# Properties

 Label 2.2025.24t65.a Dimension $2$ Group $S_3\times C_{12}$ Conductor $2025$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: 24T65 Conductor: $$2025$$$$\medspace = 3^{4} \cdot 5^{2}$$ Artin number field: Galois closure of 24.0.572565594852444156646728515625.2 Galois orbit size: $4$ Smallest permutation container: 24T65 Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.135.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $$x^{12} + 34x^{7} + 27x^{6} + 16x^{5} + 17x^{4} + 6x^{3} + 23x^{2} + 38x + 3$$
Roots:
 $r_{ 1 }$ $=$ $$36 a^{11} + 8 a^{10} + 38 a^{9} + 10 a^{7} + 3 a^{6} + 3 a^{5} + 11 a^{4} + 24 a^{3} + 6 a^{2} + 38 a + 15 + \left(31 a^{11} + 40 a^{10} + 23 a^{9} + 26 a^{8} + 13 a^{7} + a^{6} + 21 a^{5} + 22 a^{4} + a^{3} + 15 a^{2} + 9 a + 36\right)\cdot 43 + \left(30 a^{11} + 7 a^{10} + 24 a^{9} + 30 a^{8} + 38 a^{7} + 20 a^{6} + 18 a^{5} + 2 a^{4} + 36 a^{3} + 37 a^{2} + 8 a + 41\right)\cdot 43^{2} + \left(4 a^{11} + 9 a^{10} + 40 a^{9} + 15 a^{8} + 21 a^{7} + 11 a^{6} + 13 a^{5} + 16 a^{4} + 26 a^{3} + 13 a^{2} + 3\right)\cdot 43^{3} + \left(10 a^{11} + 2 a^{10} + 14 a^{9} + 2 a^{8} + 23 a^{7} + 30 a^{6} + 21 a^{5} + a^{4} + 26 a^{3} + 35 a^{2} + 39 a + 15\right)\cdot 43^{4} + \left(5 a^{11} + 21 a^{10} + 18 a^{9} + 18 a^{8} + 5 a^{7} + 22 a^{6} + 39 a^{5} + 31 a^{4} + 6 a^{3} + 17 a^{2} + 39 a + 2\right)\cdot 43^{5} + \left(5 a^{11} + 27 a^{10} + 17 a^{9} + 6 a^{8} + 26 a^{7} + 6 a^{6} + 6 a^{5} + 27 a^{4} + 40 a^{3} + 19 a^{2} + 28 a + 4\right)\cdot 43^{6} +O(43^{7})$$ 36*a^11 + 8*a^10 + 38*a^9 + 10*a^7 + 3*a^6 + 3*a^5 + 11*a^4 + 24*a^3 + 6*a^2 + 38*a + 15 + (31*a^11 + 40*a^10 + 23*a^9 + 26*a^8 + 13*a^7 + a^6 + 21*a^5 + 22*a^4 + a^3 + 15*a^2 + 9*a + 36)*43 + (30*a^11 + 7*a^10 + 24*a^9 + 30*a^8 + 38*a^7 + 20*a^6 + 18*a^5 + 2*a^4 + 36*a^3 + 37*a^2 + 8*a + 41)*43^2 + (4*a^11 + 9*a^10 + 40*a^9 + 15*a^8 + 21*a^7 + 11*a^6 + 13*a^5 + 16*a^4 + 26*a^3 + 13*a^2 + 3)*43^3 + (10*a^11 + 2*a^10 + 14*a^9 + 2*a^8 + 23*a^7 + 30*a^6 + 21*a^5 + a^4 + 26*a^3 + 35*a^2 + 39*a + 15)*43^4 + (5*a^11 + 21*a^10 + 18*a^9 + 18*a^8 + 5*a^7 + 22*a^6 + 39*a^5 + 31*a^4 + 6*a^3 + 17*a^2 + 39*a + 2)*43^5 + (5*a^11 + 27*a^10 + 17*a^9 + 6*a^8 + 26*a^7 + 6*a^6 + 6*a^5 + 27*a^4 + 40*a^3 + 19*a^2 + 28*a + 4)*43^6+O(43^7) $r_{ 2 }$ $=$ $$39 a^{11} + 33 a^{9} + 5 a^{8} + 5 a^{7} + 34 a^{6} + 3 a^{5} + 5 a^{4} + 24 a^{3} + 15 a^{2} + 27 a + 24 + \left(6 a^{10} + 16 a^{9} + 24 a^{8} + 20 a^{7} + 39 a^{6} + 40 a^{5} + 2 a^{4} + 25 a^{3} + 31 a^{2} + 37 a + 19\right)\cdot 43 + \left(40 a^{11} + a^{10} + 19 a^{9} + 20 a^{8} + 7 a^{7} + 7 a^{6} + 34 a^{5} + 11 a^{4} + 20 a^{3} + 33 a^{2} + 32 a + 15\right)\cdot 43^{2} + \left(29 a^{11} + 31 a^{10} + 12 a^{9} + 21 a^{8} + 14 a^{7} + 15 a^{6} + a^{5} + 22 a^{4} + 29 a^{3} + 14 a^{2} + 32 a + 1\right)\cdot 43^{3} + \left(18 a^{11} + 18 a^{10} + 13 a^{9} + 16 a^{8} + 26 a^{7} + 26 a^{6} + 4 a^{5} + 2 a^{4} + 5 a^{3} + 38 a^{2} + 3 a + 27\right)\cdot 43^{4} + \left(7 a^{11} + 26 a^{10} + 30 a^{9} + 14 a^{8} + 23 a^{7} + 29 a^{6} + 24 a^{5} + 9 a^{4} + 23 a^{3} + 24 a^{2} + 16 a\right)\cdot 43^{5} + \left(21 a^{11} + 40 a^{10} + 13 a^{9} + 30 a^{8} + 13 a^{7} + 41 a^{6} + 16 a^{5} + 21 a^{4} + 3 a^{3} + 9 a^{2} + 9 a + 16\right)\cdot 43^{6} +O(43^{7})$$ 39*a^11 + 33*a^9 + 5*a^8 + 5*a^7 + 34*a^6 + 3*a^5 + 5*a^4 + 24*a^3 + 15*a^2 + 27*a + 24 + (6*a^10 + 16*a^9 + 24*a^8 + 20*a^7 + 39*a^6 + 40*a^5 + 2*a^4 + 25*a^3 + 31*a^2 + 37*a + 19)*43 + (40*a^11 + a^10 + 19*a^9 + 20*a^8 + 7*a^7 + 7*a^6 + 34*a^5 + 11*a^4 + 20*a^3 + 33*a^2 + 32*a + 15)*43^2 + (29*a^11 + 31*a^10 + 12*a^9 + 21*a^8 + 14*a^7 + 15*a^6 + a^5 + 22*a^4 + 29*a^3 + 14*a^2 + 32*a + 1)*43^3 + (18*a^11 + 18*a^10 + 13*a^9 + 16*a^8 + 26*a^7 + 26*a^6 + 4*a^5 + 2*a^4 + 5*a^3 + 38*a^2 + 3*a + 27)*43^4 + (7*a^11 + 26*a^10 + 30*a^9 + 14*a^8 + 23*a^7 + 29*a^6 + 24*a^5 + 9*a^4 + 23*a^3 + 24*a^2 + 16*a)*43^5 + (21*a^11 + 40*a^10 + 13*a^9 + 30*a^8 + 13*a^7 + 41*a^6 + 16*a^5 + 21*a^4 + 3*a^3 + 9*a^2 + 9*a + 16)*43^6+O(43^7) $r_{ 3 }$ $=$ $$32 a^{11} + 8 a^{10} + 14 a^{9} + 7 a^{8} + 6 a^{7} + 35 a^{6} + 37 a^{5} + 23 a^{4} + 41 a^{3} + 19 a^{2} + 16 a + 33 + \left(39 a^{11} + 36 a^{10} + 28 a^{9} + 32 a^{8} + 21 a^{7} + 8 a^{6} + 18 a^{5} + 33 a^{4} + 36 a^{3} + 34 a^{2} + 41 a + 37\right)\cdot 43 + \left(16 a^{11} + 40 a^{10} + 7 a^{9} + 23 a^{8} + 23 a^{7} + 38 a^{6} + 26 a^{5} + 4 a^{4} + 9 a^{3} + 30 a^{2} + 33 a + 10\right)\cdot 43^{2} + \left(42 a^{11} + 38 a^{10} + 19 a^{9} + 42 a^{8} + 17 a^{7} + 6 a^{6} + 6 a^{5} + a^{4} + 32 a^{3} + 28 a^{2} + 26 a + 6\right)\cdot 43^{3} + \left(14 a^{11} + 35 a^{10} + 26 a^{9} + 29 a^{8} + 15 a^{7} + 16 a^{6} + 36 a^{5} + 15 a^{4} + 15 a^{3} + 37 a^{2} + 14 a + 36\right)\cdot 43^{4} + \left(35 a^{11} + 30 a^{10} + 41 a^{9} + 33 a^{8} + 8 a^{7} + 14 a^{6} + 37 a^{5} + 33 a^{4} + 3 a^{3} + 31 a^{2} + 7 a + 17\right)\cdot 43^{5} + \left(25 a^{11} + 17 a^{10} + 37 a^{9} + 7 a^{8} + 36 a^{7} + a^{6} + 28 a^{5} + 16 a^{4} + 24 a^{3} + 28 a^{2} + 4 a + 7\right)\cdot 43^{6} +O(43^{7})$$ 32*a^11 + 8*a^10 + 14*a^9 + 7*a^8 + 6*a^7 + 35*a^6 + 37*a^5 + 23*a^4 + 41*a^3 + 19*a^2 + 16*a + 33 + (39*a^11 + 36*a^10 + 28*a^9 + 32*a^8 + 21*a^7 + 8*a^6 + 18*a^5 + 33*a^4 + 36*a^3 + 34*a^2 + 41*a + 37)*43 + (16*a^11 + 40*a^10 + 7*a^9 + 23*a^8 + 23*a^7 + 38*a^6 + 26*a^5 + 4*a^4 + 9*a^3 + 30*a^2 + 33*a + 10)*43^2 + (42*a^11 + 38*a^10 + 19*a^9 + 42*a^8 + 17*a^7 + 6*a^6 + 6*a^5 + a^4 + 32*a^3 + 28*a^2 + 26*a + 6)*43^3 + (14*a^11 + 35*a^10 + 26*a^9 + 29*a^8 + 15*a^7 + 16*a^6 + 36*a^5 + 15*a^4 + 15*a^3 + 37*a^2 + 14*a + 36)*43^4 + (35*a^11 + 30*a^10 + 41*a^9 + 33*a^8 + 8*a^7 + 14*a^6 + 37*a^5 + 33*a^4 + 3*a^3 + 31*a^2 + 7*a + 17)*43^5 + (25*a^11 + 17*a^10 + 37*a^9 + 7*a^8 + 36*a^7 + a^6 + 28*a^5 + 16*a^4 + 24*a^3 + 28*a^2 + 4*a + 7)*43^6+O(43^7) $r_{ 4 }$ $=$ $$27 a^{11} + 4 a^{10} + 38 a^{9} + 32 a^{8} + 21 a^{7} + 30 a^{6} + 14 a^{5} + 8 a^{4} + 10 a^{3} + 30 a^{2} + a + 20 + \left(2 a^{11} + 41 a^{10} + 39 a^{9} + 29 a^{8} + 31 a^{7} + 17 a^{6} + 31 a^{5} + 22 a^{4} + 28 a^{3} + 12 a^{2} + 24 a + 4\right)\cdot 43 + \left(22 a^{11} + 32 a^{10} + 25 a^{9} + 42 a^{7} + 13 a^{5} + 26 a^{4} + 13 a^{3} + 16 a^{2} + 26 a + 7\right)\cdot 43^{2} + \left(40 a^{11} + 42 a^{10} + 5 a^{9} + 15 a^{8} + 8 a^{7} + 4 a^{6} + 42 a^{5} + 36 a^{4} + 36 a^{3} + 35 a^{2} + 8 a + 37\right)\cdot 43^{3} + \left(8 a^{11} + 16 a^{10} + 5 a^{9} + 14 a^{8} + 16 a^{7} + 3 a^{6} + 31 a^{5} + 17 a^{4} + 5 a^{3} + 2 a^{2} + 20 a + 5\right)\cdot 43^{4} + \left(15 a^{11} + 39 a^{10} + 32 a^{9} + 23 a^{8} + 34 a^{7} + 20 a^{6} + 8 a^{5} + 27 a^{4} + 36 a^{3} + 21 a^{2} + 9 a + 27\right)\cdot 43^{5} + \left(12 a^{11} + 13 a^{10} + 31 a^{9} + 10 a^{8} + 33 a^{7} + 25 a^{6} + 33 a^{5} + 27 a^{4} + 7 a^{3} + 27 a^{2} + 40 a + 4\right)\cdot 43^{6} +O(43^{7})$$ 27*a^11 + 4*a^10 + 38*a^9 + 32*a^8 + 21*a^7 + 30*a^6 + 14*a^5 + 8*a^4 + 10*a^3 + 30*a^2 + a + 20 + (2*a^11 + 41*a^10 + 39*a^9 + 29*a^8 + 31*a^7 + 17*a^6 + 31*a^5 + 22*a^4 + 28*a^3 + 12*a^2 + 24*a + 4)*43 + (22*a^11 + 32*a^10 + 25*a^9 + 42*a^7 + 13*a^5 + 26*a^4 + 13*a^3 + 16*a^2 + 26*a + 7)*43^2 + (40*a^11 + 42*a^10 + 5*a^9 + 15*a^8 + 8*a^7 + 4*a^6 + 42*a^5 + 36*a^4 + 36*a^3 + 35*a^2 + 8*a + 37)*43^3 + (8*a^11 + 16*a^10 + 5*a^9 + 14*a^8 + 16*a^7 + 3*a^6 + 31*a^5 + 17*a^4 + 5*a^3 + 2*a^2 + 20*a + 5)*43^4 + (15*a^11 + 39*a^10 + 32*a^9 + 23*a^8 + 34*a^7 + 20*a^6 + 8*a^5 + 27*a^4 + 36*a^3 + 21*a^2 + 9*a + 27)*43^5 + (12*a^11 + 13*a^10 + 31*a^9 + 10*a^8 + 33*a^7 + 25*a^6 + 33*a^5 + 27*a^4 + 7*a^3 + 27*a^2 + 40*a + 4)*43^6+O(43^7) $r_{ 5 }$ $=$ $$10 a^{11} + 36 a^{10} + 5 a^{9} + 36 a^{8} + 30 a^{7} + 8 a^{6} + 2 a^{5} + 39 a^{4} + 35 a^{3} + 10 a^{2} + 31 + \left(27 a^{11} + 18 a^{10} + 38 a^{9} + 13 a^{8} + 4 a^{7} + 10 a^{6} + 9 a^{5} + 25 a^{4} + 24 a^{3} + 21 a^{2} + 42 a\right)\cdot 43 + \left(39 a^{11} + 41 a^{10} + 26 a^{9} + 3 a^{8} + 30 a^{7} + 32 a^{6} + 30 a^{5} + 21 a^{4} + 4 a^{3} + 20 a^{2} + 34 a + 11\right)\cdot 43^{2} + \left(36 a^{11} + 23 a^{10} + 38 a^{9} + 35 a^{8} + 14 a^{7} + 42 a^{6} + 12 a^{5} + 5 a^{4} + 30 a^{3} + 41 a^{2} + 24 a + 1\right)\cdot 43^{3} + \left(5 a^{11} + 7 a^{10} + 12 a^{9} + 39 a^{8} + 35 a^{7} + 36 a^{6} + 3 a^{5} + 23 a^{4} + 2 a^{3} + 9 a^{2} + 37 a + 1\right)\cdot 43^{4} + \left(24 a^{11} + 10 a^{10} + 26 a^{9} + 35 a^{8} + 42 a^{7} + 40 a^{6} + 10 a^{5} + 36 a^{4} + 42 a^{3} + 12 a^{2} + 28 a + 4\right)\cdot 43^{5} + \left(26 a^{11} + 30 a^{10} + 38 a^{9} + 5 a^{8} + 17 a^{7} + 11 a^{6} + 12 a^{5} + 14 a^{4} + 41 a^{3} + 28 a + 40\right)\cdot 43^{6} +O(43^{7})$$ 10*a^11 + 36*a^10 + 5*a^9 + 36*a^8 + 30*a^7 + 8*a^6 + 2*a^5 + 39*a^4 + 35*a^3 + 10*a^2 + 31 + (27*a^11 + 18*a^10 + 38*a^9 + 13*a^8 + 4*a^7 + 10*a^6 + 9*a^5 + 25*a^4 + 24*a^3 + 21*a^2 + 42*a)*43 + (39*a^11 + 41*a^10 + 26*a^9 + 3*a^8 + 30*a^7 + 32*a^6 + 30*a^5 + 21*a^4 + 4*a^3 + 20*a^2 + 34*a + 11)*43^2 + (36*a^11 + 23*a^10 + 38*a^9 + 35*a^8 + 14*a^7 + 42*a^6 + 12*a^5 + 5*a^4 + 30*a^3 + 41*a^2 + 24*a + 1)*43^3 + (5*a^11 + 7*a^10 + 12*a^9 + 39*a^8 + 35*a^7 + 36*a^6 + 3*a^5 + 23*a^4 + 2*a^3 + 9*a^2 + 37*a + 1)*43^4 + (24*a^11 + 10*a^10 + 26*a^9 + 35*a^8 + 42*a^7 + 40*a^6 + 10*a^5 + 36*a^4 + 42*a^3 + 12*a^2 + 28*a + 4)*43^5 + (26*a^11 + 30*a^10 + 38*a^9 + 5*a^8 + 17*a^7 + 11*a^6 + 12*a^5 + 14*a^4 + 41*a^3 + 28*a + 40)*43^6+O(43^7) $r_{ 6 }$ $=$ $$24 a^{11} + 13 a^{10} + 17 a^{9} + 36 a^{8} + 12 a^{7} + 42 a^{6} + 8 a^{5} + 22 a^{4} + 39 a^{3} + 2 a^{2} + 31 a + 13 + \left(11 a^{11} + 10 a^{10} + 11 a^{9} + 18 a^{8} + 14 a^{7} + 12 a^{6} + 31 a^{5} + 8 a^{4} + 29 a^{3} + 23 a^{2} + 5 a + 27\right)\cdot 43 + \left(15 a^{11} + 18 a^{10} + 32 a^{9} + 28 a^{8} + 21 a^{6} + 22 a^{5} + 38 a^{4} + 16 a^{3} + 21 a^{2} + 23 a + 28\right)\cdot 43^{2} + \left(2 a^{11} + 30 a^{10} + 20 a^{9} + 20 a^{8} + 7 a^{7} + 34 a^{6} + 13 a^{5} + 30 a^{4} + 9 a^{3} + 33 a^{2} + 41 a + 14\right)\cdot 43^{3} + \left(13 a^{11} + 31 a^{10} + 20 a^{9} + 2 a^{8} + 33 a^{7} + 38 a^{6} + 27 a^{5} + 38 a^{4} + 14 a^{3} + 33 a^{2} + 8 a + 24\right)\cdot 43^{4} + \left(16 a^{11} + 14 a^{10} + 11 a^{9} + 7 a^{8} + 31 a^{7} + a^{6} + 8 a^{5} + 2 a^{4} + 33 a^{3} + 41 a^{2} + 34 a + 1\right)\cdot 43^{5} + \left(a^{11} + 3 a^{10} + 9 a^{9} + 5 a^{8} + 38 a^{7} + 22 a^{6} + 28 a^{5} + 33 a^{4} + 31 a^{3} + 9 a^{2} + 41 a + 34\right)\cdot 43^{6} +O(43^{7})$$ 24*a^11 + 13*a^10 + 17*a^9 + 36*a^8 + 12*a^7 + 42*a^6 + 8*a^5 + 22*a^4 + 39*a^3 + 2*a^2 + 31*a + 13 + (11*a^11 + 10*a^10 + 11*a^9 + 18*a^8 + 14*a^7 + 12*a^6 + 31*a^5 + 8*a^4 + 29*a^3 + 23*a^2 + 5*a + 27)*43 + (15*a^11 + 18*a^10 + 32*a^9 + 28*a^8 + 21*a^6 + 22*a^5 + 38*a^4 + 16*a^3 + 21*a^2 + 23*a + 28)*43^2 + (2*a^11 + 30*a^10 + 20*a^9 + 20*a^8 + 7*a^7 + 34*a^6 + 13*a^5 + 30*a^4 + 9*a^3 + 33*a^2 + 41*a + 14)*43^3 + (13*a^11 + 31*a^10 + 20*a^9 + 2*a^8 + 33*a^7 + 38*a^6 + 27*a^5 + 38*a^4 + 14*a^3 + 33*a^2 + 8*a + 24)*43^4 + (16*a^11 + 14*a^10 + 11*a^9 + 7*a^8 + 31*a^7 + a^6 + 8*a^5 + 2*a^4 + 33*a^3 + 41*a^2 + 34*a + 1)*43^5 + (a^11 + 3*a^10 + 9*a^9 + 5*a^8 + 38*a^7 + 22*a^6 + 28*a^5 + 33*a^4 + 31*a^3 + 9*a^2 + 41*a + 34)*43^6+O(43^7) $r_{ 7 }$ $=$ $$2 a^{11} + 36 a^{10} + 4 a^{9} + 41 a^{8} + 16 a^{7} + 15 a^{6} + 14 a^{5} + 32 a^{4} + 7 a^{3} + 27 a^{2} + 6 a + 34 + \left(18 a^{11} + 27 a^{10} + 40 a^{9} + 14 a^{8} + 39 a^{7} + 27 a^{6} + 34 a^{5} + 8 a^{4} + 3 a^{3} + 14 a^{2} + 14 a + 33\right)\cdot 43 + \left(20 a^{11} + 20 a^{10} + 30 a^{9} + 26 a^{8} + 39 a^{7} + 11 a^{6} + 16 a^{5} + 18 a^{4} + 10 a^{3} + 11 a^{2} + 10 a + 23\right)\cdot 43^{2} + \left(22 a^{11} + 30 a^{10} + 17 a^{9} + 38 a^{8} + 14 a^{7} + 21 a^{6} + 39 a^{5} + 3 a^{4} + 35 a^{3} + 23 a^{2} + 42 a + 41\right)\cdot 43^{3} + \left(12 a^{11} + 32 a^{10} + 15 a^{9} + a^{8} + 2 a^{7} + 16 a^{6} + 23 a^{5} + 31 a^{4} + 32 a^{3} + 19 a^{2} + 40 a + 4\right)\cdot 43^{4} + \left(35 a^{11} + 10 a^{10} + 32 a^{9} + 31 a^{8} + 39 a^{7} + 16 a^{6} + 24 a^{5} + 13 a^{4} + 16 a^{3} + 8 a^{2} + 30 a + 28\right)\cdot 43^{5} + \left(40 a^{11} + 6 a^{10} + 40 a^{9} + 6 a^{8} + 24 a^{7} + 31 a^{6} + 18 a^{5} + 27 a^{4} + 30 a^{3} + 11 a^{2} + 24 a + 31\right)\cdot 43^{6} +O(43^{7})$$ 2*a^11 + 36*a^10 + 4*a^9 + 41*a^8 + 16*a^7 + 15*a^6 + 14*a^5 + 32*a^4 + 7*a^3 + 27*a^2 + 6*a + 34 + (18*a^11 + 27*a^10 + 40*a^9 + 14*a^8 + 39*a^7 + 27*a^6 + 34*a^5 + 8*a^4 + 3*a^3 + 14*a^2 + 14*a + 33)*43 + (20*a^11 + 20*a^10 + 30*a^9 + 26*a^8 + 39*a^7 + 11*a^6 + 16*a^5 + 18*a^4 + 10*a^3 + 11*a^2 + 10*a + 23)*43^2 + (22*a^11 + 30*a^10 + 17*a^9 + 38*a^8 + 14*a^7 + 21*a^6 + 39*a^5 + 3*a^4 + 35*a^3 + 23*a^2 + 42*a + 41)*43^3 + (12*a^11 + 32*a^10 + 15*a^9 + a^8 + 2*a^7 + 16*a^6 + 23*a^5 + 31*a^4 + 32*a^3 + 19*a^2 + 40*a + 4)*43^4 + (35*a^11 + 10*a^10 + 32*a^9 + 31*a^8 + 39*a^7 + 16*a^6 + 24*a^5 + 13*a^4 + 16*a^3 + 8*a^2 + 30*a + 28)*43^5 + (40*a^11 + 6*a^10 + 40*a^9 + 6*a^8 + 24*a^7 + 31*a^6 + 18*a^5 + 27*a^4 + 30*a^3 + 11*a^2 + 24*a + 31)*43^6+O(43^7) $r_{ 8 }$ $=$ $$19 a^{11} + 39 a^{10} + 23 a^{9} + 37 a^{8} + 42 a^{7} + 16 a^{6} + 8 a^{5} + 12 a^{4} + 38 a^{3} + 25 a^{2} + 22 a + 1 + \left(15 a^{11} + 23 a^{10} + 17 a^{9} + 14 a^{8} + 38 a^{7} + 14 a^{6} + 35 a^{5} + 8 a^{4} + 19 a^{3} + 13 a^{2} + 32 a + 14\right)\cdot 43 + \left(26 a^{11} + 20 a^{10} + 4 a^{9} + 16 a^{8} + 6 a^{7} + 18 a^{6} + 40 a^{5} + 6 a^{4} + 10 a^{3} + 32 a^{2} + 17 a + 2\right)\cdot 43^{2} + \left(37 a^{11} + 41 a^{10} + 23 a^{9} + 42 a^{8} + 36 a^{7} + 25 a^{6} + 5 a^{5} + 14 a^{4} + 16 a^{3} + 8 a^{2} + 31 a + 6\right)\cdot 43^{3} + \left(40 a^{11} + 5 a^{10} + 4 a^{9} + 26 a^{8} + 37 a^{7} + 21 a^{6} + 5 a^{5} + 39 a^{3} + 20 a^{2} + a + 28\right)\cdot 43^{4} + \left(41 a^{11} + 34 a^{10} + 25 a^{9} + 29 a^{8} + 37 a^{7} + 14 a^{6} + 13 a^{5} + 30 a^{3} + 15\right)\cdot 43^{5} + \left(21 a^{11} + 27 a^{10} + 35 a^{9} + 24 a^{8} + 29 a^{7} + 26 a^{6} + 38 a^{5} + 32 a^{4} + 23 a^{3} + 28 a^{2} + 42 a + 17\right)\cdot 43^{6} +O(43^{7})$$ 19*a^11 + 39*a^10 + 23*a^9 + 37*a^8 + 42*a^7 + 16*a^6 + 8*a^5 + 12*a^4 + 38*a^3 + 25*a^2 + 22*a + 1 + (15*a^11 + 23*a^10 + 17*a^9 + 14*a^8 + 38*a^7 + 14*a^6 + 35*a^5 + 8*a^4 + 19*a^3 + 13*a^2 + 32*a + 14)*43 + (26*a^11 + 20*a^10 + 4*a^9 + 16*a^8 + 6*a^7 + 18*a^6 + 40*a^5 + 6*a^4 + 10*a^3 + 32*a^2 + 17*a + 2)*43^2 + (37*a^11 + 41*a^10 + 23*a^9 + 42*a^8 + 36*a^7 + 25*a^6 + 5*a^5 + 14*a^4 + 16*a^3 + 8*a^2 + 31*a + 6)*43^3 + (40*a^11 + 5*a^10 + 4*a^9 + 26*a^8 + 37*a^7 + 21*a^6 + 5*a^5 + 39*a^3 + 20*a^2 + a + 28)*43^4 + (41*a^11 + 34*a^10 + 25*a^9 + 29*a^8 + 37*a^7 + 14*a^6 + 13*a^5 + 30*a^3 + 15)*43^5 + (21*a^11 + 27*a^10 + 35*a^9 + 24*a^8 + 29*a^7 + 26*a^6 + 38*a^5 + 32*a^4 + 23*a^3 + 28*a^2 + 42*a + 17)*43^6+O(43^7) $r_{ 9 }$ $=$ $$6 a^{11} + 30 a^{10} + 35 a^{9} + 16 a^{7} + 22 a^{6} + 22 a^{5} + 9 a^{4} + 4 a^{3} + a^{2} + 35 a + 24 + \left(8 a^{11} + 13 a^{10} + 40 a^{9} + 33 a^{8} + 7 a^{7} + 22 a^{6} + 11 a^{5} + 40 a^{4} + 14 a^{3} + 40 a^{2} + 9 a + 24\right)\cdot 43 + \left(21 a^{11} + 22 a^{10} + 32 a^{9} + 2 a^{8} + 18 a^{7} + 28 a^{6} + 24 a^{5} + 38 a^{4} + 38 a^{3} + 33 a^{2} + 7 a\right)\cdot 43^{2} + \left(38 a^{11} + 21 a^{10} + 4 a^{9} + 3 a^{8} + 16 a^{7} + 33 a^{6} + 11 a^{5} + 21 a^{4} + a^{3} + 28 a^{2} + 3 a + 15\right)\cdot 43^{3} + \left(18 a^{11} + 5 a^{9} + 19 a^{8} + 8 a^{7} + 35 a^{6} + 8 a^{5} + 34 a^{4} + 4 a^{3} + 8 a^{2} + 8 a + 35\right)\cdot 43^{4} + \left(23 a^{11} + 38 a^{10} + 28 a^{9} + 10 a^{8} + 39 a^{7} + 14 a^{6} + 30 a^{5} + 8 a^{3} + 35 a^{2} + 28 a + 24\right)\cdot 43^{5} + \left(7 a^{11} + 35 a^{10} + 25 a^{9} + 11 a^{8} + 19 a^{7} + 27 a^{6} + 4 a^{5} + 26 a^{4} + 10 a^{3} + 32 a^{2} + 19 a + 18\right)\cdot 43^{6} +O(43^{7})$$ 6*a^11 + 30*a^10 + 35*a^9 + 16*a^7 + 22*a^6 + 22*a^5 + 9*a^4 + 4*a^3 + a^2 + 35*a + 24 + (8*a^11 + 13*a^10 + 40*a^9 + 33*a^8 + 7*a^7 + 22*a^6 + 11*a^5 + 40*a^4 + 14*a^3 + 40*a^2 + 9*a + 24)*43 + (21*a^11 + 22*a^10 + 32*a^9 + 2*a^8 + 18*a^7 + 28*a^6 + 24*a^5 + 38*a^4 + 38*a^3 + 33*a^2 + 7*a)*43^2 + (38*a^11 + 21*a^10 + 4*a^9 + 3*a^8 + 16*a^7 + 33*a^6 + 11*a^5 + 21*a^4 + a^3 + 28*a^2 + 3*a + 15)*43^3 + (18*a^11 + 5*a^9 + 19*a^8 + 8*a^7 + 35*a^6 + 8*a^5 + 34*a^4 + 4*a^3 + 8*a^2 + 8*a + 35)*43^4 + (23*a^11 + 38*a^10 + 28*a^9 + 10*a^8 + 39*a^7 + 14*a^6 + 30*a^5 + 8*a^3 + 35*a^2 + 28*a + 24)*43^5 + (7*a^11 + 35*a^10 + 25*a^9 + 11*a^8 + 19*a^7 + 27*a^6 + 4*a^5 + 26*a^4 + 10*a^3 + 32*a^2 + 19*a + 18)*43^6+O(43^7) $r_{ 10 }$ $=$ $$28 a^{11} + 27 a^{9} + 8 a^{8} + 8 a^{7} + 20 a^{6} + 22 a^{5} + 8 a^{4} + 4 a^{3} + 24 a^{2} + 26 a + 4 + \left(39 a^{11} + a^{10} + 40 a^{9} + 10 a^{8} + 38 a^{7} + 18 a^{6} + 7 a^{5} + 35 a^{4} + 18 a^{3} + 16 a^{2} + 6 a + 17\right)\cdot 43 + \left(19 a^{11} + 9 a^{10} + 6 a^{9} + 2 a^{8} + 10 a^{7} + 40 a^{6} + 13 a^{5} + 20 a^{4} + 42 a^{3} + 7 a^{2} + 20 a + 18\right)\cdot 43^{2} + \left(3 a^{11} + 24 a^{10} + 8 a^{9} + 3 a^{8} + 42 a^{7} + 16 a^{6} + 25 a^{5} + 19 a^{4} + 17 a^{3} + 39 a^{2} + 37 a + 19\right)\cdot 43^{3} + \left(5 a^{11} + 25 a^{10} + 41 a^{9} + 30 a^{8} + 9 a^{7} + 39 a^{6} + 13 a^{5} + 26 a^{4} + 28 a^{3} + 29 a^{2} + 26 a + 42\right)\cdot 43^{4} + \left(17 a^{11} + 31 a^{10} + 42 a^{9} + 27 a^{8} + 30 a^{7} + 25 a^{6} + 8 a^{4} + 11 a^{3} + 37 a^{2} + 14 a + 41\right)\cdot 43^{5} + \left(34 a^{11} + 36 a^{10} + 14 a^{9} + 7 a^{8} + 5 a^{7} + 33 a^{6} + 4 a^{5} + 27 a^{4} + 23 a^{3} + 17 a^{2} + a + 4\right)\cdot 43^{6} +O(43^{7})$$ 28*a^11 + 27*a^9 + 8*a^8 + 8*a^7 + 20*a^6 + 22*a^5 + 8*a^4 + 4*a^3 + 24*a^2 + 26*a + 4 + (39*a^11 + a^10 + 40*a^9 + 10*a^8 + 38*a^7 + 18*a^6 + 7*a^5 + 35*a^4 + 18*a^3 + 16*a^2 + 6*a + 17)*43 + (19*a^11 + 9*a^10 + 6*a^9 + 2*a^8 + 10*a^7 + 40*a^6 + 13*a^5 + 20*a^4 + 42*a^3 + 7*a^2 + 20*a + 18)*43^2 + (3*a^11 + 24*a^10 + 8*a^9 + 3*a^8 + 42*a^7 + 16*a^6 + 25*a^5 + 19*a^4 + 17*a^3 + 39*a^2 + 37*a + 19)*43^3 + (5*a^11 + 25*a^10 + 41*a^9 + 30*a^8 + 9*a^7 + 39*a^6 + 13*a^5 + 26*a^4 + 28*a^3 + 29*a^2 + 26*a + 42)*43^4 + (17*a^11 + 31*a^10 + 42*a^9 + 27*a^8 + 30*a^7 + 25*a^6 + 8*a^4 + 11*a^3 + 37*a^2 + 14*a + 41)*43^5 + (34*a^11 + 36*a^10 + 14*a^9 + 7*a^8 + 5*a^7 + 33*a^6 + 4*a^5 + 27*a^4 + 23*a^3 + 17*a^2 + a + 4)*43^6+O(43^7) $r_{ 11 }$ $=$ $$34 a^{11} + 30 a^{10} + 31 a^{9} + 37 a^{8} + a^{7} + 13 a^{6} + 42 a^{5} + 11 a^{4} + 14 a^{3} + 39 a^{2} + 17 a + 27 + \left(41 a^{11} + 41 a^{10} + 27 a^{9} + 16 a^{8} + 41 a^{7} + 30 a^{6} + 22 a^{5} + 38 a^{4} + 22 a^{3} + 34 a^{2} + 42 a + 22\right)\cdot 43 + \left(14 a^{11} + 9 a^{10} + 6 a^{9} + 22 a^{8} + 18 a^{7} + 15 a^{6} + 31 a^{5} + 36 a^{4} + 37 a^{3} + 25 a^{2} + 18\right)\cdot 43^{2} + \left(4 a^{11} + 3 a^{10} + 42 a^{9} + 34 a^{8} + 8 a^{7} + 41 a^{6} + 34 a^{5} + 14 a^{4} + 31 a^{3} + 23 a^{2} + 3 a + 40\right)\cdot 43^{3} + \left(35 a^{11} + 41 a^{10} + 23 a^{9} + 8 a^{8} + 15 a^{7} + 11 a^{6} + 14 a^{5} + 12 a^{4} + 14 a^{3} + 17 a^{2} + 12 a + 34\right)\cdot 43^{4} + \left(12 a^{11} + 20 a^{10} + 10 a^{9} + 4 a^{8} + 7 a^{7} + 38 a^{6} + 25 a^{5} + 28 a^{4} + 32 a^{3} + 26 a^{2} + 20 a + 36\right)\cdot 43^{5} + \left(a^{11} + 32 a^{10} + 8 a^{9} + 31 a^{8} + 33 a^{7} + 6 a^{6} + 9 a^{5} + 35 a^{4} + 21 a^{3} + 9 a^{2} + 7 a + 5\right)\cdot 43^{6} +O(43^{7})$$ 34*a^11 + 30*a^10 + 31*a^9 + 37*a^8 + a^7 + 13*a^6 + 42*a^5 + 11*a^4 + 14*a^3 + 39*a^2 + 17*a + 27 + (41*a^11 + 41*a^10 + 27*a^9 + 16*a^8 + 41*a^7 + 30*a^6 + 22*a^5 + 38*a^4 + 22*a^3 + 34*a^2 + 42*a + 22)*43 + (14*a^11 + 9*a^10 + 6*a^9 + 22*a^8 + 18*a^7 + 15*a^6 + 31*a^5 + 36*a^4 + 37*a^3 + 25*a^2 + 18)*43^2 + (4*a^11 + 3*a^10 + 42*a^9 + 34*a^8 + 8*a^7 + 41*a^6 + 34*a^5 + 14*a^4 + 31*a^3 + 23*a^2 + 3*a + 40)*43^3 + (35*a^11 + 41*a^10 + 23*a^9 + 8*a^8 + 15*a^7 + 11*a^6 + 14*a^5 + 12*a^4 + 14*a^3 + 17*a^2 + 12*a + 34)*43^4 + (12*a^11 + 20*a^10 + 10*a^9 + 4*a^8 + 7*a^7 + 38*a^6 + 25*a^5 + 28*a^4 + 32*a^3 + 26*a^2 + 20*a + 36)*43^5 + (a^11 + 32*a^10 + 8*a^9 + 31*a^8 + 33*a^7 + 6*a^6 + 9*a^5 + 35*a^4 + 21*a^3 + 9*a^2 + 7*a + 5)*43^6+O(43^7) $r_{ 12 }$ $=$ $$26 a^{11} + 15 a^{10} + 35 a^{9} + 34 a^{8} + 25 a^{7} + 5 a^{6} + 31 a^{5} + 30 a^{4} + 16 a^{3} + 5 a^{2} + 36 a + 32 + \left(36 a^{11} + 10 a^{10} + 14 a^{9} + 25 a^{8} + 32 a^{7} + 4 a^{6} + 6 a^{5} + 10 a^{4} + 31 a^{3} + 39 a^{2} + 13 a + 39\right)\cdot 43 + \left(3 a^{11} + 18 a^{10} + 11 a^{9} + 28 a^{8} + 3 a^{7} + 34 a^{6} + 37 a^{5} + 13 a^{4} + 39 a^{3} + a^{2} + 35 a + 1\right)\cdot 43^{2} + \left(24 a^{11} + 21 a^{10} + 2 a^{9} + 8 a^{8} + 35 a^{7} + 11 a^{6} + 12 a^{5} + 13 a^{4} + 28 a^{3} + 3 a^{2} + 27 a + 17\right)\cdot 43^{3} + \left(11 a^{11} + 10 a^{10} + 34 a^{9} + 36 a^{8} + 37 a^{7} + 3 a^{6} + 35 a^{5} + 36 a^{4} + 27 a^{3} + 41 a^{2} + 36 a + 22\right)\cdot 43^{4} + \left(4 a^{11} + 28 a^{10} + 20 a^{9} + 37 a^{8} + 18 a^{7} + 32 a^{6} + a^{5} + 39 a^{4} + 21 a^{3} + 30 a^{2} + 30 a + 23\right)\cdot 43^{5} + \left(30 a^{11} + 10 a^{10} + 28 a^{9} + 38 a^{8} + 25 a^{7} + 20 a^{6} + 7 a^{5} + 22 a^{4} + 28 a^{3} + 34 a^{2} + 37 a + 27\right)\cdot 43^{6} +O(43^{7})$$ 26*a^11 + 15*a^10 + 35*a^9 + 34*a^8 + 25*a^7 + 5*a^6 + 31*a^5 + 30*a^4 + 16*a^3 + 5*a^2 + 36*a + 32 + (36*a^11 + 10*a^10 + 14*a^9 + 25*a^8 + 32*a^7 + 4*a^6 + 6*a^5 + 10*a^4 + 31*a^3 + 39*a^2 + 13*a + 39)*43 + (3*a^11 + 18*a^10 + 11*a^9 + 28*a^8 + 3*a^7 + 34*a^6 + 37*a^5 + 13*a^4 + 39*a^3 + a^2 + 35*a + 1)*43^2 + (24*a^11 + 21*a^10 + 2*a^9 + 8*a^8 + 35*a^7 + 11*a^6 + 12*a^5 + 13*a^4 + 28*a^3 + 3*a^2 + 27*a + 17)*43^3 + (11*a^11 + 10*a^10 + 34*a^9 + 36*a^8 + 37*a^7 + 3*a^6 + 35*a^5 + 36*a^4 + 27*a^3 + 41*a^2 + 36*a + 22)*43^4 + (4*a^11 + 28*a^10 + 20*a^9 + 37*a^8 + 18*a^7 + 32*a^6 + a^5 + 39*a^4 + 21*a^3 + 30*a^2 + 30*a + 23)*43^5 + (30*a^11 + 10*a^10 + 28*a^9 + 38*a^8 + 25*a^7 + 20*a^6 + 7*a^5 + 22*a^4 + 28*a^3 + 34*a^2 + 37*a + 27)*43^6+O(43^7) $r_{ 13 }$ $=$ $$16 a^{11} + 6 a^{10} + 8 a^{9} + 6 a^{8} + 5 a^{7} + 30 a^{6} + 29 a^{5} + 28 a^{4} + 13 a^{3} + 16 a^{2} + 41 + \left(38 a^{11} + 13 a^{10} + 41 a^{9} + 5 a^{8} + 9 a^{7} + 8 a^{6} + 42 a^{5} + 36 a^{4} + 4 a^{3} + 37 a^{2} + 7 a + 3\right)\cdot 43 + \left(19 a^{11} + 31 a^{10} + 11 a^{9} + 33 a^{8} + 40 a^{7} + 25 a^{6} + 12 a^{5} + 11 a^{4} + 24 a^{3} + 7 a^{2} + 3 a + 29\right)\cdot 43^{2} + \left(2 a^{11} + 42 a^{10} + 24 a^{9} + 22 a^{8} + 7 a^{7} + 40 a^{6} + 40 a^{5} + 31 a^{4} + 12 a^{3} + 22 a^{2} + 23 a + 7\right)\cdot 43^{3} + \left(35 a^{11} + 10 a^{10} + 13 a^{9} + 42 a^{8} + 36 a^{7} + 32 a^{6} + 32 a^{5} + 2 a^{4} + 23 a^{3} + 31 a^{2} + 30 a + 40\right)\cdot 43^{4} + \left(31 a^{11} + 2 a^{10} + 14 a^{9} + 5 a^{8} + 21 a^{7} + 25 a^{6} + 26 a^{5} + 42 a^{4} + 34 a^{3} + 18 a^{2} + 40\right)\cdot 43^{5} + \left(29 a^{11} + 28 a^{10} + 4 a^{9} + 10 a^{8} + 21 a^{7} + 42 a^{6} + 41 a^{5} + 12 a^{4} + 13 a^{3} + 28 a^{2} + 17 a + 31\right)\cdot 43^{6} +O(43^{7})$$ 16*a^11 + 6*a^10 + 8*a^9 + 6*a^8 + 5*a^7 + 30*a^6 + 29*a^5 + 28*a^4 + 13*a^3 + 16*a^2 + 41 + (38*a^11 + 13*a^10 + 41*a^9 + 5*a^8 + 9*a^7 + 8*a^6 + 42*a^5 + 36*a^4 + 4*a^3 + 37*a^2 + 7*a + 3)*43 + (19*a^11 + 31*a^10 + 11*a^9 + 33*a^8 + 40*a^7 + 25*a^6 + 12*a^5 + 11*a^4 + 24*a^3 + 7*a^2 + 3*a + 29)*43^2 + (2*a^11 + 42*a^10 + 24*a^9 + 22*a^8 + 7*a^7 + 40*a^6 + 40*a^5 + 31*a^4 + 12*a^3 + 22*a^2 + 23*a + 7)*43^3 + (35*a^11 + 10*a^10 + 13*a^9 + 42*a^8 + 36*a^7 + 32*a^6 + 32*a^5 + 2*a^4 + 23*a^3 + 31*a^2 + 30*a + 40)*43^4 + (31*a^11 + 2*a^10 + 14*a^9 + 5*a^8 + 21*a^7 + 25*a^6 + 26*a^5 + 42*a^4 + 34*a^3 + 18*a^2 + 40)*43^5 + (29*a^11 + 28*a^10 + 4*a^9 + 10*a^8 + 21*a^7 + 42*a^6 + 41*a^5 + 12*a^4 + 13*a^3 + 28*a^2 + 17*a + 31)*43^6+O(43^7) $r_{ 14 }$ $=$ $$4 a^{11} + 38 a^{10} + 10 a^{9} + 6 a^{8} + 2 a^{7} + 7 a^{6} + 30 a^{5} + 18 a^{4} + 28 a^{3} + 29 a^{2} + 41 a + 38 + \left(30 a^{11} + 14 a^{10} + 4 a^{9} + 13 a^{8} + 20 a^{7} + 28 a^{6} + 33 a^{5} + 31 a^{4} + 8 a^{3} + 30 a^{2} + 11 a + 24\right)\cdot 43 + \left(25 a^{11} + 29 a^{10} + 38 a^{9} + 36 a^{8} + 41 a^{7} + 42 a^{6} + a^{4} + 41 a^{2} + 37 a + 40\right)\cdot 43^{2} + \left(33 a^{11} + 3 a^{10} + 16 a^{9} + 21 a^{8} + 20 a^{7} + 4 a^{6} + 9 a^{5} + 36 a^{4} + 8 a^{3} + 14 a^{2} + 9 a + 23\right)\cdot 43^{3} + \left(28 a^{11} + 30 a^{10} + 7 a^{9} + 2 a^{8} + 31 a^{7} + 21 a^{6} + 9 a^{4} + 23 a^{3} + 13 a^{2} + 4 a + 40\right)\cdot 43^{4} + \left(22 a^{11} + 20 a^{10} + 37 a^{9} + 21 a^{8} + 6 a^{7} + 27 a^{6} + 34 a^{5} + 32 a^{4} + 37 a^{3} + 3 a^{2} + 30 a + 13\right)\cdot 43^{5} + \left(17 a^{11} + 11 a^{10} + 10 a^{9} + 35 a^{8} + 41 a^{7} + 21 a^{5} + 23 a^{4} + 5 a^{3} + 36 a^{2} + 24 a\right)\cdot 43^{6} +O(43^{7})$$ 4*a^11 + 38*a^10 + 10*a^9 + 6*a^8 + 2*a^7 + 7*a^6 + 30*a^5 + 18*a^4 + 28*a^3 + 29*a^2 + 41*a + 38 + (30*a^11 + 14*a^10 + 4*a^9 + 13*a^8 + 20*a^7 + 28*a^6 + 33*a^5 + 31*a^4 + 8*a^3 + 30*a^2 + 11*a + 24)*43 + (25*a^11 + 29*a^10 + 38*a^9 + 36*a^8 + 41*a^7 + 42*a^6 + a^4 + 41*a^2 + 37*a + 40)*43^2 + (33*a^11 + 3*a^10 + 16*a^9 + 21*a^8 + 20*a^7 + 4*a^6 + 9*a^5 + 36*a^4 + 8*a^3 + 14*a^2 + 9*a + 23)*43^3 + (28*a^11 + 30*a^10 + 7*a^9 + 2*a^8 + 31*a^7 + 21*a^6 + 9*a^4 + 23*a^3 + 13*a^2 + 4*a + 40)*43^4 + (22*a^11 + 20*a^10 + 37*a^9 + 21*a^8 + 6*a^7 + 27*a^6 + 34*a^5 + 32*a^4 + 37*a^3 + 3*a^2 + 30*a + 13)*43^5 + (17*a^11 + 11*a^10 + 10*a^9 + 35*a^8 + 41*a^7 + 21*a^5 + 23*a^4 + 5*a^3 + 36*a^2 + 24*a)*43^6+O(43^7) $r_{ 15 }$ $=$ $$29 a^{11} + 6 a^{10} + 15 a^{9} + 14 a^{8} + 17 a^{7} + 24 a^{6} + 31 a^{5} + 34 a^{4} + 37 a^{3} + 26 a^{2} + a + 20 + \left(22 a^{11} + 36 a^{10} + 17 a^{9} + 4 a^{8} + 13 a^{7} + a^{6} + 28 a^{5} + 4 a^{3} + 38 a^{2} + 4 a + 17\right)\cdot 43 + \left(42 a^{11} + 8 a^{10} + 22 a^{9} + 3 a^{8} + 25 a^{7} + 36 a^{6} + 2 a^{5} + 40 a^{4} + 26 a^{2} + 41 a + 41\right)\cdot 43^{2} + \left(3 a^{11} + 5 a^{10} + 23 a^{9} + 11 a^{8} + 11 a^{7} + 20 a^{6} + 7 a^{5} + 18 a^{4} + 27 a^{3} + 24 a^{2} + 8 a + 18\right)\cdot 43^{3} + \left(11 a^{11} + 11 a^{10} + 3 a^{9} + 2 a^{8} + 39 a^{7} + 24 a^{6} + 24 a^{5} + 18 a^{4} + 39 a^{3} + 5 a^{2} + 14 a + 25\right)\cdot 43^{4} + \left(35 a^{11} + a^{10} + 16 a^{9} + 8 a^{8} + 13 a^{7} + 41 a^{6} + 34 a^{5} + 3 a^{4} + 31 a^{2} + 5 a + 5\right)\cdot 43^{5} + \left(39 a^{11} + 12 a^{10} + 16 a^{9} + 12 a^{8} + 23 a^{7} + 18 a^{6} + 39 a^{5} + 15 a^{4} + a^{3} + 11 a^{2} + 21 a + 14\right)\cdot 43^{6} +O(43^{7})$$ 29*a^11 + 6*a^10 + 15*a^9 + 14*a^8 + 17*a^7 + 24*a^6 + 31*a^5 + 34*a^4 + 37*a^3 + 26*a^2 + a + 20 + (22*a^11 + 36*a^10 + 17*a^9 + 4*a^8 + 13*a^7 + a^6 + 28*a^5 + 4*a^3 + 38*a^2 + 4*a + 17)*43 + (42*a^11 + 8*a^10 + 22*a^9 + 3*a^8 + 25*a^7 + 36*a^6 + 2*a^5 + 40*a^4 + 26*a^2 + 41*a + 41)*43^2 + (3*a^11 + 5*a^10 + 23*a^9 + 11*a^8 + 11*a^7 + 20*a^6 + 7*a^5 + 18*a^4 + 27*a^3 + 24*a^2 + 8*a + 18)*43^3 + (11*a^11 + 11*a^10 + 3*a^9 + 2*a^8 + 39*a^7 + 24*a^6 + 24*a^5 + 18*a^4 + 39*a^3 + 5*a^2 + 14*a + 25)*43^4 + (35*a^11 + a^10 + 16*a^9 + 8*a^8 + 13*a^7 + 41*a^6 + 34*a^5 + 3*a^4 + 31*a^2 + 5*a + 5)*43^5 + (39*a^11 + 12*a^10 + 16*a^9 + 12*a^8 + 23*a^7 + 18*a^6 + 39*a^5 + 15*a^4 + a^3 + 11*a^2 + 21*a + 14)*43^6+O(43^7) $r_{ 16 }$ $=$ $$39 a^{11} + 28 a^{10} + 11 a^{9} + 42 a^{8} + 7 a^{7} + 17 a^{6} + 30 a^{5} + 2 a^{4} + 35 a^{3} + 40 a^{2} + 18 a + 36 + \left(38 a^{11} + 7 a^{10} + 21 a^{9} + 7 a^{8} + 18 a^{7} + 16 a^{6} + 5 a^{5} + 19 a^{4} + 25 a^{3} + 25 a^{2} + 35 a + 40\right)\cdot 43 + \left(38 a^{11} + 42 a^{10} + 8 a^{9} + 19 a^{8} + 16 a^{7} + 12 a^{6} + 26 a^{4} + 5 a^{3} + 36 a^{2} + 40 a + 18\right)\cdot 43^{2} + \left(19 a^{11} + 21 a^{10} + 32 a^{9} + 17 a^{8} + 25 a^{6} + 38 a^{5} + 14 a^{4} + 24 a^{3} + 12 a^{2} + 16 a + 36\right)\cdot 43^{3} + \left(27 a^{11} + 30 a^{10} + 19 a^{9} + 28 a^{8} + 4 a^{7} + 23 a^{6} + 34 a^{5} + 9 a^{4} + 32 a^{3} + 25 a^{2} + 5 a + 33\right)\cdot 43^{4} + \left(41 a^{11} + 29 a^{10} + 18 a^{9} + 21 a^{8} + 18 a^{7} + 5 a^{6} + 27 a^{5} + 24 a^{4} + 25 a^{3} + 25 a^{2} + 32 a + 19\right)\cdot 43^{5} + \left(3 a^{11} + 23 a^{10} + 18 a^{9} + 39 a^{8} + 23 a^{7} + 2 a^{6} + 30 a^{5} + 8 a^{4} + 28 a^{3} + 40 a^{2} + 12 a + 18\right)\cdot 43^{6} +O(43^{7})$$ 39*a^11 + 28*a^10 + 11*a^9 + 42*a^8 + 7*a^7 + 17*a^6 + 30*a^5 + 2*a^4 + 35*a^3 + 40*a^2 + 18*a + 36 + (38*a^11 + 7*a^10 + 21*a^9 + 7*a^8 + 18*a^7 + 16*a^6 + 5*a^5 + 19*a^4 + 25*a^3 + 25*a^2 + 35*a + 40)*43 + (38*a^11 + 42*a^10 + 8*a^9 + 19*a^8 + 16*a^7 + 12*a^6 + 26*a^4 + 5*a^3 + 36*a^2 + 40*a + 18)*43^2 + (19*a^11 + 21*a^10 + 32*a^9 + 17*a^8 + 25*a^6 + 38*a^5 + 14*a^4 + 24*a^3 + 12*a^2 + 16*a + 36)*43^3 + (27*a^11 + 30*a^10 + 19*a^9 + 28*a^8 + 4*a^7 + 23*a^6 + 34*a^5 + 9*a^4 + 32*a^3 + 25*a^2 + 5*a + 33)*43^4 + (41*a^11 + 29*a^10 + 18*a^9 + 21*a^8 + 18*a^7 + 5*a^6 + 27*a^5 + 24*a^4 + 25*a^3 + 25*a^2 + 32*a + 19)*43^5 + (3*a^11 + 23*a^10 + 18*a^9 + 39*a^8 + 23*a^7 + 2*a^6 + 30*a^5 + 8*a^4 + 28*a^3 + 40*a^2 + 12*a + 18)*43^6+O(43^7) $r_{ 17 }$ $=$ $$a^{11} + 5 a^{10} + 13 a^{9} + 17 a^{7} + 18 a^{6} + 18 a^{5} + 23 a^{4} + 15 a^{3} + 36 a^{2} + 13 a + 4 + \left(3 a^{11} + 32 a^{10} + 21 a^{9} + 27 a^{8} + 22 a^{7} + 19 a^{6} + 10 a^{5} + 23 a^{4} + 27 a^{3} + 30 a^{2} + 23 a + 25\right)\cdot 43 + \left(34 a^{11} + 12 a^{10} + 28 a^{9} + 9 a^{8} + 29 a^{7} + 37 a^{6} + a^{4} + 11 a^{3} + 14 a^{2} + 27 a\right)\cdot 43^{2} + \left(42 a^{11} + 12 a^{10} + 40 a^{9} + 24 a^{8} + 4 a^{7} + 40 a^{6} + 18 a^{5} + 5 a^{4} + 14 a^{3} + 39 a + 24\right)\cdot 43^{3} + \left(13 a^{11} + 40 a^{10} + 22 a^{9} + 21 a^{8} + 11 a^{7} + 19 a^{6} + 13 a^{5} + 7 a^{4} + 12 a^{3} + 42 a^{2} + 38 a + 35\right)\cdot 43^{4} + \left(14 a^{11} + 26 a^{10} + 39 a^{9} + 14 a^{8} + 41 a^{7} + 5 a^{6} + 16 a^{5} + 11 a^{4} + 28 a^{3} + 32 a^{2} + 17 a + 15\right)\cdot 43^{5} + \left(30 a^{11} + 22 a^{10} + 42 a^{9} + 25 a^{8} + 39 a^{7} + 9 a^{6} + 31 a^{5} + 32 a^{4} + 35 a^{3} + 33 a^{2} + 37 a + 20\right)\cdot 43^{6} +O(43^{7})$$ a^11 + 5*a^10 + 13*a^9 + 17*a^7 + 18*a^6 + 18*a^5 + 23*a^4 + 15*a^3 + 36*a^2 + 13*a + 4 + (3*a^11 + 32*a^10 + 21*a^9 + 27*a^8 + 22*a^7 + 19*a^6 + 10*a^5 + 23*a^4 + 27*a^3 + 30*a^2 + 23*a + 25)*43 + (34*a^11 + 12*a^10 + 28*a^9 + 9*a^8 + 29*a^7 + 37*a^6 + a^4 + 11*a^3 + 14*a^2 + 27*a)*43^2 + (42*a^11 + 12*a^10 + 40*a^9 + 24*a^8 + 4*a^7 + 40*a^6 + 18*a^5 + 5*a^4 + 14*a^3 + 39*a + 24)*43^3 + (13*a^11 + 40*a^10 + 22*a^9 + 21*a^8 + 11*a^7 + 19*a^6 + 13*a^5 + 7*a^4 + 12*a^3 + 42*a^2 + 38*a + 35)*43^4 + (14*a^11 + 26*a^10 + 39*a^9 + 14*a^8 + 41*a^7 + 5*a^6 + 16*a^5 + 11*a^4 + 28*a^3 + 32*a^2 + 17*a + 15)*43^5 + (30*a^11 + 22*a^10 + 42*a^9 + 25*a^8 + 39*a^7 + 9*a^6 + 31*a^5 + 32*a^4 + 35*a^3 + 33*a^2 + 37*a + 20)*43^6+O(43^7) $r_{ 18 }$ $=$ $$19 a^{11} + 26 a^{9} + 30 a^{8} + 30 a^{7} + 32 a^{6} + 18 a^{5} + 30 a^{4} + 15 a^{3} + 4 a^{2} + 33 a + 15 + \left(2 a^{11} + 36 a^{10} + 28 a^{9} + 8 a^{8} + 27 a^{7} + 27 a^{6} + 38 a^{5} + 5 a^{4} + 42 a^{3} + 38 a^{2} + 41 a + 6\right)\cdot 43 + \left(26 a^{11} + 32 a^{10} + 16 a^{9} + 20 a^{8} + 24 a^{7} + 37 a^{6} + 37 a^{5} + 11 a^{4} + 22 a^{3} + a^{2} + 32 a + 9\right)\cdot 43^{2} + \left(9 a^{11} + 30 a^{10} + 22 a^{9} + 18 a^{8} + 29 a^{7} + 10 a^{6} + 15 a^{5} + a^{4} + 38 a^{3} + 32 a^{2} + 15 a + 22\right)\cdot 43^{3} + \left(19 a^{11} + 41 a^{10} + 31 a^{9} + 39 a^{8} + 6 a^{7} + 20 a^{6} + 25 a^{5} + 14 a^{4} + 8 a^{3} + 17 a^{2} + 12 a + 16\right)\cdot 43^{4} + \left(18 a^{11} + 27 a^{10} + 12 a^{9} + 32 a^{7} + 30 a^{6} + 18 a^{5} + 25 a^{4} + 8 a^{3} + 23 a^{2} + 12 a\right)\cdot 43^{5} + \left(30 a^{11} + 8 a^{10} + 14 a^{9} + 5 a^{8} + 23 a^{7} + 10 a^{6} + 22 a^{5} + 37 a^{4} + 16 a^{3} + 15 a^{2} + 32 a + 22\right)\cdot 43^{6} +O(43^{7})$$ 19*a^11 + 26*a^9 + 30*a^8 + 30*a^7 + 32*a^6 + 18*a^5 + 30*a^4 + 15*a^3 + 4*a^2 + 33*a + 15 + (2*a^11 + 36*a^10 + 28*a^9 + 8*a^8 + 27*a^7 + 27*a^6 + 38*a^5 + 5*a^4 + 42*a^3 + 38*a^2 + 41*a + 6)*43 + (26*a^11 + 32*a^10 + 16*a^9 + 20*a^8 + 24*a^7 + 37*a^6 + 37*a^5 + 11*a^4 + 22*a^3 + a^2 + 32*a + 9)*43^2 + (9*a^11 + 30*a^10 + 22*a^9 + 18*a^8 + 29*a^7 + 10*a^6 + 15*a^5 + a^4 + 38*a^3 + 32*a^2 + 15*a + 22)*43^3 + (19*a^11 + 41*a^10 + 31*a^9 + 39*a^8 + 6*a^7 + 20*a^6 + 25*a^5 + 14*a^4 + 8*a^3 + 17*a^2 + 12*a + 16)*43^4 + (18*a^11 + 27*a^10 + 12*a^9 + 32*a^7 + 30*a^6 + 18*a^5 + 25*a^4 + 8*a^3 + 23*a^2 + 12*a)*43^5 + (30*a^11 + 8*a^10 + 14*a^9 + 5*a^8 + 23*a^7 + 10*a^6 + 22*a^5 + 37*a^4 + 16*a^3 + 15*a^2 + 32*a + 22)*43^6+O(43^7) $r_{ 19 }$ $=$ $$20 a^{11} + 5 a^{10} + 41 a^{9} + 42 a^{8} + 36 a^{7} + 38 a^{6} + 7 a^{5} + 9 a^{4} + 31 a^{3} + 28 a^{2} + 10 a + 26 + \left(4 a^{11} + 8 a^{10} + 29 a^{9} + 36 a^{8} + 23 a^{7} + 3 a^{6} + a^{5} + 14 a^{4} + 26 a^{3} + 16 a^{2} + 2 a + 25\right)\cdot 43 + \left(11 a^{11} + 35 a^{10} + 28 a^{9} + 39 a^{8} + 32 a^{6} + 28 a^{5} + a^{4} + 38 a^{3} + 29 a^{2} + 8 a + 13\right)\cdot 43^{2} + \left(39 a^{11} + 24 a^{9} + 8 a^{8} + 17 a^{7} + 37 a^{6} + a^{5} + 27 a^{4} + 21 a^{3} + 33 a^{2} + 13 a + 39\right)\cdot 43^{3} + \left(35 a^{11} + 9 a^{10} + 35 a^{9} + 4 a^{8} + 12 a^{7} + 14 a^{6} + 35 a^{5} + 15 a^{4} + 12 a^{3} + 30 a^{2} + 16 a + 14\right)\cdot 43^{4} + \left(37 a^{11} + 34 a^{10} + 33 a^{9} + 5 a^{8} + 27 a^{7} + 33 a^{6} + 22 a^{5} + 24 a^{4} + 7 a^{3} + 27 a^{2} + 15 a + 31\right)\cdot 43^{5} + \left(15 a^{11} + 35 a^{10} + 39 a^{9} + 4 a^{8} + 16 a^{7} + 34 a^{6} + 4 a^{5} + 33 a^{4} + 40 a^{3} + 4 a^{2} + 31 a + 29\right)\cdot 43^{6} +O(43^{7})$$ 20*a^11 + 5*a^10 + 41*a^9 + 42*a^8 + 36*a^7 + 38*a^6 + 7*a^5 + 9*a^4 + 31*a^3 + 28*a^2 + 10*a + 26 + (4*a^11 + 8*a^10 + 29*a^9 + 36*a^8 + 23*a^7 + 3*a^6 + a^5 + 14*a^4 + 26*a^3 + 16*a^2 + 2*a + 25)*43 + (11*a^11 + 35*a^10 + 28*a^9 + 39*a^8 + 32*a^6 + 28*a^5 + a^4 + 38*a^3 + 29*a^2 + 8*a + 13)*43^2 + (39*a^11 + 24*a^9 + 8*a^8 + 17*a^7 + 37*a^6 + a^5 + 27*a^4 + 21*a^3 + 33*a^2 + 13*a + 39)*43^3 + (35*a^11 + 9*a^10 + 35*a^9 + 4*a^8 + 12*a^7 + 14*a^6 + 35*a^5 + 15*a^4 + 12*a^3 + 30*a^2 + 16*a + 14)*43^4 + (37*a^11 + 34*a^10 + 33*a^9 + 5*a^8 + 27*a^7 + 33*a^6 + 22*a^5 + 24*a^4 + 7*a^3 + 27*a^2 + 15*a + 31)*43^5 + (15*a^11 + 35*a^10 + 39*a^9 + 4*a^8 + 16*a^7 + 34*a^6 + 4*a^5 + 33*a^4 + 40*a^3 + 4*a^2 + 31*a + 29)*43^6+O(43^7) $r_{ 20 }$ $=$ $$33 a^{11} + 24 a^{10} + 13 a^{9} + 20 a^{8} + 40 a^{7} + 8 a^{6} + 41 a^{5} + 5 a^{4} + 17 a^{3} + 8 a^{2} + 6 a + 34 + \left(3 a^{11} + 34 a^{10} + 31 a^{9} + 30 a^{8} + 21 a^{7} + 21 a^{6} + 4 a^{5} + 10 a^{4} + 26 a^{3} + 34 a^{2} + 5 a + 41\right)\cdot 43 + \left(17 a^{11} + 34 a^{10} + 5 a^{9} + 13 a^{8} + 39 a^{7} + 8 a^{6} + 35 a^{5} + 3 a^{4} + 32 a^{3} + 24 a^{2} + 24 a + 33\right)\cdot 43^{2} + \left(21 a^{11} + 21 a^{10} + 35 a^{9} + 19 a^{8} + 41 a^{7} + 27 a^{6} + 30 a^{5} + 36 a^{4} + 20 a^{3} + 4 a^{2} + 6 a + 31\right)\cdot 43^{3} + \left(22 a^{11} + 15 a^{10} + 3 a^{9} + 35 a^{8} + 31 a^{7} + 36 a^{6} + 18 a^{5} + 31 a^{4} + 9 a^{3} + 42 a^{2} + 29 a + 14\right)\cdot 43^{4} + \left(23 a^{11} + 18 a^{10} + 33 a^{9} + 24 a^{8} + 32 a^{7} + 33 a^{6} + 32 a^{5} + 18 a^{4} + 28 a^{3} + 33 a^{2} + 2 a + 35\right)\cdot 43^{5} + \left(18 a^{10} + 25 a^{9} + 36 a^{8} + 26 a^{7} + 39 a^{6} + 2 a^{5} + 35 a^{4} + 6 a^{3} + 23 a^{2} + 8 a + 10\right)\cdot 43^{6} +O(43^{7})$$ 33*a^11 + 24*a^10 + 13*a^9 + 20*a^8 + 40*a^7 + 8*a^6 + 41*a^5 + 5*a^4 + 17*a^3 + 8*a^2 + 6*a + 34 + (3*a^11 + 34*a^10 + 31*a^9 + 30*a^8 + 21*a^7 + 21*a^6 + 4*a^5 + 10*a^4 + 26*a^3 + 34*a^2 + 5*a + 41)*43 + (17*a^11 + 34*a^10 + 5*a^9 + 13*a^8 + 39*a^7 + 8*a^6 + 35*a^5 + 3*a^4 + 32*a^3 + 24*a^2 + 24*a + 33)*43^2 + (21*a^11 + 21*a^10 + 35*a^9 + 19*a^8 + 41*a^7 + 27*a^6 + 30*a^5 + 36*a^4 + 20*a^3 + 4*a^2 + 6*a + 31)*43^3 + (22*a^11 + 15*a^10 + 3*a^9 + 35*a^8 + 31*a^7 + 36*a^6 + 18*a^5 + 31*a^4 + 9*a^3 + 42*a^2 + 29*a + 14)*43^4 + (23*a^11 + 18*a^10 + 33*a^9 + 24*a^8 + 32*a^7 + 33*a^6 + 32*a^5 + 18*a^4 + 28*a^3 + 33*a^2 + 2*a + 35)*43^5 + (18*a^10 + 25*a^9 + 36*a^8 + 26*a^7 + 39*a^6 + 2*a^5 + 35*a^4 + 6*a^3 + 23*a^2 + 8*a + 10)*43^6+O(43^7) $r_{ 21 }$ $=$ $$17 a^{11} + a^{10} + 30 a^{9} + a^{8} + 8 a^{7} + 5 a^{6} + 12 a^{5} + 19 a^{4} + 38 a^{3} + 17 a^{2} + 14 + \left(20 a^{11} + 11 a^{10} + 6 a^{9} + 24 a^{8} + 29 a^{7} + 24 a^{6} + 34 a^{5} + 23 a^{4} + 13 a^{3} + 27 a^{2} + 37 a + 38\right)\cdot 43 + \left(26 a^{11} + 13 a^{10} + 4 a^{9} + 6 a^{8} + 15 a^{7} + 28 a^{6} + 42 a^{5} + 9 a^{4} + 14 a^{3} + 14 a^{2} + 4 a + 2\right)\cdot 43^{2} + \left(3 a^{11} + 19 a^{10} + 23 a^{9} + 28 a^{8} + 20 a^{7} + 2 a^{6} + 32 a^{5} + 6 a^{4} + 22 a^{2} + 38 a + 34\right)\cdot 43^{3} + \left(2 a^{11} + 24 a^{10} + 16 a^{9} + 3 a^{8} + 14 a^{7} + 16 a^{6} + 6 a^{5} + 17 a^{4} + 17 a^{3} + a^{2} + 17 a + 1\right)\cdot 43^{4} + \left(30 a^{11} + 30 a^{10} + 2 a^{9} + a^{8} + 21 a^{7} + 19 a^{6} + 6 a^{5} + 7 a^{4} + 9 a^{3} + 12 a^{2} + 13 a + 41\right)\cdot 43^{5} + \left(29 a^{11} + 27 a^{10} + 27 a^{8} + 3 a^{7} + 31 a^{6} + 32 a^{5} + 15 a^{4} + 30 a^{3} + 14 a^{2} + 40 a + 13\right)\cdot 43^{6} +O(43^{7})$$ 17*a^11 + a^10 + 30*a^9 + a^8 + 8*a^7 + 5*a^6 + 12*a^5 + 19*a^4 + 38*a^3 + 17*a^2 + 14 + (20*a^11 + 11*a^10 + 6*a^9 + 24*a^8 + 29*a^7 + 24*a^6 + 34*a^5 + 23*a^4 + 13*a^3 + 27*a^2 + 37*a + 38)*43 + (26*a^11 + 13*a^10 + 4*a^9 + 6*a^8 + 15*a^7 + 28*a^6 + 42*a^5 + 9*a^4 + 14*a^3 + 14*a^2 + 4*a + 2)*43^2 + (3*a^11 + 19*a^10 + 23*a^9 + 28*a^8 + 20*a^7 + 2*a^6 + 32*a^5 + 6*a^4 + 22*a^2 + 38*a + 34)*43^3 + (2*a^11 + 24*a^10 + 16*a^9 + 3*a^8 + 14*a^7 + 16*a^6 + 6*a^5 + 17*a^4 + 17*a^3 + a^2 + 17*a + 1)*43^4 + (30*a^11 + 30*a^10 + 2*a^9 + a^8 + 21*a^7 + 19*a^6 + 6*a^5 + 7*a^4 + 9*a^3 + 12*a^2 + 13*a + 41)*43^5 + (29*a^11 + 27*a^10 + 27*a^8 + 3*a^7 + 31*a^6 + 32*a^5 + 15*a^4 + 30*a^3 + 14*a^2 + 40*a + 13)*43^6+O(43^7) $r_{ 22 }$ $=$ $$15 a^{11} + 35 a^{10} + 16 a^{9} + a^{8} + 29 a^{7} + 37 a^{6} + 5 a^{5} + 3 a^{4} + 19 a^{3} + 12 a^{2} + 14 a + 35 + \left(a^{11} + 17 a^{10} + 27 a^{9} + 11 a^{8} + 8 a^{7} + a^{6} + 21 a^{5} + 3 a^{4} + 4 a^{3} + 32 a^{2} + 25 a + 33\right)\cdot 43 + \left(2 a^{11} + 38 a^{10} + 15 a^{9} + 21 a^{8} + a^{7} + 22 a^{6} + 19 a^{5} + 3 a^{4} + 26 a^{3} + 22 a^{2} + 25 a + 16\right)\cdot 43^{2} + \left(7 a^{11} + 8 a^{10} + 5 a^{9} + 15 a^{7} + 3 a^{6} + 20 a^{5} + 19 a^{4} + 25 a^{3} + 37 a^{2} + 34 a + 4\right)\cdot 43^{3} + \left(a^{11} + 24 a^{10} + 15 a^{9} + 38 a^{8} + 21 a^{7} + 26 a^{6} + 15 a^{5} + 37 a^{4} + 5 a^{3} + 38 a^{2} + 29 a + 21\right)\cdot 43^{4} + \left(4 a^{11} + 7 a^{10} + 37 a^{9} + 14 a^{8} + 4 a^{7} + 13 a^{6} + 7 a^{4} + 15 a^{3} + 40 a^{2} + 21 a + 27\right)\cdot 43^{5} + \left(24 a^{11} + 28 a^{10} + 22 a^{9} + 2 a^{8} + 6 a^{7} + 20 a^{6} + 36 a^{5} + 29 a^{4} + 5 a^{3} + 39 a^{2} + 19 a + 8\right)\cdot 43^{6} +O(43^{7})$$ 15*a^11 + 35*a^10 + 16*a^9 + a^8 + 29*a^7 + 37*a^6 + 5*a^5 + 3*a^4 + 19*a^3 + 12*a^2 + 14*a + 35 + (a^11 + 17*a^10 + 27*a^9 + 11*a^8 + 8*a^7 + a^6 + 21*a^5 + 3*a^4 + 4*a^3 + 32*a^2 + 25*a + 33)*43 + (2*a^11 + 38*a^10 + 15*a^9 + 21*a^8 + a^7 + 22*a^6 + 19*a^5 + 3*a^4 + 26*a^3 + 22*a^2 + 25*a + 16)*43^2 + (7*a^11 + 8*a^10 + 5*a^9 + 15*a^7 + 3*a^6 + 20*a^5 + 19*a^4 + 25*a^3 + 37*a^2 + 34*a + 4)*43^3 + (a^11 + 24*a^10 + 15*a^9 + 38*a^8 + 21*a^7 + 26*a^6 + 15*a^5 + 37*a^4 + 5*a^3 + 38*a^2 + 29*a + 21)*43^4 + (4*a^11 + 7*a^10 + 37*a^9 + 14*a^8 + 4*a^7 + 13*a^6 + 7*a^4 + 15*a^3 + 40*a^2 + 21*a + 27)*43^5 + (24*a^11 + 28*a^10 + 22*a^9 + 2*a^8 + 6*a^7 + 20*a^6 + 36*a^5 + 29*a^4 + 5*a^3 + 39*a^2 + 19*a + 8)*43^6+O(43^7) $r_{ 23 }$ $=$ $$12 a^{11} + a^{10} + 24 a^{9} + 31 a^{8} + 10 a^{7} + 4 a^{6} + 41 a^{5} + 20 a^{4} + 42 a^{3} + 33 a^{2} + 36 a + 32 + \left(2 a^{11} + 22 a^{10} + 28 a^{9} + 23 a^{8} + 33 a^{7} + 14 a^{6} + 22 a^{5} + 33 a^{4} + 34 a^{3} + 32 a^{2} + 24 a + 34\right)\cdot 43 + \left(23 a^{11} + 13 a^{10} + 32 a^{9} + 13 a^{8} + 20 a^{7} + 38 a^{6} + 23 a^{5} + 27 a^{4} + 32 a^{3} + 4 a^{2} + 34 a + 20\right)\cdot 43^{2} + \left(16 a^{11} + 7 a^{10} + a^{9} + 36 a^{8} + 16 a^{7} + 39 a^{5} + 20 a^{4} + 23 a^{3} + 38 a^{2} + 34 a + 25\right)\cdot 43^{3} + \left(19 a^{11} + 42 a^{10} + 24 a^{9} + 38 a^{8} + a^{7} + 2 a^{6} + 37 a^{5} + 36 a^{4} + 13 a^{3} + 17 a^{2} + 30 a + 12\right)\cdot 43^{4} + \left(15 a^{11} + 30 a^{10} + 37 a^{9} + 3 a^{8} + 33 a^{7} + 28 a^{6} + 26 a^{5} + 25 a^{4} + 25 a^{3} + 3 a^{2} + 6 a + 9\right)\cdot 43^{5} + \left(5 a^{11} + 24 a^{10} + 28 a^{9} + 24 a^{8} + 37 a^{7} + 35 a^{6} + 27 a^{5} + 11 a^{3} + 20 a^{2} + 40 a + 40\right)\cdot 43^{6} +O(43^{7})$$ 12*a^11 + a^10 + 24*a^9 + 31*a^8 + 10*a^7 + 4*a^6 + 41*a^5 + 20*a^4 + 42*a^3 + 33*a^2 + 36*a + 32 + (2*a^11 + 22*a^10 + 28*a^9 + 23*a^8 + 33*a^7 + 14*a^6 + 22*a^5 + 33*a^4 + 34*a^3 + 32*a^2 + 24*a + 34)*43 + (23*a^11 + 13*a^10 + 32*a^9 + 13*a^8 + 20*a^7 + 38*a^6 + 23*a^5 + 27*a^4 + 32*a^3 + 4*a^2 + 34*a + 20)*43^2 + (16*a^11 + 7*a^10 + a^9 + 36*a^8 + 16*a^7 + 39*a^5 + 20*a^4 + 23*a^3 + 38*a^2 + 34*a + 25)*43^3 + (19*a^11 + 42*a^10 + 24*a^9 + 38*a^8 + a^7 + 2*a^6 + 37*a^5 + 36*a^4 + 13*a^3 + 17*a^2 + 30*a + 12)*43^4 + (15*a^11 + 30*a^10 + 37*a^9 + 3*a^8 + 33*a^7 + 28*a^6 + 26*a^5 + 25*a^4 + 25*a^3 + 3*a^2 + 6*a + 9)*43^5 + (5*a^11 + 24*a^10 + 28*a^9 + 24*a^8 + 37*a^7 + 35*a^6 + 27*a^5 + 11*a^3 + 20*a^2 + 40*a + 40)*43^6+O(43^7) $r_{ 24 }$ $=$ $$28 a^{11} + 19 a^{10} + 9 a^{9} + 7 a^{8} + 37 a^{7} + 10 a^{6} + 5 a^{5} + 29 a^{4} + 13 a^{3} + 21 a^{2} + 3 a + 6 + \left(31 a^{11} + 11 a^{10} + 4 a^{9} + 20 a^{8} + 28 a^{7} + 12 a^{6} + 2 a^{5} + 15 a^{4} + 40 a^{3} + 3 a^{2} + 18 a + 31\right)\cdot 43 + \left(20 a^{11} + 23 a^{10} + 30 a^{9} + 7 a^{8} + 19 a^{7} + 12 a^{6} + 2 a^{5} + 10 a^{4} + 26 a^{3} + 17 a^{2} + 27 a + 21\right)\cdot 43^{2} + \left(28 a^{11} + 22 a^{10} + 30 a^{9} + 26 a^{8} + 6 a^{7} + 35 a^{6} + 42 a^{5} + 14 a^{4} + 2 a^{3} + 21 a^{2} + 37 a\right)\cdot 43^{3} + \left(17 a^{11} + 6 a^{10} + 18 a^{9} + 30 a^{8} + a^{7} + 40 a^{6} + 2 a^{5} + 33 a^{4} + 14 a^{3} + 40 a^{2} + 35 a + 24\right)\cdot 43^{4} + \left(2 a^{11} + 22 a^{10} + 42 a^{9} + 34 a^{8} + 30 a^{7} + 22 a^{6} + 2 a^{5} + 18 a^{4} + 29 a^{3} + 16 a^{2} + 10 a + 7\right)\cdot 43^{5} + \left(17 a^{11} + 34 a^{10} + 31 a^{9} + 21 a^{8} + 32 a^{7} + 14 a^{6} + 17 a^{5} + 2 a^{4} + 33 a^{3} + 17 a^{2} + 31 a + 7\right)\cdot 43^{6} +O(43^{7})$$ 28*a^11 + 19*a^10 + 9*a^9 + 7*a^8 + 37*a^7 + 10*a^6 + 5*a^5 + 29*a^4 + 13*a^3 + 21*a^2 + 3*a + 6 + (31*a^11 + 11*a^10 + 4*a^9 + 20*a^8 + 28*a^7 + 12*a^6 + 2*a^5 + 15*a^4 + 40*a^3 + 3*a^2 + 18*a + 31)*43 + (20*a^11 + 23*a^10 + 30*a^9 + 7*a^8 + 19*a^7 + 12*a^6 + 2*a^5 + 10*a^4 + 26*a^3 + 17*a^2 + 27*a + 21)*43^2 + (28*a^11 + 22*a^10 + 30*a^9 + 26*a^8 + 6*a^7 + 35*a^6 + 42*a^5 + 14*a^4 + 2*a^3 + 21*a^2 + 37*a)*43^3 + (17*a^11 + 6*a^10 + 18*a^9 + 30*a^8 + a^7 + 40*a^6 + 2*a^5 + 33*a^4 + 14*a^3 + 40*a^2 + 35*a + 24)*43^4 + (2*a^11 + 22*a^10 + 42*a^9 + 34*a^8 + 30*a^7 + 22*a^6 + 2*a^5 + 18*a^4 + 29*a^3 + 16*a^2 + 10*a + 7)*43^5 + (17*a^11 + 34*a^10 + 31*a^9 + 21*a^8 + 32*a^7 + 14*a^6 + 17*a^5 + 2*a^4 + 33*a^3 + 17*a^2 + 31*a + 7)*43^6+O(43^7)

### Generators of the action on the roots $r_1, \ldots, r_{ 24 }$

 Cycle notation $(1,22)(2,20)(3,8)(4,10)(5,23)(6,9)(7,13)(11,16)(12,18)(14,17)(15,21)(19,24)$ $(1,20,14,10,9,12,22,2,17,4,6,18)(3,13,16,23,19,21,8,7,11,5,24,15)$ $(3,19,11)(5,13,21)(7,15,23)(8,24,16)$ $(1,4,22,10)(2,9,20,6)(3,5,8,23)(7,19,13,24)(11,21,16,15)(12,14,18,17)$ $(1,21,22,15)(2,3,20,8)(4,16,10,11)(5,6,23,9)(7,17,13,14)(12,24,18,19)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 24 }$ Character values $c1$ $c2$ $c3$ $c4$ $1$ $1$ $()$ $2$ $2$ $2$ $2$ $1$ $2$ $(1,22)(2,20)(3,8)(4,10)(5,23)(6,9)(7,13)(11,16)(12,18)(14,17)(15,21)(19,24)$ $-2$ $-2$ $-2$ $-2$ $3$ $2$ $(1,16)(2,5)(3,6)(4,15)(7,12)(8,9)(10,21)(11,22)(13,18)(14,19)(17,24)(20,23)$ $0$ $0$ $0$ $0$ $3$ $2$ $(1,11)(2,23)(3,9)(4,21)(5,20)(6,8)(7,18)(10,15)(12,13)(14,24)(16,22)(17,19)$ $0$ $0$ $0$ $0$ $1$ $3$ $(1,9,17)(2,18,10)(3,19,11)(4,20,12)(5,13,21)(6,14,22)(7,15,23)(8,24,16)$ $-2 \zeta_{12}^{2}$ $2 \zeta_{12}^{2} - 2$ $2 \zeta_{12}^{2} - 2$ $-2 \zeta_{12}^{2}$ $1$ $3$ $(1,17,9)(2,10,18)(3,11,19)(4,12,20)(5,21,13)(6,22,14)(7,23,15)(8,16,24)$ $2 \zeta_{12}^{2} - 2$ $-2 \zeta_{12}^{2}$ $-2 \zeta_{12}^{2}$ $2 \zeta_{12}^{2} - 2$ $2$ $3$ $(3,19,11)(5,13,21)(7,15,23)(8,24,16)$ $-\zeta_{12}^{2} + 1$ $\zeta_{12}^{2}$ $\zeta_{12}^{2}$ $-\zeta_{12}^{2} + 1$ $2$ $3$ $(3,11,19)(5,21,13)(7,23,15)(8,16,24)$ $\zeta_{12}^{2}$ $-\zeta_{12}^{2} + 1$ $-\zeta_{12}^{2} + 1$ $\zeta_{12}^{2}$ $2$ $3$ $(1,9,17)(2,18,10)(3,11,19)(4,20,12)(5,21,13)(6,14,22)(7,23,15)(8,16,24)$ $-1$ $-1$ $-1$ $-1$ $1$ $4$ $(1,10,22,4)(2,6,20,9)(3,23,8,5)(7,24,13,19)(11,15,16,21)(12,17,18,14)$ $2 \zeta_{12}^{3}$ $2 \zeta_{12}^{3}$ $-2 \zeta_{12}^{3}$ $-2 \zeta_{12}^{3}$ $1$ $4$ $(1,4,22,10)(2,9,20,6)(3,5,8,23)(7,19,13,24)(11,21,16,15)(12,14,18,17)$ $-2 \zeta_{12}^{3}$ $-2 \zeta_{12}^{3}$ $2 \zeta_{12}^{3}$ $2 \zeta_{12}^{3}$ $3$ $4$ $(1,21,22,15)(2,3,20,8)(4,16,10,11)(5,6,23,9)(7,17,13,14)(12,24,18,19)$ $0$ $0$ $0$ $0$ $3$ $4$ $(1,15,22,21)(2,8,20,3)(4,11,10,16)(5,9,23,6)(7,14,13,17)(12,19,18,24)$ $0$ $0$ $0$ $0$ $1$ $6$ $(1,14,9,22,17,6)(2,4,18,20,10,12)(3,16,19,8,11,24)(5,15,13,23,21,7)$ $-2 \zeta_{12}^{2} + 2$ $2 \zeta_{12}^{2}$ $2 \zeta_{12}^{2}$ $-2 \zeta_{12}^{2} + 2$ $1$ $6$ $(1,6,17,22,9,14)(2,12,10,20,18,4)(3,24,11,8,19,16)(5,7,21,23,13,15)$ $2 \zeta_{12}^{2}$ $-2 \zeta_{12}^{2} + 2$ $-2 \zeta_{12}^{2} + 2$ $2 \zeta_{12}^{2}$ $2$ $6$ $(1,22)(2,20)(3,24,11,8,19,16)(4,10)(5,7,21,23,13,15)(6,9)(12,18)(14,17)$ $\zeta_{12}^{2} - 1$ $-\zeta_{12}^{2}$ $-\zeta_{12}^{2}$ $\zeta_{12}^{2} - 1$ $2$ $6$ $(1,22)(2,20)(3,16,19,8,11,24)(4,10)(5,15,13,23,21,7)(6,9)(12,18)(14,17)$ $-\zeta_{12}^{2}$ $\zeta_{12}^{2} - 1$ $\zeta_{12}^{2} - 1$ $-\zeta_{12}^{2}$ $2$ $6$ $(1,14,9,22,17,6)(2,4,18,20,10,12)(3,24,11,8,19,16)(5,7,21,23,13,15)$ $1$ $1$ $1$ $1$ $3$ $6$ $(1,8,17,16,9,24)(2,13,10,5,18,21)(3,14,11,6,19,22)(4,23,12,15,20,7)$ $0$ $0$ $0$ $0$ $3$ $6$ $(1,24,9,16,17,8)(2,21,18,5,10,13)(3,22,19,6,11,14)(4,7,20,15,12,23)$ $0$ $0$ $0$ $0$ $3$ $6$ $(1,3,17,11,9,19)(2,7,10,23,18,15)(4,5,12,21,20,13)(6,24,22,8,14,16)$ $0$ $0$ $0$ $0$ $3$ $6$ $(1,19,9,11,17,3)(2,15,18,23,10,7)(4,13,20,21,12,5)(6,16,14,8,22,24)$ $0$ $0$ $0$ $0$ $1$ $12$ $(1,20,14,10,9,12,22,2,17,4,6,18)(3,13,16,23,19,21,8,7,11,5,24,15)$ $2 \zeta_{12}^{3} - 2 \zeta_{12}$ $2 \zeta_{12}$ $-2 \zeta_{12}$ $-2 \zeta_{12}^{3} + 2 \zeta_{12}$ $1$ $12$ $(1,12,6,10,17,20,22,18,9,4,14,2)(3,21,24,23,11,13,8,15,19,5,16,7)$ $2 \zeta_{12}$ $2 \zeta_{12}^{3} - 2 \zeta_{12}$ $-2 \zeta_{12}^{3} + 2 \zeta_{12}$ $-2 \zeta_{12}$ $1$ $12$ $(1,2,14,4,9,18,22,20,17,10,6,12)(3,7,16,5,19,15,8,13,11,23,24,21)$ $-2 \zeta_{12}^{3} + 2 \zeta_{12}$ $-2 \zeta_{12}$ $2 \zeta_{12}$ $2 \zeta_{12}^{3} - 2 \zeta_{12}$ $1$ $12$ $(1,18,6,4,17,2,22,12,9,10,14,20)(3,15,24,5,11,7,8,21,19,23,16,13)$ $-2 \zeta_{12}$ $-2 \zeta_{12}^{3} + 2 \zeta_{12}$ $2 \zeta_{12}^{3} - 2 \zeta_{12}$ $2 \zeta_{12}$ $2$ $12$ $(1,20,14,10,9,12,22,2,17,4,6,18)(3,21,24,23,11,13,8,15,19,5,16,7)$ $\zeta_{12}^{3}$ $\zeta_{12}^{3}$ $-\zeta_{12}^{3}$ $-\zeta_{12}^{3}$ $2$ $12$ $(1,2,14,4,9,18,22,20,17,10,6,12)(3,15,24,5,11,7,8,21,19,23,16,13)$ $-\zeta_{12}^{3}$ $-\zeta_{12}^{3}$ $\zeta_{12}^{3}$ $\zeta_{12}^{3}$ $2$ $12$ $(1,4,22,10)(2,9,20,6)(3,13,16,23,19,21,8,7,11,5,24,15)(12,14,18,17)$ $-\zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$ $\zeta_{12}$ $2$ $12$ $(1,4,22,10)(2,9,20,6)(3,21,24,23,11,13,8,15,19,5,16,7)(12,14,18,17)$ $-\zeta_{12}^{3} + \zeta_{12}$ $-\zeta_{12}$ $\zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$ $2$ $12$ $(1,10,22,4)(2,6,20,9)(3,7,16,5,19,15,8,13,11,23,24,21)(12,17,18,14)$ $\zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$ $-\zeta_{12}$ $2$ $12$ $(1,10,22,4)(2,6,20,9)(3,15,24,5,11,7,8,21,19,23,16,13)(12,17,18,14)$ $\zeta_{12}^{3} - \zeta_{12}$ $\zeta_{12}$ $-\zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$ $3$ $12$ $(1,5,6,7,17,21,22,23,9,13,14,15)(2,19,12,16,10,3,20,24,18,11,4,8)$ $0$ $0$ $0$ $0$ $3$ $12$ $(1,21,14,7,9,5,22,15,17,13,6,23)(2,3,4,16,18,19,20,8,10,11,12,24)$ $0$ $0$ $0$ $0$ $3$ $12$ $(1,23,6,13,17,15,22,5,9,7,14,21)(2,24,12,11,10,8,20,19,18,16,4,3)$ $0$ $0$ $0$ $0$ $3$ $12$ $(1,15,14,13,9,23,22,21,17,7,6,5)(2,8,4,11,18,24,20,3,10,16,12,19)$ $0$ $0$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.