099399903047426048*T^2 + 359885206292019609600
$17$
\( (T^{8} - 36 T^{7} + \cdots - 14231625894720)^{2} \)
(T^8 - 36*T^7 - 17439*T^6 + 673446*T^5 + 80245500*T^4 - 2696813856*T^3 - 99699071472*T^2 + 2613586192416*T - 14231625894720)^2
$19$
\( T^{16} + 47508 T^{14} + \cdots + 15\!\cdots\!00 \)
T^16 + 47508*T^14 + 803627040*T^12 + 5827449233088*T^10 + 17669582808795648*T^8 + 20031736244328557568*T^6 + 4063907177822249164800*T^4 + 168386611638121482240000*T^2 + 1536862595952485376000000
$23$
\( T^{16} + 77076 T^{14} + \cdots + 82\!\cdots\!00 \)
T^16 + 77076*T^14 + 2049203904*T^12 + 24857218625344*T^10 + 152491196098857216*T^8 + 473761395541959499776*T^6 + 667242015632735541723136*T^4 + 283304881859214666966761472*T^2 + 8276890629615884697600000000
$29$
\( T^{16} + 206658 T^{14} + \cdots + 66\!\cdots\!00 \)
T^16 + 206658*T^14 + 17675473473*T^12 + 809323696323004*T^10 + 21355521440644356240*T^8 + 323737135753970556868608*T^6 + 2620691796612975315049504768*T^4 + 9256293604943356607298200666112*T^2 + 6653464736266712152529155679846400
$31$
\( T^{16} + 232728 T^{14} + \cdots + 10\!\cdots\!00 \)
T^16 + 232728*T^14 + 20574555600*T^12 + 885388778034048*T^10 + 19457429187958391808*T^8 + 209153578323670578321408*T^6 + 1008238085743366492380057600*T^4 + 1979083486510451553407938560000*T^2 + 1028110594683324539990246400000000
$37$
\( (T^{8} + 406 T^{7} + \cdots + 70\!\cdots\!00)^{2} \)
(T^8 + 406*T^7 - 233492*T^6 - 125409280*T^5 + 1789385552*T^4 + 9026793332768*T^3 + 1623562622557120*T^2 + 93939842508428800*T + 701394437334784000)^2
$41$
\( (T^{8} - 468 T^{7} + \cdots + 65\!\cdots\!00)^{2} \)
(T^8 - 468*T^7 - 224988*T^6 + 155048112*T^5 - 9995711760*T^4 - 9342547363008*T^3 + 2389336748432064*T^2 - 216053191075984128*T + 6578222266765900800)^2
$43$
\( (T^{8} + 274 T^{7} + \cdots - 66\!\cdots\!40)^{2} \)
(T^8 + 274*T^7 - 244736*T^6 - 99121720*T^5 - 1794810736*T^4 + 2771188130432*T^3 + 220826612908288*T^2 - 6284591104729088*T - 666624855021731840)^2
$47$
\( (T^{8} + 456 T^{7} + \cdots + 95\!\cdots\!00)^{2} \)
(T^8 + 456*T^7 - 447207*T^6 - 217787226*T^5 + 46205699604*T^4 + 22337546899536*T^3 - 2794130106527808*T^2 - 730245757410809856*T + 95108965237210982400)^2
$53$
\( T^{16} + 1299852 T^{14} + \cdots + 10\!\cdots\!00 \)
T^16 + 1299852*T^14 + 685218029424*T^12 + 186724587839773312*T^10 + 27652619568247640430336*T^8 + 2125617260255391905829350400*T^6 + 72732535721698077115747002880000*T^4 + 863636162904639975096090851328000000*T^2 + 1078135749843211314635417856000000000000
$59$
\( (T^{8} - 276 T^{7} + \cdots + 32\!\cdots\!00)^{2} \)
(T^8 - 276*T^7 - 563664*T^6 + 127733616*T^5 + 81659372208*T^4 - 14851114314624*T^3 - 3694367062563840*T^2 + 510377933594880000*T + 32333016885926400000)^2
$61$
\( T^{16} + 1827996 T^{14} + \cdots + 29\!\cdots\!00 \)
T^16 + 1827996*T^14 + 1354175905296*T^12 + 528004601408332800*T^10 + 117643392495067565776896*T^8 + 15048345863172093258868457472*T^6 + 1027321919510077640336309511782400*T^4 + 29162843522952577009037171585187840000*T^2 + 29538649292627086751976913266278400000000
$67$
\( (T^{8} - 502 T^{7} + \cdots - 18\!\cdots\!60)^{2} \)
(T^8 - 502*T^7 - 1063784*T^6 + 448865672*T^5 + 317830459120*T^4 - 95193919625728*T^3 - 25192160408625152*T^2 + 4907247297878571008*T - 182575600157286133760)^2
$71$
\( T^{16} + 1941672 T^{14} + \cdots + 43\!\cdots\!00 \)
T^16 + 1941672*T^14 + 1280280572304*T^12 + 410211272958651520*T^10 + 71826703184786865036288*T^8 + 7069597616345485714960508928*T^6 + 373753622882696976644905311539200*T^4 + 8883320924599263506814030491504640000*T^2 + 43706232619241875900732387390694400000000
$73$
\( T^{16} + 2857944 T^{14} + \cdots + 16\!\cdots\!24 \)
T^16 + 2857944*T^14 + 3208219701168*T^12 + 1802852843163152256*T^10 + 539086553259091646572800*T^8 + 85438404104832293055973318656*T^6 + 6717332693002136142497264385196032*T^4 + 206672333328473209501989349968360505344*T^2 + 164212714701012095132879228997443229057024
$79$
\( (T^{8} - 646 T^{7} + \cdots - 13\!\cdots\!00)^{2} \)
(T^8 - 646*T^7 - 1961927*T^6 + 1489511132*T^5 + 870620077696*T^4 - 764576674485952*T^3 - 62500410900413696*T^2 + 102471652402532986880*T - 13223518835311947776000)^2
$83$
\( (T^{8} - 876 T^{7} + \cdots - 26\!\cdots\!00)^{2} \)
(T^8 - 876*T^7 - 1944540*T^6 + 1055769696*T^5 + 1321755786672*T^4 - 213919992874368*T^3 - 316856318724260352*T^2 - 55584393130700421120*T - 2663903607090021273600)^2
$89$
\( (T^{8} + 3048 T^{7} + \cdots + 38\!\cdots\!00)^{2} \)
(T^8 + 3048*T^7 + 1582032*T^6 - 2058534576*T^5 - 1250006468880*T^4 + 391676655410496*T^3 + 192397192816170240*T^2 - 19284986938452480000*T + 386834474914329600000)^2
$97$
\( T^{16} + 6322590 T^{14} + \cdots + 43\!\cdots\!36 \)
T^16 + 6322590*T^14 + 14633076197721*T^12 + 15752097847324563048*T^10 + 8575029762368476696514928*T^8 + 2315821154101830514198962972288*T^6 + 265805955160335648727509971972360448*T^4 + 7631463884302452516483031406552681054208*T^2 + 43053255883534463853748460705508295172161536
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