# Properties

 Label 5225.2.a.q Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.a (trivial)

## 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: yes Analytic conductor: $$41.7218350561$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2$$ x^9 - 2*x^8 - 11*x^7 + 11*x^6 + 34*x^5 - 20*x^4 - 36*x^3 + 13*x^2 + 10*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{5} q^{3} + ( - \beta_{8} - \beta_1 + 1) q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + 1) q^{6} + (\beta_{3} + 1) q^{7} + (\beta_{8} + \beta_{6} + \beta_1 - 2) q^{8} + (\beta_{7} + \beta_{4} + 2) q^{9}+O(q^{10})$$ q + b1 * q^2 - b5 * q^3 + (-b8 - b1 + 1) * q^4 + (b7 - b6 + b3 + 1) * q^6 + (b3 + 1) * q^7 + (b8 + b6 + b1 - 2) * q^8 + (b7 + b4 + 2) * q^9 $$q + \beta_1 q^{2} - \beta_{5} q^{3} + ( - \beta_{8} - \beta_1 + 1) q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + 1) q^{6} + (\beta_{3} + 1) q^{7} + (\beta_{8} + \beta_{6} + \beta_1 - 2) q^{8} + (\beta_{7} + \beta_{4} + 2) q^{9} - q^{11} + (\beta_{8} + \beta_{6} - \beta_{5} + \cdots - 1) q^{12}+ \cdots + ( - \beta_{7} - \beta_{4} - 2) q^{99}+O(q^{100})$$ q + b1 * q^2 - b5 * q^3 + (-b8 - b1 + 1) * q^4 + (b7 - b6 + b3 + 1) * q^6 + (b3 + 1) * q^7 + (b8 + b6 + b1 - 2) * q^8 + (b7 + b4 + 2) * q^9 - q^11 + (b8 + b6 - b5 - b3 - b2 + b1 - 1) * q^12 + (-b6 + b3 + b1 + 1) * q^13 + (b6 - b5 - b4 - b2 + b1 + 1) * q^14 + (-b6 + b5 - b4 + b3 - 2*b1 + 1) * q^16 + (-b8 + b7 - b5 + b2 + b1) * q^17 + (-b6 - 2*b5 - b4 - b3 - b2 + 2*b1) * q^18 + q^19 + (b8 - b5 - b3 - b2 + 2*b1 + 1) * q^21 - b1 * q^22 + (2*b8 + b7 - b6 + 2*b3 - b2 + b1) * q^23 + (-2*b8 - b7 - b6 + 2*b5 + b4 + 2*b2 - 3*b1 + 1) * q^24 + (b6 - 2*b5 - b3 - b2 + 3) * q^26 + (-2*b8 - b7 - 2*b5 - b3 + b1 - 1) * q^27 + (-2*b8 + b7 + 2*b5 + b4 + b3 + 2*b2 + 2) * q^28 + (b8 - b7 + 2*b6 + b5 - b3 - 1) * q^29 + (b8 - b7 + 2*b6 + b3 - b2 - b1 - 1) * q^31 + (b8 - b7 + b6 - b5 + b4 - b3 + b1 - 4) * q^32 + b5 * q^33 + (-b8 + b7 - b5 - b4 + b3 - b2 + 4) * q^34 + (-b8 - 2*b6 + b5 + 2*b4 + 2*b3 + 3*b2 - 2*b1 + 1) * q^36 + (b8 + b7 + b6 - 2*b5 - 2*b2 - b1) * q^37 + b1 * q^38 + (2*b8 + 2*b7 - b6 - b5 - b4 + b3 - 2*b2 + 4*b1 + 2) * q^39 + (b7 + b5 - b4 + b3 + 2*b2 - 2) * q^41 + (-2*b8 + b7 - 3*b6 + b5 + 2*b4 + b3 + 2*b2 - 2*b1 + 5) * q^42 + (3*b7 - b6 - b5 - b4 + b3 + b1 + 2) * q^43 + (b8 + b1 - 1) * q^44 + (-4*b5 - b3 - b2 - 3*b1 + 1) * q^46 + (-b7 + 2*b6 - b4 + b3 - 2*b2 - 2*b1 - 4) * q^47 + (2*b8 - 2*b7 + b6 + b5 - 2*b4 - 2*b3 - b2 + 4*b1 - 6) * q^48 + (-b8 - b7 + 2*b5 + 2*b3 + 1) * q^49 + (2*b7 - b6 - b5 + b4 - b3 + 2*b1 + 6) * q^51 + (-b8 + 2*b7 - b6 + 2*b5 + b4 + b3 + 2*b2 + b1) * q^52 + (b8 - b7 - b5 + b4 + b3 - b1) * q^53 + (-b8 + 2*b7 - b6 + 2*b5 + b4 + 2*b3 + b2 + 7) * q^54 + (-2*b7 + 2*b6 - b5 - 2*b4 - 3*b3 - 2*b2 + 2*b1 - 1) * q^56 - b5 * q^57 + (-2*b8 - b7 - b6 + 4*b5 - b4 + b3 + b2 - 2*b1) * q^58 + (b6 + 2*b4 - b3 + b1 + 7) * q^59 + (-b8 + 2*b7 - b6 - 3*b5 + b4 + 2*b2 + b1 - 1) * q^61 + (-b8 + 2*b5 - 2*b4 + 2*b3 - b1) * q^62 + (-3*b8 + 2*b7 - 3*b6 - b5 + b4 + b3 + 2*b2 - 3*b1 + 2) * q^63 + (-2*b8 + b7 - 2*b6 + b4 - b3 - 2*b1 + 3) * q^64 + (-b7 + b6 - b3 - 1) * q^66 + (-3*b8 - b7 + b5 - 2*b3 + 2*b2) * q^67 + (2*b8 - b7 + 2*b6 + b5 + b4 + 2*b3 - b2 + 3*b1 + 1) * q^68 + (-2*b8 - b7 - 3*b6 - b5 - b4 + 5*b1 + 1) * q^69 + (-b7 + b6 + b5 + b4 - b3 + 3*b2 + 2) * q^71 + (4*b8 - b7 + 4*b6 - 2*b5 - 3*b4 - 3*b3 - 5*b2 - 4) * q^72 + (b7 + 2*b6 - b5 + 2*b4 + b3 - b2 + 2*b1 + 5) * q^73 + (2*b7 - 3*b6 + b4 + 3*b3 + 2*b2 - 2) * q^74 + (-b8 - b1 + 1) * q^76 + (-b3 - 1) * q^77 + (-3*b8 + b7 - b6 - 3*b5 + 3*b4 + b3 + 2*b2 - 4*b1 + 8) * q^78 + (b8 - b7 - b6 + b4 + b3 + b2 - b1 - 2) * q^79 + (3*b8 + 3*b7 + 2*b5 - b4 + 3*b3 - 2*b2 - b1 + 5) * q^81 + (-b7 + 3*b6 - b5 - 2*b4 - 2*b2 - 2*b1 - 2) * q^82 + (4*b8 - b6 + 2*b5 - b4 + b2 + 3*b1 - 7) * q^83 + (3*b8 - b7 + 2*b6 - 5*b5 - 2*b4 - 4*b3 - 3*b2 + 5*b1 - 8) * q^84 + (b7 + b6 - 4*b5 + b4 + b3 + b1) * q^86 + (-2*b8 - 3*b7 + b6 + 2*b5 + b4 + 3*b2 - 8*b1 - 5) * q^87 + (-b8 - b6 - b1 + 2) * q^88 + (-4*b7 + 3*b6 + b5 + b4 - b3 - 2*b2 - 2*b1) * q^89 + (b8 - b6 + 3*b3 - b2 - b1 + 7) * q^91 + (-b8 + 2*b7 - 3*b6 + b5 + 2*b4 + 4*b2 + 2*b1 - 6) * q^92 + (-b8 - 3*b7 + b6 + 2*b5 + 2*b4 - 2*b3 + b2 - 4*b1) * q^93 + (2*b6 + 3*b5 + 3*b3 + 2*b2 - 2*b1 - 3) * q^94 + (-b8 + b7 + b6 + 3*b5 + 2*b4 + 2*b3 + b2 - 6*b1 + 6) * q^96 + (b8 - b7 - 2*b5 + 2*b3 - b2 - 5*b1) * q^97 + (-2*b7 + 5*b6 - b5 - 2*b4 - 2*b3 - 2*b2 + 2*b1 + 2) * q^98 + (-b7 - b4 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} + 6 q^{6} + 9 q^{7} - 15 q^{8} + 16 q^{9}+O(q^{10})$$ 9 * q - 3 * q^2 - 3 * q^3 + 9 * q^4 + 6 * q^6 + 9 * q^7 - 15 * q^8 + 16 * q^9 $$9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} + 6 q^{6} + 9 q^{7} - 15 q^{8} + 16 q^{9} - 9 q^{11} - 13 q^{12} + 3 q^{13} + 4 q^{14} + 17 q^{16} - 5 q^{17} - 17 q^{18} + 9 q^{19} - q^{21} + 3 q^{22} - 4 q^{23} + 21 q^{24} + 20 q^{26} - 24 q^{27} + 24 q^{28} + 3 q^{29} - q^{31} - 38 q^{32} + 3 q^{33} + 28 q^{34} + 17 q^{36} - 5 q^{37} - 3 q^{38} - 5 q^{41} + 43 q^{42} + 11 q^{43} - 9 q^{44} + 2 q^{46} - 30 q^{47} - 54 q^{48} + 12 q^{49} + 40 q^{51} + 3 q^{52} + q^{53} + 65 q^{54} - 16 q^{56} - 3 q^{57} + 15 q^{58} + 59 q^{59} - 21 q^{61} + 10 q^{62} + 12 q^{63} + 19 q^{64} - 6 q^{66} + 2 q^{67} + 9 q^{68} - 22 q^{69} + 34 q^{71} - 32 q^{72} + 34 q^{73} - 21 q^{74} + 9 q^{76} - 9 q^{77} + 65 q^{78} - 13 q^{79} + 57 q^{81} - 10 q^{82} - 51 q^{83} - 95 q^{84} - 14 q^{86} - 8 q^{87} + 15 q^{88} + 8 q^{89} + 62 q^{91} - 57 q^{92} + 18 q^{93} + 2 q^{94} + 81 q^{96} + 8 q^{97} + 20 q^{98} - 16 q^{99}+O(q^{100})$$ 9 * q - 3 * q^2 - 3 * q^3 + 9 * q^4 + 6 * q^6 + 9 * q^7 - 15 * q^8 + 16 * q^9 - 9 * q^11 - 13 * q^12 + 3 * q^13 + 4 * q^14 + 17 * q^16 - 5 * q^17 - 17 * q^18 + 9 * q^19 - q^21 + 3 * q^22 - 4 * q^23 + 21 * q^24 + 20 * q^26 - 24 * q^27 + 24 * q^28 + 3 * q^29 - q^31 - 38 * q^32 + 3 * q^33 + 28 * q^34 + 17 * q^36 - 5 * q^37 - 3 * q^38 - 5 * q^41 + 43 * q^42 + 11 * q^43 - 9 * q^44 + 2 * q^46 - 30 * q^47 - 54 * q^48 + 12 * q^49 + 40 * q^51 + 3 * q^52 + q^53 + 65 * q^54 - 16 * q^56 - 3 * q^57 + 15 * q^58 + 59 * q^59 - 21 * q^61 + 10 * q^62 + 12 * q^63 + 19 * q^64 - 6 * q^66 + 2 * q^67 + 9 * q^68 - 22 * q^69 + 34 * q^71 - 32 * q^72 + 34 * q^73 - 21 * q^74 + 9 * q^76 - 9 * q^77 + 65 * q^78 - 13 * q^79 + 57 * q^81 - 10 * q^82 - 51 * q^83 - 95 * q^84 - 14 * q^86 - 8 * q^87 + 15 * q^88 + 8 * q^89 + 62 * q^91 - 57 * q^92 + 18 * q^93 + 2 * q^94 + 81 * q^96 + 8 * q^97 + 20 * q^98 - 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -4\nu^{8} + 6\nu^{7} + 47\nu^{6} - 35\nu^{5} - 110\nu^{4} + 112\nu^{3} + 26\nu^{2} - 126\nu + 13 ) / 29$$ (-4*v^8 + 6*v^7 + 47*v^6 - 35*v^5 - 110*v^4 + 112*v^3 + 26*v^2 - 126*v + 13) / 29 $$\beta_{2}$$ $$=$$ $$( 8\nu^{8} - 12\nu^{7} - 94\nu^{6} + 41\nu^{5} + 307\nu^{4} - 50\nu^{3} - 400\nu^{2} + 78\nu + 177 ) / 29$$ (8*v^8 - 12*v^7 - 94*v^6 + 41*v^5 + 307*v^4 - 50*v^3 - 400*v^2 + 78*v + 177) / 29 $$\beta_{3}$$ $$=$$ $$( -5\nu^{8} + 22\nu^{7} + 37\nu^{6} - 196\nu^{5} - 94\nu^{4} + 488\nu^{3} + 76\nu^{2} - 288\nu + 9 ) / 29$$ (-5*v^8 + 22*v^7 + 37*v^6 - 196*v^5 - 94*v^4 + 488*v^3 + 76*v^2 - 288*v + 9) / 29 $$\beta_{4}$$ $$=$$ $$( \nu^{8} - 16\nu^{7} + 39\nu^{6} + 103\nu^{5} - 277\nu^{4} - 202\nu^{3} + 472\nu^{2} + 162\nu - 170 ) / 29$$ (v^8 - 16*v^7 + 39*v^6 + 103*v^5 - 277*v^4 - 202*v^3 + 472*v^2 + 162*v - 170) / 29 $$\beta_{5}$$ $$=$$ $$( 14\nu^{8} - 21\nu^{7} - 150\nu^{6} + 50\nu^{5} + 356\nu^{4} + 43\nu^{3} - 178\nu^{2} - 110\nu - 2 ) / 29$$ (14*v^8 - 21*v^7 - 150*v^6 + 50*v^5 + 356*v^4 + 43*v^3 - 178*v^2 - 110*v - 2) / 29 $$\beta_{6}$$ $$=$$ $$( -20\nu^{8} + 30\nu^{7} + 235\nu^{6} - 117\nu^{5} - 695\nu^{4} + 154\nu^{3} + 565\nu^{2} - 108\nu + 7 ) / 29$$ (-20*v^8 + 30*v^7 + 235*v^6 - 117*v^5 - 695*v^4 + 154*v^3 + 565*v^2 - 108*v + 7) / 29 $$\beta_{7}$$ $$=$$ $$( 21\nu^{8} - 46\nu^{7} - 196\nu^{6} + 220\nu^{5} + 418\nu^{4} - 327\nu^{3} - 180\nu^{2} + 96\nu - 3 ) / 29$$ (21*v^8 - 46*v^7 - 196*v^6 + 220*v^5 + 418*v^4 - 327*v^3 - 180*v^2 + 96*v - 3) / 29 $$\beta_{8}$$ $$=$$ $$( 16\nu^{8} - 53\nu^{7} - 130\nu^{6} + 372\nu^{5} + 324\nu^{4} - 738\nu^{3} - 249\nu^{2} + 388\nu + 35 ) / 29$$ (16*v^8 - 53*v^7 - 130*v^6 + 372*v^5 + 324*v^4 - 738*v^3 - 249*v^2 + 388*v + 35) / 29
 $$\nu$$ $$=$$ $$( -2\beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4$$ (-2*b7 + 2*b5 + b4 - b3 + b2) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta _1 + 7 ) / 2$$ (-b8 - b7 - b6 + b5 + b4 - 2*b3 + 2*b1 + 7) / 2 $$\nu^{3}$$ $$=$$ $$( -3\beta_{8} - 7\beta_{7} - \beta_{6} + 9\beta_{5} + 4\beta_{4} - 9\beta_{3} + 3\beta_{2} + 6\beta _1 + 9 ) / 2$$ (-3*b8 - 7*b7 - b6 + 9*b5 + 4*b4 - 9*b3 + 3*b2 + 6*b1 + 9) / 2 $$\nu^{4}$$ $$=$$ $$( -15\beta_{8} - 17\beta_{7} - 9\beta_{6} + 21\beta_{5} + 14\beta_{4} - 33\beta_{3} + 7\beta_{2} + 28\beta _1 + 57 ) / 2$$ (-15*b8 - 17*b7 - 9*b6 + 21*b5 + 14*b4 - 33*b3 + 7*b2 + 28*b1 + 57) / 2 $$\nu^{5}$$ $$=$$ $$( - 51 \beta_{8} - 75 \beta_{7} - 21 \beta_{6} + 99 \beta_{5} + 51 \beta_{4} - 126 \beta_{3} + 34 \beta_{2} + \cdots + 155 ) / 2$$ (-51*b8 - 75*b7 - 21*b6 + 99*b5 + 51*b4 - 126*b3 + 34*b2 + 92*b1 + 155) / 2 $$\nu^{6}$$ $$=$$ $$( - 201 \beta_{8} - 243 \beta_{7} - 99 \beta_{6} + 315 \beta_{5} + 188 \beta_{4} - 457 \beta_{3} + \cdots + 655 ) / 2$$ (-201*b8 - 243*b7 - 99*b6 + 315*b5 + 188*b4 - 457*b3 + 113*b2 + 362*b1 + 655) / 2 $$\nu^{7}$$ $$=$$ $$( - 717 \beta_{8} - 939 \beta_{7} - 315 \beta_{6} + 1239 \beta_{5} + 681 \beta_{4} - 1688 \beta_{3} + \cdots + 2203 ) / 2$$ (-717*b8 - 939*b7 - 315*b6 + 1239*b5 + 681*b4 - 1688*b3 + 438*b2 + 1270*b1 + 2203) / 2 $$\nu^{8}$$ $$=$$ $$( - 2669 \beta_{8} - 3311 \beta_{7} - 1239 \beta_{6} + 4343 \beta_{5} + 2502 \beta_{4} - 6141 \beta_{3} + \cdots + 8381 ) / 2$$ (-2669*b8 - 3311*b7 - 1239*b6 + 4343*b5 + 2502*b4 - 6141*b3 + 1563*b2 + 4750*b1 + 8381) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.22261 −1.58924 0.724133 −1.15934 0.180277 1.56050 3.65050 −1.98926 −0.600178
−2.66854 −3.14505 5.12110 0 8.39270 4.63172 −8.32878 6.89136 0
1.2 −2.64409 −0.185485 4.99120 0 0.490440 −1.87571 −7.90900 −2.96560 0
1.3 −1.79978 2.53737 1.23919 0 −4.56670 −0.104026 1.36928 3.43826 0
1.4 −0.749568 −3.38871 −1.43815 0 2.54007 −1.17594 2.57713 8.48335 0
1.5 −0.287410 0.930455 −1.91740 0 −0.267422 −0.300889 1.12590 −2.13425 0
1.6 0.0815632 0.583675 −1.99335 0 0.0476064 3.83288 −0.325710 −2.65932 0
1.7 1.31671 −2.24751 −0.266275 0 −2.95932 −2.74324 −2.98403 2.05132 0
1.8 1.53772 2.83656 0.364584 0 4.36184 2.01727 −2.51481 5.04608 0
1.9 2.21339 −0.921304 2.89909 0 −2.03920 4.71795 1.99002 −2.15120 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$11$$ $$1$$
$$19$$ $$-1$$

## 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 5225.2.a.q 9
5.b even 2 1 1045.2.a.j 9
15.d odd 2 1 9405.2.a.bi 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.j 9 5.b even 2 1
5225.2.a.q 9 1.a even 1 1 trivial
9405.2.a.bi 9 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{9} + 3T_{2}^{8} - 9T_{2}^{7} - 27T_{2}^{6} + 25T_{2}^{5} + 70T_{2}^{4} - 21T_{2}^{3} - 51T_{2}^{2} - 8T_{2} + 1$$ T2^9 + 3*T2^8 - 9*T2^7 - 27*T2^6 + 25*T2^5 + 70*T2^4 - 21*T2^3 - 51*T2^2 - 8*T2 + 1 $$T_{7}^{9} - 9T_{7}^{8} + 3T_{7}^{7} + 136T_{7}^{6} - 137T_{7}^{5} - 701T_{7}^{4} + 316T_{7}^{3} + 1236T_{7}^{2} + 432T_{7} + 32$$ T7^9 - 9*T7^8 + 3*T7^7 + 136*T7^6 - 137*T7^5 - 701*T7^4 + 316*T7^3 + 1236*T7^2 + 432*T7 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} + 3 T^{8} + \cdots + 1$$
$3$ $$T^{9} + 3 T^{8} + \cdots + 16$$
$5$ $$T^{9}$$
$7$ $$T^{9} - 9 T^{8} + \cdots + 32$$
$11$ $$(T + 1)^{9}$$
$13$ $$T^{9} - 3 T^{8} + \cdots - 4096$$
$17$ $$T^{9} + 5 T^{8} + \cdots + 69376$$
$19$ $$(T - 1)^{9}$$
$23$ $$T^{9} + 4 T^{8} + \cdots + 4067776$$
$29$ $$T^{9} - 3 T^{8} + \cdots + 476032$$
$31$ $$T^{9} + T^{8} + \cdots + 1701376$$
$37$ $$T^{9} + 5 T^{8} + \cdots + 7003648$$
$41$ $$T^{9} + 5 T^{8} + \cdots - 2331392$$
$43$ $$T^{9} - 11 T^{8} + \cdots + 12703712$$
$47$ $$T^{9} + 30 T^{8} + \cdots - 3050368$$
$53$ $$T^{9} - T^{8} + \cdots + 746752$$
$59$ $$T^{9} - 59 T^{8} + \cdots - 7130816$$
$61$ $$T^{9} + 21 T^{8} + \cdots + 40008976$$
$67$ $$T^{9} - 2 T^{8} + \cdots + 2973104$$
$71$ $$T^{9} - 34 T^{8} + \cdots + 21829696$$
$73$ $$T^{9} - 34 T^{8} + \cdots - 14909696$$
$79$ $$T^{9} + 13 T^{8} + \cdots + 1052672$$
$83$ $$T^{9} + 51 T^{8} + \cdots - 20900704$$
$89$ $$T^{9} - 8 T^{8} + \cdots - 245309648$$
$97$ $$T^{9} - 8 T^{8} + \cdots + 50205184$$