# Properties

 Label 5265.2.a.ba Level $5265$ Weight $2$ Character orbit 5265.a Self dual yes Analytic conductor $42.041$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5265 = 3^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5265.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$42.0412366642$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1$$ x^8 - 3*x^7 - 8*x^6 + 31*x^5 - x^4 - 70*x^3 + 66*x^2 - 19*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 585) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{7} + ( - \beta_{3} - \beta_1 + 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 1) * q^4 + q^5 + (-b7 + b4 + b1 - 1) * q^7 + (-b3 - b1 + 1) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{7} + ( - \beta_{3} - \beta_1 + 1) q^{8} - \beta_1 q^{10} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{11} - q^{13} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{14} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 + 2) q^{16} + (\beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{17} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{19} + (\beta_{2} + 1) q^{20} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{22} + ( - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2) q^{23} + q^{25} + \beta_1 q^{26} + (3 \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{28} + ( - \beta_{6} + \beta_{4} + 2 \beta_1 - 2) q^{29} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 - 5) q^{31} + ( - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{32} + ( - 3 \beta_{7} - \beta_{6} + 4 \beta_{5} - \beta_{3} - 2 \beta_1 - 1) q^{34} + ( - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{35} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{37} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + 2 \beta_1 - 2) q^{38} + ( - \beta_{3} - \beta_1 + 1) q^{40} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{41} + (2 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2}) q^{43} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{44} + (2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 4) q^{46} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{47} + (3 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_1 + 2) q^{49} - \beta_1 q^{50} + ( - \beta_{2} - 1) q^{52} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{53} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{55} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 6 \beta_1 - 3) q^{56} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 6) q^{58} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{59} + ( - \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 2) q^{61} + (2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_1 + 1) q^{62} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3} - 4) q^{64} - q^{65} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{67} + (5 \beta_{7} + 3 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1) q^{68} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{70} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{71} + ( - 3 \beta_{6} + \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{73} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{3} + \beta_1 + 4) q^{74} + (2 \beta_{7} - 4 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 5) q^{76} + (4 \beta_{7} - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{77} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 6 \beta_1 + 3) q^{79} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 + 2) q^{80} + (\beta_{7} - 4 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 3) q^{82} + (2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1) q^{83} + (\beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{85} + (2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 4 \beta_{4} - 3 \beta_{2} + 2 \beta_1 - 7) q^{86} + (\beta_{7} + 3 \beta_{6} - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 3) q^{88} + (3 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1) q^{89} + (\beta_{7} - \beta_{4} - \beta_1 + 1) q^{91} + ( - 5 \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 4 \beta_1 - 1) q^{92} + ( - 4 \beta_{7} + 6 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{94} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{95} + (\beta_{7} + 3 \beta_{6} + \beta_{4} + 2 \beta_{2} + 3 \beta_1 - 6) q^{97} + ( - 4 \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 5) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + 1) * q^4 + q^5 + (-b7 + b4 + b1 - 1) * q^7 + (-b3 - b1 + 1) * q^8 - b1 * q^10 + (-b7 + b6 + b5 - 1) * q^11 - q^13 + (b7 - b5 - b4 + b3 - b2 + 2*b1 - 2) * q^14 + (-b6 - b5 - b4 - b1 + 2) * q^16 + (b7 - 2*b5 + b4 + b3 + b1 + 1) * q^17 + (b7 + b6 - b5 - b4 - b3 - b2 + b1 - 2) * q^19 + (b2 + 1) * q^20 + (2*b7 + b6 - b5 + 2*b3 - b2 + b1) * q^22 + (-b6 + 2*b5 - b4 - 2) * q^23 + q^25 + b1 * q^26 + (3*b5 - b3 - 2*b2 + b1 - 6) * q^28 + (-b6 + b4 + 2*b1 - 2) * q^29 + (2*b7 - b6 + 2*b5 - b4 - b2 - 2*b1 - 5) * q^31 + (-b7 - b6 + b4 - b3 + 2*b2 + b1) * q^32 + (-3*b7 - b6 + 4*b5 - b3 - 2*b1 - 1) * q^34 + (-b7 + b4 + b1 - 1) * q^35 + (b5 + b4 + b3 - b2 - 2*b1 + 1) * q^37 + (-2*b7 - 2*b6 + 2*b5 + b2 + 2*b1 - 2) * q^38 + (-b3 - b1 + 1) * q^40 + (-2*b6 + b5 + b4 - b3 + b2 + b1 + 1) * q^41 + (2*b5 - 3*b4 + b3 - b2) * q^43 + (-b7 - b6 + 4*b5 + 2*b4 + b3 - 2*b2 - b1 - 2) * q^44 + (2*b7 + 2*b6 - 3*b5 + b4 - 2*b2 + 3*b1 - 4) * q^46 + (2*b7 - 2*b5 - 2*b4 + 2*b3 - 2*b2 - 2*b1 + 3) * q^47 + (3*b7 - b6 - b5 - b4 - 2*b1 + 2) * q^49 - b1 * q^50 + (-b2 - 1) * q^52 + (2*b7 + b6 - 2*b5 - 2*b4 - 2*b3 + b2 + 2*b1 - 2) * q^53 + (-b7 + b6 + b5 - 1) * q^55 + (b7 + 2*b6 - 2*b5 + b4 + 3*b3 - b2 + 6*b1 - 3) * q^56 + (-b5 - b4 - 2*b2 + 3*b1 - 6) * q^58 + (-2*b7 - b5 - b4 - b3 + b2 - b1 - 2) * q^59 + (-b6 - 4*b5 + 2*b4 + 3*b3 + b2 + 2) * q^61 + (2*b6 - b5 + b4 + b3 + 6*b1 + 1) * q^62 + (b7 + b6 - b5 - 2*b3 - 4) * q^64 - q^65 + (-b7 + b6 - 2*b5 - 3*b4 - 2*b3 + 2*b2 + b1 - 3) * q^67 + (5*b7 + 3*b6 - 5*b5 - 3*b4 + b3 - b2 + 3*b1) * q^68 + (b7 - b5 - b4 + b3 - b2 + 2*b1 - 2) * q^70 + (-2*b7 + b6 + b5 + b3 + 2*b2 + 2*b1 - 2) * q^71 + (-3*b6 + b5 - 2*b3 + b2 - 2*b1 + 1) * q^73 + (b7 + 2*b6 + 3*b3 + b1 + 4) * q^74 + (2*b7 - 4*b5 + 2*b4 + b3 - 2*b2 + 2*b1 - 5) * q^76 + (4*b7 - 2*b6 - b5 - 3*b4 - 2*b3 + b2 + 2) * q^77 + (-2*b7 - b5 - b4 + b3 - b2 - 6*b1 + 3) * q^79 + (-b6 - b5 - b4 - b1 + 2) * q^80 + (b7 - 4*b5 - 2*b4 - b3 - b2 - b1 - 3) * q^82 + (2*b6 + b5 - 2*b4 - b3 - b1) * q^83 + (b7 - 2*b5 + b4 + b3 + b1 + 1) * q^85 + (2*b7 + 3*b6 - b5 + 4*b4 - 3*b2 + 2*b1 - 7) * q^86 + (b7 + 3*b6 - 3*b5 - b4 + 3*b3 - 2*b2 + 6*b1 - 3) * q^88 + (3*b7 + b6 - b5 - 2*b4 - b3 - 3*b2 + b1) * q^89 + (b7 - b4 - b1 + 1) * q^91 + (-5*b7 - b6 + 3*b5 + b4 + 2*b3 + 4*b1 - 1) * q^92 + (-4*b7 + 6*b5 + 4*b4 - 2*b3 + 2*b2 - b1 + 2) * q^94 + (b7 + b6 - b5 - b4 - b3 - b2 + b1 - 2) * q^95 + (b7 + 3*b6 + b4 + 2*b2 + 3*b1 - 6) * q^97 + (-4*b7 - b6 + 3*b5 + b4 - 3*b3 + 3*b2 - 4*b1 + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10})$$ 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8 $$8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100})$$ 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 - 8 * q^13 - 10 * q^14 + 11 * q^16 + 2 * q^17 - 10 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 8 * q^25 + 3 * q^26 - 34 * q^28 - 14 * q^29 - 31 * q^31 - q^32 - 7 * q^34 - 11 * q^35 + q^37 - 9 * q^38 + 6 * q^40 + 12 * q^41 + 15 * q^43 - 16 * q^44 - 32 * q^46 + 18 * q^47 + 17 * q^49 - 3 * q^50 - 9 * q^52 - 2 * q^53 - 6 * q^55 - 16 * q^56 - 42 * q^58 - 24 * q^59 - 9 * q^61 + 20 * q^62 - 30 * q^64 - 8 * q^65 - 18 * q^67 + 14 * q^68 - 10 * q^70 - 10 * q^71 + 6 * q^73 + 37 * q^74 - 53 * q^76 + 34 * q^77 - 3 * q^79 + 11 * q^80 - 34 * q^82 + 10 * q^83 + 2 * q^85 - 60 * q^86 - 14 * q^88 + 13 * q^89 + 11 * q^91 - 5 * q^92 + 17 * q^94 - 10 * q^95 - 34 * q^97 + 30 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu + 1$$ v^3 - 5*v + 1 $$\beta_{4}$$ $$=$$ $$\nu^{7} - 2\nu^{6} - 10\nu^{5} + 21\nu^{4} + 20\nu^{3} - 51\nu^{2} + 15\nu + 1$$ v^7 - 2*v^6 - 10*v^5 + 21*v^4 + 20*v^3 - 51*v^2 + 15*v + 1 $$\beta_{5}$$ $$=$$ $$\nu^{7} - 3\nu^{6} - 9\nu^{5} + 31\nu^{4} + 9\nu^{3} - 73\nu^{2} + 43\nu - 4$$ v^7 - 3*v^6 - 9*v^5 + 31*v^4 + 9*v^3 - 73*v^2 + 43*v - 4 $$\beta_{6}$$ $$=$$ $$-2\nu^{7} + 5\nu^{6} + 19\nu^{5} - 53\nu^{4} - 29\nu^{3} + 130\nu^{2} - 59\nu + 1$$ -2*v^7 + 5*v^6 + 19*v^5 - 53*v^4 - 29*v^3 + 130*v^2 - 59*v + 1 $$\beta_{7}$$ $$=$$ $$3\nu^{7} - 7\nu^{6} - 28\nu^{5} + 74\nu^{4} + 40\nu^{3} - 179\nu^{2} + 92\nu - 7$$ 3*v^7 - 7*v^6 - 28*v^5 + 74*v^4 + 40*v^3 - 179*v^2 + 92*v - 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta _1 - 1$$ b3 + 5*b1 - 1 $$\nu^{4}$$ $$=$$ $$-\beta_{6} - \beta_{5} - \beta_{4} + 6\beta_{2} - \beta _1 + 16$$ -b6 - b5 - b4 + 6*b2 - b1 + 16 $$\nu^{5}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{4} + 9\beta_{3} - 2\beta_{2} + 27\beta _1 - 8$$ b7 + b6 - b4 + 9*b3 - 2*b2 + 27*b1 - 8 $$\nu^{6}$$ $$=$$ $$\beta_{7} - 9\beta_{6} - 11\beta_{5} - 10\beta_{4} - 2\beta_{3} + 36\beta_{2} - 10\beta _1 + 92$$ b7 - 9*b6 - 11*b5 - 10*b4 - 2*b3 + 36*b2 - 10*b1 + 92 $$\nu^{7}$$ $$=$$ $$12\beta_{7} + 13\beta_{6} - \beta_{5} - 8\beta_{4} + 66\beta_{3} - 23\beta_{2} + 156\beta _1 - 60$$ 12*b7 + 13*b6 - b5 - 8*b4 + 66*b3 - 23*b2 + 156*b1 - 60

## 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
 2.44829 2.25662 1.63404 0.655066 0.494096 0.0672022 −1.96921 −2.58610
−2.44829 0 3.99411 1.00000 0 −3.93452 −4.88214 0 −2.44829
1.2 −2.25662 0 3.09235 1.00000 0 0.706571 −2.46502 0 −2.25662
1.3 −1.63404 0 0.670078 1.00000 0 2.13012 2.17314 0 −1.63404
1.4 −0.655066 0 −1.57089 1.00000 0 −0.777184 2.33917 0 −0.655066
1.5 −0.494096 0 −1.75587 1.00000 0 −4.28539 1.85576 0 −0.494096
1.6 −0.0672022 0 −1.99548 1.00000 0 2.46357 0.268505 0 −0.0672022
1.7 1.96921 0 1.87778 1.00000 0 −3.02828 −0.240686 0 1.96921
1.8 2.58610 0 4.68793 1.00000 0 −4.27489 6.95128 0 2.58610
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 5265.2.a.ba 8
3.b odd 2 1 5265.2.a.bf 8
9.c even 3 2 1755.2.i.f 16
9.d odd 6 2 585.2.i.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.e 16 9.d odd 6 2
1755.2.i.f 16 9.c even 3 2
5265.2.a.ba 8 1.a even 1 1 trivial
5265.2.a.bf 8 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5265))$$:

 $$T_{2}^{8} + 3T_{2}^{7} - 8T_{2}^{6} - 31T_{2}^{5} - T_{2}^{4} + 70T_{2}^{3} + 66T_{2}^{2} + 19T_{2} + 1$$ T2^8 + 3*T2^7 - 8*T2^6 - 31*T2^5 - T2^4 + 70*T2^3 + 66*T2^2 + 19*T2 + 1 $$T_{7}^{8} + 11T_{7}^{7} + 24T_{7}^{6} - 106T_{7}^{5} - 385T_{7}^{4} + 232T_{7}^{3} + 1360T_{7}^{2} - 30T_{7} - 629$$ T7^8 + 11*T7^7 + 24*T7^6 - 106*T7^5 - 385*T7^4 + 232*T7^3 + 1360*T7^2 - 30*T7 - 629 $$T_{11}^{8} + 6T_{11}^{7} - 38T_{11}^{6} - 250T_{11}^{5} + 413T_{11}^{4} + 3220T_{11}^{3} - 1002T_{11}^{2} - 11999T_{11} + 634$$ T11^8 + 6*T11^7 - 38*T11^6 - 250*T11^5 + 413*T11^4 + 3220*T11^3 - 1002*T11^2 - 11999*T11 + 634 $$T_{17}^{8} - 2T_{17}^{7} - 78T_{17}^{6} + 73T_{17}^{5} + 1532T_{17}^{4} - 748T_{17}^{3} - 5408T_{17}^{2} - 1275T_{17} + 892$$ T17^8 - 2*T17^7 - 78*T17^6 + 73*T17^5 + 1532*T17^4 - 748*T17^3 - 5408*T17^2 - 1275*T17 + 892

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 3 T^{7} - 8 T^{6} - 31 T^{5} + \cdots + 1$$
$3$ $$T^{8}$$
$5$ $$(T - 1)^{8}$$
$7$ $$T^{8} + 11 T^{7} + 24 T^{6} + \cdots - 629$$
$11$ $$T^{8} + 6 T^{7} - 38 T^{6} - 250 T^{5} + \cdots + 634$$
$13$ $$(T + 1)^{8}$$
$17$ $$T^{8} - 2 T^{7} - 78 T^{6} + 73 T^{5} + \cdots + 892$$
$19$ $$T^{8} + 10 T^{7} - 37 T^{6} + \cdots - 1584$$
$23$ $$T^{8} + 6 T^{7} - 79 T^{6} + \cdots + 12015$$
$29$ $$T^{8} + 14 T^{7} + 5 T^{6} + \cdots - 1115$$
$31$ $$T^{8} + 31 T^{7} + 276 T^{6} + \cdots + 50472$$
$37$ $$T^{8} - T^{7} - 120 T^{6} + \cdots - 33660$$
$41$ $$T^{8} - 12 T^{7} - 82 T^{6} + \cdots - 19275$$
$43$ $$T^{8} - 15 T^{7} - 167 T^{6} + \cdots - 7900376$$
$47$ $$T^{8} - 18 T^{7} - 86 T^{6} + \cdots + 651523$$
$53$ $$T^{8} + 2 T^{7} - 282 T^{6} + \cdots + 111438$$
$59$ $$T^{8} + 24 T^{7} + 60 T^{6} + \cdots + 1877418$$
$61$ $$T^{8} + 9 T^{7} - 376 T^{6} + \cdots + 10312831$$
$67$ $$T^{8} + 18 T^{7} - 346 T^{6} + \cdots - 31217363$$
$71$ $$T^{8} + 10 T^{7} - 211 T^{6} + \cdots - 127950$$
$73$ $$T^{8} - 6 T^{7} - 300 T^{6} + \cdots + 1196532$$
$79$ $$T^{8} + 3 T^{7} - 498 T^{6} + \cdots + 131224698$$
$83$ $$T^{8} - 10 T^{7} - 115 T^{6} + \cdots + 685$$
$89$ $$T^{8} - 13 T^{7} - 228 T^{6} + \cdots + 1032219$$
$97$ $$T^{8} + 34 T^{7} + 162 T^{6} + \cdots - 11067218$$