# Properties

 Label 2695.2.a.y 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: $$x^{10} - 3x^{9} - 13x^{8} + 41x^{7} + 51x^{6} - 184x^{5} - 45x^{4} + 297x^{3} - 59x^{2} - 109x + 21$$ x^10 - 3*x^9 - 13*x^8 + 41*x^7 + 51*x^6 - 184*x^5 - 45*x^4 + 297*x^3 - 59*x^2 - 109*x + 21 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 385) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + ( - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{6} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{8} + 2) q^{9}+O(q^{10})$$ q + b1 * q^2 + b5 * q^3 + (b2 + 2) * q^4 - q^5 + (-b8 + b7 - b4 + b2 - b1 + 1) * q^6 + (b3 + b2 + b1 + 1) * q^8 + (-b8 + 2) * q^9 $$q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + ( - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{6} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{8} + 2) q^{9} - \beta_1 q^{10} + q^{11} + ( - \beta_{9} - \beta_{7} + 2 \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{12} + (\beta_{6} - 1) q^{13} - \beta_{5} q^{15} + ( - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{16} + (\beta_{6} - \beta_{4} + \beta_{2} - \beta_1) q^{17} + ( - \beta_{7} + 2 \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{18} + \beta_{9} q^{19} + ( - \beta_{2} - 2) q^{20} + \beta_1 q^{22} + ( - \beta_{4} - \beta_{3} + 2) q^{23} + ( - \beta_{9} - \beta_{8} - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{24} + q^{25} + (\beta_{7} + \beta_{4} - \beta_1 + 1) q^{26} + (\beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_1 - 2) q^{27} + (\beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{29} + (\beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{30} + ( - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{2} + 2) q^{31} + (\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{32} + \beta_{5} q^{33} + ( - \beta_{9} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{34} + ( - \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots + 5) q^{36}+ \cdots + ( - \beta_{8} + 2) q^{99}+O(q^{100})$$ q + b1 * q^2 + b5 * q^3 + (b2 + 2) * q^4 - q^5 + (-b8 + b7 - b4 + b2 - b1 + 1) * q^6 + (b3 + b2 + b1 + 1) * q^8 + (-b8 + 2) * q^9 - b1 * q^10 + q^11 + (-b9 - b7 + 2*b5 + b3 - b2 + b1) * q^12 + (b6 - 1) * q^13 - b5 * q^15 + (-b8 + b7 - b5 + b3 + 2*b2 + b1 + 2) * q^16 + (b6 - b4 + b2 - b1) * q^17 + (-b7 + 2*b5 + b3 - b2 + 2*b1) * q^18 + b9 * q^19 + (-b2 - 2) * q^20 + b1 * q^22 + (-b4 - b3 + 2) * q^23 + (-b9 - b8 - 2*b4 + 2*b2 - 2*b1 + 3) * q^24 + q^25 + (b7 + b4 - b1 + 1) * q^26 + (b9 + b7 - b6 + b5 + 2*b1 - 2) * q^27 + (b8 - b5 + b4 - b3 + 1) * q^29 + (b8 - b7 + b4 - b2 + b1 - 1) * q^30 + (-b8 + b7 - b6 + 2*b2 + 2) * q^31 + (b6 + b5 + b4 + b3 + 2*b2 + b1 + 5) * q^32 + b5 * q^33 + (-b9 - b6 + 2*b5 + b4 - b2 + b1) * q^34 + (-b8 + 2*b7 - b6 - b5 - 2*b4 - b3 + 4*b2 - 3*b1 + 5) * q^36 + (b7 + b5 - b3 + b2 - b1 + 2) * q^37 + (b9 - b6 - b5 + 2*b4 - b3 - b2 + b1) * q^38 + (b9 + b7 - b6 - b5 - b4 - b3) * q^39 + (-b3 - b2 - b1 - 1) * q^40 + (b8 - b7 + b4 - 2*b3 - b2 + b1 - 1) * q^41 + (-b9 + 2*b8 - 3*b7 + b6 + b5 + b4 - b2 + b1 + 1) * q^43 + (b2 + 2) * q^44 + (b8 - 2) * q^45 + (-b9 + b8 - 2*b7 - b6 + 3*b5 - b3 - 3*b2 + 3*b1 + 1) * q^46 + (b7 - b6) * q^47 + (-b9 - b7 - b6 + 3*b5 - 2*b4 + 2*b1 - 2) * q^48 + b1 * q^50 + (-b5 - b4 + b3 + 2*b1) * q^51 + (b9 + 2*b7 - 2*b5 + b3 + b2 - 3) * q^52 + (-b9 + b5 + b4 - b3 + 2) * q^53 + (b9 - b8 + b7 - b5 - b3 + 3*b2 - 3*b1 + 9) * q^54 - q^55 + (b9 - 2*b8 + b7 + b6 - 2*b2 - 2*b1) * q^57 + (b9 + 2*b8 + b6 - 3*b5 + b4 - b2 + 3*b1 - 4) * q^58 + (b9 - b8 + b5 + b4 + b3 - 2*b1 + 2) * q^59 + (b9 + b7 - 2*b5 - b3 + b2 - b1) * q^60 + (b8 - b7 + 2*b6 - b5 - b3 - b2 - b1 + 1) * q^61 + (-b7 + b6 + 2*b5 - b4 + 3*b3 + 2*b2 + 3*b1 + 2) * q^62 + (b9 + 2*b7 + b6 - b5 + b3 + 3*b2 + 3*b1 + 3) * q^64 + (-b6 + 1) * q^65 + (-b8 + b7 - b4 + b2 - b1 + 1) * q^66 + (b7 + b5 - b4 - b2 + b1 + 2) * q^67 + (-2*b8 + 2*b7 - b5 - 3*b4 + b3 + 2*b2 - 2*b1 + 2) * q^68 + (-b8 + b7 + b6 + 2*b5 + 2*b3 + 2*b1 - 1) * q^69 + (-b8 - b7 - b6 + b5 + b4 + b3 - 2*b2 + 2) * q^71 + (-2*b9 + 2*b8 - 2*b7 + 3*b5 + b3 - b2 + 5*b1 - 5) * q^72 + (-b9 + 2*b8 - 3*b7 + b5 + b4 - b3 - 2*b2 + 2*b1 - 3) * q^73 + (b7 + b6 + b5 - b4 + b3 + 2*b1 - 2) * q^74 + b5 * q^75 + (b9 + 2*b8 - b7 + b6 - 4*b5 + 2*b4 - 2*b2 + 2*b1 - 4) * q^76 + (2*b8 - 3*b7 - b6 + 2*b5 + 2*b4 - 2*b3 - 4*b2 + 2*b1) * q^78 + (b9 - b7 - b5 - b3 - b2 - b1) * q^79 + (b8 - b7 + b5 - b3 - 2*b2 - b1 - 2) * q^80 + (-2*b8 + b7 + 2*b6 - 2*b5 - b4 - b2 - 3*b1 + 3) * q^81 + (b9 + 2*b8 - b7 - 2*b5 - b3 - 3*b2 + b1 - 2) * q^82 + (b9 - b3 + 3*b2 + b1 + 1) * q^83 + (-b6 + b4 - b2 + b1) * q^85 + (-b8 + 2*b7 - b6 - 5*b5 - 2*b4 - b3 + 2*b2 + b1) * q^86 + (-b9 + 2*b8 - 2*b7 + b5 + 2*b4 - 4*b2 - 6) * q^87 + (b3 + b2 + b1 + 1) * q^88 + (-b9 + 2*b8 - 4*b7 + b6 + b5 + 2*b4 + b3 - b2 + b1 + 3) * q^89 + (b7 - 2*b5 - b3 + b2 - 2*b1) * q^90 + (-b9 - 2*b8 - b6 - 4*b4 - b3 + b2 - 3*b1 + 4) * q^92 + (-2*b9 - 3*b7 + 4*b5 + b4 + 2*b3 - 3*b2 + 5*b1) * q^93 + (b6 - b4 + b2 - b1) * q^94 - b9 * q^95 + (-b9 - b8 - b7 - 2*b6 + 5*b5 - 2*b4 - b3 - b2 - b1 + 7) * q^96 + (-b9 + 2*b8 - b7 + b4 - 2*b3 - b2 - b1 - 4) * q^97 + (-b8 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 3 q^{2} - 3 q^{3} + 15 q^{4} - 10 q^{5} + 5 q^{6} + 9 q^{8} + 19 q^{9}+O(q^{10})$$ 10 * q + 3 * q^2 - 3 * q^3 + 15 * q^4 - 10 * q^5 + 5 * q^6 + 9 * q^8 + 19 * q^9 $$10 q + 3 q^{2} - 3 q^{3} + 15 q^{4} - 10 q^{5} + 5 q^{6} + 9 q^{8} + 19 q^{9} - 3 q^{10} + 10 q^{11} - 3 q^{12} - 6 q^{13} + 3 q^{15} + 21 q^{16} - 5 q^{17} + q^{18} + q^{19} - 15 q^{20} + 3 q^{22} + 18 q^{23} + 10 q^{24} + 10 q^{25} + 13 q^{26} - 15 q^{27} + 14 q^{29} - 5 q^{30} + 10 q^{31} + 46 q^{32} - 3 q^{33} - 2 q^{34} + 26 q^{36} + 13 q^{37} + 9 q^{38} + 3 q^{39} - 9 q^{40} - 7 q^{41} + 6 q^{43} + 15 q^{44} - 19 q^{45} + 10 q^{46} + q^{47} - 35 q^{48} + 3 q^{50} + 9 q^{51} - 17 q^{52} + 16 q^{53} + 73 q^{54} - 10 q^{55} + 12 q^{57} - 9 q^{58} + 13 q^{59} + 3 q^{60} + 18 q^{61} + 14 q^{62} + 43 q^{64} + 6 q^{65} + 5 q^{66} + 29 q^{67} + 13 q^{68} + 19 q^{71} - 48 q^{72} - 31 q^{73} - 8 q^{74} - 3 q^{75} - 8 q^{76} + 3 q^{78} - 21 q^{80} + 42 q^{81} + q^{82} - 2 q^{83} + 5 q^{85} + 10 q^{86} - 50 q^{87} + 9 q^{88} + 23 q^{89} - q^{90} + 14 q^{92} + 4 q^{93} - 5 q^{94} - q^{95} + 39 q^{96} - 43 q^{97} + 19 q^{99}+O(q^{100})$$ 10 * q + 3 * q^2 - 3 * q^3 + 15 * q^4 - 10 * q^5 + 5 * q^6 + 9 * q^8 + 19 * q^9 - 3 * q^10 + 10 * q^11 - 3 * q^12 - 6 * q^13 + 3 * q^15 + 21 * q^16 - 5 * q^17 + q^18 + q^19 - 15 * q^20 + 3 * q^22 + 18 * q^23 + 10 * q^24 + 10 * q^25 + 13 * q^26 - 15 * q^27 + 14 * q^29 - 5 * q^30 + 10 * q^31 + 46 * q^32 - 3 * q^33 - 2 * q^34 + 26 * q^36 + 13 * q^37 + 9 * q^38 + 3 * q^39 - 9 * q^40 - 7 * q^41 + 6 * q^43 + 15 * q^44 - 19 * q^45 + 10 * q^46 + q^47 - 35 * q^48 + 3 * q^50 + 9 * q^51 - 17 * q^52 + 16 * q^53 + 73 * q^54 - 10 * q^55 + 12 * q^57 - 9 * q^58 + 13 * q^59 + 3 * q^60 + 18 * q^61 + 14 * q^62 + 43 * q^64 + 6 * q^65 + 5 * q^66 + 29 * q^67 + 13 * q^68 + 19 * q^71 - 48 * q^72 - 31 * q^73 - 8 * q^74 - 3 * q^75 - 8 * q^76 + 3 * q^78 - 21 * q^80 + 42 * q^81 + q^82 - 2 * q^83 + 5 * q^85 + 10 * q^86 - 50 * q^87 + 9 * q^88 + 23 * q^89 - q^90 + 14 * q^92 + 4 * q^93 - 5 * q^94 - q^95 + 39 * q^96 - 43 * q^97 + 19 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 3x^{9} - 13x^{8} + 41x^{7} + 51x^{6} - 184x^{5} - 45x^{4} + 297x^{3} - 59x^{2} - 109x + 21$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5\nu + 3$$ v^3 - v^2 - 5*v + 3 $$\beta_{4}$$ $$=$$ $$( 3\nu^{9} - 5\nu^{8} - 37\nu^{7} + 52\nu^{6} + 153\nu^{5} - 140\nu^{4} - 287\nu^{3} + 36\nu^{2} + 235\nu + 21 ) / 39$$ (3*v^9 - 5*v^8 - 37*v^7 + 52*v^6 + 153*v^5 - 140*v^4 - 287*v^3 + 36*v^2 + 235*v + 21) / 39 $$\beta_{5}$$ $$=$$ $$( 4\nu^{9} - 11\nu^{8} - 58\nu^{7} + 143\nu^{6} + 295\nu^{5} - 594\nu^{4} - 595\nu^{3} + 841\nu^{2} + 335\nu - 219 ) / 39$$ (4*v^9 - 11*v^8 - 58*v^7 + 143*v^6 + 295*v^5 - 594*v^4 - 595*v^3 + 841*v^2 + 335*v - 219) / 39 $$\beta_{6}$$ $$=$$ $$( -7\nu^{9} + 16\nu^{8} + 95\nu^{7} - 195\nu^{6} - 409\nu^{5} + 734\nu^{4} + 531\nu^{3} - 916\nu^{2} + 54\nu + 198 ) / 39$$ (-7*v^9 + 16*v^8 + 95*v^7 - 195*v^6 - 409*v^5 + 734*v^4 + 531*v^3 - 916*v^2 + 54*v + 198) / 39 $$\beta_{7}$$ $$=$$ $$( - 8 \nu^{9} + 9 \nu^{8} + 129 \nu^{7} - 104 \nu^{6} - 707 \nu^{5} + 356 \nu^{4} + 1450 \nu^{3} - 395 \nu^{2} - 800 \nu + 87 ) / 39$$ (-8*v^9 + 9*v^8 + 129*v^7 - 104*v^6 - 707*v^5 + 356*v^4 + 1450*v^3 - 395*v^2 - 800*v + 87) / 39 $$\beta_{8}$$ $$=$$ $$( - 12 \nu^{9} + 20 \nu^{8} + 187 \nu^{7} - 247 \nu^{6} - 1002 \nu^{5} + 911 \nu^{4} + 2084 \nu^{3} - 963 \nu^{2} - 1291 \nu + 33 ) / 39$$ (-12*v^9 + 20*v^8 + 187*v^7 - 247*v^6 - 1002*v^5 + 911*v^4 + 2084*v^3 - 963*v^2 - 1291*v + 33) / 39 $$\beta_{9}$$ $$=$$ $$( 9 \nu^{9} - 15 \nu^{8} - 137 \nu^{7} + 195 \nu^{6} + 706 \nu^{5} - 810 \nu^{4} - 1355 \nu^{3} + 1135 \nu^{2} + 653 \nu - 223 ) / 13$$ (9*v^9 - 15*v^8 - 137*v^7 + 195*v^6 + 706*v^5 - 810*v^4 - 1355*v^3 + 1135*v^2 + 653*v - 223) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5\beta _1 + 1$$ b3 + b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$-\beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} + 8\beta_{2} + \beta _1 + 22$$ -b8 + b7 - b5 + b3 + 8*b2 + b1 + 22 $$\nu^{5}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} + 9\beta_{3} + 10\beta_{2} + 29\beta _1 + 13$$ b6 + b5 + b4 + 9*b3 + 10*b2 + 29*b1 + 13 $$\nu^{6}$$ $$=$$ $$\beta_{9} - 10\beta_{8} + 12\beta_{7} + \beta_{6} - 11\beta_{5} + 11\beta_{3} + 59\beta_{2} + 13\beta _1 + 135$$ b9 - 10*b8 + 12*b7 + b6 - 11*b5 + 11*b3 + 59*b2 + 13*b1 + 135 $$\nu^{7}$$ $$=$$ $$\beta_{9} + 3\beta_{7} + 11\beta_{6} + 8\beta_{5} + 14\beta_{4} + 68\beta_{3} + 84\beta_{2} + 183\beta _1 + 124$$ b9 + 3*b7 + 11*b6 + 8*b5 + 14*b4 + 68*b3 + 84*b2 + 183*b1 + 124 $$\nu^{8}$$ $$=$$ $$15 \beta_{9} - 76 \beta_{8} + 104 \beta_{7} + 16 \beta_{6} - 97 \beta_{5} + 5 \beta_{4} + 97 \beta_{3} + 427 \beta_{2} + 130 \beta _1 + 878$$ 15*b9 - 76*b8 + 104*b7 + 16*b6 - 97*b5 + 5*b4 + 97*b3 + 427*b2 + 130*b1 + 878 $$\nu^{9}$$ $$=$$ $$20 \beta_{9} + 49 \beta_{7} + 94 \beta_{6} + 30 \beta_{5} + 143 \beta_{4} + 493 \beta_{3} + 672 \beta_{2} + 1216 \beta _1 + 1057$$ 20*b9 + 49*b7 + 94*b6 + 30*b5 + 143*b4 + 493*b3 + 672*b2 + 1216*b1 + 1057

## 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
 −2.47520 −2.10414 −1.66564 −0.650789 0.190943 1.06607 1.29002 1.93394 2.63370 2.78109
−2.47520 −3.19739 4.12660 −1.00000 7.91417 0 −5.26377 7.22329 2.47520
1.2 −2.10414 1.86644 2.42740 −1.00000 −3.92725 0 −0.899305 0.483603 2.10414
1.3 −1.66564 0.562253 0.774372 −1.00000 −0.936513 0 2.04146 −2.68387 1.66564
1.4 −0.650789 −1.14162 −1.57647 −1.00000 0.742952 0 2.32753 −1.69671 0.650789
1.5 0.190943 −3.31339 −1.96354 −1.00000 −0.632670 0 −0.756812 7.97855 −0.190943
1.6 1.06607 3.07491 −0.863486 −1.00000 3.27808 0 −3.05269 6.45507 −1.06607
1.7 1.29002 0.350856 −0.335851 −1.00000 0.452611 0 −3.01329 −2.87690 −1.29002
1.8 1.93394 −2.14512 1.74012 −1.00000 −4.14853 0 −0.502587 1.60154 −1.93394
1.9 2.63370 2.48035 4.93639 −1.00000 6.53251 0 7.73356 3.15215 −2.63370
1.10 2.78109 −1.53729 5.73447 −1.00000 −4.27536 0 10.3859 −0.636727 −2.78109
 $$n$$: 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.y 10
7.b odd 2 1 2695.2.a.z 10
7.c even 3 2 385.2.i.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.d 20 7.c even 3 2
2695.2.a.y 10 1.a even 1 1 trivial
2695.2.a.z 10 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2695))$$:

 $$T_{2}^{10} - 3T_{2}^{9} - 13T_{2}^{8} + 41T_{2}^{7} + 51T_{2}^{6} - 184T_{2}^{5} - 45T_{2}^{4} + 297T_{2}^{3} - 59T_{2}^{2} - 109T_{2} + 21$$ T2^10 - 3*T2^9 - 13*T2^8 + 41*T2^7 + 51*T2^6 - 184*T2^5 - 45*T2^4 + 297*T2^3 - 59*T2^2 - 109*T2 + 21 $$T_{3}^{10} + 3 T_{3}^{9} - 20 T_{3}^{8} - 58 T_{3}^{7} + 128 T_{3}^{6} + 357 T_{3}^{5} - 281 T_{3}^{4} - 768 T_{3}^{3} + 148 T_{3}^{2} + 368 T_{3} - 112$$ T3^10 + 3*T3^9 - 20*T3^8 - 58*T3^7 + 128*T3^6 + 357*T3^5 - 281*T3^4 - 768*T3^3 + 148*T3^2 + 368*T3 - 112

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 3 T^{9} - 13 T^{8} + 41 T^{7} + \cdots + 21$$
$3$ $$T^{10} + 3 T^{9} - 20 T^{8} - 58 T^{7} + \cdots - 112$$
$5$ $$(T + 1)^{10}$$
$7$ $$T^{10}$$
$11$ $$(T - 1)^{10}$$
$13$ $$T^{10} + 6 T^{9} - 52 T^{8} + \cdots - 56056$$
$17$ $$T^{10} + 5 T^{9} - 73 T^{8} + \cdots + 52416$$
$19$ $$T^{10} - T^{9} - 128 T^{8} + \cdots + 57024$$
$23$ $$T^{10} - 18 T^{9} + 17 T^{8} + \cdots + 1087536$$
$29$ $$T^{10} - 14 T^{9} - 60 T^{8} + \cdots - 2074896$$
$31$ $$T^{10} - 10 T^{9} - 164 T^{8} + \cdots - 166824$$
$37$ $$T^{10} - 13 T^{9} - 90 T^{8} + \cdots + 6559424$$
$41$ $$T^{10} + 7 T^{9} - 204 T^{8} + \cdots - 4402944$$
$43$ $$T^{10} - 6 T^{9} - 294 T^{8} + \cdots - 13653189$$
$47$ $$T^{10} - T^{9} - 87 T^{8} - 131 T^{7} + \cdots + 2496$$
$53$ $$T^{10} - 16 T^{9} - 155 T^{8} + \cdots + 2303616$$
$59$ $$T^{10} - 13 T^{9} - 271 T^{8} + \cdots - 60061224$$
$61$ $$T^{10} - 18 T^{9} - 185 T^{8} + \cdots - 7149584$$
$67$ $$T^{10} - 29 T^{9} + 113 T^{8} + \cdots - 2847728$$
$71$ $$T^{10} - 19 T^{9} - 167 T^{8} + \cdots - 16214508$$
$73$ $$T^{10} + 31 T^{9} + 68 T^{8} + \cdots + 4541548$$
$79$ $$T^{10} - 392 T^{8} + \cdots - 120362368$$
$83$ $$T^{10} + 2 T^{9} - 559 T^{8} + \cdots - 423475857$$
$89$ $$T^{10} - 23 T^{9} + \cdots + 1911845607$$
$97$ $$T^{10} + 43 T^{9} + \cdots - 326335744$$