# Properties

 Label 625.2.e.i Level $625$ Weight $2$ Character orbit 625.e Analytic conductor $4.991$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(124,625)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("625.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.e (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.58140625.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{5} - \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{4}+ \cdots + (\beta_{6} - \beta_{3} - \beta_1) q^{9}+O(q^{10})$$ q + (-b5 - b4) * q^2 + (-b5 - b2 - b1 + 1) * q^3 + (b7 + b6 + b5 + b2 + b1) * q^4 + (b6 - b5 - 2*b4 - b2 + 1) * q^6 + (-b6 - b4 - 2*b2 + 1) * q^7 + (b4 - 2*b3 - 2) * q^8 + (b6 - b3 - b1) * q^9 $$q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{5} - \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{4}+ \cdots + (2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{99}+O(q^{100})$$ q + (-b5 - b4) * q^2 + (-b5 - b2 - b1 + 1) * q^3 + (b7 + b6 + b5 + b2 + b1) * q^4 + (b6 - b5 - 2*b4 - b2 + 1) * q^6 + (-b6 - b4 - 2*b2 + 1) * q^7 + (b4 - 2*b3 - 2) * q^8 + (b6 - b3 - b1) * q^9 - 2*b2 * q^11 + (b7 + b5 - 2*b4 + b3 - 2*b2 + b1) * q^12 + (-b6 + b4 - b1 + 1) * q^13 + (-b7 + 2*b6 - 2*b4 + b3 + b2 - 1) * q^14 + (b7 - b5 - b1) * q^16 + (-b4 + 2*b3 - 3*b2 - b1) * q^17 + (b7 + b4 - 3*b3 - b2 + b1 - 1) * q^18 + (b6 - b5 - b2 + 1) * q^19 + (b7 + b6 + b4 - b3 + 3*b2 + 2*b1 - 2) * q^21 + (-2*b7 + 2*b6 - 2) * q^22 + (-b7 + 2*b4 - b3 + b2 - 3) * q^23 + (b7 + 2*b5 + b4 - 3*b3 + 3*b2 + b1 - 4) * q^24 + (-b6 - 2*b5 - b4 - b3 + b2 - 2*b1 + 1) * q^26 + (-b7 + 2*b5 - b3 - b2 + 1) * q^27 + (b7 - b6 + 3*b5 + 3*b4 - 3*b3 + 3*b1 + 1) * q^28 + (-2*b7 - 2*b6 - b5 - 2*b4 + 5*b3 - 5*b2 - 3*b1 + 6) * q^29 + (2*b7 - 3*b6 + 3*b5 - b4 - b2 + b1 + 2) * q^31 + (b7 + b6 - 6*b4 + 3*b3 - b2 + b1 + 2) * q^32 + (2*b6 + 2*b5 + 2*b4 - 4*b3 + 2*b2 + 2*b1 - 4) * q^33 + (-b7 + b5 + 3*b4 - 2*b3 + b1 - 3) * q^34 + (-b7 + 2*b6 - 4*b4 + 3*b3 - 1) * q^36 + (-b7 - b6 - b5 - 3*b4 + 4*b3 - 8*b2 - 2*b1 + 5) * q^37 + (-b7 + 2*b6 - b5 - 2*b4 + b3 - 2*b2 + b1) * q^38 + (-2*b6 - b5 + b4 + b2 - 2*b1 + 2) * q^39 + (-b7 - b6 + b5 + b4 + 3*b3 + 2*b1 - 1) * q^41 + (b6 + b5 - b4 + 2*b3 - b2 + b1) * q^42 + (-b7 + b6 + 6*b4 - 5*b3 + b2 - b1 - 3) * q^43 + (-2*b6 + 2*b5 + 2*b4 - 2*b2 + 2) * q^44 + (-b7 - b6 + b5 + 3*b4 + b3 - 3*b1 + 3) * q^46 + (3*b7 - 3*b6 - 2*b5 - b4 + b3 - b2 - 2*b1 + 4) * q^47 + (b7 + b5 - 3*b4 - 3*b3 + 1) * q^48 + (-b7 - b6 - 4*b5 - 2*b4 + 3*b3 - 3*b2 - 3*b1 + 4) * q^49 + (2*b6 + 4*b5 + 2*b4 - 4*b3 + 4*b2 + 4*b1 - 4) * q^51 + (2*b7 - 3*b4 + b3 - b2 + 4) * q^52 + (-3*b7 + 3*b6 - 2*b5 + 3*b3 - 2*b2 - 2*b1 - 4) * q^53 + (-b7 - b6 - b5 + 3*b4 - 3*b3 + 2*b2 - b1 - 3) * q^54 + (-2*b7 + 2*b6 - 2*b5 - b4 + 2*b2 - b1 - 3) * q^56 + (-b7 + 3*b6 - 5*b3 - 3*b2 - b1 - 1) * q^57 + (-4*b6 - 4*b5 + b2 - 2*b1 + 3) * q^58 + (b7 - 3*b6 - b5 - 2*b4 - b3 + 2*b1 + 2) * q^59 + (4*b7 - 6*b6 + b5 + 2*b4 + b3 + 2*b1 + 2) * q^61 + (-b7 + b6 - b5 - 2*b4 - 3*b3 + 6*b2 - 8) * q^62 + (2*b7 + 2*b5 + 2*b4 + 2*b1 + 2) * q^63 + (4*b7 - 4*b6 + 2*b5 - 2*b4 + 4*b3 - 5*b2 + 4*b1) * q^64 + (-2*b7 + 2*b6 + 2*b5 + 4*b4 - 2*b3 - 4) * q^66 + (2*b4 - 4*b3 + 4*b2 + 4*b1 - 4) * q^67 + (-4*b7 - b6 - 3*b4 + 3*b3 - 5*b2 - 4*b1 + 4) * q^68 + (-2*b7 - b6 + b5 + 3*b4 + b2 - b1 - 2) * q^69 + (-3*b7 - 3*b6 - 2*b5 + 3*b4 - 5*b3 - b2 - 4*b1 - 4) * q^71 + (4*b7 - 4*b6 + 3*b5 + b4 + b3 + 2*b2 + 3*b1) * q^72 + (2*b7 - b5 + 2*b4 + b3 + 2*b2 - 2) * q^73 + (-3*b7 + 2*b6 - 2*b5 - b4 + b1 - 3) * q^74 + (b7 + 2*b5 + b4 + b3 - b2 + b1 - 2) * q^76 + (-2*b5 + 2*b4 + 2*b2 - 4) * q^77 + (2*b7 - 2*b6 - 3*b5 - 4*b4 + b2 - 3*b1 + 5) * q^78 + (3*b7 + 3*b6 - b5 - 3*b4 - b3 + 4*b2 + 7*b1 - 5) * q^79 + (-4*b7 + 2*b6 - 2*b5 + 5*b4 + 2*b2 - 2*b1 - 4) * q^81 + (-3*b7 - 3*b6 + 4*b4 + 4*b3 + 2*b2 - 3*b1 + 1) * q^82 + (-b6 - b5 - 5*b4 + 10*b3 - 9*b2 - 5*b1 + 6) * q^83 + (b7 - b6 - b5 + 4*b4 + b3 - 4) * q^84 + (-4*b7 + 2*b6 - 3*b5 + b4 - 2*b3 - b2 - 6*b1 + 2) * q^86 + (-3*b7 + b6 - 3*b5 + 6*b4 - b3 + 2*b2 - 2*b1 + 4) * q^87 + (-2*b3 + 4*b2 - 4) * q^88 + (-2*b6 - b5 + 6*b4 - 5*b3 + 5*b2 - 2*b1 + 2) * q^89 + (b7 - b6 - b5 - b3) * q^91 + (-4*b6 - 4*b5 + b4 - 2*b3 - 3*b1 + 1) * q^92 + (5*b7 - 4*b6 - b4 + 7*b3 + 7*b2 + 5*b1) * q^93 + (2*b7 + 3*b6 - 3*b5 - 11*b4 + b2 + b1) * q^94 + (3*b7 + 3*b6 - 7*b4 + 5*b3 - b2 + 6*b1 + 2) * q^96 + (-4*b7 + 4*b6 + 4*b5 + 8*b4 - 5*b3 - 4*b2 + 4*b1 - 3) * q^97 + (2*b7 - 2*b5 - 6*b4 + 4*b3 - 3*b2 + 8) * q^98 + (2*b7 + 4*b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 5 q^{3} + 4 q^{4} + 6 q^{6} - 10 q^{8} + q^{9}+O(q^{10})$$ 8 * q + 5 * q^3 + 4 * q^4 + 6 * q^6 - 10 * q^8 + q^9 $$8 q + 5 q^{3} + 4 q^{4} + 6 q^{6} - 10 q^{8} + q^{9} - 4 q^{11} - 10 q^{12} + 5 q^{13} - 7 q^{14} - 2 q^{16} - 15 q^{17} + 10 q^{19} + q^{21} - 10 q^{22} - 15 q^{23} - 20 q^{24} + 6 q^{26} + 5 q^{27} + 20 q^{28} + 15 q^{29} + q^{31} - 10 q^{33} - 12 q^{34} - 17 q^{36} + 5 q^{37} + 12 q^{39} - 9 q^{41} - 5 q^{42} + 8 q^{44} + 16 q^{46} + 15 q^{47} + 5 q^{48} + 14 q^{49} - 4 q^{51} + 20 q^{52} - 35 q^{53} - 10 q^{54} - 15 q^{56} + 20 q^{58} + 15 q^{59} + 6 q^{61} - 45 q^{62} + 20 q^{63} - 26 q^{64} - 18 q^{66} - 13 q^{69} - 29 q^{71} - 5 q^{72} - 10 q^{73} - 12 q^{74} - 20 q^{76} - 20 q^{77} + 25 q^{78} - 10 q^{79} - 12 q^{81} - 15 q^{83} - 27 q^{84} + 16 q^{86} + 55 q^{87} - 20 q^{88} + 40 q^{89} + q^{91} + 5 q^{92} - 7 q^{94} + 11 q^{96} + 10 q^{97} + 40 q^{98} - 8 q^{99}+O(q^{100})$$ 8 * q + 5 * q^3 + 4 * q^4 + 6 * q^6 - 10 * q^8 + q^9 - 4 * q^11 - 10 * q^12 + 5 * q^13 - 7 * q^14 - 2 * q^16 - 15 * q^17 + 10 * q^19 + q^21 - 10 * q^22 - 15 * q^23 - 20 * q^24 + 6 * q^26 + 5 * q^27 + 20 * q^28 + 15 * q^29 + q^31 - 10 * q^33 - 12 * q^34 - 17 * q^36 + 5 * q^37 + 12 * q^39 - 9 * q^41 - 5 * q^42 + 8 * q^44 + 16 * q^46 + 15 * q^47 + 5 * q^48 + 14 * q^49 - 4 * q^51 + 20 * q^52 - 35 * q^53 - 10 * q^54 - 15 * q^56 + 20 * q^58 + 15 * q^59 + 6 * q^61 - 45 * q^62 + 20 * q^63 - 26 * q^64 - 18 * q^66 - 13 * q^69 - 29 * q^71 - 5 * q^72 - 10 * q^73 - 12 * q^74 - 20 * q^76 - 20 * q^77 + 25 * q^78 - 10 * q^79 - 12 * q^81 - 15 * q^83 - 27 * q^84 + 16 * q^86 + 55 * q^87 - 20 * q^88 + 40 * q^89 + q^91 + 5 * q^92 - 7 * q^94 + 11 * q^96 + 10 * q^97 + 40 * q^98 - 8 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355$$ (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 $$\beta_{4}$$ $$=$$ $$( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355$$ (728*v^7 - 1327*v^6 + 1246*v^5 - 3353*v^4 + 3584*v^3 + 8547*v^2 + 2190*v - 15715) / 1355 $$\beta_{5}$$ $$=$$ $$( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355$$ (-857*v^7 + 1666*v^6 - 1743*v^5 + 4424*v^4 - 4907*v^3 - 9470*v^2 - 2485*v + 18200) / 1355 $$\beta_{6}$$ $$=$$ $$( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355$$ (891*v^7 - 1623*v^6 + 1624*v^5 - 4492*v^4 + 4991*v^3 + 9520*v^2 + 3235*v - 18960) / 1355 $$\beta_{7}$$ $$=$$ $$( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355$$ (955*v^7 - 1829*v^6 + 1942*v^5 - 4891*v^4 + 5723*v^3 + 9646*v^2 + 2415*v - 20550) / 1355
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}$$ -b7 - b5 - b4 + b3 + b2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3$$ b7 + b5 - 3*b4 + 4*b3 + b2 + b1 + 3 $$\nu^{4}$$ $$=$$ $$5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2$$ 5*b7 - 5*b6 + 4*b5 + 2*b4 + 2*b3 + 2*b2 + 4*b1 + 2 $$\nu^{5}$$ $$=$$ $$4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13$$ 4*b7 - 6*b6 - 7*b3 + 11*b2 + 4*b1 - 13 $$\nu^{6}$$ $$=$$ $$7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21$$ 7*b6 + 7*b5 - 8*b4 - 7*b3 + 21*b2 - 2*b1 - 21 $$\nu^{7}$$ $$=$$ $$23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2}$$ 23*b7 + 38*b5 + 23*b4 - 23*b3 + 12*b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$1 - \beta_{2} + \beta_{3} - \beta_{4}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 −0.357358 − 1.86824i 1.66637 + 0.917186i 1.17421 + 0.0566033i −0.983224 − 0.644389i 1.17421 − 0.0566033i −0.983224 + 0.644389i −0.357358 + 1.86824i 1.66637 − 0.917186i
−2.19625 0.713605i 0.279141 0.384204i 2.69625 + 1.95894i 0 −0.887234 + 0.644613i 3.03582i −1.80902 2.48990i 0.857358 + 2.63868i 0
124.2 1.07822 + 0.350334i 1.52988 2.10569i −0.578217 0.420099i 0 2.38723 1.73443i 0.407162i −1.80902 2.48990i −1.16637 3.58973i 0
249.1 −0.107666 + 0.148189i −1.39991 + 0.454857i 0.607666 + 1.87020i 0 0.0833172 0.256424i 3.26086i −0.690983 0.224514i −0.674207 + 0.489840i 0
249.2 1.22570 1.68703i 2.09089 0.679371i −0.725700 2.23347i 0 1.41668 4.36010i 0.992398i −0.690983 0.224514i 1.48322 1.07763i 0
374.1 −0.107666 0.148189i −1.39991 0.454857i 0.607666 1.87020i 0 0.0833172 + 0.256424i 3.26086i −0.690983 + 0.224514i −0.674207 0.489840i 0
374.2 1.22570 + 1.68703i 2.09089 + 0.679371i −0.725700 + 2.23347i 0 1.41668 + 4.36010i 0.992398i −0.690983 + 0.224514i 1.48322 + 1.07763i 0
499.1 −2.19625 + 0.713605i 0.279141 + 0.384204i 2.69625 1.95894i 0 −0.887234 0.644613i 3.03582i −1.80902 + 2.48990i 0.857358 2.63868i 0
499.2 1.07822 0.350334i 1.52988 + 2.10569i −0.578217 + 0.420099i 0 2.38723 + 1.73443i 0.407162i −1.80902 + 2.48990i −1.16637 + 3.58973i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.2 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.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.i 8
5.b even 2 1 625.2.e.a 8
5.c odd 4 2 625.2.d.o 16
25.d even 5 1 25.2.e.a 8
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.b.c 8
25.d even 5 1 625.2.e.a 8
25.e even 10 1 25.2.e.a 8
25.e even 10 1 125.2.e.b 8
25.e even 10 1 625.2.b.c 8
25.e even 10 1 inner 625.2.e.i 8
25.f odd 20 4 125.2.d.b 16
25.f odd 20 2 625.2.a.f 8
25.f odd 20 2 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.j odd 10 1 225.2.m.a 8
75.l even 20 2 5625.2.a.x 8
100.h odd 10 1 400.2.y.c 8
100.j odd 10 1 400.2.y.c 8
100.l even 20 2 10000.2.a.bj 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 25.d even 5 1
25.2.e.a 8 25.e even 10 1
125.2.d.b 16 25.f odd 20 4
125.2.e.b 8 25.d even 5 1
125.2.e.b 8 25.e even 10 1
225.2.m.a 8 75.h odd 10 1
225.2.m.a 8 75.j odd 10 1
400.2.y.c 8 100.h odd 10 1
400.2.y.c 8 100.j odd 10 1
625.2.a.f 8 25.f odd 20 2
625.2.b.c 8 25.d even 5 1
625.2.b.c 8 25.e even 10 1
625.2.d.o 16 5.c odd 4 2
625.2.d.o 16 25.f odd 20 2
625.2.e.a 8 5.b even 2 1
625.2.e.a 8 25.d even 5 1
625.2.e.i 8 1.a even 1 1 trivial
625.2.e.i 8 25.e even 10 1 inner
5625.2.a.x 8 75.l even 20 2
10000.2.a.bj 8 100.l even 20 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{8} - 4T_{2}^{6} + 10T_{2}^{5} + 11T_{2}^{4} - 40T_{2}^{3} + 21T_{2}^{2} + 5T_{2} + 1$$ T2^8 - 4*T2^6 + 10*T2^5 + 11*T2^4 - 40*T2^3 + 21*T2^2 + 5*T2 + 1 $$T_{3}^{8} - 5T_{3}^{7} + 9T_{3}^{6} + 5T_{3}^{5} - 39T_{3}^{4} + 20T_{3}^{3} + 64T_{3}^{2} - 40T_{3} + 16$$ T3^8 - 5*T3^7 + 9*T3^6 + 5*T3^5 - 39*T3^4 + 20*T3^3 + 64*T3^2 - 40*T3 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 4 T^{6} + \cdots + 1$$
$3$ $$T^{8} - 5 T^{7} + \cdots + 16$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 21 T^{6} + \cdots + 16$$
$11$ $$(T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2}$$
$13$ $$T^{8} - 5 T^{7} + \cdots + 1$$
$17$ $$T^{8} + 15 T^{7} + \cdots + 1936$$
$19$ $$T^{8} - 10 T^{7} + \cdots + 400$$
$23$ $$T^{8} + 15 T^{7} + \cdots + 256$$
$29$ $$T^{8} - 15 T^{7} + \cdots + 483025$$
$31$ $$T^{8} - T^{7} + \cdots + 1936$$
$37$ $$T^{8} - 5 T^{7} + \cdots + 116281$$
$41$ $$T^{8} + 9 T^{7} + \cdots + 13456$$
$43$ $$T^{8} + 129 T^{6} + \cdots + 246016$$
$47$ $$T^{8} - 15 T^{7} + \cdots + 65536$$
$53$ $$T^{8} + 35 T^{7} + \cdots + 8755681$$
$59$ $$T^{8} - 15 T^{7} + \cdots + 4080400$$
$61$ $$T^{8} - 6 T^{7} + \cdots + 116281$$
$67$ $$T^{8} - 4 T^{6} + \cdots + 246016$$
$71$ $$T^{8} + 29 T^{7} + \cdots + 24245776$$
$73$ $$T^{8} + 10 T^{7} + \cdots + 1$$
$79$ $$T^{8} + 10 T^{7} + \cdots + 33408400$$
$83$ $$T^{8} + 15 T^{7} + \cdots + 99856$$
$89$ $$T^{8} - 40 T^{7} + \cdots + 1392400$$
$97$ $$T^{8} - 10 T^{7} + \cdots + 301334881$$