Exact Lattice Identification of the Singular Hecke Order

Order the three rational 3-adic characters as (V7, V1, V5). In these coordinates, the Hecke lattice basis {1, T_3, T_11} has column matrix

['1 + O(3^120)', '1 + 3 + 3^3 + 2*3^4 + 3^5 + 2*3^7 + 2*3^8 + 2*3^11 + 3^12 + 2*3^14 + 2*3^15 + 3^16 + 3^19 + 2*3^22 + 2*3^23 + 2*3^24 + 3^25 + 2*3^26 + 2*3^27 + 3^28 + 3^29 + 3^30 + 2*3^31 + 2*3^32 + 2*3^33 + 2*3^34 + 3^35 + 2*3^36 + 2*3^37 + 2*3^38 + 3^39 + 3^41 + 2*3^42 + 3^43 + 2*3^44 + 2*3^46 + 2*3^49 + 3^50 + 2*3^52 + 3^53 + 3^54 + 3^56 + 2*3^57 + 2*3^58 + 3^59 + 3^60 + 3^62 + 3^63 + 3^64 + 3^66 + 3^68 + 2*3^70 + 3^71 + 3^74 + 2*3^77 + 2*3^80 + 2*3^81 + 2*3^82 + 3^83 + 3^84 + 2*3^86 + 2*3^87 + 3^89 + 2*3^92 + 3^95 + 2*3^96 + 3^97 + 3^99 + 3^101 + 2*3^102 + 2*3^103 + 3^104 + 3^105 + 3^107 + 3^108 + 2*3^109 + 3^110 + 2*3^111 + 3^112 + 2*3^114 + 2*3^115 + 3^116 + O(3^118)', '3 + 3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 3^7 + 3^8 + 3^10 + 3^12 + 3^13 + 3^16 + 3^18 + 3^19 + 2*3^20 + 2*3^22 + 2*3^23 + 2*3^25 + 3^28 + 3^30 + 2*3^31 + 2*3^33 + 2*3^34 + 3^35 + 3^36 + 3^38 + 3^39 + 3^41 + 3^43 + 2*3^49 + 3^54 + 2*3^55 + 3^56 + 3^57 + 3^58 + 3^61 + 2*3^63 + 2*3^65 + 3^66 + 2*3^68 + 3^69 + 2*3^70 + 3^71 + 3^72 + 3^73 + 3^74 + 3^75 + 2*3^76 + 2*3^77 + 2*3^79 + 2*3^80 + 3^81 + 3^82 + 3^83 + 3^85 + 2*3^86 + 2*3^87 + 3^88 + 3^90 + 3^91 + 3^94 + 3^95 + 3^97 + 2*3^98 + 2*3^99 + 2*3^100 + 2*3^101 + 3^102 + 3^103 + 2*3^104 + 3^107 + 2*3^112 + 2*3^113 + 2*3^116 + 3^118 + O(3^119)']
['1 + O(3^120)', 'O(3^118)', '3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + 2*3^12 + 2*3^13 + 2*3^14 + 2*3^15 + 2*3^16 + 2*3^17 + 2*3^18 + 2*3^19 + 2*3^20 + 2*3^21 + 2*3^22 + 2*3^23 + 2*3^24 + 2*3^25 + 2*3^26 + 2*3^27 + 2*3^28 + 2*3^29 + 2*3^30 + 2*3^31 + 2*3^32 + 2*3^33 + 2*3^34 + 2*3^35 + 2*3^36 + 2*3^37 + 2*3^38 + 2*3^39 + 2*3^40 + 2*3^41 + 2*3^42 + 2*3^43 + 2*3^44 + 2*3^45 + 2*3^46 + 2*3^47 + 2*3^48 + 2*3^49 + 2*3^50 + 2*3^51 + 2*3^52 + 2*3^53 + 2*3^54 + 2*3^55 + 2*3^56 + 2*3^57 + 2*3^58 + 2*3^59 + 2*3^60 + 2*3^61 + 2*3^62 + 2*3^63 + 2*3^64 + 2*3^65 + 2*3^66 + 2*3^67 + 2*3^68 + 2*3^69 + 2*3^70 + 2*3^71 + 2*3^72 + 2*3^73 + 2*3^74 + 2*3^75 + 2*3^76 + 2*3^77 + 2*3^78 + 2*3^79 + 2*3^80 + 2*3^81 + 2*3^82 + 2*3^83 + 2*3^84 + 2*3^85 + 2*3^86 + 2*3^87 + 2*3^88 + 2*3^89 + 2*3^90 + 2*3^91 + 2*3^92 + 2*3^93 + 2*3^94 + 2*3^95 + 2*3^96 + 2*3^97 + 2*3^98 + 2*3^99 + 2*3^100 + 2*3^101 + 2*3^102 + 2*3^103 + 2*3^104 + 2*3^105 + 2*3^106 + 2*3^107 + 2*3^108 + 2*3^109 + 2*3^110 + 2*3^111 + 2*3^112 + 2*3^113 + 2*3^114 + 2*3^115 + 2*3^116 + 2*3^117 + 2*3^118 + O(3^119)']
['1 + O(3^120)', '2*3^2 + 2*3^3 + 2*3^5 + 3^9 + 3^10 + 3^13 + 2*3^15 + 2*3^16 + 2*3^17 + 3^18 + 2*3^19 + 2*3^20 + 3^21 + 2*3^23 + 3^24 + 3^26 + 2*3^28 + 2*3^29 + 2*3^33 + 3^35 + 3^36 + 3^37 + 3^38 + 2*3^39 + 3^40 + 3^42 + 2*3^43 + 2*3^45 + 3^46 + 2*3^49 + 3^50 + 3^51 + 2*3^52 + 2*3^53 + 2*3^54 + 3^55 + 2*3^58 + 2*3^59 + 2*3^61 + 3^62 + 3^63 + 2*3^64 + 3^66 + 3^67 + 3^69 + 2*3^70 + 2*3^72 + 2*3^73 + 3^77 + 2*3^78 + 2*3^80 + 2*3^81 + 2*3^82 + 2*3^83 + 3^85 + 2*3^87 + 3^88 + 3^89 + 2*3^90 + 2*3^93 + 3^94 + 2*3^95 + 2*3^97 + 2*3^99 + 2*3^100 + 3^101 + 3^102 + 3^103 + 2*3^104 + 3^106 + 3^107 + 3^109 + 3^110 + 3^111 + 3^112 + 3^113 + 3^115 + 3^116 + 3^117 + O(3^118)', '2*3 + 3^3 + 2*3^4 + 2*3^7 + 3^9 + 3^10 + 2*3^12 + 2*3^13 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + 2*3^20 + 3^21 + 3^24 + 2*3^25 + 2*3^26 + 3^27 + 2*3^28 + 3^29 + 2*3^30 + 2*3^31 + 3^32 + 3^34 + 3^35 + 3^36 + 2*3^37 + 3^38 + 2*3^39 + 2*3^42 + 3^43 + 2*3^44 + 2*3^45 + 3^49 + 2*3^50 + 2*3^51 + 3^52 + 3^54 + 2*3^55 + 2*3^56 + 3^57 + 3^58 + 3^61 + 3^63 + 3^64 + 3^65 + 3^67 + 3^68 + 3^69 + 2*3^70 + 3^71 + 2*3^74 + 3^75 + 3^76 + 2*3^77 + 2*3^78 + 2*3^79 + 3^80 + 3^84 + 3^90 + 2*3^91 + 2*3^92 + 3^94 + 3^95 + 3^96 + 3^98 + 3^99 + 2*3^100 + 3^101 + 3^102 + 3^103 + 3^106 + 2*3^107 + 2*3^108 + 3^109 + 2*3^110 + 3^111 + 3^112 + 3^114 + 3^117 + 3^118 + O(3^119)']

Its determinant has 3-adic valuation 1.

Compare with the standard congruence-order basis

u1 = (1,0,0),   u2 = (0,1,1),   u3 = (0,3,0),

whose determinant also has valuation 1.

The change-of-basis matrix from {u1,u2,u3} to {1,T_3,T_11} has determinant of valuation 0, so it is a GL_3(Z_3)-matrix.

Therefore the two lattices are equal:

Span_{Z_3}{1, T_3, T_11}
 = { (x_7,x_1,x_5) in Z_3^3 : x_1 ≡ x_5 mod 3 }.

Heegner congruence checks

operator	pair congruent mod 3?	v3(V1-V5)
`T_1`	`True`	`None`
`T_3`	`True`	`2`
`T_11`	`True`	`1`
`T_43`	`True`	`1`
`T_67`	`True`	`1`
`T_163`	`True`	`116`

So the congruence condition is not just a pattern on a few generators; it is the exact defining lattice condition on the singular Hecke order.