# Properties

 Label 22.12.a.b Level $22$ Weight $12$ Character orbit 22.a Self dual yes Analytic conductor $16.904$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 22.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.9035499723$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 331687x - 40657734$$ x^3 - 331687*x - 40657734 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - 32 q^{2} + (\beta_{2} - 23) q^{3} + 1024 q^{4} + ( - \beta_{2} - \beta_1 - 1875) q^{5} + ( - 32 \beta_{2} + 736) q^{6} + ( - 54 \beta_{2} + 21 \beta_1 - 10210) q^{7} - 32768 q^{8} + ( - 343 \beta_{2} - 105 \beta_1 + 177830) q^{9}+O(q^{10})$$ q - 32 * q^2 + (b2 - 23) * q^3 + 1024 * q^4 + (-b2 - b1 - 1875) * q^5 + (-32*b2 + 736) * q^6 + (-54*b2 + 21*b1 - 10210) * q^7 - 32768 * q^8 + (-343*b2 - 105*b1 + 177830) * q^9 $$q - 32 q^{2} + (\beta_{2} - 23) q^{3} + 1024 q^{4} + ( - \beta_{2} - \beta_1 - 1875) q^{5} + ( - 32 \beta_{2} + 736) q^{6} + ( - 54 \beta_{2} + 21 \beta_1 - 10210) q^{7} - 32768 q^{8} + ( - 343 \beta_{2} - 105 \beta_1 + 177830) q^{9} + (32 \beta_{2} + 32 \beta_1 + 60000) q^{10} + 161051 q^{11} + (1024 \beta_{2} - 23552) q^{12} + ( - 242 \beta_{2} + 823 \beta_1 + 140936) q^{13} + (1728 \beta_{2} - 672 \beta_1 + 326720) q^{14} + ( - 703 \beta_{2} - 168 \beta_1 - 193915) q^{15} + 1048576 q^{16} + (4364 \beta_{2} - 2383 \beta_1 - 5606526) q^{17} + (10976 \beta_{2} + 3360 \beta_1 - 5690560) q^{18} + ( - 13564 \beta_{2} - 2713 \beta_1 - 4250212) q^{19} + ( - 1024 \beta_{2} - 1024 \beta_1 - 1920000) q^{20} + ( - 10822 \beta_{2} + 11403 \beta_1 - 21370930) q^{21} - 5153632 q^{22} + ( - 54647 \beta_{2} - 12026 \beta_1 - 3817227) q^{23} + ( - 32768 \beta_{2} + 753664) q^{24} + (5269 \beta_{2} + 5089 \beta_1 - 41653700) q^{25} + (7744 \beta_{2} - 26336 \beta_1 - 4509952) q^{26} + (199903 \beta_{2} + 7350 \beta_1 - 109263533) q^{27} + ( - 55296 \beta_{2} + 21504 \beta_1 - 10455040) q^{28} + (74250 \beta_{2} + 47208 \beta_1 - 45591624) q^{29} + (22496 \beta_{2} + 5376 \beta_1 + 6205280) q^{30} + (11905 \beta_{2} - 96374 \beta_1 + 80684309) q^{31} - 33554432 q^{32} + (161051 \beta_{2} - 3704173) q^{33} + ( - 139648 \beta_{2} + 76256 \beta_1 + 179408832) q^{34} + ( - 6614 \beta_{2} - 42959 \beta_1 - 39913050) q^{35} + ( - 351232 \beta_{2} - 107520 \beta_1 + 182097920) q^{36} + ( - 353015 \beta_{2} + 85897 \beta_1 + 150102743) q^{37} + (434048 \beta_{2} + 86816 \beta_1 + 136006784) q^{38} + ( - 482820 \beta_{2} + 250089 \beta_1 - 185644728) q^{39} + (32768 \beta_{2} + 32768 \beta_1 + 61440000) q^{40} + ( - 567234 \beta_{2} - 513561 \beta_1 + 32386284) q^{41} + (346304 \beta_{2} - 364896 \beta_1 + 683869760) q^{42} + ( - 647878 \beta_{2} + 336686 \beta_1 - 237709042) q^{43} + 164916224 q^{44} + (351328 \beta_{2} + 205098 \beta_1 + 107158270) q^{45} + (1748704 \beta_{2} + 384832 \beta_1 + 122151264) q^{46} + (361512 \beta_{2} + 250158 \beta_1 - 1110062592) q^{47} + (1048576 \beta_{2} - 24117248) q^{48} + (3721284 \beta_{2} - 1030596 \beta_1 + 987542157) q^{49} + ( - 168608 \beta_{2} - 162848 \beta_1 + 1332918400) q^{50} + ( - 4972690 \beta_{2} - 1108779 \beta_1 + 1955544434) q^{51} + ( - 247808 \beta_{2} + 842752 \beta_1 + 144318464) q^{52} + (4007732 \beta_{2} + 2244158 \beta_1 - 1257725718) q^{53} + ( - 6396896 \beta_{2} - 235200 \beta_1 + 3496433056) q^{54} + ( - 161051 \beta_{2} - 161051 \beta_1 - 301970625) q^{55} + (1769472 \beta_{2} - 688128 \beta_1 + 334561280) q^{56} + (2401744 \beta_{2} + 683571 \beta_1 - 4391449892) q^{57} + ( - 2376000 \beta_{2} - 1510656 \beta_1 + 1458931968) q^{58} + ( - 4601133 \beta_{2} - 844614 \beta_1 - 3099318045) q^{59} + ( - 719872 \beta_{2} - 172032 \beta_1 - 198568960) q^{60} + (6282758 \beta_{2} - 912928 \beta_1 + 884673188) q^{61} + ( - 380960 \beta_{2} + 3083968 \beta_1 - 2581897888) q^{62} + ( - 18057308 \beta_{2} + 529242 \beta_1 - 2874437420) q^{63} + 1073741824 q^{64} + ( - 2161008 \beta_{2} - 2582643 \beta_1 - 3022999800) q^{65} + ( - 5153632 \beta_{2} + 118533536) q^{66} + (13772063 \beta_{2} + 1920422 \beta_1 + 548609255) q^{67} + (4468736 \beta_{2} - 2440192 \beta_1 - 5741082624) q^{68} + (23915965 \beta_{2} + 2454837 \beta_1 - 17869775027) q^{69} + (211648 \beta_{2} + 1374688 \beta_1 + 1277217600) q^{70} + (13052763 \beta_{2} + 2282196 \beta_1 - 702164889) q^{71} + (11239424 \beta_{2} + 3440640 \beta_1 - 5827133440) q^{72} + (21271114 \beta_{2} - 5577017 \beta_1 - 1754453560) q^{73} + (11296480 \beta_{2} - 2748704 \beta_1 - 4803287776) q^{74} + ( - 47675608 \beta_{2} + 836052 \beta_1 + 2228132300) q^{75} + ( - 13889536 \beta_{2} - 2778112 \beta_1 - 4352217088) q^{76} + ( - 8696754 \beta_{2} + 3382071 \beta_1 - 1644330710) q^{77} + (15450240 \beta_{2} - 8002848 \beta_1 + 5940631296) q^{78} + ( - 26194422 \beta_{2} - 5435430 \beta_1 + 2985566042) q^{79} + ( - 1048576 \beta_{2} - 1048576 \beta_1 - 1966080000) q^{80} + ( - 118733272 \beta_{2} - 382830 \beta_1 + 41003279993) q^{81} + (18151488 \beta_{2} + 16433952 \beta_1 - 1036361088) q^{82} + (91042834 \beta_{2} - 3456056 \beta_1 + 3860084742) q^{83} + ( - 11081728 \beta_{2} + 11676672 \beta_1 - 21883832320) q^{84} + (8796106 \beta_{2} + 11976811 \beta_1 + 17631847770) q^{85} + (20732096 \beta_{2} - 10773952 \beta_1 + 7606689344) q^{86} + ( - 109572840 \beta_{2} + 5091534 \beta_1 + 21823774488) q^{87} - 5277319168 q^{88} + (58147597 \beta_{2} + 264751 \beta_1 + 60369007695) q^{89} + ( - 11242496 \beta_{2} - 6563136 \beta_1 - 3429064640) q^{90} + (94009688 \beta_{2} - 6749302 \beta_1 + 70174924240) q^{91} + ( - 55958528 \beta_{2} - 12314624 \beta_1 - 3908840448) q^{92} + (158985357 \beta_{2} - 27560127 \beta_1 + 13679042925) q^{93} + ( - 11568384 \beta_{2} - 8005056 \beta_1 + 35522002944) q^{94} + (21335960 \beta_{2} + 15042335 \beta_1 + 20467592940) q^{95} + ( - 33554432 \beta_{2} + 771751936) q^{96} + (12329735 \beta_{2} - 3199237 \beta_1 - 5715614935) q^{97} + ( - 119081088 \beta_{2} + 32979072 \beta_1 - 31601349024) q^{98} + ( - 55240493 \beta_{2} - 16910355 \beta_1 + 28639699330) q^{99}+O(q^{100})$$ q - 32 * q^2 + (b2 - 23) * q^3 + 1024 * q^4 + (-b2 - b1 - 1875) * q^5 + (-32*b2 + 736) * q^6 + (-54*b2 + 21*b1 - 10210) * q^7 - 32768 * q^8 + (-343*b2 - 105*b1 + 177830) * q^9 + (32*b2 + 32*b1 + 60000) * q^10 + 161051 * q^11 + (1024*b2 - 23552) * q^12 + (-242*b2 + 823*b1 + 140936) * q^13 + (1728*b2 - 672*b1 + 326720) * q^14 + (-703*b2 - 168*b1 - 193915) * q^15 + 1048576 * q^16 + (4364*b2 - 2383*b1 - 5606526) * q^17 + (10976*b2 + 3360*b1 - 5690560) * q^18 + (-13564*b2 - 2713*b1 - 4250212) * q^19 + (-1024*b2 - 1024*b1 - 1920000) * q^20 + (-10822*b2 + 11403*b1 - 21370930) * q^21 - 5153632 * q^22 + (-54647*b2 - 12026*b1 - 3817227) * q^23 + (-32768*b2 + 753664) * q^24 + (5269*b2 + 5089*b1 - 41653700) * q^25 + (7744*b2 - 26336*b1 - 4509952) * q^26 + (199903*b2 + 7350*b1 - 109263533) * q^27 + (-55296*b2 + 21504*b1 - 10455040) * q^28 + (74250*b2 + 47208*b1 - 45591624) * q^29 + (22496*b2 + 5376*b1 + 6205280) * q^30 + (11905*b2 - 96374*b1 + 80684309) * q^31 - 33554432 * q^32 + (161051*b2 - 3704173) * q^33 + (-139648*b2 + 76256*b1 + 179408832) * q^34 + (-6614*b2 - 42959*b1 - 39913050) * q^35 + (-351232*b2 - 107520*b1 + 182097920) * q^36 + (-353015*b2 + 85897*b1 + 150102743) * q^37 + (434048*b2 + 86816*b1 + 136006784) * q^38 + (-482820*b2 + 250089*b1 - 185644728) * q^39 + (32768*b2 + 32768*b1 + 61440000) * q^40 + (-567234*b2 - 513561*b1 + 32386284) * q^41 + (346304*b2 - 364896*b1 + 683869760) * q^42 + (-647878*b2 + 336686*b1 - 237709042) * q^43 + 164916224 * q^44 + (351328*b2 + 205098*b1 + 107158270) * q^45 + (1748704*b2 + 384832*b1 + 122151264) * q^46 + (361512*b2 + 250158*b1 - 1110062592) * q^47 + (1048576*b2 - 24117248) * q^48 + (3721284*b2 - 1030596*b1 + 987542157) * q^49 + (-168608*b2 - 162848*b1 + 1332918400) * q^50 + (-4972690*b2 - 1108779*b1 + 1955544434) * q^51 + (-247808*b2 + 842752*b1 + 144318464) * q^52 + (4007732*b2 + 2244158*b1 - 1257725718) * q^53 + (-6396896*b2 - 235200*b1 + 3496433056) * q^54 + (-161051*b2 - 161051*b1 - 301970625) * q^55 + (1769472*b2 - 688128*b1 + 334561280) * q^56 + (2401744*b2 + 683571*b1 - 4391449892) * q^57 + (-2376000*b2 - 1510656*b1 + 1458931968) * q^58 + (-4601133*b2 - 844614*b1 - 3099318045) * q^59 + (-719872*b2 - 172032*b1 - 198568960) * q^60 + (6282758*b2 - 912928*b1 + 884673188) * q^61 + (-380960*b2 + 3083968*b1 - 2581897888) * q^62 + (-18057308*b2 + 529242*b1 - 2874437420) * q^63 + 1073741824 * q^64 + (-2161008*b2 - 2582643*b1 - 3022999800) * q^65 + (-5153632*b2 + 118533536) * q^66 + (13772063*b2 + 1920422*b1 + 548609255) * q^67 + (4468736*b2 - 2440192*b1 - 5741082624) * q^68 + (23915965*b2 + 2454837*b1 - 17869775027) * q^69 + (211648*b2 + 1374688*b1 + 1277217600) * q^70 + (13052763*b2 + 2282196*b1 - 702164889) * q^71 + (11239424*b2 + 3440640*b1 - 5827133440) * q^72 + (21271114*b2 - 5577017*b1 - 1754453560) * q^73 + (11296480*b2 - 2748704*b1 - 4803287776) * q^74 + (-47675608*b2 + 836052*b1 + 2228132300) * q^75 + (-13889536*b2 - 2778112*b1 - 4352217088) * q^76 + (-8696754*b2 + 3382071*b1 - 1644330710) * q^77 + (15450240*b2 - 8002848*b1 + 5940631296) * q^78 + (-26194422*b2 - 5435430*b1 + 2985566042) * q^79 + (-1048576*b2 - 1048576*b1 - 1966080000) * q^80 + (-118733272*b2 - 382830*b1 + 41003279993) * q^81 + (18151488*b2 + 16433952*b1 - 1036361088) * q^82 + (91042834*b2 - 3456056*b1 + 3860084742) * q^83 + (-11081728*b2 + 11676672*b1 - 21883832320) * q^84 + (8796106*b2 + 11976811*b1 + 17631847770) * q^85 + (20732096*b2 - 10773952*b1 + 7606689344) * q^86 + (-109572840*b2 + 5091534*b1 + 21823774488) * q^87 - 5277319168 * q^88 + (58147597*b2 + 264751*b1 + 60369007695) * q^89 + (-11242496*b2 - 6563136*b1 - 3429064640) * q^90 + (94009688*b2 - 6749302*b1 + 70174924240) * q^91 + (-55958528*b2 - 12314624*b1 - 3908840448) * q^92 + (158985357*b2 - 27560127*b1 + 13679042925) * q^93 + (-11568384*b2 - 8005056*b1 + 35522002944) * q^94 + (21335960*b2 + 15042335*b1 + 20467592940) * q^95 + (-33554432*b2 + 771751936) * q^96 + (12329735*b2 - 3199237*b1 - 5715614935) * q^97 + (-119081088*b2 + 32979072*b1 - 31601349024) * q^98 + (-55240493*b2 - 16910355*b1 + 28639699330) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 96 q^{2} - 70 q^{3} + 3072 q^{4} - 5624 q^{5} + 2240 q^{6} - 30576 q^{7} - 98304 q^{8} + 533833 q^{9}+O(q^{10})$$ 3 * q - 96 * q^2 - 70 * q^3 + 3072 * q^4 - 5624 * q^5 + 2240 * q^6 - 30576 * q^7 - 98304 * q^8 + 533833 * q^9 $$3 q - 96 q^{2} - 70 q^{3} + 3072 q^{4} - 5624 q^{5} + 2240 q^{6} - 30576 q^{7} - 98304 q^{8} + 533833 q^{9} + 179968 q^{10} + 483153 q^{11} - 71680 q^{12} + 423050 q^{13} + 978432 q^{14} - 581042 q^{15} + 3145728 q^{16} - 16823942 q^{17} - 17082656 q^{18} - 12737072 q^{19} - 5758976 q^{20} - 64101968 q^{21} - 15460896 q^{22} - 11397034 q^{23} + 2293760 q^{24} - 124966369 q^{25} - 13537600 q^{26} - 327990502 q^{27} - 31309824 q^{28} - 136849122 q^{29} + 18593344 q^{30} + 242041022 q^{31} - 100663296 q^{32} - 11273570 q^{33} + 538366144 q^{34} - 119732536 q^{35} + 546644992 q^{36} + 450661244 q^{37} + 407586304 q^{38} - 556451364 q^{39} + 184287232 q^{40} + 97726086 q^{41} + 2051262976 q^{42} - 712479248 q^{43} + 494748672 q^{44} + 321123482 q^{45} + 364705088 q^{46} - 3330549288 q^{47} - 73400320 q^{48} + 2958905187 q^{49} + 3998923808 q^{50} + 5871605992 q^{51} + 433203200 q^{52} - 3777184886 q^{53} + 10495696064 q^{54} - 905750824 q^{55} + 1001914368 q^{56} - 13176751420 q^{57} + 4379171904 q^{58} - 9293353002 q^{59} - 594987008 q^{60} + 2647736806 q^{61} - 7745312704 q^{62} - 8605254952 q^{63} + 3221225472 q^{64} - 9066838392 q^{65} + 360754240 q^{66} + 1632055702 q^{67} - 17227716608 q^{68} - 53633241046 q^{69} + 3831441152 q^{70} - 2119547430 q^{71} - 17492639744 q^{72} - 5284631794 q^{73} - 14421159808 q^{74} + 6732072508 q^{75} - 13042761728 q^{76} - 4924295376 q^{77} + 17806443648 q^{78} + 8982892548 q^{79} - 5897191424 q^{80} + 123128573251 q^{81} - 3127234752 q^{82} + 11489211392 q^{83} - 65640415232 q^{84} + 52886747204 q^{85} + 22799335936 q^{86} + 65580896304 q^{87} - 15831957504 q^{88} + 181048875488 q^{89} - 10275951424 q^{90} + 210430763032 q^{91} - 11670562816 q^{92} + 40878143418 q^{93} + 106577577216 q^{94} + 61381442860 q^{95} + 2348810240 q^{96} - 17159174540 q^{97} - 94684965984 q^{98} + 85974338483 q^{99}+O(q^{100})$$ 3 * q - 96 * q^2 - 70 * q^3 + 3072 * q^4 - 5624 * q^5 + 2240 * q^6 - 30576 * q^7 - 98304 * q^8 + 533833 * q^9 + 179968 * q^10 + 483153 * q^11 - 71680 * q^12 + 423050 * q^13 + 978432 * q^14 - 581042 * q^15 + 3145728 * q^16 - 16823942 * q^17 - 17082656 * q^18 - 12737072 * q^19 - 5758976 * q^20 - 64101968 * q^21 - 15460896 * q^22 - 11397034 * q^23 + 2293760 * q^24 - 124966369 * q^25 - 13537600 * q^26 - 327990502 * q^27 - 31309824 * q^28 - 136849122 * q^29 + 18593344 * q^30 + 242041022 * q^31 - 100663296 * q^32 - 11273570 * q^33 + 538366144 * q^34 - 119732536 * q^35 + 546644992 * q^36 + 450661244 * q^37 + 407586304 * q^38 - 556451364 * q^39 + 184287232 * q^40 + 97726086 * q^41 + 2051262976 * q^42 - 712479248 * q^43 + 494748672 * q^44 + 321123482 * q^45 + 364705088 * q^46 - 3330549288 * q^47 - 73400320 * q^48 + 2958905187 * q^49 + 3998923808 * q^50 + 5871605992 * q^51 + 433203200 * q^52 - 3777184886 * q^53 + 10495696064 * q^54 - 905750824 * q^55 + 1001914368 * q^56 - 13176751420 * q^57 + 4379171904 * q^58 - 9293353002 * q^59 - 594987008 * q^60 + 2647736806 * q^61 - 7745312704 * q^62 - 8605254952 * q^63 + 3221225472 * q^64 - 9066838392 * q^65 + 360754240 * q^66 + 1632055702 * q^67 - 17227716608 * q^68 - 53633241046 * q^69 + 3831441152 * q^70 - 2119547430 * q^71 - 17492639744 * q^72 - 5284631794 * q^73 - 14421159808 * q^74 + 6732072508 * q^75 - 13042761728 * q^76 - 4924295376 * q^77 + 17806443648 * q^78 + 8982892548 * q^79 - 5897191424 * q^80 + 123128573251 * q^81 - 3127234752 * q^82 + 11489211392 * q^83 - 65640415232 * q^84 + 52886747204 * q^85 + 22799335936 * q^86 + 65580896304 * q^87 - 15831957504 * q^88 + 181048875488 * q^89 - 10275951424 * q^90 + 210430763032 * q^91 - 11670562816 * q^92 + 40878143418 * q^93 + 106577577216 * q^94 + 61381442860 * q^95 + 2348810240 * q^96 - 17159174540 * q^97 - 94684965984 * q^98 + 85974338483 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 331687x - 40657734$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu$$ 4*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 213\nu - 221198 ) / 220$$ (v^2 - 213*v - 221198) / 220
 $$\nu$$ $$=$$ $$( \beta_1 ) / 4$$ (b1) / 4 $$\nu^{2}$$ $$=$$ $$( 880\beta_{2} + 213\beta _1 + 884792 ) / 4$$ (880*b2 + 213*b1 + 884792) / 4

## 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
 −129.060 629.503 −500.443
−32.0000 −827.781 1024.00 −553.980 26489.0 22407.2 −32768.0 508075. 17727.4
1.2 −32.0000 163.327 1024.00 −4579.34 −5226.46 32606.6 −32768.0 −150471. 146539.
1.3 −32.0000 594.455 1024.00 −490.681 −19022.5 −85589.8 −32768.0 176229. 15701.8
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$11$$ $$-1$$

## 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 22.12.a.b 3
3.b odd 2 1 198.12.a.l 3
4.b odd 2 1 176.12.a.d 3
11.b odd 2 1 242.12.a.c 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.12.a.b 3 1.a even 1 1 trivial
176.12.a.d 3 4.b odd 2 1
198.12.a.l 3 3.b odd 2 1
242.12.a.c 3 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 70T_{3}^{2} - 530187T_{3} + 80369604$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(22))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 32)^{3}$$
$3$ $$T^{3} + 70 T^{2} - 530187 T + 80369604$$
$5$ $$T^{3} + 5624 T^{2} + \cdots + 1244790450$$
$7$ $$T^{3} + 30576 T^{2} + \cdots + 62533813132000$$
$11$ $$(T - 161051)^{3}$$
$13$ $$T^{3} - 423050 T^{2} + \cdots - 32\!\cdots\!88$$
$17$ $$T^{3} + 16823942 T^{2} + \cdots - 15\!\cdots\!72$$
$19$ $$T^{3} + 12737072 T^{2} + \cdots - 78\!\cdots\!60$$
$23$ $$T^{3} + 11397034 T^{2} + \cdots - 27\!\cdots\!16$$
$29$ $$T^{3} + 136849122 T^{2} + \cdots - 10\!\cdots\!00$$
$31$ $$T^{3} - 242041022 T^{2} + \cdots + 54\!\cdots\!88$$
$37$ $$T^{3} - 450661244 T^{2} + \cdots + 28\!\cdots\!94$$
$41$ $$T^{3} - 97726086 T^{2} + \cdots + 73\!\cdots\!64$$
$43$ $$T^{3} + 712479248 T^{2} + \cdots + 70\!\cdots\!00$$
$47$ $$T^{3} + 3330549288 T^{2} + \cdots + 87\!\cdots\!12$$
$53$ $$T^{3} + 3777184886 T^{2} + \cdots - 94\!\cdots\!00$$
$59$ $$T^{3} + 9293353002 T^{2} + \cdots - 26\!\cdots\!00$$
$61$ $$T^{3} - 2647736806 T^{2} + \cdots - 59\!\cdots\!20$$
$67$ $$T^{3} - 1632055702 T^{2} + \cdots + 47\!\cdots\!72$$
$71$ $$T^{3} + 2119547430 T^{2} + \cdots + 25\!\cdots\!00$$
$73$ $$T^{3} + 5284631794 T^{2} + \cdots - 42\!\cdots\!16$$
$79$ $$T^{3} - 8982892548 T^{2} + \cdots - 96\!\cdots\!00$$
$83$ $$T^{3} - 11489211392 T^{2} + \cdots + 54\!\cdots\!28$$
$89$ $$T^{3} - 181048875488 T^{2} + \cdots - 92\!\cdots\!30$$
$97$ $$T^{3} + 17159174540 T^{2} + \cdots - 13\!\cdots\!02$$