# Properties

 Label 49.10.c.e Level $49$ Weight $10$ Character orbit 49.c Analytic conductor $25.237$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.2367559720$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} + 427x^{4} - 3606x^{3} + 183492x^{2} - 858816x + 4064256$$ x^6 - x^5 + 427*x^4 - 3606*x^3 + 183492*x^2 - 858816*x + 4064256 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + 7 \beta_{3}) q^{2} + ( - \beta_{5} - \beta_{4} + 28 \beta_{3} - \beta_{2} - \beta_1 + 28) q^{3} + (8 \beta_{5} - 7 \beta_{4} - 519 \beta_{3} - 7 \beta_{2} + 8 \beta_1 - 519) q^{4} + ( - 43 \beta_{5} + 13 \beta_{4} - 518 \beta_{3}) q^{5} + (6 \beta_{2} - 36 \beta_1 - 1638) q^{6} + (147 \beta_{2} - 470 \beta_1 + 4685) q^{8} + (90 \beta_{5} - 126 \beta_{4} - 8667 \beta_{3}) q^{9}+O(q^{10})$$ q + (-b5 + 7*b3) * q^2 + (-b5 - b4 + 28*b3 - b2 - b1 + 28) * q^3 + (8*b5 - 7*b4 - 519*b3 - 7*b2 + 8*b1 - 519) * q^4 + (-43*b5 + 13*b4 - 518*b3) * q^5 + (6*b2 - 36*b1 - 1638) * q^6 + (147*b2 - 470*b1 + 4685) * q^8 + (90*b5 - 126*b4 - 8667*b3) * q^9 $$q + ( - \beta_{5} + 7 \beta_{3}) q^{2} + ( - \beta_{5} - \beta_{4} + 28 \beta_{3} - \beta_{2} - \beta_1 + 28) q^{3} + (8 \beta_{5} - 7 \beta_{4} - 519 \beta_{3} - 7 \beta_{2} + 8 \beta_1 - 519) q^{4} + ( - 43 \beta_{5} + 13 \beta_{4} - 518 \beta_{3}) q^{5} + (6 \beta_{2} - 36 \beta_1 - 1638) q^{6} + (147 \beta_{2} - 470 \beta_1 + 4685) q^{8} + (90 \beta_{5} - 126 \beta_{4} - 8667 \beta_{3}) q^{9} + (370 \beta_{5} - 470 \beta_{4} - 32620 \beta_{3} - 470 \beta_{2} + \cdots - 32620) q^{10}+ \cdots + (376362 \beta_{2} - 7689150 \beta_1 - 633659724) q^{99}+O(q^{100})$$ q + (-b5 + 7*b3) * q^2 + (-b5 - b4 + 28*b3 - b2 - b1 + 28) * q^3 + (8*b5 - 7*b4 - 519*b3 - 7*b2 + 8*b1 - 519) * q^4 + (-43*b5 + 13*b4 - 518*b3) * q^5 + (6*b2 - 36*b1 - 1638) * q^6 + (147*b2 - 470*b1 + 4685) * q^8 + (90*b5 - 126*b4 - 8667*b3) * q^9 + (370*b5 - 470*b4 - 32620*b3 - 470*b2 + 370*b1 - 32620) * q^10 + (-650*b5 - 658*b4 + 1148*b3 - 658*b2 - 650*b1 + 1148) * q^11 + (700*b5 - 182*b4 + 35462*b3) * q^12 + (175*b2 - 3017*b1 + 6594) * q^13 + (-1848*b2 - 1392*b1 + 66768) * q^15 + (-10614*b5 + 1617*b4 + 160987*b3) * q^16 + (3030*b5 + 1574*b4 + 338898*b3 + 1574*b2 + 3030*b1 + 338898) * q^17 + (-16947*b5 + 2268*b4 + 91089*b3 + 2268*b2 - 16947*b1 + 91089) * q^18 + (15371*b5 - 2437*b4 - 74284*b3) * q^19 + (2044*b2 - 41524*b1 + 640696) * q^20 + (4004*b2 - 40972*b1 - 949016) * q^22 + (-24200*b5 - 3808*b4 + 628544*b3) * q^23 + (4500*b5 + 10338*b4 - 483210*b3 + 10338*b2 + 4500*b1 - 483210) * q^24 + (15338*b5 + 4802*b4 - 1024407*b3 + 4802*b2 + 15338*b1 - 1024407) * q^25 + (-20986*b5 + 23394*b4 + 2928352*b3) * q^26 + (-15786*b2 - 33930*b1 - 183960) * q^27 + (18914*b2 + 54866*b1 + 1360606) * q^29 + (51960*b5 - 14280*b4 + 2684400*b3) * q^30 + (55698*b5 - 70302*b4 + 956480*b3 - 70302*b2 + 55698*b1 + 956480) * q^31 + (36066*b5 - 20055*b4 - 8407317*b3 - 20055*b2 + 36066*b1 - 8407317) * q^32 + (76152*b5 - 65640*b4 + 6753264*b3) * q^33 + (748*b2 + 438178*b1 + 1327214) * q^34 + (-83601*b2 + 209376*b1 - 11798793) * q^36 + (209418*b5 + 60522*b4 + 465206*b3) * q^37 + (-248060*b5 + 139278*b4 + 14493290*b3 + 139278*b2 - 248060*b1 + 14493290) * q^38 + (110012*b5 + 13748*b4 + 2996896*b3 + 13748*b2 + 110012*b1 + 2996896) * q^39 + (-625640*b5 + 76600*b4 + 27619760*b3) * q^40 + (131894*b2 + 163478*b1 + 4806886) * q^41 + (65366*b2 + 121982*b1 - 20543724) * q^43 + (314984*b5 + 1960*b4 + 32337328*b3) * q^44 + (-714681*b5 + 7551*b4 + 9924894*b3 + 7551*b2 - 714681*b1 + 9924894) * q^45 + (405224*b5 - 119896*b4 - 29915888*b3 - 119896*b2 + 405224*b1 - 29915888) * q^46 + (534778*b5 - 83238*b4 + 3456320*b3) * q^47 + (-9710*b2 - 174140*b1 - 5599594) * q^48 + (44940*b2 - 727615*b1 + 24441685) * q^50 + (-587238*b5 - 186102*b4 - 8715576*b3) * q^51 + (2925244*b5 - 361424*b4 - 26969348*b3 - 361424*b2 + 2925244*b1 - 26969348) * q^52 + (-1553376*b5 + 450352*b4 - 22500870*b3 + 450352*b2 - 1553376*b1 - 22500870) * q^53 + (1176120*b5 + 32292*b4 + 39293100*b3) * q^54 + (-988364*b2 - 1659356*b1 + 35274344) * q^55 + (182350*b2 + 403894*b1 + 2823704) * q^57 + (-2535150*b5 - 138180*b4 - 53054610*b3) * q^58 + (-2231195*b5 - 49659*b4 - 14196700*b3 - 49659*b2 - 2231195*b1 - 14196700) * q^59 + (991536*b5 - 396816*b4 + 59850336*b3 - 396816*b2 + 991536*b1 + 59850336) * q^60 + (-589107*b5 + 844773*b4 - 63915614*b3) * q^61 + (1303812*b2 - 3668848*b1 + 15661156) * q^62 + (-314727*b2 - 4312590*b1 + 2617387) * q^64 + (-958090*b5 + 1035790*b4 + 121427740*b3) * q^65 + (2410512*b5 + 1386384*b4 - 2685984*b3 + 1386384*b2 + 2410512*b1 - 2685984) * q^66 + (-3939816*b5 - 47712*b4 + 85058596*b3 - 47712*b2 - 3939816*b1 + 85058596) * q^67 + (613704*b5 - 2251634*b4 - 247828602*b3) * q^68 + (981336*b2 - 697656*b1 - 85967952) * q^69 + (-526260*b2 + 3499356*b1 + 98838168) * q^71 + (8765370*b5 - 1391229*b4 - 203104755*b3) * q^72 + (-9544844*b5 + 118516*b4 + 114737770*b3 + 118516*b2 - 9544844*b1 + 114737770) * q^73 + (4189718*b5 + 679140*b4 + 230232154*b3 + 679140*b2 + 4189718*b1 + 230232154) * q^74 + (4723*b5 + 1484467*b4 - 93010372*b3) * q^75 + (-2299290*b2 + 15924468*b1 - 242946662) * q^76 + (591360*b2 + 3780504*b1 + 93377592) * q^78 + (-7475532*b5 + 1679412*b4 - 320137552*b3) * q^79 + (11964112*b5 - 4328752*b4 - 444444448*b3 - 4328752*b2 + 11964112*b1 - 444444448) * q^80 + (4598370*b5 - 3824982*b4 + 11942559*b3 - 3824982*b2 + 4598370*b1 + 11942559) * q^81 + (-13216518*b5 + 570276*b4 - 187558434*b3) * q^82 + (1977367*b2 + 4479559*b1 + 366839060) * q^83 + (-1518034*b2 + 18558254*b1 + 146059804) * q^85 + (16416916*b5 - 4116*b4 - 293660752*b3) * q^86 + (-5138394*b5 + 457014*b4 - 207273864*b3 + 457014*b2 - 5138394*b1 - 207273864) * q^87 + (11172080*b5 + 4229456*b4 - 402041600*b3 + 4229456*b2 + 11172080*b1 - 402041600) * q^88 + (10612104*b5 - 1815976*b4 - 168938826*b3) * q^89 + (-5100930*b2 + 11130390*b1 - 767817540) * q^90 + (6344912*b2 - 25723952*b1 + 230374496) * q^92 + (1921156*b5 - 7972076*b4 + 564419504*b3) * q^93 + (-2488928*b5 + 4825540*b4 + 462668276*b3 + 4825540*b2 - 2488928*b1 + 462668276) * q^94 + (-10749416*b5 + 4098416*b4 + 734357024*b3 + 4098416*b2 - 10749416*b1 + 734357024) * q^95 + (8360604*b5 + 6385806*b4 - 111128598*b3) * q^96 + (864850*b2 - 33276782*b1 + 215832750) * q^97 + (376362*b2 - 7689150*b1 - 633659724) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 21 q^{2} + 84 q^{3} - 1557 q^{4} + 1554 q^{5} - 9828 q^{6} + 28110 q^{8} + 26001 q^{9}+O(q^{10})$$ 6 * q - 21 * q^2 + 84 * q^3 - 1557 * q^4 + 1554 * q^5 - 9828 * q^6 + 28110 * q^8 + 26001 * q^9 $$6 q - 21 q^{2} + 84 q^{3} - 1557 q^{4} + 1554 q^{5} - 9828 q^{6} + 28110 q^{8} + 26001 q^{9} - 97860 q^{10} + 3444 q^{11} - 106386 q^{12} + 39564 q^{13} + 400608 q^{15} - 482961 q^{16} + 1016694 q^{17} + 273267 q^{18} + 222852 q^{19} + 3844176 q^{20} - 5694096 q^{22} - 1885632 q^{23} - 1449630 q^{24} - 3073221 q^{25} - 8785056 q^{26} - 1103760 q^{27} + 8163636 q^{29} - 8053200 q^{30} + 2869440 q^{31} - 25221951 q^{32} - 20259792 q^{33} + 7963284 q^{34} - 70792758 q^{36} - 1395618 q^{37} + 43479870 q^{38} + 8990688 q^{39} - 82859280 q^{40} + 28841316 q^{41} - 123262344 q^{43} - 97011984 q^{44} + 29774682 q^{45} - 89747664 q^{46} - 10368960 q^{47} - 33597564 q^{48} + 146650110 q^{50} + 26146728 q^{51} - 80908044 q^{52} - 67502610 q^{53} - 117879300 q^{54} + 211646064 q^{55} + 16942224 q^{57} + 159163830 q^{58} - 42590100 q^{59} + 179551008 q^{60} + 191746842 q^{61} + 93966936 q^{62} + 15704322 q^{64} - 364283220 q^{65} - 8057952 q^{66} + 255175788 q^{67} + 743485806 q^{68} - 515807712 q^{69} + 593029008 q^{71} + 609314265 q^{72} + 344213310 q^{73} + 690696462 q^{74} + 279031116 q^{75} - 1457679972 q^{76} + 560265552 q^{78} + 960412656 q^{79} - 1333333344 q^{80} + 35827677 q^{81} + 562675302 q^{82} + 2201034360 q^{83} + 876358824 q^{85} + 880982256 q^{86} - 621821592 q^{87} - 1206124800 q^{88} + 506816478 q^{89} - 4606905240 q^{90} + 1382246976 q^{92} - 1693258512 q^{93} + 1388004828 q^{94} + 2203071072 q^{95} + 333385794 q^{96} + 1294996500 q^{97} - 3801958344 q^{99}+O(q^{100})$$ 6 * q - 21 * q^2 + 84 * q^3 - 1557 * q^4 + 1554 * q^5 - 9828 * q^6 + 28110 * q^8 + 26001 * q^9 - 97860 * q^10 + 3444 * q^11 - 106386 * q^12 + 39564 * q^13 + 400608 * q^15 - 482961 * q^16 + 1016694 * q^17 + 273267 * q^18 + 222852 * q^19 + 3844176 * q^20 - 5694096 * q^22 - 1885632 * q^23 - 1449630 * q^24 - 3073221 * q^25 - 8785056 * q^26 - 1103760 * q^27 + 8163636 * q^29 - 8053200 * q^30 + 2869440 * q^31 - 25221951 * q^32 - 20259792 * q^33 + 7963284 * q^34 - 70792758 * q^36 - 1395618 * q^37 + 43479870 * q^38 + 8990688 * q^39 - 82859280 * q^40 + 28841316 * q^41 - 123262344 * q^43 - 97011984 * q^44 + 29774682 * q^45 - 89747664 * q^46 - 10368960 * q^47 - 33597564 * q^48 + 146650110 * q^50 + 26146728 * q^51 - 80908044 * q^52 - 67502610 * q^53 - 117879300 * q^54 + 211646064 * q^55 + 16942224 * q^57 + 159163830 * q^58 - 42590100 * q^59 + 179551008 * q^60 + 191746842 * q^61 + 93966936 * q^62 + 15704322 * q^64 - 364283220 * q^65 - 8057952 * q^66 + 255175788 * q^67 + 743485806 * q^68 - 515807712 * q^69 + 593029008 * q^71 + 609314265 * q^72 + 344213310 * q^73 + 690696462 * q^74 + 279031116 * q^75 - 1457679972 * q^76 + 560265552 * q^78 + 960412656 * q^79 - 1333333344 * q^80 + 35827677 * q^81 + 562675302 * q^82 + 2201034360 * q^83 + 876358824 * q^85 + 880982256 * q^86 - 621821592 * q^87 - 1206124800 * q^88 + 506816478 * q^89 - 4606905240 * q^90 + 1382246976 * q^92 - 1693258512 * q^93 + 1388004828 * q^94 + 2203071072 * q^95 + 333385794 * q^96 + 1294996500 * q^97 - 3801958344 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 427x^{4} - 3606x^{3} + 183492x^{2} - 858816x + 4064256$$ :

 $$\beta_{1}$$ $$=$$ $$( 415\nu^{5} - 177205\nu^{4} - 1825733\nu^{3} - 76149180\nu^{2} + 356408640\nu - 18279850176 ) / 464953608$$ (415*v^5 - 177205*v^4 - 1825733*v^3 - 76149180*v^2 + 356408640*v - 18279850176) / 464953608 $$\beta_{2}$$ $$=$$ $$( -451\nu^{5} + 192577\nu^{4} - 4738111\nu^{3} + 82754892\nu^{2} - 387326016\nu + 30551662512 ) / 464953608$$ (-451*v^5 + 192577*v^4 - 4738111*v^3 + 82754892*v^2 - 387326016*v + 30551662512) / 464953608 $$\beta_{3}$$ $$=$$ $$( -4331\nu^{5} + 4283\nu^{4} - 1828841\nu^{3} + 6865794\nu^{2} - 785896236\nu - 41319936 ) / 3719628864$$ (-4331*v^5 + 4283*v^4 - 1828841*v^3 + 6865794*v^2 - 785896236*v - 41319936) / 3719628864 $$\beta_{4}$$ $$=$$ $$( -97909\nu^{5} - 629099\nu^{4} - 41343799\nu^{3} + 155211966\nu^{2} - 24361417764\nu - 934101504 ) / 1859814432$$ (-97909*v^5 - 629099*v^4 - 41343799*v^3 + 155211966*v^2 - 24361417764*v - 934101504) / 1859814432 $$\beta_{5}$$ $$=$$ $$( 710\nu^{5} + 4339\nu^{4} + 299810\nu^{3} - 1125540\nu^{2} + 97141857\nu + 6773760 ) / 12915378$$ (710*v^5 + 4339*v^4 + 299810*v^3 - 1125540*v^2 + 97141857*v + 6773760) / 12915378
 $$\nu$$ $$=$$ $$( -\beta_{5} - \beta_{4} - 2\beta_{3} ) / 6$$ (-b5 - b4 - 2*b3) / 6 $$\nu^{2}$$ $$=$$ $$( -25\beta_{5} + 11\beta_{4} - 1706\beta_{3} + 11\beta_{2} - 25\beta _1 - 1706 ) / 6$$ (-25*b5 + 11*b4 - 1706*b3 + 11*b2 - 25*b1 - 1706) / 6 $$\nu^{3}$$ $$=$$ $$( -415\beta_{2} - 451\beta _1 + 9538 ) / 6$$ (-415*b2 - 451*b1 + 9538) / 6 $$\nu^{4}$$ $$=$$ $$( 9085\beta_{5} - 6287\beta_{4} + 713186\beta_{3} ) / 6$$ (9085*b5 - 6287*b4 + 713186*b3) / 6 $$\nu^{5}$$ $$=$$ $$( 150811\beta_{5} + 192679\beta_{4} - 6789298\beta_{3} + 192679\beta_{2} + 150811\beta _1 - 6789298 ) / 6$$ (150811*b5 + 192679*b4 - 6789298*b3 + 192679*b2 + 150811*b1 - 6789298) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{3}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 2.48064 + 4.29659i −11.1179 − 19.2567i 9.13724 + 15.8262i 2.48064 − 4.29659i −11.1179 + 19.2567i 9.13724 − 15.8262i
−20.9010 36.2015i 0.116170 0.201212i −617.701 + 1069.89i −895.944 1551.82i −9.71222 0 30239.6 9841.47 + 17045.9i −37452.2 + 64869.1i
18.2 −6.68036 11.5707i 81.7073 141.521i 166.746 288.812i 961.094 + 1664.66i −2183.34 0 −11296.4 −3510.66 6080.64i 12840.9 22241.1i
18.3 17.0813 + 29.5857i −39.8234 + 68.9762i −327.544 + 567.323i 711.850 + 1232.96i −2720.95 0 −4888.28 6669.69 + 11552.2i −24318.7 + 42121.2i
30.1 −20.9010 + 36.2015i 0.116170 + 0.201212i −617.701 1069.89i −895.944 + 1551.82i −9.71222 0 30239.6 9841.47 17045.9i −37452.2 64869.1i
30.2 −6.68036 + 11.5707i 81.7073 + 141.521i 166.746 + 288.812i 961.094 1664.66i −2183.34 0 −11296.4 −3510.66 + 6080.64i 12840.9 + 22241.1i
30.3 17.0813 29.5857i −39.8234 68.9762i −327.544 567.323i 711.850 1232.96i −2720.95 0 −4888.28 6669.69 11552.2i −24318.7 42121.2i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 30.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.10.c.e 6
7.b odd 2 1 49.10.c.d 6
7.c even 3 1 49.10.a.c 3
7.c even 3 1 inner 49.10.c.e 6
7.d odd 6 1 7.10.a.b 3
7.d odd 6 1 49.10.c.d 6
21.g even 6 1 63.10.a.e 3
28.f even 6 1 112.10.a.h 3
35.i odd 6 1 175.10.a.d 3
35.k even 12 2 175.10.b.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.b 3 7.d odd 6 1
49.10.a.c 3 7.c even 3 1
49.10.c.d 6 7.b odd 2 1
49.10.c.d 6 7.d odd 6 1
49.10.c.e 6 1.a even 1 1 trivial
49.10.c.e 6 7.c even 3 1 inner
63.10.a.e 3 21.g even 6 1
112.10.a.h 3 28.f even 6 1
175.10.a.d 3 35.i odd 6 1
175.10.b.d 6 35.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{6} + 21T_{2}^{5} + 1767T_{2}^{4} + 10314T_{2}^{3} + 2158956T_{2}^{2} + 25300080T_{2} + 364046400$$ T2^6 + 21*T2^5 + 1767*T2^4 + 10314*T2^3 + 2158956*T2^2 + 25300080*T2 + 364046400 $$T_{3}^{6} - 84T_{3}^{5} + 20052T_{3}^{4} + 1085616T_{3}^{3} + 169150032T_{3}^{2} - 39299904T_{3} + 9144576$$ T3^6 - 84*T3^5 + 20052*T3^4 + 1085616*T3^3 + 169150032*T3^2 - 39299904*T3 + 9144576

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 21 T^{5} + \cdots + 364046400$$
$3$ $$T^{6} - 84 T^{5} + 20052 T^{4} + \cdots + 9144576$$
$5$ $$T^{6} - 1554 T^{5} + \cdots + 24\!\cdots\!00$$
$7$ $$T^{6}$$
$11$ $$T^{6} - 3444 T^{5} + \cdots + 11\!\cdots\!04$$
$13$ $$(T^{3} - 19782 T^{2} + \cdots + 41548412541440)^{2}$$
$17$ $$T^{6} - 1016694 T^{5} + \cdots + 48\!\cdots\!24$$
$19$ $$T^{6} - 222852 T^{5} + \cdots + 18\!\cdots\!00$$
$23$ $$T^{6} + 1885632 T^{5} + \cdots + 94\!\cdots\!96$$
$29$ $$(T^{3} - 4081818 T^{2} + \cdots + 44\!\cdots\!00)^{2}$$
$31$ $$T^{6} - 2869440 T^{5} + \cdots + 55\!\cdots\!56$$
$37$ $$T^{6} + 1395618 T^{5} + \cdots + 11\!\cdots\!84$$
$41$ $$(T^{3} - 14420658 T^{2} + \cdots + 19\!\cdots\!12)^{2}$$
$43$ $$(T^{3} + 61631172 T^{2} + \cdots + 68\!\cdots\!80)^{2}$$
$47$ $$T^{6} + 10368960 T^{5} + \cdots + 19\!\cdots\!56$$
$53$ $$T^{6} + 67502610 T^{5} + \cdots + 57\!\cdots\!84$$
$59$ $$T^{6} + 42590100 T^{5} + \cdots + 17\!\cdots\!00$$
$61$ $$T^{6} - 191746842 T^{5} + \cdots + 26\!\cdots\!64$$
$67$ $$T^{6} - 255175788 T^{5} + \cdots + 41\!\cdots\!96$$
$71$ $$(T^{3} - 296514504 T^{2} + \cdots + 16\!\cdots\!80)^{2}$$
$73$ $$T^{6} - 344213310 T^{5} + \cdots + 38\!\cdots\!04$$
$79$ $$T^{6} - 960412656 T^{5} + \cdots + 12\!\cdots\!00$$
$83$ $$(T^{3} - 1100517180 T^{2} + \cdots - 18\!\cdots\!48)^{2}$$
$89$ $$T^{6} - 506816478 T^{5} + \cdots + 39\!\cdots\!00$$
$97$ $$(T^{3} - 647498250 T^{2} + \cdots + 49\!\cdots\!16)^{2}$$
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