# Properties

 Label 2695.2.a.x Level $2695$ Weight $2$ Character orbit 2695.a Self dual yes Analytic conductor $21.520$ Analytic rank $0$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2695 = 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2695.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.5196833447$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 22x + 1$$ x^10 - 2*x^9 - 13*x^8 + 24*x^7 + 56*x^6 - 92*x^5 - 86*x^4 + 116*x^3 + 31*x^2 - 22*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{4} + \beta_{2}) q^{6} + (\beta_{9} + \beta_{8} + \beta_{3} + \beta_1) q^{8} + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{2}) q^{9}+O(q^{10})$$ q + b1 * q^2 + b8 * q^3 + (b2 + 1) * q^4 + q^5 + (b4 + b2) * q^6 + (b9 + b8 + b3 + b1) * q^8 + (b9 - b6 + b5 - b2) * q^9 $$q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{4} + \beta_{2}) q^{6} + (\beta_{9} + \beta_{8} + \beta_{3} + \beta_1) q^{8} + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{2}) q^{9} + \beta_1 q^{10} - q^{11} + (\beta_{9} + \beta_{7} + \beta_1) q^{12} + (\beta_{9} - \beta_{7} + \beta_1) q^{13} + \beta_{8} q^{15} + (\beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} + \beta_1) q^{16} + (\beta_{9} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{17} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{18} + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{19} + (\beta_{2} + 1) q^{20} - \beta_1 q^{22} + ( - \beta_{9} - \beta_{7} - \beta_{6} - 1) q^{23} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + 3) q^{24} + q^{25} + (2 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{26} + ( - 2 \beta_{9} - \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{27} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5}) q^{29} + (\beta_{4} + \beta_{2}) q^{30} + ( - 2 \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{31} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{32} - \beta_{8} q^{33} + (2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{34} + ( - 2 \beta_{9} + \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{36}+ \cdots + ( - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{2}) q^{99}+O(q^{100})$$ q + b1 * q^2 + b8 * q^3 + (b2 + 1) * q^4 + q^5 + (b4 + b2) * q^6 + (b9 + b8 + b3 + b1) * q^8 + (b9 - b6 + b5 - b2) * q^9 + b1 * q^10 - q^11 + (b9 + b7 + b1) * q^12 + (b9 - b7 + b1) * q^13 + b8 * q^15 + (b9 + b8 + b6 + b4 + b1) * q^16 + (b9 + b5 + b3 - b2 + b1 + 2) * q^17 + (-b9 + b7 + b6 - b5 + b4 - 2*b3 + b2 - b1) * q^18 + (b9 - b6 + b5 - b4 - b2 + b1) * q^19 + (b2 + 1) * q^20 - b1 * q^22 + (-b9 - b7 - b6 - 1) * q^23 + (b7 - b6 + b3 + b2 + 3) * q^24 + q^25 + (2*b9 + 2*b8 - b7 + b6 - 2*b4 - b3 - b2 + 2*b1 + 1) * q^26 + (-2*b9 - b6 - b4 + b2 - b1 + 1) * q^27 + (-b9 + b8 - b7 + b6 - b5) * q^29 + (b4 + b2) * q^30 + (-2*b9 - b5 + b4 + b3 + b2 + b1) * q^31 + (b5 + b4 + b2 - b1 + 1) * q^32 - b8 * q^33 + (2*b6 + b4 + b3 + b2 + 2*b1 + 2) * q^34 + (-2*b9 + b7 - 2*b6 - b5 + b4 + 2*b3 + 2*b2 - b1 - 2) * q^36 + (b9 - b7 - b6 + b5 - b4 - b2 + 2) * q^37 + (-b9 - b8 + b6 - b5 + b4 - b3 + 2*b2 + 3) * q^38 + (-2*b9 + b7 + 2*b6 - b5 + b4 + b2 + b1 - 1) * q^39 + (b9 + b8 + b3 + b1) * q^40 + (-2*b8 - 2*b6 + b5 - b4 - b2 - b1 + 4) * q^41 + (-b9 + b8 + 2*b7 - b5 + b4 + b3 + b2 - 2*b1 + 3) * q^43 + (-b2 - 1) * q^44 + (b9 - b6 + b5 - b2) * q^45 + (-b9 + b6 - b5 - 2*b4 - 4*b3 - b2 - 2*b1 + 1) * q^46 + (-2*b9 - 3*b8 + b7 - b6 - b5 - b3 + b2 + 2) * q^47 + (-3*b9 - b5 + 2*b4 - b3 + 3*b2 + b1 + 2) * q^48 + b1 * q^50 + (-b9 + 4*b8 - b7 - b6 - b4 + b2 - b1) * q^51 + (b9 - 2*b7 + b5 + 3*b3 + b2 + 3*b1 + 2) * q^52 + (2*b9 + 2*b8 - b7 + 3*b6 + b5 - 2*b2) * q^53 + (-2*b9 - 2*b8 - b5 - b3 + b1) * q^54 - q^55 + (2*b8 - 2*b6 + b5 - b4 + 2*b3 + b2 - 2*b1 + 1) * q^57 + (b9 - 2*b7 + b5 - 2*b4 - 2*b2 - 1) * q^58 + (-b9 + b7 + 2*b6 - 2*b5 + b4 - 2*b3 + 2*b2 + 3) * q^59 + (b9 + b7 + b1) * q^60 + (b9 + 2*b8 - 2*b7 + b6 - 2*b4 - b3 - 2*b2 + 2) * q^61 + (-b9 + b7 - b4 - 2*b3 + b2 - 2*b1 + 5) * q^62 + (-b9 + b7 - b6 - b4 + 2*b3 - b2 + b1 - 3) * q^64 + (b9 - b7 + b1) * q^65 + (-b4 - b2) * q^66 + (-b7 + b6 + b5 - 2*b3 + 2) * q^67 + (b9 + 2*b8 - b7 + b6 + 4*b3 + 2*b2 + 3*b1) * q^68 + (2*b9 + 2*b6 + b5 - 2*b3 - 4*b2 + b1) * q^69 + (b9 + b8 - b6 - b4 + 2*b3 + b1 - 1) * q^71 + (-b9 + 2*b7 - 2*b6 - b4 - 2*b3 + b2 - 3*b1 + 2) * q^72 + (-4*b9 - b8 - 4*b3 + 2*b2 - 2*b1 + 4) * q^73 + (-b7 + 2*b6 - b5 - b4 - 2*b3 - b2 + 3*b1 - 1) * q^74 + b8 * q^75 + (2*b8 - b5 + 2*b3 + 3*b1) * q^76 + (-b8 - 2*b6 + 2*b5 + b4 + 5*b3 + b2 - 2*b1 + 4) * q^78 + (-b9 - b6 - b5 - 2*b4 - 2*b3 - 3*b2 - 1) * q^79 + (b9 + b8 + b6 + b4 + b1) * q^80 + (4*b9 + 2*b6 - 2*b4 - 3*b2 + 1) * q^81 + (-3*b9 - 2*b8 + b7 + b6 - 2*b5 - b4 - 4*b3 - b2 + 2*b1 - 1) * q^82 + (3*b4 - 2*b3 + b2 - 3*b1 + 1) * q^83 + (b9 + b5 + b3 - b2 + b1 + 2) * q^85 + (-2*b9 - b8 + 3*b7 - 2*b6 + 4*b4 + 3*b2 - 3) * q^86 + (2*b9 - 2*b8 - b7 + 2*b6 + b5 + b4 - b3 - 4*b2 + b1 + 1) * q^87 + (-b9 - b8 - b3 - b1) * q^88 + (3*b9 - b6 + b5 + b4 + 4*b3 + b2 + b1 + 2) * q^89 + (-b9 + b7 + b6 - b5 + b4 - 2*b3 + b2 - b1) * q^90 + (b9 - 4*b8 - b7 - 3*b6 + b5 - b4 + 2*b3 - 3*b2 - 4) * q^92 + (3*b9 - b8 + 2*b6 + 2*b4 - 3*b3 - 2*b2 + 2*b1) * q^93 + (-3*b9 - 2*b8 + 2*b7 - 3*b6 - b5 - 2*b4 - 3*b3 - b1 + 4) * q^94 + (b9 - b6 + b5 - b4 - b2 + b1) * q^95 + (2*b8 - b4 - 3*b3 - b2 + 2*b1) * q^96 + (2*b7 - b6 - 2*b5 - b4 + 2*b3 - b2 - 3*b1 + 1) * q^97 + (-b9 + b6 - b5 + b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 2 q^{2} + 10 q^{4} + 10 q^{5} + 4 q^{6} + 6 q^{8} + 10 q^{9}+O(q^{10})$$ 10 * q + 2 * q^2 + 10 * q^4 + 10 * q^5 + 4 * q^6 + 6 * q^8 + 10 * q^9 $$10 q + 2 q^{2} + 10 q^{4} + 10 q^{5} + 4 q^{6} + 6 q^{8} + 10 q^{9} + 2 q^{10} - 10 q^{11} + 4 q^{12} + 8 q^{13} + 6 q^{16} + 28 q^{17} - 10 q^{18} + 8 q^{19} + 10 q^{20} - 2 q^{22} - 8 q^{23} + 32 q^{24} + 10 q^{25} + 12 q^{26} - 8 q^{29} + 4 q^{30} - 4 q^{31} + 14 q^{32} + 20 q^{34} - 22 q^{36} + 28 q^{37} + 24 q^{38} - 24 q^{39} + 6 q^{40} + 44 q^{41} + 20 q^{43} - 10 q^{44} + 10 q^{45} - 12 q^{46} + 12 q^{47} + 16 q^{48} + 2 q^{50} - 4 q^{51} + 36 q^{52} - 8 q^{54} - 10 q^{55} + 12 q^{57} - 8 q^{58} + 16 q^{59} + 4 q^{60} + 16 q^{61} + 36 q^{62} - 34 q^{64} + 8 q^{65} - 4 q^{66} + 20 q^{67} + 8 q^{68} + 4 q^{69} - 4 q^{71} + 10 q^{72} + 20 q^{73} - 16 q^{74} + 4 q^{76} + 52 q^{78} - 20 q^{79} + 6 q^{80} + 10 q^{81} - 32 q^{82} + 16 q^{83} + 28 q^{85} - 20 q^{86} + 20 q^{87} - 6 q^{88} + 44 q^{89} - 10 q^{90} - 24 q^{92} + 16 q^{93} + 24 q^{94} + 8 q^{95} - 4 q^{97} - 10 q^{99}+O(q^{100})$$ 10 * q + 2 * q^2 + 10 * q^4 + 10 * q^5 + 4 * q^6 + 6 * q^8 + 10 * q^9 + 2 * q^10 - 10 * q^11 + 4 * q^12 + 8 * q^13 + 6 * q^16 + 28 * q^17 - 10 * q^18 + 8 * q^19 + 10 * q^20 - 2 * q^22 - 8 * q^23 + 32 * q^24 + 10 * q^25 + 12 * q^26 - 8 * q^29 + 4 * q^30 - 4 * q^31 + 14 * q^32 + 20 * q^34 - 22 * q^36 + 28 * q^37 + 24 * q^38 - 24 * q^39 + 6 * q^40 + 44 * q^41 + 20 * q^43 - 10 * q^44 + 10 * q^45 - 12 * q^46 + 12 * q^47 + 16 * q^48 + 2 * q^50 - 4 * q^51 + 36 * q^52 - 8 * q^54 - 10 * q^55 + 12 * q^57 - 8 * q^58 + 16 * q^59 + 4 * q^60 + 16 * q^61 + 36 * q^62 - 34 * q^64 + 8 * q^65 - 4 * q^66 + 20 * q^67 + 8 * q^68 + 4 * q^69 - 4 * q^71 + 10 * q^72 + 20 * q^73 - 16 * q^74 + 4 * q^76 + 52 * q^78 - 20 * q^79 + 6 * q^80 + 10 * q^81 - 32 * q^82 + 16 * q^83 + 28 * q^85 - 20 * q^86 + 20 * q^87 - 6 * q^88 + 44 * q^89 - 10 * q^90 - 24 * q^92 + 16 * q^93 + 24 * q^94 + 8 * q^95 - 4 * q^97 - 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 22x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( -2\nu^{9} - 3\nu^{8} + 27\nu^{7} + 35\nu^{6} - 116\nu^{5} - 130\nu^{4} + 177\nu^{3} + 169\nu^{2} - 80\nu - 29 ) / 23$$ (-2*v^9 - 3*v^8 + 27*v^7 + 35*v^6 - 116*v^5 - 130*v^4 + 177*v^3 + 169*v^2 - 80*v - 29) / 23 $$\beta_{4}$$ $$=$$ $$( 5\nu^{9} - 4\nu^{8} - 56\nu^{7} + 39\nu^{6} + 198\nu^{5} - 112\nu^{4} - 247\nu^{3} + 49\nu^{2} + 85\nu + 61 ) / 23$$ (5*v^9 - 4*v^8 - 56*v^7 + 39*v^6 + 198*v^5 - 112*v^4 - 247*v^3 + 49*v^2 + 85*v + 61) / 23 $$\beta_{5}$$ $$=$$ $$( -5\nu^{9} + 4\nu^{8} + 56\nu^{7} - 39\nu^{6} - 175\nu^{5} + 112\nu^{4} + 63\nu^{3} - 72\nu^{2} + 214\nu - 15 ) / 23$$ (-5*v^9 + 4*v^8 + 56*v^7 - 39*v^6 - 175*v^5 + 112*v^4 + 63*v^3 - 72*v^2 + 214*v - 15) / 23 $$\beta_{6}$$ $$=$$ $$( -7\nu^{9} + \nu^{8} + 83\nu^{7} - 4\nu^{6} - 314\nu^{5} + 5\nu^{4} + 401\nu^{3} - 18\nu^{2} - 73\nu + 2 ) / 23$$ (-7*v^9 + v^8 + 83*v^7 - 4*v^6 - 314*v^5 + 5*v^4 + 401*v^3 - 18*v^2 - 73*v + 2) / 23 $$\beta_{7}$$ $$=$$ $$( -4\nu^{9} + 17\nu^{8} + 54\nu^{7} - 183\nu^{6} - 255\nu^{5} + 591\nu^{4} + 446\nu^{3} - 582\nu^{2} - 206\nu + 57 ) / 23$$ (-4*v^9 + 17*v^8 + 54*v^7 - 183*v^6 - 255*v^5 + 591*v^4 + 446*v^3 - 582*v^2 - 206*v + 57) / 23 $$\beta_{8}$$ $$=$$ $$( 8\nu^{9} - 11\nu^{8} - 108\nu^{7} + 136\nu^{6} + 487\nu^{5} - 538\nu^{4} - 800\nu^{3} + 681\nu^{2} + 320\nu - 91 ) / 23$$ (8*v^9 - 11*v^8 - 108*v^7 + 136*v^6 + 487*v^5 - 538*v^4 - 800*v^3 + 681*v^2 + 320*v - 91) / 23 $$\beta_{9}$$ $$=$$ $$( - 6 \nu^{9} + 14 \nu^{8} + 81 \nu^{7} - 171 \nu^{6} - 371 \nu^{5} + 668 \nu^{4} + 646 \nu^{3} - 850 \nu^{2} - 355 \nu + 120 ) / 23$$ (-6*v^9 + 14*v^8 + 81*v^7 - 171*v^6 - 371*v^5 + 668*v^4 + 646*v^3 - 850*v^2 - 355*v + 120) / 23
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{8} + \beta_{3} + 5\beta_1$$ b9 + b8 + b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} + 6\beta_{2} + \beta _1 + 14$$ b9 + b8 + b6 + b4 + 6*b2 + b1 + 14 $$\nu^{5}$$ $$=$$ $$8\beta_{9} + 8\beta_{8} + \beta_{5} + \beta_{4} + 8\beta_{3} + \beta_{2} + 27\beta _1 + 1$$ 8*b9 + 8*b8 + b5 + b4 + 8*b3 + b2 + 27*b1 + 1 $$\nu^{6}$$ $$=$$ $$9\beta_{9} + 10\beta_{8} + \beta_{7} + 9\beta_{6} + 9\beta_{4} + 2\beta_{3} + 35\beta_{2} + 11\beta _1 + 73$$ 9*b9 + 10*b8 + b7 + 9*b6 + 9*b4 + 2*b3 + 35*b2 + 11*b1 + 73 $$\nu^{7}$$ $$=$$ $$52 \beta_{9} + 52 \beta_{8} + \beta_{7} + \beta_{6} + 9 \beta_{5} + 12 \beta_{4} + 54 \beta_{3} + 14 \beta_{2} + 151 \beta _1 + 16$$ 52*b9 + 52*b8 + b7 + b6 + 9*b5 + 12*b4 + 54*b3 + 14*b2 + 151*b1 + 16 $$\nu^{8}$$ $$=$$ $$66 \beta_{9} + 77 \beta_{8} + 12 \beta_{7} + 62 \beta_{6} + \beta_{5} + 63 \beta_{4} + 24 \beta_{3} + 204 \beta_{2} + 93 \beta _1 + 401$$ 66*b9 + 77*b8 + 12*b7 + 62*b6 + b5 + 63*b4 + 24*b3 + 204*b2 + 93*b1 + 401 $$\nu^{9}$$ $$=$$ $$320 \beta_{9} + 321 \beta_{8} + 13 \beta_{7} + 13 \beta_{6} + 62 \beta_{5} + 102 \beta_{4} + 341 \beta_{3} + 132 \beta_{2} + 863 \beta _1 + 163$$ 320*b9 + 321*b8 + 13*b7 + 13*b6 + 62*b5 + 102*b4 + 341*b3 + 132*b2 + 863*b1 + 163

## 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
 −2.29554 −2.15225 −1.47696 −0.568445 0.0495267 0.330732 1.29753 1.81378 2.48590 2.51572
−2.29554 −0.293572 3.26952 1.00000 0.673908 0 −2.91425 −2.91382 −2.29554
1.2 −2.15225 −2.37643 2.63217 1.00000 5.11466 0 −1.36058 2.64742 −2.15225
1.3 −1.47696 2.93275 0.181403 1.00000 −4.33154 0 2.68599 5.60100 −1.47696
1.4 −0.568445 0.674153 −1.67687 1.00000 −0.383219 0 2.09010 −2.54552 −0.568445
1.5 0.0495267 −3.19919 −1.99755 1.00000 −0.158445 0 −0.197985 7.23480 0.0495267
1.6 0.330732 2.43493 −1.89062 1.00000 0.805311 0 −1.28675 2.92889 0.330732
1.7 1.29753 −1.54034 −0.316404 1.00000 −1.99865 0 −3.00561 −0.627346 1.29753
1.8 1.81378 −1.20588 1.28980 1.00000 −2.18721 0 −1.28815 −1.54584 1.81378
1.9 2.48590 0.309874 4.17970 1.00000 0.770315 0 5.41853 −2.90398 2.48590
1.10 2.51572 2.26371 4.32884 1.00000 5.69486 0 5.85871 2.12439 2.51572
 $$n$$: 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
$$5$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## 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 2695.2.a.x yes 10
7.b odd 2 1 2695.2.a.w 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2695.2.a.w 10 7.b odd 2 1
2695.2.a.x yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2695))$$:

 $$T_{2}^{10} - 2T_{2}^{9} - 13T_{2}^{8} + 24T_{2}^{7} + 56T_{2}^{6} - 92T_{2}^{5} - 86T_{2}^{4} + 116T_{2}^{3} + 31T_{2}^{2} - 22T_{2} + 1$$ T2^10 - 2*T2^9 - 13*T2^8 + 24*T2^7 + 56*T2^6 - 92*T2^5 - 86*T2^4 + 116*T2^3 + 31*T2^2 - 22*T2 + 1 $$T_{3}^{10} - 20T_{3}^{8} + 130T_{3}^{6} + 12T_{3}^{5} - 308T_{3}^{4} - 68T_{3}^{3} + 182T_{3}^{2} + 4T_{3} - 14$$ T3^10 - 20*T3^8 + 130*T3^6 + 12*T3^5 - 308*T3^4 - 68*T3^3 + 182*T3^2 + 4*T3 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 2 T^{9} - 13 T^{8} + 24 T^{7} + \cdots + 1$$
$3$ $$T^{10} - 20 T^{8} + 130 T^{6} + \cdots - 14$$
$5$ $$(T - 1)^{10}$$
$7$ $$T^{10}$$
$11$ $$(T + 1)^{10}$$
$13$ $$T^{10} - 8 T^{9} - 42 T^{8} + 448 T^{7} + \cdots + 34$$
$17$ $$T^{10} - 28 T^{9} + 298 T^{8} + \cdots - 9486$$
$19$ $$T^{10} - 8 T^{9} - 42 T^{8} + \cdots + 14536$$
$23$ $$T^{10} + 8 T^{9} - 70 T^{8} + \cdots + 58052$$
$29$ $$T^{10} + 8 T^{9} - 100 T^{8} + \cdots - 17648$$
$31$ $$T^{10} + 4 T^{9} - 130 T^{8} + \cdots + 106642$$
$37$ $$T^{10} - 28 T^{9} + 280 T^{8} + \cdots - 65596$$
$41$ $$T^{10} - 44 T^{9} + 704 T^{8} + \cdots - 1704254$$
$43$ $$T^{10} - 20 T^{9} - 36 T^{8} + \cdots + 7268$$
$47$ $$T^{10} - 12 T^{9} - 134 T^{8} + \cdots + 223634$$
$53$ $$T^{10} - 360 T^{8} + 500 T^{7} + \cdots - 889916$$
$59$ $$T^{10} - 16 T^{9} - 128 T^{8} + \cdots + 21298$$
$61$ $$T^{10} - 16 T^{9} + \cdots + 127321954$$
$67$ $$T^{10} - 20 T^{9} + 44 T^{8} + \cdots - 485852$$
$71$ $$T^{10} + 4 T^{9} - 124 T^{8} + \cdots - 349808$$
$73$ $$T^{10} - 20 T^{9} - 200 T^{8} + \cdots - 29120686$$
$79$ $$T^{10} + 20 T^{9} - 232 T^{8} + \cdots - 7623452$$
$83$ $$T^{10} - 16 T^{9} + \cdots + 3533457544$$
$89$ $$T^{10} - 44 T^{9} + \cdots + 344547208$$
$97$ $$T^{10} + 4 T^{9} - 480 T^{8} + \cdots - 706936$$