# Properties

 Label 121.6.a.d Level $121$ Weight $6$ Character orbit 121.a Self dual yes Analytic conductor $19.406$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,6,Mod(1,121)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 121.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$19.4064421974$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.54492.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x - 38$$ x^3 - x^2 - 52*x - 38 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{2} + \beta_1 + 11) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 + 30) q^{4} + ( - \beta_{2} + 9 \beta_1 + 5) q^{5} + ( - 13 \beta_{2} - 10 \beta_1 + 72) q^{6} + (10 \beta_{2} + 30 \beta_1 - 38) q^{7} + ( - 26 \beta_{2} + 188) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 - 6) q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b2 + b1 + 11) * q^3 + (-4*b2 - 6*b1 + 30) * q^4 + (-b2 + 9*b1 + 5) * q^5 + (-13*b2 - 10*b1 + 72) * q^6 + (10*b2 + 30*b1 - 38) * q^7 + (-26*b2 + 188) * q^8 + (-19*b2 + 11*b1 - 6) * q^9 $$q - \beta_{2} q^{2} + ( - \beta_{2} + \beta_1 + 11) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 + 30) q^{4} + ( - \beta_{2} + 9 \beta_1 + 5) q^{5} + ( - 13 \beta_{2} - 10 \beta_1 + 72) q^{6} + (10 \beta_{2} + 30 \beta_1 - 38) q^{7} + ( - 26 \beta_{2} + 188) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 - 6) q^{9} + (9 \beta_{2} - 42 \beta_1 + 152) q^{10} + ( - 112 \beta_{2} - 70 \beta_1 + 354) q^{12} + ( - 70 \beta_{2} - 18 \beta_1 - 156) q^{13} + (138 \beta_{2} - 60 \beta_1 - 320) q^{14} + (27 \beta_{2} + 85 \beta_1 + 523) q^{15} + ( - 164 \beta_{2} + 36 \beta_1 + 652) q^{16} + ( - 132 \beta_{2} + 60 \beta_1 - 382) q^{17} + ( - 48 \beta_{2} - 158 \beta_1 + 1288) q^{18} + (60 \beta_{2} + 60 \beta_1 - 480) q^{19} + ( - 168 \beta_{2} - 66 \beta_1 - 1138) q^{20} + (318 \beta_{2} + 362 \beta_1 + 182) q^{21} + ( - 21 \beta_{2} + 501 \beta_1 - 1189) q^{23} + ( - 526 \beta_{2} - 72 \beta_1 + 3940) q^{24} + (265 \beta_{2} + 255 \beta_1 - 104) q^{25} + ( - 160 \beta_{2} - 348 \beta_1 + 4160) q^{26} + (57 \beta_{2} - 329 \beta_1 - 887) q^{27} + (432 \beta_{2} + 108 \beta_1 - 7940) q^{28} + (182 \beta_{2} + 138 \beta_1 + 1096) q^{29} + ( - 245 \beta_{2} - 178 \beta_1 - 824) q^{30} + ( - 101 \beta_{2} + 69 \beta_1 - 1389) q^{31} + ( - 404 \beta_{2} - 1128 \beta_1 + 4512) q^{32} + ( - 26 \beta_{2} - 1032 \beta_1 + 8784) q^{34} + (818 \beta_{2} + 918 \beta_1 + 7770) q^{35} + ( - 1188 \beta_{2} - 8 \beta_1 + 1588) q^{36} + (937 \beta_{2} + 591 \beta_1 + 5711) q^{37} + (840 \beta_{2} + 120 \beta_1 - 3120) q^{38} + ( - 844 \beta_{2} - 1036 \beta_1 + 2532) q^{39} + (46 \beta_{2} + 600 \beta_1 + 4892) q^{40} + (1378 \beta_{2} - 282 \beta_1 - 1904) q^{41} + (1814 \beta_{2} + 460 \beta_1 - 16096) q^{42} + ( - 1190 \beta_{2} + 1110 \beta_1 + 8366) q^{43} + (496 \beta_{2} - 544 \beta_1 + 6334) q^{45} + (2107 \beta_{2} - 2130 \beta_1 + 6312) q^{46} + ( - 600 \beta_{2} - 1272 \beta_1 - 5320) q^{47} + ( - 2604 \beta_{2} - 628 \beta_1 + 20564) q^{48} + (340 \beta_{2} + 2220 \beta_1 + 15437) q^{49} + (1674 \beta_{2} + 570 \beta_1 - 13880) q^{50} + ( - 1034 \beta_{2} - 1102 \beta_1 + 7942) q^{51} + ( - 3256 \beta_{2} + 1008 \beta_1 + 11432) q^{52} + ( - 476 \beta_{2} - 1620 \beta_1 + 17402) q^{53} + (457 \beta_{2} + 1658 \beta_1 - 6824) q^{54} + (5468 \beta_{2} + 4080 \beta_1 - 15464) q^{56} + (1560 \beta_{2} + 720 \beta_1 - 6960) q^{57} + ( - 92 \beta_{2} + 540 \beta_1 - 9904) q^{58} + (3141 \beta_{2} + 747 \beta_1 - 1495) q^{59} + ( - 1376 \beta_{2} - 3478 \beta_1 - 3326) q^{60} + (1466 \beta_{2} + 4038 \beta_1 - 7508) q^{61} + (1123 \beta_{2} - 882 \beta_1 + 6952) q^{62} + (3332 \beta_{2} - 308 \beta_1 + 4268) q^{63} + ( - 3136 \beta_{2} + 936 \beta_1 - 7096) q^{64} + (264 \beta_{2} - 4848 \beta_1 + 4172) q^{65} + ( - 6575 \beta_{2} - 2721 \beta_1 - 15011) q^{67} + ( - 6728 \beta_{2} + 2052 \beta_1 + 3516) q^{68} + (3421 \beta_{2} + 3611 \beta_1 + 10477) q^{69} + ( - 2662 \beta_{2} + 1236 \beta_1 - 41536) q^{70} + ( - 3935 \beta_{2} - 2673 \beta_1 + 13985) q^{71} + ( - 4820 \beta_{2} - 2040 \beta_1 + 32360) q^{72} + ( - 5370 \beta_{2} + 3522 \beta_1 - 6316) q^{73} + ( - 781 \beta_{2} + 3258 \beta_1 - 52184) q^{74} + (4824 \beta_{2} + 5096 \beta_1 - 9004) q^{75} + (4800 \beta_{2} + 2640 \beta_1 - 35520) q^{76} + ( - 7980 \beta_{2} - 920 \beta_1 + 41968) q^{78} + ( - 3902 \beta_{2} - 1362 \beta_1 - 41262) q^{79} + (1868 \beta_{2} - 12 \beta_1 + 39564) q^{80} + (4600 \beta_{2} - 6280 \beta_1 - 26879) q^{81} + (6852 \beta_{2} + 9396 \beta_1 - 88256) q^{82} + ( - 3150 \beta_{2} - 11250 \beta_1 + 51726) q^{83} + (14096 \beta_{2} - 2540 \beta_1 - 113692) q^{84} + (3310 \beta_{2} - 7302 \beta_1 + 37114) q^{85} + ( - 10906 \beta_{2} - 11580 \beta_1 + 84880) q^{86} + (1960 \beta_{2} + 4296 \beta_1 + 5024) q^{87} + ( - 5167 \beta_{2} - 4569 \beta_1 - 34085) q^{89} + ( - 5438 \beta_{2} + 5152 \beta_1 - 36192) q^{90} + (6840 \beta_{2} - 10536 \beta_1 - 33032) q^{91} + ( - 1472 \beta_{2} + 5130 \beta_1 - 113886) q^{92} + (421 \beta_{2} - 1709 \beta_1 - 4971) q^{93} + (376 \beta_{2} + 1488 \beta_1 + 24480) q^{94} + (1680 \beta_{2} - 120 \beta_1 + 7440) q^{95} + ( - 15404 \beta_{2} - 10808 \beta_1 + 29088) q^{96} + (123 \beta_{2} + 6429 \beta_1 + 1085) q^{97} + ( - 9637 \beta_{2} - 6840 \beta_1 + 1120) q^{98}+O(q^{100})$$ q - b2 * q^2 + (-b2 + b1 + 11) * q^3 + (-4*b2 - 6*b1 + 30) * q^4 + (-b2 + 9*b1 + 5) * q^5 + (-13*b2 - 10*b1 + 72) * q^6 + (10*b2 + 30*b1 - 38) * q^7 + (-26*b2 + 188) * q^8 + (-19*b2 + 11*b1 - 6) * q^9 + (9*b2 - 42*b1 + 152) * q^10 + (-112*b2 - 70*b1 + 354) * q^12 + (-70*b2 - 18*b1 - 156) * q^13 + (138*b2 - 60*b1 - 320) * q^14 + (27*b2 + 85*b1 + 523) * q^15 + (-164*b2 + 36*b1 + 652) * q^16 + (-132*b2 + 60*b1 - 382) * q^17 + (-48*b2 - 158*b1 + 1288) * q^18 + (60*b2 + 60*b1 - 480) * q^19 + (-168*b2 - 66*b1 - 1138) * q^20 + (318*b2 + 362*b1 + 182) * q^21 + (-21*b2 + 501*b1 - 1189) * q^23 + (-526*b2 - 72*b1 + 3940) * q^24 + (265*b2 + 255*b1 - 104) * q^25 + (-160*b2 - 348*b1 + 4160) * q^26 + (57*b2 - 329*b1 - 887) * q^27 + (432*b2 + 108*b1 - 7940) * q^28 + (182*b2 + 138*b1 + 1096) * q^29 + (-245*b2 - 178*b1 - 824) * q^30 + (-101*b2 + 69*b1 - 1389) * q^31 + (-404*b2 - 1128*b1 + 4512) * q^32 + (-26*b2 - 1032*b1 + 8784) * q^34 + (818*b2 + 918*b1 + 7770) * q^35 + (-1188*b2 - 8*b1 + 1588) * q^36 + (937*b2 + 591*b1 + 5711) * q^37 + (840*b2 + 120*b1 - 3120) * q^38 + (-844*b2 - 1036*b1 + 2532) * q^39 + (46*b2 + 600*b1 + 4892) * q^40 + (1378*b2 - 282*b1 - 1904) * q^41 + (1814*b2 + 460*b1 - 16096) * q^42 + (-1190*b2 + 1110*b1 + 8366) * q^43 + (496*b2 - 544*b1 + 6334) * q^45 + (2107*b2 - 2130*b1 + 6312) * q^46 + (-600*b2 - 1272*b1 - 5320) * q^47 + (-2604*b2 - 628*b1 + 20564) * q^48 + (340*b2 + 2220*b1 + 15437) * q^49 + (1674*b2 + 570*b1 - 13880) * q^50 + (-1034*b2 - 1102*b1 + 7942) * q^51 + (-3256*b2 + 1008*b1 + 11432) * q^52 + (-476*b2 - 1620*b1 + 17402) * q^53 + (457*b2 + 1658*b1 - 6824) * q^54 + (5468*b2 + 4080*b1 - 15464) * q^56 + (1560*b2 + 720*b1 - 6960) * q^57 + (-92*b2 + 540*b1 - 9904) * q^58 + (3141*b2 + 747*b1 - 1495) * q^59 + (-1376*b2 - 3478*b1 - 3326) * q^60 + (1466*b2 + 4038*b1 - 7508) * q^61 + (1123*b2 - 882*b1 + 6952) * q^62 + (3332*b2 - 308*b1 + 4268) * q^63 + (-3136*b2 + 936*b1 - 7096) * q^64 + (264*b2 - 4848*b1 + 4172) * q^65 + (-6575*b2 - 2721*b1 - 15011) * q^67 + (-6728*b2 + 2052*b1 + 3516) * q^68 + (3421*b2 + 3611*b1 + 10477) * q^69 + (-2662*b2 + 1236*b1 - 41536) * q^70 + (-3935*b2 - 2673*b1 + 13985) * q^71 + (-4820*b2 - 2040*b1 + 32360) * q^72 + (-5370*b2 + 3522*b1 - 6316) * q^73 + (-781*b2 + 3258*b1 - 52184) * q^74 + (4824*b2 + 5096*b1 - 9004) * q^75 + (4800*b2 + 2640*b1 - 35520) * q^76 + (-7980*b2 - 920*b1 + 41968) * q^78 + (-3902*b2 - 1362*b1 - 41262) * q^79 + (1868*b2 - 12*b1 + 39564) * q^80 + (4600*b2 - 6280*b1 - 26879) * q^81 + (6852*b2 + 9396*b1 - 88256) * q^82 + (-3150*b2 - 11250*b1 + 51726) * q^83 + (14096*b2 - 2540*b1 - 113692) * q^84 + (3310*b2 - 7302*b1 + 37114) * q^85 + (-10906*b2 - 11580*b1 + 84880) * q^86 + (1960*b2 + 4296*b1 + 5024) * q^87 + (-5167*b2 - 4569*b1 - 34085) * q^89 + (-5438*b2 + 5152*b1 - 36192) * q^90 + (6840*b2 - 10536*b1 - 33032) * q^91 + (-1472*b2 + 5130*b1 - 113886) * q^92 + (421*b2 - 1709*b1 - 4971) * q^93 + (376*b2 + 1488*b1 + 24480) * q^94 + (1680*b2 - 120*b1 + 7440) * q^95 + (-15404*b2 - 10808*b1 + 29088) * q^96 + (123*b2 + 6429*b1 + 1085) * q^97 + (-9637*b2 - 6840*b1 + 1120) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 34 q^{3} + 84 q^{4} + 24 q^{5} + 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9}+O(q^{10})$$ 3 * q + 34 * q^3 + 84 * q^4 + 24 * q^5 + 206 * q^6 - 84 * q^7 + 564 * q^8 - 7 * q^9 $$3 q + 34 q^{3} + 84 q^{4} + 24 q^{5} + 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9} + 414 q^{10} + 992 q^{12} - 486 q^{13} - 1020 q^{14} + 1654 q^{15} + 1992 q^{16} - 1086 q^{17} + 3706 q^{18} - 1380 q^{19} - 3480 q^{20} + 908 q^{21} - 3066 q^{23} + 11748 q^{24} - 57 q^{25} + 12132 q^{26} - 2990 q^{27} - 23712 q^{28} + 3426 q^{29} - 2650 q^{30} - 4098 q^{31} + 12408 q^{32} + 25320 q^{34} + 24228 q^{35} + 4756 q^{36} + 17724 q^{37} - 9240 q^{38} + 6560 q^{39} + 15276 q^{40} - 5994 q^{41} - 47828 q^{42} + 26208 q^{43} + 18458 q^{45} + 16806 q^{46} - 17232 q^{47} + 61064 q^{48} + 48531 q^{49} - 41070 q^{50} + 22724 q^{51} + 35304 q^{52} + 50586 q^{53} - 18814 q^{54} - 42312 q^{56} - 20160 q^{57} - 29172 q^{58} - 3738 q^{59} - 13456 q^{60} - 18486 q^{61} + 19974 q^{62} + 12496 q^{63} - 20352 q^{64} + 7668 q^{65} - 47754 q^{67} + 12600 q^{68} + 35042 q^{69} - 123372 q^{70} + 39282 q^{71} + 95040 q^{72} - 15426 q^{73} - 153294 q^{74} - 21916 q^{75} - 103920 q^{76} + 124984 q^{78} - 125148 q^{79} + 118680 q^{80} - 86917 q^{81} - 255372 q^{82} + 143928 q^{83} - 343616 q^{84} + 104040 q^{85} + 243060 q^{86} + 19368 q^{87} - 106824 q^{89} - 103424 q^{90} - 109632 q^{91} - 336528 q^{92} - 16622 q^{93} + 74928 q^{94} + 22200 q^{95} + 76456 q^{96} + 9684 q^{97} - 3480 q^{98}+O(q^{100})$$ 3 * q + 34 * q^3 + 84 * q^4 + 24 * q^5 + 206 * q^6 - 84 * q^7 + 564 * q^8 - 7 * q^9 + 414 * q^10 + 992 * q^12 - 486 * q^13 - 1020 * q^14 + 1654 * q^15 + 1992 * q^16 - 1086 * q^17 + 3706 * q^18 - 1380 * q^19 - 3480 * q^20 + 908 * q^21 - 3066 * q^23 + 11748 * q^24 - 57 * q^25 + 12132 * q^26 - 2990 * q^27 - 23712 * q^28 + 3426 * q^29 - 2650 * q^30 - 4098 * q^31 + 12408 * q^32 + 25320 * q^34 + 24228 * q^35 + 4756 * q^36 + 17724 * q^37 - 9240 * q^38 + 6560 * q^39 + 15276 * q^40 - 5994 * q^41 - 47828 * q^42 + 26208 * q^43 + 18458 * q^45 + 16806 * q^46 - 17232 * q^47 + 61064 * q^48 + 48531 * q^49 - 41070 * q^50 + 22724 * q^51 + 35304 * q^52 + 50586 * q^53 - 18814 * q^54 - 42312 * q^56 - 20160 * q^57 - 29172 * q^58 - 3738 * q^59 - 13456 * q^60 - 18486 * q^61 + 19974 * q^62 + 12496 * q^63 - 20352 * q^64 + 7668 * q^65 - 47754 * q^67 + 12600 * q^68 + 35042 * q^69 - 123372 * q^70 + 39282 * q^71 + 95040 * q^72 - 15426 * q^73 - 153294 * q^74 - 21916 * q^75 - 103920 * q^76 + 124984 * q^78 - 125148 * q^79 + 118680 * q^80 - 86917 * q^81 - 255372 * q^82 + 143928 * q^83 - 343616 * q^84 + 104040 * q^85 + 243060 * q^86 + 19368 * q^87 - 106824 * q^89 - 103424 * q^90 - 109632 * q^91 - 336528 * q^92 - 16622 * q^93 + 74928 * q^94 + 22200 * q^95 + 76456 * q^96 + 9684 * q^97 - 3480 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x - 38$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 3\nu - 34 ) / 3$$ (v^2 - 3*v - 34) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2} + 3\beta _1 + 34$$ 3*b2 + 3*b1 + 34

## Embeddings

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −6.29828 8.04796 −0.749680
−8.18772 −3.48600 35.0388 −59.8722 28.5424 −145.071 −24.8808 −230.848 490.217
1.2 −2.20859 16.8394 −27.1221 75.2230 −37.1913 225.525 130.577 40.5643 −166.137
1.3 10.3963 20.6466 76.0833 8.64919 214.649 −164.454 458.304 183.283 89.9197
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.6.a.d 3
3.b odd 2 1 1089.6.a.r 3
11.b odd 2 1 11.6.a.b 3
33.d even 2 1 99.6.a.g 3
44.c even 2 1 176.6.a.i 3
55.d odd 2 1 275.6.a.b 3
55.e even 4 2 275.6.b.b 6
77.b even 2 1 539.6.a.e 3
88.b odd 2 1 704.6.a.q 3
88.g even 2 1 704.6.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 11.b odd 2 1
99.6.a.g 3 33.d even 2 1
121.6.a.d 3 1.a even 1 1 trivial
176.6.a.i 3 44.c even 2 1
275.6.a.b 3 55.d odd 2 1
275.6.b.b 6 55.e even 4 2
539.6.a.e 3 77.b even 2 1
704.6.a.q 3 88.b odd 2 1
704.6.a.t 3 88.g even 2 1
1089.6.a.r 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 90T_{2} - 188$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(121))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 90T - 188$$
$3$ $$T^{3} - 34 T^{2} + \cdots + 1212$$
$5$ $$T^{3} - 24 T^{2} + \cdots + 38954$$
$7$ $$T^{3} + 84 T^{2} + \cdots - 5380448$$
$11$ $$T^{3}$$
$13$ $$T^{3} + 486 T^{2} + \cdots - 164136608$$
$17$ $$T^{3} + 1086 T^{2} + \cdots - 331752056$$
$19$ $$T^{3} + 1380 T^{2} + \cdots - 57024000$$
$23$ $$T^{3} + \cdots - 17004325928$$
$29$ $$T^{3} + \cdots + 4029189120$$
$31$ $$T^{3} + \cdots + 1094344400$$
$37$ $$T^{3} + \cdots + 541788167034$$
$41$ $$T^{3} + \cdots - 201929821568$$
$43$ $$T^{3} + \cdots + 2443875098544$$
$47$ $$T^{3} + \cdots - 70174939136$$
$53$ $$T^{3} + \cdots - 1850911309656$$
$59$ $$T^{3} + \cdots + 7759637437060$$
$61$ $$T^{3} + \cdots - 15233874751008$$
$67$ $$T^{3} + \cdots - 147288561330212$$
$71$ $$T^{3} + \cdots - 1290398551704$$
$73$ $$T^{3} + \cdots + 34539701265952$$
$79$ $$T^{3} + \cdots - 1279883216320$$
$83$ $$T^{3} + \cdots + 411597824719824$$
$89$ $$T^{3} + \cdots - 90320980174650$$
$97$ $$T^{3} + \cdots - 10221902527106$$