Properties

 Label 400.2.u.d Level $400$ Weight $2$ Character orbit 400.u Analytic conductor $3.194$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.u (of order $$5$$, degree $$4$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.19401608085$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.58140625.2 Defining polynomial: $$x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25$$ x^8 - 3*x^7 + 4*x^6 - 7*x^5 + 11*x^4 + 5*x^3 - 10*x^2 - 25*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 50) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2}) q^{3} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{7} + 3 \beta_{6} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10})$$ q + (b6 + b5 + b3 + b2) * q^3 + (b3 + b1) * q^5 + (-b5 - b4 + b1 - 1) * q^7 + (b7 + 3*b6 - b4 + 3*b3 + 2*b2 + b1 - 2) * q^9 $$q + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2}) q^{3} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{7} + 3 \beta_{6} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{9} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{11} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{13} + (\beta_{7} + \beta_{6} + 5 \beta_{3}) q^{15} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{2} - 1) q^{17} + ( - \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 3) q^{19} + ( - 5 \beta_{6} - \beta_{5} - \beta_{3} - 5 \beta_{2}) q^{21} + ( - 2 \beta_{7} - 3 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_1 + 2) q^{23} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{25} + (5 \beta_{6} + \beta_{5} + 4 \beta_{3} - \beta_1 - 4) q^{27} + ( - 2 \beta_{6} - \beta_{5} - 5 \beta_{3} - 2 \beta_{2}) q^{29} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 5 \beta_{2} + 2) q^{31} + ( - 5 \beta_{6} + \beta_{4} - 4 \beta_{2} + 5) q^{33} + ( - \beta_{7} + \beta_{5} - 5 \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{35} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 3 \beta_{2} - 3) q^{37} + (\beta_{7} - 5 \beta_{6} - 3 \beta_{5} - \beta_{4} - 3 \beta_{3} + 3 \beta_1 + 3) q^{39} + (\beta_{7} + 3 \beta_{6} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{41} + (\beta_{5} + \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - \beta_1 - 3) q^{43} + (2 \beta_{7} + 7 \beta_{6} - \beta_{4} + 5 \beta_{3} - 1) q^{45} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_1) q^{47} + (\beta_{5} + \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - \beta_1 + 2) q^{49} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 5) q^{51} + (4 \beta_{6} + 3 \beta_{5} + 4 \beta_{2}) q^{53} + ( - \beta_{7} + \beta_{5} - 4 \beta_{3} + \beta_{2} + 4) q^{55} + ( - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{57} + (4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (2 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_1 + 1) q^{61} + ( - \beta_{7} - 7 \beta_{6} + 2 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 6) q^{63} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - 6 \beta_{3} - \beta_1 + 2) q^{65} + ( - 3 \beta_{7} + 3 \beta_{5} + 7 \beta_{2}) q^{67} + ( - 2 \beta_{7} - 7 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 6 \beta_{2} + 7) q^{69} + (\beta_{7} + 4 \beta_{6} - \beta_{5} - 3 \beta_{3} + 4 \beta_{2} + \beta_1) q^{71} + ( - 3 \beta_{7} - 8 \beta_{6} + \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_1 + 2) q^{73} + (\beta_{7} + 9 \beta_{6} + 3 \beta_{5} - \beta_{4} + 9 \beta_{3} + 8 \beta_{2} + \beta_1 - 9) q^{75} + (5 \beta_{6} + \beta_{5} - 4 \beta_{3} - \beta_1 + 4) q^{77} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{79} + (\beta_{7} + 6 \beta_{6} - \beta_{5} - 2 \beta_{4} + 6 \beta_{2} - 6) q^{81} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - 5 \beta_{2} - 2) q^{83} + (\beta_{7} - 4 \beta_{6} - 3 \beta_{4} - \beta_{3} - 5 \beta_{2} - \beta_1 + 2) q^{85} + ( - 5 \beta_{7} - 11 \beta_{6} + 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + \cdots + 6) q^{87}+ \cdots + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 6) q^{99}+O(q^{100})$$ q + (b6 + b5 + b3 + b2) * q^3 + (b3 + b1) * q^5 + (-b5 - b4 + b1 - 1) * q^7 + (b7 + 3*b6 - b4 + 3*b3 + 2*b2 + b1 - 2) * q^9 + (-b6 - b5 + b1) * q^11 + (-b7 + b6 + b4 + b3 + 3*b2 - b1 - 3) * q^13 + (b7 + b6 + 5*b3) * q^15 + (-b7 + b6 + b5 - 2*b2 - 1) * q^17 + (-b7 + 3*b6 + b5 + b4 + b2 - 3) * q^19 + (-5*b6 - b5 - b3 - 5*b2) * q^21 + (-2*b7 - 3*b6 + b5 + 2*b4 - 2*b3 - b1 + 2) * q^23 + (b6 + b5 - b4 + b3 + b2 + 3) * q^25 + (5*b6 + b5 + 4*b3 - b1 - 4) * q^27 + (-2*b6 - b5 - 5*b3 - 2*b2) * q^29 + (b7 - 2*b6 - b5 - 2*b4 - 5*b2 + 2) * q^31 + (-5*b6 + b4 - 4*b2 + 5) * q^33 + (-b7 + b5 - 5*b3 - 4*b2 - b1 + 4) * q^35 + (-2*b7 + 2*b6 + 2*b3 + 3*b2 - 3) * q^37 + (b7 - 5*b6 - 3*b5 - b4 - 3*b3 + 3*b1 + 3) * q^39 + (b7 + 3*b6 + b4 + 3*b3 + 3*b2 - b1 - 3) * q^41 + (b5 + b4 + 4*b3 + 4*b2 - b1 - 3) * q^43 + (2*b7 + 7*b6 - b4 + 5*b3 - 1) * q^45 + (b7 - b5 + b3 + b1) * q^47 + (b5 + b4 - 4*b3 - 4*b2 - b1 + 2) * q^49 + (-b5 - b4 + b3 + b2 - 2*b1 - 5) * q^51 + (4*b6 + 3*b5 + 4*b2) * q^53 + (-b7 + b5 - 4*b3 + b2 + 4) * q^55 + (-3*b5 - 3*b4 + 2*b3 + 2*b2 + b1 - 3) * q^57 + (4*b7 + 2*b6 - 2*b4 + 2*b3 + 2*b2 + 2*b1 - 2) * q^59 + (2*b7 - 3*b6 - 4*b5 - 2*b4 - b3 + 4*b1 + 1) * q^61 + (-b7 - 7*b6 + 2*b4 - 7*b3 - 6*b2 - 2*b1 + 6) * q^63 + (2*b7 - 3*b6 + 2*b4 - 6*b3 - b1 + 2) * q^65 + (-3*b7 + 3*b5 + 7*b2) * q^67 + (-2*b7 - 7*b6 + 2*b5 + 3*b4 + 6*b2 + 7) * q^69 + (b7 + 4*b6 - b5 - 3*b3 + 4*b2 + b1) * q^71 + (-3*b7 - 8*b6 + b5 + 3*b4 - 2*b3 - b1 + 2) * q^73 + (b7 + 9*b6 + 3*b5 - b4 + 9*b3 + 8*b2 + b1 - 9) * q^75 + (5*b6 + b5 - 4*b3 - b1 + 4) * q^77 + (2*b6 - 2*b5 - 2*b3 + 2*b2) * q^79 + (b7 + 6*b6 - b5 - 2*b4 + 6*b2 - 6) * q^81 + (b7 + 2*b6 - b5 + 2*b4 - 5*b2 - 2) * q^83 + (b7 - 4*b6 - 3*b4 - b3 - 5*b2 - b1 + 2) * q^85 + (-5*b7 - 11*b6 + 2*b4 - 11*b3 - 6*b2 - 2*b1 + 6) * q^87 + (b7 + 2*b6 - 4*b5 - b4 + 4*b3 + 4*b1 - 4) * q^89 + (2*b7 + 3*b6 + 3*b4 + 3*b3 + b2 - 3*b1 - 1) * q^91 + (2*b5 + 2*b4 - 7*b3 - 7*b2 - 5*b1 - 2) * q^93 + (4*b7 - 4*b6 - 3*b5 - 3*b4 - 3*b3 - 3*b2 + 5) * q^95 + (-5*b6 - 8*b3 - 5*b2) * q^97 + (2*b5 + 2*b4 + b3 + b2 - b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 3 q^{3} - 4 q^{7} - q^{9}+O(q^{10})$$ 8 * q + 3 * q^3 - 4 * q^7 - q^9 $$8 q + 3 q^{3} - 4 q^{7} - q^{9} - q^{11} - 13 q^{13} + 10 q^{15} - 11 q^{17} - 20 q^{19} - 19 q^{21} + 3 q^{23} + 30 q^{25} - 15 q^{27} - 15 q^{29} + 9 q^{31} + 19 q^{33} + 15 q^{35} - 6 q^{37} + 12 q^{39} - 9 q^{41} - 12 q^{43} + 15 q^{45} + q^{47} - 4 q^{49} - 26 q^{51} + 7 q^{53} + 25 q^{55} - 10 q^{59} + 6 q^{61} + 8 q^{63} - 10 q^{65} + 11 q^{67} + 43 q^{69} + 9 q^{71} - 8 q^{73} - 30 q^{75} + 33 q^{77} + 10 q^{79} - 17 q^{81} - 27 q^{83} + 5 q^{85} - 15 q^{89} - q^{91} - 46 q^{93} + 30 q^{95} - 36 q^{97} + 42 q^{99}+O(q^{100})$$ 8 * q + 3 * q^3 - 4 * q^7 - q^9 - q^11 - 13 * q^13 + 10 * q^15 - 11 * q^17 - 20 * q^19 - 19 * q^21 + 3 * q^23 + 30 * q^25 - 15 * q^27 - 15 * q^29 + 9 * q^31 + 19 * q^33 + 15 * q^35 - 6 * q^37 + 12 * q^39 - 9 * q^41 - 12 * q^43 + 15 * q^45 + q^47 - 4 * q^49 - 26 * q^51 + 7 * q^53 + 25 * q^55 - 10 * q^59 + 6 * q^61 + 8 * q^63 - 10 * q^65 + 11 * q^67 + 43 * q^69 + 9 * q^71 - 8 * q^73 - 30 * q^75 + 33 * q^77 + 10 * q^79 - 17 * q^81 - 27 * q^83 + 5 * q^85 - 15 * q^89 - q^91 - 46 * q^93 + 30 * q^95 - 36 * q^97 + 42 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( -64\nu^{7} + 206\nu^{6} - 318\nu^{5} + 399\nu^{4} - 732\nu^{3} - 126\nu^{2} + 2175\nu + 235 ) / 1355$$ (-64*v^7 + 206*v^6 - 318*v^5 + 399*v^4 - 732*v^3 - 126*v^2 + 2175*v + 235) / 1355 $$\beta_{2}$$ $$=$$ $$( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355$$ (406*v^7 - 714*v^6 + 747*v^5 - 1896*v^4 + 2103*v^3 + 4949*v^2 + 1065*v - 7800) / 1355 $$\beta_{3}$$ $$=$$ $$( -420\nu^{7} + 776\nu^{6} - 698\nu^{5} + 1924\nu^{4} - 2297\nu^{3} - 5129\nu^{2} - 1055\nu + 10265 ) / 1355$$ (-420*v^7 + 776*v^6 - 698*v^5 + 1924*v^4 - 2297*v^3 - 5129*v^2 - 1055*v + 10265) / 1355 $$\beta_{4}$$ $$=$$ $$( 504\nu^{7} - 877\nu^{6} + 946\nu^{5} - 2363\nu^{4} + 2919\nu^{3} + 5125\nu^{2} + 3705\nu - 11505 ) / 1355$$ (504*v^7 - 877*v^6 + 946*v^5 - 2363*v^4 + 2919*v^3 + 5125*v^2 + 3705*v - 11505) / 1355 $$\beta_{5}$$ $$=$$ $$( -613\nu^{7} + 1321\nu^{6} - 1513\nu^{5} + 3394\nu^{4} - 4352\nu^{3} - 6178\nu^{2} - 530\nu + 12985 ) / 1355$$ (-613*v^7 + 1321*v^6 - 1513*v^5 + 3394*v^4 - 4352*v^3 - 6178*v^2 - 530*v + 12985) / 1355 $$\beta_{6}$$ $$=$$ $$( -714\nu^{7} + 1265\nu^{6} - 1295\nu^{5} + 3325\nu^{4} - 3390\nu^{3} - 8367\nu^{2} - 2200\nu + 14605 ) / 1355$$ (-714*v^7 + 1265*v^6 - 1295*v^5 + 3325*v^4 - 3390*v^3 - 8367*v^2 - 2200*v + 14605) / 1355 $$\beta_{7}$$ $$=$$ $$( -1020\nu^{7} + 1962\nu^{6} - 2121\nu^{5} + 5563\nu^{4} - 6314\nu^{3} - 10443\nu^{2} - 3530\nu + 21445 ) / 1355$$ (-1020*v^7 + 1962*v^6 - 2121*v^5 + 5563*v^4 - 6314*v^3 - 10443*v^2 - 3530*v + 21445) / 1355
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 5$$ (b7 - b6 - 2*b5 + b4 + b3 - 2*b2 + 3*b1 + 3) / 5 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} - 3\beta_{4} + 2\beta_{3} + 11\beta_{2} + 6\beta _1 - 4 ) / 5$$ (2*b7 + 3*b6 - 4*b5 - 3*b4 + 2*b3 + 11*b2 + 6*b1 - 4) / 5 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} + 16\beta_{6} + 2\beta_{5} + 4\beta_{4} - 6\beta_{3} + 17\beta_{2} - 3\beta _1 + 2 ) / 5$$ (-b7 + 16*b6 + 2*b5 + 4*b4 - 6*b3 + 17*b2 - 3*b1 + 2) / 5 $$\nu^{4}$$ $$=$$ $$( 13\beta_{7} - 3\beta_{6} - 6\beta_{5} + 13\beta_{4} - 7\beta_{3} - 6\beta_{2} - 6\beta _1 + 14 ) / 5$$ (13*b7 - 3*b6 - 6*b5 + 13*b4 - 7*b3 - 6*b2 - 6*b1 + 14) / 5 $$\nu^{5}$$ $$=$$ $$( 16\beta_{7} - 6\beta_{6} - 22\beta_{5} + 6\beta_{4} + 51\beta_{3} + 43\beta_{2} + 8\beta _1 - 67 ) / 5$$ (16*b7 - 6*b6 - 22*b5 + 6*b4 + 51*b3 + 43*b2 + 8*b1 - 67) / 5 $$\nu^{6}$$ $$=$$ $$( -23\beta_{7} + 63\beta_{6} + 46\beta_{5} + 12\beta_{4} + 52\beta_{3} + 156\beta_{2} - 34\beta _1 - 144 ) / 5$$ (-23*b7 + 63*b6 + 46*b5 + 12*b4 + 52*b3 + 156*b2 - 34*b1 - 144) / 5 $$\nu^{7}$$ $$=$$ $$( -31\beta_{7} - 9\beta_{6} + 137\beta_{5} + 84\beta_{4} - 31\beta_{3} - 33\beta_{2} - 168\beta _1 + 62 ) / 5$$ (-31*b7 - 9*b6 + 137*b5 + 84*b4 - 31*b3 - 33*b2 - 168*b1 + 62) / 5

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 −0.983224 − 0.644389i 1.17421 + 0.0566033i 1.66637 − 0.917186i −0.357358 + 1.86824i 1.66637 + 0.917186i −0.357358 − 1.86824i −0.983224 + 0.644389i 1.17421 − 0.0566033i
0 −1.09089 + 0.792578i 0 −2.15743 0.587785i 0 0.833366 0 −0.365190 + 1.12394i 0
81.2 0 2.39991 1.74363i 0 2.15743 0.587785i 0 −1.83337 0 1.79224 5.51595i 0
161.1 0 −0.529876 + 1.63079i 0 2.02373 + 0.951057i 0 2.77447 0 0.0483405 + 0.0351215i 0
161.2 0 0.720859 2.21858i 0 −2.02373 + 0.951057i 0 −3.77447 0 −1.97539 1.43521i 0
241.1 0 −0.529876 1.63079i 0 2.02373 0.951057i 0 2.77447 0 0.0483405 0.0351215i 0
241.2 0 0.720859 + 2.21858i 0 −2.02373 0.951057i 0 −3.77447 0 −1.97539 + 1.43521i 0
321.1 0 −1.09089 0.792578i 0 −2.15743 + 0.587785i 0 0.833366 0 −0.365190 1.12394i 0
321.2 0 2.39991 + 1.74363i 0 2.15743 + 0.587785i 0 −1.83337 0 1.79224 + 5.51595i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 321.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.u.d 8
4.b odd 2 1 50.2.d.b 8
12.b even 2 1 450.2.h.e 8
20.d odd 2 1 250.2.d.d 8
20.e even 4 2 250.2.e.c 16
25.d even 5 1 inner 400.2.u.d 8
25.d even 5 1 10000.2.a.t 4
25.e even 10 1 10000.2.a.x 4
100.h odd 10 1 250.2.d.d 8
100.h odd 10 1 1250.2.a.f 4
100.j odd 10 1 50.2.d.b 8
100.j odd 10 1 1250.2.a.l 4
100.l even 20 2 250.2.e.c 16
100.l even 20 2 1250.2.b.e 8
300.n even 10 1 450.2.h.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b 8 4.b odd 2 1
50.2.d.b 8 100.j odd 10 1
250.2.d.d 8 20.d odd 2 1
250.2.d.d 8 100.h odd 10 1
250.2.e.c 16 20.e even 4 2
250.2.e.c 16 100.l even 20 2
400.2.u.d 8 1.a even 1 1 trivial
400.2.u.d 8 25.d even 5 1 inner
450.2.h.e 8 12.b even 2 1
450.2.h.e 8 300.n even 10 1
1250.2.a.f 4 100.h odd 10 1
1250.2.a.l 4 100.j odd 10 1
1250.2.b.e 8 100.l even 20 2
10000.2.a.t 4 25.d even 5 1
10000.2.a.x 4 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 3T_{3}^{7} + 8T_{3}^{6} - 6T_{3}^{5} + 25T_{3}^{4} + 24T_{3}^{3} + 128T_{3}^{2} + 192T_{3} + 256$$ acting on $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 3 T^{7} + 8 T^{6} - 6 T^{5} + \cdots + 256$$
$5$ $$T^{8} - 15 T^{6} + 105 T^{4} + \cdots + 625$$
$7$ $$(T^{4} + 2 T^{3} - 11 T^{2} - 12 T + 16)^{2}$$
$11$ $$T^{8} + T^{7} - 3 T^{6} - 7 T^{5} + \cdots + 256$$
$13$ $$T^{8} + 13 T^{7} + 93 T^{6} + \cdots + 39601$$
$17$ $$T^{8} + 11 T^{7} + 72 T^{6} + \cdots + 11881$$
$19$ $$T^{8} + 20 T^{7} + 190 T^{6} + \cdots + 6400$$
$23$ $$T^{8} - 3 T^{7} + 63 T^{6} + 49 T^{5} + \cdots + 256$$
$29$ $$T^{8} + 15 T^{7} + 105 T^{6} + \cdots + 25$$
$31$ $$T^{8} - 9 T^{7} + 87 T^{6} + \cdots + 1597696$$
$37$ $$T^{8} + 6 T^{7} + 47 T^{6} + \cdots + 5041$$
$41$ $$T^{8} + 9 T^{7} + 27 T^{6} + \cdots + 7921$$
$43$ $$(T^{4} + 6 T^{3} - 39 T^{2} - 64 T + 176)^{2}$$
$47$ $$T^{8} - T^{7} + 17 T^{6} - 33 T^{5} + \cdots + 256$$
$53$ $$T^{8} - 7 T^{7} + 108 T^{6} + \cdots + 65536$$
$59$ $$T^{8} + 10 T^{7} + 260 T^{6} + \cdots + 102400$$
$61$ $$T^{8} - 6 T^{7} - 18 T^{6} + \cdots + 2920681$$
$67$ $$T^{8} - 11 T^{7} + 237 T^{6} + \cdots + 891136$$
$71$ $$T^{8} - 9 T^{7} + 217 T^{6} + \cdots + 4096$$
$73$ $$T^{8} + 8 T^{7} + 208 T^{6} + \cdots + 1175056$$
$79$ $$T^{8} - 10 T^{7} + 120 T^{6} + \cdots + 102400$$
$83$ $$T^{8} + 27 T^{7} + 403 T^{6} + \cdots + 13424896$$
$89$ $$T^{8} + 15 T^{7} + 70 T^{6} + \cdots + 9610000$$
$97$ $$(T^{4} + 18 T^{3} + 124 T^{2} + 7 T + 1)^{2}$$