gp: [N,k,chi] = [575,4,Mod(24,575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(575, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("575.24");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [14,0,0,-54,0,-82]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == 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 13 1,\beta_1,\ldots,\beta_{13} 1 , β 1 , … , β 1 3 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 14 + 83 x 12 + 2715 x 10 + 44273 x 8 + 372280 x 6 + 1482448 x 4 + 2136384 x 2 + 746496 x^{14} + 83x^{12} + 2715x^{10} + 44273x^{8} + 372280x^{6} + 1482448x^{4} + 2136384x^{2} + 746496 x 1 4 + 8 3 x 1 2 + 2 7 1 5 x 1 0 + 4 4 2 7 3 x 8 + 3 7 2 2 8 0 x 6 + 1 4 8 2 4 4 8 x 4 + 2 1 3 6 3 8 4 x 2 + 7 4 6 4 9 6
x^14 + 83*x^12 + 2715*x^10 + 44273*x^8 + 372280*x^6 + 1482448*x^4 + 2136384*x^2 + 746496
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( 2761 ν 12 + 193316 ν 10 + 4988175 ν 8 + 57142094 ν 6 + 264366298 ν 4 + ⋯ − 132164352 ) / 21505248 ( 2761 \nu^{12} + 193316 \nu^{10} + 4988175 \nu^{8} + 57142094 \nu^{6} + 264366298 \nu^{4} + \cdots - 132164352 ) / 21505248 ( 2 7 6 1 ν 1 2 + 1 9 3 3 1 6 ν 1 0 + 4 9 8 8 1 7 5 ν 8 + 5 7 1 4 2 0 9 4 ν 6 + 2 6 4 3 6 6 2 9 8 ν 4 + ⋯ − 1 3 2 1 6 4 3 5 2 ) / 2 1 5 0 5 2 4 8
(2761*v^12 + 193316*v^10 + 4988175*v^8 + 57142094*v^6 + 264366298*v^4 + 248540320*v^2 - 132164352) / 21505248
β 3 \beta_{3} β 3 = = =
( 2899 ν 12 + 208820 ν 10 + 5584101 ν 8 + 67767626 ν 6 + 365072926 ν 4 + ⋯ + 573138432 ) / 21505248 ( 2899 \nu^{12} + 208820 \nu^{10} + 5584101 \nu^{8} + 67767626 \nu^{6} + 365072926 \nu^{4} + \cdots + 573138432 ) / 21505248 ( 2 8 9 9 ν 1 2 + 2 0 8 8 2 0 ν 1 0 + 5 5 8 4 1 0 1 ν 8 + 6 7 7 6 7 6 2 6 ν 6 + 3 6 5 0 7 2 9 2 6 ν 4 + ⋯ + 5 7 3 1 3 8 4 3 2 ) / 2 1 5 0 5 2 4 8
(2899*v^12 + 208820*v^10 + 5584101*v^8 + 67767626*v^6 + 365072926*v^4 + 758000056*v^2 + 573138432) / 21505248
β 4 \beta_{4} β 4 = = =
( 673 ν 12 + 46391 ν 10 + 1172553 ν 8 + 13181345 ν 6 + 62463766 ν 4 + ⋯ + 29332800 ) / 3584208 ( 673 \nu^{12} + 46391 \nu^{10} + 1172553 \nu^{8} + 13181345 \nu^{6} + 62463766 \nu^{4} + \cdots + 29332800 ) / 3584208 ( 6 7 3 ν 1 2 + 4 6 3 9 1 ν 1 0 + 1 1 7 2 5 5 3 ν 8 + 1 3 1 8 1 3 4 5 ν 6 + 6 2 4 6 3 7 6 6 ν 4 + ⋯ + 2 9 3 3 2 8 0 0 ) / 3 5 8 4 2 0 8
(673*v^12 + 46391*v^10 + 1172553*v^8 + 13181345*v^6 + 62463766*v^4 + 91418008*v^2 + 29332800) / 3584208
β 5 \beta_{5} β 5 = = =
( − 673 ν 12 − 46391 ν 10 − 1172553 ν 8 − 13181345 ν 6 − 62463766 ν 4 + ⋯ + 13677696 ) / 3584208 ( - 673 \nu^{12} - 46391 \nu^{10} - 1172553 \nu^{8} - 13181345 \nu^{6} - 62463766 \nu^{4} + \cdots + 13677696 ) / 3584208 ( − 6 7 3 ν 1 2 − 4 6 3 9 1 ν 1 0 − 1 1 7 2 5 5 3 ν 8 − 1 3 1 8 1 3 4 5 ν 6 − 6 2 4 6 3 7 6 6 ν 4 + ⋯ + 1 3 6 7 7 6 9 6 ) / 3 5 8 4 2 0 8
(-673*v^12 - 46391*v^10 - 1172553*v^8 - 13181345*v^6 - 62463766*v^4 - 87833800*v^2 + 13677696) / 3584208
β 6 \beta_{6} β 6 = = =
( − 16975 ν 13 − 1118189 ν 11 − 26046213 ν 9 − 244991279 ν 7 + ⋯ + 3227461056 ν ) / 1548377856 ( - 16975 \nu^{13} - 1118189 \nu^{11} - 26046213 \nu^{9} - 244991279 \nu^{7} + \cdots + 3227461056 \nu ) / 1548377856 ( − 1 6 9 7 5 ν 1 3 − 1 1 1 8 1 8 9 ν 1 1 − 2 6 0 4 6 2 1 3 ν 9 − 2 4 4 9 9 1 2 7 9 ν 7 + ⋯ + 3 2 2 7 4 6 1 0 5 6 ν ) / 1 5 4 8 3 7 7 8 5 6
(-16975*v^13 - 1118189*v^11 - 26046213*v^9 - 244991279*v^7 - 625111960*v^5 + 1819792112*v^3 + 3227461056*v) / 1548377856
β 7 \beta_{7} β 7 = = =
( 7196 ν 12 + 487045 ν 10 + 12004194 ν 8 + 129952549 ν 6 + 577259876 ν 4 + ⋯ + 189432288 ) / 21505248 ( 7196 \nu^{12} + 487045 \nu^{10} + 12004194 \nu^{8} + 129952549 \nu^{6} + 577259876 \nu^{4} + \cdots + 189432288 ) / 21505248 ( 7 1 9 6 ν 1 2 + 4 8 7 0 4 5 ν 1 0 + 1 2 0 0 4 1 9 4 ν 8 + 1 2 9 9 5 2 5 4 9 ν 6 + 5 7 7 2 5 9 8 7 6 ν 4 + ⋯ + 1 8 9 4 3 2 2 8 8 ) / 2 1 5 0 5 2 4 8
(7196*v^12 + 487045*v^10 + 12004194*v^8 + 129952549*v^6 + 577259876*v^4 + 736925240*v^2 + 189432288) / 21505248
β 8 \beta_{8} β 8 = = =
( − 26717 ν 13 − 2037367 ν 11 − 58609191 ν 9 − 779396941 ν 7 + ⋯ + 1827097920 ν ) / 1548377856 ( - 26717 \nu^{13} - 2037367 \nu^{11} - 58609191 \nu^{9} - 779396941 \nu^{7} + \cdots + 1827097920 \nu ) / 1548377856 ( − 2 6 7 1 7 ν 1 3 − 2 0 3 7 3 6 7 ν 1 1 − 5 8 6 0 9 1 9 1 ν 9 − 7 7 9 3 9 6 9 4 1 ν 7 + ⋯ + 1 8 2 7 0 9 7 9 2 0 ν ) / 1 5 4 8 3 7 7 8 5 6
(-26717*v^13 - 2037367*v^11 - 58609191*v^9 - 779396941*v^7 - 4614947408*v^5 - 8865303248*v^3 + 1827097920*v) / 1548377856
β 9 \beta_{9} β 9 = = =
( 27685 ν 12 + 1912367 ν 10 + 48919983 ν 8 + 569920133 ν 6 + 2981644672 ν 4 + ⋯ + 3235561344 ) / 43010496 ( 27685 \nu^{12} + 1912367 \nu^{10} + 48919983 \nu^{8} + 569920133 \nu^{6} + 2981644672 \nu^{4} + \cdots + 3235561344 ) / 43010496 ( 2 7 6 8 5 ν 1 2 + 1 9 1 2 3 6 7 ν 1 0 + 4 8 9 1 9 9 8 3 ν 8 + 5 6 9 9 2 0 1 3 3 ν 6 + 2 9 8 1 6 4 4 6 7 2 ν 4 + ⋯ + 3 2 3 5 5 6 1 3 4 4 ) / 4 3 0 1 0 4 9 6
(27685*v^12 + 1912367*v^10 + 48919983*v^8 + 569920133*v^6 + 2981644672*v^4 + 5942605792*v^2 + 3235561344) / 43010496
β 10 \beta_{10} β 1 0 = = =
( − 1077 ν 13 − 65807 ν 11 − 1326331 ν 9 − 8251101 ν 7 + 27581252 ν 5 + ⋯ + 537403648 ν ) / 28673664 ( - 1077 \nu^{13} - 65807 \nu^{11} - 1326331 \nu^{9} - 8251101 \nu^{7} + 27581252 \nu^{5} + \cdots + 537403648 \nu ) / 28673664 ( − 1 0 7 7 ν 1 3 − 6 5 8 0 7 ν 1 1 − 1 3 2 6 3 3 1 ν 9 − 8 2 5 1 1 0 1 ν 7 + 2 7 5 8 1 2 5 2 ν 5 + ⋯ + 5 3 7 4 0 3 6 4 8 ν ) / 2 8 6 7 3 6 6 4
(-1077*v^13 - 65807*v^11 - 1326331*v^9 - 8251101*v^7 + 27581252*v^5 + 347008384*v^3 + 537403648*v) / 28673664
β 11 \beta_{11} β 1 1 = = =
( 7253 ν 13 + 551887 ν 11 + 16165695 ν 9 + 229537141 ν 7 + 1623583616 ν 5 + ⋯ + 4412473344 ν ) / 129031488 ( 7253 \nu^{13} + 551887 \nu^{11} + 16165695 \nu^{9} + 229537141 \nu^{7} + 1623583616 \nu^{5} + \cdots + 4412473344 \nu ) / 129031488 ( 7 2 5 3 ν 1 3 + 5 5 1 8 8 7 ν 1 1 + 1 6 1 6 5 6 9 5 ν 9 + 2 2 9 5 3 7 1 4 1 ν 7 + 1 6 2 3 5 8 3 6 1 6 ν 5 + ⋯ + 4 4 1 2 4 7 3 3 4 4 ν ) / 1 2 9 0 3 1 4 8 8
(7253*v^13 + 551887*v^11 + 16165695*v^9 + 229537141*v^7 + 1623583616*v^5 + 5110840400*v^3 + 4412473344*v) / 129031488
β 12 \beta_{12} β 1 2 = = =
( 20576 ν 13 + 1347439 ν 11 + 31418445 ν 9 + 302528533 ν 7 + 890890415 ν 5 + ⋯ − 6944411376 ν ) / 193547232 ( 20576 \nu^{13} + 1347439 \nu^{11} + 31418445 \nu^{9} + 302528533 \nu^{7} + 890890415 \nu^{5} + \cdots - 6944411376 \nu ) / 193547232 ( 2 0 5 7 6 ν 1 3 + 1 3 4 7 4 3 9 ν 1 1 + 3 1 4 1 8 4 4 5 ν 9 + 3 0 2 5 2 8 5 3 3 ν 7 + 8 9 0 8 9 0 4 1 5 ν 5 + ⋯ − 6 9 4 4 4 1 1 3 7 6 ν ) / 1 9 3 5 4 7 2 3 2
(20576*v^13 + 1347439*v^11 + 31418445*v^9 + 302528533*v^7 + 890890415*v^5 - 1808414776*v^3 - 6944411376*v) / 193547232
β 13 \beta_{13} β 1 3 = = =
( 183923 ν 13 + 12861853 ν 11 + 334482897 ν 9 + 3987741031 ν 7 + ⋯ + 32486214528 ν ) / 774188928 ( 183923 \nu^{13} + 12861853 \nu^{11} + 334482897 \nu^{9} + 3987741031 \nu^{7} + \cdots + 32486214528 \nu ) / 774188928 ( 1 8 3 9 2 3 ν 1 3 + 1 2 8 6 1 8 5 3 ν 1 1 + 3 3 4 4 8 2 8 9 7 ν 9 + 3 9 8 7 7 4 1 0 3 1 ν 7 + ⋯ + 3 2 4 8 6 2 1 4 5 2 8 ν ) / 7 7 4 1 8 8 9 2 8
(183923*v^13 + 12861853*v^11 + 334482897*v^9 + 3987741031*v^7 + 21645631184*v^5 + 46700459024*v^3 + 32486214528*v) / 774188928
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 5 + β 4 − 12 \beta_{5} + \beta_{4} - 12 β 5 + β 4 − 1 2
b5 + b4 - 12
ν 3 \nu^{3} ν 3 = = =
2 β 10 + 2 β 8 − 10 β 6 − 19 β 1 2\beta_{10} + 2\beta_{8} - 10\beta_{6} - 19\beta_1 2 β 1 0 + 2 β 8 − 1 0 β 6 − 1 9 β 1
2*b10 + 2*b8 - 10*b6 - 19*b1
ν 4 \nu^{4} ν 4 = = =
4 β 7 − 23 β 5 − 33 β 4 + 4 β 3 + 216 4\beta_{7} - 23\beta_{5} - 33\beta_{4} + 4\beta_{3} + 216 4 β 7 − 2 3 β 5 − 3 3 β 4 + 4 β 3 + 2 1 6
4*b7 - 23*b5 - 33*b4 + 4*b3 + 216
ν 5 \nu^{5} ν 5 = = =
8 β 13 − 8 β 12 − 14 β 11 − 62 β 10 − 62 β 8 + 334 β 6 + 395 β 1 8\beta_{13} - 8\beta_{12} - 14\beta_{11} - 62\beta_{10} - 62\beta_{8} + 334\beta_{6} + 395\beta_1 8 β 1 3 − 8 β 1 2 − 1 4 β 1 1 − 6 2 β 1 0 − 6 2 β 8 + 3 3 4 β 6 + 3 9 5 β 1
8*b13 - 8*b12 - 14*b11 - 62*b10 - 62*b8 + 334*b6 + 395*b1
ν 6 \nu^{6} ν 6 = = =
− 160 β 7 + 505 β 5 + 921 β 4 − 148 β 3 − 36 β 2 − 4332 -160\beta_{7} + 505\beta_{5} + 921\beta_{4} - 148\beta_{3} - 36\beta_{2} - 4332 − 1 6 0 β 7 + 5 0 5 β 5 + 9 2 1 β 4 − 1 4 8 β 3 − 3 6 β 2 − 4 3 3 2
-160*b7 + 505*b5 + 921*b4 - 148*b3 - 36*b2 - 4332
ν 7 \nu^{7} ν 7 = = =
− 344 β 13 + 296 β 12 + 636 β 11 + 1578 β 10 + 1698 β 8 − 9402 β 6 − 8675 β 1 -344\beta_{13} + 296\beta_{12} + 636\beta_{11} + 1578\beta_{10} + 1698\beta_{8} - 9402\beta_{6} - 8675\beta_1 − 3 4 4 β 1 3 + 2 9 6 β 1 2 + 6 3 6 β 1 1 + 1 5 7 8 β 1 0 + 1 6 9 8 β 8 − 9 4 0 2 β 6 − 8 6 7 5 β 1
-344*b13 + 296*b12 + 636*b11 + 1578*b10 + 1698*b8 - 9402*b6 - 8675*b1
ν 8 \nu^{8} ν 8 = = =
96 β 9 + 4844 β 7 − 11243 β 5 − 24481 β 4 + 4212 β 3 + 1832 β 2 + 92368 96\beta_{9} + 4844\beta_{7} - 11243\beta_{5} - 24481\beta_{4} + 4212\beta_{3} + 1832\beta_{2} + 92368 9 6 β 9 + 4 8 4 4 β 7 − 1 1 2 4 3 β 5 − 2 4 4 8 1 β 4 + 4 2 1 2 β 3 + 1 8 3 2 β 2 + 9 2 3 6 8
96*b9 + 4844*b7 - 11243*b5 - 24481*b4 + 4212*b3 + 1832*b2 + 92368
ν 9 \nu^{9} ν 9 = = =
11080 β 13 − 8232 β 12 − 21018 β 11 − 37942 β 10 − 43814 β 8 + ⋯ + 197115 β 1 11080 \beta_{13} - 8232 \beta_{12} - 21018 \beta_{11} - 37942 \beta_{10} - 43814 \beta_{8} + \cdots + 197115 \beta_1 1 1 0 8 0 β 1 3 − 8 2 3 2 β 1 2 − 2 1 0 1 8 β 1 1 − 3 7 9 4 2 β 1 0 − 4 3 8 1 4 β 8 + ⋯ + 1 9 7 1 1 5 β 1
11080*b13 - 8232*b12 - 21018*b11 - 37942*b10 - 43814*b8 + 251462*b6 + 197115*b1
ν 10 \nu^{10} ν 1 0 = = =
− 5696 β 9 − 134520 β 7 + 254829 β 5 + 636777 β 4 − 109300 β 3 + ⋯ − 2054900 - 5696 \beta_{9} - 134520 \beta_{7} + 254829 \beta_{5} + 636777 \beta_{4} - 109300 \beta_{3} + \cdots - 2054900 − 5 6 9 6 β 9 − 1 3 4 5 2 0 β 7 + 2 5 4 8 2 9 β 5 + 6 3 6 7 7 7 β 4 − 1 0 9 3 0 0 β 3 + ⋯ − 2 0 5 4 9 0 0
-5696*b9 - 134520*b7 + 254829*b5 + 636777*b4 - 109300*b3 - 64684*b2 - 2054900
ν 11 \nu^{11} ν 1 1 = = =
− 319896 β 13 + 207208 β 12 + 614920 β 11 + 896002 β 10 + 1092906 β 8 + ⋯ − 4579603 β 1 - 319896 \beta_{13} + 207208 \beta_{12} + 614920 \beta_{11} + 896002 \beta_{10} + 1092906 \beta_{8} + \cdots - 4579603 \beta_1 − 3 1 9 8 9 6 β 1 3 + 2 0 7 2 0 8 β 1 2 + 6 1 4 9 2 0 β 1 1 + 8 9 6 0 0 2 β 1 0 + 1 0 9 2 9 0 6 β 8 + ⋯ − 4 5 7 9 6 0 3 β 1
-319896*b13 + 207208*b12 + 614920*b11 + 896002*b10 + 1092906*b8 - 6559826*b6 - 4579603*b1
ν 12 \nu^{12} ν 1 2 = = =
225376 β 9 + 3595588 β 7 − 5869375 β 5 − 16347697 β 4 + 2723220 β 3 + ⋯ + 47101912 225376 \beta_{9} + 3595588 \beta_{7} - 5869375 \beta_{5} - 16347697 \beta_{4} + 2723220 \beta_{3} + \cdots + 47101912 2 2 5 3 7 6 β 9 + 3 5 9 5 5 8 8 β 7 − 5 8 6 9 3 7 5 β 5 − 1 6 3 4 7 6 9 7 β 4 + 2 7 2 3 2 2 0 β 3 + ⋯ + 4 7 1 0 1 9 1 2
225376*b9 + 3595588*b7 - 5869375*b5 - 16347697*b4 + 2723220*b3 + 1972016*b2 + 47101912
ν 13 \nu^{13} ν 1 3 = = =
8741576 β 13 − 4995688 β 12 − 16920198 β 11 − 21081230 β 10 + ⋯ + 108029099 β 1 8741576 \beta_{13} - 4995688 \beta_{12} - 16920198 \beta_{11} - 21081230 \beta_{10} + \cdots + 108029099 \beta_1 8 7 4 1 5 7 6 β 1 3 − 4 9 9 5 6 8 8 β 1 2 − 1 6 9 2 0 1 9 8 β 1 1 − 2 1 0 8 1 2 3 0 β 1 0 + ⋯ + 1 0 8 0 2 9 0 9 9 β 1
8741576*b13 - 4995688*b12 - 16920198*b11 - 21081230*b10 - 26773806*b8 + 168504670*b6 + 108029099*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 575 Z ) × \left(\mathbb{Z}/575\mathbb{Z}\right)^\times ( Z / 5 7 5 Z ) × .
n n n
51 51 5 1
277 277 2 7 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 4 n e w ( 575 , [ χ ] ) S_{4}^{\mathrm{new}}(575, [\chi]) S 4 n e w ( 5 7 5 , [ χ ] ) :
T 2 14 + 83 T 2 12 + 2715 T 2 10 + 44273 T 2 8 + 372280 T 2 6 + 1482448 T 2 4 + 2136384 T 2 2 + 746496 T_{2}^{14} + 83T_{2}^{12} + 2715T_{2}^{10} + 44273T_{2}^{8} + 372280T_{2}^{6} + 1482448T_{2}^{4} + 2136384T_{2}^{2} + 746496 T 2 1 4 + 8 3 T 2 1 2 + 2 7 1 5 T 2 1 0 + 4 4 2 7 3 T 2 8 + 3 7 2 2 8 0 T 2 6 + 1 4 8 2 4 4 8 T 2 4 + 2 1 3 6 3 8 4 T 2 2 + 7 4 6 4 9 6
T2^14 + 83*T2^12 + 2715*T2^10 + 44273*T2^8 + 372280*T2^6 + 1482448*T2^4 + 2136384*T2^2 + 746496
T 3 14 + 199 T 3 12 + 14475 T 3 10 + 490274 T 3 8 + 8194505 T 3 6 + 65475300 T 3 4 + ⋯ + 84640000 T_{3}^{14} + 199 T_{3}^{12} + 14475 T_{3}^{10} + 490274 T_{3}^{8} + 8194505 T_{3}^{6} + 65475300 T_{3}^{4} + \cdots + 84640000 T 3 1 4 + 1 9 9 T 3 1 2 + 1 4 4 7 5 T 3 1 0 + 4 9 0 2 7 4 T 3 8 + 8 1 9 4 5 0 5 T 3 6 + 6 5 4 7 5 3 0 0 T 3 4 + ⋯ + 8 4 6 4 0 0 0 0
T3^14 + 199*T3^12 + 14475*T3^10 + 490274*T3^8 + 8194505*T3^6 + 65475300*T3^4 + 207534400*T3^2 + 84640000
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 14 + 83 T 12 + ⋯ + 746496 T^{14} + 83 T^{12} + \cdots + 746496 T 1 4 + 8 3 T 1 2 + ⋯ + 7 4 6 4 9 6
T^14 + 83*T^12 + 2715*T^10 + 44273*T^8 + 372280*T^6 + 1482448*T^4 + 2136384*T^2 + 746496
3 3 3
T 14 + 199 T 12 + ⋯ + 84640000 T^{14} + 199 T^{12} + \cdots + 84640000 T 1 4 + 1 9 9 T 1 2 + ⋯ + 8 4 6 4 0 0 0 0
T^14 + 199*T^12 + 14475*T^10 + 490274*T^8 + 8194505*T^6 + 65475300*T^4 + 207534400*T^2 + 84640000
5 5 5
T 14 T^{14} T 1 4
T^14
7 7 7
T 14 + ⋯ + 610305007739904 T^{14} + \cdots + 610305007739904 T 1 4 + ⋯ + 6 1 0 3 0 5 0 0 7 7 3 9 9 0 4
T^14 + 1599*T^12 + 909279*T^10 + 242269253*T^8 + 33007610308*T^6 + 2279182331232*T^4 + 70286168937024*T^2 + 610305007739904
11 11 1 1
( T 7 + 52 T 6 + ⋯ − 935052376 ) 2 (T^{7} + 52 T^{6} + \cdots - 935052376)^{2} ( T 7 + 5 2 T 6 + ⋯ − 9 3 5 0 5 2 3 7 6 ) 2
(T^7 + 52*T^6 - 902*T^5 - 124012*T^4 - 3642303*T^3 - 50494376*T^2 - 345017588*T - 935052376)^2
13 13 1 3
T 14 + ⋯ + 22 ⋯ 76 T^{14} + \cdots + 22\!\cdots\!76 T 1 4 + ⋯ + 2 2 ⋯ 7 6
T^14 + 10909*T^12 + 43080050*T^10 + 79208556759*T^8 + 70860404955608*T^6 + 29241912667567276*T^4 + 4543721912014557561*T^2 + 224112589506872461476
17 17 1 7
T 14 + ⋯ + 74 ⋯ 00 T^{14} + \cdots + 74\!\cdots\!00 T 1 4 + ⋯ + 7 4 ⋯ 0 0
T^14 + 33853*T^12 + 397306700*T^10 + 2154977085296*T^8 + 5815992173400448*T^6 + 7691913130099775744*T^4 + 4438741683873274200064*T^2 + 747092134644818175590400
19 19 1 9
( T 7 − 10 T 6 + ⋯ − 451105530840 ) 2 (T^{7} - 10 T^{6} + \cdots - 451105530840)^{2} ( T 7 − 1 0 T 6 + ⋯ − 4 5 1 1 0 5 5 3 0 8 4 0 ) 2
(T^7 - 10*T^6 - 8810*T^5 - 18328*T^4 + 24816017*T^3 + 295698466*T^2 - 22881873104*T - 451105530840)^2
23 23 2 3
( T 2 + 529 ) 7 (T^{2} + 529)^{7} ( T 2 + 5 2 9 ) 7
(T^2 + 529)^7
29 29 2 9
( T 7 + ⋯ − 323393919122280 ) 2 (T^{7} + \cdots - 323393919122280)^{2} ( T 7 + ⋯ − 3 2 3 3 9 3 9 1 9 1 2 2 2 8 0 ) 2
(T^7 - 455*T^6 + 27704*T^5 + 15559525*T^4 - 2928172396*T^3 + 138054555094*T^2 + 3937666746343*T - 323393919122280)^2
31 31 3 1
( T 7 + ⋯ + 38841367049280 ) 2 (T^{7} + \cdots + 38841367049280)^{2} ( T 7 + ⋯ + 3 8 8 4 1 3 6 7 0 4 9 2 8 0 ) 2
(T^7 + 690*T^6 + 121061*T^5 - 8870581*T^4 - 3503927942*T^3 - 134097336300*T^2 + 3937277103720*T + 38841367049280)^2
37 37 3 7
T 14 + ⋯ + 59 ⋯ 04 T^{14} + \cdots + 59\!\cdots\!04 T 1 4 + ⋯ + 5 9 ⋯ 0 4
T^14 + 289784*T^12 + 29953122544*T^10 + 1380878577556928*T^8 + 28901480358685453568*T^6 + 245143564857890060480512*T^4 + 721330767125786489733136384*T^2 + 599522962411051111623010811904
41 41 4 1
( T 7 + ⋯ − 15 ⋯ 85 ) 2 (T^{7} + \cdots - 15\!\cdots\!85)^{2} ( T 7 + ⋯ − 1 5 ⋯ 8 5 ) 2
(T^7 + 230*T^6 - 218075*T^5 - 52827543*T^4 + 10018678129*T^3 + 2862441725210*T^2 + 139133372922953*T - 1510156295246985)^2
43 43 4 3
T 14 + ⋯ + 31 ⋯ 00 T^{14} + \cdots + 31\!\cdots\!00 T 1 4 + ⋯ + 3 1 ⋯ 0 0
T^14 + 554323*T^12 + 123626235515*T^10 + 14165854562407825*T^8 + 884437813718329597496*T^6 + 29583170329101310360278800*T^4 + 491733744777197726959655880000*T^2 + 3175703585404320160514465296000000
47 47 4 7
T 14 + ⋯ + 19 ⋯ 00 T^{14} + \cdots + 19\!\cdots\!00 T 1 4 + ⋯ + 1 9 ⋯ 0 0
T^14 + 325550*T^12 + 38128839949*T^10 + 2112957779305001*T^8 + 58560991920733058624*T^6 + 765880107445991438992480*T^4 + 3845804304940754448533344000*T^2 + 1906659990105917820670727942400
53 53 5 3
T 14 + ⋯ + 29 ⋯ 16 T^{14} + \cdots + 29\!\cdots\!16 T 1 4 + ⋯ + 2 9 ⋯ 1 6
T^14 + 1116048*T^12 + 502937993376*T^10 + 118034785091569984*T^8 + 15367875505904879444736*T^6 + 1076783643065742206630003712*T^4 + 34853220276806234279551556616192*T^2 + 299826393121296936067867324052668416
59 59 5 9
( T 7 + ⋯ − 771684969846528 ) 2 (T^{7} + \cdots - 771684969846528)^{2} ( T 7 + ⋯ − 7 7 1 6 8 4 9 6 9 8 4 6 5 2 8 ) 2
(T^7 - 551*T^6 - 334593*T^5 + 103679463*T^4 + 29866045440*T^3 - 18377324504*T^2 - 198507309485856*T - 771684969846528)^2
61 61 6 1
( T 7 + ⋯ − 67 ⋯ 12 ) 2 (T^{7} + \cdots - 67\!\cdots\!12)^{2} ( T 7 + ⋯ − 6 7 ⋯ 1 2 ) 2
(T^7 + 1242*T^6 + 80872*T^5 - 377109608*T^4 - 131371688416*T^3 - 5331039914944*T^2 + 1343548846899712*T - 6786221819789312)^2
67 67 6 7
T 14 + ⋯ + 19 ⋯ 00 T^{14} + \cdots + 19\!\cdots\!00 T 1 4 + ⋯ + 1 9 ⋯ 0 0
T^14 + 2298265*T^12 + 1896192053880*T^10 + 691578149209682192*T^8 + 107532831775938986884416*T^6 + 5615862283510345664011125760*T^4 + 79268078462484482237960709734400*T^2 + 192807850818385450229542816422297600
71 71 7 1
( T 7 + ⋯ + 30 ⋯ 00 ) 2 (T^{7} + \cdots + 30\!\cdots\!00)^{2} ( T 7 + ⋯ + 3 0 ⋯ 0 0 ) 2
(T^7 - 186*T^6 - 415889*T^5 - 27041343*T^4 + 30766250960*T^3 + 3449611622180*T^2 - 266167838093440*T + 3045392502059200)^2
73 73 7 3
T 14 + ⋯ + 13 ⋯ 69 T^{14} + \cdots + 13\!\cdots\!69 T 1 4 + ⋯ + 1 3 ⋯ 6 9
T^14 + 3258034*T^12 + 3234050965247*T^10 + 979088601723418565*T^8 + 127490898890782232259859*T^6 + 7605369604433016388348823672*T^4 + 186642928793846134335584759376373*T^2 + 1376401685189280441263616993686937369
79 79 7 9
( T 7 + ⋯ − 30 ⋯ 36 ) 2 (T^{7} + \cdots - 30\!\cdots\!36)^{2} ( T 7 + ⋯ − 3 0 ⋯ 3 6 ) 2
(T^7 - 2003*T^6 + 701001*T^5 + 849422091*T^4 - 728119482110*T^3 + 168945438461028*T^2 + 504221297568000*T - 3027491396524334736)^2
83 83 8 3
T 14 + ⋯ + 13 ⋯ 16 T^{14} + \cdots + 13\!\cdots\!16 T 1 4 + ⋯ + 1 3 ⋯ 1 6
T^14 + 6014076*T^12 + 14316886937174*T^10 + 17082109387166918196*T^8 + 10650641384440831416643297*T^6 + 3274681780431780899404303210072*T^4 + 408174166100120975726059030410202128*T^2 + 13580163444785857799593830138708010698816
89 89 8 9
( T 7 + ⋯ − 28 ⋯ 00 ) 2 (T^{7} + \cdots - 28\!\cdots\!00)^{2} ( T 7 + ⋯ − 2 8 ⋯ 0 0 ) 2
(T^7 - 2629*T^6 + 1688646*T^5 + 637340660*T^4 - 840251596496*T^3 + 127347150202048*T^2 + 19806712749873888*T - 2810758158951888000)^2
97 97 9 7
T 14 + ⋯ + 36 ⋯ 24 T^{14} + \cdots + 36\!\cdots\!24 T 1 4 + ⋯ + 3 6 ⋯ 2 4
T^14 + 6459432*T^12 + 12796623253040*T^10 + 8732059686227246912*T^8 + 2679658722590802941034240*T^6 + 390625695603172254069851635712*T^4 + 24160304302575392556820983350677504*T^2 + 367274502773172920594411476041812557824
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