# Properties

 Label 5.16.a.b Level $5$ Weight $16$ Character orbit 5.a Self dual yes Analytic conductor $7.135$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 5.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.13467525500$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 1972x + 21070$$ x^3 - x^2 - 1972*x + 21070 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 5$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (8 \beta_{2} - 16 \beta_1 + 1170) q^{3} + ( - 9 \beta_{2} - 143 \beta_1 + 6744) q^{4} - 78125 q^{5} + ( - 1256 \beta_{2} - 3362 \beta_1 + 690186) q^{6} + (4968 \beta_{2} + 10960 \beta_1 - 296426) q^{7} + ( - 36 \beta_{2} + 5628 \beta_1 + 5560104) q^{8} + ( - 18848 \beta_{2} - 10496 \beta_1 + 15960729) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (8*b2 - 16*b1 + 1170) * q^3 + (-9*b2 - 143*b1 + 6744) * q^4 - 78125 * q^5 + (-1256*b2 - 3362*b1 + 690186) * q^6 + (4968*b2 + 10960*b1 - 296426) * q^7 + (-36*b2 + 5628*b1 + 5560104) * q^8 + (-18848*b2 - 10496*b1 + 15960729) * q^9 $$q + ( - \beta_1 + 1) q^{2} + (8 \beta_{2} - 16 \beta_1 + 1170) q^{3} + ( - 9 \beta_{2} - 143 \beta_1 + 6744) q^{4} - 78125 q^{5} + ( - 1256 \beta_{2} - 3362 \beta_1 + 690186) q^{6} + (4968 \beta_{2} + 10960 \beta_1 - 296426) q^{7} + ( - 36 \beta_{2} + 5628 \beta_1 + 5560104) q^{8} + ( - 18848 \beta_{2} - 10496 \beta_1 + 15960729) q^{9} + (78125 \beta_1 - 78125) q^{10} + (55440 \beta_{2} - 161920 \beta_1 + 40922992) q^{11} + ( - 117818 \beta_{2} - 655862 \beta_1 + 86263728) q^{12} + (468000 \beta_{2} + 311168 \beta_1 - 30044118) q^{13} + ( - 591912 \beta_{2} + 1902426 \beta_1 - 398039346) q^{14} + ( - 625000 \beta_{2} + 1250000 \beta_1 - 91406250) q^{15} + (350568 \beta_{2} - 75464 \beta_1 - 438050976) q^{16} + (1907424 \beta_{2} - 1452928 \beta_1 - 1289506310) q^{17} + (2525408 \beta_{2} - 17639641 \beta_1 + 296753145) q^{18} + ( - 7803936 \beta_{2} - 3098272 \beta_1 + 1219994004) q^{19} + (703125 \beta_{2} + 11171875 \beta_1 - 526875000) q^{20} + (6523968 \beta_{2} + 79767936 \beta_1 + 3227700300) q^{21} + ( - 9163440 \beta_{2} - 63361232 \beta_1 + 6832445312) q^{22} + ( - 11330568 \beta_{2} + 18859632 \beta_1 + 8901862434) q^{23} + (51630552 \beta_{2} - 70408296 \beta_1 + 2546915472) q^{24} + 6103515625 q^{25} + ( - 62251488 \beta_{2} + 78909974 \beta_1 - 8999462966) q^{26} + (18437936 \beta_{2} - 206415328 \beta_1 - 33093483204) q^{27} + ( - 63393822 \beta_{2} + 303127438 \beta_1 - 70057040624) q^{28} + (15846336 \beta_{2} - 449184128 \beta_1 + 48370531646) q^{29} + (98125000 \beta_{2} + 262656250 \beta_1 - 53920781250) q^{30} + (108915120 \beta_{2} + 471103840 \beta_1 - 8431121228) q^{31} + ( - 48228480 \beta_{2} + 246422464 \beta_1 - 177159095104) q^{32} + (67387936 \beta_{2} - 768555392 \beta_1 + 283606577760) q^{33} + ( - 278208288 \beta_{2} + 1102264774 \beta_1 + 69669379418) q^{34} + ( - 388125000 \beta_{2} - 856250000 \beta_1 + 23158281250) q^{35} + (107822783 \beta_{2} - 2432395159 \beta_1 + 192198464664) q^{36} + (163094976 \beta_{2} + 906686976 \beta_1 + 140183343962) q^{37} + (1056862656 \beta_{2} - 1737987988 \beta_1 + 68188853716) q^{38} + ( - 314100592 \beta_{2} + 5134834016 \beta_1 + 798562731252) q^{39} + (2812500 \beta_{2} - 439687500 \beta_1 - 434383125000) q^{40} + ( - 191270880 \beta_{2} - 3845980160 \beta_1 + 89989506422) q^{41} + ( - 188920128 \beta_{2} + 8164586292 \beta_1 - 3102130426356) q^{42} + ( - 1476432504 \beta_{2} + 5073312560 \beta_1 + 784554249738) q^{43} + ( - 1113190848 \beta_{2} - 10615580096 \beta_1 + 1104227239808) q^{44} + (1472500000 \beta_{2} + 820000000 \beta_1 - 1246931953125) q^{45} + (1744685640 \beta_{2} - 6337100370 \beta_1 - 816764743158) q^{46} + (801928872 \beta_{2} - 13707131632 \beta_1 - 2967991804082) q^{47} + ( - 3949661168 \beta_{2} + 9462698032 \beta_1 + 324594331584) q^{48} + (11441544480 \beta_{2} - 1200221440 \beta_1 + 5871319044973) q^{49} + ( - 6103515625 \beta_1 + 6103515625) q^{50} + ( - 14056160624 \beta_{2} + \cdots + 3764158749588) q^{51}+ \cdots + (434642042384 \beta_{2} + \cdots + 425085777531888) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (8*b2 - 16*b1 + 1170) * q^3 + (-9*b2 - 143*b1 + 6744) * q^4 - 78125 * q^5 + (-1256*b2 - 3362*b1 + 690186) * q^6 + (4968*b2 + 10960*b1 - 296426) * q^7 + (-36*b2 + 5628*b1 + 5560104) * q^8 + (-18848*b2 - 10496*b1 + 15960729) * q^9 + (78125*b1 - 78125) * q^10 + (55440*b2 - 161920*b1 + 40922992) * q^11 + (-117818*b2 - 655862*b1 + 86263728) * q^12 + (468000*b2 + 311168*b1 - 30044118) * q^13 + (-591912*b2 + 1902426*b1 - 398039346) * q^14 + (-625000*b2 + 1250000*b1 - 91406250) * q^15 + (350568*b2 - 75464*b1 - 438050976) * q^16 + (1907424*b2 - 1452928*b1 - 1289506310) * q^17 + (2525408*b2 - 17639641*b1 + 296753145) * q^18 + (-7803936*b2 - 3098272*b1 + 1219994004) * q^19 + (703125*b2 + 11171875*b1 - 526875000) * q^20 + (6523968*b2 + 79767936*b1 + 3227700300) * q^21 + (-9163440*b2 - 63361232*b1 + 6832445312) * q^22 + (-11330568*b2 + 18859632*b1 + 8901862434) * q^23 + (51630552*b2 - 70408296*b1 + 2546915472) * q^24 + 6103515625 * q^25 + (-62251488*b2 + 78909974*b1 - 8999462966) * q^26 + (18437936*b2 - 206415328*b1 - 33093483204) * q^27 + (-63393822*b2 + 303127438*b1 - 70057040624) * q^28 + (15846336*b2 - 449184128*b1 + 48370531646) * q^29 + (98125000*b2 + 262656250*b1 - 53920781250) * q^30 + (108915120*b2 + 471103840*b1 - 8431121228) * q^31 + (-48228480*b2 + 246422464*b1 - 177159095104) * q^32 + (67387936*b2 - 768555392*b1 + 283606577760) * q^33 + (-278208288*b2 + 1102264774*b1 + 69669379418) * q^34 + (-388125000*b2 - 856250000*b1 + 23158281250) * q^35 + (107822783*b2 - 2432395159*b1 + 192198464664) * q^36 + (163094976*b2 + 906686976*b1 + 140183343962) * q^37 + (1056862656*b2 - 1737987988*b1 + 68188853716) * q^38 + (-314100592*b2 + 5134834016*b1 + 798562731252) * q^39 + (2812500*b2 - 439687500*b1 - 434383125000) * q^40 + (-191270880*b2 - 3845980160*b1 + 89989506422) * q^41 + (-188920128*b2 + 8164586292*b1 - 3102130426356) * q^42 + (-1476432504*b2 + 5073312560*b1 + 784554249738) * q^43 + (-1113190848*b2 - 10615580096*b1 + 1104227239808) * q^44 + (1472500000*b2 + 820000000*b1 - 1246931953125) * q^45 + (1744685640*b2 - 6337100370*b1 - 816764743158) * q^46 + (801928872*b2 - 13707131632*b1 - 2967991804082) * q^47 + (-3949661168*b2 + 9462698032*b1 + 324594331584) * q^48 + (11441544480*b2 - 1200221440*b1 + 5871319044973) * q^49 + (-6103515625*b1 + 6103515625) * q^50 + (-14056160624*b2 + 30495917152*b1 + 3764158749588) * q^51 + (-5972277402*b2 + 9385811370*b1 - 2584622609296) * q^52 + (5342178528*b2 - 1397858176*b1 + 2917729043890) * q^53 + (-4420611056*b2 + 3966885988*b1 + 8253584076684) * q^54 + (-4331250000*b2 + 12650000000*b1 - 3197108750000) * q^55 + (30935660616*b2 + 50128503432*b1 + 545614940976) * q^56 + (13632064352*b2 - 90133108096*b1 - 13915230943128) * q^57 + (-6245297856*b2 - 111996214462*b1 + 17908672830334) * q^58 + (-38237401728*b2 - 61570663456*b1 - 4757774834108) * q^59 + (9204531250*b2 + 51239218750*b1 - 6739353750000) * q^60 + (-14013100800*b2 + 54667750400*b1 - 12675394357398) * q^61 + (-10899267120*b2 + 76417017708*b1 - 17848373015868) * q^62 + (48851983464*b2 + 108361321296*b1 - 32320767959514) * q^63 + (-2565851328*b2 + 214141604544*b1 + 4097833960960) * q^64 + (-36562500000*b2 - 24310000000*b1 + 2347196718750) * q^65 + (-16283921632*b2 - 392067564064*b1 + 31128789956352) * q^66 + (27059954904*b2 + 3434774032*b1 + 48140710931470) * q^67 + (-13911134634*b2 + 131679680314*b1 - 3204041226256) * q^68 + (106361381184*b2 - 166797455232*b1 - 27954896024412) * q^69 + (46243125000*b2 - 148627031250*b1 + 31096823906250) * q^70 + (-12951392400*b2 + 185221511200*b1 - 42113355733348) * q^71 + (-119631492612*b2 + 41495406876*b1 + 87340637409768) * q^72 + (-186900192288*b2 - 38377998208*b1 + 66526497962514) * q^73 + (-14510018880*b2 - 9802843610*b1 - 34525135960294) * q^74 + (48828125000*b2 - 97656250000*b1 + 7141113281250) * q^75 + (93173573772*b2 - 202890344556*b1 + 36270077895392) * q^76 + (177770726496*b2 + 1008983437440*b1 - 7660022733792) * q^77 + (89873488432*b2 - 72557306900*b1 - 204315548781084) * q^78 + (27048434976*b2 - 431477536448*b1 - 24733997065064) * q^79 + (-27388125000*b2 + 5895625000*b1 + 34222732500000) * q^80 + (-273609701216*b2 + 131777037568*b1 - 84222063836475) * q^81 + (-8027169120*b2 - 638031397942*b1 + 150689532005782) * q^82 + (76077264744*b2 - 209783995344*b1 - 73080917645334) * q^83 + (-114036209004*b2 + 1645776751692*b1 - 432800660349408) * q^84 + (-149017500000*b2 + 113510000000*b1 + 100742680468750) * q^85 + (250883931096*b2 - 78908191258*b1 - 210157151249342) * q^86 + (-196619127952*b2 - 2154564884704*b1 + 401577393416028) * q^87 + (359460908928*b2 - 546550671744*b1 + 188741623454208) * q^88 + (-353173148352*b2 - 1597935151104*b1 - 120905615348022) * q^89 + (-197297500000*b2 + 1378096953125*b1 - 23183839453125) * q^90 + (636767825616*b2 + 1228007654432*b1 + 714067444892828) * q^91 + (71734844934*b2 - 683649074358*b1 - 29731828789200) * q^92 + (419328994176*b2 + 2552036114688*b1 - 90540549111000) * q^93 + (-234832297896*b2 + 1029598401058*b1 + 544312190743430) * q^94 + (609682500000*b2 + 242052500000*b1 - 95312031562500) * q^95 + (-1057662743296*b2 + 3286751220608*b1 - 485075736395904) * q^96 + (-272479015008*b2 + 1603946375808*b1 - 169292821431262) * q^97 + (-1601176675680*b2 - 5927335044653*b1 + 134585441891213) * q^98 + (434642042384*b2 - 4265681698432*b1 + 425085777531888) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{2} + 3518 q^{3} + 20384 q^{4} - 234375 q^{5} + 2075176 q^{6} - 905206 q^{7} + 16674720 q^{8} + 47911531 q^{9}+O(q^{10})$$ 3 * q + 4 * q^2 + 3518 * q^3 + 20384 * q^4 - 234375 * q^5 + 2075176 * q^6 - 905206 * q^7 + 16674720 * q^8 + 47911531 * q^9 $$3 q + 4 q^{2} + 3518 q^{3} + 20384 q^{4} - 234375 q^{5} + 2075176 q^{6} - 905206 q^{7} + 16674720 q^{8} + 47911531 q^{9} - 312500 q^{10} + 122875456 q^{11} + 259564864 q^{12} - 90911522 q^{13} - 1195428552 q^{14} - 274843750 q^{15} - 1314428032 q^{16} - 3868973426 q^{17} + 905373668 q^{18} + 3670884220 q^{19} - 1592500000 q^{20} + 9596808996 q^{21} + 20569860608 q^{22} + 26698058238 q^{23} + 7659524160 q^{24} + 18310546875 q^{25} - 27015047384 q^{26} - 99092472220 q^{27} - 210410855488 q^{28} + 145544932730 q^{29} - 162123125000 q^{30} - 25873382644 q^{31} - 531675479296 q^{32} + 851520900736 q^{33} + 208184081768 q^{34} + 70719218750 q^{35} + 578919966368 q^{36} + 419480249934 q^{37} + 205247686480 q^{38} + 2390867460332 q^{39} - 1302712500000 q^{40} + 274005770306 q^{41} - 9314366945232 q^{42} + 2350065869158 q^{43} + 3324410490368 q^{44} - 3743088359375 q^{45} - 2445701814744 q^{46} - 8891070209486 q^{47} + 968269957888 q^{48} + 17603715811879 q^{49} + 24414062500 q^{50} + 11276036492236 q^{51} - 7757281361856 q^{52} + 8749242811318 q^{53} + 24761205955120 q^{54} - 9599645000000 q^{55} + 1555780658880 q^{56} - 41669191785640 q^{57} + 53844260003320 q^{58} - 14173516437140 q^{59} - 20278505000000 q^{60} - 38066837721794 q^{61} - 53610636798192 q^{62} - 97119517183302 q^{63} + 12081926129664 q^{64} + 7102462656250 q^{65} + 93794721354752 q^{66} + 144391638065474 q^{67} - 9729892224448 q^{68} - 83804251999188 q^{69} + 93392855625000 q^{70} - 126512337318844 q^{71} + 262100048315040 q^{72} + 199804772078038 q^{73} - 103551095018392 q^{74} + 21472167968750 q^{75} + 108919950456960 q^{76} - 24166822365312 q^{77} - 612963962524784 q^{78} - 73797562093720 q^{79} + 102689690000000 q^{80} - 252524358845777 q^{81} + 452714654584408 q^{82} - 219109046205402 q^{83} - 12\!\cdots\!12 q^{84}+ \cdots + 12\!\cdots\!12 q^{99}+O(q^{100})$$ 3 * q + 4 * q^2 + 3518 * q^3 + 20384 * q^4 - 234375 * q^5 + 2075176 * q^6 - 905206 * q^7 + 16674720 * q^8 + 47911531 * q^9 - 312500 * q^10 + 122875456 * q^11 + 259564864 * q^12 - 90911522 * q^13 - 1195428552 * q^14 - 274843750 * q^15 - 1314428032 * q^16 - 3868973426 * q^17 + 905373668 * q^18 + 3670884220 * q^19 - 1592500000 * q^20 + 9596808996 * q^21 + 20569860608 * q^22 + 26698058238 * q^23 + 7659524160 * q^24 + 18310546875 * q^25 - 27015047384 * q^26 - 99092472220 * q^27 - 210410855488 * q^28 + 145544932730 * q^29 - 162123125000 * q^30 - 25873382644 * q^31 - 531675479296 * q^32 + 851520900736 * q^33 + 208184081768 * q^34 + 70719218750 * q^35 + 578919966368 * q^36 + 419480249934 * q^37 + 205247686480 * q^38 + 2390867460332 * q^39 - 1302712500000 * q^40 + 274005770306 * q^41 - 9314366945232 * q^42 + 2350065869158 * q^43 + 3324410490368 * q^44 - 3743088359375 * q^45 - 2445701814744 * q^46 - 8891070209486 * q^47 + 968269957888 * q^48 + 17603715811879 * q^49 + 24414062500 * q^50 + 11276036492236 * q^51 - 7757281361856 * q^52 + 8749242811318 * q^53 + 24761205955120 * q^54 - 9599645000000 * q^55 + 1555780658880 * q^56 - 41669191785640 * q^57 + 53844260003320 * q^58 - 14173516437140 * q^59 - 20278505000000 * q^60 - 38066837721794 * q^61 - 53610636798192 * q^62 - 97119517183302 * q^63 + 12081926129664 * q^64 + 7102462656250 * q^65 + 93794721354752 * q^66 + 144391638065474 * q^67 - 9729892224448 * q^68 - 83804251999188 * q^69 + 93392855625000 * q^70 - 126512337318844 * q^71 + 262100048315040 * q^72 + 199804772078038 * q^73 - 103551095018392 * q^74 + 21472167968750 * q^75 + 108919950456960 * q^76 - 24166822365312 * q^77 - 612963962524784 * q^78 - 73797562093720 * q^79 + 102689690000000 * q^80 - 252524358845777 * q^81 + 452714654584408 * q^82 - 219109046205402 * q^83 - 1299933721590912 * q^84 + 302263548906250 * q^85 - 630643429487864 * q^86 + 1207083364260740 * q^87 + 566411960125440 * q^88 - 360765737744610 * q^89 - 70732317812500 * q^90 + 2140337559198436 * q^91 - 88583572138176 * q^92 - 274593012441864 * q^93 + 1632141806127128 * q^94 - 286787829687500 * q^95 - 1457456297665024 * q^96 - 509209931654586 * q^97 + 411284837393972 * q^98 + 1279088372251712 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 1972x + 21070$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{2} + 78\nu + 2597 ) / 21$$ (-2*v^2 + 78*v + 2597) / 21 $$\beta_{2}$$ $$=$$ $$( 2\nu^{2} + 42\nu - 2645 ) / 3$$ (2*v^2 + 42*v - 2645) / 3
 $$\nu$$ $$=$$ $$( \beta_{2} + 7\beta _1 + 16 ) / 40$$ (b2 + 7*b1 + 16) / 40 $$\nu^{2}$$ $$=$$ $$( 39\beta_{2} - 147\beta _1 + 52564 ) / 40$$ (39*b2 - 147*b1 + 52564) / 40

## 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
 11.3631 38.1900 −48.5531
−152.575 −6379.22 −9488.76 −78125.0 973313. −1.77538e6 6.44734e6 2.63456e7 1.19200e7
1.2 −125.613 4146.67 −16989.4 −78125.0 −520875. 4.19779e6 6.25017e6 2.84597e6 9.81350e6
1.3 282.188 5750.55 46862.2 −78125.0 1.62274e6 −3.32761e6 3.97721e6 1.87200e7 −2.20460e7
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$

## 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 5.16.a.b 3
3.b odd 2 1 45.16.a.f 3
4.b odd 2 1 80.16.a.g 3
5.b even 2 1 25.16.a.c 3
5.c odd 4 2 25.16.b.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.16.a.b 3 1.a even 1 1 trivial
25.16.a.c 3 5.b even 2 1
25.16.b.c 6 5.c odd 4 2
45.16.a.f 3 3.b odd 2 1
80.16.a.g 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 4T_{2}^{2} - 59336T_{2} - 5408256$$ acting on $$S_{16}^{\mathrm{new}}(\Gamma_0(5))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 4 T^{2} - 59336 T - 5408256$$
$3$ $$T^{3} - 3518 T^{2} + \cdots + 152116768152$$
$5$ $$(T + 78125)^{3}$$
$7$ $$T^{3} + 905206 T^{2} + \cdots - 24\!\cdots\!16$$
$11$ $$T^{3} - 122875456 T^{2} + \cdots + 92\!\cdots\!92$$
$13$ $$T^{3} + 90911522 T^{2} + \cdots - 95\!\cdots\!08$$
$17$ $$T^{3} + 3868973426 T^{2} + \cdots + 65\!\cdots\!64$$
$19$ $$T^{3} - 3670884220 T^{2} + \cdots + 46\!\cdots\!00$$
$23$ $$T^{3} - 26698058238 T^{2} + \cdots - 27\!\cdots\!68$$
$29$ $$T^{3} - 145544932730 T^{2} + \cdots + 75\!\cdots\!00$$
$31$ $$T^{3} + 25873382644 T^{2} + \cdots - 90\!\cdots\!08$$
$37$ $$T^{3} - 419480249934 T^{2} + \cdots + 70\!\cdots\!24$$
$41$ $$T^{3} - 274005770306 T^{2} + \cdots - 22\!\cdots\!08$$
$43$ $$T^{3} - 2350065869158 T^{2} + \cdots + 82\!\cdots\!12$$
$47$ $$T^{3} + 8891070209486 T^{2} + \cdots - 20\!\cdots\!96$$
$53$ $$T^{3} - 8749242811318 T^{2} + \cdots + 14\!\cdots\!52$$
$59$ $$T^{3} + 14173516437140 T^{2} + \cdots + 44\!\cdots\!00$$
$61$ $$T^{3} + 38066837721794 T^{2} + \cdots - 18\!\cdots\!08$$
$67$ $$T^{3} - 144391638065474 T^{2} + \cdots - 97\!\cdots\!36$$
$71$ $$T^{3} + 126512337318844 T^{2} + \cdots + 13\!\cdots\!92$$
$73$ $$T^{3} - 199804772078038 T^{2} + \cdots + 74\!\cdots\!32$$
$79$ $$T^{3} + 73797562093720 T^{2} + \cdots - 65\!\cdots\!00$$
$83$ $$T^{3} + 219109046205402 T^{2} + \cdots + 10\!\cdots\!72$$
$89$ $$T^{3} + 360765737744610 T^{2} + \cdots - 25\!\cdots\!00$$
$97$ $$T^{3} + 509209931654586 T^{2} + \cdots - 21\!\cdots\!96$$