# Properties

 Label 98.10.a.l Level $98$ Weight $10$ Character orbit 98.a Self dual yes Analytic conductor $50.474$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 98.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.4735119441$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 7165x^{4} + 16168x^{3} + 12807408x^{2} - 32180328x - 5372164$$ x^6 - 2*x^5 - 7165*x^4 + 16168*x^3 + 12807408*x^2 - 32180328*x - 5372164 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}\cdot 7^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + (\beta_{3} + \beta_{2}) q^{3} + 256 q^{4} + ( - \beta_{5} - 3 \beta_{3} - 2 \beta_{2}) q^{5} + (16 \beta_{3} + 16 \beta_{2}) q^{6} + 4096 q^{8} + ( - \beta_{4} + 6 \beta_1 + 12094) q^{9}+O(q^{10})$$ q + 16 * q^2 + (b3 + b2) * q^3 + 256 * q^4 + (-b5 - 3*b3 - 2*b2) * q^5 + (16*b3 + 16*b2) * q^6 + 4096 * q^8 + (-b4 + 6*b1 + 12094) * q^9 $$q + 16 q^{2} + (\beta_{3} + \beta_{2}) q^{3} + 256 q^{4} + ( - \beta_{5} - 3 \beta_{3} - 2 \beta_{2}) q^{5} + (16 \beta_{3} + 16 \beta_{2}) q^{6} + 4096 q^{8} + ( - \beta_{4} + 6 \beta_1 + 12094) q^{9} + ( - 16 \beta_{5} - 48 \beta_{3} - 32 \beta_{2}) q^{10} + ( - \beta_{4} + 11 \beta_1 + 18448) q^{11} + (256 \beta_{3} + 256 \beta_{2}) q^{12} + ( - 25 \beta_{5} + 237 \beta_{3} + 140 \beta_{2}) q^{13} + (16 \beta_{4} - 130 \beta_1 - 87170) q^{15} + 65536 q^{16} + ( - 100 \beta_{5} + 984 \beta_{3} + 3271 \beta_{2}) q^{17} + ( - 16 \beta_{4} + 96 \beta_1 + 193504) q^{18} + (3009 \beta_{3} - 995 \beta_{2}) q^{19} + ( - 256 \beta_{5} - 768 \beta_{3} - 512 \beta_{2}) q^{20} + ( - 16 \beta_{4} + 176 \beta_1 + 295168) q^{22} + (112 \beta_{4} + 178 \beta_1 + 40586) q^{23} + (4096 \beta_{3} + 4096 \beta_{2}) q^{24} + ( - 287 \beta_{4} - 50 \beta_1 + 2854320) q^{25} + ( - 400 \beta_{5} + 3792 \beta_{3} + 2240 \beta_{2}) q^{26} + (1580 \beta_{5} + 1976 \beta_{3} + 22876 \beta_{2}) q^{27} + (207 \beta_{4} + 498 \beta_1 + 1744037) q^{29} + (256 \beta_{4} - 2080 \beta_1 - 1394720) q^{30} + (2260 \beta_{5} + 11430 \beta_{3} - 51446 \beta_{2}) q^{31} + 1048576 q^{32} + (2670 \beta_{5} + 36958 \beta_{3} + 53218 \beta_{2}) q^{33} + ( - 1600 \beta_{5} + 15744 \beta_{3} + 52336 \beta_{2}) q^{34} + ( - 256 \beta_{4} + 1536 \beta_1 + 3096064) q^{36} + ( - 653 \beta_{4} + 2938 \beta_1 + 4056625) q^{37} + (48144 \beta_{3} - 15920 \beta_{2}) q^{38} + (210 \beta_{4} - 1500 \beta_1 + 7105410) q^{39} + ( - 4096 \beta_{5} - 12288 \beta_{3} - 8192 \beta_{2}) q^{40} + ( - 900 \beta_{5} - 26244 \beta_{3} + 151337 \beta_{2}) q^{41} + (1153 \beta_{4} - 7883 \beta_1 + 12456752) q^{43} + ( - 256 \beta_{4} + 2816 \beta_1 + 4722688) q^{44} + ( - 13009 \beta_{5} - 241987 \beta_{3} - 564518 \beta_{2}) q^{45} + (1792 \beta_{4} + 2848 \beta_1 + 649376) q^{46} + ( - 6920 \beta_{5} - 258162 \beta_{3} - 16242 \beta_{2}) q^{47} + (65536 \beta_{3} + 65536 \beta_{2}) q^{48} + ( - 4592 \beta_{4} - 800 \beta_1 + 45669120) q^{50} + ( - 1871 \beta_{4} - 3109 \beta_1 + 43373962) q^{51} + ( - 6400 \beta_{5} + 60672 \beta_{3} + 35840 \beta_{2}) q^{52} + (1310 \beta_{4} + 10700 \beta_1 + 22972348) q^{53} + (25280 \beta_{5} + 31616 \beta_{3} + 366016 \beta_{2}) q^{54} + ( - 14268 \beta_{5} - 383064 \beta_{3} - 1048636 \beta_{2}) q^{55} + (995 \beta_{4} + 14050 \beta_1 + 74948345) q^{57} + (3312 \beta_{4} + 7968 \beta_1 + 27904592) q^{58} + (38940 \beta_{5} - 168267 \beta_{3} - 277999 \beta_{2}) q^{59} + (4096 \beta_{4} - 33280 \beta_1 - 22315520) q^{60} + ( - 17575 \beta_{5} - 260397 \beta_{3} + 1618678 \beta_{2}) q^{61} + (36160 \beta_{5} + 182880 \beta_{3} - 823136 \beta_{2}) q^{62} + 16777216 q^{64} + ( - 2305 \beta_{4} - 35710 \beta_1 + 94318275) q^{65} + (42720 \beta_{5} + 591328 \beta_{3} + 851488 \beta_{2}) q^{66} + ( - 6542 \beta_{4} - 21068 \beta_1 - 33828738) q^{67} + ( - 25600 \beta_{5} + 251904 \beta_{3} + 837376 \beta_{2}) q^{68} + (8340 \beta_{5} + 489956 \beta_{3} - 2639644 \beta_{2}) q^{69} + (10270 \beta_{4} + 56350 \beta_1 + 102397488) q^{71} + ( - 4096 \beta_{4} + 24576 \beta_1 + 49537024) q^{72} + ( - 70010 \beta_{5} + 113934 \beta_{3} - 2404929 \beta_{2}) q^{73} + ( - 10448 \beta_{4} + 47008 \beta_1 + 64906000) q^{74} + (67164 \beta_{5} + 2429367 \beta_{3} + 10072083 \beta_{2}) q^{75} + (770304 \beta_{3} - 254720 \beta_{2}) q^{76} + (3360 \beta_{4} - 24000 \beta_1 + 113686560) q^{78} + (14942 \beta_{4} + 27998 \beta_1 - 30390664) q^{79} + ( - 65536 \beta_{5} - 196608 \beta_{3} - 131072 \beta_{2}) q^{80} + ( - 25313 \beta_{4} + 93198 \beta_1 - 72107470) q^{81} + ( - 14400 \beta_{5} - 419904 \beta_{3} + 2421392 \beta_{2}) q^{82} + ( - 43180 \beta_{5} - 119799 \beta_{3} - 6327087 \beta_{2}) q^{83} + ( - 5969 \beta_{4} - 281270 \beta_1 + 344990855) q^{85} + (18448 \beta_{4} - 126128 \beta_1 + 199308032) q^{86} + (52260 \beta_{5} + 2876942 \beta_{3} - 3064078 \beta_{2}) q^{87} + ( - 4096 \beta_{4} + 45056 \beta_1 + 75563008) q^{88} + ( - 113580 \beta_{5} - 1840248 \beta_{3} + 8660755 \beta_{2}) q^{89} + ( - 208144 \beta_{5} - 3871792 \beta_{3} - 9032288 \beta_{2}) q^{90} + (28672 \beta_{4} + 45568 \beta_1 + 10390016) q^{92} + (19806 \beta_{4} + 261084 \beta_1 + 31867458) q^{93} + ( - 110720 \beta_{5} - 4130592 \beta_{3} - 259872 \beta_{2}) q^{94} + (44140 \beta_{4} - 190970 \beta_1 - 218670950) q^{95} + (1048576 \beta_{3} + 1048576 \beta_{2}) q^{96} + (182930 \beta_{5} + 1014690 \beta_{3} + 3117303 \beta_{2}) q^{97} + ( - 70915 \beta_{4} + 323205 \beta_1 + 887229172) q^{99}+O(q^{100})$$ q + 16 * q^2 + (b3 + b2) * q^3 + 256 * q^4 + (-b5 - 3*b3 - 2*b2) * q^5 + (16*b3 + 16*b2) * q^6 + 4096 * q^8 + (-b4 + 6*b1 + 12094) * q^9 + (-16*b5 - 48*b3 - 32*b2) * q^10 + (-b4 + 11*b1 + 18448) * q^11 + (256*b3 + 256*b2) * q^12 + (-25*b5 + 237*b3 + 140*b2) * q^13 + (16*b4 - 130*b1 - 87170) * q^15 + 65536 * q^16 + (-100*b5 + 984*b3 + 3271*b2) * q^17 + (-16*b4 + 96*b1 + 193504) * q^18 + (3009*b3 - 995*b2) * q^19 + (-256*b5 - 768*b3 - 512*b2) * q^20 + (-16*b4 + 176*b1 + 295168) * q^22 + (112*b4 + 178*b1 + 40586) * q^23 + (4096*b3 + 4096*b2) * q^24 + (-287*b4 - 50*b1 + 2854320) * q^25 + (-400*b5 + 3792*b3 + 2240*b2) * q^26 + (1580*b5 + 1976*b3 + 22876*b2) * q^27 + (207*b4 + 498*b1 + 1744037) * q^29 + (256*b4 - 2080*b1 - 1394720) * q^30 + (2260*b5 + 11430*b3 - 51446*b2) * q^31 + 1048576 * q^32 + (2670*b5 + 36958*b3 + 53218*b2) * q^33 + (-1600*b5 + 15744*b3 + 52336*b2) * q^34 + (-256*b4 + 1536*b1 + 3096064) * q^36 + (-653*b4 + 2938*b1 + 4056625) * q^37 + (48144*b3 - 15920*b2) * q^38 + (210*b4 - 1500*b1 + 7105410) * q^39 + (-4096*b5 - 12288*b3 - 8192*b2) * q^40 + (-900*b5 - 26244*b3 + 151337*b2) * q^41 + (1153*b4 - 7883*b1 + 12456752) * q^43 + (-256*b4 + 2816*b1 + 4722688) * q^44 + (-13009*b5 - 241987*b3 - 564518*b2) * q^45 + (1792*b4 + 2848*b1 + 649376) * q^46 + (-6920*b5 - 258162*b3 - 16242*b2) * q^47 + (65536*b3 + 65536*b2) * q^48 + (-4592*b4 - 800*b1 + 45669120) * q^50 + (-1871*b4 - 3109*b1 + 43373962) * q^51 + (-6400*b5 + 60672*b3 + 35840*b2) * q^52 + (1310*b4 + 10700*b1 + 22972348) * q^53 + (25280*b5 + 31616*b3 + 366016*b2) * q^54 + (-14268*b5 - 383064*b3 - 1048636*b2) * q^55 + (995*b4 + 14050*b1 + 74948345) * q^57 + (3312*b4 + 7968*b1 + 27904592) * q^58 + (38940*b5 - 168267*b3 - 277999*b2) * q^59 + (4096*b4 - 33280*b1 - 22315520) * q^60 + (-17575*b5 - 260397*b3 + 1618678*b2) * q^61 + (36160*b5 + 182880*b3 - 823136*b2) * q^62 + 16777216 * q^64 + (-2305*b4 - 35710*b1 + 94318275) * q^65 + (42720*b5 + 591328*b3 + 851488*b2) * q^66 + (-6542*b4 - 21068*b1 - 33828738) * q^67 + (-25600*b5 + 251904*b3 + 837376*b2) * q^68 + (8340*b5 + 489956*b3 - 2639644*b2) * q^69 + (10270*b4 + 56350*b1 + 102397488) * q^71 + (-4096*b4 + 24576*b1 + 49537024) * q^72 + (-70010*b5 + 113934*b3 - 2404929*b2) * q^73 + (-10448*b4 + 47008*b1 + 64906000) * q^74 + (67164*b5 + 2429367*b3 + 10072083*b2) * q^75 + (770304*b3 - 254720*b2) * q^76 + (3360*b4 - 24000*b1 + 113686560) * q^78 + (14942*b4 + 27998*b1 - 30390664) * q^79 + (-65536*b5 - 196608*b3 - 131072*b2) * q^80 + (-25313*b4 + 93198*b1 - 72107470) * q^81 + (-14400*b5 - 419904*b3 + 2421392*b2) * q^82 + (-43180*b5 - 119799*b3 - 6327087*b2) * q^83 + (-5969*b4 - 281270*b1 + 344990855) * q^85 + (18448*b4 - 126128*b1 + 199308032) * q^86 + (52260*b5 + 2876942*b3 - 3064078*b2) * q^87 + (-4096*b4 + 45056*b1 + 75563008) * q^88 + (-113580*b5 - 1840248*b3 + 8660755*b2) * q^89 + (-208144*b5 - 3871792*b3 - 9032288*b2) * q^90 + (28672*b4 + 45568*b1 + 10390016) * q^92 + (19806*b4 + 261084*b1 + 31867458) * q^93 + (-110720*b5 - 4130592*b3 - 259872*b2) * q^94 + (44140*b4 - 190970*b1 - 218670950) * q^95 + (1048576*b3 + 1048576*b2) * q^96 + (182930*b5 + 1014690*b3 + 3117303*b2) * q^97 + (-70915*b4 + 323205*b1 + 887229172) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 96 q^{2} + 1536 q^{4} + 24576 q^{8} + 72550 q^{9}+O(q^{10})$$ 6 * q + 96 * q^2 + 1536 * q^4 + 24576 * q^8 + 72550 * q^9 $$6 q + 96 q^{2} + 1536 q^{4} + 24576 q^{8} + 72550 q^{9} + 110664 q^{11} - 522728 q^{15} + 393216 q^{16} + 1160800 q^{18} + 1770624 q^{22} + 243384 q^{23} + 17125446 q^{25} + 10463640 q^{29} - 8363648 q^{30} + 6291456 q^{32} + 18572800 q^{36} + 24332568 q^{37} + 42635880 q^{39} + 74758584 q^{43} + 28329984 q^{44} + 3894144 q^{46} + 274007136 q^{50} + 260246248 q^{51} + 137815308 q^{53} + 449663960 q^{57} + 167418240 q^{58} - 133818368 q^{60} + 100663296 q^{64} + 565976460 q^{65} - 202943376 q^{67} + 614292768 q^{71} + 297164800 q^{72} + 389321088 q^{74} + 682174080 q^{78} - 182370096 q^{79} - 432881842 q^{81} + 2070495732 q^{85} + 1196137344 q^{86} + 453279744 q^{88} + 62306304 q^{92} + 190722192 q^{93} - 1311555480 q^{95} + 5322586792 q^{99}+O(q^{100})$$ 6 * q + 96 * q^2 + 1536 * q^4 + 24576 * q^8 + 72550 * q^9 + 110664 * q^11 - 522728 * q^15 + 393216 * q^16 + 1160800 * q^18 + 1770624 * q^22 + 243384 * q^23 + 17125446 * q^25 + 10463640 * q^29 - 8363648 * q^30 + 6291456 * q^32 + 18572800 * q^36 + 24332568 * q^37 + 42635880 * q^39 + 74758584 * q^43 + 28329984 * q^44 + 3894144 * q^46 + 274007136 * q^50 + 260246248 * q^51 + 137815308 * q^53 + 449663960 * q^57 + 167418240 * q^58 - 133818368 * q^60 + 100663296 * q^64 + 565976460 * q^65 - 202943376 * q^67 + 614292768 * q^71 + 297164800 * q^72 + 389321088 * q^74 + 682174080 * q^78 - 182370096 * q^79 - 432881842 * q^81 + 2070495732 * q^85 + 1196137344 * q^86 + 453279744 * q^88 + 62306304 * q^92 + 190722192 * q^93 - 1311555480 * q^95 + 5322586792 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 7165x^{4} + 16168x^{3} + 12807408x^{2} - 32180328x - 5372164$$ :

 $$\beta_{1}$$ $$=$$ $$( 16557 \nu^{5} + 1210854 \nu^{4} - 100542547 \nu^{3} - 8636476956 \nu^{2} + 22368923366 \nu + 10243942375796 ) / 3101451260$$ (16557*v^5 + 1210854*v^4 - 100542547*v^3 - 8636476956*v^2 + 22368923366*v + 10243942375796) / 3101451260 $$\beta_{2}$$ $$=$$ $$( -10717\nu^{5} + 2982\nu^{4} + 64030099\nu^{3} - 46012148\nu^{2} - 91498140874\nu + 114577250368 ) / 1860870756$$ (-10717*v^5 + 2982*v^4 + 64030099*v^3 - 46012148*v^2 - 91498140874*v + 114577250368) / 1860870756 $$\beta_{3}$$ $$=$$ $$( - 142991 \nu^{5} - 3910461 \nu^{4} + 1169730299 \nu^{3} + 13263914945 \nu^{2} - 2355511160234 \nu + 2902579423454 ) / 18608707560$$ (-142991*v^5 - 3910461*v^4 + 1169730299*v^3 + 13263914945*v^2 - 2355511160234*v + 2902579423454) / 18608707560 $$\beta_{4}$$ $$=$$ $$( 26311 \nu^{5} - 1222782 \nu^{4} - 155577849 \nu^{3} + 8820525548 \nu^{2} - 21107028046 \nu - 10581294778128 ) / 1240580504$$ (26311*v^5 - 1222782*v^4 - 155577849*v^3 + 8820525548*v^2 - 21107028046*v - 10581294778128) / 1240580504 $$\beta_{5}$$ $$=$$ $$( - 374123 \nu^{5} + 15601227 \nu^{4} + 3145018667 \nu^{3} - 58197203515 \nu^{2} - 6442323297722 \nu + 8276645371622 ) / 4652176890$$ (-374123*v^5 + 15601227*v^4 + 3145018667*v^3 - 58197203515*v^2 - 6442323297722*v + 8276645371622) / 4652176890
 $$\nu$$ $$=$$ $$( -2\beta_{4} - 12\beta_{2} - 5\beta _1 + 195 ) / 588$$ (-2*b4 - 12*b2 - 5*b1 + 195) / 588 $$\nu^{2}$$ $$=$$ $$( 28\beta_{5} + 58\beta_{4} - 336\beta_{3} + 16\beta_{2} - 275\beta _1 + 1404621 ) / 588$$ (28*b5 + 58*b4 - 336*b3 + 16*b2 - 275*b1 + 1404621) / 588 $$\nu^{3}$$ $$=$$ $$( 1302\beta_{5} - 7114\beta_{4} + 19656\beta_{3} - 87480\beta_{2} - 18205\beta _1 - 543285 ) / 588$$ (1302*b5 - 7114*b4 + 19656*b3 - 87480*b2 - 18205*b1 - 543285) / 588 $$\nu^{4}$$ $$=$$ $$( 202328\beta_{5} + 210234\beta_{4} - 2380896\beta_{3} + 214016\beta_{2} - 983475\beta _1 + 5039754837 ) / 588$$ (202328*b5 + 210234*b4 - 2380896*b3 + 214016*b2 - 983475*b1 + 5039754837) / 588 $$\nu^{5}$$ $$=$$ $$( 7715050 \beta_{5} - 25618702 \beta_{4} + 118217400 \beta_{3} - 522316448 \beta_{2} - 65172775 \beta _1 - 3252625455 ) / 588$$ (7715050*b5 - 25618702*b4 + 118217400*b3 - 522316448*b2 - 65172775*b1 - 3252625455) / 588

## 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
 2.67131 −61.3733 61.1162 58.2878 −58.5449 −0.157117
16.0000 −245.181 256.000 2662.60 −3922.90 0 4096.00 40431.0 42601.6
1.2 16.0000 −143.106 256.000 −474.347 −2289.70 0 4096.00 796.313 −7589.55
1.3 16.0000 −121.370 256.000 −2666.02 −1941.92 0 4096.00 −4952.27 −42656.3
1.4 16.0000 121.370 256.000 2666.02 1941.92 0 4096.00 −4952.27 42656.3
1.5 16.0000 143.106 256.000 474.347 2289.70 0 4096.00 796.313 7589.55
1.6 16.0000 245.181 256.000 −2662.60 3922.90 0 4096.00 40431.0 −42601.6
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.10.a.l 6
7.b odd 2 1 inner 98.10.a.l 6
7.c even 3 2 98.10.c.n 12
7.d odd 6 2 98.10.c.n 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.10.a.l 6 1.a even 1 1 trivial
98.10.a.l 6 7.b odd 2 1 inner
98.10.c.n 12 7.c even 3 2
98.10.c.n 12 7.d odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 95324T_{3}^{4} + 2418290244T_{3}^{2} - 18134891776800$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 16)^{6}$$
$3$ $$T^{6} - 95324 T^{4} + \cdots - 18134891776800$$
$5$ $$T^{6} - 14422098 T^{4} + \cdots - 11\!\cdots\!00$$
$7$ $$T^{6}$$
$11$ $$(T^{3} - 55332 T^{2} + \cdots + 13071858183056)^{2}$$
$13$ $$T^{6} - 13450472658 T^{4} + \cdots - 85\!\cdots\!00$$
$17$ $$T^{6} - 375850017702 T^{4} + \cdots - 83\!\cdots\!72$$
$19$ $$T^{6} - 720924860892 T^{4} + \cdots - 81\!\cdots\!00$$
$23$ $$(T^{3} - 121692 T^{2} + \cdots - 95\!\cdots\!60)^{2}$$
$29$ $$(T^{3} - 5231820 T^{2} + \cdots + 15\!\cdots\!76)^{2}$$
$31$ $$T^{6} - 116478776411856 T^{4} + \cdots - 11\!\cdots\!08$$
$37$ $$(T^{3} - 12166284 T^{2} + \cdots - 24\!\cdots\!00)^{2}$$
$41$ $$T^{6} - 386100075738678 T^{4} + \cdots - 53\!\cdots\!68$$
$43$ $$(T^{3} - 37379292 T^{2} + \cdots + 20\!\cdots\!48)^{2}$$
$47$ $$T^{6} + \cdots - 28\!\cdots\!00$$
$53$ $$(T^{3} - 68907654 T^{2} + \cdots + 73\!\cdots\!68)^{2}$$
$59$ $$T^{6} + \cdots - 50\!\cdots\!92$$
$61$ $$T^{6} + \cdots - 50\!\cdots\!00$$
$67$ $$(T^{3} + 101471688 T^{2} + \cdots - 72\!\cdots\!12)^{2}$$
$71$ $$(T^{3} - 307146384 T^{2} + \cdots + 12\!\cdots\!48)^{2}$$
$73$ $$T^{6} + \cdots - 11\!\cdots\!32$$
$79$ $$(T^{3} + 91185048 T^{2} + \cdots - 37\!\cdots\!20)^{2}$$
$83$ $$T^{6} + \cdots - 61\!\cdots\!52$$
$89$ $$T^{6} + \cdots - 13\!\cdots\!00$$
$97$ $$T^{6} + \cdots - 13\!\cdots\!72$$
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