diff --git a/aux-files/aut.txt b/aux-files/aut.txt
new file mode 100644
index 0000000000000000000000000000000000000000..bd8ac2fa99a893ba859924d542c586339d364306
--- /dev/null
+++ b/aux-files/aut.txt
@@ -0,0 +1,8065 @@
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+ 1  -3    6  -15  -6    3
+-3  10  -18   45  18  -12
+-1   4   -8   20   8   -5
+-1   1    0    0   0   -1
+ 0  -1    1   -3  -1    0
+--
+ 1   2   -2    6   3   0
+ 1  -3    6  -15  -6   3
+-3  10  -18   45  18  -6
+-1   4   -8   20   8  -2
+-1   1    0    0   0  -2
+ 0  -1    1   -3  -1   1
+--
+ 1   2  -2    6   3  0
+ 1   0   0    0   0  0
+-3  -6   8  -21  -9  3
+-1  -2   4  -10  -4  1
+-1  -1  -2    4   1  0
+ 0  -1   1   -3  -1  0
+--
+ 1   2  -2    6    3  0
+ 1   0   0    0    0  0
+-3  -5  12  -30  -12  3
+-1  -2   4  -10   -4  1
+-1   0   2   -5   -2  0
+ 0  -1   1   -3   -1  0
+--
+ 1   2  -2   6   3  0
+ 1   0   0   0   0  0
+-3   0  -4   9   3  3
+-1   1  -2   5   2  1
+-1  -2   0  -1  -1  0
+ 0   0  -1   2   1  0
+--
+ 1   2  -2    6   3  0
+ 1   0   0    0   0  0
+-3   1   0    0   0  3
+-1   1  -2    5   2  1
+-1  -1   4  -10  -4  0
+ 0   0  -1    2   1  0
+--
+ 1   2  -2    6   3   0
+ 1   0   0    0   0   3
+-3  -6   8  -21  -9  -6
+-1  -2   4  -10  -4  -2
+-1  -1  -2    4   1  -2
+ 0  -1   1   -3  -1   1
+--
+ 1   2  -2    6    3   0
+ 1   0   0    0    0   3
+-3  -5  12  -30  -12  -6
+-1  -2   4  -10   -4  -2
+-1   0   2   -5   -2  -2
+ 0  -1   1   -3   -1   1
+--
+ 1   2  -2   6   3    0
+ 1   0   0   0   0    3
+-3   0  -4   9   3  -12
+-1   1  -2   5   2   -5
+-1  -2   0  -1  -1   -1
+ 0   0  -1   2   1    0
+--
+ 1   2  -2    6   3    0
+ 1   0   0    0   0    3
+-3   1   0    0   0  -12
+-1   1  -2    5   2   -5
+-1  -1   4  -10  -4   -1
+ 0   0  -1    2   1    0
diff --git a/aux-files/correction_matrices.sage b/aux-files/correction_matrices.sage
new file mode 100755
index 0000000000000000000000000000000000000000..f4dedece739ea68ba1b31d929005015c8e44c597
--- /dev/null
+++ b/aux-files/correction_matrices.sage
@@ -0,0 +1,79 @@
+#!/usr/bin/env sage
+# SPDX-FileCopyrightText: © 2022 Maik Herbers
+# SPDX-License-Identifier: GPL-3.0-or-later
+
+def left_matrix(width, height, replacement_spec):
+    r"""
+    Output a projection that maps basis vectors onto combinations of others
+
+    INPUT:
+
+    - ``width`` - dimension of the input space
+    - ``height`` - dimension of the output space
+    - ``replacement_spec`` - dictionary that maps indices of input basis
+      vectors to lists of pairs of replacement basis indices and their
+      multiplicities; all basis vectors not specified are mapped onto
+      themselves.
+    """
+
+    m = zero_matrix(height, width)
+    offset = 0
+    for idx in range(width):
+        if idx in replacement_spec:
+            # map `idx' onto specified linear combination
+            for replacement_idx, multiplicity in replacement_spec[idx]:
+                row = replacement_idx - offset
+                if replacement_idx > idx:
+                    row -= 1
+                m[row, idx] = multiplicity
+            offset += 1
+        else:
+            m[idx - offset, idx] = 1
+    return m
+
+def right_matrix(width, height, skip):
+    r"""
+    Output an embedding where into a specified subspace
+
+    INPUT:
+
+    - ``width`` - dimension of the input space
+    - ``height`` - dimension of the output space
+    - ``skip`` - list of basis vectors in the output space that are not in the
+      image
+    """
+
+    m = zero_matrix(height, width)
+    offset = 0
+    for i in range(height):
+        if i in skip:
+            offset += 1
+        else:
+            m[i, i - offset] = 1
+    return m
+
+def get_matrices(n, replacement_spec):
+    r"""
+    Compute the "change of basis" matrices
+
+    INPUT:
+
+    - ``n`` - size of the input basis
+    - ``replacement_spec`` - dictionary suitable for ``left_matrix``
+    """
+
+    new_size = n - len(replacement_spec)
+    left = left_matrix(n, new_size, replacement_spec)
+    right = right_matrix(new_size, n, replacement_spec.keys())
+    return left, right
+
+# 6A - 6E and 10A - 10C have indices 20 - 24 and 33 - 35 resp. in our
+# list of component functions
+replacement_spec = {
+    22: [(20, 2), (21, 2), (23, 1), (24, -4)],
+    34: [(33, -2), (35, 3)]
+}
+
+l, r = get_matrices(38, replacement_spec)
+
+print(f"left matrix:\n{l}\n\nright matrix:\n{r}")
diff --git a/aux-files/gram_matrix.txt b/aux-files/gram_matrix.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e48bdaf21ac75ee8ab76c0082daad58f31d41813
--- /dev/null
+++ b/aux-files/gram_matrix.txt
@@ -0,0 +1,6 @@
+ 6  2  2  2  2  0
+ 2  6  4  4 -2  2
+ 2  4  6  4 -2  2
+ 2  4  4  6  0  2
+ 2 -2 -2  0  6  2
+ 0  2  2  2  2  6
diff --git a/aux-files/smith_gram_matrix.txt b/aux-files/smith_gram_matrix.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e21142cd40ece764945cf291bff9df75331a0902
--- /dev/null
+++ b/aux-files/smith_gram_matrix.txt
@@ -0,0 +1,6 @@
+   4   -2    2   -6    0    6
+  -2   16  -20   54   24  -12
+   2  -20   58 -144  -60   18
+  -6   54 -144  360  150  -48
+   0   24  -60  150   66  -18
+   6  -12   18  -48  -18   18