BUG: DatetimeIndex.is_year_start breaks on custom business days frequ…

…encies bigger then `1C` (#58665)

* fix bug dti.is_year_start breaks on freq custom business day with digit

* rename the variable freqstr

* delete double space

doc/source/whatsnew/v3.0.0.rst

@@ -384,6 +384,8 @@ Datetimelike
 - Bug in :func:`date_range` where the last valid timestamp would sometimes not be produced (:issue:`56134`)
 - Bug in :func:`date_range` where using a negative frequency value would not include all points between the start and end values (:issue:`56382`)
 - Bug in :func:`tseries.api.guess_datetime_format` would fail to infer time format when "%Y" == "%H%M" (:issue:`57452`)
+- Bug in :meth:`DatetimeIndex.is_year_start` and :meth:`DatetimeIndex.is_quarter_start` does not raise on Custom business days frequencies bigger then "1C" (:issue:`58664`)
+- Bug in :meth:`DatetimeIndex.is_year_start` and :meth:`DatetimeIndex.is_quarter_start` returning ``False`` on double-digit frequencies (:issue:`58523`)
 - Bug in setting scalar values with mismatched resolution into arrays with non-nanosecond ``datetime64``, ``timedelta64`` or :class:`DatetimeTZDtype` incorrectly truncating those scalars (:issue:`56410`)
 
 Timedelta
@@ -421,7 +423,6 @@ Interval
 Indexing
 ^^^^^^^^
 - Bug in :meth:`DataFrame.__getitem__` returning modified columns when called with ``slice`` in Python 3.12 (:issue:`57500`)
-- Bug in :meth:`DatetimeIndex.is_year_start` and :meth:`DatetimeIndex.is_quarter_start` returning ``False`` on double-digit frequencies (:issue:`58523`)
 -
 
 Missing

pandas/_libs/tslibs/fields.pyx

@@ -253,7 +253,7 @@ def get_start_end_field(
         # month of year. Other offsets use month, startingMonth as ending
         # month of year.
 
-        if freq_name.lstrip("B")[0:2]  in ["MS", "QS", "YS"]:
+        if freq_name.lstrip("B")[0:2] in ["MS", "QS", "YS"]:
             end_month = 12 if month_kw == 1 else month_kw - 1
             start_month = month_kw
         else:

pandas/tests/indexes/datetimes/test_scalar_compat.py

@@ -336,3 +336,10 @@ def test_dti_is_year_quarter_start_doubledigit_freq(self):
 
         dr = date_range("2017-01-01", periods=2, freq="10QS")
         assert all(dr.is_quarter_start)
+
+    def test_dti_is_year_start_freq_custom_business_day_with_digit(self):
+        # GH#58664
+        dr = date_range("2020-01-01", periods=2, freq="2C")
+        msg = "Custom business days is not supported by is_year_start"
+        with pytest.raises(ValueError, match=msg):
+            dr.is_year_start