# Properties

 Label 7623.2.a.cy Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7623 = 3^{2} \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7623.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.8699614608$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11$$ x^10 - 19*x^8 - x^7 + 124*x^6 + 6*x^5 - 316*x^4 + 17*x^3 + 253*x^2 - 70*x - 11 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{6} q^{5} + q^{7} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 2) * q^4 + b6 * q^5 + q^7 + (b4 + b3 + 2*b1 + 1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{6} q^{5} + q^{7} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{8} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2}) q^{10} + ( - \beta_{9} + \beta_{5} - \beta_{3}) q^{13} + \beta_1 q^{14} + (\beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{16} + ( - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{7} + \beta_{4} + \beta_{3} + 1) q^{19} + (\beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{3} + 2) q^{20} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2}) q^{23} + (\beta_{9} + \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{25} + ( - \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - \beta_1 - 3) q^{26} + (\beta_{2} + 2) q^{28} + ( - \beta_{9} + \beta_{8} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{29} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{31} + (2 \beta_{9} + \beta_{8} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_1 + 7) q^{32} + (\beta_{9} + \beta_{6} - \beta_{4} + 3 \beta_{3} - 3 \beta_1 + 4) q^{34} + \beta_{6} q^{35} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{37} + (\beta_{9} + 2 \beta_{8} - 2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{38} + (2 \beta_{9} + \beta_{8} - 5 \beta_{5} + 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{40} + ( - \beta_{9} - \beta_{6} + \beta_{5} + \beta_{3} - 2) q^{41} + ( - \beta_{9} - \beta_{8} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{9} + 2 \beta_{7} + \beta_{6} + \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{46} + ( - 2 \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{47} + q^{49} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - 6 \beta_{3} - \beta_{2} + 3 \beta_1 - 4) q^{50} + ( - 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - 3 \beta_{6} + 5 \beta_{5} - 7 \beta_{3} + \cdots - 7) q^{52}+ \cdots + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 2) * q^4 + b6 * q^5 + q^7 + (b4 + b3 + 2*b1 + 1) * q^8 + (b9 + b8 + b7 + b6 - b5 + b3 - b2) * q^10 + (-b9 + b5 - b3) * q^13 + b1 * q^14 + (b8 - b7 - b5 - b3 + 2*b2 + 2*b1 + 4) * q^16 + (-b5 - b2 + b1 - 1) * q^17 + (-b7 + b4 + b3 + 1) * q^19 + (b9 - 2*b7 + b6 - b5 + 3*b3 + 2) * q^20 + (b6 - b4 - b3 + b2) * q^23 + (b9 + b6 + 2*b5 - b4 - b3 - b1 + 3) * q^25 + (-b9 + b7 - b6 + 2*b5 - 4*b3 - b1 - 3) * q^26 + (b2 + 2) * q^28 + (-b9 + b8 - b3 + b2 + b1 + 1) * q^29 + (-b9 - b8 - b7 - b6 - b5 - b3 + b2 + 2) * q^31 + (2*b9 + b8 + 2*b6 + b5 + b4 + 3*b3 + 3*b1 + 7) * q^32 + (b9 + b6 - b4 + 3*b3 - 3*b1 + 4) * q^34 + b6 * q^35 + (-b9 - b8 + b7 + b4 + 2*b3 - b2 + b1 + 3) * q^37 + (b9 + 2*b8 - 2*b5 - b3 + b2 + b1 - 1) * q^38 + (2*b9 + b8 - 5*b5 + 3*b3 - b2 + b1 + 1) * q^40 + (-b9 - b6 + b5 + b3 - 2) * q^41 + (-b9 - b8 - b3 + b2 + b1 - 1) * q^43 + (b9 + 2*b7 + b6 + b4 + 3*b3 - 3*b2 + b1 + 1) * q^46 + (-2*b9 - b7 - b6 - b5 + b3 + 2*b2 + b1 - 1) * q^47 + q^49 + (-b9 + b7 - b6 - b5 - 6*b3 - b2 + 3*b1 - 4) * q^50 + (-2*b9 - 2*b8 - b7 - 3*b6 + 5*b5 - 7*b3 + 2*b2 - 3*b1 - 7) * q^52 + (-b8 + b7 + b6 + 2*b5 - b3 + b2) * q^53 + (b4 + b3 + 2*b1 + 1) * q^56 + (b6 + 3*b5 + 2*b4 + 2*b1 + 2) * q^58 + (b9 + b7 + b6 + 2*b4 + 2*b3 - 2*b2 - b1 - 1) * q^59 + (-b9 - 2*b7 - 2*b6 + b5 + b3 + 2*b1) * q^61 + (b9 + 2*b7 - b6 + b5 + 5*b3 - 2*b2 + 2*b1) * q^62 + (b9 + b8 + 2*b6 - 4*b5 + b4 - 2*b3 + 2*b2 + 6*b1 + 6) * q^64 + (2*b9 + b7 + 3*b6 + b5 - 2*b4 - b3 - 2*b2 - 3*b1 + 3) * q^65 + (b8 + 2*b7 + 2*b6 - b5 + 2*b3 - b2 - b1 + 5) * q^67 + (b9 + b7 + b6 - 3*b5 + 2*b3 - 3*b2 + 2*b1 - 8) * q^68 + (b9 + b8 + b7 + b6 - b5 + b3 - b2) * q^70 + (-b9 - b8 - 2*b5 - 5*b3 + b2 - b1 - 5) * q^71 + (-b9 + b7 + b6 - 2*b5 + 2*b3 + 3*b1 + 1) * q^73 + (-b9 - b6 - b5 - 2*b4 - b3 + 3*b2 + 2*b1 + 3) * q^74 + (2*b9 - b7 + 4*b6 + 2*b5 + b4 + 5*b3 + 2*b1 + 5) * q^76 + (-b8 - b7 - b6 - b3 + b2 + 2*b1) * q^79 + (3*b9 + b7 + 4*b6 - 2*b5 + 12*b3 - 2*b2 + b1 + 7) * q^80 + (-2*b9 - b8 - 2*b6 + b5 - 5*b3 + b2 - 3*b1 - 3) * q^82 + (-2*b7 - 2*b5 - 4*b3 - 2) * q^83 + (b8 + 3*b7 + b5 + b4 - 2*b3 - b2 + b1 - 2) * q^85 + (2*b7 - b6 + b5 + 2*b3 + 4) * q^86 + (b9 + b7 + b6 + 3*b5 + 2*b3 - b2 + 2) * q^89 + (-b9 + b5 - b3) * q^91 + (-b9 - 3*b7 - b6 - 3*b5 - b4 - b3 + 3*b2 - b1 + 5) * q^92 + (b9 + 2*b7 + b5 + 2*b4 + 5*b3 - 2*b2 + 2) * q^94 + (-b9 - 2*b8 - b7 - 2*b6 - b5 + 4*b3 + 3*b2 - 2*b1 + 6) * q^95 + (b9 - b5 - b3 - 2*b1 + 2) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10})$$ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 $$10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100})$$ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$( - 25 \nu^{9} + 31 \nu^{8} + 558 \nu^{7} - 505 \nu^{6} - 4093 \nu^{5} + 2537 \nu^{4} + 11069 \nu^{3} - 4152 \nu^{2} - 8301 \nu + 385 ) / 2024$$ (-25*v^9 + 31*v^8 + 558*v^7 - 505*v^6 - 4093*v^5 + 2537*v^4 + 11069*v^3 - 4152*v^2 - 8301*v + 385) / 2024 $$\beta_{4}$$ $$=$$ $$( 25 \nu^{9} - 31 \nu^{8} - 558 \nu^{7} + 505 \nu^{6} + 4093 \nu^{5} - 2537 \nu^{4} - 9045 \nu^{3} + 4152 \nu^{2} - 3843 \nu - 2409 ) / 2024$$ (25*v^9 - 31*v^8 - 558*v^7 + 505*v^6 + 4093*v^5 - 2537*v^4 - 9045*v^3 + 4152*v^2 - 3843*v - 2409) / 2024 $$\beta_{5}$$ $$=$$ $$( - 31 \nu^{9} - 83 \nu^{8} + 530 \nu^{7} + 993 \nu^{6} - 2687 \nu^{5} - 3169 \nu^{4} + 3727 \nu^{3} + 1976 \nu^{2} + 1365 \nu + 275 ) / 2024$$ (-31*v^9 - 83*v^8 + 530*v^7 + 993*v^6 - 2687*v^5 - 3169*v^4 + 3727*v^3 + 1976*v^2 + 1365*v + 275) / 2024 $$\beta_{6}$$ $$=$$ $$( - 97 \nu^{9} - 325 \nu^{8} + 2246 \nu^{7} + 5327 \nu^{6} - 16569 \nu^{5} - 27479 \nu^{4} + 43393 \nu^{3} + 43072 \nu^{2} - 29941 \nu - 5995 ) / 2024$$ (-97*v^9 - 325*v^8 + 2246*v^7 + 5327*v^6 - 16569*v^5 - 27479*v^4 + 43393*v^3 + 43072*v^2 - 29941*v - 5995) / 2024 $$\beta_{7}$$ $$=$$ $$( - 229 \nu^{9} + 203 \nu^{8} + 3654 \nu^{7} - 2197 \nu^{6} - 19033 \nu^{5} + 4861 \nu^{4} + 34681 \nu^{3} + 5848 \nu^{2} - 13617 \nu - 7403 ) / 2024$$ (-229*v^9 + 203*v^8 + 3654*v^7 - 2197*v^6 - 19033*v^5 + 4861*v^4 + 34681*v^3 + 5848*v^2 - 13617*v - 7403) / 2024 $$\beta_{8}$$ $$=$$ $$( - 285 \nu^{9} + 151 \nu^{8} + 4742 \nu^{7} - 1709 \nu^{6} - 25813 \nu^{5} + 6253 \nu^{4} + 49477 \nu^{3} - 12520 \nu^{2} - 24601 \nu + 9449 ) / 2024$$ (-285*v^9 + 151*v^8 + 4742*v^7 - 1709*v^6 - 25813*v^5 + 6253*v^4 + 49477*v^3 - 12520*v^2 - 24601*v + 9449) / 2024 $$\beta_{9}$$ $$=$$ $$( 70 \nu^{9} + 65 \nu^{8} - 1360 \nu^{7} - 1116 \nu^{6} + 8981 \nu^{5} + 5850 \nu^{4} - 22543 \nu^{3} - 8412 \nu^{2} + 16260 \nu - 1331 ) / 506$$ (70*v^9 + 65*v^8 - 1360*v^7 - 1116*v^6 + 8981*v^5 + 5850*v^4 - 22543*v^3 - 8412*v^2 + 16260*v - 1331) / 506
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 6\beta _1 + 1$$ b4 + b3 + 6*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 8\beta_{2} + 2\beta _1 + 24$$ b8 - b7 - b5 - b3 + 8*b2 + 2*b1 + 24 $$\nu^{5}$$ $$=$$ $$2\beta_{9} + \beta_{8} + 2\beta_{6} + \beta_{5} + 9\beta_{4} + 11\beta_{3} + 39\beta _1 + 15$$ 2*b9 + b8 + 2*b6 + b5 + 9*b4 + 11*b3 + 39*b1 + 15 $$\nu^{6}$$ $$=$$ $$\beta_{9} + 11 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} - 14 \beta_{5} + \beta_{4} - 12 \beta_{3} + 58 \beta_{2} + 26 \beta _1 + 158$$ b9 + 11*b8 - 10*b7 + 2*b6 - 14*b5 + b4 - 12*b3 + 58*b2 + 26*b1 + 158 $$\nu^{7}$$ $$=$$ $$26 \beta_{9} + 13 \beta_{8} - \beta_{7} + 27 \beta_{6} + 10 \beta_{5} + 69 \beta_{4} + 104 \beta_{3} + 3 \beta_{2} + 266 \beta _1 + 157$$ 26*b9 + 13*b8 - b7 + 27*b6 + 10*b5 + 69*b4 + 104*b3 + 3*b2 + 266*b1 + 157 $$\nu^{8}$$ $$=$$ $$18 \beta_{9} + 97 \beta_{8} - 80 \beta_{7} + 30 \beta_{6} - 145 \beta_{5} + 16 \beta_{4} - 92 \beta_{3} + 412 \beta_{2} + 257 \beta _1 + 1095$$ 18*b9 + 97*b8 - 80*b7 + 30*b6 - 145*b5 + 16*b4 - 92*b3 + 412*b2 + 257*b1 + 1095 $$\nu^{9}$$ $$=$$ $$255 \beta_{9} + 126 \beta_{8} - 21 \beta_{7} + 272 \beta_{6} + 61 \beta_{5} + 509 \beta_{4} + 909 \beta_{3} + 52 \beta_{2} + 1873 \beta _1 + 1444$$ 255*b9 + 126*b8 - 21*b7 + 272*b6 + 61*b5 + 509*b4 + 909*b3 + 52*b2 + 1873*b1 + 1444

## 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
 −2.65195 −2.39396 −2.09767 −1.33330 −0.112481 0.473713 0.871604 1.80545 2.63994 2.79866
−2.65195 0 5.03286 −1.71311 0 1.00000 −8.04301 0 4.54308
1.2 −2.39396 0 3.73106 3.93829 0 1.00000 −4.14409 0 −9.42812
1.3 −2.09767 0 2.40021 −3.15947 0 1.00000 −0.839503 0 6.62751
1.4 −1.33330 0 −0.222305 0.873210 0 1.00000 2.96300 0 −1.16425
1.5 −0.112481 0 −1.98735 −1.06131 0 1.00000 0.448501 0 0.119378
1.6 0.473713 0 −1.77560 −3.75881 0 1.00000 −1.78855 0 −1.78060
1.7 0.871604 0 −1.24031 4.06436 0 1.00000 −2.82426 0 3.54252
1.8 1.80545 0 1.25966 −2.77715 0 1.00000 −1.33666 0 −5.01400
1.9 2.63994 0 4.96928 −3.08369 0 1.00000 7.83871 0 −8.14075
1.10 2.79866 0 5.83249 1.67767 0 1.00000 10.7258 0 4.69523
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 7623.2.a.cy 10
3.b odd 2 1 2541.2.a.br 10
11.b odd 2 1 7623.2.a.cx 10
11.d odd 10 2 693.2.m.j 20
33.d even 2 1 2541.2.a.bq 10
33.f even 10 2 231.2.j.g 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.g 20 33.f even 10 2
693.2.m.j 20 11.d odd 10 2
2541.2.a.bq 10 33.d even 2 1
2541.2.a.br 10 3.b odd 2 1
7623.2.a.cx 10 11.b odd 2 1
7623.2.a.cy 10 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{10} - 19T_{2}^{8} - T_{2}^{7} + 124T_{2}^{6} + 6T_{2}^{5} - 316T_{2}^{4} + 17T_{2}^{3} + 253T_{2}^{2} - 70T_{2} - 11$$ T2^10 - 19*T2^8 - T2^7 + 124*T2^6 + 6*T2^5 - 316*T2^4 + 17*T2^3 + 253*T2^2 - 70*T2 - 11 $$T_{5}^{10} + 5 T_{5}^{9} - 28 T_{5}^{8} - 179 T_{5}^{7} + 108 T_{5}^{6} + 1873 T_{5}^{5} + 1751 T_{5}^{4} - 4812 T_{5}^{3} - 6768 T_{5}^{2} + 2392 T_{5} + 4336$$ T5^10 + 5*T5^9 - 28*T5^8 - 179*T5^7 + 108*T5^6 + 1873*T5^5 + 1751*T5^4 - 4812*T5^3 - 6768*T5^2 + 2392*T5 + 4336 $$T_{13}^{10} - 6 T_{13}^{9} - 65 T_{13}^{8} + 334 T_{13}^{7} + 1540 T_{13}^{6} - 4808 T_{13}^{5} - 19424 T_{13}^{4} + 12448 T_{13}^{3} + 90048 T_{13}^{2} + 74752 T_{13} + 2816$$ T13^10 - 6*T13^9 - 65*T13^8 + 334*T13^7 + 1540*T13^6 - 4808*T13^5 - 19424*T13^4 + 12448*T13^3 + 90048*T13^2 + 74752*T13 + 2816

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 19 T^{8} - T^{7} + 124 T^{6} + \cdots - 11$$
$3$ $$T^{10}$$
$5$ $$T^{10} + 5 T^{9} - 28 T^{8} + \cdots + 4336$$
$7$ $$(T - 1)^{10}$$
$11$ $$T^{10}$$
$13$ $$T^{10} - 6 T^{9} - 65 T^{8} + \cdots + 2816$$
$17$ $$T^{10} + 8 T^{9} - 34 T^{8} + \cdots - 8336$$
$19$ $$T^{10} - 121 T^{8} + 136 T^{7} + \cdots - 50944$$
$23$ $$T^{10} - 106 T^{8} - 38 T^{7} + \cdots - 600380$$
$29$ $$T^{10} - 14 T^{9} - 90 T^{8} + \cdots - 3572144$$
$31$ $$T^{10} - 26 T^{9} + 172 T^{8} + \cdots + 532400$$
$37$ $$T^{10} - 24 T^{9} + 70 T^{8} + \cdots - 1675684$$
$41$ $$T^{10} + 19 T^{9} + 74 T^{8} + \cdots - 10096$$
$43$ $$T^{10} + 6 T^{9} - 118 T^{8} + \cdots - 21296$$
$47$ $$T^{10} + 15 T^{9} - 137 T^{8} + \cdots - 2694400$$
$53$ $$T^{10} - T^{9} - 276 T^{8} + \cdots - 240496$$
$59$ $$T^{10} + 23 T^{9} + \cdots - 159106816$$
$61$ $$T^{10} - 395 T^{8} + \cdots - 87904256$$
$67$ $$T^{10} - 38 T^{9} + 269 T^{8} + \cdots + 52960256$$
$71$ $$T^{10} + 26 T^{9} - 78 T^{8} + \cdots + 17073920$$
$73$ $$T^{10} + T^{9} - 337 T^{8} + \cdots - 7615744$$
$79$ $$T^{10} - 5 T^{9} - 238 T^{8} + \cdots + 1785296$$
$83$ $$T^{10} + 6 T^{9} - 340 T^{8} + \cdots - 779264$$
$89$ $$T^{10} - 9 T^{9} - 250 T^{8} + \cdots + 106384$$
$97$ $$T^{10} - 24 T^{9} + 99 T^{8} + \cdots + 481024$$