Properties

 Label 138.8.a.f Level $138$ Weight $8$ Character orbit 138.a Self dual yes Analytic conductor $43.109$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

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Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 138.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$43.1091335168$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 62310x^{2} - 465075x + 877928625$$ x^4 - x^3 - 62310*x^2 - 465075*x + 877928625 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 q^{2} + 27 q^{3} + 64 q^{4} + ( - \beta_1 - 40) q^{5} - 216 q^{6} + (\beta_{2} + \beta_1 + 304) q^{7} - 512 q^{8} + 729 q^{9}+O(q^{10})$$ q - 8 * q^2 + 27 * q^3 + 64 * q^4 + (-b1 - 40) * q^5 - 216 * q^6 + (b2 + b1 + 304) * q^7 - 512 * q^8 + 729 * q^9 $$q - 8 q^{2} + 27 q^{3} + 64 q^{4} + ( - \beta_1 - 40) q^{5} - 216 q^{6} + (\beta_{2} + \beta_1 + 304) q^{7} - 512 q^{8} + 729 q^{9} + (8 \beta_1 + 320) q^{10} + (2 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 456) q^{11} + 1728 q^{12} + ( - 4 \beta_{3} + 4 \beta_{2} - 14 \beta_1 - 1618) q^{13} + ( - 8 \beta_{2} - 8 \beta_1 - 2432) q^{14} + ( - 27 \beta_1 - 1080) q^{15} + 4096 q^{16} + (\beta_{3} - 10 \beta_{2} - 48 \beta_1 + 3241) q^{17} - 5832 q^{18} + (3 \beta_{3} - 3 \beta_{2} + 20 \beta_1 + 18607) q^{19} + ( - 64 \beta_1 - 2560) q^{20} + (27 \beta_{2} + 27 \beta_1 + 8208) q^{21} + ( - 16 \beta_{3} - 24 \beta_{2} + 32 \beta_1 - 3648) q^{22} + 12167 q^{23} - 13824 q^{24} + (15 \beta_{3} + 15 \beta_{2} + 97 \beta_1 + 48080) q^{25} + (32 \beta_{3} - 32 \beta_{2} + 112 \beta_1 + 12944) q^{26} + 19683 q^{27} + (64 \beta_{2} + 64 \beta_1 + 19456) q^{28} + ( - 5 \beta_{3} - 85 \beta_{2} + 5 \beta_1 - 23613) q^{29} + (216 \beta_1 + 8640) q^{30} + ( - 29 \beta_{3} - 81 \beta_{2} + 69 \beta_1 - 9875) q^{31} - 32768 q^{32} + (54 \beta_{3} + 81 \beta_{2} - 108 \beta_1 + 12312) q^{33} + ( - 8 \beta_{3} + 80 \beta_{2} + 384 \beta_1 - 25928) q^{34} + ( - 150 \beta_{3} + 80 \beta_{2} - 546 \beta_1 - 196750) q^{35} + 46656 q^{36} + (66 \beta_{3} - 87 \beta_{2} + 462 \beta_1 + 18698) q^{37} + ( - 24 \beta_{3} + 24 \beta_{2} - 160 \beta_1 - 148856) q^{38} + ( - 108 \beta_{3} + 108 \beta_{2} - 378 \beta_1 - 43686) q^{39} + (512 \beta_1 + 20480) q^{40} + (23 \beta_{3} + 167 \beta_{2} - 177 \beta_1 - 129681) q^{41} + ( - 216 \beta_{2} - 216 \beta_1 - 65664) q^{42} + (159 \beta_{3} + 331 \beta_{2} + 252 \beta_1 + 179031) q^{43} + (128 \beta_{3} + 192 \beta_{2} - 256 \beta_1 + 29184) q^{44} + ( - 729 \beta_1 - 29160) q^{45} - 97336 q^{46} + (39 \beta_{3} - 949 \beta_{2} + 299 \beta_1 - 3227) q^{47} + 110592 q^{48} + (350 \beta_{3} + 336 \beta_{2} + 196 \beta_1 + 413771) q^{49} + ( - 120 \beta_{3} - 120 \beta_{2} - 776 \beta_1 - 384640) q^{50} + (27 \beta_{3} - 270 \beta_{2} - 1296 \beta_1 + 87507) q^{51} + ( - 256 \beta_{3} + 256 \beta_{2} - 896 \beta_1 - 103552) q^{52} + (20 \beta_{3} + 2 \beta_{2} + 1309 \beta_1 + 262684) q^{53} - 157464 q^{54} + ( - 665 \beta_{3} - 1155 \beta_{2} - 1337 \beta_1 + 289785) q^{55} + ( - 512 \beta_{2} - 512 \beta_1 - 155648) q^{56} + (81 \beta_{3} - 81 \beta_{2} + 540 \beta_1 + 502389) q^{57} + (40 \beta_{3} + 680 \beta_{2} - 40 \beta_1 + 188904) q^{58} + (90 \beta_{3} - 70 \beta_{2} - 106 \beta_1 + 1099594) q^{59} + ( - 1728 \beta_1 - 69120) q^{60} + ( - 384 \beta_{3} - 791 \beta_{2} + 60 \beta_1 + 1332240) q^{61} + (232 \beta_{3} + 648 \beta_{2} - 552 \beta_1 + 79000) q^{62} + (729 \beta_{2} + 729 \beta_1 + 221616) q^{63} + 262144 q^{64} + (310 \beta_{3} + 3590 \beta_{2} + 2784 \beta_1 + 1590130) q^{65} + ( - 432 \beta_{3} - 648 \beta_{2} + 864 \beta_1 - 98496) q^{66} + (313 \beta_{3} - 703 \beta_{2} + 2020 \beta_1 + 2824825) q^{67} + (64 \beta_{3} - 640 \beta_{2} - 3072 \beta_1 + 207424) q^{68} + 328509 q^{69} + (1200 \beta_{3} - 640 \beta_{2} + 4368 \beta_1 + 1574000) q^{70} + ( - 107 \beta_{3} + 653 \beta_{2} + 4681 \beta_1 + 2096543) q^{71} - 373248 q^{72} + ( - 1137 \beta_{3} - 435 \beta_{2} + 1915 \beta_1 + 1776363) q^{73} + ( - 528 \beta_{3} + 696 \beta_{2} - 3696 \beta_1 - 149584) q^{74} + (405 \beta_{3} + 405 \beta_{2} + 2619 \beta_1 + 1298160) q^{75} + (192 \beta_{3} - 192 \beta_{2} + 1280 \beta_1 + 1190848) q^{76} + (512 \beta_{3} + 2918 \beta_{2} + 8396 \beta_1 + 2442836) q^{77} + (864 \beta_{3} - 864 \beta_{2} + 3024 \beta_1 + 349488) q^{78} + ( - 258 \beta_{3} + 1853 \beta_{2} - 3555 \beta_1 + 2448178) q^{79} + ( - 4096 \beta_1 - 163840) q^{80} + 531441 q^{81} + ( - 184 \beta_{3} - 1336 \beta_{2} + 1416 \beta_1 + 1037448) q^{82} + (430 \beta_{3} - 2439 \beta_{2} - 2304 \beta_1 + 607016) q^{83} + (1728 \beta_{2} + 1728 \beta_1 + 525312) q^{84} + (1910 \beta_{3} - 980 \beta_{2} + 1068 \beta_1 + 6446030) q^{85} + ( - 1272 \beta_{3} - 2648 \beta_{2} - 2016 \beta_1 - 1432248) q^{86} + ( - 135 \beta_{3} - 2295 \beta_{2} + 135 \beta_1 - 637551) q^{87} + ( - 1024 \beta_{3} - 1536 \beta_{2} + 2048 \beta_1 - 233472) q^{88} + ( - 1507 \beta_{3} - 3266 \beta_{2} + 9098 \beta_1 + 751009) q^{89} + (5832 \beta_1 + 233280) q^{90} + ( - 2324 \beta_{3} - 2382 \beta_{2} - 34770 \beta_1 + 441804) q^{91} + 778688 q^{92} + ( - 783 \beta_{3} - 2187 \beta_{2} + 1863 \beta_1 - 266625) q^{93} + ( - 312 \beta_{3} + 7592 \beta_{2} - 2392 \beta_1 + 25816) q^{94} + ( - 375 \beta_{3} - 2835 \beta_{2} - 20023 \beta_1 - 3072085) q^{95} - 884736 q^{96} + (2772 \beta_{3} + 1096 \beta_{2} - 14170 \beta_1 + 8062418) q^{97} + ( - 2800 \beta_{3} - 2688 \beta_{2} - 1568 \beta_1 - 3310168) q^{98} + (1458 \beta_{3} + 2187 \beta_{2} - 2916 \beta_1 + 332424) q^{99}+O(q^{100})$$ q - 8 * q^2 + 27 * q^3 + 64 * q^4 + (-b1 - 40) * q^5 - 216 * q^6 + (b2 + b1 + 304) * q^7 - 512 * q^8 + 729 * q^9 + (8*b1 + 320) * q^10 + (2*b3 + 3*b2 - 4*b1 + 456) * q^11 + 1728 * q^12 + (-4*b3 + 4*b2 - 14*b1 - 1618) * q^13 + (-8*b2 - 8*b1 - 2432) * q^14 + (-27*b1 - 1080) * q^15 + 4096 * q^16 + (b3 - 10*b2 - 48*b1 + 3241) * q^17 - 5832 * q^18 + (3*b3 - 3*b2 + 20*b1 + 18607) * q^19 + (-64*b1 - 2560) * q^20 + (27*b2 + 27*b1 + 8208) * q^21 + (-16*b3 - 24*b2 + 32*b1 - 3648) * q^22 + 12167 * q^23 - 13824 * q^24 + (15*b3 + 15*b2 + 97*b1 + 48080) * q^25 + (32*b3 - 32*b2 + 112*b1 + 12944) * q^26 + 19683 * q^27 + (64*b2 + 64*b1 + 19456) * q^28 + (-5*b3 - 85*b2 + 5*b1 - 23613) * q^29 + (216*b1 + 8640) * q^30 + (-29*b3 - 81*b2 + 69*b1 - 9875) * q^31 - 32768 * q^32 + (54*b3 + 81*b2 - 108*b1 + 12312) * q^33 + (-8*b3 + 80*b2 + 384*b1 - 25928) * q^34 + (-150*b3 + 80*b2 - 546*b1 - 196750) * q^35 + 46656 * q^36 + (66*b3 - 87*b2 + 462*b1 + 18698) * q^37 + (-24*b3 + 24*b2 - 160*b1 - 148856) * q^38 + (-108*b3 + 108*b2 - 378*b1 - 43686) * q^39 + (512*b1 + 20480) * q^40 + (23*b3 + 167*b2 - 177*b1 - 129681) * q^41 + (-216*b2 - 216*b1 - 65664) * q^42 + (159*b3 + 331*b2 + 252*b1 + 179031) * q^43 + (128*b3 + 192*b2 - 256*b1 + 29184) * q^44 + (-729*b1 - 29160) * q^45 - 97336 * q^46 + (39*b3 - 949*b2 + 299*b1 - 3227) * q^47 + 110592 * q^48 + (350*b3 + 336*b2 + 196*b1 + 413771) * q^49 + (-120*b3 - 120*b2 - 776*b1 - 384640) * q^50 + (27*b3 - 270*b2 - 1296*b1 + 87507) * q^51 + (-256*b3 + 256*b2 - 896*b1 - 103552) * q^52 + (20*b3 + 2*b2 + 1309*b1 + 262684) * q^53 - 157464 * q^54 + (-665*b3 - 1155*b2 - 1337*b1 + 289785) * q^55 + (-512*b2 - 512*b1 - 155648) * q^56 + (81*b3 - 81*b2 + 540*b1 + 502389) * q^57 + (40*b3 + 680*b2 - 40*b1 + 188904) * q^58 + (90*b3 - 70*b2 - 106*b1 + 1099594) * q^59 + (-1728*b1 - 69120) * q^60 + (-384*b3 - 791*b2 + 60*b1 + 1332240) * q^61 + (232*b3 + 648*b2 - 552*b1 + 79000) * q^62 + (729*b2 + 729*b1 + 221616) * q^63 + 262144 * q^64 + (310*b3 + 3590*b2 + 2784*b1 + 1590130) * q^65 + (-432*b3 - 648*b2 + 864*b1 - 98496) * q^66 + (313*b3 - 703*b2 + 2020*b1 + 2824825) * q^67 + (64*b3 - 640*b2 - 3072*b1 + 207424) * q^68 + 328509 * q^69 + (1200*b3 - 640*b2 + 4368*b1 + 1574000) * q^70 + (-107*b3 + 653*b2 + 4681*b1 + 2096543) * q^71 - 373248 * q^72 + (-1137*b3 - 435*b2 + 1915*b1 + 1776363) * q^73 + (-528*b3 + 696*b2 - 3696*b1 - 149584) * q^74 + (405*b3 + 405*b2 + 2619*b1 + 1298160) * q^75 + (192*b3 - 192*b2 + 1280*b1 + 1190848) * q^76 + (512*b3 + 2918*b2 + 8396*b1 + 2442836) * q^77 + (864*b3 - 864*b2 + 3024*b1 + 349488) * q^78 + (-258*b3 + 1853*b2 - 3555*b1 + 2448178) * q^79 + (-4096*b1 - 163840) * q^80 + 531441 * q^81 + (-184*b3 - 1336*b2 + 1416*b1 + 1037448) * q^82 + (430*b3 - 2439*b2 - 2304*b1 + 607016) * q^83 + (1728*b2 + 1728*b1 + 525312) * q^84 + (1910*b3 - 980*b2 + 1068*b1 + 6446030) * q^85 + (-1272*b3 - 2648*b2 - 2016*b1 - 1432248) * q^86 + (-135*b3 - 2295*b2 + 135*b1 - 637551) * q^87 + (-1024*b3 - 1536*b2 + 2048*b1 - 233472) * q^88 + (-1507*b3 - 3266*b2 + 9098*b1 + 751009) * q^89 + (5832*b1 + 233280) * q^90 + (-2324*b3 - 2382*b2 - 34770*b1 + 441804) * q^91 + 778688 * q^92 + (-783*b3 - 2187*b2 + 1863*b1 - 266625) * q^93 + (-312*b3 + 7592*b2 - 2392*b1 + 25816) * q^94 + (-375*b3 - 2835*b2 - 20023*b1 - 3072085) * q^95 - 884736 * q^96 + (2772*b3 + 1096*b2 - 14170*b1 + 8062418) * q^97 + (-2800*b3 - 2688*b2 - 1568*b1 - 3310168) * q^98 + (1458*b3 + 2187*b2 - 2916*b1 + 332424) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 32 q^{2} + 108 q^{3} + 256 q^{4} - 162 q^{5} - 864 q^{6} + 1218 q^{7} - 2048 q^{8} + 2916 q^{9}+O(q^{10})$$ 4 * q - 32 * q^2 + 108 * q^3 + 256 * q^4 - 162 * q^5 - 864 * q^6 + 1218 * q^7 - 2048 * q^8 + 2916 * q^9 $$4 q - 32 q^{2} + 108 q^{3} + 256 q^{4} - 162 q^{5} - 864 q^{6} + 1218 q^{7} - 2048 q^{8} + 2916 q^{9} + 1296 q^{10} + 1820 q^{11} + 6912 q^{12} - 6508 q^{13} - 9744 q^{14} - 4374 q^{15} + 16384 q^{16} + 12870 q^{17} - 23328 q^{18} + 74474 q^{19} - 10368 q^{20} + 32886 q^{21} - 14560 q^{22} + 48668 q^{23} - 55296 q^{24} + 192544 q^{25} + 52064 q^{26} + 78732 q^{27} + 77952 q^{28} - 94452 q^{29} + 34992 q^{30} - 39420 q^{31} - 131072 q^{32} + 49140 q^{33} - 102960 q^{34} - 788392 q^{35} + 186624 q^{36} + 75848 q^{37} - 595792 q^{38} - 175716 q^{39} + 82944 q^{40} - 519032 q^{41} - 263088 q^{42} + 716946 q^{43} + 116480 q^{44} - 118098 q^{45} - 389344 q^{46} - 12232 q^{47} + 442368 q^{48} + 1656176 q^{49} - 1540352 q^{50} + 347490 q^{51} - 416512 q^{52} + 1053394 q^{53} - 629856 q^{54} + 1155136 q^{55} - 623616 q^{56} + 2010798 q^{57} + 755616 q^{58} + 4398344 q^{59} - 279936 q^{60} + 5328312 q^{61} + 315360 q^{62} + 887922 q^{63} + 1048576 q^{64} + 6366708 q^{65} - 393120 q^{66} + 11303966 q^{67} + 823680 q^{68} + 1314036 q^{69} + 6307136 q^{70} + 8395320 q^{71} - 1492992 q^{72} + 7107008 q^{73} - 606784 q^{74} + 5198688 q^{75} + 4766336 q^{76} + 9789160 q^{77} + 1405728 q^{78} + 9785086 q^{79} - 663552 q^{80} + 2125764 q^{81} + 4152256 q^{82} + 2424316 q^{83} + 2104704 q^{84} + 25790076 q^{85} - 5735568 q^{86} - 2550204 q^{87} - 931840 q^{88} + 3019218 q^{89} + 944784 q^{90} + 1693028 q^{91} + 3114752 q^{92} - 1064340 q^{93} + 97856 q^{94} - 12329136 q^{95} - 3538944 q^{96} + 32226876 q^{97} - 13249408 q^{98} + 1326780 q^{99}+O(q^{100})$$ 4 * q - 32 * q^2 + 108 * q^3 + 256 * q^4 - 162 * q^5 - 864 * q^6 + 1218 * q^7 - 2048 * q^8 + 2916 * q^9 + 1296 * q^10 + 1820 * q^11 + 6912 * q^12 - 6508 * q^13 - 9744 * q^14 - 4374 * q^15 + 16384 * q^16 + 12870 * q^17 - 23328 * q^18 + 74474 * q^19 - 10368 * q^20 + 32886 * q^21 - 14560 * q^22 + 48668 * q^23 - 55296 * q^24 + 192544 * q^25 + 52064 * q^26 + 78732 * q^27 + 77952 * q^28 - 94452 * q^29 + 34992 * q^30 - 39420 * q^31 - 131072 * q^32 + 49140 * q^33 - 102960 * q^34 - 788392 * q^35 + 186624 * q^36 + 75848 * q^37 - 595792 * q^38 - 175716 * q^39 + 82944 * q^40 - 519032 * q^41 - 263088 * q^42 + 716946 * q^43 + 116480 * q^44 - 118098 * q^45 - 389344 * q^46 - 12232 * q^47 + 442368 * q^48 + 1656176 * q^49 - 1540352 * q^50 + 347490 * q^51 - 416512 * q^52 + 1053394 * q^53 - 629856 * q^54 + 1155136 * q^55 - 623616 * q^56 + 2010798 * q^57 + 755616 * q^58 + 4398344 * q^59 - 279936 * q^60 + 5328312 * q^61 + 315360 * q^62 + 887922 * q^63 + 1048576 * q^64 + 6366708 * q^65 - 393120 * q^66 + 11303966 * q^67 + 823680 * q^68 + 1314036 * q^69 + 6307136 * q^70 + 8395320 * q^71 - 1492992 * q^72 + 7107008 * q^73 - 606784 * q^74 + 5198688 * q^75 + 4766336 * q^76 + 9789160 * q^77 + 1405728 * q^78 + 9785086 * q^79 - 663552 * q^80 + 2125764 * q^81 + 4152256 * q^82 + 2424316 * q^83 + 2104704 * q^84 + 25790076 * q^85 - 5735568 * q^86 - 2550204 * q^87 - 931840 * q^88 + 3019218 * q^89 + 944784 * q^90 + 1693028 * q^91 + 3114752 * q^92 - 1064340 * q^93 + 97856 * q^94 - 12329136 * q^95 - 3538944 * q^96 + 32226876 * q^97 - 13249408 * q^98 + 1326780 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 62310x^{2} - 465075x + 877928625$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 136\nu^{2} - 31800\nu + 3849525 ) / 675$$ (v^3 - 136*v^2 - 31800*v + 3849525) / 675 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 316\nu^{2} + 30270\nu - 9456750 ) / 675$$ (-v^3 + 316*v^2 + 30270*v - 9456750) / 675
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( 15\beta_{3} + 15\beta_{2} + 17\beta _1 + 124605 ) / 4$$ (15*b3 + 15*b2 + 17*b1 + 124605) / 4 $$\nu^{3}$$ $$=$$ $$510\beta_{3} + 1185\beta_{2} + 16478\beta _1 + 387045$$ 510*b3 + 1185*b2 + 16478*b1 + 387045

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
 212.947 136.241 −167.177 −181.010
−8.00000 27.0000 64.0000 −465.894 −216.000 1570.02 −512.000 729.000 3727.15
1.2 −8.00000 27.0000 64.0000 −312.481 −216.000 −132.348 −512.000 729.000 2499.85
1.3 −8.00000 27.0000 64.0000 294.355 −216.000 995.541 −512.000 729.000 −2354.84
1.4 −8.00000 27.0000 64.0000 322.021 −216.000 −1215.22 −512.000 729.000 −2576.17
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$23$$ $$-1$$

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 138.8.a.f 4
3.b odd 2 1 414.8.a.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.f 4 1.a even 1 1 trivial
414.8.a.l 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 162T_{5}^{3} - 239400T_{5}^{2} - 15953000T_{5} + 13799586000$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(138))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 8)^{4}$$
$3$ $$(T - 27)^{4}$$
$5$ $$T^{4} + 162 T^{3} + \cdots + 13799586000$$
$7$ $$T^{4} - 1218 T^{3} + \cdots + 251384354880$$
$11$ $$T^{4} + \cdots - 243981790840800$$
$13$ $$T^{4} + 6508 T^{3} + \cdots + 60\!\cdots\!72$$
$17$ $$T^{4} - 12870 T^{3} + \cdots + 33\!\cdots\!88$$
$19$ $$T^{4} - 74474 T^{3} + \cdots + 50\!\cdots\!36$$
$23$ $$(T - 12167)^{4}$$
$29$ $$T^{4} + 94452 T^{3} + \cdots + 36\!\cdots\!36$$
$31$ $$T^{4} + 39420 T^{3} + \cdots + 12\!\cdots\!40$$
$37$ $$T^{4} - 75848 T^{3} + \cdots - 91\!\cdots\!08$$
$41$ $$T^{4} + 519032 T^{3} + \cdots - 41\!\cdots\!40$$
$43$ $$T^{4} - 716946 T^{3} + \cdots - 19\!\cdots\!28$$
$47$ $$T^{4} + 12232 T^{3} + \cdots + 44\!\cdots\!52$$
$53$ $$T^{4} - 1053394 T^{3} + \cdots + 17\!\cdots\!48$$
$59$ $$T^{4} - 4398344 T^{3} + \cdots + 13\!\cdots\!68$$
$61$ $$T^{4} - 5328312 T^{3} + \cdots - 12\!\cdots\!88$$
$67$ $$T^{4} - 11303966 T^{3} + \cdots + 38\!\cdots\!12$$
$71$ $$T^{4} - 8395320 T^{3} + \cdots - 10\!\cdots\!60$$
$73$ $$T^{4} - 7107008 T^{3} + \cdots + 56\!\cdots\!32$$
$79$ $$T^{4} - 9785086 T^{3} + \cdots + 10\!\cdots\!16$$
$83$ $$T^{4} - 2424316 T^{3} + \cdots + 53\!\cdots\!04$$
$89$ $$T^{4} - 3019218 T^{3} + \cdots - 10\!\cdots\!84$$
$97$ $$T^{4} - 32226876 T^{3} + \cdots - 79\!\cdots\!20$$
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