gp: [N,k,chi] = [448,4,Mod(255,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.255");
S:= CuspForms(chi, 4);
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 − 2 x 18 − 24 x 17 + 28 x 16 + 56 x 15 − 192 x 14 + 352 x 13 − 448 x 12 + ⋯ + 1073741824 x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + \cdots + 1073741824 x 2 0 − 2 x 1 8 − 2 4 x 1 7 + 2 8 x 1 6 + 5 6 x 1 5 − 1 9 2 x 1 4 + 3 5 2 x 1 3 − 4 4 8 x 1 2 + ⋯ + 1 0 7 3 7 4 1 8 2 4
x^20 - 2*x^18 - 24*x^17 + 28*x^16 + 56*x^15 - 192*x^14 + 352*x^13 - 448*x^12 + 5376*x^11 - 41472*x^10 + 43008*x^9 - 28672*x^8 + 180224*x^7 - 786432*x^6 + 1835008*x^5 + 7340032*x^4 - 50331648*x^3 - 33554432*x^2 + 1073741824
:
β 1 \beta_{1} β 1 = = =
( − 793 ν 19 − 7760 ν 18 − 70462 ν 17 + 82360 ν 16 − 36188 ν 15 + ⋯ + 940060966912 ) / 2280224980992 ( - 793 \nu^{19} - 7760 \nu^{18} - 70462 \nu^{17} + 82360 \nu^{16} - 36188 \nu^{15} + \cdots + 940060966912 ) / 2280224980992 ( − 7 9 3 ν 1 9 − 7 7 6 0 ν 1 8 − 7 0 4 6 2 ν 1 7 + 8 2 3 6 0 ν 1 6 − 3 6 1 8 8 ν 1 5 + ⋯ + 9 4 0 0 6 0 9 6 6 9 1 2 ) / 2 2 8 0 2 2 4 9 8 0 9 9 2
(-793*v^19 - 7760*v^18 - 70462*v^17 + 82360*v^16 - 36188*v^15 + 29384*v^14 - 1591040*v^13 + 2565664*v^12 + 4427712*v^11 - 24367872*v^10 + 49737216*v^9 + 1005568*v^8 + 274354176*v^7 - 2761342976*v^6 + 9211805696*v^5 - 2364801024*v^4 - 4386193408*v^3 - 76466356224*v^2 + 466305941504*v + 940060966912) / 2280224980992
β 2 \beta_{2} β 2 = = =
( − 253 ν 19 + 720 ν 18 − 11414 ν 17 − 65832 ν 16 − 109324 ν 15 + ⋯ + 1201651318784 ) / 325746425856 ( - 253 \nu^{19} + 720 \nu^{18} - 11414 \nu^{17} - 65832 \nu^{16} - 109324 \nu^{15} + \cdots + 1201651318784 ) / 325746425856 ( − 2 5 3 ν 1 9 + 7 2 0 ν 1 8 − 1 1 4 1 4 ν 1 7 − 6 5 8 3 2 ν 1 6 − 1 0 9 3 2 4 ν 1 5 + ⋯ + 1 2 0 1 6 5 1 3 1 8 7 8 4 ) / 3 2 5 7 4 6 4 2 5 8 5 6
(-253*v^19 + 720*v^18 - 11414*v^17 - 65832*v^16 - 109324*v^15 - 1487000*v^14 - 2116736*v^13 + 814496*v^12 + 26694336*v^11 - 105614080*v^10 - 109507072*v^9 + 131422208*v^8 - 1184673792*v^7 - 1735704576*v^6 + 6138822656*v^5 + 21521235968*v^4 + 73925656576*v^3 - 92949970944*v^2 + 276958281728*v + 1201651318784) / 325746425856
β 3 \beta_{3} β 3 = = =
( 647 ν 19 − 13656 ν 18 − 111550 ν 17 + 323784 ν 16 − 943772 ν 15 + ⋯ + 3620321886208 ) / 1140112490496 ( 647 \nu^{19} - 13656 \nu^{18} - 111550 \nu^{17} + 323784 \nu^{16} - 943772 \nu^{15} + \cdots + 3620321886208 ) / 1140112490496 ( 6 4 7 ν 1 9 − 1 3 6 5 6 ν 1 8 − 1 1 1 5 5 0 ν 1 7 + 3 2 3 7 8 4 ν 1 6 − 9 4 3 7 7 2 ν 1 5 + ⋯ + 3 6 2 0 3 2 1 8 8 6 2 0 8 ) / 1 1 4 0 1 1 2 4 9 0 4 9 6
(647*v^19 - 13656*v^18 - 111550*v^17 + 323784*v^16 - 943772*v^15 + 2117352*v^14 - 21505984*v^13 + 22798880*v^12 - 67722048*v^11 + 20075264*v^10 - 7834112*v^9 - 1015257088*v^8 + 4428140544*v^7 - 17098948608*v^6 + 57840173056*v^5 + 16812605440*v^4 + 291370958848*v^3 - 524946505728*v^2 + 1932701728768*v + 3620321886208) / 1140112490496
β 4 \beta_{4} β 4 = = =
( − 13969 ν 19 + 83816 ν 18 − 322254 ν 17 + 133768 ν 16 + ⋯ + 10557834919936 ) / 2280224980992 ( - 13969 \nu^{19} + 83816 \nu^{18} - 322254 \nu^{17} + 133768 \nu^{16} + \cdots + 10557834919936 ) / 2280224980992 ( − 1 3 9 6 9 ν 1 9 + 8 3 8 1 6 ν 1 8 − 3 2 2 2 5 4 ν 1 7 + 1 3 3 7 6 8 ν 1 6 + ⋯ + 1 0 5 5 7 8 3 4 9 1 9 9 3 6 ) / 2 2 8 0 2 2 4 9 8 0 9 9 2
(-13969*v^19 + 83816*v^18 - 322254*v^17 + 133768*v^16 - 2181820*v^15 + 699048*v^14 - 16759488*v^13 - 11335904*v^12 + 165664448*v^11 - 299425024*v^10 + 883013120*v^9 - 2908362752*v^8 + 11807854592*v^7 + 3505963008*v^6 + 25277693952*v^5 + 256363986944*v^4 - 262463815680*v^3 + 504167923712*v^2 - 3284072923136*v + 10557834919936) / 2280224980992
β 5 \beta_{5} β 5 = = =
( − 7197 ν 19 + 68984 ν 18 + 122498 ν 17 + 1668776 ν 16 + ⋯ − 9654952263680 ) / 1140112490496 ( - 7197 \nu^{19} + 68984 \nu^{18} + 122498 \nu^{17} + 1668776 \nu^{16} + \cdots - 9654952263680 ) / 1140112490496 ( − 7 1 9 7 ν 1 9 + 6 8 9 8 4 ν 1 8 + 1 2 2 4 9 8 ν 1 7 + 1 6 6 8 7 7 6 ν 1 6 + ⋯ − 9 6 5 4 9 5 2 2 6 3 6 8 0 ) / 1 1 4 0 1 1 2 4 9 0 4 9 6
(-7197*v^19 + 68984*v^18 + 122498*v^17 + 1668776*v^16 - 2718076*v^15 + 3043400*v^14 - 571424*v^13 + 2594400*v^12 - 26932032*v^11 - 130411520*v^10 + 295655424*v^9 - 2928101376*v^8 + 4377501696*v^7 + 360202240*v^6 + 48066101248*v^5 - 253988175872*v^4 + 220231368704*v^3 + 366752038912*v^2 + 8890850738176*v - 9654952263680) / 1140112490496
β 6 \beta_{6} β 6 = = =
( − 4809 ν 19 + 488 ν 18 − 47870 ν 17 + 103368 ν 16 − 318364 ν 15 + ⋯ + 562506498048 ) / 325746425856 ( - 4809 \nu^{19} + 488 \nu^{18} - 47870 \nu^{17} + 103368 \nu^{16} - 318364 \nu^{15} + \cdots + 562506498048 ) / 325746425856 ( − 4 8 0 9 ν 1 9 + 4 8 8 ν 1 8 − 4 7 8 7 0 ν 1 7 + 1 0 3 3 6 8 ν 1 6 − 3 1 8 3 6 4 ν 1 5 + ⋯ + 5 6 2 5 0 6 4 9 8 0 4 8 ) / 3 2 5 7 4 6 4 2 5 8 5 6
(-4809*v^19 + 488*v^18 - 47870*v^17 + 103368*v^16 - 318364*v^15 + 2036072*v^14 - 2027072*v^13 + 7705376*v^12 - 29861696*v^11 - 110637312*v^10 + 162176512*v^9 - 475445248*v^8 + 284053504*v^7 - 2084585472*v^6 + 13384613888*v^5 - 52310048768*v^4 - 33484177408*v^3 - 87270883328*v^2 + 806447218688*v + 562506498048) / 325746425856
β 7 \beta_{7} β 7 = = =
( 39171 ν 19 + 88016 ν 18 + 123178 ν 17 − 159272 ν 16 + 596340 ν 15 + ⋯ − 10618098679808 ) / 2280224980992 ( 39171 \nu^{19} + 88016 \nu^{18} + 123178 \nu^{17} - 159272 \nu^{16} + 596340 \nu^{15} + \cdots - 10618098679808 ) / 2280224980992 ( 3 9 1 7 1 ν 1 9 + 8 8 0 1 6 ν 1 8 + 1 2 3 1 7 8 ν 1 7 − 1 5 9 2 7 2 ν 1 6 + 5 9 6 3 4 0 ν 1 5 + ⋯ − 1 0 6 1 8 0 9 8 6 7 9 8 0 8 ) / 2 2 8 0 2 2 4 9 8 0 9 9 2
(39171*v^19 + 88016*v^18 + 123178*v^17 - 159272*v^16 + 596340*v^15 + 927592*v^14 - 1870720*v^13 + 39336864*v^12 + 15506112*v^11 + 177052928*v^10 - 527573504*v^9 - 984066048*v^8 - 2413375488*v^7 - 6866649088*v^6 - 48440344576*v^5 - 4242276352*v^4 + 193131970560*v^3 - 1463598186496*v^2 - 4227120234496*v - 10618098679808) / 2280224980992
β 8 \beta_{8} β 8 = = =
( 1751 ν 19 + 1586 ν 18 + 12018 ν 17 + 98900 ν 16 − 115692 ν 15 + ⋯ − 932611883008 ) / 71257030656 ( 1751 \nu^{19} + 1586 \nu^{18} + 12018 \nu^{17} + 98900 \nu^{16} - 115692 \nu^{15} + \cdots - 932611883008 ) / 71257030656 ( 1 7 5 1 ν 1 9 + 1 5 8 6 ν 1 8 + 1 2 0 1 8 ν 1 7 + 9 8 9 0 0 ν 1 6 − 1 1 5 6 9 2 ν 1 5 + ⋯ − 9 3 2 6 1 1 8 8 3 0 0 8 ) / 7 1 2 5 7 0 3 0 6 5 6
(1751*v^19 + 1586*v^18 + 12018*v^17 + 98900*v^16 - 115692*v^15 + 170432*v^14 - 394960*v^13 + 3798432*v^12 - 5915776*v^11 + 557952*v^10 - 23881728*v^9 - 24167424*v^8 - 52215808*v^7 - 233136128*v^6 + 4145643520*v^5 - 15210512384*v^4 + 17581998080*v^3 - 79358328832*v^2 - 475877343232*v - 932611883008) / 71257030656
β 9 \beta_{9} β 9 = = =
( − 1894 ν 19 + 58567 ν 18 + 387356 ν 17 − 54686 ν 16 − 826352 ν 15 + ⋯ − 20322543730688 ) / 285028122624 ( - 1894 \nu^{19} + 58567 \nu^{18} + 387356 \nu^{17} - 54686 \nu^{16} - 826352 \nu^{15} + \cdots - 20322543730688 ) / 285028122624 ( − 1 8 9 4 ν 1 9 + 5 8 5 6 7 ν 1 8 + 3 8 7 3 5 6 ν 1 7 − 5 4 6 8 6 ν 1 6 − 8 2 6 3 5 2 ν 1 5 + ⋯ − 2 0 3 2 2 5 4 3 7 3 0 6 8 8 ) / 2 8 5 0 2 8 1 2 2 6 2 4
(-1894*v^19 + 58567*v^18 + 387356*v^17 - 54686*v^16 - 826352*v^15 + 1219892*v^14 + 6845128*v^13 + 6709888*v^12 - 47992032*v^11 + 476655552*v^10 - 281508096*v^9 - 738248192*v^8 - 8388421632*v^7 + 10845171712*v^6 - 16203759616*v^5 - 127953666048*v^4 + 144358768640*v^3 + 11875123200*v^2 - 1069086146560*v - 20322543730688) / 285028122624
β 10 \beta_{10} β 1 0 = = =
( − 35213 ν 19 + 43608 ν 18 − 69790 ν 17 + 86376 ν 16 − 867004 ν 15 + ⋯ + 5266837864448 ) / 1140112490496 ( - 35213 \nu^{19} + 43608 \nu^{18} - 69790 \nu^{17} + 86376 \nu^{16} - 867004 \nu^{15} + \cdots + 5266837864448 ) / 1140112490496 ( − 3 5 2 1 3 ν 1 9 + 4 3 6 0 8 ν 1 8 − 6 9 7 9 0 ν 1 7 + 8 6 3 7 6 ν 1 6 − 8 6 7 0 0 4 ν 1 5 + ⋯ + 5 2 6 6 8 3 7 8 6 4 4 4 8 ) / 1 1 4 0 1 1 2 4 9 0 4 9 6
(-35213*v^19 + 43608*v^18 - 69790*v^17 + 86376*v^16 - 867004*v^15 + 316488*v^14 + 5747936*v^13 - 58180512*v^12 + 67720384*v^11 - 139338752*v^10 + 677763072*v^9 - 2541422592*v^8 + 5212954624*v^7 + 4090380288*v^6 - 18264195072*v^5 - 10619977728*v^4 - 61080600576*v^3 + 1636485300224*v^2 - 1736911618048*v + 5266837864448) / 1140112490496
β 11 \beta_{11} β 1 1 = = =
( 7973 ν 19 − 27196 ν 18 + 132966 ν 17 − 40640 ν 16 + 118860 ν 15 + ⋯ − 4866399272960 ) / 190018748416 ( 7973 \nu^{19} - 27196 \nu^{18} + 132966 \nu^{17} - 40640 \nu^{16} + 118860 \nu^{15} + \cdots - 4866399272960 ) / 190018748416 ( 7 9 7 3 ν 1 9 − 2 7 1 9 6 ν 1 8 + 1 3 2 9 6 6 ν 1 7 − 4 0 6 4 0 ν 1 6 + 1 1 8 8 6 0 ν 1 5 + ⋯ − 4 8 6 6 3 9 9 2 7 2 9 6 0 ) / 1 9 0 0 1 8 7 4 8 4 1 6
(7973*v^19 - 27196*v^18 + 132966*v^17 - 40640*v^16 + 118860*v^15 - 1798008*v^14 + 869472*v^13 + 15967584*v^12 - 66126144*v^11 + 118768640*v^10 - 433992192*v^9 + 1072005120*v^8 - 5621231616*v^7 + 2759262208*v^6 + 987758592*v^5 - 23105110016*v^4 - 135078608896*v^3 - 269903462400*v^2 + 1563972075520*v - 4866399272960) / 190018748416
β 12 \beta_{12} β 1 2 = = =
( 14861 ν 19 + 21795 ν 18 + 209574 ν 17 − 243230 ν 16 − 861276 ν 15 + ⋯ − 10412292571136 ) / 285028122624 ( 14861 \nu^{19} + 21795 \nu^{18} + 209574 \nu^{17} - 243230 \nu^{16} - 861276 \nu^{15} + \cdots - 10412292571136 ) / 285028122624 ( 1 4 8 6 1 ν 1 9 + 2 1 7 9 5 ν 1 8 + 2 0 9 5 7 4 ν 1 7 − 2 4 3 2 3 0 ν 1 6 − 8 6 1 2 7 6 ν 1 5 + ⋯ − 1 0 4 1 2 2 9 2 5 7 1 1 3 6 ) / 2 8 5 0 2 8 1 2 2 6 2 4
(14861*v^19 + 21795*v^18 + 209574*v^17 - 243230*v^16 - 861276*v^15 - 137748*v^14 + 2624488*v^13 + 1155104*v^12 - 53040544*v^11 + 268729280*v^10 + 100088064*v^9 - 37201408*v^8 - 6479583232*v^7 + 2404429824*v^6 - 5926207488*v^5 - 53524168704*v^4 + 2503737344*v^3 - 94601478144*v^2 + 43679481856*v - 10412292571136) / 285028122624
β 13 \beta_{13} β 1 3 = = =
( − 4095 ν 19 − 10250 ν 18 + 27770 ν 17 + 32460 ν 16 − 150956 ν 15 + ⋯ − 1487602188288 ) / 81436606464 ( - 4095 \nu^{19} - 10250 \nu^{18} + 27770 \nu^{17} + 32460 \nu^{16} - 150956 \nu^{15} + \cdots - 1487602188288 ) / 81436606464 ( − 4 0 9 5 ν 1 9 − 1 0 2 5 0 ν 1 8 + 2 7 7 7 0 ν 1 7 + 3 2 4 6 0 ν 1 6 − 1 5 0 9 5 6 ν 1 5 + ⋯ − 1 4 8 7 6 0 2 1 8 8 2 8 8 ) / 8 1 4 3 6 6 0 6 4 6 4
(-4095*v^19 - 10250*v^18 + 27770*v^17 + 32460*v^16 - 150956*v^15 - 231200*v^14 + 577184*v^13 + 2242240*v^12 - 24113152*v^11 + 49678080*v^10 + 78960896*v^9 + 141718528*v^8 - 344504320*v^7 + 2562809856*v^6 + 3281403904*v^5 - 30313676800*v^4 + 5142740992*v^3 + 177902452736*v^2 + 743566213120*v - 1487602188288) / 81436606464
β 14 \beta_{14} β 1 4 = = =
( − 9811 ν 19 − 11616 ν 18 + 66782 ν 17 − 1880 ν 16 − 190468 ν 15 + ⋯ − 2344381054976 ) / 162873212928 ( - 9811 \nu^{19} - 11616 \nu^{18} + 66782 \nu^{17} - 1880 \nu^{16} - 190468 \nu^{15} + \cdots - 2344381054976 ) / 162873212928 ( − 9 8 1 1 ν 1 9 − 1 1 6 1 6 ν 1 8 + 6 6 7 8 2 ν 1 7 − 1 8 8 0 ν 1 6 − 1 9 0 4 6 8 ν 1 5 + ⋯ − 2 3 4 4 3 8 1 0 5 4 9 7 6 ) / 1 6 2 8 7 3 2 1 2 9 2 8
(-9811*v^19 - 11616*v^18 + 66782*v^17 - 1880*v^16 - 190468*v^15 - 1487912*v^14 + 4819808*v^13 - 8768096*v^12 - 19183552*v^11 + 49680896*v^10 + 173129728*v^9 + 108795904*v^8 - 2055675904*v^7 + 6642941952*v^6 - 6944227328*v^5 + 10918821888*v^4 + 15639511040*v^3 + 635476901888*v^2 + 565861941248*v - 2344381054976) / 162873212928
β 15 \beta_{15} β 1 5 = = =
( 74377 ν 19 − 27452 ν 18 − 793194 ν 17 + 594432 ν 16 + 1813708 ν 15 + ⋯ + 34979287400448 ) / 1140112490496 ( 74377 \nu^{19} - 27452 \nu^{18} - 793194 \nu^{17} + 594432 \nu^{16} + 1813708 \nu^{15} + \cdots + 34979287400448 ) / 1140112490496 ( 7 4 3 7 7 ν 1 9 − 2 7 4 5 2 ν 1 8 − 7 9 3 1 9 4 ν 1 7 + 5 9 4 4 3 2 ν 1 6 + 1 8 1 3 7 0 8 ν 1 5 + ⋯ + 3 4 9 7 9 2 8 7 4 0 0 4 4 8 ) / 1 1 4 0 1 1 2 4 9 0 4 9 6
(74377*v^19 - 27452*v^18 - 793194*v^17 + 594432*v^16 + 1813708*v^15 + 11576552*v^14 - 53398912*v^13 + 69042336*v^12 + 241217216*v^11 - 803999488*v^10 - 1358860288*v^9 + 4139667456*v^8 + 13462298624*v^7 - 84038942720*v^6 - 54386917376*v^5 + 179542949888*v^4 + 17232297984*v^3 - 2528244662272*v^2 + 290514272256*v + 34979287400448) / 1140112490496
β 16 \beta_{16} β 1 6 = = =
( − 16233 ν 19 + 65704 ν 18 − 77422 ν 17 + 26120 ν 16 − 86716 ν 15 + ⋯ + 2878970265600 ) / 285028122624 ( - 16233 \nu^{19} + 65704 \nu^{18} - 77422 \nu^{17} + 26120 \nu^{16} - 86716 \nu^{15} + \cdots + 2878970265600 ) / 285028122624 ( − 1 6 2 3 3 ν 1 9 + 6 5 7 0 4 ν 1 8 − 7 7 4 2 2 ν 1 7 + 2 6 1 2 0 ν 1 6 − 8 6 7 1 6 ν 1 5 + ⋯ + 2 8 7 8 9 7 0 2 6 5 6 0 0 ) / 2 8 5 0 2 8 1 2 2 6 2 4
(-16233*v^19 + 65704*v^18 - 77422*v^17 + 26120*v^16 - 86716*v^15 + 110184*v^14 + 1996928*v^13 - 25246304*v^12 + 51378880*v^11 - 110414592*v^10 + 764983808*v^9 - 885610496*v^8 + 3314397184*v^7 - 10717446144*v^6 - 2312241152*v^5 + 28232646656*v^4 - 78648442880*v^3 + 732820406272*v^2 - 746720329728*v + 2878970265600) / 285028122624
β 17 \beta_{17} β 1 7 = = =
( − 151077 ν 19 + 463976 ν 18 − 229926 ν 17 − 371992 ν 16 + ⋯ − 7711211126784 ) / 2280224980992 ( - 151077 \nu^{19} + 463976 \nu^{18} - 229926 \nu^{17} - 371992 \nu^{16} + \cdots - 7711211126784 ) / 2280224980992 ( − 1 5 1 0 7 7 ν 1 9 + 4 6 3 9 7 6 ν 1 8 − 2 2 9 9 2 6 ν 1 7 − 3 7 1 9 9 2 ν 1 6 + ⋯ − 7 7 1 1 2 1 1 1 2 6 7 8 4 ) / 2 2 8 0 2 2 4 9 8 0 9 9 2
(-151077*v^19 + 463976*v^18 - 229926*v^17 - 371992*v^16 - 12417004*v^15 + 14604616*v^14 + 65545792*v^13 - 64959072*v^12 + 161418688*v^11 + 503660288*v^10 + 2711924224*v^9 - 20351932416*v^8 + 51193925632*v^7 - 26389725184*v^6 + 145507942400*v^5 - 402981126144*v^4 + 962193260544*v^3 + 2888310980608*v^2 - 20365862502400*v - 7711211126784) / 2280224980992
β 18 \beta_{18} β 1 8 = = =
( − 158821 ν 19 − 488368 ν 18 + 901626 ν 17 + 2990744 ν 16 + ⋯ − 3806280548352 ) / 2280224980992 ( - 158821 \nu^{19} - 488368 \nu^{18} + 901626 \nu^{17} + 2990744 \nu^{16} + \cdots - 3806280548352 ) / 2280224980992 ( − 1 5 8 8 2 1 ν 1 9 − 4 8 8 3 6 8 ν 1 8 + 9 0 1 6 2 6 ν 1 7 + 2 9 9 0 7 4 4 ν 1 6 + ⋯ − 3 8 0 6 2 8 0 5 4 8 3 5 2 ) / 2 2 8 0 2 2 4 9 8 0 9 9 2
(-158821*v^19 - 488368*v^18 + 901626*v^17 + 2990744*v^16 - 2618860*v^15 - 20629848*v^14 + 46982784*v^13 + 24075424*v^12 - 84442944*v^11 - 418810624*v^10 + 5389856256*v^9 - 4816431104*v^8 - 19463909376*v^7 + 37344067584*v^6 + 51168083968*v^5 - 242166267904*v^4 - 1735501283328*v^3 + 8894654971904*v^2 + 26095617310720*v - 3806280548352) / 2280224980992
β 19 \beta_{19} β 1 9 = = =
( 25722 ν 19 − 28761 ν 18 + 6684 ν 17 − 453598 ν 16 + 507664 ν 15 + ⋯ + 1721912786944 ) / 285028122624 ( 25722 \nu^{19} - 28761 \nu^{18} + 6684 \nu^{17} - 453598 \nu^{16} + 507664 \nu^{15} + \cdots + 1721912786944 ) / 285028122624 ( 2 5 7 2 2 ν 1 9 − 2 8 7 6 1 ν 1 8 + 6 6 8 4 ν 1 7 − 4 5 3 5 9 8 ν 1 6 + 5 0 7 6 6 4 ν 1 5 + ⋯ + 1 7 2 1 9 1 2 7 8 6 9 4 4 ) / 2 8 5 0 2 8 1 2 2 6 2 4
(25722*v^19 - 28761*v^18 + 6684*v^17 - 453598*v^16 + 507664*v^15 - 2000972*v^14 - 132408*v^13 + 7628416*v^12 - 25635040*v^11 + 7919552*v^10 - 558029056*v^9 + 344570368*v^8 - 3676923904*v^7 + 2844078080*v^6 - 158580736*v^5 + 54793732096*v^4 - 177504780288*v^3 + 127906349056*v^2 + 592864870400*v + 1721912786944) / 285028122624
ν \nu ν = = =
( − β 10 − β 8 + β 5 ) / 16 ( -\beta_{10} - \beta_{8} + \beta_{5} ) / 16 ( − β 1 0 − β 8 + β 5 ) / 1 6
(-b10 - b8 + b5) / 16
ν 2 \nu^{2} ν 2 = = =
( 2 β 19 + β 16 + β 15 + β 14 + β 13 − β 12 − β 11 − β 10 + ⋯ − 1 ) / 8 ( 2 \beta_{19} + \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \cdots - 1 ) / 8 ( 2 β 1 9 + β 1 6 + β 1 5 + β 1 4 + β 1 3 − β 1 2 − β 1 1 − β 1 0 + ⋯ − 1 ) / 8
(2*b19 + b16 + b15 + b14 + b13 - b12 - b11 - b10 + b9 + b8 - 2*b7 - 4*b1 - 1) / 8
ν 3 \nu^{3} ν 3 = = =
( − β 18 − β 16 + 4 β 14 − 4 β 11 − 5 β 10 + 5 β 7 + β 5 + ⋯ + 28 ) / 8 ( - \beta_{18} - \beta_{16} + 4 \beta_{14} - 4 \beta_{11} - 5 \beta_{10} + 5 \beta_{7} + \beta_{5} + \cdots + 28 ) / 8 ( − β 1 8 − β 1 6 + 4 β 1 4 − 4 β 1 1 − 5 β 1 0 + 5 β 7 + β 5 + ⋯ + 2 8 ) / 8
(-b18 - b16 + 4*b14 - 4*b11 - 5*b10 + 5*b7 + b5 - 8*b4 + 4*b3 + 4*b2 + 28) / 8
ν 4 \nu^{4} ν 4 = = =
( − β 19 + β 17 + 8 β 14 − 8 β 13 − β 12 + β 11 − 15 β 10 + ⋯ − 37 ) / 4 ( - \beta_{19} + \beta_{17} + 8 \beta_{14} - 8 \beta_{13} - \beta_{12} + \beta_{11} - 15 \beta_{10} + \cdots - 37 ) / 4 ( − β 1 9 + β 1 7 + 8 β 1 4 − 8 β 1 3 − β 1 2 + β 1 1 − 1 5 β 1 0 + ⋯ − 3 7 ) / 4
(-b19 + b17 + 8*b14 - 8*b13 - b12 + b11 - 15*b10 - 2*b8 - 7*b7 + b6 + b5 + 8*b4 + 8*b3 - 7*b2 - 36*b1 - 37) / 4
ν 5 \nu^{5} ν 5 = = =
( 12 β 19 + β 18 + 11 β 16 − 10 β 15 − 10 β 14 − 6 β 13 − 6 β 12 + ⋯ − 6 ) / 4 ( 12 \beta_{19} + \beta_{18} + 11 \beta_{16} - 10 \beta_{15} - 10 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + \cdots - 6 ) / 4 ( 1 2 β 1 9 + β 1 8 + 1 1 β 1 6 − 1 0 β 1 5 − 1 0 β 1 4 − 6 β 1 3 − 6 β 1 2 + ⋯ − 6 ) / 4
(12*b19 + b18 + 11*b16 - 10*b15 - 10*b14 - 6*b13 - 6*b12 - 10*b11 - 50*b10 + 10*b9 + 11*b8 - 101*b7 - 4*b6 - 4*b4 + 8*b3 + 4*b2 + 12*b1 - 6) / 4
ν 6 \nu^{6} ν 6 = = =
( 3 β 19 − 8 β 18 − 13 β 17 − 11 β 16 − 13 β 15 − β 14 + 11 β 13 + ⋯ + 274 ) / 2 ( 3 \beta_{19} - 8 \beta_{18} - 13 \beta_{17} - 11 \beta_{16} - 13 \beta_{15} - \beta_{14} + 11 \beta_{13} + \cdots + 274 ) / 2 ( 3 β 1 9 − 8 β 1 8 − 1 3 β 1 7 − 1 1 β 1 6 − 1 3 β 1 5 − β 1 4 + 1 1 β 1 3 + ⋯ + 2 7 4 ) / 2
(3*b19 - 8*b18 - 13*b17 - 11*b16 - 13*b15 - b14 + 11*b13 - 6*b12 - 12*b11 - 9*b10 + 3*b9 + 9*b7 + 11*b6 + 8*b5 + 56*b4 - 28*b3 - b2 + 274) / 2
ν 7 \nu^{7} ν 7 = = =
( − 8 β 19 + 24 β 17 − 44 β 14 + 44 β 13 − 8 β 12 + 8 β 11 + ⋯ − 1436 ) / 2 ( - 8 \beta_{19} + 24 \beta_{17} - 44 \beta_{14} + 44 \beta_{13} - 8 \beta_{12} + 8 \beta_{11} + \cdots - 1436 ) / 2 ( − 8 β 1 9 + 2 4 β 1 7 − 4 4 β 1 4 + 4 4 β 1 3 − 8 β 1 2 + 8 β 1 1 + ⋯ − 1 4 3 6 ) / 2
(-8*b19 + 24*b17 - 44*b14 + 44*b13 - 8*b12 + 8*b11 - 17*b10 - 36*b9 - 7*b8 + 4*b7 - 68*b6 + 25*b5 + 12*b4 + 12*b3 - 24*b2 - 1428*b1 - 1436) / 2
ν 8 \nu^{8} ν 8 = = =
− 26 β 19 − 55 β 18 + 46 β 16 + 51 β 15 + 27 β 14 + 59 β 13 + ⋯ + 13 - 26 \beta_{19} - 55 \beta_{18} + 46 \beta_{16} + 51 \beta_{15} + 27 \beta_{14} + 59 \beta_{13} + \cdots + 13 − 2 6 β 1 9 − 5 5 β 1 8 + 4 6 β 1 6 + 5 1 β 1 5 + 2 7 β 1 4 + 5 9 β 1 3 + ⋯ + 1 3
-26*b19 - 55*b18 + 46*b16 + 51*b15 + 27*b14 + 59*b13 + 13*b12 + 5*b11 - 211*b10 - 5*b9 + 46*b8 - 367*b7 - 32*b6 + 80*b4 - 160*b3 + 32*b2 - 780*b1 + 13
ν 9 \nu^{9} ν 9 = = =
− 114 β 19 + 97 β 18 − 98 β 17 + 59 β 16 − 98 β 15 − 46 β 14 + ⋯ + 1128 - 114 \beta_{19} + 97 \beta_{18} - 98 \beta_{17} + 59 \beta_{16} - 98 \beta_{15} - 46 \beta_{14} + \cdots + 1128 − 1 1 4 β 1 9 + 9 7 β 1 8 − 9 8 β 1 7 + 5 9 β 1 6 − 9 8 β 1 5 − 4 6 β 1 4 + ⋯ + 1 1 2 8
-114*b19 + 97*b18 - 98*b17 + 59*b16 - 98*b15 - 46*b14 + 62*b13 + 228*b12 - 180*b11 - 341*b10 - 114*b9 + 341*b7 + 62*b6 - 97*b5 + 88*b4 - 44*b3 - 46*b2 + 1128
ν 10 \nu^{10} ν 1 0 = = =
− 6 β 19 + 38 β 17 − 152 β 14 + 152 β 13 − 6 β 12 + 6 β 11 + ⋯ + 14250 - 6 \beta_{19} + 38 \beta_{17} - 152 \beta_{14} + 152 \beta_{13} - 6 \beta_{12} + 6 \beta_{11} + \cdots + 14250 − 6 β 1 9 + 3 8 β 1 7 − 1 5 2 β 1 4 + 1 5 2 β 1 3 − 6 β 1 2 + 6 β 1 1 + ⋯ + 1 4 2 5 0
-6*b19 + 38*b17 - 152*b14 + 152*b13 - 6*b12 + 6*b11 - 20*b10 + 296*b9 - 234*b8 - 66*b7 - 1026*b6 - 112*b5 + 312*b4 + 312*b3 - 874*b2 + 14256*b1 + 14250
ν 11 \nu^{11} ν 1 1 = = =
288 β 19 + 1138 β 18 − 334 β 16 + 624 β 15 − 1840 β 14 + 232 β 13 + ⋯ − 144 288 \beta_{19} + 1138 \beta_{18} - 334 \beta_{16} + 624 \beta_{15} - 1840 \beta_{14} + 232 \beta_{13} + \cdots - 144 2 8 8 β 1 9 + 1 1 3 8 β 1 8 − 3 3 4 β 1 6 + 6 2 4 β 1 5 − 1 8 4 0 β 1 4 + 2 3 2 β 1 3 + ⋯ − 1 4 4
288*b19 + 1138*b18 - 334*b16 + 624*b15 - 1840*b14 + 232*b13 - 144*b12 - 1544*b11 - 2264*b10 + 1544*b9 - 334*b8 - 5666*b7 - 2072*b6 + 1576*b4 - 3152*b3 + 2072*b2 + 14360*b1 - 144
ν 12 \nu^{12} ν 1 2 = = =
2316 β 19 + 1892 β 18 + 2316 β 17 + 1400 β 16 + 2316 β 15 + 2476 β 14 + ⋯ + 75160 2316 \beta_{19} + 1892 \beta_{18} + 2316 \beta_{17} + 1400 \beta_{16} + 2316 \beta_{15} + 2476 \beta_{14} + \cdots + 75160 2 3 1 6 β 1 9 + 1 8 9 2 β 1 8 + 2 3 1 6 β 1 7 + 1 4 0 0 β 1 6 + 2 3 1 6 β 1 5 + 2 4 7 6 β 1 4 + ⋯ + 7 5 1 6 0
2316*b19 + 1892*b18 + 2316*b17 + 1400*b16 + 2316*b15 + 2476*b14 + 4012*b13 - 4632*b12 - 1184*b11 - 16016*b10 + 2316*b9 + 16016*b7 + 4012*b6 - 1892*b5 + 9088*b4 - 4544*b3 + 2476*b2 + 75160
ν 13 \nu^{13} ν 1 3 = = =
− 184 β 19 + 13048 β 17 + 13392 β 14 − 13392 β 13 − 184 β 12 + ⋯ + 76120 - 184 \beta_{19} + 13048 \beta_{17} + 13392 \beta_{14} - 13392 \beta_{13} - 184 \beta_{12} + \cdots + 76120 − 1 8 4 β 1 9 + 1 3 0 4 8 β 1 7 + 1 3 3 9 2 β 1 4 − 1 3 3 9 2 β 1 3 − 1 8 4 β 1 2 + ⋯ + 7 6 1 2 0
-184*b19 + 13048*b17 + 13392*b14 - 13392*b13 - 184*b12 + 184*b11 - 41644*b10 - 3792*b9 - 1516*b8 - 16472*b7 + 1992*b6 + 8700*b5 - 22864*b4 - 22864*b3 - 11400*b2 + 76304*b1 + 76120
ν 14 \nu^{14} ν 1 4 = = =
8144 β 19 + 960 β 18 − 7976 β 16 + 20328 β 15 + 21928 β 14 − 38904 β 13 + ⋯ − 4072 8144 \beta_{19} + 960 \beta_{18} - 7976 \beta_{16} + 20328 \beta_{15} + 21928 \beta_{14} - 38904 \beta_{13} + \cdots - 4072 8 1 4 4 β 1 9 + 9 6 0 β 1 8 − 7 9 7 6 β 1 6 + 2 0 3 2 8 β 1 5 + 2 1 9 2 8 β 1 4 − 3 8 9 0 4 β 1 3 + ⋯ − 4 0 7 2
8144*b19 + 960*b18 - 7976*b16 + 20328*b15 + 21928*b14 - 38904*b13 - 4072*b12 - 31880*b11 - 102664*b10 + 31880*b9 - 7976*b8 - 206288*b7 + 60832*b6 + 2848*b4 - 5696*b3 - 60832*b2 - 340288*b1 - 4072
ν 15 \nu^{15} ν 1 5 = = =
65856 β 19 + 30664 β 18 − 89920 β 17 + 90056 β 16 − 89920 β 15 + ⋯ − 388832 65856 \beta_{19} + 30664 \beta_{18} - 89920 \beta_{17} + 90056 \beta_{16} - 89920 \beta_{15} + \cdots - 388832 6 5 8 5 6 β 1 9 + 3 0 6 6 4 β 1 8 − 8 9 9 2 0 β 1 7 + 9 0 0 5 6 β 1 6 − 8 9 9 2 0 β 1 5 + ⋯ − 3 8 8 8 3 2
65856*b19 + 30664*b18 - 89920*b17 + 90056*b16 - 89920*b15 - 77152*b14 - 5568*b13 - 131712*b12 - 77280*b11 - 103896*b10 + 65856*b9 + 103896*b7 - 5568*b6 - 30664*b5 - 237248*b4 + 118624*b3 - 77152*b2 - 388832
ν 16 \nu^{16} ν 1 6 = = =
− 94256 β 19 − 102352 β 17 + 209920 β 14 − 209920 β 13 − 94256 β 12 + ⋯ + 3601040 - 94256 \beta_{19} - 102352 \beta_{17} + 209920 \beta_{14} - 209920 \beta_{13} - 94256 \beta_{12} + \cdots + 3601040 − 9 4 2 5 6 β 1 9 − 1 0 2 3 5 2 β 1 7 + 2 0 9 9 2 0 β 1 4 − 2 0 9 9 2 0 β 1 3 − 9 4 2 5 6 β 1 2 + ⋯ + 3 6 0 1 0 4 0
-94256*b19 - 102352*b17 + 209920*b14 - 209920*b13 - 94256*b12 + 94256*b11 + 137968*b10 + 384*b9 + 237856*b8 + 190512*b7 - 188752*b6 + 243056*b5 + 64256*b4 + 64256*b3 - 398672*b2 + 3695296*b1 + 3601040
ν 17 \nu^{17} ν 1 7 = = =
− 209344 β 19 + 616944 β 18 + 137296 β 16 − 41440 β 15 − 717792 β 14 + ⋯ + 104672 - 209344 \beta_{19} + 616944 \beta_{18} + 137296 \beta_{16} - 41440 \beta_{15} - 717792 \beta_{14} + \cdots + 104672 − 2 0 9 3 4 4 β 1 9 + 6 1 6 9 4 4 β 1 8 + 1 3 7 2 9 6 β 1 6 − 4 1 4 4 0 β 1 5 − 7 1 7 7 9 2 β 1 4 + ⋯ + 1 0 4 6 7 2
-209344*b19 + 616944*b18 + 137296*b16 - 41440*b15 - 717792*b14 - 328992*b13 + 104672*b12 - 72672*b11 - 1010528*b10 + 72672*b9 + 137296*b8 - 2638000*b7 - 388800*b6 - 506560*b4 + 1013120*b3 + 388800*b2 - 35356608*b1 + 104672
ν 18 \nu^{18} ν 1 8 = = =
789408 β 19 − 2061056 β 18 − 513632 β 17 + 3020128 β 16 − 513632 β 15 + ⋯ + 58623424 789408 \beta_{19} - 2061056 \beta_{18} - 513632 \beta_{17} + 3020128 \beta_{16} - 513632 \beta_{15} + \cdots + 58623424 7 8 9 4 0 8 β 1 9 − 2 0 6 1 0 5 6 β 1 8 − 5 1 3 6 3 2 β 1 7 + 3 0 2 0 1 2 8 β 1 6 − 5 1 3 6 3 2 β 1 5 + ⋯ + 5 8 6 2 3 4 2 4
789408*b19 - 2061056*b18 - 513632*b17 + 3020128*b16 - 513632*b15 + 746528*b14 + 1147552*b13 - 1578816*b12 + 750976*b11 - 3562208*b10 + 789408*b9 + 3562208*b7 + 1147552*b6 + 2061056*b5 + 3942656*b4 - 1971328*b3 + 746528*b2 + 58623424
ν 19 \nu^{19} ν 1 9 = = =
5736192 β 19 − 1460480 β 17 − 396928 β 14 + 396928 β 13 + 5736192 β 12 + ⋯ − 6007424 5736192 \beta_{19} - 1460480 \beta_{17} - 396928 \beta_{14} + 396928 \beta_{13} + 5736192 \beta_{12} + \cdots - 6007424 5 7 3 6 1 9 2 β 1 9 − 1 4 6 0 4 8 0 β 1 7 − 3 9 6 9 2 8 β 1 4 + 3 9 6 9 2 8 β 1 3 + 5 7 3 6 1 9 2 β 1 2 + ⋯ − 6 0 0 7 4 2 4
5736192*b19 - 1460480*b17 - 396928*b14 + 396928*b13 + 5736192*b12 - 5736192*b11 - 4566496*b10 - 2959232*b9 + 4054240*b8 + 699776*b7 - 4264320*b6 + 5966048*b5 - 3329408*b4 - 3329408*b3 - 3867392*b2 - 11743616*b1 - 6007424
Character values
We give the values of χ \chi χ on generators for ( Z / 448 Z ) × \left(\mathbb{Z}/448\mathbb{Z}\right)^\times ( Z / 4 4 8 Z ) × .
n n n
127 127 1 2 7
129 129 1 2 9
197 197 1 9 7
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
1 + β 1 1 + \beta_{1} 1 + β 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 kernel of the linear operator
T 3 20 + 163 T 3 18 + 18379 T 3 16 + 1043398 T 3 14 + 42494101 T 3 12 + ⋯ + 51883209 T_{3}^{20} + 163 T_{3}^{18} + 18379 T_{3}^{16} + 1043398 T_{3}^{14} + 42494101 T_{3}^{12} + \cdots + 51883209 T 3 2 0 + 1 6 3 T 3 1 8 + 1 8 3 7 9 T 3 1 6 + 1 0 4 3 3 9 8 T 3 1 4 + 4 2 4 9 4 1 0 1 T 3 1 2 + ⋯ + 5 1 8 8 3 2 0 9
T3^20 + 163*T3^18 + 18379*T3^16 + 1043398*T3^14 + 42494101*T3^12 + 927039337*T3^10 + 14545748317*T3^8 + 119263400326*T3^6 + 669515995603*T3^4 + 5898399843*T3^2 + 51883209
acting on S 4 n e w ( 448 , [ χ ] ) S_{4}^{\mathrm{new}}(448, [\chi]) S 4 n e w ( 4 4 8 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 20 T^{20} T 2 0
T^20
3 3 3
T 20 + 163 T 18 + ⋯ + 51883209 T^{20} + 163 T^{18} + \cdots + 51883209 T 2 0 + 1 6 3 T 1 8 + ⋯ + 5 1 8 8 3 2 0 9
T^20 + 163*T^18 + 18379*T^16 + 1043398*T^14 + 42494101*T^12 + 927039337*T^10 + 14545748317*T^8 + 119263400326*T^6 + 669515995603*T^4 + 5898399843*T^2 + 51883209
5 5 5
( T 10 − 3 T 9 + ⋯ + 81588675 ) 2 (T^{10} - 3 T^{9} + \cdots + 81588675)^{2} ( T 1 0 − 3 T 9 + ⋯ + 8 1 5 8 8 6 7 5 ) 2
(T^10 - 3*T^9 - 299*T^8 + 906*T^7 + 84761*T^6 + 261591*T^5 - 1926469*T^4 - 6757398*T^3 + 49396951*T^2 + 115162845*T + 81588675)^2
7 7 7
T 20 + ⋯ + 22 ⋯ 49 T^{20} + \cdots + 22\!\cdots\!49 T 2 0 + ⋯ + 2 2 ⋯ 4 9
T^20 + 322*T^18 + 86093*T^16 + 107501688*T^14 + 36728466754*T^12 + 5950089233676*T^10 + 4321067385141346*T^8 + 1487961738200295288*T^6 + 140195011884904285757*T^4 + 61689156504542385437122*T^2 + 22539340290692258087863249
11 11 1 1
T 20 + ⋯ + 20 ⋯ 25 T^{20} + \cdots + 20\!\cdots\!25 T 2 0 + ⋯ + 2 0 ⋯ 2 5
T^20 - 5985*T^18 + 22516499*T^16 - 53599840914*T^14 + 94391335342469*T^12 - 116554315700537683*T^10 + 106447974876197279077*T^8 - 61455340568615486895346*T^6 + 22311112540224699024471379*T^4 - 2129100977323504363493825*T^2 + 203122091044435930950625
13 13 1 3
( T 10 + ⋯ + 10072189747200 ) 2 (T^{10} + \cdots + 10072189747200)^{2} ( T 1 0 + ⋯ + 1 0 0 7 2 1 8 9 7 4 7 2 0 0 ) 2
(T^10 + 5924*T^8 + 10544432*T^6 + 5983444672*T^4 + 940129122304*T^2 + 10072189747200)^2
17 17 1 7
( T 10 + ⋯ + 60 ⋯ 75 ) 2 (T^{10} + \cdots + 60\!\cdots\!75)^{2} ( T 1 0 + ⋯ + 6 0 ⋯ 7 5 ) 2
(T^10 + 3*T^9 - 10723*T^8 - 32178*T^7 + 110389001*T^6 + 7626792369*T^5 + 145581511771*T^4 - 2933689388994*T^3 - 91515356083889*T^2 + 1667036543705715*T + 60677626846401075)^2
19 19 1 9
T 20 + ⋯ + 12 ⋯ 25 T^{20} + \cdots + 12\!\cdots\!25 T 2 0 + ⋯ + 1 2 ⋯ 2 5
T^20 + 34667*T^18 + 739321531*T^16 + 10090710253910*T^14 + 101688106164103285*T^12 + 747016569289277381057*T^10 + 4189450762831190153743405*T^8 + 17156476125088887621551832854*T^6 + 51971349967897761973579194517939*T^4 + 101318780657900256349317431050226475*T^2 + 121222934536537213239964283765026205625
23 23 2 3
T 20 + ⋯ + 17 ⋯ 61 T^{20} + \cdots + 17\!\cdots\!61 T 2 0 + ⋯ + 1 7 ⋯ 6 1
T^20 - 38721*T^18 + 1195146467*T^16 - 10322924418402*T^14 + 64011688751415749*T^12 - 194865156517234237603*T^10 + 423682714455045954555781*T^8 - 222780759728919608786770018*T^6 + 86010595531544060890211189155*T^4 - 14305109795437641868422002167361*T^2 + 1756572989134534697546441841293761
29 29 2 9
( T 5 − 88 T 4 + ⋯ − 5066614016 ) 4 (T^{5} - 88 T^{4} + \cdots - 5066614016)^{4} ( T 5 − 8 8 T 4 + ⋯ − 5 0 6 6 6 1 4 0 1 6 ) 4
(T^5 - 88*T^4 - 40796*T^3 + 4923568*T^2 - 2408128*T - 5066614016)^4
31 31 3 1
T 20 + ⋯ + 43 ⋯ 25 T^{20} + \cdots + 43\!\cdots\!25 T 2 0 + ⋯ + 4 3 ⋯ 2 5
T^20 + 129411*T^18 + 14481693531*T^16 + 271342758337398*T^14 + 3712965269092946901*T^12 + 23071824100500556516569*T^10 + 104558455162145102186308221*T^8 + 67383120326858256904286896278*T^6 + 34337061272404917326024721244899*T^4 + 4239671506785868958346529602778275*T^2 + 432901017499524744723793471385555625
37 37 3 7
( T 10 + ⋯ + 48 ⋯ 25 ) 2 (T^{10} + \cdots + 48\!\cdots\!25)^{2} ( T 1 0 + ⋯ + 4 8 ⋯ 2 5 ) 2
(T^10 + 129*T^9 + 47239*T^8 + 4625430*T^7 + 1364598141*T^6 + 120978951207*T^5 + 19350195624517*T^4 + 810307003720758*T^3 + 109671722097362959*T^2 + 2736352858160219745*T + 482519402593780761225)^2
41 41 4 1
( T 10 + ⋯ + 14 ⋯ 72 ) 2 (T^{10} + \cdots + 14\!\cdots\!72)^{2} ( T 1 0 + ⋯ + 1 4 ⋯ 7 2 ) 2
(T^10 + 371044*T^8 + 45452939696*T^6 + 2106458435887808*T^4 + 31103998837585985536*T^2 + 140803883847964082307072)^2
43 43 4 3
( T 10 + ⋯ + 55 ⋯ 56 ) 2 (T^{10} + \cdots + 55\!\cdots\!56)^{2} ( T 1 0 + ⋯ + 5 5 ⋯ 5 6 ) 2
(T^10 + 481188*T^8 + 76172353504*T^6 + 4759034131340160*T^4 + 109792582980774068480*T^2 + 550304061763914439238656)^2
47 47 4 7
T 20 + ⋯ + 90 ⋯ 25 T^{20} + \cdots + 90\!\cdots\!25 T 2 0 + ⋯ + 9 0 ⋯ 2 5
T^20 + 548571*T^18 + 229434171691*T^16 + 35108367873452838*T^14 + 3974465219081383020021*T^12 + 136825662287451566854199649*T^10 + 3563745911126847288431753319661*T^8 + 18941722733844734810337562917093318*T^6 + 84134220860617896775540448827876713859*T^4 + 8831367017348108071923180543346024964475*T^2 + 905934593789940138239144193914106698405625
53 53 5 3
( T 10 + ⋯ + 23 ⋯ 25 ) 2 (T^{10} + \cdots + 23\!\cdots\!25)^{2} ( T 1 0 + ⋯ + 2 3 ⋯ 2 5 ) 2
(T^10 + 285*T^9 + 385439*T^8 - 7091442*T^7 + 92220981965*T^6 + 6987261591931*T^5 + 4698483953203645*T^4 + 466225986823511566*T^3 + 197068551005473270351*T^2 + 17851214676172869265565*T + 2340067246925087308090225)^2
59 59 5 9
T 20 + ⋯ + 53 ⋯ 25 T^{20} + \cdots + 53\!\cdots\!25 T 2 0 + ⋯ + 5 3 ⋯ 2 5
T^20 + 814891*T^18 + 432699299611*T^16 + 134460135107508598*T^14 + 30315504424429292696821*T^12 + 4340227330047808197683578849*T^10 + 452096353071456707740355597838061*T^8 + 28476916865990330226986293117067504758*T^6 + 1191451481700732600555473374763509586499859*T^4 + 8613454989343535801566726420984770486145298475*T^2 + 53413274518394041156481785192287565341823737605625
61 61 6 1
( T 10 + ⋯ + 25 ⋯ 47 ) 2 (T^{10} + \cdots + 25\!\cdots\!47)^{2} ( T 1 0 + ⋯ + 2 5 ⋯ 4 7 ) 2
(T^10 + 147*T^9 - 622255*T^8 - 92530326*T^7 + 318077925929*T^6 + 123028492312221*T^5 - 24061957047836909*T^4 - 15895203677629438566*T^3 + 2206275605299001185363*T^2 + 1834843738581090841026339*T + 254442110188891525325439147)^2
67 67 6 7
T 20 + ⋯ + 16 ⋯ 21 T^{20} + \cdots + 16\!\cdots\!21 T 2 0 + ⋯ + 1 6 ⋯ 2 1
T^20 - 1823609*T^18 + 2113861632803*T^16 - 1494540359131738850*T^14 + 772593603058462076582629*T^12 - 275662385232141826652383474939*T^10 + 72715981276686478582550036287224437*T^8 - 12453660818603582693694478280881385068162*T^6 + 1436893798110874145311771468088725730237011027*T^4 - 56094530016365454099014743859253909075845260262169*T^2 + 1658417760319435758355949528165800894501015332493765521
71 71 7 1
( T 10 + ⋯ + 16 ⋯ 00 ) 2 (T^{10} + \cdots + 16\!\cdots\!00)^{2} ( T 1 0 + ⋯ + 1 6 ⋯ 0 0 ) 2
(T^10 + 1630916*T^8 + 350254218720*T^6 + 21869277178736512*T^4 + 382835409624793751808*T^2 + 1698224008550476213273600)^2
73 73 7 3
( T 10 + ⋯ + 39 ⋯ 75 ) 2 (T^{10} + \cdots + 39\!\cdots\!75)^{2} ( T 1 0 + ⋯ + 3 9 ⋯ 7 5 ) 2
(T^10 - 483*T^9 - 1004063*T^8 + 522521958*T^7 + 811349980841*T^6 - 393556648905309*T^5 - 253040332806530029*T^4 + 132050619613873078134*T^3 + 63860205841108298709571*T^2 - 31012062068088558161771955*T + 3969458312725371848805237675)^2
79 79 7 9
T 20 + ⋯ + 14 ⋯ 25 T^{20} + \cdots + 14\!\cdots\!25 T 2 0 + ⋯ + 1 4 ⋯ 2 5
T^20 - 2364593*T^18 + 3655335450275*T^16 - 3268164132038273570*T^14 + 2113059516916069044934405*T^12 - 862158946836260413837811605651*T^10 + 252114012418879548009016384463490149*T^8 - 41816185499530560119942623326125069463778*T^6 + 4978926600682118533401487770601891941484071619*T^4 - 332265380681025030932239217287348441012299012435825*T^2 + 14734867887370417989695364597555135836178855569050550625
83 83 8 3
( T 10 + ⋯ − 41 ⋯ 08 ) 2 (T^{10} + \cdots - 41\!\cdots\!08)^{2} ( T 1 0 + ⋯ − 4 1 ⋯ 0 8 ) 2
(T^10 - 1929696*T^8 + 1383233136896*T^6 - 457320228679630848*T^4 + 70638001277968517693440*T^2 - 4131577363857790011973828608)^2
89 89 8 9
( T 10 + ⋯ + 99 ⋯ 27 ) 2 (T^{10} + \cdots + 99\!\cdots\!27)^{2} ( T 1 0 + ⋯ + 9 9 ⋯ 2 7 ) 2
(T^10 + 1593*T^9 + 456121*T^8 - 620890866*T^7 - 187812567367*T^6 + 319508558427303*T^5 + 253640187212613659*T^4 + 70991208413939653182*T^3 + 9232417389159998129107*T^2 + 481985295424015915679049*T + 9974778865395417866421027)^2
97 97 9 7
( T 10 + ⋯ + 12 ⋯ 00 ) 2 (T^{10} + \cdots + 12\!\cdots\!00)^{2} ( T 1 0 + ⋯ + 1 2 ⋯ 0 0 ) 2
(T^10 + 2197892*T^8 + 1444821395120*T^6 + 349801161086178496*T^4 + 34935762410783908962304*T^2 + 1209195120125848936533196800)^2
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