# Properties

 Label 1620.4.i.w Level $1620$ Weight $4$ Character orbit 1620.i Analytic conductor $95.583$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1620.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$95.5830942093$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904$$ x^12 + 306*x^10 + 30777*x^8 + 1381040*x^6 + 28918584*x^4 + 243888288*x^2 + 366645904 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}\cdot 3^{13}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 \beta_{6} q^{5} + (\beta_{8} - 2 \beta_{6} - \beta_{2} - 2) q^{7}+O(q^{10})$$ q + 5*b6 * q^5 + (b8 - 2*b6 - b2 - 2) * q^7 $$q + 5 \beta_{6} q^{5} + (\beta_{8} - 2 \beta_{6} - \beta_{2} - 2) q^{7} + (\beta_{10} - \beta_{9} - \beta_{7}) q^{11} + ( - \beta_{11} + \beta_{10} - 2 \beta_{8} - \beta_{7} - 14 \beta_{6} - \beta_{5} + \beta_{4}) q^{13} + ( - \beta_{5} - 5 \beta_{2} - 3 \beta_1 - 2) q^{17} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_1 - 19) q^{19} + (\beta_{11} + \beta_{10} + 7 \beta_{8} - 2 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} + \beta_{4}) q^{23} + ( - 25 \beta_{6} - 25) q^{25} + (4 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 3 \beta_{7} + 28 \beta_{6} - 4 \beta_1 + 28) q^{29} + (5 \beta_{11} - 2 \beta_{10} - \beta_{9} + 5 \beta_{8} - 6 \beta_{7} - 54 \beta_{6} - 6 \beta_{5} + \cdots + \beta_{3}) q^{31}+ \cdots + (19 \beta_{11} + 4 \beta_{10} - 16 \beta_{9} + 27 \beta_{8} + 13 \beta_{7} + \cdots + 198) q^{97}+O(q^{100})$$ q + 5*b6 * q^5 + (b8 - 2*b6 - b2 - 2) * q^7 + (b10 - b9 - b7) * q^11 + (-b11 + b10 - 2*b8 - b7 - 14*b6 - b5 + b4) * q^13 + (-b5 - 5*b2 - 3*b1 - 2) * q^17 + (-b4 - 3*b3 + 2*b1 - 19) * q^19 + (b11 + b10 + 7*b8 - 2*b7 - 5*b6 - 2*b5 + b4) * q^23 + (-25*b6 - 25) * q^25 + (4*b11 + 3*b10 + 2*b9 + 3*b7 + 28*b6 - 4*b1 + 28) * q^29 + (5*b11 - 2*b10 - b9 + 5*b8 - 6*b7 - 54*b6 - 6*b5 - 2*b4 + b3) * q^31 + (5*b2 + 10) * q^35 + (5*b5 - 2*b4 + 4*b3 - 5*b2 - b1 - 82) * q^37 + (-3*b11 - 4*b9 + 2*b8 + 10*b7 - 52*b6 + 10*b5 + 4*b3) * q^41 + (-b11 - 5*b10 + 2*b9 - 7*b8 - 9*b7 + 26*b6 + 7*b2 + b1 + 26) * q^43 + (4*b11 - 8*b10 - 2*b9 - 10*b8 - 3*b7 + 77*b6 + 10*b2 - 4*b1 + 77) * q^47 + (5*b11 + b10 - 2*b9 + 5*b8 + 8*b7 - 98*b6 + 8*b5 + b4 + 2*b3) * q^49 + (7*b5 - 5*b4 - 6*b3 - 8*b2 - 6*b1 - 169) * q^53 + (-5*b5 + 5*b4 + 5*b3) * q^55 + (-13*b11 + 2*b10 + 5*b9 - 16*b8 - 10*b7 - 168*b6 - 10*b5 + 2*b4 - 5*b3) * q^59 + (12*b10 - 4*b9 + 2*b8 - 14*b7 - 6*b6 - 2*b2 - 6) * q^61 + (5*b11 - 5*b10 + 10*b8 + 5*b7 + 70*b6 - 10*b2 - 5*b1 + 70) * q^65 + (12*b11 - 2*b10 + 10*b9 - 24*b8 + 2*b7 + 24*b6 + 2*b5 - 2*b4 - 10*b3) * q^67 + (5*b5 + 13*b4 + 17*b3 + 12*b2 - 14*b1 - 202) * q^71 + (7*b5 + 16*b4 + 4*b3 + 49*b2 + 31*b1 - 150) * q^73 + (15*b11 - 18*b10 + 16*b9 - 9*b8 - 9*b7 - 112*b6 - 9*b5 - 18*b4 - 16*b3) * q^77 + (26*b11 - 4*b10 + 6*b9 + 4*b8 + 32*b7 + 156*b6 - 4*b2 - 26*b1 + 156) * q^79 + (-17*b11 - 5*b10 + 16*b9 - 23*b8 - 5*b7 + 48*b6 + 23*b2 + 17*b1 + 48) * q^83 + (15*b11 + 25*b8 + 5*b7 - 10*b6 + 5*b5) * q^85 + (-25*b5 - 7*b4 - 4*b3 + 14*b2 + 10*b1 + 20) * q^89 + (-18*b5 - 15*b4 - 14*b3 - 45*b2 + 3*b1 + 381) * q^91 + (-10*b11 + 5*b10 - 15*b9 - 95*b6 + 5*b4 + 15*b3) * q^95 + (19*b11 + 4*b10 - 16*b9 + 27*b8 + 13*b7 + 198*b6 - 27*b2 - 19*b1 + 198) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 30 q^{5} - 12 q^{7}+O(q^{10})$$ 12 * q - 30 * q^5 - 12 * q^7 $$12 q - 30 q^{5} - 12 q^{7} + 84 q^{13} - 24 q^{17} - 228 q^{19} + 30 q^{23} - 150 q^{25} + 168 q^{29} + 324 q^{31} + 120 q^{35} - 984 q^{37} + 312 q^{41} + 156 q^{43} + 462 q^{47} + 588 q^{49} - 2028 q^{53} + 1008 q^{59} - 36 q^{61} + 420 q^{65} - 144 q^{67} - 2424 q^{71} - 1800 q^{73} + 672 q^{77} + 936 q^{79} + 288 q^{83} + 60 q^{85} + 240 q^{89} + 4572 q^{91} + 570 q^{95} + 1188 q^{97}+O(q^{100})$$ 12 * q - 30 * q^5 - 12 * q^7 + 84 * q^13 - 24 * q^17 - 228 * q^19 + 30 * q^23 - 150 * q^25 + 168 * q^29 + 324 * q^31 + 120 * q^35 - 984 * q^37 + 312 * q^41 + 156 * q^43 + 462 * q^47 + 588 * q^49 - 2028 * q^53 + 1008 * q^59 - 36 * q^61 + 420 * q^65 - 144 * q^67 - 2424 * q^71 - 1800 * q^73 + 672 * q^77 + 936 * q^79 + 288 * q^83 + 60 * q^85 + 240 * q^89 + 4572 * q^91 + 570 * q^95 + 1188 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904$$ :

 $$\beta_{1}$$ $$=$$ $$( - 1291 \nu^{10} - 383792 \nu^{8} - 35790499 \nu^{6} - 1330336354 \nu^{4} - 18065389388 \nu^{2} - 45652500136 ) / 1027615680$$ (-1291*v^10 - 383792*v^8 - 35790499*v^6 - 1330336354*v^4 - 18065389388*v^2 - 45652500136) / 1027615680 $$\beta_{2}$$ $$=$$ $$( 5102693 \nu^{10} + 1398025756 \nu^{8} + 112247897357 \nu^{6} + 3452478375842 \nu^{4} + 38122728094324 \nu^{2} + \cdots + 83013902159528 ) / 1103659240320$$ (5102693*v^10 + 1398025756*v^8 + 112247897357*v^6 + 3452478375842*v^4 + 38122728094324*v^2 + 83013902159528) / 1103659240320 $$\beta_{3}$$ $$=$$ $$( 44857 \nu^{10} + 12522644 \nu^{8} + 1045133377 \nu^{6} + 34081870402 \nu^{4} + 393573716612 \nu^{2} + 423568473640 ) / 8489686464$$ (44857*v^10 + 12522644*v^8 + 1045133377*v^6 + 34081870402*v^4 + 393573716612*v^2 + 423568473640) / 8489686464 $$\beta_{4}$$ $$=$$ $$( 119695 \nu^{10} + 32147810 \nu^{8} + 2466092659 \nu^{6} + 69043953676 \nu^{4} + 581554032092 \nu^{2} - 136972654160 ) / 18394320672$$ (119695*v^10 + 32147810*v^8 + 2466092659*v^6 + 69043953676*v^4 + 581554032092*v^2 - 136972654160) / 18394320672 $$\beta_{5}$$ $$=$$ $$( - 10810771 \nu^{10} - 2947200752 \nu^{8} - 234983924419 \nu^{6} - 7244547031114 \nu^{4} - 80864075604908 \nu^{2} + \cdots - 153225963267016 ) / 1103659240320$$ (-10810771*v^10 - 2947200752*v^8 - 234983924419*v^6 - 7244547031114*v^4 - 80864075604908*v^2 - 153225963267016) / 1103659240320 $$\beta_{6}$$ $$=$$ $$( - 23357 \nu^{11} - 6338239 \nu^{9} - 500676503 \nu^{7} - 15279946733 \nu^{5} - 175329177676 \nu^{3} - 573920985812 \nu - 233206555680 ) / 466413111360$$ (-23357*v^11 - 6338239*v^9 - 500676503*v^7 - 15279946733*v^5 - 175329177676*v^3 - 573920985812*v - 233206555680) / 466413111360 $$\beta_{7}$$ $$=$$ $$( 3422391862 \nu^{11} + 155253482331 \nu^{10} + 678724169054 \nu^{9} + 42324749999472 \nu^{8} + 8571410900578 \nu^{7} + \cdots + 22\!\cdots\!76 ) / 31\!\cdots\!40$$ (3422391862*v^11 + 155253482331*v^10 + 678724169054*v^9 + 42324749999472*v^8 + 8571410900578*v^7 + 3374604138581259*v^6 - 2317989495878582*v^5 + 104038939913828154*v^4 - 80772383343930664*v^3 + 1161288989762083788*v^2 - 591211552618868408*v + 2200478058477616776) / 31699300700471040 $$\beta_{8}$$ $$=$$ $$( 1657489291 \nu^{11} + 24426591391 \nu^{10} + 498195552242 \nu^{9} + 6692349293972 \nu^{8} + 47646796509679 \nu^{7} + \cdots + 39\!\cdots\!36 ) / 10\!\cdots\!80$$ (1657489291*v^11 + 24426591391*v^10 + 498195552242*v^9 + 6692349293972*v^8 + 47646796509679*v^7 + 537330684647959*v^6 + 1885300499238244*v^5 + 16527013985155654*v^4 + 31285410315898988*v^3 + 182493499387528988*v^2 + 188670337960980496*v + 397387549637660536) / 10566433566823680 $$\beta_{9}$$ $$=$$ $$( - 867417539 \nu^{11} + 3220956885 \nu^{10} - 242242649518 \nu^{9} + 899188452420 \nu^{8} - 20179270104191 \nu^{7} + \cdots + 30\!\cdots\!00 ) / 12\!\cdots\!40$$ (-867417539*v^11 + 3220956885*v^10 - 242242649518*v^9 + 899188452420*v^8 - 20179270104191*v^7 + 75045802135485*v^6 - 646208074534496*v^5 + 2447248704215610*v^4 - 6715236806064652*v^3 + 28260560721324660*v^2 + 8074723128389056*v + 30414334249720200) / 1219203873095040 $$\beta_{10}$$ $$=$$ $$( 4899606623 \nu^{11} - 17189398950 \nu^{10} + 1275959537596 \nu^{9} - 4616746994100 \nu^{8} + 90300640751687 \nu^{7} + \cdots + 19\!\cdots\!00 ) / 52\!\cdots\!40$$ (4899606623*v^11 - 17189398950*v^10 + 1275959537596*v^9 - 4616746994100*v^8 + 90300640751687*v^7 - 354155566758990*v^6 + 2027571824356622*v^5 - 9915402187410360*v^4 + 2175231711636604*v^3 - 83516974548732120*v^2 - 184512033378602152*v + 19670642863917600) / 5283216783411840 $$\beta_{11}$$ $$=$$ $$( - 1866498795 \nu^{11} - 1106223043 \nu^{10} - 516891193560 \nu^{9} - 328861002416 \nu^{8} - 42548567539275 \nu^{7} + \cdots - 39\!\cdots\!28 ) / 17\!\cdots\!80$$ (-1866498795*v^11 - 1106223043*v^10 - 516891193560*v^9 - 328861002416*v^8 - 42548567539275*v^7 - 30667912249627*v^6 - 1375919491743090*v^5 - 1139929302661042*v^4 - 16698652541954460*v^3 - 15479744401063724*v^2 - 44222535099900360*v - 39118394749034728) / 1761072261137280
 $$\nu$$ $$=$$ $$( 2\beta_{10} + 2\beta_{9} - 4\beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{3} ) / 18$$ (2*b10 + 2*b9 - 4*b7 - 2*b5 + b4 - b3) / 18 $$\nu^{2}$$ $$=$$ $$( -6\beta_{5} + 3\beta_{4} - 21\beta_{3} - 26\beta _1 - 918 ) / 18$$ (-6*b5 + 3*b4 - 21*b3 - 26*b1 - 918) / 18 $$\nu^{3}$$ $$=$$ $$( 408 \beta_{11} - 164 \beta_{10} - 314 \beta_{9} + 300 \beta_{8} + 304 \beta_{7} - 5616 \beta_{6} + 152 \beta_{5} - 82 \beta_{4} + 157 \beta_{3} - 150 \beta_{2} - 204 \beta _1 - 2808 ) / 18$$ (408*b11 - 164*b10 - 314*b9 + 300*b8 + 304*b7 - 5616*b6 + 152*b5 - 82*b4 + 157*b3 - 150*b2 - 204*b1 - 2808) / 18 $$\nu^{4}$$ $$=$$ $$( 1398\beta_{5} - 1131\beta_{4} + 3507\beta_{3} + 1980\beta_{2} + 5278\beta _1 + 96246 ) / 18$$ (1398*b5 - 1131*b4 + 3507*b3 + 1980*b2 + 5278*b1 + 96246) / 18 $$\nu^{5}$$ $$=$$ $$( - 79160 \beta_{11} + 20624 \beta_{10} + 51938 \beta_{9} - 49668 \beta_{8} - 33208 \beta_{7} + 1092240 \beta_{6} - 16604 \beta_{5} + 10312 \beta_{4} - 25969 \beta_{3} + 24834 \beta_{2} + \cdots + 546120 ) / 18$$ (-79160*b11 + 20624*b10 + 51938*b9 - 49668*b8 - 33208*b7 + 1092240*b6 - 16604*b5 + 10312*b4 - 25969*b3 + 24834*b2 + 39580*b1 + 546120) / 18 $$\nu^{6}$$ $$=$$ $$( -88698\beta_{5} + 68253\beta_{4} - 190725\beta_{3} - 147780\beta_{2} - 300898\beta _1 - 4542450 ) / 6$$ (-88698*b5 + 68253*b4 - 190725*b3 - 147780*b2 - 300898*b1 - 4542450) / 6 $$\nu^{7}$$ $$=$$ $$( 13312824 \beta_{11} - 3150776 \beta_{10} - 8464034 \beta_{9} + 7729524 \beta_{8} + 4620712 \beta_{7} - 185682672 \beta_{6} + 2310356 \beta_{5} - 1575388 \beta_{4} + \cdots - 92841336 ) / 18$$ (13312824*b11 - 3150776*b10 - 8464034*b9 + 7729524*b8 + 4620712*b7 - 185682672*b6 + 2310356*b5 - 1575388*b4 + 4232017*b3 - 3864762*b2 - 6656412*b1 - 92841336) / 18 $$\nu^{8}$$ $$=$$ $$( 45823038 \beta_{5} - 33969423 \beta_{4} + 93192375 \beta_{3} + 78683724 \beta_{2} + 148284326 \beta _1 + 2128577670 ) / 18$$ (45823038*b5 - 33969423*b4 + 93192375*b3 + 78683724*b2 + 148284326*b1 + 2128577670) / 18 $$\nu^{9}$$ $$=$$ $$( - 2181126936 \beta_{11} + 505095320 \beta_{10} + 1375227746 \beta_{9} - 1225681140 \beta_{8} - 713179240 \beta_{7} + 30601018512 \beta_{6} + \cdots + 15300509256 ) / 18$$ (-2181126936*b11 + 505095320*b10 + 1375227746*b9 - 1225681140*b8 - 713179240*b7 + 30601018512*b6 - 356589620*b5 + 252547660*b4 - 687613873*b3 + 612840570*b2 + 1090563468*b1 + 15300509256) / 18 $$\nu^{10}$$ $$=$$ $$( - 7602088494 \beta_{5} + 5545453335 \beta_{4} - 15162026055 \beta_{3} - 13140890748 \beta_{2} - 24146228950 \beta _1 - 341966456118 ) / 18$$ (-7602088494*b5 + 5545453335*b4 - 15162026055*b3 - 13140890748*b2 - 24146228950*b1 - 341966456118) / 18 $$\nu^{11}$$ $$=$$ $$( 355230206072 \beta_{11} - 81835116872 \beta_{10} - 223422680258 \beta_{9} + 197156937876 \beta_{8} + 114022803976 \beta_{7} - 4996477881648 \beta_{6} + \cdots - 2498238940824 ) / 18$$ (355230206072*b11 - 81835116872*b10 - 223422680258*b9 + 197156937876*b8 + 114022803976*b7 - 4996477881648*b6 + 57011401988*b5 - 40917558436*b4 + 111711340129*b3 - 98578468938*b2 - 177615103036*b1 - 2498238940824) / 18

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 1.37492i 5.63924i 4.35846i − 12.7486i − 6.01416i 7.39017i − 1.37492i − 5.63924i − 4.35846i 12.7486i 6.01416i − 7.39017i
0 0 0 −2.50000 4.33013i 0 −11.2133 + 19.4221i 0 0 0
541.2 0 0 0 −2.50000 4.33013i 0 −8.28565 + 14.3512i 0 0 0
541.3 0 0 0 −2.50000 4.33013i 0 −3.10707 + 5.38160i 0 0 0
541.4 0 0 0 −2.50000 4.33013i 0 −0.292449 + 0.506536i 0 0 0
541.5 0 0 0 −2.50000 4.33013i 0 5.24065 9.07707i 0 0 0
541.6 0 0 0 −2.50000 4.33013i 0 11.6578 20.1920i 0 0 0
1081.1 0 0 0 −2.50000 + 4.33013i 0 −11.2133 19.4221i 0 0 0
1081.2 0 0 0 −2.50000 + 4.33013i 0 −8.28565 14.3512i 0 0 0
1081.3 0 0 0 −2.50000 + 4.33013i 0 −3.10707 5.38160i 0 0 0
1081.4 0 0 0 −2.50000 + 4.33013i 0 −0.292449 0.506536i 0 0 0
1081.5 0 0 0 −2.50000 + 4.33013i 0 5.24065 + 9.07707i 0 0 0
1081.6 0 0 0 −2.50000 + 4.33013i 0 11.6578 + 20.1920i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1081.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.4.i.w 12
3.b odd 2 1 1620.4.i.x 12
9.c even 3 1 1620.4.a.j yes 6
9.c even 3 1 inner 1620.4.i.w 12
9.d odd 6 1 1620.4.a.i 6
9.d odd 6 1 1620.4.i.x 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.4.a.i 6 9.d odd 6 1
1620.4.a.j yes 6 9.c even 3 1
1620.4.i.w 12 1.a even 1 1 trivial
1620.4.i.w 12 9.c even 3 1 inner
1620.4.i.x 12 3.b odd 2 1
1620.4.i.x 12 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{12} + 12 T_{7}^{11} + 807 T_{7}^{10} + 7612 T_{7}^{9} + 465309 T_{7}^{8} + 4143240 T_{7}^{7} + 98836980 T_{7}^{6} + 281425968 T_{7}^{5} + 9518654844 T_{7}^{4} + 46178285440 T_{7}^{3} + \cdots + 108966010000$$ T7^12 + 12*T7^11 + 807*T7^10 + 7612*T7^9 + 465309*T7^8 + 4143240*T7^7 + 98836980*T7^6 + 281425968*T7^5 + 9518654844*T7^4 + 46178285440*T7^3 + 345480783600*T7^2 + 200199048000*T7 + 108966010000 $$T_{11}^{12} + 5826 T_{11}^{10} - 120960 T_{11}^{9} + 27242307 T_{11}^{8} - 421485120 T_{11}^{7} + 39574013994 T_{11}^{6} - 400272788160 T_{11}^{5} + 39988229062761 T_{11}^{4} + \cdots + 24\!\cdots\!00$$ T11^12 + 5826*T11^10 - 120960*T11^9 + 27242307*T11^8 - 421485120*T11^7 + 39574013994*T11^6 - 400272788160*T11^5 + 39988229062761*T11^4 - 274593035828160*T11^3 + 15223470471794700*T11^2 + 107765874298656000*T11 + 2430225018940410000

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$(T^{2} + 5 T + 25)^{6}$$
$7$ $$T^{12} + 12 T^{11} + \cdots + 108966010000$$
$11$ $$T^{12} + 5826 T^{10} + \cdots + 24\!\cdots\!00$$
$13$ $$T^{12} - 84 T^{11} + \cdots + 66\!\cdots\!00$$
$17$ $$(T^{6} + 12 T^{5} - 14628 T^{4} + \cdots + 20589584400)^{2}$$
$19$ $$(T^{6} + 114 T^{5} + \cdots - 171382733975)^{2}$$
$23$ $$T^{12} - 30 T^{11} + \cdots + 55\!\cdots\!00$$
$29$ $$T^{12} - 168 T^{11} + \cdots + 13\!\cdots\!44$$
$31$ $$T^{12} - 324 T^{11} + \cdots + 15\!\cdots\!44$$
$37$ $$(T^{6} + 492 T^{5} + \cdots + 147577423801600)^{2}$$
$41$ $$T^{12} - 312 T^{11} + \cdots + 30\!\cdots\!21$$
$43$ $$T^{12} - 156 T^{11} + \cdots + 15\!\cdots\!00$$
$47$ $$T^{12} - 462 T^{11} + \cdots + 38\!\cdots\!00$$
$53$ $$(T^{6} + 1014 T^{5} + \cdots + 24472129608900)^{2}$$
$59$ $$T^{12} - 1008 T^{11} + \cdots + 35\!\cdots\!69$$
$61$ $$T^{12} + 36 T^{11} + \cdots + 75\!\cdots\!00$$
$67$ $$T^{12} + 144 T^{11} + \cdots + 58\!\cdots\!00$$
$71$ $$(T^{6} + 1212 T^{5} + \cdots - 36\!\cdots\!24)^{2}$$
$73$ $$(T^{6} + 900 T^{5} + \cdots + 68\!\cdots\!00)^{2}$$
$79$ $$T^{12} - 936 T^{11} + \cdots + 22\!\cdots\!64$$
$83$ $$T^{12} - 288 T^{11} + \cdots + 56\!\cdots\!00$$
$89$ $$(T^{6} - 120 T^{5} + \cdots + 157896529778448)^{2}$$
$97$ $$T^{12} - 1188 T^{11} + \cdots + 42\!\cdots\!00$$