1 | initial version |

**Short answer:**

```
sage: Gamma(1).random_element()
[-13 40]
[-27 83]
```

**Some comments:**

Never walk through the long corridors of the department where people are studying modular forms and elliptic curves with a visible book on *numerical approximation*. (They will immediately notice it, and kindly help you find the right department. It is so important to keep things clean and structural!)

So why use that `CDF`

to make things uncertainly complicated?

The generators are living in a beautiful ring, `ZZ`

, and here is the best place to make computations. Moreover, the corresponding group is the modular group `SL2Z`

. Mention it!

Let us type some piece of code to investigate the landscape. First of all let us consider the matrices:

```
L = SL2Z( [1,1,0,1] )
R = SL2Z( [1,0,1,1] )
L, R
```

This gives:

```
(
[1 1] [1 0]
[0 1], [1 1]
)
```

The posted generators correspond to:

```
sage: R^(-1), L
(
[ 1 0] [1 1]
[-1 1], [0 1]
)
```

Do $L,R$ generate indeed the full modular group `Gamma(1)`

or `SL2Z`

? We only have to generate the (standard) generators of this group using $L,R$. For this:

```
from sage.modular.arithgroup.arithgroup_perm import eval_sl2z_word, sl2z_word_problem
S, T = SL2Z.gens()
S, T
wT = sl2z_word_problem( T )
wS = sl2z_word_problem( S )
```

This gives `wT`

and `wS`

and the representations:

```
sage: wT
[(0, 1)]
sage: wS
[(0, 1), (1, -1), (0, 1), (1, -1), (0, 1), (1, -1), (0, 1), (1, -1), (0, 1)]
sage: T == L
True
sage: S == (L*R^-1)^4*L
True
```

The `0`

stays on the first place of the tuples in the list for the usage of `L`

, the `1`

for `R`

. So `(1, -1)`

means one should use the `R`

, namely to the power `-1`

. The evaluation does the job directly:

```
sage: T == eval_sl2z_word( wT )
True
sage: S == eval_sl2z_word( wS )
True
```

The random element generated above has the representation as follows...

```
sage: Z = SL2Z( [ -13, 40, -27, 83 ] )
sage: wZ = sl2z_word_problem( Z )
sage: wZ
[(1, 2),
(0, 12),
(1, 1),
(0, -4),
(0, 1),
(1, -1),
(0, 1),
(1, -1),
(0, 1),
(1, -1)]
sage: R^2 * L^12 * R * L^-4 * (R * L^-1)^3 == Z
True
```

Let us generate many elements, say twelve here:

```
sage: [ SL2Z.random_element() for _ in [1..12] ]
[
[ 9 77] [-58 79] [-72 23] [11 25] [73 53] [ 80 -11]
[-11 -94], [ 11 -15], [ 97 -31], [ 7 16], [84 61], [-29 4],
[ -6 7] [ 15 62] [-53 55] [-80 73] [ -1 -5] [-83 -55]
[ 77 -90], [-23 -95], [-80 83], [-57 52], [-16 -81], [ 80 53]
]
```

They never go beyond $\pm 100$? Strange randomizer...

Let us take a closer look at the implemented method:

```
sage: SL2Z.random_element?
Signature: SL2Z.random_element(bound=100, *args, **kwds)
```

and so on, and the first line in the doc tells us what to do. Let us do it six times!

```
sage: [ SL2Z.random_element( bound=2017) for _ in [1..6] ]
[
[-607 182] [ 577 -1998] [ 627 -1291] [ 567 1340] [ 1093 -663]
[1831 -549], [ -255 883], [ -932 1919], [-278 -657], [ 1670 -1013],
[ 29 -1269]
[ 33 -1444]
]
```

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