# Properties

 Label 585.2.j.i Level $585$ Weight $2$ Character orbit 585.j Analytic conductor $4.671$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 50x^{6} - 42x^{5} + 124x^{4} - 12x^{3} + 96x^{2} - 36x + 36$$ x^10 - 2*x^9 + 10*x^8 - 8*x^7 + 50*x^6 - 42*x^5 + 124*x^4 - 12*x^3 + 96*x^2 - 36*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_1) q^{2} + ( - \beta_{8} + \beta_{4} - 1) q^{4} - q^{5} + \beta_{7} q^{7} + (\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{8}+O(q^{10})$$ q + (-b3 - b1) * q^2 + (-b8 + b4 - 1) * q^4 - q^5 + b7 * q^7 + (b6 + b5 + b3 - b2 - 1) * q^8 $$q + ( - \beta_{3} - \beta_1) q^{2} + ( - \beta_{8} + \beta_{4} - 1) q^{4} - q^{5} + \beta_{7} q^{7} + (\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{8} + (\beta_{3} + \beta_1) q^{10} + (\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} - \beta_1) q^{11} + ( - \beta_{8} + \beta_{6} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{9} + \beta_{6} - \beta_{2} + 1) q^{14} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{16} + (2 \beta_{8} - \beta_1) q^{17} + ( - \beta_{8} - \beta_{4} + 1) q^{19} + (\beta_{8} - \beta_{4} + 1) q^{20} + ( - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_1 - 2) q^{22} + (\beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{23} + q^{25} + ( - \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{26}+ \cdots + ( - 3 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 7 \beta_{4} + 7) q^{98}+O(q^{100})$$ q + (-b3 - b1) * q^2 + (-b8 + b4 - 1) * q^4 - q^5 + b7 * q^7 + (b6 + b5 + b3 - b2 - 1) * q^8 + (b3 + b1) * q^10 + (b9 + b7 + b4 - b3 - b1) * q^11 + (-b8 + b6 - b2 + b1) * q^13 + (-b9 + b6 - b2 + 1) * q^14 + (-b9 + b8 - b7 + b5 + 2*b3 - b2 + b1) * q^16 + (2*b8 - b1) * q^17 + (-b8 - b4 + 1) * q^19 + (b8 - b4 + 1) * q^20 + (-b8 + b7 + b6 + 2*b4 - b1 - 2) * q^22 + (b8 + b5 - b4 + b3 - b2) * q^23 + q^25 + (-b9 + 2*b8 - b7 + b6 + 2*b5 - b4 + 2*b3 - b2 - b1 + 2) * q^26 + (2*b8 + 2*b5 - b2 - 2*b1) * q^28 + (b9 + b7 + 3*b4 - b2) * q^29 + (b9 + b6 + b3 - b2 + 2) * q^31 + (2*b8 - 2*b7 - 2*b4 + 2) * q^32 + (-2*b6 - b5 - 3*b3 + 2*b2 - 1) * q^34 - b7 * q^35 + (b9 - b8 + b7 - b5 - 2*b3 + b2 - b1) * q^37 + (b6 + b5 + b3 - b2 - 1) * q^38 + (-b6 - b5 - b3 + b2 + 1) * q^40 + (-b9 - 2*b8 - b7 - 2*b5 + b4 - 2*b3 - b2) * q^41 + (b8 + 2*b4 - 3*b1 - 2) * q^43 + (2*b6 + 4*b5 + 4*b3 - 2*b2 - 2) * q^44 + (3*b8 - b7 - b6 - 2*b4 + b1 + 2) * q^46 + (2*b9 - b6 - b5 - 2*b3 + b2 - 3) * q^47 + (-3*b8 - 3*b5 - b4 + b3 + 2*b2 + 4*b1) * q^49 + (-b3 - b1) * q^50 + (-b9 + 2*b8 - 2*b7 - b6 + b5 - 2*b4 + 2*b1 - 2) * q^52 + (-2*b9 - 2*b5 - 2*b3 + 4) * q^53 + (-b9 - b7 - b4 + b3 + b1) * q^55 + (2*b8 + b7 + b4 + b1 - 1) * q^56 + (2*b8 + b6 - 2*b4 - 4*b1 + 2) * q^58 + (-2*b8 + b7 - b6 - 3*b4 + 3) * q^59 + (-b7 + b6 + 2*b4 - b1 - 2) * q^61 + (3*b8 + 3*b5 - 5*b4 + b2 - 3*b1) * q^62 + (-2*b6 - 2*b3 + 2*b2) * q^64 + (b8 - b6 + b2 - b1) * q^65 + (b9 + b7 - 2*b4 - 2*b3 - b2 - 2*b1) * q^67 + (2*b9 - 3*b8 + 2*b7 - 3*b5 + 9*b4 - 3*b3 + b2) * q^68 + (b9 - b6 + b2 - 1) * q^70 + (-2*b8 + b6 - 2*b4 + b1 + 2) * q^71 + (b6 - b5 - 6*b3 - b2 - 2) * q^73 + (-4*b8 + 2*b7 + 2*b6 + 4*b4 - 4) * q^74 + (-b9 + b8 - b7 + b5 - 4*b4 + 2*b3 - b2 + b1) * q^76 + (-2*b9 - b6 - 3*b5 + b3 + b2 - 7) * q^77 + (-2*b6 - b5 + 3*b3 + 2*b2 - 2) * q^79 + (b9 - b8 + b7 - b5 - 2*b3 + b2 - b1) * q^80 + (-2*b7 + b6 + 2*b4 - 4*b1 - 2) * q^82 + (-3*b6 + b5 + b3 + 3*b2 - 3) * q^83 + (-2*b8 + b1) * q^85 + (-b6 + 2*b5 + 3*b3 + b2 - 8) * q^86 + (6*b8 + 6*b5 - 2*b4 + 8*b3 - 2*b2 + 2*b1) * q^88 + (2*b4 + 3*b3 - b2 + 3*b1) * q^89 + (2*b9 + 2*b8 + 3*b7 + b6 + 2*b5 + b1 - 1) * q^91 + (2*b9 - 2*b6 - 4*b5 - 8*b3 + 2*b2 + 4) * q^92 + (3*b9 - 4*b8 + 3*b7 - 4*b5 + 3*b4 - b3 + 3*b2 + 3*b1) * q^94 + (b8 + b4 - 1) * q^95 + (b8 - 2*b7 - 2*b6 - 2*b4 - b1 + 2) * q^97 + (-3*b8 + 2*b7 + 3*b6 - 7*b4 + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 2 q^{2} - 6 q^{4} - 10 q^{5} - q^{7} - 12 q^{8}+O(q^{10})$$ 10 * q + 2 * q^2 - 6 * q^4 - 10 * q^5 - q^7 - 12 * q^8 $$10 q + 2 q^{2} - 6 q^{4} - 10 q^{5} - q^{7} - 12 q^{8} - 2 q^{10} + 8 q^{11} + q^{13} + 8 q^{14} - 4 q^{16} + 4 q^{19} + 6 q^{20} - 14 q^{22} - 6 q^{23} + 10 q^{25} + 10 q^{26} + 2 q^{28} + 16 q^{29} + 18 q^{31} + 14 q^{32} + q^{35} + 4 q^{37} - 12 q^{38} + 12 q^{40} + 6 q^{41} - 15 q^{43} - 28 q^{44} + 16 q^{46} - 20 q^{47} - 10 q^{49} + 2 q^{50} - 22 q^{52} + 40 q^{53} - 8 q^{55} - 2 q^{56} + 4 q^{58} + 12 q^{59} - 11 q^{61} - 22 q^{62} + 8 q^{64} - q^{65} - 5 q^{67} + 50 q^{68} - 8 q^{70} + 10 q^{71} + 2 q^{73} - 26 q^{74} - 24 q^{76} - 84 q^{77} - 34 q^{79} + 4 q^{80} - 16 q^{82} - 32 q^{83} - 88 q^{86} - 20 q^{88} + 4 q^{89} - q^{91} + 68 q^{92} + 16 q^{94} - 4 q^{95} + 11 q^{97} + 30 q^{98}+O(q^{100})$$ 10 * q + 2 * q^2 - 6 * q^4 - 10 * q^5 - q^7 - 12 * q^8 - 2 * q^10 + 8 * q^11 + q^13 + 8 * q^14 - 4 * q^16 + 4 * q^19 + 6 * q^20 - 14 * q^22 - 6 * q^23 + 10 * q^25 + 10 * q^26 + 2 * q^28 + 16 * q^29 + 18 * q^31 + 14 * q^32 + q^35 + 4 * q^37 - 12 * q^38 + 12 * q^40 + 6 * q^41 - 15 * q^43 - 28 * q^44 + 16 * q^46 - 20 * q^47 - 10 * q^49 + 2 * q^50 - 22 * q^52 + 40 * q^53 - 8 * q^55 - 2 * q^56 + 4 * q^58 + 12 * q^59 - 11 * q^61 - 22 * q^62 + 8 * q^64 - q^65 - 5 * q^67 + 50 * q^68 - 8 * q^70 + 10 * q^71 + 2 * q^73 - 26 * q^74 - 24 * q^76 - 84 * q^77 - 34 * q^79 + 4 * q^80 - 16 * q^82 - 32 * q^83 - 88 * q^86 - 20 * q^88 + 4 * q^89 - q^91 + 68 * q^92 + 16 * q^94 - 4 * q^95 + 11 * q^97 + 30 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 50x^{6} - 42x^{5} + 124x^{4} - 12x^{3} + 96x^{2} - 36x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 355 \nu^{9} - 2199 \nu^{8} + 7398 \nu^{7} - 18528 \nu^{6} + 39184 \nu^{5} - 87134 \nu^{4} + 143476 \nu^{3} - 156238 \nu^{2} + 124656 \nu - 55404 ) / 40746$$ (355*v^9 - 2199*v^8 + 7398*v^7 - 18528*v^6 + 39184*v^5 - 87134*v^4 + 143476*v^3 - 156238*v^2 + 124656*v - 55404) / 40746 $$\beta_{3}$$ $$=$$ $$( 4282 \nu^{9} - 12311 \nu^{8} + 50191 \nu^{7} - 64307 \nu^{6} + 231068 \nu^{5} - 318192 \nu^{4} + 721480 \nu^{3} - 289782 \nu^{2} + 136368 \nu - 235800 ) / 448206$$ (4282*v^9 - 12311*v^8 + 50191*v^7 - 64307*v^6 + 231068*v^5 - 318192*v^4 + 721480*v^3 - 289782*v^2 + 136368*v - 235800) / 448206 $$\beta_{4}$$ $$=$$ $$( - 6550 \nu^{9} + 8818 \nu^{8} - 53189 \nu^{7} + 2209 \nu^{6} - 263193 \nu^{5} + 44032 \nu^{4} - 494008 \nu^{3} - 642880 \nu^{2} - 339018 \nu + 99432 ) / 448206$$ (-6550*v^9 + 8818*v^8 - 53189*v^7 + 2209*v^6 - 263193*v^5 + 44032*v^4 - 494008*v^3 - 642880*v^2 - 339018*v + 99432) / 448206 $$\beta_{5}$$ $$=$$ $$( - 8029 \nu^{9} + 19682 \nu^{8} - 80242 \nu^{7} + 81275 \nu^{6} - 369416 \nu^{5} + 508704 \nu^{4} - 959878 \nu^{3} + 463284 \nu^{2} - 218016 \nu + 1426266 ) / 448206$$ (-8029*v^9 + 19682*v^8 - 80242*v^7 + 81275*v^6 - 369416*v^5 + 508704*v^4 - 959878*v^3 + 463284*v^2 - 218016*v + 1426266) / 448206 $$\beta_{6}$$ $$=$$ $$( - 9476 \nu^{9} + 17684 \nu^{8} - 89335 \nu^{7} + 36452 \nu^{6} - 354900 \nu^{5} + 123782 \nu^{4} - 621080 \nu^{3} - 732992 \nu^{2} + 907392 \nu - 408504 ) / 448206$$ (-9476*v^9 + 17684*v^8 - 89335*v^7 + 36452*v^6 - 354900*v^5 + 123782*v^4 - 621080*v^3 - 732992*v^2 + 907392*v - 408504) / 448206 $$\beta_{7}$$ $$=$$ $$( - 10145 \nu^{9} - 7555 \nu^{8} - 72631 \nu^{7} - 139765 \nu^{6} - 519015 \nu^{5} - 700936 \nu^{4} - 1168190 \nu^{3} - 1844240 \nu^{2} - 2260776 \nu - 953850 ) / 448206$$ (-10145*v^9 - 7555*v^8 - 72631*v^7 - 139765*v^6 - 519015*v^5 - 700936*v^4 - 1168190*v^3 - 1844240*v^2 - 2260776*v - 953850) / 448206 $$\beta_{8}$$ $$=$$ $$( - 5301 \nu^{9} + 6361 \nu^{8} - 43172 \nu^{7} - 3447 \nu^{6} - 217077 \nu^{5} - 19472 \nu^{4} - 414542 \nu^{3} - 551312 \nu^{2} - 311802 \nu - 297390 ) / 149402$$ (-5301*v^9 + 6361*v^8 - 43172*v^7 - 3447*v^6 - 217077*v^5 - 19472*v^4 - 414542*v^3 - 551312*v^2 - 311802*v - 297390) / 149402 $$\beta_{9}$$ $$=$$ $$( - 9385 \nu^{9} + 23127 \nu^{8} - 94287 \nu^{7} + 111339 \nu^{6} - 434076 \nu^{5} + 597744 \nu^{4} - 956678 \nu^{3} + 544374 \nu^{2} - 256176 \nu + 675982 ) / 149402$$ (-9385*v^9 + 23127*v^8 - 94287*v^7 + 111339*v^6 - 434076*v^5 + 597744*v^4 - 956678*v^3 + 544374*v^2 - 256176*v + 675982) / 149402
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{5} - 3\beta_{4} + \beta_{3}$$ b8 + b5 - 3*b4 + b3 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + 5\beta_{3} - \beta_{2} - 1$$ b6 + b5 + 5*b3 - b2 - 1 $$\nu^{4}$$ $$=$$ $$-7\beta_{8} + \beta_{7} + \beta_{6} + 14\beta_{4} - \beta _1 - 14$$ -7*b8 + b7 + b6 + 14*b4 - b1 - 14 $$\nu^{5}$$ $$=$$ $$2\beta_{9} - 10\beta_{8} + 2\beta_{7} - 10\beta_{5} + 10\beta_{4} - 30\beta_{3} + 8\beta_{2} - 20\beta_1$$ 2*b9 - 10*b8 + 2*b7 - 10*b5 + 10*b4 - 30*b3 + 8*b2 - 20*b1 $$\nu^{6}$$ $$=$$ $$10\beta_{9} - 12\beta_{6} - 46\beta_{5} - 58\beta_{3} + 12\beta_{2} + 76$$ 10*b9 - 12*b6 - 46*b5 - 58*b3 + 12*b2 + 76 $$\nu^{7}$$ $$=$$ $$82\beta_{8} - 22\beta_{7} - 56\beta_{6} - 84\beta_{4} + 110\beta _1 + 84$$ 82*b8 - 22*b7 - 56*b6 - 84*b4 + 110*b1 + 84 $$\nu^{8}$$ $$=$$ $$-78\beta_{9} + 304\beta_{8} - 78\beta_{7} + 304\beta_{5} - 446\beta_{4} + 414\beta_{3} - 104\beta_{2} + 110\beta_1$$ -78*b9 + 304*b8 - 78*b7 + 304*b5 - 446*b4 + 414*b3 - 104*b2 + 110*b1 $$\nu^{9}$$ $$=$$ $$-182\beta_{9} + 382\beta_{6} + 622\beta_{5} + 1268\beta_{3} - 382\beta_{2} - 660$$ -182*b9 + 382*b6 + 622*b5 + 1268*b3 - 382*b2 - 660

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
406.1
 −1.06141 + 1.83842i −0.473183 + 0.819577i 0.313396 − 0.542817i 0.905157 − 1.56778i 1.31604 − 2.27945i −1.06141 − 1.83842i −0.473183 − 0.819577i 0.313396 + 0.542817i 0.905157 + 1.56778i 1.31604 + 2.27945i
−1.06141 1.83842i 0 −1.25320 + 2.17061i −1.00000 0 −0.733534 + 1.27052i 1.07500 0 1.06141 + 1.83842i
406.2 −0.473183 0.819577i 0 0.552196 0.956432i −1.00000 0 0.781529 1.35365i −2.93789 0 0.473183 + 0.819577i
406.3 0.313396 + 0.542817i 0 0.803566 1.39182i −1.00000 0 −2.21563 + 3.83759i 2.26092 0 −0.313396 0.542817i
406.4 0.905157 + 1.56778i 0 −0.638619 + 1.10612i −1.00000 0 2.21251 3.83219i 1.30843 0 −0.905157 1.56778i
406.5 1.31604 + 2.27945i 0 −2.46394 + 4.26767i −1.00000 0 −0.544875 + 0.943751i −7.70645 0 −1.31604 2.27945i
451.1 −1.06141 + 1.83842i 0 −1.25320 2.17061i −1.00000 0 −0.733534 1.27052i 1.07500 0 1.06141 1.83842i
451.2 −0.473183 + 0.819577i 0 0.552196 + 0.956432i −1.00000 0 0.781529 + 1.35365i −2.93789 0 0.473183 0.819577i
451.3 0.313396 0.542817i 0 0.803566 + 1.39182i −1.00000 0 −2.21563 3.83759i 2.26092 0 −0.313396 + 0.542817i
451.4 0.905157 1.56778i 0 −0.638619 1.10612i −1.00000 0 2.21251 + 3.83219i 1.30843 0 −0.905157 + 1.56778i
451.5 1.31604 2.27945i 0 −2.46394 4.26767i −1.00000 0 −0.544875 0.943751i −7.70645 0 −1.31604 + 2.27945i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.5 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.i yes 10
3.b odd 2 1 585.2.j.h 10
13.c even 3 1 inner 585.2.j.i yes 10
13.c even 3 1 7605.2.a.cm 5
13.e even 6 1 7605.2.a.cn 5
39.h odd 6 1 7605.2.a.cl 5
39.i odd 6 1 585.2.j.h 10
39.i odd 6 1 7605.2.a.co 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.j.h 10 3.b odd 2 1
585.2.j.h 10 39.i odd 6 1
585.2.j.i yes 10 1.a even 1 1 trivial
585.2.j.i yes 10 13.c even 3 1 inner
7605.2.a.cl 5 39.h odd 6 1
7605.2.a.cm 5 13.c even 3 1
7605.2.a.cn 5 13.e even 6 1
7605.2.a.co 5 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - 2T_{2}^{9} + 10T_{2}^{8} - 8T_{2}^{7} + 50T_{2}^{6} - 42T_{2}^{5} + 124T_{2}^{4} - 12T_{2}^{3} + 96T_{2}^{2} - 36T_{2} + 36$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 2 T^{9} + 10 T^{8} - 8 T^{7} + \cdots + 36$$
$3$ $$T^{10}$$
$5$ $$(T + 1)^{10}$$
$7$ $$T^{10} + T^{9} + 23 T^{8} + 22 T^{7} + \cdots + 2401$$
$11$ $$T^{10} - 8 T^{9} + 64 T^{8} - 128 T^{7} + \cdots + 576$$
$13$ $$T^{10} - T^{9} - 21 T^{8} + \cdots + 371293$$
$17$ $$T^{10} + 50 T^{8} + 28 T^{7} + \cdots + 412164$$
$19$ $$T^{10} - 4 T^{9} + 24 T^{8} - 40 T^{7} + \cdots + 144$$
$23$ $$T^{10} + 6 T^{9} + 56 T^{8} + \cdots + 5184$$
$29$ $$T^{10} - 16 T^{9} + 198 T^{8} + \cdots + 66564$$
$31$ $$(T^{5} - 9 T^{4} - 30 T^{3} + 178 T^{2} + \cdots + 603)^{2}$$
$37$ $$T^{10} - 4 T^{9} + 80 T^{8} + \cdots + 20736$$
$41$ $$T^{10} - 6 T^{9} + 134 T^{8} + \cdots + 141562404$$
$43$ $$T^{10} + 15 T^{9} + 195 T^{8} + \cdots + 4239481$$
$47$ $$(T^{5} + 10 T^{4} - 90 T^{3} - 752 T^{2} + \cdots + 7434)^{2}$$
$53$ $$(T^{5} - 20 T^{4} + 1872 T^{2} + \cdots + 15552)^{2}$$
$59$ $$T^{10} - 12 T^{9} + 186 T^{8} + \cdots + 7584516$$
$61$ $$T^{10} + 11 T^{9} + 135 T^{8} + \cdots + 9138529$$
$67$ $$T^{10} + 5 T^{9} + 69 T^{8} - 212 T^{7} + \cdots + 289$$
$71$ $$T^{10} - 10 T^{9} + 122 T^{8} + \cdots + 26244$$
$73$ $$(T^{5} - T^{4} - 236 T^{3} + 144 T^{2} + \cdots + 9693)^{2}$$
$79$ $$(T^{5} + 17 T^{4} - 26 T^{3} - 1086 T^{2} + \cdots + 2349)^{2}$$
$83$ $$(T^{5} + 16 T^{4} - 116 T^{3} - 2060 T^{2} + \cdots + 6264)^{2}$$
$89$ $$T^{10} - 4 T^{9} + 118 T^{8} + \cdots + 2916$$
$97$ $$T^{10} - 11 T^{9} + 251 T^{8} + \cdots + 58967041$$