[N,k,chi] = [576,4,Mod(145,576)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.145");
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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).
\(n\)
\(65\)
\(127\)
\(325\)
\(\chi(n)\)
\(1\)
\(1\)
\(-\beta_{1}\)
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
\( T_{5}^{10} - 2 T_{5}^{9} + 2 T_{5}^{8} + 1216 T_{5}^{7} + 70152 T_{5}^{6} + 238960 T_{5}^{5} + 121104 T_{5}^{4} - 16403712 T_{5}^{3} + 303177744 T_{5}^{2} - 350050848 T_{5} + 202085408 \)
T5^10 - 2*T5^9 + 2*T5^8 + 1216*T5^7 + 70152*T5^6 + 238960*T5^5 + 121104*T5^4 - 16403712*T5^3 + 303177744*T5^2 - 350050848*T5 + 202085408
acting on \(S_{4}^{\mathrm{new}}(576, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{10} \)
T^10
$3$
\( T^{10} \)
T^10
$5$
\( T^{10} - 2 T^{9} + 2 T^{8} + \cdots + 202085408 \)
T^10 - 2*T^9 + 2*T^8 + 1216*T^7 + 70152*T^6 + 238960*T^5 + 121104*T^4 - 16403712*T^3 + 303177744*T^2 - 350050848*T + 202085408
$7$
\( T^{10} + 1668 T^{8} + \cdots + 1936000000 \)
T^10 + 1668*T^8 + 822752*T^6 + 108889984*T^4 + 1007165696*T^2 + 1936000000
$11$
\( T^{10} - 18 T^{9} + \cdots + 3810412010528 \)
T^10 - 18*T^9 + 162*T^8 - 98976*T^7 + 10893448*T^6 - 470191184*T^5 + 11596827024*T^4 - 161036931712*T^3 + 1239116280336*T^2 - 3072960643104*T + 3810412010528
$13$
\( T^{10} + 2 T^{9} + \cdots + 11\!\cdots\!28 \)
T^10 + 2*T^9 + 2*T^8 - 49600*T^7 + 16525704*T^6 - 250009712*T^5 + 697009168*T^4 + 381714028800*T^3 + 14003671653904*T^2 + 177772297473568*T + 1128382274666528
$17$
\( (T^{5} - 2 T^{4} - 11912 T^{3} + \cdots + 556317664)^{2} \)
(T^5 - 2*T^4 - 11912*T^3 - 63216*T^2 + 32655888*T + 556317664)^2
$19$
\( T^{10} - 26 T^{9} + \cdots + 19\!\cdots\!12 \)
T^10 - 26*T^9 + 338*T^8 + 518240*T^7 + 171199496*T^6 + 859995248*T^5 + 54061042704*T^4 + 29494616138112*T^3 + 3781733909606416*T^2 + 121686632109503328*T + 1957784020253121312
$23$
\( T^{10} + 45284 T^{8} + \cdots + 25\!\cdots\!84 \)
T^10 + 45284*T^8 + 659906016*T^6 + 3356415204224*T^4 + 5853829992751360*T^2 + 2515997041083638784
$29$
\( T^{10} - 202 T^{9} + \cdots + 71\!\cdots\!12 \)
T^10 - 202*T^9 + 20402*T^8 + 6104512*T^7 + 805743752*T^6 + 5081279536*T^5 + 1167330884496*T^4 + 308671955027712*T^3 + 39119271732533776*T^2 + 749235198907452768*T + 7174895625869198112
$31$
\( (T^{5} + 184 T^{4} - 14912 T^{3} + \cdots - 678952960)^{2} \)
(T^5 + 184*T^4 - 14912*T^3 - 2117120*T^2 + 96370688*T - 678952960)^2
$37$
\( T^{10} + 10 T^{9} + \cdots + 69\!\cdots\!32 \)
T^10 + 10*T^9 + 50*T^8 + 1466432*T^7 + 3533346568*T^6 + 113532054224*T^5 + 2033864619152*T^4 + 781155377762560*T^3 + 768345771890009104*T^2 + 32597513787698387616*T + 691484188508881657632
$41$
\( T^{10} + 248192 T^{8} + \cdots + 58\!\cdots\!00 \)
T^10 + 248192*T^8 + 20256591872*T^6 + 717520447209472*T^4 + 11078490196063289344*T^2 + 58457884687352620646400
$43$
\( T^{10} - 838 T^{9} + \cdots + 54\!\cdots\!32 \)
T^10 - 838*T^9 + 351122*T^8 - 65506784*T^7 + 7618050760*T^6 - 980593787632*T^5 + 292441750094224*T^4 - 53245925430785920*T^3 + 5527320337404144400*T^2 - 244960046971056772960*T + 5428075536530540678432
$47$
\( (T^{5} + 472 T^{4} + \cdots + 154359955456)^{2} \)
(T^5 + 472*T^4 - 56896*T^3 - 24501760*T^2 + 34766848*T + 154359955456)^2
$53$
\( T^{10} - 378 T^{9} + \cdots + 63\!\cdots\!52 \)
T^10 - 378*T^9 + 71442*T^8 + 109029056*T^7 + 157841071368*T^6 - 17679454284880*T^5 + 1350019425137552*T^4 + 723927781227277056*T^3 + 423327820385245240336*T^2 - 23160738099080854505888*T + 633574931132629058776352
$59$
\( T^{10} - 1706 T^{9} + \cdots + 18\!\cdots\!48 \)
T^10 - 1706*T^9 + 1455218*T^8 - 638938784*T^7 + 196471556680*T^6 - 86422664782480*T^5 + 65649505199853968*T^4 - 28360317550212729472*T^3 + 6734120632813889472784*T^2 - 505204797539837414719392*T + 18950647112971160088706848
$61$
\( T^{10} - 910 T^{9} + \cdots + 96\!\cdots\!00 \)
T^10 - 910*T^9 + 414050*T^8 + 252493120*T^7 + 84699231112*T^6 - 6220933337840*T^5 + 2467720519178000*T^4 + 1439598946129982720*T^3 + 419191471593829393936*T^2 + 284043552892632210720*T + 96233756418169927200
$67$
\( T^{10} + 1942 T^{9} + \cdots + 15\!\cdots\!08 \)
T^10 + 1942*T^9 + 1885682*T^8 + 946883552*T^7 + 279012575112*T^6 + 42102307991536*T^5 + 3927921981284880*T^4 + 481113229077495680*T^3 + 151802546431036300816*T^2 + 21555616974221911467616*T + 1530424337612976871762208
$71$
\( T^{10} + 1078692 T^{8} + \cdots + 10\!\cdots\!56 \)
T^10 + 1078692*T^8 + 396610003424*T^6 + 64222301440447360*T^4 + 4550304717312899080448*T^2 + 107586737247866277669446656
$73$
\( T^{10} + 755888 T^{8} + \cdots + 42\!\cdots\!24 \)
T^10 + 755888*T^8 + 169546275584*T^6 + 15235670414888960*T^4 + 530927871865054035968*T^2 + 4274086953481210874036224
$79$
\( (T^{5} - 2208 T^{4} + \cdots + 10448447471616)^{2} \)
(T^5 - 2208*T^4 + 1350912*T^3 + 33652736*T^2 - 183347445760*T + 10448447471616)^2
$83$
\( T^{10} + 2562 T^{9} + \cdots + 16\!\cdots\!72 \)
T^10 + 2562*T^9 + 3281922*T^8 + 1426542624*T^7 + 179373287752*T^6 - 21397913819056*T^5 + 374001334554361872*T^4 + 104542516635194882176*T^3 + 7795754246940374828304*T^2 - 15843886076057777581043424*T + 16100348859099784083392471072
$89$
\( T^{10} + 3406512 T^{8} + \cdots + 13\!\cdots\!96 \)
T^10 + 3406512*T^8 + 3353163782912*T^6 + 1101931785064321024*T^4 + 61204888317079786618880*T^2 + 1368303035475100482666496
$97$
\( (T^{5} + 2 T^{4} + \cdots + 65755091474464)^{2} \)
(T^5 + 2*T^4 - 1997576*T^3 - 730301968*T^2 + 370826423312*T + 65755091474464)^2
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