Exact Presentation of the Singular Local Hecke Order

Using the exact lattice identification

O_sing = { (x_7,x_1,x_5) in Z_3^3 : x_1 ≡ x_5 mod 3 },

choose the basis

e7    = (1,0,0),
epair = (0,1,1),
n     = (0,3,0).

Then multiplication is coordinatewise in Z_3^3, so the basis satisfies

e7^2 = e7,
epair^2 = epair,
e7*epair = 0,
e7*n = 0,
epair*n = n,
n^2 = 3n.

Hence

O_sing ≅ Z_3 e7 ⊕ Z_3 epair ⊕ Z_3 n
       ≅ Z_3 × Z_3[eta]/(eta^2 - 3 eta),

with eta = n on the pair factor.

Reducing modulo 3 gives

O_sing / 3 O_sing ≅ F_3 × F_3[eps]/(eps^2),

since n^2 = 3n collapses to eps^2 = 0.

Left regular representation

In the basis [e7, epair, n], left multiplication by the basis elements is

L(e7)    = [[1, 0, 0], [0, 0, 0], [0, 0, 0]]
L(epair) = [[0, 0, 0], [0, 1, 0], [0, 0, 1]]
L(n)     = [[0, 0, 0], [0, 0, 0], [0, 1, 3]]

Modulo 3, the pair block becomes a Jordan block:

epair ↦ identity on the pair block,
n     ↦ nilpotent with n^2 = 0.

Heegner operators in the exact basis

operator	residue coords in [e7, epair, n]	matrix mod 3
T_1	[1, 1, 0]	[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
T_3	[1, 0, 0]	[[1, 0, 0], [0, 0, 0], [0, 0, 0]]
T_7	[2, 2, 2]	[[2, 0, 0], [0, 2, 0], [0, 2, 2]]
T_11	[0, 0, 2]	[[0, 0, 0], [0, 0, 0], [0, 2, 0]]
T_19	[2, 0, 0]	[[2, 0, 0], [0, 0, 0], [0, 0, 0]]
T_43	[2, 1, 1]	[[2, 0, 0], [0, 1, 0], [0, 1, 1]]
T_67	[2, 1, 1]	[[2, 0, 0], [0, 1, 0], [0, 1, 1]]
T_163	[1, 2, 0]	[[1, 0, 0], [0, 2, 0], [0, 0, 2]]

If one writes

I = e7 + epair,   E = e7,   N = -n,

then the residue formulas become exactly

T_3 = E,
T_19 = -E,
T_11 = N,
T_7 = -I + N,
T_43 = T_67 = I + E - N,
T_163 = -I - E.

So the singular Hecke node is exactly one split idempotent branch e7 together with one rank-2 pair factor whose deformation parameter n satisfies n^2 = 3n.