The AMS Fraktur font has glyphs for digits, the font newtxmath uses with the libertine option hasn't them and has blackboard bold digits instead (but not in the standard places, because \mathbb works differently than in amsfonts).
Here's the Fraktur font used by libertine:

Here's the AMS Euler Fraktur:

They're not really alike, so you may want to keep the libertine ones and just replace the digits. In order to blend with the heaviness of the libertine fonts I use the bold version.
\documentclass{sokendai_thesis} % https://github.com/koikezlemma/phd_thesis_template/blob/master/sokendai_thesis.cls
\AtBeginDocument{%
\NewCommandCopy\TXmathfrak\mathfrak
\RenewCommandCopy\mathfrak\ZLINmathfrak
}
\ExplSyntaxOn
\NewDocumentCommand{\ZLINmathfrak}{m}
{
\tl_map_function:nN { #1 } \zlin_mathfrak:n
}
\cs_new_protected:Nn \zlin_mathfrak:n
{
\str_case:nnF { #1 }
{
{0}{__zlin_mathfrak_digit:n { 0 }}
{1}{__zlin_mathfrak_digit:n { 1 }}
{2}{__zlin_mathfrak_digit:n { 2 }}
{3}{__zlin_mathfrak_digit:n { 3 }}
{4}{__zlin_mathfrak_digit:n { 4 }}
{5}{__zlin_mathfrak_digit:n { 5 }}
{6}{__zlin_mathfrak_digit:n { 6 }}
{7}{__zlin_mathfrak_digit:n { 7 }}
{8}{__zlin_mathfrak_digit:n { 8 }}
{9}{__zlin_mathfrak_digit:n { 9 }}
}
{ \TXmathfrak{#1} }
}
\cs_new_protected:Nn __zlin_mathfrak_digit:n
{
\text{\usefont{U}{euf}{b}{n}#1}
}
\ExplSyntaxOff
\begin{document}
${f_\rho^{\mathfrak{0}}, f_\rho^{\mathfrak{1}}, \cdots, f_\rho^{\mathfrak{K}} }$
$\forall{\mathfrak{k}}\in{\mathfrak{0},\cdots, \mathfrak{K}}$.
$\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz}$
$\mathfrak{0123456789}$
\end{document}

Alternatively, use the AMS Euler Fraktur:
\documentclass{sokendai_thesis} % https://github.com/koikezlemma/phd_thesis_template/blob/master/sokendai_thesis.cls
\DeclareMathAlphabet{\mathfrak}{U}{euf}{m}{n}
\begin{document}
${f_\rho^{\mathfrak{0}}, f_\rho^{\mathfrak{1}}, \cdots, f_\rho^{\mathfrak{K}} }$
$\forall{\mathfrak{k}}\in{\mathfrak{0},\cdots, \mathfrak{K}}$.
$\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz}$
$\mathfrak{0123456789}$
\end{document}

\documentclass...and end\end{document}for LaTeX. In that sense, you've not provided enough; but you've also given too much. Likely you don't need all of those packages to reproduce the problem, so you should also remove the ones you don't need before posting your edited example here. – cfr Nov 04 '23 at 00:46\usepackage{amssymb}to the preamble. – Mico Nov 04 '23 at 00:51LaTeX Error: Command `\Bbbk' already defined.I also tried 'amsmath' and 'amsfonts', but doesn't help. – zlin Nov 04 '23 at 01:07\usepackage{amssymb}? Toward the beginning, or toward the end? Incidentally, theamssymbpackage loads theamsfontspackage automatically. By the way, would you mind revealing which document class you employ? (It's the main argument of the\documentclassinstruction.) – Mico Nov 04 '23 at 01:43\documentclass{...}and ending with\end{document}. – dexteritas Nov 04 '23 at 12:52amssymbtowards the beginning (as did in the line 30 of sokendai_thesis.cls). and the document class I am using is from https://github.com/koikezlemma/phd_thesis_template/blob/master/sokendai_thesis.cls – zlin Nov 04 '23 at 19:00\mathfrak{0}or\mathfrak{1}even look like? Fraktur style in math mode is typically only used for letters, not so much for numbers, at least to my knowledge. (Also, numbers in Fraktur do not look much differently from numbers in Antiqua, so it might not be the best choice if disambiguation is important.) – Jasper Habicht Nov 04 '23 at 22:56