# Properties

 Label 1.23.22t1.a.f Dimension $1$ Group $C_{22}$ Conductor $23$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: 22T1 Conductor: $$23$$ Artin field: Galois closure of $$\Q(\zeta_{23})$$ Galois orbit size: $10$ Smallest permutation container: 22T1 Parity: odd Dirichlet character: $$\chi_{23}(17,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{11} + 28x^{2} + 8x + 27$$

Roots:
 $r_{ 1 }$ $=$ $$3 a^{10} + 12 a^{9} + 26 a^{8} + 4 a^{7} + 16 a^{6} + 22 a^{5} + 14 a^{4} + 25 a^{3} + 21 a^{2} + 22 a + 19 + \left(3 a^{10} + 23 a^{9} + 15 a^{8} + 7 a^{7} + 14 a^{6} + 8 a^{5} + 7 a^{4} + 8 a^{3} + a^{2} + 4 a + 27\right)\cdot 29 + \left(18 a^{10} + 16 a^{9} + 17 a^{8} + 22 a^{7} + 18 a^{6} + 9 a^{5} + 18 a^{4} + 5 a^{3} + 27 a^{2} + 5 a + 24\right)\cdot 29^{2} + \left(17 a^{10} + 4 a^{9} + 5 a^{8} + 8 a^{7} + 17 a^{6} + 12 a^{5} + 22 a^{4} + 28 a^{3} + 17 a^{2} + 11\right)\cdot 29^{3} + \left(16 a^{10} + 19 a^{9} + 11 a^{8} + 8 a^{7} + 12 a^{6} + 18 a^{5} + 16 a^{4} + 28 a^{3} + 28 a^{2} + 28 a + 28\right)\cdot 29^{4} +O(29^{5})$$ 3*a^10 + 12*a^9 + 26*a^8 + 4*a^7 + 16*a^6 + 22*a^5 + 14*a^4 + 25*a^3 + 21*a^2 + 22*a + 19 + (3*a^10 + 23*a^9 + 15*a^8 + 7*a^7 + 14*a^6 + 8*a^5 + 7*a^4 + 8*a^3 + a^2 + 4*a + 27)*29 + (18*a^10 + 16*a^9 + 17*a^8 + 22*a^7 + 18*a^6 + 9*a^5 + 18*a^4 + 5*a^3 + 27*a^2 + 5*a + 24)*29^2 + (17*a^10 + 4*a^9 + 5*a^8 + 8*a^7 + 17*a^6 + 12*a^5 + 22*a^4 + 28*a^3 + 17*a^2 + 11)*29^3 + (16*a^10 + 19*a^9 + 11*a^8 + 8*a^7 + 12*a^6 + 18*a^5 + 16*a^4 + 28*a^3 + 28*a^2 + 28*a + 28)*29^4+O(29^5) $r_{ 2 }$ $=$ $$5 a^{10} + 3 a^{9} + 13 a^{8} + 18 a^{7} + 18 a^{6} + 7 a^{5} + 21 a^{4} + 18 a^{3} + 18 a^{2} + 22 a + 4 + \left(26 a^{10} + 6 a^{9} + 25 a^{8} + 21 a^{7} + 23 a^{6} + 3 a^{5} + 23 a^{4} + 26 a^{3} + 16 a^{2} + 19 a + 7\right)\cdot 29 + \left(9 a^{10} + 6 a^{9} + 21 a^{8} + a^{7} + 13 a^{6} + 3 a^{5} + 17 a^{4} + 15 a^{3} + 15 a^{2} + 22 a + 7\right)\cdot 29^{2} + \left(23 a^{10} + 27 a^{9} + 21 a^{8} + 10 a^{7} + 3 a^{6} + 11 a^{5} + 15 a^{4} + 25 a^{2} + 28 a + 26\right)\cdot 29^{3} + \left(21 a^{10} + 25 a^{9} + 22 a^{8} + 5 a^{7} + 12 a^{5} + 4 a^{3} + 13 a^{2} + 15 a + 15\right)\cdot 29^{4} +O(29^{5})$$ 5*a^10 + 3*a^9 + 13*a^8 + 18*a^7 + 18*a^6 + 7*a^5 + 21*a^4 + 18*a^3 + 18*a^2 + 22*a + 4 + (26*a^10 + 6*a^9 + 25*a^8 + 21*a^7 + 23*a^6 + 3*a^5 + 23*a^4 + 26*a^3 + 16*a^2 + 19*a + 7)*29 + (9*a^10 + 6*a^9 + 21*a^8 + a^7 + 13*a^6 + 3*a^5 + 17*a^4 + 15*a^3 + 15*a^2 + 22*a + 7)*29^2 + (23*a^10 + 27*a^9 + 21*a^8 + 10*a^7 + 3*a^6 + 11*a^5 + 15*a^4 + 25*a^2 + 28*a + 26)*29^3 + (21*a^10 + 25*a^9 + 22*a^8 + 5*a^7 + 12*a^5 + 4*a^3 + 13*a^2 + 15*a + 15)*29^4+O(29^5) $r_{ 3 }$ $=$ $$5 a^{10} + 13 a^{9} + 14 a^{8} + 25 a^{7} + 7 a^{6} + 9 a^{5} + 4 a^{4} + a^{3} + 3 a + 3 + \left(4 a^{10} + 6 a^{9} + a^{8} + 10 a^{7} + 17 a^{6} + 18 a^{5} + 11 a^{4} + 8 a^{3} + 13 a^{2} + 6 a + 8\right)\cdot 29 + \left(18 a^{10} + 10 a^{9} + 21 a^{8} + 21 a^{7} + 24 a^{6} + 5 a^{5} + 3 a^{4} + 23 a^{3} + 9 a^{2} + 4 a + 14\right)\cdot 29^{2} + \left(22 a^{10} + 11 a^{9} + 17 a^{8} + 15 a^{7} + 13 a^{6} + 16 a^{5} + 20 a^{4} + 17 a^{3} + 4 a^{2} + 17 a + 10\right)\cdot 29^{3} + \left(24 a^{10} + 5 a^{9} + 19 a^{8} + 18 a^{7} + 8 a^{6} + 16 a^{5} + 2 a^{4} + 27 a^{3} + 2 a^{2} + 22 a + 20\right)\cdot 29^{4} +O(29^{5})$$ 5*a^10 + 13*a^9 + 14*a^8 + 25*a^7 + 7*a^6 + 9*a^5 + 4*a^4 + a^3 + 3*a + 3 + (4*a^10 + 6*a^9 + a^8 + 10*a^7 + 17*a^6 + 18*a^5 + 11*a^4 + 8*a^3 + 13*a^2 + 6*a + 8)*29 + (18*a^10 + 10*a^9 + 21*a^8 + 21*a^7 + 24*a^6 + 5*a^5 + 3*a^4 + 23*a^3 + 9*a^2 + 4*a + 14)*29^2 + (22*a^10 + 11*a^9 + 17*a^8 + 15*a^7 + 13*a^6 + 16*a^5 + 20*a^4 + 17*a^3 + 4*a^2 + 17*a + 10)*29^3 + (24*a^10 + 5*a^9 + 19*a^8 + 18*a^7 + 8*a^6 + 16*a^5 + 2*a^4 + 27*a^3 + 2*a^2 + 22*a + 20)*29^4+O(29^5) $r_{ 4 }$ $=$ $$5 a^{10} + 21 a^{9} + 16 a^{8} + 10 a^{7} + 10 a^{6} + 7 a^{5} + 22 a^{4} + 21 a^{3} + 4 a^{2} + 23 a + 13 + \left(23 a^{10} + 2 a^{9} + 20 a^{8} + 6 a^{7} + 23 a^{6} + 14 a^{5} + 10 a^{4} + 23 a^{3} + 23 a^{2} + 19 a + 5\right)\cdot 29 + \left(5 a^{10} + 24 a^{9} + 19 a^{8} + 5 a^{7} + 21 a^{6} + 10 a^{5} + 11 a^{4} + 16 a^{3} + 15 a^{2} + 16 a + 2\right)\cdot 29^{2} + \left(14 a^{10} + 14 a^{9} + 26 a^{8} + 7 a^{7} + 15 a^{6} + 14 a^{5} + 13 a^{4} + 3 a^{3} + 10 a^{2} + 20 a + 24\right)\cdot 29^{3} + \left(11 a^{10} + a^{9} + 28 a^{8} + 10 a^{7} + 8 a^{6} + a^{5} + 6 a^{4} + 22 a^{3} + 16 a^{2} + 6 a\right)\cdot 29^{4} +O(29^{5})$$ 5*a^10 + 21*a^9 + 16*a^8 + 10*a^7 + 10*a^6 + 7*a^5 + 22*a^4 + 21*a^3 + 4*a^2 + 23*a + 13 + (23*a^10 + 2*a^9 + 20*a^8 + 6*a^7 + 23*a^6 + 14*a^5 + 10*a^4 + 23*a^3 + 23*a^2 + 19*a + 5)*29 + (5*a^10 + 24*a^9 + 19*a^8 + 5*a^7 + 21*a^6 + 10*a^5 + 11*a^4 + 16*a^3 + 15*a^2 + 16*a + 2)*29^2 + (14*a^10 + 14*a^9 + 26*a^8 + 7*a^7 + 15*a^6 + 14*a^5 + 13*a^4 + 3*a^3 + 10*a^2 + 20*a + 24)*29^3 + (11*a^10 + a^9 + 28*a^8 + 10*a^7 + 8*a^6 + a^5 + 6*a^4 + 22*a^3 + 16*a^2 + 6*a)*29^4+O(29^5) $r_{ 5 }$ $=$ $$6 a^{10} + 9 a^{9} + 18 a^{8} + 18 a^{6} + 17 a^{5} + 5 a^{4} + 16 a^{3} + 7 a^{2} + 26 a + 3 + \left(23 a^{10} + 10 a^{8} + 9 a^{7} + 17 a^{6} + 12 a^{5} + 2 a^{4} + 3 a^{3} + 23 a^{2} + 27 a + 19\right)\cdot 29 + \left(11 a^{10} + 20 a^{9} + 15 a^{8} + 21 a^{7} + 20 a^{6} + 5 a^{5} + 23 a^{4} + 11 a^{3} + 10 a^{2} + 6 a + 23\right)\cdot 29^{2} + \left(3 a^{10} + 15 a^{9} + 4 a^{8} + 10 a^{7} + 3 a^{6} + 26 a^{5} + 16 a^{4} + 4 a^{3} + 23 a^{2} + 15 a + 22\right)\cdot 29^{3} + \left(8 a^{10} + 22 a^{9} + 17 a^{8} + 16 a^{7} + 12 a^{6} + 18 a^{5} + 27 a^{4} + 28 a^{3} + 7 a^{2} + 17 a + 25\right)\cdot 29^{4} +O(29^{5})$$ 6*a^10 + 9*a^9 + 18*a^8 + 18*a^6 + 17*a^5 + 5*a^4 + 16*a^3 + 7*a^2 + 26*a + 3 + (23*a^10 + 10*a^8 + 9*a^7 + 17*a^6 + 12*a^5 + 2*a^4 + 3*a^3 + 23*a^2 + 27*a + 19)*29 + (11*a^10 + 20*a^9 + 15*a^8 + 21*a^7 + 20*a^6 + 5*a^5 + 23*a^4 + 11*a^3 + 10*a^2 + 6*a + 23)*29^2 + (3*a^10 + 15*a^9 + 4*a^8 + 10*a^7 + 3*a^6 + 26*a^5 + 16*a^4 + 4*a^3 + 23*a^2 + 15*a + 22)*29^3 + (8*a^10 + 22*a^9 + 17*a^8 + 16*a^7 + 12*a^6 + 18*a^5 + 27*a^4 + 28*a^3 + 7*a^2 + 17*a + 25)*29^4+O(29^5) $r_{ 6 }$ $=$ $$7 a^{10} + 6 a^{9} + 27 a^{8} + 12 a^{7} + 23 a^{6} + a^{5} + 19 a^{4} + 4 a^{3} + 28 a + 18 + \left(3 a^{10} + 17 a^{9} + 24 a^{8} + 19 a^{6} + 26 a^{5} + 15 a^{4} + 5 a^{3} + 2 a^{2} + 25 a + 2\right)\cdot 29 + \left(27 a^{10} + 15 a^{9} + 5 a^{7} + 15 a^{6} + 27 a^{5} + 16 a^{4} + 14 a^{3} + 26 a^{2} + 21 a + 10\right)\cdot 29^{2} + \left(28 a^{10} + 18 a^{9} + 15 a^{8} + 28 a^{6} + 21 a^{5} + 2 a^{4} + 17 a^{3} + 20 a^{2} + 27 a + 15\right)\cdot 29^{3} + \left(28 a^{10} + 6 a^{9} + 28 a^{8} + 20 a^{7} + 26 a^{6} + 27 a^{5} + 20 a^{4} + 25 a^{3} + 3 a^{2} + 15 a\right)\cdot 29^{4} +O(29^{5})$$ 7*a^10 + 6*a^9 + 27*a^8 + 12*a^7 + 23*a^6 + a^5 + 19*a^4 + 4*a^3 + 28*a + 18 + (3*a^10 + 17*a^9 + 24*a^8 + 19*a^6 + 26*a^5 + 15*a^4 + 5*a^3 + 2*a^2 + 25*a + 2)*29 + (27*a^10 + 15*a^9 + 5*a^7 + 15*a^6 + 27*a^5 + 16*a^4 + 14*a^3 + 26*a^2 + 21*a + 10)*29^2 + (28*a^10 + 18*a^9 + 15*a^8 + 28*a^6 + 21*a^5 + 2*a^4 + 17*a^3 + 20*a^2 + 27*a + 15)*29^3 + (28*a^10 + 6*a^9 + 28*a^8 + 20*a^7 + 26*a^6 + 27*a^5 + 20*a^4 + 25*a^3 + 3*a^2 + 15*a)*29^4+O(29^5) $r_{ 7 }$ $=$ $$7 a^{10} + 27 a^{9} + 25 a^{8} + 17 a^{7} + 14 a^{6} + 15 a^{5} + 9 a^{4} + 24 a^{3} + 27 a^{2} + 8 a + 14 + \left(5 a^{10} + 7 a^{9} + 20 a^{8} + 21 a^{7} + 28 a^{6} + 3 a^{5} + 27 a^{4} + 16 a^{3} + 23 a^{2} + 6 a + 23\right)\cdot 29 + \left(13 a^{10} + 26 a^{9} + 15 a^{8} + 27 a^{7} + 23 a^{6} + 17 a^{5} + 12 a^{4} + 20 a^{3} + 18 a^{2} + 9 a + 5\right)\cdot 29^{2} + \left(2 a^{10} + 23 a^{9} + 20 a^{8} + 26 a^{7} + 15 a^{6} + 23 a^{5} + 8 a^{4} + a^{3} + a^{2} + 2 a + 22\right)\cdot 29^{3} + \left(28 a^{10} + 7 a^{9} + 2 a^{8} + 24 a^{7} + 19 a^{6} + 16 a^{5} + 13 a^{4} + 22 a^{3} + 18 a^{2} + 12 a + 15\right)\cdot 29^{4} +O(29^{5})$$ 7*a^10 + 27*a^9 + 25*a^8 + 17*a^7 + 14*a^6 + 15*a^5 + 9*a^4 + 24*a^3 + 27*a^2 + 8*a + 14 + (5*a^10 + 7*a^9 + 20*a^8 + 21*a^7 + 28*a^6 + 3*a^5 + 27*a^4 + 16*a^3 + 23*a^2 + 6*a + 23)*29 + (13*a^10 + 26*a^9 + 15*a^8 + 27*a^7 + 23*a^6 + 17*a^5 + 12*a^4 + 20*a^3 + 18*a^2 + 9*a + 5)*29^2 + (2*a^10 + 23*a^9 + 20*a^8 + 26*a^7 + 15*a^6 + 23*a^5 + 8*a^4 + a^3 + a^2 + 2*a + 22)*29^3 + (28*a^10 + 7*a^9 + 2*a^8 + 24*a^7 + 19*a^6 + 16*a^5 + 13*a^4 + 22*a^3 + 18*a^2 + 12*a + 15)*29^4+O(29^5) $r_{ 8 }$ $=$ $$8 a^{10} + 6 a^{9} + 24 a^{8} + 23 a^{7} + 12 a^{6} + 12 a^{5} + 25 a^{3} + 18 a^{2} + 21 a + 20 + \left(3 a^{10} + 9 a^{9} + 10 a^{8} + 26 a^{7} + 9 a^{6} + 22 a^{5} + 19 a^{4} + 13 a^{3} + 14 a^{2} + 24 a + 14\right)\cdot 29 + \left(9 a^{10} + 20 a^{9} + 23 a^{8} + 13 a^{7} + 9 a^{6} + 24 a^{5} + 28 a^{4} + 21 a^{3} + 24 a^{2} + 9 a + 8\right)\cdot 29^{2} + \left(22 a^{10} + 23 a^{9} + 3 a^{8} + 16 a^{7} + 26 a^{6} + 17 a^{5} + 28 a^{4} + 27 a^{3} + 26 a^{2} + 15 a + 14\right)\cdot 29^{3} + \left(14 a^{10} + 12 a^{9} + 10 a^{8} + 25 a^{7} + 9 a^{6} + 24 a^{5} + 10 a^{4} + 4 a^{3} + 11 a^{2} + 18 a + 17\right)\cdot 29^{4} +O(29^{5})$$ 8*a^10 + 6*a^9 + 24*a^8 + 23*a^7 + 12*a^6 + 12*a^5 + 25*a^3 + 18*a^2 + 21*a + 20 + (3*a^10 + 9*a^9 + 10*a^8 + 26*a^7 + 9*a^6 + 22*a^5 + 19*a^4 + 13*a^3 + 14*a^2 + 24*a + 14)*29 + (9*a^10 + 20*a^9 + 23*a^8 + 13*a^7 + 9*a^6 + 24*a^5 + 28*a^4 + 21*a^3 + 24*a^2 + 9*a + 8)*29^2 + (22*a^10 + 23*a^9 + 3*a^8 + 16*a^7 + 26*a^6 + 17*a^5 + 28*a^4 + 27*a^3 + 26*a^2 + 15*a + 14)*29^3 + (14*a^10 + 12*a^9 + 10*a^8 + 25*a^7 + 9*a^6 + 24*a^5 + 10*a^4 + 4*a^3 + 11*a^2 + 18*a + 17)*29^4+O(29^5) $r_{ 9 }$ $=$ $$13 a^{10} + 10 a^{9} + 15 a^{8} + 18 a^{7} + 25 a^{6} + 22 a^{5} + 27 a^{4} + 13 a^{3} + 14 a^{2} + 13 a + 9 + \left(11 a^{10} + 18 a^{9} + 17 a^{8} + 4 a^{7} + 21 a^{6} + 27 a^{5} + 21 a^{4} + 25 a^{3} + 7 a^{2} + 5 a + 22\right)\cdot 29 + \left(28 a^{10} + 14 a^{9} + 12 a^{8} + 3 a^{7} + 22 a^{6} + 4 a^{5} + 27 a^{3} + 28 a^{2} + 4 a + 7\right)\cdot 29^{2} + \left(23 a^{10} + 28 a^{9} + 14 a^{8} + 19 a^{7} + 11 a^{6} + 25 a^{5} + 18 a^{4} + 17 a^{3} + 3 a^{2} + 20 a + 18\right)\cdot 29^{3} + \left(a^{10} + 2 a^{9} + 14 a^{8} + 12 a^{7} + 4 a^{6} + 26 a^{5} + 24 a^{4} + 13 a^{3} + 13 a^{2} + a + 19\right)\cdot 29^{4} +O(29^{5})$$ 13*a^10 + 10*a^9 + 15*a^8 + 18*a^7 + 25*a^6 + 22*a^5 + 27*a^4 + 13*a^3 + 14*a^2 + 13*a + 9 + (11*a^10 + 18*a^9 + 17*a^8 + 4*a^7 + 21*a^6 + 27*a^5 + 21*a^4 + 25*a^3 + 7*a^2 + 5*a + 22)*29 + (28*a^10 + 14*a^9 + 12*a^8 + 3*a^7 + 22*a^6 + 4*a^5 + 27*a^3 + 28*a^2 + 4*a + 7)*29^2 + (23*a^10 + 28*a^9 + 14*a^8 + 19*a^7 + 11*a^6 + 25*a^5 + 18*a^4 + 17*a^3 + 3*a^2 + 20*a + 18)*29^3 + (a^10 + 2*a^9 + 14*a^8 + 12*a^7 + 4*a^6 + 26*a^5 + 24*a^4 + 13*a^3 + 13*a^2 + a + 19)*29^4+O(29^5) $r_{ 10 }$ $=$ $$16 a^{10} + 5 a^{9} + 2 a^{8} + 18 a^{7} + 5 a^{5} + 22 a^{4} + 9 a^{3} + a^{2} + 17 a + 21 + \left(10 a^{10} + 20 a^{9} + 26 a^{8} + 18 a^{7} + 16 a^{6} + 17 a^{5} + 26 a^{4} + 8 a^{3} + 26 a^{2} + 7\right)\cdot 29 + \left(21 a^{10} + 25 a^{9} + 4 a^{8} + 7 a^{7} + 15 a^{6} + 6 a^{5} + 27 a^{4} + 17 a^{3} + 15 a^{2} + 7\right)\cdot 29^{2} + \left(18 a^{10} + 4 a^{9} + 20 a^{8} + 21 a^{7} + 7 a^{6} + 26 a^{5} + 12 a^{4} + 17 a^{3} + 4 a^{2} + 15 a + 21\right)\cdot 29^{3} + \left(26 a^{10} + 14 a^{9} + 26 a^{8} + 20 a^{7} + 13 a^{6} + 23 a^{5} + 4 a^{4} + 2 a^{3} + 20 a^{2} + 25 a + 26\right)\cdot 29^{4} +O(29^{5})$$ 16*a^10 + 5*a^9 + 2*a^8 + 18*a^7 + 5*a^5 + 22*a^4 + 9*a^3 + a^2 + 17*a + 21 + (10*a^10 + 20*a^9 + 26*a^8 + 18*a^7 + 16*a^6 + 17*a^5 + 26*a^4 + 8*a^3 + 26*a^2 + 7)*29 + (21*a^10 + 25*a^9 + 4*a^8 + 7*a^7 + 15*a^6 + 6*a^5 + 27*a^4 + 17*a^3 + 15*a^2 + 7)*29^2 + (18*a^10 + 4*a^9 + 20*a^8 + 21*a^7 + 7*a^6 + 26*a^5 + 12*a^4 + 17*a^3 + 4*a^2 + 15*a + 21)*29^3 + (26*a^10 + 14*a^9 + 26*a^8 + 20*a^7 + 13*a^6 + 23*a^5 + 4*a^4 + 2*a^3 + 20*a^2 + 25*a + 26)*29^4+O(29^5) $r_{ 11 }$ $=$ $$16 a^{10} + 13 a^{9} + 17 a^{8} + a^{7} + 26 a^{6} + 25 a^{5} + 3 a^{4} + 23 a^{3} + 18 a^{2} + 13 a + 2 + \left(21 a^{10} + 9 a^{9} + a^{8} + 20 a^{7} + 6 a^{6} + 19 a^{5} + 21 a^{4} + a^{3} + 19 a^{2} + a\right)\cdot 29 + \left(10 a^{10} + 2 a^{9} + 7 a^{8} + 3 a^{7} + a^{6} + 5 a^{5} + 3 a^{4} + 28 a^{3} + 28 a^{2} + 9 a + 14\right)\cdot 29^{2} + \left(17 a^{10} + 24 a^{9} + a^{8} + 2 a^{7} + 12 a^{6} + 3 a^{5} + 10 a^{4} + 14 a^{3} + 18 a^{2} + 5 a + 16\right)\cdot 29^{3} + \left(9 a^{10} + 2 a^{9} + 10 a^{8} + 2 a^{7} + 23 a^{6} + 24 a^{5} + 13 a^{4} + 14 a^{3} + 9 a^{2} + 11 a + 22\right)\cdot 29^{4} +O(29^{5})$$ 16*a^10 + 13*a^9 + 17*a^8 + a^7 + 26*a^6 + 25*a^5 + 3*a^4 + 23*a^3 + 18*a^2 + 13*a + 2 + (21*a^10 + 9*a^9 + a^8 + 20*a^7 + 6*a^6 + 19*a^5 + 21*a^4 + a^3 + 19*a^2 + a)*29 + (10*a^10 + 2*a^9 + 7*a^8 + 3*a^7 + a^6 + 5*a^5 + 3*a^4 + 28*a^3 + 28*a^2 + 9*a + 14)*29^2 + (17*a^10 + 24*a^9 + a^8 + 2*a^7 + 12*a^6 + 3*a^5 + 10*a^4 + 14*a^3 + 18*a^2 + 5*a + 16)*29^3 + (9*a^10 + 2*a^9 + 10*a^8 + 2*a^7 + 23*a^6 + 24*a^5 + 13*a^4 + 14*a^3 + 9*a^2 + 11*a + 22)*29^4+O(29^5) $r_{ 12 }$ $=$ $$17 a^{10} + 6 a^{9} + 25 a^{8} + 8 a^{7} + a^{6} + 5 a^{5} + 19 a^{3} + 3 a^{2} + 9 a + 15 + \left(8 a^{10} + 14 a^{9} + 14 a^{8} + 11 a^{7} + 7 a^{6} + a^{5} + 8 a^{4} + 18 a^{2} + 27 a + 9\right)\cdot 29 + \left(20 a^{10} + 3 a^{9} + 17 a^{8} + 25 a^{7} + 11 a^{6} + a^{5} + 12 a^{4} + 16 a^{3} + 28 a^{2} + 8 a + 7\right)\cdot 29^{2} + \left(16 a^{10} + 9 a^{9} + 19 a^{8} + 3 a^{7} + 25 a^{6} + 2 a^{5} + 27 a^{4} + 20 a^{3} + 6 a^{2} + 12 a + 14\right)\cdot 29^{3} + \left(5 a^{10} + 26 a^{9} + 8 a^{8} + 22 a^{7} + 21 a^{6} + 6 a^{5} + 22 a^{4} + 4 a^{3} + 12 a^{2} + 26 a + 22\right)\cdot 29^{4} +O(29^{5})$$ 17*a^10 + 6*a^9 + 25*a^8 + 8*a^7 + a^6 + 5*a^5 + 19*a^3 + 3*a^2 + 9*a + 15 + (8*a^10 + 14*a^9 + 14*a^8 + 11*a^7 + 7*a^6 + a^5 + 8*a^4 + 18*a^2 + 27*a + 9)*29 + (20*a^10 + 3*a^9 + 17*a^8 + 25*a^7 + 11*a^6 + a^5 + 12*a^4 + 16*a^3 + 28*a^2 + 8*a + 7)*29^2 + (16*a^10 + 9*a^9 + 19*a^8 + 3*a^7 + 25*a^6 + 2*a^5 + 27*a^4 + 20*a^3 + 6*a^2 + 12*a + 14)*29^3 + (5*a^10 + 26*a^9 + 8*a^8 + 22*a^7 + 21*a^6 + 6*a^5 + 22*a^4 + 4*a^3 + 12*a^2 + 26*a + 22)*29^4+O(29^5) $r_{ 13 }$ $=$ $$20 a^{10} + 11 a^{9} + 7 a^{8} + 23 a^{7} + 5 a^{6} + 16 a^{5} + 14 a^{3} + 26 a^{2} + 7 a + 9 + \left(10 a^{10} + 9 a^{9} + 28 a^{7} + 2 a^{6} + 13 a^{5} + 24 a^{4} + 13 a^{3} + 24 a^{2} + 25 a + 22\right)\cdot 29 + \left(27 a^{10} + 20 a^{9} + 14 a^{8} + 13 a^{7} + 20 a^{6} + 20 a^{5} + 23 a^{4} + 14 a^{3} + 5 a^{2} + 14 a + 6\right)\cdot 29^{2} + \left(10 a^{10} + 3 a^{9} + 3 a^{8} + 15 a^{7} + 5 a^{6} + 21 a^{5} + 10 a^{4} + 12 a^{3} + 18 a^{2} + 5 a + 22\right)\cdot 29^{3} + \left(8 a^{10} + 6 a^{9} + 27 a^{8} + 14 a^{7} + 6 a^{6} + 14 a^{5} + 28 a^{4} + 2 a^{3} + 21 a^{2} + 3 a + 12\right)\cdot 29^{4} +O(29^{5})$$ 20*a^10 + 11*a^9 + 7*a^8 + 23*a^7 + 5*a^6 + 16*a^5 + 14*a^3 + 26*a^2 + 7*a + 9 + (10*a^10 + 9*a^9 + 28*a^7 + 2*a^6 + 13*a^5 + 24*a^4 + 13*a^3 + 24*a^2 + 25*a + 22)*29 + (27*a^10 + 20*a^9 + 14*a^8 + 13*a^7 + 20*a^6 + 20*a^5 + 23*a^4 + 14*a^3 + 5*a^2 + 14*a + 6)*29^2 + (10*a^10 + 3*a^9 + 3*a^8 + 15*a^7 + 5*a^6 + 21*a^5 + 10*a^4 + 12*a^3 + 18*a^2 + 5*a + 22)*29^3 + (8*a^10 + 6*a^9 + 27*a^8 + 14*a^7 + 6*a^6 + 14*a^5 + 28*a^4 + 2*a^3 + 21*a^2 + 3*a + 12)*29^4+O(29^5) $r_{ 14 }$ $=$ $$20 a^{10} + 20 a^{9} + 18 a^{8} + 21 a^{7} + 5 a^{6} + 15 a^{5} + 3 a^{4} + 16 a^{3} + 27 a^{2} + a + 22 + \left(24 a^{10} + 7 a^{9} + 24 a^{8} + 16 a^{7} + 4 a^{6} + 21 a^{5} + 20 a^{4} + 3 a^{3} + 28 a^{2} + 18 a + 11\right)\cdot 29 + \left(5 a^{10} + 21 a^{9} + 2 a^{8} + 5 a^{7} + 19 a^{6} + 22 a^{5} + 2 a^{4} + 8 a^{3} + a^{2} + 14\right)\cdot 29^{2} + \left(19 a^{10} + 28 a^{9} + 18 a^{8} + 17 a^{7} + 5 a^{6} + 14 a^{5} + 16 a^{4} + 7 a^{3} + 13 a^{2} + 3 a + 1\right)\cdot 29^{3} + \left(18 a^{10} + 5 a^{9} + 23 a^{8} + 14 a^{7} + 23 a^{6} + 22 a^{5} + 6 a^{4} + 28 a^{3} + 24 a^{2} + 8\right)\cdot 29^{4} +O(29^{5})$$ 20*a^10 + 20*a^9 + 18*a^8 + 21*a^7 + 5*a^6 + 15*a^5 + 3*a^4 + 16*a^3 + 27*a^2 + a + 22 + (24*a^10 + 7*a^9 + 24*a^8 + 16*a^7 + 4*a^6 + 21*a^5 + 20*a^4 + 3*a^3 + 28*a^2 + 18*a + 11)*29 + (5*a^10 + 21*a^9 + 2*a^8 + 5*a^7 + 19*a^6 + 22*a^5 + 2*a^4 + 8*a^3 + a^2 + 14)*29^2 + (19*a^10 + 28*a^9 + 18*a^8 + 17*a^7 + 5*a^6 + 14*a^5 + 16*a^4 + 7*a^3 + 13*a^2 + 3*a + 1)*29^3 + (18*a^10 + 5*a^9 + 23*a^8 + 14*a^7 + 23*a^6 + 22*a^5 + 6*a^4 + 28*a^3 + 24*a^2 + 8)*29^4+O(29^5) $r_{ 15 }$ $=$ $$22 a^{10} + 22 a^{9} + 9 a^{8} + 13 a^{7} + 20 a^{5} + 26 a^{4} + 23 a^{3} + 23 a^{2} + 17 a + 4 + \left(26 a^{10} + 11 a^{9} + 25 a^{8} + 7 a^{7} + 9 a^{6} + 10 a^{5} + 22 a^{4} + 28 a^{3} + 11 a + 9\right)\cdot 29 + \left(17 a^{10} + a^{9} + 18 a^{8} + 17 a^{7} + 13 a^{6} + 9 a^{5} + 24 a^{4} + 8 a^{3} + 7 a^{2} + 14 a\right)\cdot 29^{2} + \left(20 a^{10} + 26 a^{9} + 12 a^{8} + 23 a^{7} + 12 a^{6} + 6 a^{5} + 2 a^{4} + 7 a^{3} + 2 a^{2} + 9 a + 14\right)\cdot 29^{3} + \left(11 a^{10} + 22 a^{9} + a^{8} + 23 a^{7} + 25 a^{6} + 19 a^{5} + 26 a^{4} + 21 a^{3} + 9 a^{2} + 13 a + 20\right)\cdot 29^{4} +O(29^{5})$$ 22*a^10 + 22*a^9 + 9*a^8 + 13*a^7 + 20*a^5 + 26*a^4 + 23*a^3 + 23*a^2 + 17*a + 4 + (26*a^10 + 11*a^9 + 25*a^8 + 7*a^7 + 9*a^6 + 10*a^5 + 22*a^4 + 28*a^3 + 11*a + 9)*29 + (17*a^10 + a^9 + 18*a^8 + 17*a^7 + 13*a^6 + 9*a^5 + 24*a^4 + 8*a^3 + 7*a^2 + 14*a)*29^2 + (20*a^10 + 26*a^9 + 12*a^8 + 23*a^7 + 12*a^6 + 6*a^5 + 2*a^4 + 7*a^3 + 2*a^2 + 9*a + 14)*29^3 + (11*a^10 + 22*a^9 + a^8 + 23*a^7 + 25*a^6 + 19*a^5 + 26*a^4 + 21*a^3 + 9*a^2 + 13*a + 20)*29^4+O(29^5) $r_{ 16 }$ $=$ $$23 a^{10} + 24 a^{9} + 5 a^{8} + 23 a^{7} + 20 a^{6} + 15 a^{5} + 12 a^{4} + 16 a^{3} + 9 a^{2} + 18 a + 7 + \left(25 a^{10} + 20 a^{9} + 9 a^{8} + 25 a^{7} + 12 a^{6} + 18 a^{5} + 13 a^{4} + 4 a^{3} + 14 a^{2} + 15 a + 4\right)\cdot 29 + \left(19 a^{9} + 6 a^{8} + 26 a^{7} + a^{6} + 26 a^{5} + 9 a^{4} + 25 a^{3} + 21 a^{2} + 27 a + 19\right)\cdot 29^{2} + \left(17 a^{10} + 26 a^{9} + 16 a^{8} + 22 a^{7} + 6 a^{6} + 10 a^{4} + 7 a^{3} + 26 a^{2} + 28 a + 7\right)\cdot 29^{3} + \left(15 a^{10} + 24 a^{9} + 2 a^{8} + 3 a^{7} + 3 a^{6} + 3 a^{5} + 23 a^{4} + 10 a^{3} + 7 a^{2} + 25 a + 13\right)\cdot 29^{4} +O(29^{5})$$ 23*a^10 + 24*a^9 + 5*a^8 + 23*a^7 + 20*a^6 + 15*a^5 + 12*a^4 + 16*a^3 + 9*a^2 + 18*a + 7 + (25*a^10 + 20*a^9 + 9*a^8 + 25*a^7 + 12*a^6 + 18*a^5 + 13*a^4 + 4*a^3 + 14*a^2 + 15*a + 4)*29 + (19*a^9 + 6*a^8 + 26*a^7 + a^6 + 26*a^5 + 9*a^4 + 25*a^3 + 21*a^2 + 27*a + 19)*29^2 + (17*a^10 + 26*a^9 + 16*a^8 + 22*a^7 + 6*a^6 + 10*a^4 + 7*a^3 + 26*a^2 + 28*a + 7)*29^3 + (15*a^10 + 24*a^9 + 2*a^8 + 3*a^7 + 3*a^6 + 3*a^5 + 23*a^4 + 10*a^3 + 7*a^2 + 25*a + 13)*29^4+O(29^5) $r_{ 17 }$ $=$ $$25 a^{10} + 3 a^{9} + a^{8} + 21 a^{7} + 5 a^{6} + 11 a^{5} + 9 a^{4} + 26 a^{3} + 21 a^{2} + 7 a + 15 + \left(17 a^{10} + 9 a^{9} + 9 a^{8} + 28 a^{7} + 9 a^{6} + 10 a^{5} + 22 a^{4} + 5 a^{3} + 27 a^{2} + 7\right)\cdot 29 + \left(19 a^{10} + a^{9} + 10 a^{8} + 25 a^{6} + 9 a^{5} + 3 a^{4} + 10 a^{3} + 20 a^{2} + 26 a + 5\right)\cdot 29^{2} + \left(12 a^{10} + 14 a^{9} + 3 a^{8} + 4 a^{7} + 26 a^{6} + 6 a^{5} + 11 a^{4} + 3 a^{3} + 17 a^{2} + 19 a + 8\right)\cdot 29^{3} + \left(5 a^{10} + a^{9} + 2 a^{8} + a^{7} + 6 a^{6} + 25 a^{5} + 22 a^{3} + 14 a^{2} + 3 a + 6\right)\cdot 29^{4} +O(29^{5})$$ 25*a^10 + 3*a^9 + a^8 + 21*a^7 + 5*a^6 + 11*a^5 + 9*a^4 + 26*a^3 + 21*a^2 + 7*a + 15 + (17*a^10 + 9*a^9 + 9*a^8 + 28*a^7 + 9*a^6 + 10*a^5 + 22*a^4 + 5*a^3 + 27*a^2 + 7)*29 + (19*a^10 + a^9 + 10*a^8 + 25*a^6 + 9*a^5 + 3*a^4 + 10*a^3 + 20*a^2 + 26*a + 5)*29^2 + (12*a^10 + 14*a^9 + 3*a^8 + 4*a^7 + 26*a^6 + 6*a^5 + 11*a^4 + 3*a^3 + 17*a^2 + 19*a + 8)*29^3 + (5*a^10 + a^9 + 2*a^8 + a^7 + 6*a^6 + 25*a^5 + 22*a^3 + 14*a^2 + 3*a + 6)*29^4+O(29^5) $r_{ 18 }$ $=$ $$25 a^{10} + 8 a^{9} + a^{8} + 12 a^{7} + 10 a^{6} + a^{5} + 10 a^{4} + 20 a^{3} + 13 a^{2} + 22 a + 26 + \left(9 a^{10} + 16 a^{9} + 22 a^{8} + 4 a^{7} + 16 a^{6} + 10 a^{5} + 20 a^{4} + 15 a^{3} + 18 a^{2} + 11 a + 18\right)\cdot 29 + \left(13 a^{10} + 2 a^{9} + 14 a^{8} + 10 a^{7} + 5 a^{6} + 17 a^{5} + a^{4} + 15 a^{3} + 19 a^{2} + 24 a + 22\right)\cdot 29^{2} + \left(11 a^{10} + 11 a^{9} + 23 a^{8} + 16 a^{7} + 26 a^{6} + 21 a^{5} + 9 a^{4} + 19 a^{3} + a^{2} + 21 a + 20\right)\cdot 29^{3} + \left(26 a^{9} + 14 a^{8} + 10 a^{7} + a^{6} + 12 a^{5} + 19 a^{4} + 6 a^{3} + 12 a^{2} + 21 a + 20\right)\cdot 29^{4} +O(29^{5})$$ 25*a^10 + 8*a^9 + a^8 + 12*a^7 + 10*a^6 + a^5 + 10*a^4 + 20*a^3 + 13*a^2 + 22*a + 26 + (9*a^10 + 16*a^9 + 22*a^8 + 4*a^7 + 16*a^6 + 10*a^5 + 20*a^4 + 15*a^3 + 18*a^2 + 11*a + 18)*29 + (13*a^10 + 2*a^9 + 14*a^8 + 10*a^7 + 5*a^6 + 17*a^5 + a^4 + 15*a^3 + 19*a^2 + 24*a + 22)*29^2 + (11*a^10 + 11*a^9 + 23*a^8 + 16*a^7 + 26*a^6 + 21*a^5 + 9*a^4 + 19*a^3 + a^2 + 21*a + 20)*29^3 + (26*a^9 + 14*a^8 + 10*a^7 + a^6 + 12*a^5 + 19*a^4 + 6*a^3 + 12*a^2 + 21*a + 20)*29^4+O(29^5) $r_{ 19 }$ $=$ $$25 a^{10} + 20 a^{9} + 11 a^{8} + 6 a^{7} + 19 a^{6} + 24 a^{5} + 11 a^{4} + 5 a^{3} + 17 a^{2} + 23 a + 9 + \left(27 a^{10} + 19 a^{9} + 5 a^{8} + 12 a^{7} + 14 a^{6} + 16 a^{5} + 27 a^{4} + 5 a^{3} + 19 a^{2} + 13 a + 14\right)\cdot 29 + \left(14 a^{10} + 5 a^{9} + 28 a^{8} + 24 a^{7} + 24 a^{6} + 15 a^{5} + 18 a^{4} + 5 a^{3} + 3 a^{2} + 24 a + 5\right)\cdot 29^{2} + \left(28 a^{10} + 24 a^{9} + 19 a^{8} + 27 a^{7} + 7 a^{6} + 12 a^{5} + 2 a^{4} + 28 a^{3} + 25 a^{2} + 22 a + 13\right)\cdot 29^{3} + \left(7 a^{10} + 10 a^{9} + 15 a^{8} + 11 a^{7} + 3 a^{6} + 22 a^{5} + 11 a^{4} + 26 a^{2} + 8 a + 25\right)\cdot 29^{4} +O(29^{5})$$ 25*a^10 + 20*a^9 + 11*a^8 + 6*a^7 + 19*a^6 + 24*a^5 + 11*a^4 + 5*a^3 + 17*a^2 + 23*a + 9 + (27*a^10 + 19*a^9 + 5*a^8 + 12*a^7 + 14*a^6 + 16*a^5 + 27*a^4 + 5*a^3 + 19*a^2 + 13*a + 14)*29 + (14*a^10 + 5*a^9 + 28*a^8 + 24*a^7 + 24*a^6 + 15*a^5 + 18*a^4 + 5*a^3 + 3*a^2 + 24*a + 5)*29^2 + (28*a^10 + 24*a^9 + 19*a^8 + 27*a^7 + 7*a^6 + 12*a^5 + 2*a^4 + 28*a^3 + 25*a^2 + 22*a + 13)*29^3 + (7*a^10 + 10*a^9 + 15*a^8 + 11*a^7 + 3*a^6 + 22*a^5 + 11*a^4 + 26*a^2 + 8*a + 25)*29^4+O(29^5) $r_{ 20 }$ $=$ $$26 a^{10} + 5 a^{8} + 25 a^{7} + 14 a^{6} + 14 a^{5} + 21 a^{4} + 19 a^{3} + 8 a^{2} + 10 a + 24 + \left(23 a^{10} + 3 a^{9} + 12 a^{8} + 3 a^{7} + 19 a^{6} + 22 a^{5} + 7 a^{4} + 17 a^{3} + 5 a^{2} + 6 a + 19\right)\cdot 29 + \left(17 a^{10} + 8 a^{9} + 28 a^{8} + 11 a^{7} + 2 a^{6} + 25 a^{5} + 23 a^{4} + 9 a^{3} + 15 a^{2} + 17 a + 26\right)\cdot 29^{2} + \left(28 a^{10} + 24 a^{9} + 10 a^{8} + 23 a^{7} + 14 a^{6} + 22 a^{5} + 12 a^{4} + 15 a^{3} + 25 a^{2} + 18 a + 25\right)\cdot 29^{3} + \left(15 a^{10} + 23 a^{9} + 20 a^{8} + 22 a^{7} + 28 a^{6} + a^{5} + 17 a^{4} + 10 a^{3} + 17 a^{2} + 22 a + 9\right)\cdot 29^{4} +O(29^{5})$$ 26*a^10 + 5*a^8 + 25*a^7 + 14*a^6 + 14*a^5 + 21*a^4 + 19*a^3 + 8*a^2 + 10*a + 24 + (23*a^10 + 3*a^9 + 12*a^8 + 3*a^7 + 19*a^6 + 22*a^5 + 7*a^4 + 17*a^3 + 5*a^2 + 6*a + 19)*29 + (17*a^10 + 8*a^9 + 28*a^8 + 11*a^7 + 2*a^6 + 25*a^5 + 23*a^4 + 9*a^3 + 15*a^2 + 17*a + 26)*29^2 + (28*a^10 + 24*a^9 + 10*a^8 + 23*a^7 + 14*a^6 + 22*a^5 + 12*a^4 + 15*a^3 + 25*a^2 + 18*a + 25)*29^3 + (15*a^10 + 23*a^9 + 20*a^8 + 22*a^7 + 28*a^6 + a^5 + 17*a^4 + 10*a^3 + 17*a^2 + 22*a + 9)*29^4+O(29^5) $r_{ 21 }$ $=$ $$26 a^{10} + 16 a^{9} + 3 a^{8} + 13 a^{7} + 14 a^{6} + 20 a^{5} + 21 a^{4} + 22 a^{3} + 17 a^{2} + 18 a + 3 + \left(20 a^{8} + 14 a^{7} + 18 a^{6} + 11 a^{5} + 27 a^{4} + 7 a^{3} + 25 a^{2} + 11 a + 2\right)\cdot 29 + \left(14 a^{10} + 21 a^{9} + 24 a^{8} + 18 a^{7} + 21 a^{6} + a^{5} + 19 a^{4} + 14 a^{3} + 7 a^{2} + 19 a + 26\right)\cdot 29^{2} + \left(22 a^{10} + 24 a^{9} + 7 a^{8} + 3 a^{7} + 28 a^{6} + 20 a^{5} + 28 a^{4} + 15 a^{3} + 28 a^{2} + 6 a + 17\right)\cdot 29^{3} + \left(23 a^{10} + 25 a^{9} + 20 a^{8} + 15 a^{7} + 7 a^{6} + 18 a^{5} + a^{4} + 14 a^{3} + 15 a^{2} + 10 a + 4\right)\cdot 29^{4} +O(29^{5})$$ 26*a^10 + 16*a^9 + 3*a^8 + 13*a^7 + 14*a^6 + 20*a^5 + 21*a^4 + 22*a^3 + 17*a^2 + 18*a + 3 + (20*a^8 + 14*a^7 + 18*a^6 + 11*a^5 + 27*a^4 + 7*a^3 + 25*a^2 + 11*a + 2)*29 + (14*a^10 + 21*a^9 + 24*a^8 + 18*a^7 + 21*a^6 + a^5 + 19*a^4 + 14*a^3 + 7*a^2 + 19*a + 26)*29^2 + (22*a^10 + 24*a^9 + 7*a^8 + 3*a^7 + 28*a^6 + 20*a^5 + 28*a^4 + 15*a^3 + 28*a^2 + 6*a + 17)*29^3 + (23*a^10 + 25*a^9 + 20*a^8 + 15*a^7 + 7*a^6 + 18*a^5 + a^4 + 14*a^3 + 15*a^2 + 10*a + 4)*29^4+O(29^5) $r_{ 22 }$ $=$ $$28 a^{10} + 6 a^{9} + 8 a^{8} + 8 a^{7} + 28 a^{6} + 7 a^{5} + 2 a^{4} + 18 a^{3} + 27 a^{2} + 20 a + 2 + \left(6 a^{10} + 27 a^{9} + 18 a^{7} + 6 a^{6} + 8 a^{5} + 25 a^{4} + 15 a^{3} + 21 a^{2} + 5 a + 3\right)\cdot 29 + \left(22 a^{10} + 2 a^{9} + 22 a^{8} + 2 a^{7} + 15 a^{6} + 19 a^{5} + 13 a^{4} + 18 a^{3} + 23 a^{2} + 2 a + 1\right)\cdot 29^{2} + \left(22 a^{10} + 16 a^{9} + 2 a^{8} + 22 a^{7} + 3 a^{6} + 20 a^{5} + 17 a^{4} + 23 a^{2} + 2 a + 28\right)\cdot 29^{3} + \left(12 a^{10} + 22 a^{9} + 19 a^{8} + 13 a^{7} + 22 a^{6} + 18 a^{5} + 20 a^{4} + 3 a^{3} + 10 a^{2} + 7 a + 9\right)\cdot 29^{4} +O(29^{5})$$ 28*a^10 + 6*a^9 + 8*a^8 + 8*a^7 + 28*a^6 + 7*a^5 + 2*a^4 + 18*a^3 + 27*a^2 + 20*a + 2 + (6*a^10 + 27*a^9 + 18*a^7 + 6*a^6 + 8*a^5 + 25*a^4 + 15*a^3 + 21*a^2 + 5*a + 3)*29 + (22*a^10 + 2*a^9 + 22*a^8 + 2*a^7 + 15*a^6 + 19*a^5 + 13*a^4 + 18*a^3 + 23*a^2 + 2*a + 1)*29^2 + (22*a^10 + 16*a^9 + 2*a^8 + 22*a^7 + 3*a^6 + 20*a^5 + 17*a^4 + 23*a^2 + 2*a + 28)*29^3 + (12*a^10 + 22*a^9 + 19*a^8 + 13*a^7 + 22*a^6 + 18*a^5 + 20*a^4 + 3*a^3 + 10*a^2 + 7*a + 9)*29^4+O(29^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 22 }$

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 22 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,21)(2,6)(3,12)(4,20)(5,9)(7,17)(8,19)(10,16)(11,14)(13,18)(15,22)$ $-1$ $1$ $11$ $(1,9,11,2,15,16,13,12,4,19,17)(3,20,8,7,21,5,14,6,22,10,18)$ $\zeta_{11}^{6}$ $1$ $11$ $(1,11,15,13,4,17,9,2,16,12,19)(3,8,21,14,22,18,20,7,5,6,10)$ $\zeta_{11}$ $1$ $11$ $(1,2,13,19,9,15,12,17,11,16,4)(3,7,14,10,20,21,6,18,8,5,22)$ $\zeta_{11}^{7}$ $1$ $11$ $(1,15,4,9,16,19,11,13,17,2,12)(3,21,22,20,5,10,8,14,18,7,6)$ $\zeta_{11}^{2}$ $1$ $11$ $(1,16,17,15,19,2,4,11,12,9,13)(3,5,18,21,10,7,22,8,6,20,14)$ $\zeta_{11}^{8}$ $1$ $11$ $(1,13,9,12,11,4,2,19,15,17,16)(3,14,20,6,8,22,7,10,21,18,5)$ $\zeta_{11}^{3}$ $1$ $11$ $(1,12,2,17,13,11,19,16,9,4,15)(3,6,7,18,14,8,10,5,20,22,21)$ $\zeta_{11}^{9}$ $1$ $11$ $(1,4,16,11,17,12,15,9,19,13,2)(3,22,5,8,18,6,21,20,10,14,7)$ $\zeta_{11}^{4}$ $1$ $11$ $(1,19,12,16,2,9,17,4,13,15,11)(3,10,6,5,7,20,18,22,14,21,8)$ $-\zeta_{11}^{9} - \zeta_{11}^{8} - \zeta_{11}^{7} - \zeta_{11}^{6} - \zeta_{11}^{5} - \zeta_{11}^{4} - \zeta_{11}^{3} - \zeta_{11}^{2} - \zeta_{11} - 1$ $1$ $11$ $(1,17,19,4,12,13,16,15,2,11,9)(3,18,10,22,6,14,5,21,7,8,20)$ $\zeta_{11}^{5}$ $1$ $22$ $(1,18,9,3,11,20,2,8,15,7,16,21,13,5,12,14,4,6,19,22,17,10)$ $-\zeta_{11}^{3}$ $1$ $22$ $(1,3,2,7,13,14,19,10,9,20,15,21,12,6,17,18,11,8,16,5,4,22)$ $-\zeta_{11}^{9}$ $1$ $22$ $(1,20,16,14,17,3,15,5,19,18,2,21,4,10,11,7,12,22,9,8,13,6)$ $-\zeta_{11}^{4}$ $1$ $22$ $(1,8,12,10,2,5,17,20,13,22,11,21,19,3,16,6,9,7,4,18,15,14)$ $\zeta_{11}^{9} + \zeta_{11}^{8} + \zeta_{11}^{7} + \zeta_{11}^{6} + \zeta_{11}^{5} + \zeta_{11}^{4} + \zeta_{11}^{3} + \zeta_{11}^{2} + \zeta_{11} + 1$ $1$ $22$ $(1,7,19,20,12,18,16,22,2,14,9,21,17,8,4,3,13,10,15,6,11,5)$ $-\zeta_{11}^{5}$ $1$ $22$ $(1,5,11,6,15,10,13,3,4,8,17,21,9,14,2,22,16,18,12,20,19,7)$ $-\zeta_{11}^{6}$ $1$ $22$ $(1,14,15,18,4,7,9,6,16,3,19,21,11,22,13,20,17,5,2,10,12,8)$ $-\zeta_{11}$ $1$ $22$ $(1,6,13,8,9,22,12,7,11,10,4,21,2,18,19,5,15,3,17,14,16,20)$ $-\zeta_{11}^{7}$ $1$ $22$ $(1,22,4,5,16,8,11,18,17,6,12,21,15,20,9,10,19,14,13,7,2,3)$ $-\zeta_{11}^{2}$ $1$ $22$ $(1,10,17,22,19,6,4,14,12,5,13,21,16,7,15,8,2,20,11,3,9,18)$ $-\zeta_{11}^{8}$

The blue line marks the conjugacy class containing complex conjugation.