# Properties

 Label 9025.2.a.bz Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.41289040.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5$$ x^6 - 3*x^5 - 8*x^4 + 21*x^3 + 18*x^2 - 25*x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 475) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - \nu^{4} - 10\nu^{3} + 6\nu^{2} + 20\nu - 5 ) / 5$$ (v^5 - v^4 - 10*v^3 + 6*v^2 + 20*v - 5) / 5 $$\beta_{4}$$ $$=$$ $$( 2\nu^{5} - 2\nu^{4} - 15\nu^{3} + 7\nu^{2} + 15\nu - 5 ) / 5$$ (2*v^5 - 2*v^4 - 15*v^3 + 7*v^2 + 15*v - 5) / 5 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} + 3\nu^{4} - 20\nu^{3} - 28\nu^{2} + 25\nu + 20 ) / 5$$ (2*v^5 + 3*v^4 - 20*v^3 - 28*v^2 + 25*v + 20) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 2\beta_{3} + \beta_{2} + 6\beta _1 + 3$$ b4 - 2*b3 + b2 + 6*b1 + 3 $$\nu^{4}$$ $$=$$ $$\beta_{5} - 2\beta_{3} + 8\beta_{2} + 11\beta _1 + 26$$ b5 - 2*b3 + 8*b2 + 11*b1 + 26 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 10\beta_{4} - 17\beta_{3} + 12\beta_{2} + 45\beta _1 + 37$$ b5 + 10*b4 - 17*b3 + 12*b2 + 45*b1 + 37

## 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
 2.85674 −0.181608 1.02983 −2.24415 2.95188 −1.41269
−1.48119 −0.181608 0.193937 0 0.268996 −1.30422 2.67513 −2.96702 0
1.2 −1.48119 2.85674 0.193937 0 −4.23138 3.78541 2.67513 5.16096 0
1.3 0.311108 −2.24415 −1.90321 0 −0.698174 3.96928 −1.21432 2.03623 0
1.4 0.311108 1.02983 −1.90321 0 0.320390 −3.28038 −1.21432 −1.93944 0
1.5 2.17009 −1.41269 2.70928 0 −3.06566 −1.76171 1.53919 −1.00431 0
1.6 2.17009 2.95188 2.70928 0 6.40583 0.591620 1.53919 5.71358 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 9025.2.a.bz 6
5.b even 2 1 9025.2.a.br 6
19.b odd 2 1 9025.2.a.bs 6
19.c even 3 2 475.2.e.f 12
95.d odd 2 1 9025.2.a.by 6
95.i even 6 2 475.2.e.h yes 12
95.m odd 12 4 475.2.j.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.e.f 12 19.c even 3 2
475.2.e.h yes 12 95.i even 6 2
475.2.j.d 24 95.m odd 12 4
9025.2.a.br 6 5.b even 2 1
9025.2.a.bs 6 19.b odd 2 1
9025.2.a.by 6 95.d odd 2 1
9025.2.a.bz 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1$$ T2^3 - T2^2 - 3*T2 + 1 $$T_{3}^{6} - 3T_{3}^{5} - 8T_{3}^{4} + 21T_{3}^{3} + 18T_{3}^{2} - 25T_{3} - 5$$ T3^6 - 3*T3^5 - 8*T3^4 + 21*T3^3 + 18*T3^2 - 25*T3 - 5 $$T_{7}^{6} - 2T_{7}^{5} - 21T_{7}^{4} + 20T_{7}^{3} + 123T_{7}^{2} + 38T_{7} - 67$$ T7^6 - 2*T7^5 - 21*T7^4 + 20*T7^3 + 123*T7^2 + 38*T7 - 67 $$T_{11}^{6} + T_{11}^{5} - 46T_{11}^{4} + 9T_{11}^{3} + 522T_{11}^{2} - 871T_{11} + 247$$ T11^6 + T11^5 - 46*T11^4 + 9*T11^3 + 522*T11^2 - 871*T11 + 247 $$T_{29}^{6} - 3T_{29}^{5} - 108T_{29}^{4} + 97T_{29}^{3} + 3140T_{29}^{2} + 2693T_{29} - 12781$$ T29^6 - 3*T29^5 - 108*T29^4 + 97*T29^3 + 3140*T29^2 + 2693*T29 - 12781

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{3} - T^{2} - 3 T + 1)^{2}$$
$3$ $$T^{6} - 3 T^{5} - 8 T^{4} + 21 T^{3} + \cdots - 5$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 2 T^{5} - 21 T^{4} + 20 T^{3} + \cdots - 67$$
$11$ $$T^{6} + T^{5} - 46 T^{4} + 9 T^{3} + \cdots + 247$$
$13$ $$T^{6} - 5 T^{5} - 36 T^{4} + 139 T^{3} + \cdots - 317$$
$17$ $$T^{6} + 3 T^{5} - 24 T^{4} - 53 T^{3} + \cdots - 1$$
$19$ $$T^{6}$$
$23$ $$T^{6} + 6 T^{5} - 57 T^{4} + \cdots + 1073$$
$29$ $$T^{6} - 3 T^{5} - 108 T^{4} + \cdots - 12781$$
$31$ $$T^{6} + 3 T^{5} - 64 T^{4} - 189 T^{3} + \cdots - 631$$
$37$ $$T^{6} + 6 T^{5} - 32 T^{4} - 168 T^{3} + \cdots - 64$$
$41$ $$T^{6} - 11 T^{5} - 24 T^{4} + \cdots + 1319$$
$43$ $$T^{6} - 13 T^{5} + 38 T^{4} + \cdots - 911$$
$47$ $$T^{6} + 6 T^{5} - 131 T^{4} + \cdots + 23179$$
$53$ $$T^{6} - 18 T^{5} - 69 T^{4} + \cdots + 341861$$
$59$ $$T^{6} - 4 T^{5} - 237 T^{4} + \cdots + 11593$$
$61$ $$T^{6} - 25 T^{5} + 98 T^{4} + \cdots - 17491$$
$67$ $$T^{6} - 6 T^{5} - 164 T^{4} + \cdots - 12080$$
$71$ $$T^{6} - 18 T^{5} - 139 T^{4} + \cdots - 11657$$
$73$ $$T^{6} - T^{5} - 166 T^{4} + \cdots - 68555$$
$79$ $$T^{6} - 3 T^{5} - 322 T^{4} + \cdots + 326351$$
$83$ $$T^{6} + 23 T^{5} - 74 T^{4} + \cdots + 378053$$
$89$ $$T^{6} - 12 T^{5} - 307 T^{4} + \cdots - 349609$$
$97$ $$T^{6} - 3 T^{5} - 322 T^{4} + \cdots - 159631$$