Properties

 Label 49.10.a.c Level $49$ Weight $10$ Character orbit 49.a Self dual yes Analytic conductor $25.237$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$25.2367559720$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 426x + 2016$$ x^3 - x^2 - 426*x + 2016 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 7) q^{2} + (\beta_{2} + \beta_1 - 28) q^{3} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + ( - 43 \beta_{2} + 13 \beta_1 - 518) q^{5} + ( - 36 \beta_{2} + 6 \beta_1 - 1638) q^{6} + ( - 470 \beta_{2} + 147 \beta_1 + 4685) q^{8} + (90 \beta_{2} - 126 \beta_1 - 8667) q^{9}+O(q^{10})$$ q + (-b2 + 7) * q^2 + (b2 + b1 - 28) * q^3 + (-8*b2 + 7*b1 + 519) * q^4 + (-43*b2 + 13*b1 - 518) * q^5 + (-36*b2 + 6*b1 - 1638) * q^6 + (-470*b2 + 147*b1 + 4685) * q^8 + (90*b2 - 126*b1 - 8667) * q^9 $$q + ( - \beta_{2} + 7) q^{2} + (\beta_{2} + \beta_1 - 28) q^{3} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + ( - 43 \beta_{2} + 13 \beta_1 - 518) q^{5} + ( - 36 \beta_{2} + 6 \beta_1 - 1638) q^{6} + ( - 470 \beta_{2} + 147 \beta_1 + 4685) q^{8} + (90 \beta_{2} - 126 \beta_1 - 8667) q^{9} + ( - 370 \beta_{2} + 470 \beta_1 + 32620) q^{10} + (650 \beta_{2} + 658 \beta_1 - 1148) q^{11} + (700 \beta_{2} - 182 \beta_1 + 35462) q^{12} + ( - 3017 \beta_{2} + 175 \beta_1 + 6594) q^{13} + ( - 1392 \beta_{2} - 1848 \beta_1 + 66768) q^{15} + ( - 10614 \beta_{2} + 1617 \beta_1 + 160987) q^{16} + ( - 3030 \beta_{2} - 1574 \beta_1 - 338898) q^{17} + (16947 \beta_{2} - 2268 \beta_1 - 91089) q^{18} + (15371 \beta_{2} - 2437 \beta_1 - 74284) q^{19} + ( - 41524 \beta_{2} + 2044 \beta_1 + 640696) q^{20} + ( - 40972 \beta_{2} + 4004 \beta_1 - 949016) q^{22} + ( - 24200 \beta_{2} - 3808 \beta_1 + 628544) q^{23} + ( - 4500 \beta_{2} - 10338 \beta_1 + 483210) q^{24} + ( - 15338 \beta_{2} - 4802 \beta_1 + 1024407) q^{25} + ( - 20986 \beta_{2} + 23394 \beta_1 + 2928352) q^{26} + ( - 33930 \beta_{2} - 15786 \beta_1 - 183960) q^{27} + (54866 \beta_{2} + 18914 \beta_1 + 1360606) q^{29} + (51960 \beta_{2} - 14280 \beta_1 + 2684400) q^{30} + ( - 55698 \beta_{2} + 70302 \beta_1 - 956480) q^{31} + ( - 36066 \beta_{2} + 20055 \beta_1 + 8407317) q^{32} + (76152 \beta_{2} - 65640 \beta_1 + 6753264) q^{33} + (438178 \beta_{2} + 748 \beta_1 + 1327214) q^{34} + (209376 \beta_{2} - 83601 \beta_1 - 11798793) q^{36} + (209418 \beta_{2} + 60522 \beta_1 + 465206) q^{37} + (248060 \beta_{2} - 139278 \beta_1 - 14493290) q^{38} + ( - 110012 \beta_{2} - 13748 \beta_1 - 2996896) q^{39} + ( - 625640 \beta_{2} + 76600 \beta_1 + 27619760) q^{40} + (163478 \beta_{2} + 131894 \beta_1 + 4806886) q^{41} + (121982 \beta_{2} + 65366 \beta_1 - 20543724) q^{43} + (314984 \beta_{2} + 1960 \beta_1 + 32337328) q^{44} + (714681 \beta_{2} - 7551 \beta_1 - 9924894) q^{45} + ( - 405224 \beta_{2} + 119896 \beta_1 + 29915888) q^{46} + (534778 \beta_{2} - 83238 \beta_1 + 3456320) q^{47} + ( - 174140 \beta_{2} - 9710 \beta_1 - 5599594) q^{48} + ( - 727615 \beta_{2} + 44940 \beta_1 + 24441685) q^{50} + ( - 587238 \beta_{2} - 186102 \beta_1 - 8715576) q^{51} + ( - 2925244 \beta_{2} + 361424 \beta_1 + 26969348) q^{52} + (1553376 \beta_{2} - 450352 \beta_1 + 22500870) q^{53} + (1176120 \beta_{2} + 32292 \beta_1 + 39293100) q^{54} + ( - 1659356 \beta_{2} - 988364 \beta_1 + 35274344) q^{55} + (403894 \beta_{2} + 182350 \beta_1 + 2823704) q^{57} + ( - 2535150 \beta_{2} - 138180 \beta_1 - 53054610) q^{58} + (2231195 \beta_{2} + 49659 \beta_1 + 14196700) q^{59} + ( - 991536 \beta_{2} + 396816 \beta_1 - 59850336) q^{60} + ( - 589107 \beta_{2} + 844773 \beta_1 - 63915614) q^{61} + ( - 3668848 \beta_{2} + 1303812 \beta_1 + 15661156) q^{62} + ( - 4312590 \beta_{2} - 314727 \beta_1 + 2617387) q^{64} + ( - 958090 \beta_{2} + 1035790 \beta_1 + 121427740) q^{65} + ( - 2410512 \beta_{2} - 1386384 \beta_1 + 2685984) q^{66} + (3939816 \beta_{2} + 47712 \beta_1 - 85058596) q^{67} + (613704 \beta_{2} - 2251634 \beta_1 - 247828602) q^{68} + ( - 697656 \beta_{2} + 981336 \beta_1 - 85967952) q^{69} + (3499356 \beta_{2} - 526260 \beta_1 + 98838168) q^{71} + (8765370 \beta_{2} - 1391229 \beta_1 - 203104755) q^{72} + (9544844 \beta_{2} - 118516 \beta_1 - 114737770) q^{73} + ( - 4189718 \beta_{2} - 679140 \beta_1 - 230232154) q^{74} + (4723 \beta_{2} + 1484467 \beta_1 - 93010372) q^{75} + (15924468 \beta_{2} - 2299290 \beta_1 - 242946662) q^{76} + (3780504 \beta_{2} + 591360 \beta_1 + 93377592) q^{78} + ( - 7475532 \beta_{2} + 1679412 \beta_1 - 320137552) q^{79} + ( - 11964112 \beta_{2} + 4328752 \beta_1 + 444444448) q^{80} + ( - 4598370 \beta_{2} + 3824982 \beta_1 - 11942559) q^{81} + ( - 13216518 \beta_{2} + 570276 \beta_1 - 187558434) q^{82} + (4479559 \beta_{2} + 1977367 \beta_1 + 366839060) q^{83} + (18558254 \beta_{2} - 1518034 \beta_1 + 146059804) q^{85} + (16416916 \beta_{2} - 4116 \beta_1 - 293660752) q^{86} + (5138394 \beta_{2} - 457014 \beta_1 + 207273864) q^{87} + ( - 11172080 \beta_{2} - 4229456 \beta_1 + 402041600) q^{88} + (10612104 \beta_{2} - 1815976 \beta_1 - 168938826) q^{89} + (11130390 \beta_{2} - 5100930 \beta_1 - 767817540) q^{90} + ( - 25723952 \beta_{2} + 6344912 \beta_1 + 230374496) q^{92} + (1921156 \beta_{2} - 7972076 \beta_1 + 564419504) q^{93} + (2488928 \beta_{2} - 4825540 \beta_1 - 462668276) q^{94} + (10749416 \beta_{2} - 4098416 \beta_1 - 734357024) q^{95} + (8360604 \beta_{2} + 6385806 \beta_1 - 111128598) q^{96} + ( - 33276782 \beta_{2} + 864850 \beta_1 + 215832750) q^{97} + ( - 7689150 \beta_{2} + 376362 \beta_1 - 633659724) q^{99}+O(q^{100})$$ q + (-b2 + 7) * q^2 + (b2 + b1 - 28) * q^3 + (-8*b2 + 7*b1 + 519) * q^4 + (-43*b2 + 13*b1 - 518) * q^5 + (-36*b2 + 6*b1 - 1638) * q^6 + (-470*b2 + 147*b1 + 4685) * q^8 + (90*b2 - 126*b1 - 8667) * q^9 + (-370*b2 + 470*b1 + 32620) * q^10 + (650*b2 + 658*b1 - 1148) * q^11 + (700*b2 - 182*b1 + 35462) * q^12 + (-3017*b2 + 175*b1 + 6594) * q^13 + (-1392*b2 - 1848*b1 + 66768) * q^15 + (-10614*b2 + 1617*b1 + 160987) * q^16 + (-3030*b2 - 1574*b1 - 338898) * q^17 + (16947*b2 - 2268*b1 - 91089) * q^18 + (15371*b2 - 2437*b1 - 74284) * q^19 + (-41524*b2 + 2044*b1 + 640696) * q^20 + (-40972*b2 + 4004*b1 - 949016) * q^22 + (-24200*b2 - 3808*b1 + 628544) * q^23 + (-4500*b2 - 10338*b1 + 483210) * q^24 + (-15338*b2 - 4802*b1 + 1024407) * q^25 + (-20986*b2 + 23394*b1 + 2928352) * q^26 + (-33930*b2 - 15786*b1 - 183960) * q^27 + (54866*b2 + 18914*b1 + 1360606) * q^29 + (51960*b2 - 14280*b1 + 2684400) * q^30 + (-55698*b2 + 70302*b1 - 956480) * q^31 + (-36066*b2 + 20055*b1 + 8407317) * q^32 + (76152*b2 - 65640*b1 + 6753264) * q^33 + (438178*b2 + 748*b1 + 1327214) * q^34 + (209376*b2 - 83601*b1 - 11798793) * q^36 + (209418*b2 + 60522*b1 + 465206) * q^37 + (248060*b2 - 139278*b1 - 14493290) * q^38 + (-110012*b2 - 13748*b1 - 2996896) * q^39 + (-625640*b2 + 76600*b1 + 27619760) * q^40 + (163478*b2 + 131894*b1 + 4806886) * q^41 + (121982*b2 + 65366*b1 - 20543724) * q^43 + (314984*b2 + 1960*b1 + 32337328) * q^44 + (714681*b2 - 7551*b1 - 9924894) * q^45 + (-405224*b2 + 119896*b1 + 29915888) * q^46 + (534778*b2 - 83238*b1 + 3456320) * q^47 + (-174140*b2 - 9710*b1 - 5599594) * q^48 + (-727615*b2 + 44940*b1 + 24441685) * q^50 + (-587238*b2 - 186102*b1 - 8715576) * q^51 + (-2925244*b2 + 361424*b1 + 26969348) * q^52 + (1553376*b2 - 450352*b1 + 22500870) * q^53 + (1176120*b2 + 32292*b1 + 39293100) * q^54 + (-1659356*b2 - 988364*b1 + 35274344) * q^55 + (403894*b2 + 182350*b1 + 2823704) * q^57 + (-2535150*b2 - 138180*b1 - 53054610) * q^58 + (2231195*b2 + 49659*b1 + 14196700) * q^59 + (-991536*b2 + 396816*b1 - 59850336) * q^60 + (-589107*b2 + 844773*b1 - 63915614) * q^61 + (-3668848*b2 + 1303812*b1 + 15661156) * q^62 + (-4312590*b2 - 314727*b1 + 2617387) * q^64 + (-958090*b2 + 1035790*b1 + 121427740) * q^65 + (-2410512*b2 - 1386384*b1 + 2685984) * q^66 + (3939816*b2 + 47712*b1 - 85058596) * q^67 + (613704*b2 - 2251634*b1 - 247828602) * q^68 + (-697656*b2 + 981336*b1 - 85967952) * q^69 + (3499356*b2 - 526260*b1 + 98838168) * q^71 + (8765370*b2 - 1391229*b1 - 203104755) * q^72 + (9544844*b2 - 118516*b1 - 114737770) * q^73 + (-4189718*b2 - 679140*b1 - 230232154) * q^74 + (4723*b2 + 1484467*b1 - 93010372) * q^75 + (15924468*b2 - 2299290*b1 - 242946662) * q^76 + (3780504*b2 + 591360*b1 + 93377592) * q^78 + (-7475532*b2 + 1679412*b1 - 320137552) * q^79 + (-11964112*b2 + 4328752*b1 + 444444448) * q^80 + (-4598370*b2 + 3824982*b1 - 11942559) * q^81 + (-13216518*b2 + 570276*b1 - 187558434) * q^82 + (4479559*b2 + 1977367*b1 + 366839060) * q^83 + (18558254*b2 - 1518034*b1 + 146059804) * q^85 + (16416916*b2 - 4116*b1 - 293660752) * q^86 + (5138394*b2 - 457014*b1 + 207273864) * q^87 + (-11172080*b2 - 4229456*b1 + 402041600) * q^88 + (10612104*b2 - 1815976*b1 - 168938826) * q^89 + (11130390*b2 - 5100930*b1 - 767817540) * q^90 + (-25723952*b2 + 6344912*b1 + 230374496) * q^92 + (1921156*b2 - 7972076*b1 + 564419504) * q^93 + (2488928*b2 - 4825540*b1 - 462668276) * q^94 + (10749416*b2 - 4098416*b1 - 734357024) * q^95 + (8360604*b2 + 6385806*b1 - 111128598) * q^96 + (-33276782*b2 + 864850*b1 + 215832750) * q^97 + (-7689150*b2 + 376362*b1 - 633659724) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 21 q^{2} - 84 q^{3} + 1557 q^{4} - 1554 q^{5} - 4914 q^{6} + 14055 q^{8} - 26001 q^{9}+O(q^{10})$$ 3 * q + 21 * q^2 - 84 * q^3 + 1557 * q^4 - 1554 * q^5 - 4914 * q^6 + 14055 * q^8 - 26001 * q^9 $$3 q + 21 q^{2} - 84 q^{3} + 1557 q^{4} - 1554 q^{5} - 4914 q^{6} + 14055 q^{8} - 26001 q^{9} + 97860 q^{10} - 3444 q^{11} + 106386 q^{12} + 19782 q^{13} + 200304 q^{15} + 482961 q^{16} - 1016694 q^{17} - 273267 q^{18} - 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22} + 1885632 q^{23} + 1449630 q^{24} + 3073221 q^{25} + 8785056 q^{26} - 551880 q^{27} + 4081818 q^{29} + 8053200 q^{30} - 2869440 q^{31} + 25221951 q^{32} + 20259792 q^{33} + 3981642 q^{34} - 35396379 q^{36} + 1395618 q^{37} - 43479870 q^{38} - 8990688 q^{39} + 82859280 q^{40} + 14420658 q^{41} - 61631172 q^{43} + 97011984 q^{44} - 29774682 q^{45} + 89747664 q^{46} + 10368960 q^{47} - 16798782 q^{48} + 73325055 q^{50} - 26146728 q^{51} + 80908044 q^{52} + 67502610 q^{53} + 117879300 q^{54} + 105823032 q^{55} + 8471112 q^{57} - 159163830 q^{58} + 42590100 q^{59} - 179551008 q^{60} - 191746842 q^{61} + 46983468 q^{62} + 7852161 q^{64} + 364283220 q^{65} + 8057952 q^{66} - 255175788 q^{67} - 743485806 q^{68} - 257903856 q^{69} + 296514504 q^{71} - 609314265 q^{72} - 344213310 q^{73} - 690696462 q^{74} - 279031116 q^{75} - 728839986 q^{76} + 280132776 q^{78} - 960412656 q^{79} + 1333333344 q^{80} - 35827677 q^{81} - 562675302 q^{82} + 1100517180 q^{83} + 438179412 q^{85} - 880982256 q^{86} + 621821592 q^{87} + 1206124800 q^{88} - 506816478 q^{89} - 2303452620 q^{90} + 691123488 q^{92} + 1693258512 q^{93} - 1388004828 q^{94} - 2203071072 q^{95} - 333385794 q^{96} + 647498250 q^{97} - 1900979172 q^{99}+O(q^{100})$$ 3 * q + 21 * q^2 - 84 * q^3 + 1557 * q^4 - 1554 * q^5 - 4914 * q^6 + 14055 * q^8 - 26001 * q^9 + 97860 * q^10 - 3444 * q^11 + 106386 * q^12 + 19782 * q^13 + 200304 * q^15 + 482961 * q^16 - 1016694 * q^17 - 273267 * q^18 - 222852 * q^19 + 1922088 * q^20 - 2847048 * q^22 + 1885632 * q^23 + 1449630 * q^24 + 3073221 * q^25 + 8785056 * q^26 - 551880 * q^27 + 4081818 * q^29 + 8053200 * q^30 - 2869440 * q^31 + 25221951 * q^32 + 20259792 * q^33 + 3981642 * q^34 - 35396379 * q^36 + 1395618 * q^37 - 43479870 * q^38 - 8990688 * q^39 + 82859280 * q^40 + 14420658 * q^41 - 61631172 * q^43 + 97011984 * q^44 - 29774682 * q^45 + 89747664 * q^46 + 10368960 * q^47 - 16798782 * q^48 + 73325055 * q^50 - 26146728 * q^51 + 80908044 * q^52 + 67502610 * q^53 + 117879300 * q^54 + 105823032 * q^55 + 8471112 * q^57 - 159163830 * q^58 + 42590100 * q^59 - 179551008 * q^60 - 191746842 * q^61 + 46983468 * q^62 + 7852161 * q^64 + 364283220 * q^65 + 8057952 * q^66 - 255175788 * q^67 - 743485806 * q^68 - 257903856 * q^69 + 296514504 * q^71 - 609314265 * q^72 - 344213310 * q^73 - 690696462 * q^74 - 279031116 * q^75 - 728839986 * q^76 + 280132776 * q^78 - 960412656 * q^79 + 1333333344 * q^80 - 35827677 * q^81 - 562675302 * q^82 + 1100517180 * q^83 + 438179412 * q^85 - 880982256 * q^86 + 621821592 * q^87 + 1206124800 * q^88 - 506816478 * q^89 - 2303452620 * q^90 + 691123488 * q^92 + 1693258512 * q^93 - 1388004828 * q^94 - 2203071072 * q^95 - 333385794 * q^96 + 647498250 * q^97 - 1900979172 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 426x + 2016$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{2} + 25\nu + 276 ) / 6$$ (-v^2 + 25*v + 276) / 6 $$\beta_{2}$$ $$=$$ $$( \nu^{2} + 11\nu - 288 ) / 6$$ (v^2 + 11*v - 288) / 6
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 2 ) / 6$$ (b2 + b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( 25\beta_{2} - 11\beta _1 + 1706 ) / 6$$ (25*b2 - 11*b1 + 1706) / 6

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 18.2745 −22.2358 4.96128
−34.1627 79.6469 655.088 −1423.70 −2720.95 0 −4888.28 −13339.4 48637.4
1.2 13.3607 −163.415 −333.491 −1922.19 −2183.34 0 −11296.4 7021.32 −25681.8
1.3 41.8019 −0.232339 1235.40 1791.89 −9.71222 0 30239.6 −19682.9 74904.4
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$

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 49.10.a.c 3
7.b odd 2 1 7.10.a.b 3
7.c even 3 2 49.10.c.e 6
7.d odd 6 2 49.10.c.d 6
21.c even 2 1 63.10.a.e 3
28.d even 2 1 112.10.a.h 3
35.c odd 2 1 175.10.a.d 3
35.f even 4 2 175.10.b.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.b 3 7.b odd 2 1
49.10.a.c 3 1.a even 1 1 trivial
49.10.c.d 6 7.d odd 6 2
49.10.c.e 6 7.c even 3 2
63.10.a.e 3 21.c even 2 1
112.10.a.h 3 28.d even 2 1
175.10.a.d 3 35.c odd 2 1
175.10.b.d 6 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2}^{3} - 21T_{2}^{2} - 1326T_{2} + 19080$$ T2^3 - 21*T2^2 - 1326*T2 + 19080 $$T_{3}^{3} + 84T_{3}^{2} - 12996T_{3} - 3024$$ T3^3 + 84*T3^2 - 12996*T3 - 3024

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 21 T^{2} - 1326 T + 19080$$
$3$ $$T^{3} + 84 T^{2} - 12996 T - 3024$$
$5$ $$T^{3} + 1554 T^{2} + \cdots - 4903718400$$
$7$ $$T^{3}$$
$11$ $$T^{3} + \cdots + 108859759460352$$
$13$ $$T^{3} - 19782 T^{2} + \cdots + 41548412541440$$
$17$ $$T^{3} + 1016694 T^{2} + \cdots + 21\!\cdots\!32$$
$19$ $$T^{3} + 222852 T^{2} + \cdots + 43\!\cdots\!60$$
$23$ $$T^{3} - 1885632 T^{2} + \cdots + 97\!\cdots\!36$$
$29$ $$T^{3} - 4081818 T^{2} + \cdots + 44\!\cdots\!00$$
$31$ $$T^{3} + 2869440 T^{2} + \cdots + 74\!\cdots\!84$$
$37$ $$T^{3} - 1395618 T^{2} + \cdots - 34\!\cdots\!28$$
$41$ $$T^{3} - 14420658 T^{2} + \cdots + 19\!\cdots\!12$$
$43$ $$T^{3} + 61631172 T^{2} + \cdots + 68\!\cdots\!80$$
$47$ $$T^{3} - 10368960 T^{2} + \cdots + 43\!\cdots\!16$$
$53$ $$T^{3} - 67502610 T^{2} + \cdots + 23\!\cdots\!28$$
$59$ $$T^{3} - 42590100 T^{2} + \cdots - 42\!\cdots\!00$$
$61$ $$T^{3} + 191746842 T^{2} + \cdots - 51\!\cdots\!08$$
$67$ $$T^{3} + 255175788 T^{2} + \cdots - 20\!\cdots\!64$$
$71$ $$T^{3} - 296514504 T^{2} + \cdots + 16\!\cdots\!80$$
$73$ $$T^{3} + 344213310 T^{2} + \cdots - 19\!\cdots\!48$$
$79$ $$T^{3} + 960412656 T^{2} + \cdots - 11\!\cdots\!00$$
$83$ $$T^{3} - 1100517180 T^{2} + \cdots - 18\!\cdots\!48$$
$89$ $$T^{3} + 506816478 T^{2} + \cdots - 19\!\cdots\!40$$
$97$ $$T^{3} - 647498250 T^{2} + \cdots + 49\!\cdots\!16$$