= Solution
Only states within an <energy> window of order $k_BT$ around the <Fermi energy> can change occupation in a degenerate metal. A <Sommerfeld expansion> gives electronic <energy> above its zero-temperature value proportional to $g(E_F)(k_BT)^2$, hence $C_e=\gamma T$. Three-dimensional linear-dispersion <acoustic phonons> have density of states proportional to $\omega^2$; integrating their Bose occupations gives thermal <energy> proportional to $T^4$ and the <Debye heat capacity> $C_{
m ph}=\beta T^3$. Thus $C/T=\gamma+\beta T^2$ at sufficiently low <temperature> in the normal state.
Reading the straight low-temperature fit gives approximately $\gamma\simeq33\ \mathrm{mJ\,mol^{-1}K^{-2}}$ and $\beta\simeq3.8\ \mathrm{mJ\,mol^{-1}K^{-4}}$. These estimates agree with the reported fit to this material in the original experiment.<Superconductivity in the Filled Cage Compounds Ba6Ge25 and Ba4Na2Ge25 by Friedrich Malte Grosche and others>[Figure 3 caption: low-temperature normal-state coefficients for Ba4Na2Ge25]{ref} Treating the graph values as approximate, the molar <density of states> is
$$
\boxed{g(E_F)=\frac{3\gamma}{\pi^2k_B^2}\simeq5.3\times10^{43}\ \mathrm{J^{-1}mol^{-1}}\simeq14\ \mathrm{eV^{-1}}\text{ per formula unit}.}
$$
The stipulated two electrons per formula unit give $n=2N_A/V_m\simeq2.62\times10^{27}\ \mathrm{m^{-3}}$. For a single spin-degenerate parabolic band, the molar density of states is $g(E_F)=3(2N_A)/(2E_F)=3N_A/E_F$. Hence
$$
\boxed{E_F\simeq3.4\times10^{-20}\ \mathrm J\simeq0.21\ \mathrm{eV},\qquad
k_F=(3\pi^2n)^{1/3}\simeq4.3\times10^9\ \mathrm{m^{-1}},\qquad
m^*=\frac{\hbar^2k_F^2}{2E_F}\simeq3.2m_e.}
$$
These are model-dependent effective parameters inferred from the measured heat capacity and assumed carrier count, rather than an independent determination of a unique band structure.
Using the Debye prefactor stated in the question gives
$$
\boxed{\Theta_D=\left(\frac{12\pi^4N_Ak_B}{5\beta}\right)^{1/3}\simeq80\ \mathrm K.}
$$
This prefactor counts three effective modes per formula unit. Consistently choosing the same count for the <Debye wavevector> gives $k_D=(6\pi^2N_A/V_m)^{1/3}\simeq4.3\times10^9\ \mathrm{m^{-1}}$. The corresponding average <sound speed> is
$$
\boxed{v_D=\frac{k_B\Theta_D}{\hbar k_D}\simeq2.5\times10^3\ \mathrm{m\,s^{-1}}.}
$$
There is a normalization choice here: a formula unit has 31 atoms. With the conventional full atomic-mode count, the molar Debye coefficient contains an additional factor 31, and both $\Theta_D$ and $k_D$ increase by $31^{1/3}$. That convention gives $\Theta_D\simeq251\ \mathrm K$ and $k_D\simeq1.34\times10^{10}\ \mathrm{m^{-1}}$, while leaving the inferred <sound speed> unchanged. The <mode count in the Debye model> must be used consistently; mixing the printed <temperature> with an atomic-count <wavevector> gives an erroneous <sound speed>.
The upward curvature above about $3\ \mathrm K$ indicates low-energy vibrations missing from a pure acoustic $T^3$ model. In this cage compound, the original analysis fits an additional localized mode with $\hbar\Omega/k_B\simeq32\ \mathrm K$, assigned to a <rattling mode> of a barium guest atom.<Superconductivity in the Filled Cage Compounds Ba6Ge25 and Ba4Na2Ge25 by Friedrich Malte Grosche and others>[Page 247003-3 and Figure 3: extra Einstein-mode contribution]{ref} At temperatures high compared with all relevant <phonon> energies, the full lattice approaches the <Dulong-Petit law>, $C_{
m lattice}\to3(31)R=93R\simeq773\ \mathrm{J\,mol^{-1}K^{-1}}$. The reduced three-mode model alone approaches $3R$ and omits the other branches. An approximately linear electronic term can still be added while the carriers remain degenerate, but cannot be extrapolated beyond their <Fermi temperature>.
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