# Properties

 Label 4025.2.a.v Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3$$ x^8 - x^7 - 10*x^6 + 9*x^5 + 28*x^4 - 22*x^3 - 16*x^2 + 7*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{6} - q^{7} + \beta_{3} q^{8} + ( - \beta_{6} + \beta_{4}) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b4 + 1) * q^3 + (b2 + 1) * q^4 + (-b7 + b6 + b3 + b1) * q^6 - q^7 + b3 * q^8 + (-b6 + b4) * q^9 $$q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{6} - q^{7} + \beta_{3} q^{8} + ( - \beta_{6} + \beta_{4}) q^{9} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{11} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 2) q^{12} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_1) q^{13} - \beta_1 q^{14} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{16} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2}) q^{17} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{18} + \beta_{5} q^{19} + ( - \beta_{4} - 1) q^{21} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 4) q^{22} + q^{23} + (\beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{24} + (\beta_{6} + \beta_{2} + \beta_1 + 3) q^{26} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{27} + ( - \beta_{2} - 1) q^{28} + (\beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{29} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_1) q^{31} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{32} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{33} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{34} + (2 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_1 + 3) q^{36} + ( - 3 \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{37} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{38} + (\beta_{6} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{39} + (\beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} - 3 \beta_{2} - 1) q^{41} + (\beta_{7} - \beta_{6} - \beta_{3} - \beta_1) q^{42} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{43} + (2 \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{44} + \beta_1 q^{46} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 3) q^{47} + ( - \beta_{5} + \beta_{2} + 2 \beta_1 + 1) q^{48} + q^{49} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{51} + (2 \beta_{7} - \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_1 + 3) q^{52} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 2 \beta_1 + 2) q^{53} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 3) q^{54} - \beta_{3} q^{56} + (\beta_{7} + \beta_{5} + \beta_{2} - \beta_1 + 2) q^{57} + (\beta_{7} - \beta_{6} - \beta_{5} - 4 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1) q^{58} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{59} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + 1) q^{61} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{62} + (\beta_{6} - \beta_{4}) q^{63} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{64} + ( - \beta_{6} + \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \beta_{2} + 9) q^{66} + (3 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 4) q^{67} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{68} + (\beta_{4} + 1) q^{69} + (3 \beta_{6} - \beta_{3} - \beta_1 + 3) q^{71} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1) q^{72} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{73} + (\beta_{5} + 5 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \beta_1 - 4) q^{74} + (\beta_{7} - \beta_{6} + 4 \beta_{4} + \beta_{3} - \beta_{2} + 4) q^{76} + (\beta_{7} - \beta_{3} - \beta_1) q^{77} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_1 + 3) q^{78} + ( - 3 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{79} + (\beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{81} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_1) q^{82} + (2 \beta_{7} - 2 \beta_{6} - \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{83} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 2) q^{84} + ( - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots + 1) q^{86}+ \cdots + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{3} + 3 \beta_1 + 2) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b4 + 1) * q^3 + (b2 + 1) * q^4 + (-b7 + b6 + b3 + b1) * q^6 - q^7 + b3 * q^8 + (-b6 + b4) * q^9 + (-b7 + b3 + b1) * q^11 + (b5 + b4 + b3 - b1 + 2) * q^12 + (-b7 + b5 - b4 + b1) * q^13 - b1 * q^14 + (b4 + b3 - b2) * q^16 + (-b7 - b5 - b4 + b2) * q^17 + (-b7 + b6 - b5 - b4 + b3 + b2 + b1) * q^18 + b5 * q^19 + (-b4 - 1) * q^21 + (2*b4 + b3 + b2 + 4) * q^22 + q^23 + (b6 + b3 + b2 + b1) * q^24 + (b6 + b2 + b1 + 3) * q^26 + (-b6 - b4 - b1) * q^27 + (-b2 - 1) * q^28 + (b7 - b6 - b4 - 2*b3 + b1 - 2) * q^29 + (-b7 + b6 + b5 + b1) * q^31 + (-b7 + b6 + b4 - b3 + b2 - b1 + 2) * q^32 + (-b7 + b6 - b5 + b4 + b3 + b2 + 3*b1 + 1) * q^33 + (2*b7 - 3*b6 + 2*b4 - b3 - 2*b2 - 2) * q^34 + (2*b7 - b6 + b5 + 2*b4 - b2 - b1 + 3) * q^36 + (-3*b7 + b6 + b3 + b2 - b1 + 1) * q^37 + (-b7 + 2*b6 - b4 + b3 + b2 + b1 + 1) * q^38 + (b6 + b4 + b2 + b1 + 1) * q^39 + (b7 - b6 - b5 + 3*b4 - 3*b2 - 1) * q^41 + (b7 - b6 - b3 - b1) * q^42 + (b7 - 2*b6 + 2*b5 - b3 - 2*b2) * q^43 + (2*b6 + b4 + 2*b3 + b2 + 3*b1 + 2) * q^44 + b1 * q^46 + (2*b7 - 2*b6 - b5 + 2*b4 + 3) * q^47 + (-b5 + b2 + 2*b1 + 1) * q^48 + q^49 + (-b7 - b5 + b4 - b2 + b1 - 2) * q^51 + (2*b7 - b5 + 3*b4 + b3 + b1 + 3) * q^52 + (-b6 + b4 - b2 + 2*b1 + 2) * q^53 + (b7 - b6 - b5 - b4 - b3 + b1 - 3) * q^54 - b3 * q^56 + (b7 + b5 + b2 - b1 + 2) * q^57 + (b7 - b6 - b5 - 4*b4 - 3*b3 + b2 - b1) * q^58 + (-b7 - 2*b6 - 2*b5 - b3 + 3*b2 + b1 + 1) * q^59 + (-b6 + b4 - 2*b3 + 1) * q^61 + (-b7 + 2*b6 + b5 + b4 + b3 + 3) * q^62 + (b6 - b4) * q^63 + (-b7 + b6 + b5 - b4 - b3 - 2*b2 + 2*b1 - 6) * q^64 + (-b6 + b5 + 4*b4 + 2*b3 + b2 + 9) * q^66 + (3*b7 - b6 - b5 + 2*b4 - b3 + b2 + 4) * q^67 + (2*b6 - b5 - 4*b4 - b3 + 2*b2 - b1) * q^68 + (b4 + 1) * q^69 + (3*b6 - b3 - b1 + 3) * q^71 + (-b7 + 2*b6 + b5 - 2*b4 + b2 + 2*b1) * q^72 + (b6 + 2*b5 - b4 - b3 + b2 + b1 + 4) * q^73 + (b5 + 5*b4 + 2*b3 - 4*b2 + b1 - 4) * q^74 + (b7 - b6 + 4*b4 + b3 - b2 + 4) * q^76 + (b7 - b3 - b1) * q^77 + (-b7 + b6 + b5 + b4 + 2*b3 + b1 + 3) * q^78 + (-3*b7 + 2*b6 + b5 - 3*b4 - 2*b3 + b2 + 3*b1 - 3) * q^79 + (b7 + 3*b6 - b4 - b3 - 2*b1 - 1) * q^81 + (-2*b7 + b6 - b5 - b4 - b3 + b2 - 4*b1) * q^82 + (2*b7 - 2*b6 - b4 - 3*b3 - b2 + b1 + 1) * q^83 + (-b5 - b4 - b3 + b1 - 2) * q^84 + (-2*b7 + 4*b6 - 2*b5 - 6*b4 - b3 + 4*b2 + 2*b1 + 1) * q^86 + (-b7 + b6 + b5 - b4 - 2*b2 - 3*b1 - 4) * q^87 + (-b7 + b6 + 2*b5 + 2*b3 + b2 + b1 + 5) * q^88 + (2*b7 + b6 - b5 - b4 + b3 + 3*b2 + 2*b1 + 3) * q^89 + (b7 - b5 + b4 - b1) * q^91 + (b2 + 1) * q^92 + (-b4 + b2 + 2*b1 + 2) * q^93 + (-b7 - 2*b5 - 3*b4 + b3 + 3*b2 + 4*b1 + 1) * q^94 + (b7 - 4*b6 + b4 - 2*b3 - b2 - b1 + 5) * q^96 + (2*b7 + b6 + 2*b5 - b4 - b3 - b2 - 2*b1 + 1) * q^97 + b1 * q^98 + (-b7 + 2*b6 - b5 + b3 + 3*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7}+O(q^{10})$$ 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7 $$8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7} + 3 q^{11} + 9 q^{12} + 5 q^{13} - q^{14} - q^{16} + 5 q^{17} + 2 q^{18} - 2 q^{19} - 4 q^{21} + 21 q^{22} + 8 q^{23} - 6 q^{24} + 18 q^{26} + 7 q^{27} - 5 q^{28} - 9 q^{29} - 3 q^{31} + 6 q^{32} + 4 q^{33} - 10 q^{34} + 16 q^{36} + 6 q^{37} + 4 q^{38} - 2 q^{39} - 7 q^{41} + q^{42} + 8 q^{43} + 4 q^{44} + q^{46} + 22 q^{47} + 9 q^{48} + 8 q^{49} - 12 q^{51} + 11 q^{52} + 21 q^{53} - 15 q^{54} + 8 q^{57} + 16 q^{58} + 14 q^{59} + 8 q^{61} + 12 q^{62} - 40 q^{64} + 55 q^{66} + 21 q^{67} + 3 q^{68} + 4 q^{69} + 11 q^{71} - q^{72} + 26 q^{73} - 41 q^{74} + 21 q^{76} - 3 q^{77} + 17 q^{78} - 16 q^{79} - 20 q^{81} - q^{82} + 20 q^{83} - 9 q^{84} + 14 q^{86} - 29 q^{87} + 32 q^{88} + 15 q^{89} - 5 q^{91} + 5 q^{92} + 19 q^{93} + 21 q^{94} + 52 q^{96} + q^{97} + q^{98} + 15 q^{99}+O(q^{100})$$ 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7 + 3 * q^11 + 9 * q^12 + 5 * q^13 - q^14 - q^16 + 5 * q^17 + 2 * q^18 - 2 * q^19 - 4 * q^21 + 21 * q^22 + 8 * q^23 - 6 * q^24 + 18 * q^26 + 7 * q^27 - 5 * q^28 - 9 * q^29 - 3 * q^31 + 6 * q^32 + 4 * q^33 - 10 * q^34 + 16 * q^36 + 6 * q^37 + 4 * q^38 - 2 * q^39 - 7 * q^41 + q^42 + 8 * q^43 + 4 * q^44 + q^46 + 22 * q^47 + 9 * q^48 + 8 * q^49 - 12 * q^51 + 11 * q^52 + 21 * q^53 - 15 * q^54 + 8 * q^57 + 16 * q^58 + 14 * q^59 + 8 * q^61 + 12 * q^62 - 40 * q^64 + 55 * q^66 + 21 * q^67 + 3 * q^68 + 4 * q^69 + 11 * q^71 - q^72 + 26 * q^73 - 41 * q^74 + 21 * q^76 - 3 * q^77 + 17 * q^78 - 16 * q^79 - 20 * q^81 - q^82 + 20 * q^83 - 9 * q^84 + 14 * q^86 - 29 * q^87 + 32 * q^88 + 15 * q^89 - 5 * q^91 + 5 * q^92 + 19 * q^93 + 21 * q^94 + 52 * q^96 + q^97 + q^98 + 15 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5\nu^{2} + 4\nu + 1$$ v^4 - v^3 - 5*v^2 + 4*v + 1 $$\beta_{5}$$ $$=$$ $$\nu^{6} - \nu^{5} - 8\nu^{4} + 6\nu^{3} + 17\nu^{2} - 7\nu - 7$$ v^6 - v^5 - 8*v^4 + 6*v^3 + 17*v^2 - 7*v - 7 $$\beta_{6}$$ $$=$$ $$\nu^{7} - \nu^{6} - 9\nu^{5} + 8\nu^{4} + 21\nu^{3} - 17\nu^{2} - 5\nu + 3$$ v^7 - v^6 - 9*v^5 + 8*v^4 + 21*v^3 - 17*v^2 - 5*v + 3 $$\beta_{7}$$ $$=$$ $$\nu^{7} - \nu^{6} - 10\nu^{5} + 9\nu^{4} + 27\nu^{3} - 21\nu^{2} - 10\nu + 3$$ v^7 - v^6 - 10*v^5 + 9*v^4 + 27*v^3 - 21*v^2 - 10*v + 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5\beta_{2} + 14$$ b4 + b3 + 5*b2 + 14 $$\nu^{5}$$ $$=$$ $$-\beta_{7} + \beta_{6} + \beta_{4} + 7\beta_{3} + \beta_{2} + 19\beta _1 + 2$$ -b7 + b6 + b4 + 7*b3 + b2 + 19*b1 + 2 $$\nu^{6}$$ $$=$$ $$-\beta_{7} + \beta_{6} + \beta_{5} + 9\beta_{4} + 9\beta_{3} + 24\beta_{2} + 2\beta _1 + 70$$ -b7 + b6 + b5 + 9*b4 + 9*b3 + 24*b2 + 2*b1 + 70 $$\nu^{7}$$ $$=$$ $$-10\beta_{7} + 11\beta_{6} + \beta_{5} + 10\beta_{4} + 43\beta_{3} + 10\beta_{2} + 94\beta _1 + 24$$ -10*b7 + 11*b6 + b5 + 10*b4 + 43*b3 + 10*b2 + 94*b1 + 24

## 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
 −2.18675 −2.07739 −0.571357 −0.333224 0.651939 1.09801 2.02014 2.39863
−2.18675 2.66680 2.78189 0 −5.83163 −1.00000 −1.70979 4.11181 0
1.2 −2.07739 −0.298210 2.31556 0 0.619500 −1.00000 −0.655538 −2.91107 0
1.3 −0.571357 −1.62458 −1.67355 0 0.928216 −1.00000 2.09891 −0.360734 0
1.4 −0.333224 0.161242 −1.88896 0 −0.0537298 −1.00000 1.29590 −2.97400 0
1.5 0.651939 2.38619 −1.57498 0 1.55565 −1.00000 −2.33067 2.69390 0
1.6 1.09801 0.493654 −0.794374 0 0.542037 −1.00000 −3.06825 −2.75631 0
1.7 2.02014 −1.91409 2.08098 0 −3.86674 −1.00000 0.163592 0.663753 0
1.8 2.39863 2.12900 3.75344 0 5.10670 −1.00000 4.20585 1.53265 0
 $$n$$: 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
$$5$$ $$1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 4025.2.a.v yes 8
5.b even 2 1 4025.2.a.u 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.u 8 5.b even 2 1
4025.2.a.v yes 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{8} - T_{2}^{7} - 10T_{2}^{6} + 9T_{2}^{5} + 28T_{2}^{4} - 22T_{2}^{3} - 16T_{2}^{2} + 7T_{2} + 3$$ T2^8 - T2^7 - 10*T2^6 + 9*T2^5 + 28*T2^4 - 22*T2^3 - 16*T2^2 + 7*T2 + 3 $$T_{3}^{8} - 4T_{3}^{7} - 4T_{3}^{6} + 27T_{3}^{5} - 3T_{3}^{4} - 47T_{3}^{3} + 15T_{3}^{2} + 5T_{3} - 1$$ T3^8 - 4*T3^7 - 4*T3^6 + 27*T3^5 - 3*T3^4 - 47*T3^3 + 15*T3^2 + 5*T3 - 1 $$T_{11}^{8} - 3T_{11}^{7} - 30T_{11}^{6} + 43T_{11}^{5} + 297T_{11}^{4} - 28T_{11}^{3} - 982T_{11}^{2} - 908T_{11} - 183$$ T11^8 - 3*T11^7 - 30*T11^6 + 43*T11^5 + 297*T11^4 - 28*T11^3 - 982*T11^2 - 908*T11 - 183

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} - 10 T^{6} + 9 T^{5} + \cdots + 3$$
$3$ $$T^{8} - 4 T^{7} - 4 T^{6} + 27 T^{5} + \cdots - 1$$
$5$ $$T^{8}$$
$7$ $$(T + 1)^{8}$$
$11$ $$T^{8} - 3 T^{7} - 30 T^{6} + 43 T^{5} + \cdots - 183$$
$13$ $$T^{8} - 5 T^{7} - 30 T^{6} + 193 T^{5} + \cdots + 19$$
$17$ $$T^{8} - 5 T^{7} - 43 T^{6} + \cdots + 4083$$
$19$ $$T^{8} + 2 T^{7} - 29 T^{6} - 44 T^{5} + \cdots + 675$$
$23$ $$(T - 1)^{8}$$
$29$ $$T^{8} + 9 T^{7} - 55 T^{6} + \cdots + 7311$$
$31$ $$T^{8} + 3 T^{7} - 46 T^{6} - 157 T^{5} + \cdots + 97$$
$37$ $$T^{8} - 6 T^{7} - 168 T^{6} + \cdots - 210905$$
$41$ $$T^{8} + 7 T^{7} - 182 T^{6} + \cdots + 209409$$
$43$ $$T^{8} - 8 T^{7} - 158 T^{6} + \cdots + 356869$$
$47$ $$T^{8} - 22 T^{7} + 101 T^{6} + \cdots - 269835$$
$53$ $$T^{8} - 21 T^{7} + 121 T^{6} + \cdots + 957$$
$59$ $$T^{8} - 14 T^{7} - 207 T^{6} + \cdots - 2725305$$
$61$ $$T^{8} - 8 T^{7} - 84 T^{6} + \cdots + 23917$$
$67$ $$T^{8} - 21 T^{7} + 23 T^{6} + \cdots + 46337$$
$71$ $$T^{8} - 11 T^{7} - 175 T^{6} + \cdots + 88041$$
$73$ $$T^{8} - 26 T^{7} + 132 T^{6} + \cdots + 40969$$
$79$ $$T^{8} + 16 T^{7} - 164 T^{6} + \cdots - 1052537$$
$83$ $$T^{8} - 20 T^{7} - 59 T^{6} + \cdots + 195159$$
$89$ $$T^{8} - 15 T^{7} - 354 T^{6} + \cdots + 19143423$$
$97$ $$T^{8} - T^{7} - 230 T^{6} + \cdots + 124907$$