gp: [N,k,chi] = [51,5,Mod(50,51)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(51, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 5, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("51.50");
S:= CuspForms(chi, 5);
N := Newforms(S);
Newform invariants
sage: traces = [20,0,0]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
gp: f = lf[1] \\ Warning: the index may be different
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the q q q -expansion are expressed in terms of a basis 1 , β 1 , … , β 19 1,\beta_1,\ldots,\beta_{19} 1 , β 1 , … , β 1 9 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 20 + 200 x 18 + 22051 x 16 + 1226808 x 14 + 5013252 x 12 − 3569195664 x 10 + ⋯ + 12 ⋯ 01 x^{20} + 200 x^{18} + 22051 x^{16} + 1226808 x^{14} + 5013252 x^{12} - 3569195664 x^{10} + \cdots + 12\!\cdots\!01 x 2 0 + 2 0 0 x 1 8 + 2 2 0 5 1 x 1 6 + 1 2 2 6 8 0 8 x 1 4 + 5 0 1 3 2 5 2 x 1 2 − 3 5 6 9 1 9 5 6 6 4 x 1 0 + ⋯ + 1 2 ⋯ 0 1
x^20 + 200*x^18 + 22051*x^16 + 1226808*x^14 + 5013252*x^12 - 3569195664*x^10 + 32891946372*x^8 + 52810061696568*x^6 + 6227853708942531*x^4 + 370604037770368200*x^2 + 12157665459056928801
:
β 1 \beta_{1} β 1 = = =
( 9676288307 ν 18 + 2066932568335 ν 16 + 303759546502772 ν 14 + ⋯ + 16 ⋯ 61 ) / 74 ⋯ 84 ( 9676288307 \nu^{18} + 2066932568335 \nu^{16} + 303759546502772 \nu^{14} + \cdots + 16\!\cdots\!61 ) / 74\!\cdots\!84 ( 9 6 7 6 2 8 8 3 0 7 ν 1 8 + 2 0 6 6 9 3 2 5 6 8 3 3 5 ν 1 6 + 3 0 3 7 5 9 5 4 6 5 0 2 7 7 2 ν 1 4 + ⋯ + 1 6 ⋯ 6 1 ) / 7 4 ⋯ 8 4
(9676288307*v^18 + 2066932568335*v^16 + 303759546502772*v^14 + 3485041911614124*v^12 - 219692907030623256*v^10 - 49278903139361409912*v^8 + 7400754685964758217508*v^6 + 921063050403594472427580*v^4 + 62454736469043303525874341*v^2 + 1681819857660255626563582761) / 743834282901416629234053984
β 2 \beta_{2} β 2 = = =
( − 22296163853 ν 18 − 62252999998105 ν 16 + ⋯ − 14 ⋯ 07 ) / 74 ⋯ 84 ( - 22296163853 \nu^{18} - 62252999998105 \nu^{16} + \cdots - 14\!\cdots\!07 ) / 74\!\cdots\!84 ( − 2 2 2 9 6 1 6 3 8 5 3 ν 1 8 − 6 2 2 5 2 9 9 9 9 9 8 1 0 5 ν 1 6 + ⋯ − 1 4 ⋯ 0 7 ) / 7 4 ⋯ 8 4
(-22296163853*v^18 - 62252999998105*v^16 - 9539425358537420*v^14 - 764795374095616884*v^12 - 5711772791490170328*v^10 + 3688279714553772056472*v^8 + 115003998680602616433252*v^6 - 24335094945916965164220900*v^4 - 1303013681244386489687379867*v^2 - 142121027329157246240251537407) / 743834282901416629234053984
β 3 \beta_{3} β 3 = = =
( 10596916 ν 18 − 823861717 ν 16 − 222708163958 ν 14 − 20897213752134 ν 12 + ⋯ − 50 ⋯ 57 ) / 53 ⋯ 48 ( 10596916 \nu^{18} - 823861717 \nu^{16} - 222708163958 \nu^{14} - 20897213752134 \nu^{12} + \cdots - 50\!\cdots\!57 ) / 53\!\cdots\!48 ( 1 0 5 9 6 9 1 6 ν 1 8 − 8 2 3 8 6 1 7 1 7 ν 1 6 − 2 2 2 7 0 8 1 6 3 9 5 8 ν 1 4 − 2 0 8 9 7 2 1 3 7 5 2 1 3 4 ν 1 2 + ⋯ − 5 0 ⋯ 5 7 ) / 5 3 ⋯ 4 8
(10596916*v^18 - 823861717*v^16 - 222708163958*v^14 - 20897213752134*v^12 + 269025952289922*v^10 + 180173450068836750*v^8 + 5434395504716573586*v^6 - 952263524249973328518*v^4 - 116888991228213611067450*v^2 - 5032280794514936248797957) / 53482476481263778345848
β 4 \beta_{4} β 4 = = =
( 47899676 ν 18 + 5914967161 ν 16 + 454247668586 ν 14 + 3196707526746 ν 12 + ⋯ + 33 ⋯ 13 ) / 16 ⋯ 44 ( 47899676 \nu^{18} + 5914967161 \nu^{16} + 454247668586 \nu^{14} + 3196707526746 \nu^{12} + \cdots + 33\!\cdots\!13 ) / 16\!\cdots\!44 ( 4 7 8 9 9 6 7 6 ν 1 8 + 5 9 1 4 9 6 7 1 6 1 ν 1 6 + 4 5 4 2 4 7 6 6 8 5 8 6 ν 1 4 + 3 1 9 6 7 0 7 5 2 6 7 4 6 ν 1 2 + ⋯ + 3 3 ⋯ 1 3 ) / 1 6 ⋯ 4 4
(47899676*v^18 + 5914967161*v^16 + 454247668586*v^14 + 3196707526746*v^12 - 1557654650191350*v^10 - 43362912412519182*v^8 + 9755191163110883994*v^6 + 1981620632161486869786*v^4 + 114039857971754495823294*v^2 + 3331800701351645165995113) / 160447429443791335037544
β 5 \beta_{5} β 5 = = =
( − ν 18 − 200 ν 16 − 22051 ν 14 − 1226808 ν 12 − 5013252 ν 10 + ⋯ − 33 ⋯ 80 ) / 18 ⋯ 41 ( - \nu^{18} - 200 \nu^{16} - 22051 \nu^{14} - 1226808 \nu^{12} - 5013252 \nu^{10} + \cdots - 33\!\cdots\!80 ) / 18\!\cdots\!41 ( − ν 1 8 − 2 0 0 ν 1 6 − 2 2 0 5 1 ν 1 4 − 1 2 2 6 8 0 8 ν 1 2 − 5 0 1 3 2 5 2 ν 1 0 + ⋯ − 3 3 ⋯ 8 0 ) / 1 8 ⋯ 4 1
(-v^18 - 200*v^16 - 22051*v^14 - 1226808*v^12 - 5013252*v^10 + 3569195664*v^8 - 32891946372*v^6 - 52810061696568*v^4 - 6227853708942531*v^2 - 333543633993331380) / 1853020188851841
β 6 \beta_{6} β 6 = = =
( ν 19 + 200 ν 17 + 22051 ν 15 + 1226808 ν 13 + 5013252 ν 11 + ⋯ + 37 ⋯ 00 ν ) / 15 ⋯ 21 ( \nu^{19} + 200 \nu^{17} + 22051 \nu^{15} + 1226808 \nu^{13} + 5013252 \nu^{11} + \cdots + 37\!\cdots\!00 \nu ) / 15\!\cdots\!21 ( ν 1 9 + 2 0 0 ν 1 7 + 2 2 0 5 1 ν 1 5 + 1 2 2 6 8 0 8 ν 1 3 + 5 0 1 3 2 5 2 ν 1 1 + ⋯ + 3 7 ⋯ 0 0 ν ) / 1 5 ⋯ 2 1
(v^19 + 200*v^17 + 22051*v^15 + 1226808*v^13 + 5013252*v^11 - 3569195664*v^9 + 32891946372*v^7 + 52810061696568*v^5 + 6227853708942531*v^3 + 370604037770368200*v) / 150094635296999121
β 7 \beta_{7} β 7 = = =
( − ν 19 − 200 ν 17 − 22051 ν 15 − 1226808 ν 13 − 5013252 ν 11 + ⋯ + 79 ⋯ 63 ν ) / 15 ⋯ 21 ( - \nu^{19} - 200 \nu^{17} - 22051 \nu^{15} - 1226808 \nu^{13} - 5013252 \nu^{11} + \cdots + 79\!\cdots\!63 \nu ) / 15\!\cdots\!21 ( − ν 1 9 − 2 0 0 ν 1 7 − 2 2 0 5 1 ν 1 5 − 1 2 2 6 8 0 8 ν 1 3 − 5 0 1 3 2 5 2 ν 1 1 + ⋯ + 7 9 ⋯ 6 3 ν ) / 1 5 ⋯ 2 1
(-v^19 - 200*v^17 - 22051*v^15 - 1226808*v^13 - 5013252*v^11 + 3569195664*v^9 - 32891946372*v^7 - 52810061696568*v^5 - 6227853708942531*v^3 + 79679868120629163*v) / 150094635296999121
β 8 \beta_{8} β 8 = = =
( 158730996001 ν 19 − 122002712855335 ν 17 + ⋯ − 13 ⋯ 69 ν ) / 20 ⋯ 68 ( 158730996001 \nu^{19} - 122002712855335 \nu^{17} + \cdots - 13\!\cdots\!69 \nu ) / 20\!\cdots\!68 ( 1 5 8 7 3 0 9 9 6 0 0 1 ν 1 9 − 1 2 2 0 0 2 7 1 2 8 5 5 3 3 5 ν 1 7 + ⋯ − 1 3 ⋯ 6 9 ν ) / 2 0 ⋯ 6 8
(158730996001*v^19 - 122002712855335*v^17 - 13377039759251276*v^15 - 553565694422584980*v^13 + 21455780096246052240*v^11 + 2171905519573846022256*v^9 - 56196745458300135861084*v^7 - 44146522674123536500227684*v^5 - 2929621907039142379646548161*v^3 - 132630776482277276002069130169*v) / 20083525638338248989319457568
β 9 \beta_{9} β 9 = = =
( − 174866532347 ν 18 − 6054603354571 ν 16 − 7355327030372 ν 14 + ⋯ + 23 ⋯ 07 ) / 24 ⋯ 28 ( - 174866532347 \nu^{18} - 6054603354571 \nu^{16} - 7355327030372 \nu^{14} + \cdots + 23\!\cdots\!07 ) / 24\!\cdots\!28 ( − 1 7 4 8 6 6 5 3 2 3 4 7 ν 1 8 − 6 0 5 4 6 0 3 3 5 4 5 7 1 ν 1 6 − 7 3 5 5 3 2 7 0 3 0 3 7 2 ν 1 4 + ⋯ + 2 3 ⋯ 0 7 ) / 2 4 ⋯ 2 8
(-174866532347*v^18 - 6054603354571*v^16 - 7355327030372*v^14 + 47708172656579412*v^12 + 472402914703746408*v^10 - 28178080393047863976*v^8 - 35251861075263297780948*v^6 - 496353686425488551724732*v^4 - 69815881780577229748693581*v^2 + 2367713381785095177365221107) / 247944760967138876411351328
β 10 \beta_{10} β 1 0 = = =
( − 907610110121 ν 19 − 118035894441973 ν 17 + ⋯ + 73 ⋯ 01 ν ) / 60 ⋯ 04 ( - 907610110121 \nu^{19} - 118035894441973 \nu^{17} + \cdots + 73\!\cdots\!01 \nu ) / 60\!\cdots\!04 ( − 9 0 7 6 1 0 1 1 0 1 2 1 ν 1 9 − 1 1 8 0 3 5 8 9 4 4 4 1 9 7 3 ν 1 7 + ⋯ + 7 3 ⋯ 0 1 ν ) / 6 0 ⋯ 0 4
(-907610110121*v^19 - 118035894441973*v^17 - 6452565957432236*v^15 + 879503040627363324*v^13 + 18315281782315944072*v^11 + 1798032906618516532728*v^9 - 353171946566135156863644*v^7 + 625405582695892097305260*v^5 + 390631683107174490226896129*v^3 + 73401554441341989120544999101*v) / 60250576915014746967958372704
β 11 \beta_{11} β 1 1 = = =
( − 229167004 ν 18 − 46766958929 ν 16 − 2796425671090 ν 14 + ⋯ − 21 ⋯ 89 ) / 16 ⋯ 44 ( - 229167004 \nu^{18} - 46766958929 \nu^{16} - 2796425671090 \nu^{14} + \cdots - 21\!\cdots\!89 ) / 16\!\cdots\!44 ( − 2 2 9 1 6 7 0 0 4 ν 1 8 − 4 6 7 6 6 9 5 8 9 2 9 ν 1 6 − 2 7 9 6 4 2 5 6 7 1 0 9 0 ν 1 4 + ⋯ − 2 1 ⋯ 8 9 ) / 1 6 ⋯ 4 4
(-229167004*v^18 - 46766958929*v^16 - 2796425671090*v^14 - 29163491103006*v^12 + 8358064033405374*v^10 + 462780138953797110*v^8 - 78687483012711865458*v^6 - 11439346540154354964702*v^4 - 726881407568687287896678*v^2 - 21170767014793020897933489) / 160447429443791335037544
β 12 \beta_{12} β 1 2 = = =
( − 828988309543 ν 18 − 183250833369281 ν 16 + ⋯ − 16 ⋯ 75 ) / 37 ⋯ 92 ( - 828988309543 \nu^{18} - 183250833369281 \nu^{16} + \cdots - 16\!\cdots\!75 ) / 37\!\cdots\!92 ( − 8 2 8 9 8 8 3 0 9 5 4 3 ν 1 8 − 1 8 3 2 5 0 8 3 3 3 6 9 2 8 1 ν 1 6 + ⋯ − 1 6 ⋯ 7 5 ) / 3 7 ⋯ 9 2
(-828988309543*v^18 - 183250833369281*v^16 - 18007963849715836*v^14 - 560362841193341844*v^12 + 48122322831969717744*v^10 + 4618020038962651406568*v^8 - 155274132952158252666732*v^6 - 50215718747880986451725844*v^4 - 4648279606072335934590133017*v^2 - 165970244091911800557504680775) / 371917141450708314617026992
β 13 \beta_{13} β 1 3 = = =
( − 633534294115 ν 18 − 93776242199123 ν 16 + ⋯ − 26 ⋯ 57 ) / 24 ⋯ 28 ( - 633534294115 \nu^{18} - 93776242199123 \nu^{16} + \cdots - 26\!\cdots\!57 ) / 24\!\cdots\!28 ( − 6 3 3 5 3 4 2 9 4 1 1 5 ν 1 8 − 9 3 7 7 6 2 4 2 1 9 9 1 2 3 ν 1 6 + ⋯ − 2 6 ⋯ 5 7 ) / 2 4 ⋯ 2 8
(-633534294115*v^18 - 93776242199123*v^16 - 6346200790170676*v^14 - 105064013799727596*v^12 + 33074245053623532168*v^10 + 672751245698597315016*v^8 - 228526805652094483218180*v^6 - 26961626873300359801978044*v^4 - 1670842051040689369047590373*v^2 - 26045733589212095560990167957) / 247944760967138876411351328
β 14 \beta_{14} β 1 4 = = =
( − 58464901223 ν 19 − 7551515335633 ν 17 − 434954518032500 ν 15 + ⋯ − 45 ⋯ 35 ν ) / 15 ⋯ 08 ( - 58464901223 \nu^{19} - 7551515335633 \nu^{17} - 434954518032500 \nu^{15} + \cdots - 45\!\cdots\!35 \nu ) / 15\!\cdots\!08 ( − 5 8 4 6 4 9 0 1 2 2 3 ν 1 9 − 7 5 5 1 5 1 5 3 3 5 6 3 3 ν 1 7 − 4 3 4 9 5 4 5 1 8 0 3 2 5 0 0 ν 1 5 + ⋯ − 4 5 ⋯ 3 5 ν ) / 1 5 ⋯ 0 8
(-58464901223*v^19 - 7551515335633*v^17 - 434954518032500*v^15 - 577587343958604*v^13 + 2673788016830430096*v^11 - 14025333163525202520*v^9 - 20246928485216837426244*v^7 - 1900586250613556082081468*v^5 - 94822346377155540248002521*v^3 - 458338712878730317871739735*v) / 1585541497763545972841009808
β 15 \beta_{15} β 1 5 = = =
( − 18885109733 ν 19 − 3863813826733 ν 17 − 170035349894360 ν 15 + ⋯ − 15 ⋯ 11 ν ) / 49 ⋯ 32 ( - 18885109733 \nu^{19} - 3863813826733 \nu^{17} - 170035349894360 \nu^{15} + \cdots - 15\!\cdots\!11 \nu ) / 49\!\cdots\!32 ( − 1 8 8 8 5 1 0 9 7 3 3 ν 1 9 − 3 8 6 3 8 1 3 8 2 6 7 3 3 ν 1 7 − 1 7 0 0 3 5 3 4 9 8 9 4 3 6 0 ν 1 5 + ⋯ − 1 5 ⋯ 1 1 ν ) / 4 9 ⋯ 3 2
(-18885109733*v^19 - 3863813826733*v^17 - 170035349894360*v^15 - 3435221896233984*v^13 + 563763631013808084*v^11 + 33350664954745254564*v^9 - 8936184238450595146440*v^7 - 959637281158804481499744*v^5 - 62964201321662418192715623*v^3 - 1564899018858844042015482111*v) / 493857187827989729245560432
β 16 \beta_{16} β 1 6 = = =
( 347883676277 ν 19 + 69424256467225 ν 17 + ⋯ + 50 ⋯ 35 ν ) / 66 ⋯ 56 ( 347883676277 \nu^{19} + 69424256467225 \nu^{17} + \cdots + 50\!\cdots\!35 \nu ) / 66\!\cdots\!56 ( 3 4 7 8 8 3 6 7 6 2 7 7 ν 1 9 + 6 9 4 2 4 2 5 6 4 6 7 2 2 5 ν 1 7 + ⋯ + 5 0 ⋯ 3 5 ν ) / 6 6 ⋯ 5 6
(347883676277*v^19 + 69424256467225*v^17 + 4917636166032140*v^15 + 142194335047179444*v^13 - 18896009381860590216*v^11 - 1079931830665132451880*v^9 + 45835001960998744634076*v^7 + 17481633836190624725285124*v^5 + 1599128328652914423014102835*v^3 + 50862439479789423260672027535*v) / 6694508546112749663106485856
β 17 \beta_{17} β 1 7 = = =
( − 7780409851645 ν 19 + ⋯ − 85 ⋯ 47 ν ) / 60 ⋯ 04 ( - 7780409851645 \nu^{19} + \cdots - 85\!\cdots\!47 \nu ) / 60\!\cdots\!04 ( − 7 7 8 0 4 0 9 8 5 1 6 4 5 ν 1 9 + ⋯ − 8 5 ⋯ 4 7 ν ) / 6 0 ⋯ 0 4
(-7780409851645*v^19 - 1187573745942677*v^17 - 106895683601289028*v^15 - 2132707269494524716*v^13 + 312843329081498657952*v^11 + 21667060213416961267824*v^9 - 2088447846129235780406388*v^7 - 385474697708077400739708348*v^5 - 33062608881746991101502454995*v^3 - 850948504461878571214573918347*v) / 60250576915014746967958372704
β 18 \beta_{18} β 1 8 = = =
( − 129290273341 ν 19 − 27416167122101 ν 17 + ⋯ − 30 ⋯ 55 ν ) / 98 ⋯ 64 ( - 129290273341 \nu^{19} - 27416167122101 \nu^{17} + \cdots - 30\!\cdots\!55 \nu ) / 98\!\cdots\!64 ( − 1 2 9 2 9 0 2 7 3 3 4 1 ν 1 9 − 2 7 4 1 6 1 6 7 1 2 2 1 0 1 ν 1 7 + ⋯ − 3 0 ⋯ 5 5 ν ) / 9 8 ⋯ 6 4
(-129290273341*v^19 - 27416167122101*v^17 - 2806290005667172*v^15 - 112646874708554652*v^13 + 6054037761204821952*v^11 + 793910162811922722864*v^9 - 21360357762445344736404*v^7 - 9081156895928303869993356*v^5 - 699295760175674930286087603*v^3 - 30369696153664390440853742955*v) / 987714375655979458491120864
β 19 \beta_{19} β 1 9 = = =
( 234547778099 ν 19 + 21510636793915 ν 17 + ⋯ + 21 ⋯ 69 ν ) / 10 ⋯ 72 ( 234547778099 \nu^{19} + 21510636793915 \nu^{17} + \cdots + 21\!\cdots\!69 \nu ) / 10\!\cdots\!72 ( 2 3 4 5 4 7 7 7 8 0 9 9 ν 1 9 + 2 1 5 1 0 6 3 6 7 9 3 9 1 5 ν 1 7 + ⋯ + 2 1 ⋯ 6 9 ν ) / 1 0 ⋯ 7 2
(234547778099*v^19 + 21510636793915*v^17 + 1053467519986964*v^15 - 41825718705333060*v^13 - 5198250325873753080*v^11 + 92485505472841084248*v^9 + 54081833228764670946852*v^7 + 6959631101905572078102156*v^5 + 196716051938939070130157637*v^3 + 2112829711158168298172094669*v) / 1057027665175697315227339872
ν \nu ν = = =
( β 7 + β 6 ) / 3 ( \beta_{7} + \beta_{6} ) / 3 ( β 7 + β 6 ) / 3
(b7 + b6) / 3
ν 2 \nu^{2} ν 2 = = =
( − β 5 − β 4 − β 3 + β 2 − β 1 − 60 ) / 3 ( -\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 60 ) / 3 ( − β 5 − β 4 − β 3 + β 2 − β 1 − 6 0 ) / 3
(-b5 - b4 - b3 + b2 - b1 - 60) / 3
ν 3 \nu^{3} ν 3 = = =
( − 3 β 19 + 9 β 18 − 15 β 17 + 9 β 16 − 3 β 15 + 18 β 14 + ⋯ + 5 β 6 ) / 3 ( - 3 \beta_{19} + 9 \beta_{18} - 15 \beta_{17} + 9 \beta_{16} - 3 \beta_{15} + 18 \beta_{14} + \cdots + 5 \beta_{6} ) / 3 ( − 3 β 1 9 + 9 β 1 8 − 1 5 β 1 7 + 9 β 1 6 − 3 β 1 5 + 1 8 β 1 4 + ⋯ + 5 β 6 ) / 3
(-3*b19 + 9*b18 - 15*b17 + 9*b16 - 3*b15 + 18*b14 + 9*b10 + 3*b8 - 10*b7 + 5*b6) / 3
ν 4 \nu^{4} ν 4 = = =
( 45 β 12 − 9 β 11 + 63 β 9 + 28 β 5 + 523 β 4 − 80 β 3 − 28 β 2 + ⋯ − 1425 ) / 3 ( 45 \beta_{12} - 9 \beta_{11} + 63 \beta_{9} + 28 \beta_{5} + 523 \beta_{4} - 80 \beta_{3} - 28 \beta_{2} + \cdots - 1425 ) / 3 ( 4 5 β 1 2 − 9 β 1 1 + 6 3 β 9 + 2 8 β 5 + 5 2 3 β 4 − 8 0 β 3 − 2 8 β 2 + ⋯ − 1 4 2 5 ) / 3
(45*b12 - 9*b11 + 63*b9 + 28*b5 + 523*b4 - 80*b3 - 28*b2 + 451*b1 - 1425) / 3
ν 5 \nu^{5} ν 5 = = =
( 651 β 19 − 171 β 18 − 102 β 17 − 486 β 16 − 321 β 15 + ⋯ − 5306 β 6 ) / 3 ( 651 \beta_{19} - 171 \beta_{18} - 102 \beta_{17} - 486 \beta_{16} - 321 \beta_{15} + \cdots - 5306 \beta_{6} ) / 3 ( 6 5 1 β 1 9 − 1 7 1 β 1 8 − 1 0 2 β 1 7 − 4 8 6 β 1 6 − 3 2 1 β 1 5 + ⋯ − 5 3 0 6 β 6 ) / 3
(651*b19 - 171*b18 - 102*b17 - 486*b16 - 321*b15 + 2574*b14 + 1980*b10 - 930*b8 - 62*b7 - 5306*b6) / 3
ν 6 \nu^{6} ν 6 = = =
( − 4482 β 13 + 1764 β 12 − 540 β 11 − 4950 β 9 − 667 β 5 − 47458 β 4 + ⋯ + 483618 ) / 3 ( - 4482 \beta_{13} + 1764 \beta_{12} - 540 \beta_{11} - 4950 \beta_{9} - 667 \beta_{5} - 47458 \beta_{4} + \cdots + 483618 ) / 3 ( − 4 4 8 2 β 1 3 + 1 7 6 4 β 1 2 − 5 4 0 β 1 1 − 4 9 5 0 β 9 − 6 6 7 β 5 − 4 7 4 5 8 β 4 + ⋯ + 4 8 3 6 1 8 ) / 3
(-4482*b13 + 1764*b12 - 540*b11 - 4950*b9 - 667*b5 - 47458*b4 - 1558*b3 - 656*b2 + 177920*b1 + 483618) / 3
ν 7 \nu^{7} ν 7 = = =
( − 42474 β 19 − 10116 β 18 − 48030 β 17 − 193680 β 16 − 113592 β 15 + ⋯ + 112559 β 6 ) / 3 ( - 42474 \beta_{19} - 10116 \beta_{18} - 48030 \beta_{17} - 193680 \beta_{16} - 113592 \beta_{15} + \cdots + 112559 \beta_{6} ) / 3 ( − 4 2 4 7 4 β 1 9 − 1 0 1 1 6 β 1 8 − 4 8 0 3 0 β 1 7 − 1 9 3 6 8 0 β 1 6 − 1 1 3 5 9 2 β 1 5 + ⋯ + 1 1 2 5 5 9 β 6 ) / 3
(-42474*b19 - 10116*b18 - 48030*b17 - 193680*b16 - 113592*b15 - 202482*b14 - 11178*b10 + 29586*b8 + 149984*b7 + 112559*b6) / 3
ν 8 \nu^{8} ν 8 = = =
( − 85320 β 13 + 117360 β 12 + 196056 β 11 + 937692 β 9 − 713276 β 5 + ⋯ + 753171 ) / 3 ( - 85320 \beta_{13} + 117360 \beta_{12} + 196056 \beta_{11} + 937692 \beta_{9} - 713276 \beta_{5} + \cdots + 753171 ) / 3 ( − 8 5 3 2 0 β 1 3 + 1 1 7 3 6 0 β 1 2 + 1 9 6 0 5 6 β 1 1 + 9 3 7 6 9 2 β 9 − 7 1 3 2 7 6 β 5 + ⋯ + 7 5 3 1 7 1 ) / 3
(-85320*b13 + 117360*b12 + 196056*b11 + 937692*b9 - 713276*b5 + 1844236*b4 + 424396*b3 + 329066*b2 - 1905902*b1 + 753171) / 3
ν 9 \nu^{9} ν 9 = = =
( 1025922 β 19 + 6022260 β 18 − 1097454 β 17 + 2422242 β 16 + ⋯ + 41402443 β 6 ) / 3 ( 1025922 \beta_{19} + 6022260 \beta_{18} - 1097454 \beta_{17} + 2422242 \beta_{16} + \cdots + 41402443 \beta_{6} ) / 3 ( 1 0 2 5 9 2 2 β 1 9 + 6 0 2 2 2 6 0 β 1 8 − 1 0 9 7 4 5 4 β 1 7 + 2 4 2 2 2 4 2 β 1 6 + ⋯ + 4 1 4 0 2 4 4 3 β 6 ) / 3
(1025922*b19 + 6022260*b18 - 1097454*b17 + 2422242*b16 + 7612032*b15 - 12906234*b14 + 6541524*b10 - 9022998*b8 - 339515*b7 + 41402443*b6) / 3
ν 10 \nu^{10} ν 1 0 = = =
( 19939770 β 13 + 16676316 β 12 − 32255190 β 11 − 29438298 β 9 − 28540027 β 5 + ⋯ − 3412104702 ) / 3 ( 19939770 \beta_{13} + 16676316 \beta_{12} - 32255190 \beta_{11} - 29438298 \beta_{9} - 28540027 \beta_{5} + \cdots - 3412104702 ) / 3 ( 1 9 9 3 9 7 7 0 β 1 3 + 1 6 6 7 6 3 1 6 β 1 2 − 3 2 2 5 5 1 9 0 β 1 1 − 2 9 4 3 8 2 9 8 β 9 − 2 8 5 4 0 0 2 7 β 5 + ⋯ − 3 4 1 2 1 0 4 7 0 2 ) / 3
(19939770*b13 + 16676316*b12 - 32255190*b11 - 29438298*b9 - 28540027*b5 - 6096043*b4 - 10496521*b3 - 6858107*b2 + 736443395*b1 - 3412104702) / 3
ν 11 \nu^{11} ν 1 1 = = =
( 207475341 β 19 + 96119307 β 18 − 17584647 β 17 − 560202507 β 16 + ⋯ + 2135081579 β 6 ) / 3 ( 207475341 \beta_{19} + 96119307 \beta_{18} - 17584647 \beta_{17} - 560202507 \beta_{16} + \cdots + 2135081579 \beta_{6} ) / 3 ( 2 0 7 4 7 5 3 4 1 β 1 9 + 9 6 1 1 9 3 0 7 β 1 8 − 1 7 5 8 4 6 4 7 β 1 7 − 5 6 0 2 0 2 5 0 7 β 1 6 + ⋯ + 2 1 3 5 0 8 1 5 7 9 β 6 ) / 3
(207475341*b19 + 96119307*b18 - 17584647*b17 - 560202507*b16 - 703822287*b15 + 1737561150*b14 - 502847847*b10 - 143607507*b8 - 1482385744*b7 + 2135081579*b6) / 3
ν 12 \nu^{12} ν 1 2 = = =
( − 3049074144 β 13 + 1282313043 β 12 + 1412735175 β 11 − 261088263 β 9 + ⋯ − 183613046097 ) / 3 ( - 3049074144 \beta_{13} + 1282313043 \beta_{12} + 1412735175 \beta_{11} - 261088263 \beta_{9} + \cdots - 183613046097 ) / 3 ( − 3 0 4 9 0 7 4 1 4 4 β 1 3 + 1 2 8 2 3 1 3 0 4 3 β 1 2 + 1 4 1 2 7 3 5 1 7 5 β 1 1 − 2 6 1 0 8 8 2 6 3 β 9 + ⋯ − 1 8 3 6 1 3 0 4 6 0 9 7 ) / 3
(-3049074144*b13 + 1282313043*b12 + 1412735175*b11 - 261088263*b9 - 3056845076*b5 - 13722047009*b4 - 641990780*b3 - 2098334368*b2 - 45179005247*b1 - 183613046097) / 3
ν 13 \nu^{13} ν 1 3 = = =
( − 14516668581 β 19 − 24712389801 β 18 + 35536646076 β 17 + ⋯ + 41511557974 β 6 ) / 3 ( - 14516668581 \beta_{19} - 24712389801 \beta_{18} + 35536646076 \beta_{17} + \cdots + 41511557974 \beta_{6} ) / 3 ( − 1 4 5 1 6 6 6 8 5 8 1 β 1 9 − 2 4 7 1 2 3 8 9 8 0 1 β 1 8 + 3 5 5 3 6 6 4 6 0 7 6 β 1 7 + ⋯ + 4 1 5 1 1 5 5 7 9 7 4 β 6 ) / 3
(-14516668581*b19 - 24712389801*b18 + 35536646076*b17 - 29075569614*b16 - 26473593741*b15 - 140086004382*b14 + 91239663816*b10 + 44503759806*b8 - 98242587638*b7 + 41511557974*b6) / 3
ν 14 \nu^{14} ν 1 4 = = =
( 104328211032 β 13 − 153044906400 β 12 + 195191011512 β 11 + 129975854544 β 9 + ⋯ − 4524796184604 ) / 3 ( 104328211032 \beta_{13} - 153044906400 \beta_{12} + 195191011512 \beta_{11} + 129975854544 \beta_{9} + \cdots - 4524796184604 ) / 3 ( 1 0 4 3 2 8 2 1 1 0 3 2 β 1 3 − 1 5 3 0 4 4 9 0 6 4 0 0 β 1 2 + 1 9 5 1 9 1 0 1 1 5 1 2 β 1 1 + 1 2 9 9 7 5 8 5 4 5 4 4 β 9 + ⋯ − 4 5 2 4 7 9 6 1 8 4 6 0 4 ) / 3
(104328211032*b13 - 153044906400*b12 + 195191011512*b11 + 129975854544*b9 + 64555036739*b5 + 480039109424*b4 + 471878880056*b3 - 7674739136*b2 + 7203019707224*b1 - 4524796184604) / 3
ν 15 \nu^{15} ν 1 5 = = =
( − 117333258888 β 19 + 259120246032 β 18 − 1101385341240 β 17 + ⋯ + 12750738869741 β 6 ) / 3 ( - 117333258888 \beta_{19} + 259120246032 \beta_{18} - 1101385341240 \beta_{17} + \cdots + 12750738869741 \beta_{6} ) / 3 ( − 1 1 7 3 3 3 2 5 8 8 8 8 β 1 9 + 2 5 9 1 2 0 2 4 6 0 3 2 β 1 8 − 1 1 0 1 3 8 5 3 4 1 2 4 0 β 1 7 + ⋯ + 1 2 7 5 0 7 3 8 8 6 9 7 4 1 β 6 ) / 3
(-117333258888*b19 + 259120246032*b18 - 1101385341240*b17 - 4095572054400*b16 + 7787303163816*b15 - 6381547917048*b14 - 7887094942032*b10 - 3994581189120*b8 - 3128016608416*b7 + 12750738869741*b6) / 3
ν 16 \nu^{16} ν 1 6 = = =
( 6337818083376 β 13 − 1189044059040 β 12 − 30691850959800 β 11 + 9229689217080 β 9 + ⋯ − 814310579928765 ) / 3 ( 6337818083376 \beta_{13} - 1189044059040 \beta_{12} - 30691850959800 \beta_{11} + 9229689217080 \beta_{9} + \cdots - 814310579928765 ) / 3 ( 6 3 3 7 8 1 8 0 8 3 3 7 6 β 1 3 − 1 1 8 9 0 4 4 0 5 9 0 4 0 β 1 2 − 3 0 6 9 1 8 5 0 9 5 9 8 0 0 β 1 1 + 9 2 2 9 6 8 9 2 1 7 0 8 0 β 9 + ⋯ − 8 1 4 3 1 0 5 7 9 9 2 8 7 6 5 ) / 3
(6337818083376*b13 - 1189044059040*b12 - 30691850959800*b11 + 9229689217080*b9 - 3140757933488*b5 - 68771843420264*b4 + 10913375019520*b3 + 338729172056*b2 - 545806797909920*b1 - 814310579928765) / 3
ν 17 \nu^{17} ν 1 7 = = =
( 687231001272 β 19 + 45116932842000 β 18 + 362332230399384 β 17 + ⋯ + 282679244400001 β 6 ) / 3 ( 687231001272 \beta_{19} + 45116932842000 \beta_{18} + 362332230399384 \beta_{17} + \cdots + 282679244400001 \beta_{6} ) / 3 ( 6 8 7 2 3 1 0 0 1 2 7 2 β 1 9 + 4 5 1 1 6 9 3 2 8 4 2 0 0 0 β 1 8 + 3 6 2 3 3 2 2 3 0 3 9 9 3 8 4 β 1 7 + ⋯ + 2 8 2 6 7 9 2 4 4 4 0 0 0 0 1 β 6 ) / 3
(687231001272*b19 + 45116932842000*b18 + 362332230399384*b17 + 573008254369848*b16 - 453471590913432*b15 + 128669229013752*b14 - 430716463411752*b10 - 120525975219360*b8 - 412875408335975*b7 + 282679244400001*b6) / 3
ν 18 \nu^{18} ν 1 8 = = =
( − 84541607934696 β 13 − 59745797980704 β 12 + ⋯ − 59 ⋯ 48 ) / 3 ( - 84541607934696 \beta_{13} - 59745797980704 \beta_{12} + \cdots - 59\!\cdots\!48 ) / 3 ( − 8 4 5 4 1 6 0 7 9 3 4 6 9 6 β 1 3 − 5 9 7 4 5 7 9 7 9 8 0 7 0 4 β 1 2 + ⋯ − 5 9 ⋯ 4 8 ) / 3
(-84541607934696*b13 - 59745797980704*b12 + 1455576209726544*b11 - 4061561167661040*b9 - 235882450320913*b5 + 6785520896075831*b4 + 270801601842167*b3 - 842969278000727*b2 - 28182595604946193*b1 - 59922172948781148) / 3
ν 19 \nu^{19} ν 1 9 = = =
( 85 ⋯ 65 β 19 + ⋯ + 73 ⋯ 29 β 6 ) / 3 ( 85\!\cdots\!65 \beta_{19} + \cdots + 73\!\cdots\!29 \beta_{6} ) / 3 ( 8 5 ⋯ 6 5 β 1 9 + ⋯ + 7 3 ⋯ 2 9 β 6 ) / 3
(8581888318353765*b19 - 10094563536112647*b18 + 4778920700538681*b17 - 1180639335823407*b16 + 21523635707879229*b15 - 9305846644417758*b14 + 13750016470597881*b10 + 55563886694498619*b8 - 31688996356021474*b7 + 73489117273116629*b6) / 3
Character values
We give the values of χ \chi χ on generators for ( Z / 51 Z ) × \left(\mathbb{Z}/51\mathbb{Z}\right)^\times ( Z / 5 1 Z ) × .
n n n
35 35 3 5
37 37 3 7
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
− 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.
gp: mfembed(f)
Refresh table
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 5 n e w ( 51 , [ χ ] ) S_{5}^{\mathrm{new}}(51, [\chi]) S 5 n e w ( 5 1 , [ χ ] ) :
T 2 10 + 122 T 2 8 + 5079 T 2 6 + 86726 T 2 4 + 555892 T 2 2 + 1107720 T_{2}^{10} + 122T_{2}^{8} + 5079T_{2}^{6} + 86726T_{2}^{4} + 555892T_{2}^{2} + 1107720 T 2 1 0 + 1 2 2 T 2 8 + 5 0 7 9 T 2 6 + 8 6 7 2 6 T 2 4 + 5 5 5 8 9 2 T 2 2 + 1 1 0 7 7 2 0
T2^10 + 122*T2^8 + 5079*T2^6 + 86726*T2^4 + 555892*T2^2 + 1107720
T 5 10 − 3925 T 5 8 + 4807776 T 5 6 − 1882685412 T 5 4 + 240880604544 T 5 2 − 4339677233664 T_{5}^{10} - 3925T_{5}^{8} + 4807776T_{5}^{6} - 1882685412T_{5}^{4} + 240880604544T_{5}^{2} - 4339677233664 T 5 1 0 − 3 9 2 5 T 5 8 + 4 8 0 7 7 7 6 T 5 6 − 1 8 8 2 6 8 5 4 1 2 T 5 4 + 2 4 0 8 8 0 6 0 4 5 4 4 T 5 2 − 4 3 3 9 6 7 7 2 3 3 6 6 4
T5^10 - 3925*T5^8 + 4807776*T5^6 - 1882685412*T5^4 + 240880604544*T5^2 - 4339677233664
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 10 + 122 T 8 + ⋯ + 1107720 ) 2 (T^{10} + 122 T^{8} + \cdots + 1107720)^{2} ( T 1 0 + 1 2 2 T 8 + ⋯ + 1 1 0 7 7 2 0 ) 2
(T^10 + 122*T^8 + 5079*T^6 + 86726*T^4 + 555892*T^2 + 1107720)^2
3 3 3
T 20 + ⋯ + 12 ⋯ 01 T^{20} + \cdots + 12\!\cdots\!01 T 2 0 + ⋯ + 1 2 ⋯ 0 1
T^20 + 200*T^18 + 22051*T^16 + 1226808*T^14 + 5013252*T^12 - 3569195664*T^10 + 32891946372*T^8 + 52810061696568*T^6 + 6227853708942531*T^4 + 370604037770368200*T^2 + 12157665459056928801
5 5 5
( T 10 + ⋯ − 4339677233664 ) 2 (T^{10} + \cdots - 4339677233664)^{2} ( T 1 0 + ⋯ − 4 3 3 9 6 7 7 2 3 3 6 6 4 ) 2
(T^10 - 3925*T^8 + 4807776*T^6 - 1882685412*T^4 + 240880604544*T^2 - 4339677233664)^2
7 7 7
( T 10 + ⋯ + 63 ⋯ 00 ) 2 (T^{10} + \cdots + 63\!\cdots\!00)^{2} ( T 1 0 + ⋯ + 6 3 ⋯ 0 0 ) 2
(T^10 + 13342*T^8 + 60357702*T^6 + 112885436788*T^4 + 78471419718328*T^2 + 6364383118632000)^2
11 11 1 1
( T 10 + ⋯ − 5675378866696 ) 2 (T^{10} + \cdots - 5675378866696)^{2} ( T 1 0 + ⋯ − 5 6 7 5 3 7 8 8 6 6 6 9 6 ) 2
(T^10 - 36879*T^8 + 281911556*T^6 - 287307249122*T^4 + 26326088984076*T^2 - 5675378866696)^2
13 13 1 3
( T 5 + 77 T 4 + ⋯ + 29775471712 ) 4 (T^{5} + 77 T^{4} + \cdots + 29775471712)^{4} ( T 5 + 7 7 T 4 + ⋯ + 2 9 7 7 5 4 7 1 7 1 2 ) 4
(T^5 + 77*T^4 - 38990*T^3 - 3068522*T^2 + 371033096*T + 29775471712)^4
17 17 1 7
T 20 + ⋯ + 16 ⋯ 01 T^{20} + \cdots + 16\!\cdots\!01 T 2 0 + ⋯ + 1 6 ⋯ 0 1
T^20 + 217534*T^18 + 24097582637*T^16 + 2714389342203496*T^14 + 275377564728662515282*T^12 + 23067393555895749965433780*T^10 + 1920967096240426686956187713362*T^8 + 132085420606229494849272834570409576*T^6 + 8179892408023711797730804487573024028077*T^4 + 515101280854623481857451819281415808643427774*T^2 + 16517976926780506002833800829531584028976727363201
19 19 1 9
( T 5 + ⋯ − 2581305358016 ) 4 (T^{5} + \cdots - 2581305358016)^{4} ( T 5 + ⋯ − 2 5 8 1 3 0 5 3 5 8 0 1 6 ) 4
(T^5 + 137*T^4 - 455510*T^3 - 27426284*T^2 + 49080430784*T - 2581305358016)^4
23 23 2 3
( T 10 + ⋯ − 91 ⋯ 96 ) 2 (T^{10} + \cdots - 91\!\cdots\!96)^{2} ( T 1 0 + ⋯ − 9 1 ⋯ 9 6 ) 2
(T^10 - 1613175*T^8 + 853070848268*T^6 - 159096517045025810*T^4 + 7183816008090916396860*T^2 - 91396037036501275118521096)^2
29 29 2 9
( T 10 + ⋯ − 22 ⋯ 76 ) 2 (T^{10} + \cdots - 22\!\cdots\!76)^{2} ( T 1 0 + ⋯ − 2 2 ⋯ 7 6 ) 2
(T^10 - 3523072*T^8 + 4392415575156*T^6 - 2271621056966462736*T^4 + 424046217206841885057024*T^2 - 22858741926871700975883749376)^2
31 31 3 1
( T 10 + ⋯ + 20 ⋯ 20 ) 2 (T^{10} + \cdots + 20\!\cdots\!20)^{2} ( T 1 0 + ⋯ + 2 0 ⋯ 2 0 ) 2
(T^10 + 5748022*T^8 + 11214995429046*T^6 + 8984687693799616036*T^4 + 2665637363653081272222808*T^2 + 206234953360108031286896728320)^2
37 37 3 7
( T 10 + ⋯ + 13 ⋯ 00 ) 2 (T^{10} + \cdots + 13\!\cdots\!00)^{2} ( T 1 0 + ⋯ + 1 3 ⋯ 0 0 ) 2
(T^10 + 3191584*T^8 + 3245541145068*T^6 + 1097140644079385344*T^4 + 93945797073392142538240*T^2 + 1344251567109232105777152000)^2
41 41 4 1
( T 10 + ⋯ − 26 ⋯ 16 ) 2 (T^{10} + \cdots - 26\!\cdots\!16)^{2} ( T 1 0 + ⋯ − 2 6 ⋯ 1 6 ) 2
(T^10 - 12493137*T^8 + 55370768382344*T^6 - 106027567113124696640*T^4 + 88869817371848616260861952*T^2 - 26503451166070908529137376854016)^2
43 43 4 3
( T 5 + ⋯ − 14 ⋯ 44 ) 4 (T^{5} + \cdots - 14\!\cdots\!44)^{4} ( T 5 + ⋯ − 1 4 ⋯ 4 4 ) 4
(T^5 - 2257*T^4 - 8094452*T^3 + 25077372148*T^2 - 15874408191616*T - 1461505400034944)^4
47 47 4 7
( T 10 + ⋯ + 86 ⋯ 00 ) 2 (T^{10} + \cdots + 86\!\cdots\!00)^{2} ( T 1 0 + ⋯ + 8 6 ⋯ 0 0 ) 2
(T^10 + 31089308*T^8 + 280434158013072*T^6 + 700922183625545209472*T^4 + 556441492668348174734049280*T^2 + 86582185447524345277954117632000)^2
53 53 5 3
( T 10 + ⋯ + 19 ⋯ 80 ) 2 (T^{10} + \cdots + 19\!\cdots\!80)^{2} ( T 1 0 + ⋯ + 1 9 ⋯ 8 0 ) 2
(T^10 + 13007244*T^8 + 46567522994688*T^6 + 52722456207491472192*T^4 + 6178327806970762225627392*T^2 + 191219789797365583600811182080)^2
59 59 5 9
( T 10 + ⋯ + 13 ⋯ 80 ) 2 (T^{10} + \cdots + 13\!\cdots\!80)^{2} ( T 1 0 + ⋯ + 1 3 ⋯ 8 0 ) 2
(T^10 + 53324256*T^8 + 959645439106992*T^6 + 6833684760157378050048*T^4 + 16974461934570604809053011968*T^2 + 13179919304272979256723049524756480)^2
61 61 6 1
( T 10 + ⋯ + 35 ⋯ 00 ) 2 (T^{10} + \cdots + 35\!\cdots\!00)^{2} ( T 1 0 + ⋯ + 3 5 ⋯ 0 0 ) 2
(T^10 + 66680488*T^8 + 1497682054070124*T^6 + 13675072477577916477856*T^4 + 45340214909442730363544220160*T^2 + 35710296761729670053747949660672000)^2
67 67 6 7
( T 5 + ⋯ + 11 ⋯ 48 ) 4 (T^{5} + \cdots + 11\!\cdots\!48)^{4} ( T 5 + ⋯ + 1 1 ⋯ 4 8 ) 4
(T^5 - 2852*T^4 - 46903560*T^3 + 242791409832*T^2 - 339039821216448*T + 118878902671909248)^4
71 71 7 1
( T 10 + ⋯ − 51 ⋯ 04 ) 2 (T^{10} + \cdots - 51\!\cdots\!04)^{2} ( T 1 0 + ⋯ − 5 1 ⋯ 0 4 ) 2
(T^10 - 95350858*T^8 + 2492582286239526*T^6 - 23326139826310713307740*T^4 + 64213052153470620340737862296*T^2 - 51639631706648278840368099859900704)^2
73 73 7 3
( T 10 + ⋯ + 98 ⋯ 20 ) 2 (T^{10} + \cdots + 98\!\cdots\!20)^{2} ( T 1 0 + ⋯ + 9 8 ⋯ 2 0 ) 2
(T^10 + 106125720*T^8 + 3162707733417984*T^6 + 20562110344013493362688*T^4 + 35694508231847029641877389312*T^2 + 982955968575807198342586218577920)^2
79 79 7 9
( T 10 + ⋯ + 20 ⋯ 80 ) 2 (T^{10} + \cdots + 20\!\cdots\!80)^{2} ( T 1 0 + ⋯ + 2 0 ⋯ 8 0 ) 2
(T^10 + 228702454*T^8 + 15286003960778646*T^6 + 292116556763662819430308*T^4 + 1370122017275495116077847030552*T^2 + 202899014678227113526246322303566080)^2
83 83 8 3
( T 10 + ⋯ + 11 ⋯ 80 ) 2 (T^{10} + \cdots + 11\!\cdots\!80)^{2} ( T 1 0 + ⋯ + 1 1 ⋯ 8 0 ) 2
(T^10 + 441778968*T^8 + 70559097090338736*T^6 + 4847152461145804463948928*T^4 + 132269930419193400824861819461632*T^2 + 1186886146057680479876401190434251079680)^2
89 89 8 9
( T 10 + ⋯ + 56 ⋯ 20 ) 2 (T^{10} + \cdots + 56\!\cdots\!20)^{2} ( T 1 0 + ⋯ + 5 6 ⋯ 2 0 ) 2
(T^10 + 321807296*T^8 + 29317318426755204*T^6 + 807914414718074839539632*T^4 + 5669511807698393684990104524352*T^2 + 5689874519900902933611737973973701120)^2
97 97 9 7
( T 10 + ⋯ + 23 ⋯ 80 ) 2 (T^{10} + \cdots + 23\!\cdots\!80)^{2} ( T 1 0 + ⋯ + 2 3 ⋯ 8 0 ) 2
(T^10 + 567150736*T^8 + 111317410733090304*T^6 + 8926611007905920553132544*T^4 + 256652102129730094608278411640832*T^2 + 2322308755798312149849947888400346644480)^2
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