diff --git a/special_element.c b/special_element.c
new file mode 100644
index 0000000..f88a1a8
--- /dev/null
+++ b/special_element.c
@@ -0,0 +1,277 @@
+#include "coxeter.h"
+#include "linalg.h"
+#include "mat.h"
+
+#include <gsl/gsl_poly.h>
+#include <mps/mps.h>
+
+#define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0);
+#define DEBUG(msg, ...)
+
+int solve_characteristic_polynomial(mps_context *solv, mpq_t tr, mpq_t trinv, double *eigenvalues)
+{
+	mpq_t coeff1, coeff2, zero;
+	cplx_t *roots;
+	double radii[3];
+	double *radii_p[3];
+	mps_monomial_poly *poly;
+	mps_boolean errors;
+	int result = 0;
+
+	mpq_inits(coeff1, coeff2, zero, NULL);
+	mpq_set(coeff1, trinv);
+	mpq_sub(coeff2, zero, tr);
+
+	poly = mps_monomial_poly_new(solv, 3);
+	mps_monomial_poly_set_coefficient_int(solv, poly, 0, -1, 0);
+	mps_monomial_poly_set_coefficient_q(solv, poly, 1, coeff1, zero);
+	mps_monomial_poly_set_coefficient_q(solv, poly, 2, coeff2, zero);
+	mps_monomial_poly_set_coefficient_int(solv, poly, 3, 1, 0);
+
+	mps_context_set_input_poly(solv, (mps_polynomial*)poly);
+	mps_mpsolve(solv);
+
+	roots = cplx_valloc(3);
+	for(int i = 0; i < 3; i++)
+		radii_p[i] = &(radii[i]);
+	mps_context_get_roots_d(solv, &roots, radii_p);
+	errors = mps_context_has_errors(solv);
+
+	if(errors) {
+		result = 1;
+	} else {
+		for(int i = 0; i < 3; i++) {
+			eigenvalues[i] = cplx_Re(roots[i]);
+			if(fabs(cplx_Im(roots[i])) > radii[i]) // non-real root
+				result = 2;
+		}
+	}
+
+	cplx_vfree(roots);
+	mpq_clears(coeff1, coeff2, zero, NULL);
+
+	return result;
+}
+
+// this version is only for the (4,4,4) group
+void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t m, mpq_t t)
+{
+	mpq_t s,sinv,q,x,y;
+	mpq_t zero, one, two;
+	mpq_t tmp;
+
+	mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
+	mpq_set_ui(zero, 0, 1);
+	mpq_set_ui(one, 1, 1);
+	mpq_set_ui(two, 2, 1);
+
+	// s = (1-m^2)/2m
+	mpq_mul(s, m, m);
+	mpq_sub(s, one, s);
+	mpq_div(s, s, m);
+	mpq_div(s, s, two);
+	mpq_div(sinv, one, s);
+
+	// q = (1+m^2)/(1-m^2) = 2/(1-m^2) - 1
+	mpq_mul(q, m, m);
+	mpq_sub(q, one, q);
+	mpq_div(q, two, q);
+	mpq_sub(q, q, one);
+
+	// x = -tq, y = -q/t
+	mpq_mul(x, q, t);
+	mpq_sub(x, zero, x);
+	mpq_div(y, q, t);
+	mpq_sub(y, zero, y);
+
+	// q^2 = xy = 1 + 1/s^2
+	// [         -s         s*y           0]
+	// [       -s*x s*x*y - 1/s           0]
+	// [       -s*y   s*y^2 - x           1]
+	LOOP(i,3) {
+		mat_zero(gen[i]);
+		mpq_sub(tmp, zero, s);
+		mat_set(gen[i%3], i%3, i%3, tmp);
+		mpq_mul(tmp, s, y);
+		mat_set(gen[i%3], i%3, (i+1)%3, tmp);
+		mpq_mul(tmp, s, x);
+		mpq_sub(tmp, zero, tmp);
+		mat_set(gen[i%3], (i+1)%3, i%3, tmp);
+		mpq_mul(tmp, s, x);
+		mpq_mul(tmp, tmp, y);
+		mpq_sub(tmp, tmp, sinv);
+		mat_set(gen[i%3], (i+1)%3, (i+1)%3, tmp);
+		mpq_mul(tmp, s, y);
+		mpq_sub(tmp, zero, tmp);
+		mat_set(gen[i%3], (i+2)%3, i%3, tmp);
+		mpq_mul(tmp, s, y);
+		mpq_mul(tmp, tmp, y);
+		mpq_sub(tmp, tmp, x);
+		mat_set(gen[i%3], (i+2)%3, (i+1)%3, tmp);
+		mat_set(gen[i%3], (i+2)%3, (i+2)%3, one);
+	}
+
+	LOOP(i,3) mat_pseudoinverse(ws, gen[i+3], gen[i]);
+
+	mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
+}
+
+char *print_word(groupelement_t *g, char *str)
+{
+  int i = g->length - 1;
+
+  str[g->length] = 0;
+  while(g->parent) {
+    str[i--] = 'a' + g->letter;
+    g = g->parent;
+  }
+
+  return str;
+}
+
+void enumerate(group_t *group, mat *matrices, mpq_t m, mpq_t t)
+{
+	mat_workspace *ws;
+	mat tmp;
+	mat gen[6];
+	char buf[100], buf2[100], buf3[100];
+
+	// allocate stuff
+	ws = mat_workspace_init(3);
+	for(int i = 0; i < 6; i++)
+		mat_init(gen[i], 3);
+	mat_init(tmp, 3);
+
+	initialize_triangle_generators(ws, gen, m, t);
+
+	mat_identity(matrices[0]);
+	for(int i = 1; i < group->size; i++) {
+		if(group->elements[i].length % 2 != 0)
+			continue;
+		if(!group->elements[i].inverse)
+			continue;
+
+		int parent = group->elements[i].parent->id;
+		int grandparent = group->elements[i].parent->parent->id;
+		int letter;
+
+		if(group->elements[parent].letter == 1 && group->elements[i].letter == 2)
+			letter = 0; // p = bc
+		else if(group->elements[parent].letter == 2 && group->elements[i].letter == 0)
+			letter = 1; // q = ca
+		else if(group->elements[parent].letter == 0 && group->elements[i].letter == 1)
+			letter = 2; // r = ab
+		if(group->elements[parent].letter == 2 && group->elements[i].letter == 1)
+			letter = 3; // p^{-1} = cb
+		else if(group->elements[parent].letter == 0 && group->elements[i].letter == 2)
+			letter = 4; // q^{-1} = ac
+		else if(group->elements[parent].letter == 1 && group->elements[i].letter == 0)
+			letter = 5; // r^{-1} = ba
+
+		mat_multiply(ws, matrices[i], matrices[grandparent], gen[letter]);
+	}
+
+	// free stuff
+	for(int i = 0; i < 6; i++)
+		mat_clear(gen[i]);
+	mat_clear(tmp);
+	mat_workspace_clear(ws);
+}
+
+int main(int argc, char *argv[])
+{
+	mpq_t m, t, tmp;
+	double s;
+	mat_workspace *ws;
+	mat gen[6];
+	mps_context *solver;
+	mat element, inverse;
+	int letter1, letter2, letter;
+	mpq_t tr, trinv;
+	double x, y;
+	int retval;
+	double evs[3];
+	char buf[100];
+
+	DEBUG("Allocate\n");
+
+	mpq_inits(m, t, tmp, tr, trinv, NULL);
+	ws = mat_workspace_init(3);
+	for(int i = 0; i < 6; i++)
+		mat_init(gen[i], 3);
+	mat_init(element, 3);
+	mat_init(inverse, 3);
+
+	solver = mps_context_new();
+	mps_context_set_output_prec(solver, 20); // relative precision
+	mps_context_set_output_goal(solver, MPS_OUTPUT_GOAL_APPROXIMATE);
+
+	for(int i = 1; i <= 99; i++) {
+		for(int j = 1; j <= 100; j++) {
+			mpq_set_ui(t, j, 100);
+			mpq_set_ui(m, i, 100); // 414/1000 ~ sqrt(2)-1 <-> s=1
+			s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
+
+			DEBUG("Compute matrix\n");
+			initialize_triangle_generators(ws, gen, m, t);
+
+			mat_identity(element);
+			mat_identity(inverse);
+			for(int i = 0; i < strlen(argv[1]); i+=2) {
+				letter1 = argv[1][i] - 'a';
+				letter2 = argv[1][i+1] - 'a';
+
+				if(letter1 == 1 && letter2 == 2)
+					letter = 0; // p = bc
+				else if(letter1 == 2 && letter2 == 0)
+					letter = 1; // q = ca
+				else if(letter1 == 0 && letter2 == 1)
+					letter = 2; // r = ab
+				else if(letter1 == 2 && letter2 == 1)
+					letter = 3; // p^{-1} = cb
+				else if(letter1 == 0 && letter2 == 2)
+					letter = 4; // q^{-1} = ac
+				else if(letter1 == 1 && letter2 == 0)
+					letter = 5; // r^{-1} = ba
+
+				mat_multiply(ws, element, element, gen[letter]);
+				mat_multiply(ws, inverse, gen[(letter+3)%6], inverse);
+			}
+
+			DEBUG("Compute traces\n");
+
+			mat_trace(tr, element);
+			mat_trace(trinv, inverse);
+
+			DEBUG("Solve characteristic polynomials\n");
+			retval = solve_characteristic_polynomial(solver, tr, trinv, evs);
+			if(retval == 1) {
+				fprintf(stderr, "Error! Could not solve polynomial.\n");
+				return 1;
+			}
+
+			if(fabs(evs[0]) < fabs(evs[1]))
+				SWAP(double, evs[0], evs[1]);
+			if(fabs(evs[1]) < fabs(evs[2]))
+				SWAP(double, evs[1], evs[2]);
+			if(fabs(evs[0]) < fabs(evs[1]))
+				SWAP(double, evs[0], evs[1]);
+
+			x = log(fabs(evs[0]));
+			y = -log(fabs(evs[2]));
+
+//	gmp_printf("%Qd %Qd %f %f %f\n", tr, trinv, x, y, y/x);
+			gmp_printf("%.5f %.5f %.5f %.5f\n", mpq_get_d(t), mpq_get_d(m), s, y/x);
+
+		}
+	}
+
+	DEBUG("Clean up\n");
+	mpq_clears(m, t, tmp, tr, trinv, NULL);
+	mat_workspace_clear(ws);
+	for(int i = 0; i < 6; i++)
+		mat_clear(gen[i]);
+	mat_clear(element);
+	mat_clear(inverse);
+	mps_context_free(solver);
+}