# Properties

 Label 117.4.q.e Level $117$ Weight $4$ Character orbit 117.q Analytic conductor $6.903$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648$$ x^10 + 70*x^8 + 1645*x^6 + 14700*x^4 + 44100*x^2 + 27648 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{5} + 6 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} - \beta_1 - 1) q^{5} + (\beta_{8} - 2 \beta_{2} + 4) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + 7 \beta_1) q^{8}+O(q^{10})$$ q + (-b3 + b1) * q^2 + (-b5 + 6*b2) * q^4 + (-b7 + b6 + 2*b2 - b1 - 1) * q^5 + (b8 - 2*b2 + 4) * q^7 + (-b9 + b7 - b6 - 2*b5 + b4 + 7*b1) * q^8 $$q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{5} + 6 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} - \beta_1 - 1) q^{5} + (\beta_{8} - 2 \beta_{2} + 4) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + 7 \beta_1) q^{8} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 3 \beta_{3} - 8 \beta_{2} + \cdots + 8) q^{10}+ \cdots + ( - 33 \beta_{8} + 21 \beta_{7} - 15 \beta_{5} + 30 \beta_{4} - 11 \beta_{3} + \cdots - 24) q^{98}+O(q^{100})$$ q + (-b3 + b1) * q^2 + (-b5 + 6*b2) * q^4 + (-b7 + b6 + 2*b2 - b1 - 1) * q^5 + (b8 - 2*b2 + 4) * q^7 + (-b9 + b7 - b6 - 2*b5 + b4 + 7*b1) * q^8 + (-b9 - b8 + 2*b7 - b6 + b5 - b4 + 3*b3 - 8*b2 + 3*b1 + 8) * q^10 + (-b9 + b8 + b6 + 2*b5 + 2*b4 + 5*b3 - 4*b2 - 5*b1 - 4) * q^11 + (-b9 + b8 - b7 + 2*b6 + 2*b5 - 2*b4 - 3*b3 - 17*b2 + 6*b1 + 11) * q^13 + (2*b7 + 2*b6 - 3*b4 - 2*b3 + b1 + 6) * q^14 + (-4*b7 + 2*b6 - 5*b5 + 5*b4 + 6*b3 + 50*b2 + 6*b1 - 50) * q^16 + (2*b9 - b8 - 4*b3 - 21*b2 + 8*b1) * q^17 + (-b8 + 3*b7 - 4*b5 + 8*b4 - 9*b3 - 12*b2 + 24) * q^19 + (4*b8 - 2*b7 - 5*b5 + 10*b4 - 18*b3 + 34*b2 - 68) * q^20 + (2*b9 - b8 - 3*b7 + 6*b6 + 8*b5 + 21*b3 - 58*b2 - 42*b1) * q^22 + (-3*b9 - 3*b8 + 2*b7 - b6 + b3 - 12*b2 + b1 + 12) * q^23 + (-b7 - b6 - 2*b4 - 42*b3 + 21*b1 - 96) * q^25 + (-3*b9 + 2*b8 - b7 + b6 - 7*b5 - b4 - 28*b3 + 102*b2 - 2*b1 - 48) * q^26 + (-b9 + b8 - 9*b6 - 4*b5 - 4*b4 - 45*b3 + 10*b2 + 45*b1 + 10) * q^28 + (-6*b7 + 3*b6 + 6*b5 - 6*b4 - 11*b3 - 99*b2 - 11*b1 + 99) * q^29 + (2*b9 + 3*b7 - 3*b6 + 12*b5 - 6*b4 + 44*b2 - 9*b1 - 22) * q^31 + (-3*b8 + 3*b7 - 3*b5 + 6*b4 + 51*b3 + 96*b2 - 192) * q^32 + (6*b7 - 6*b6 - 2*b5 + b4 + 100*b2 - 18*b1 - 50) * q^34 + (6*b9 - 3*b8 - 5*b7 + 10*b6 + 18*b5 - 35*b3 - 12*b2 + 70*b1) * q^35 + (b9 - b8 + 12*b6 + 36*b3 - 27*b2 - 36*b1 - 27) * q^37 + (b9 - 2*b8 - b7 - b6 + 6*b4 + 34*b3 - 17*b1 + 138) * q^38 + (-b9 + 2*b8 + 7*b7 + 7*b6 - 7*b4 + 126*b3 - 63*b1 + 200) * q^40 + (-3*b9 + 3*b8 - 8*b6 - 6*b5 - 6*b4 + 32*b3 - 71*b2 - 32*b1 - 71) * q^41 + (2*b9 - b8 + 10*b7 - 20*b6 - 6*b5 - 18*b3 - 74*b2 + 36*b1) * q^43 + (7*b9 + 15*b7 - 15*b6 + 32*b5 - 16*b4 - 500*b2 - 69*b1 + 250) * q^44 + (3*b8 - 21*b7 - 10*b5 + 20*b4 - 27*b3 + 26*b2 - 52) * q^46 + (-b9 + 3*b7 - 3*b6 - 8*b5 + 4*b4 + 8*b2 + 47*b1 - 4) * q^47 + (-2*b9 - 2*b8 - 14*b7 + 7*b6 - 12*b5 + 12*b4 - 3*b3 - 155*b2 - 3*b1 + 155) * q^49 + (b9 - b8 + b6 - 23*b5 - 23*b4 + 84*b3 + 288*b2 - 84*b1 + 288) * q^50 + (-b9 + 17*b7 - 7*b6 - 22*b5 - 5*b4 + 18*b3 + 44*b2 + 99*b1 + 272) * q^52 + (-4*b9 + 8*b8 - 9*b7 - 9*b6 - 6*b4 + 6*b3 - 3*b1 - 33) * q^53 + (-b9 - b8 - 22*b7 + 11*b6 + 46*b5 - 46*b4 + 33*b3 + 52*b2 + 33*b1 - 52) * q^55 + (10*b9 - 5*b8 + 9*b7 - 18*b6 - 36*b5 - 19*b3 + 534*b2 + 38*b1) * q^56 + (-3*b8 + 3*b7 + 17*b5 - 34*b4 - 135*b3 - 136*b2 + 272) * q^58 + (-4*b8 - 6*b7 - 16*b5 + 32*b4 - 26*b3 + 52*b2 - 104) * q^59 + (2*b9 - b8 - 3*b7 + 6*b6 - 4*b5 - 9*b3 - 275*b2 + 18*b1) * q^61 + (9*b9 + 9*b8 - 10*b7 + 5*b6 + 33*b5 - 33*b4 - 36*b3 - 156*b2 - 36*b1 + 156) * q^62 + (10*b7 + 10*b6 + 29*b4 + 132*b3 - 66*b1 - 314) * q^64 + (11*b9 - 17*b8 + 12*b7 + 6*b6 + 14*b5 - 40*b4 + 54*b3 + 229*b2 + 2*b1 - 275) * q^65 + (-3*b9 + 3*b8 - 12*b6 + 14*b5 + 14*b4 - 36*b3 + 106*b2 + 36*b1 + 106) * q^67 + (-3*b9 - 3*b8 - 10*b7 + 5*b6 + 15*b5 - 15*b4 + 15*b3 - 120*b2 + 15*b1 + 120) * q^68 + (3*b9 + 15*b7 - 15*b6 - 16*b5 + 8*b4 + 1004*b2 - 135*b1 - 502) * q^70 + (5*b8 + b7 - 4*b5 + 8*b4 - 95*b3 + 108*b2 - 216) * q^71 + (-20*b5 + 10*b4 + 114*b2 + 72*b1 - 57) * q^73 + (-24*b9 + 12*b8 + 14*b7 - 28*b6 + 39*b5 + 2*b3 - 438*b2 - 4*b1) * q^74 + (b9 - b8 - 21*b6 - 6*b5 - 6*b4 - 9*b3 - 346*b2 + 9*b1 - 346) * q^76 + (2*b9 - 4*b8 + 8*b7 + 8*b6 + 66*b4 + 112*b3 - 56*b1 + 432) * q^77 + (-2*b9 + 4*b8 - 13*b7 - 13*b6 - 2*b4 + 126*b3 - 63*b1 + 110) * q^79 + (4*b9 - 4*b8 - 6*b6 + 13*b5 + 13*b4 - 158*b3 - 562*b2 + 158*b1 - 562) * q^80 + (4*b9 - 2*b8 - 8*b7 + 16*b6 + 5*b5 + 36*b3 - 478*b2 - 72*b1) * q^82 + (-19*b9 - 37*b7 + 37*b6 - 20*b5 + 10*b4 - 344*b2 + 35*b1 + 172) * q^83 + (-21*b8 + 42*b7 - 14*b5 + 28*b4 - 54*b3 - 35*b2 + 70) * q^85 + (24*b9 - 18*b7 + 18*b6 - 42*b5 + 21*b4 + 372*b2 - 77*b1 - 186) * q^86 + (23*b9 + 23*b8 + 14*b7 - 7*b6 + 74*b5 - 74*b4 - 249*b3 - 634*b2 - 249*b1 + 634) * q^88 + (10*b9 - 10*b8 + 6*b6 + 4*b5 + 4*b4 - 118*b3 - 136*b2 + 118*b1 - 136) * q^89 + (23*b9 - 26*b8 - 25*b7 + 50*b6 + 22*b5 + 10*b4 - 123*b3 - 42*b2 + 30*b1 + 498) * q^91 + (7*b9 - 14*b8 + 29*b7 + 29*b6 - 6*b4 + 306*b3 - 153*b1 + 174) * q^92 + (-b9 - b8 - 2*b7 + b6 - 62*b5 + 62*b4 + 33*b3 + 646*b2 + 33*b1 - 646) * q^94 + (-62*b9 + 31*b8 - 13*b7 + 26*b6 + 60*b5 + 25*b3 + 276*b2 - 50*b1) * q^95 + (22*b8 - 3*b7 - 32*b5 + 64*b4 + 81*b3 + 250*b2 - 500) * q^97 + (-33*b8 + 21*b7 - 15*b5 + 30*b4 - 11*b3 + 12*b2 - 24) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 30 q^{4} + 30 q^{7}+O(q^{10})$$ 10 * q + 30 * q^4 + 30 * q^7 $$10 q + 30 q^{4} + 30 q^{7} + 40 q^{10} - 60 q^{11} + 25 q^{13} + 60 q^{14} - 250 q^{16} - 105 q^{17} + 180 q^{19} - 510 q^{20} - 290 q^{22} + 60 q^{23} - 960 q^{25} + 30 q^{26} + 150 q^{28} + 495 q^{29} - 1440 q^{32} - 60 q^{35} - 405 q^{37} + 1380 q^{38} + 2000 q^{40} - 1065 q^{41} - 370 q^{43} - 390 q^{46} + 775 q^{49} + 4320 q^{50} + 2940 q^{52} - 330 q^{53} - 260 q^{55} + 2670 q^{56} + 2040 q^{58} - 780 q^{59} - 1375 q^{61} + 780 q^{62} - 3140 q^{64} - 1605 q^{65} + 1590 q^{67} + 600 q^{68} - 1620 q^{71} - 2190 q^{74} - 5190 q^{76} + 4320 q^{77} + 1100 q^{79} - 8430 q^{80} - 2390 q^{82} + 525 q^{85} + 3170 q^{88} - 2040 q^{89} + 4770 q^{91} + 1740 q^{92} - 3230 q^{94} + 1380 q^{95} - 3750 q^{97} - 180 q^{98}+O(q^{100})$$ 10 * q + 30 * q^4 + 30 * q^7 + 40 * q^10 - 60 * q^11 + 25 * q^13 + 60 * q^14 - 250 * q^16 - 105 * q^17 + 180 * q^19 - 510 * q^20 - 290 * q^22 + 60 * q^23 - 960 * q^25 + 30 * q^26 + 150 * q^28 + 495 * q^29 - 1440 * q^32 - 60 * q^35 - 405 * q^37 + 1380 * q^38 + 2000 * q^40 - 1065 * q^41 - 370 * q^43 - 390 * q^46 + 775 * q^49 + 4320 * q^50 + 2940 * q^52 - 330 * q^53 - 260 * q^55 + 2670 * q^56 + 2040 * q^58 - 780 * q^59 - 1375 * q^61 + 780 * q^62 - 3140 * q^64 - 1605 * q^65 + 1590 * q^67 + 600 * q^68 - 1620 * q^71 - 2190 * q^74 - 5190 * q^76 + 4320 * q^77 + 1100 * q^79 - 8430 * q^80 - 2390 * q^82 + 525 * q^85 + 3170 * q^88 - 2040 * q^89 + 4770 * q^91 + 1740 * q^92 - 3230 * q^94 + 1380 * q^95 - 3750 * q^97 - 180 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 35\nu^{3} + 210\nu + 96 ) / 192$$ (v^5 + 35*v^3 + 210*v + 96) / 192 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 35\nu^{4} + 210\nu^{2} + 96\nu ) / 192$$ (v^6 + 35*v^4 + 210*v^2 + 96*v) / 192 $$\beta_{4}$$ $$=$$ $$\nu^{2} + 14$$ v^2 + 14 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 49\nu^{5} + 700\nu^{3} + 96\nu^{2} + 2940\nu + 1344 ) / 192$$ (v^7 + 49*v^5 + 700*v^3 + 96*v^2 + 2940*v + 1344) / 192 $$\beta_{6}$$ $$=$$ $$( \nu^{9} + 64\nu^{7} + 18\nu^{6} + 1309\nu^{5} + 918\nu^{4} + 9030\nu^{3} + 12132\nu^{2} + 15912\nu + 24192 ) / 1152$$ (v^9 + 64*v^7 + 18*v^6 + 1309*v^5 + 918*v^4 + 9030*v^3 + 12132*v^2 + 15912*v + 24192) / 1152 $$\beta_{7}$$ $$=$$ $$( -\nu^{9} - 64\nu^{7} + 18\nu^{6} - 1309\nu^{5} + 918\nu^{4} - 9030\nu^{3} + 12132\nu^{2} - 15912\nu + 24192 ) / 1152$$ (-v^9 - 64*v^7 + 18*v^6 - 1309*v^5 + 918*v^4 - 9030*v^3 + 12132*v^2 - 15912*v + 24192) / 1152 $$\beta_{8}$$ $$=$$ $$( - \nu^{9} + 6 \nu^{8} - 70 \nu^{7} + 366 \nu^{6} - 1603 \nu^{5} + 7008 \nu^{4} - 12654 \nu^{3} + 42840 \nu^{2} - 20304 \nu + 48384 ) / 1152$$ (-v^9 + 6*v^8 - 70*v^7 + 366*v^6 - 1603*v^5 + 7008*v^4 - 12654*v^3 + 42840*v^2 - 20304*v + 48384) / 1152 $$\beta_{9}$$ $$=$$ $$( -\nu^{9} - 70\nu^{7} - 1603\nu^{5} - 12654\nu^{3} - 20304\nu ) / 576$$ (-v^9 - 70*v^7 - 1603*v^5 - 12654*v^3 - 20304*v) / 576
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - 14$$ b4 - 14 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} - 23\beta_1$$ b9 - b7 + b6 + 2*b5 - b4 - 23*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{7} + 2\beta_{6} - 29\beta_{4} - 12\beta_{3} + 6\beta _1 + 322$$ 2*b7 + 2*b6 - 29*b4 - 12*b3 + 6*b1 + 322 $$\nu^{5}$$ $$=$$ $$-35\beta_{9} + 35\beta_{7} - 35\beta_{6} - 70\beta_{5} + 35\beta_{4} + 192\beta_{2} + 595\beta _1 - 96$$ -35*b9 + 35*b7 - 35*b6 - 70*b5 + 35*b4 + 192*b2 + 595*b1 - 96 $$\nu^{6}$$ $$=$$ $$-70\beta_{7} - 70\beta_{6} + 805\beta_{4} + 612\beta_{3} - 306\beta _1 - 8330$$ -70*b7 - 70*b6 + 805*b4 + 612*b3 - 306*b1 - 8330 $$\nu^{7}$$ $$=$$ $$1015 \beta_{9} - 1015 \beta_{7} + 1015 \beta_{6} + 2222 \beta_{5} - 1111 \beta_{4} - 9408 \beta_{2} - 15995 \beta _1 + 4704$$ 1015*b9 - 1015*b7 + 1015*b6 + 2222*b5 - 1111*b4 - 9408*b2 - 15995*b1 + 4704 $$\nu^{8}$$ $$=$$ $$- 96 \beta_{9} + 192 \beta_{8} + 1934 \beta_{7} + 1934 \beta_{6} - 22373 \beta_{4} - 23316 \beta_{3} + 11658 \beta _1 + 223930$$ -96*b9 + 192*b8 + 1934*b7 + 1934*b6 - 22373*b4 - 23316*b3 + 11658*b1 + 223930 $$\nu^{9}$$ $$=$$ $$- 28175 \beta_{9} + 27599 \beta_{7} - 27599 \beta_{6} - 68638 \beta_{5} + 34319 \beta_{4} + 350784 \beta_{2} + 436603 \beta _1 - 175392$$ -28175*b9 + 27599*b7 - 27599*b6 - 68638*b5 + 34319*b4 + 350784*b2 + 436603*b1 - 175392

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 5.04537i 3.27897i − 0.917374i − 2.04224i − 5.36472i − 5.04537i − 3.27897i 0.917374i 2.04224i 5.36472i
−4.36942 + 2.52268i 0 8.72787 15.1171i 20.1174i 0 −13.3609 7.71395i 47.7076i 0 50.7498 + 87.9013i
10.2 −2.83967 + 1.63949i 0 1.37583 2.38302i 17.5414i 0 23.1228 + 13.3499i 17.2091i 0 −28.7589 49.8119i
10.3 0.794469 0.458687i 0 −3.57921 + 6.19938i 15.4704i 0 17.8257 + 10.2917i 13.9059i 0 −7.09608 12.2908i
10.4 1.76863 1.02112i 0 −1.91462 + 3.31622i 12.0825i 0 −25.7533 14.8686i 24.1582i 0 12.3377 + 21.3694i
10.5 4.64599 2.68236i 0 10.3901 17.9962i 2.69631i 0 13.1657 + 7.60123i 68.5626i 0 −7.23249 12.5270i
82.1 −4.36942 2.52268i 0 8.72787 + 15.1171i 20.1174i 0 −13.3609 + 7.71395i 47.7076i 0 50.7498 87.9013i
82.2 −2.83967 1.63949i 0 1.37583 + 2.38302i 17.5414i 0 23.1228 13.3499i 17.2091i 0 −28.7589 + 49.8119i
82.3 0.794469 + 0.458687i 0 −3.57921 6.19938i 15.4704i 0 17.8257 10.2917i 13.9059i 0 −7.09608 + 12.2908i
82.4 1.76863 + 1.02112i 0 −1.91462 3.31622i 12.0825i 0 −25.7533 + 14.8686i 24.1582i 0 12.3377 21.3694i
82.5 4.64599 + 2.68236i 0 10.3901 + 17.9962i 2.69631i 0 13.1657 7.60123i 68.5626i 0 −7.23249 + 12.5270i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 82.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.q.e 10
3.b odd 2 1 39.4.j.c 10
12.b even 2 1 624.4.bv.h 10
13.e even 6 1 inner 117.4.q.e 10
13.f odd 12 2 1521.4.a.bk 10
39.h odd 6 1 39.4.j.c 10
39.h odd 6 1 507.4.b.i 10
39.i odd 6 1 507.4.b.i 10
39.k even 12 2 507.4.a.r 10
156.r even 6 1 624.4.bv.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c 10 3.b odd 2 1
39.4.j.c 10 39.h odd 6 1
117.4.q.e 10 1.a even 1 1 trivial
117.4.q.e 10 13.e even 6 1 inner
507.4.a.r 10 39.k even 12 2
507.4.b.i 10 39.h odd 6 1
507.4.b.i 10 39.i odd 6 1
624.4.bv.h 10 12.b even 2 1
624.4.bv.h 10 156.r even 6 1
1521.4.a.bk 10 13.f odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - 35T_{2}^{8} + 1015T_{2}^{6} + 288T_{2}^{5} - 7350T_{2}^{4} + 44100T_{2}^{2} - 60480T_{2} + 27648$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 35 T^{8} + 1015 T^{6} + \cdots + 27648$$
$3$ $$T^{10}$$
$5$ $$T^{10} + 1105 T^{8} + \cdots + 31632011568$$
$7$ $$T^{10} - 30 T^{9} + \cdots + 14692478786352$$
$11$ $$T^{10} + 60 T^{9} + \cdots + 50\!\cdots\!32$$
$13$ $$T^{10} - 25 T^{9} + \cdots + 51\!\cdots\!57$$
$17$ $$T^{10} + \cdots + 332127005819904$$
$19$ $$T^{10} - 180 T^{9} + \cdots + 19\!\cdots\!68$$
$23$ $$T^{10} - 60 T^{9} + \cdots + 66\!\cdots\!04$$
$29$ $$T^{10} - 495 T^{9} + \cdots + 18\!\cdots\!96$$
$31$ $$T^{10} + 116790 T^{8} + \cdots + 35\!\cdots\!00$$
$37$ $$T^{10} + 405 T^{9} + \cdots + 48\!\cdots\!32$$
$41$ $$T^{10} + 1065 T^{9} + \cdots + 36\!\cdots\!52$$
$43$ $$T^{10} + 370 T^{9} + \cdots + 51\!\cdots\!16$$
$47$ $$T^{10} + 181660 T^{8} + \cdots + 21\!\cdots\!28$$
$53$ $$(T^{5} + 165 T^{4} + \cdots + 46733997168)^{2}$$
$59$ $$T^{10} + 780 T^{9} + \cdots + 41\!\cdots\!68$$
$61$ $$T^{10} + 1375 T^{9} + \cdots + 87\!\cdots\!25$$
$67$ $$T^{10} - 1590 T^{9} + \cdots + 13\!\cdots\!88$$
$71$ $$T^{10} + 1620 T^{9} + \cdots + 71\!\cdots\!00$$
$73$ $$T^{10} + 600615 T^{8} + \cdots + 20\!\cdots\!75$$
$79$ $$(T^{5} - 550 T^{4} + \cdots - 920208867136)^{2}$$
$83$ $$T^{10} + 3406900 T^{8} + \cdots + 16\!\cdots\!68$$
$89$ $$T^{10} + 2040 T^{9} + \cdots + 28\!\cdots\!28$$
$97$ $$T^{10} + 3750 T^{9} + \cdots + 15\!\cdots\!68$$