= Solution
A weight-$2k$ <modular form> gives an invariant tensor differential
$$
\omega_f=f(\tau)(d\tau)^k,
$$
because $d(\gamma\tau)=(c\tau+d)^{-2}d\tau$ cancels its <automorphy factor>. Assume first $k\geq0$. At an ordinary quotient point this differential is <holomorphic>. At an <elliptic point> of order $e$, use $u$ and $w=u^e$ from part (a), and write the pullback as $A(u)(du)^k$, with $A$ <holomorphic>. Its invariance says $A(\zeta u)\zeta^k=A(u)$, so a nonzero term $a_ru^r$ has $r+k\equiv0\pmod e$. Since
$$
u^r(du)^k=\frac1{e^k}u^{r-k(e-1)}(dw)^k,
$$
the exponent in the descended coefficient is the integer $(r-k(e-1))/e$. With $r\geq0$, it is at least $-\lfloor k(e-1)/e\rfloor$. At the <modular cusp>, $d\tau=dq/(2\pi iq)$ and $f$ has a power <series> with nonnegative exponents, so $\omega_f$ has a <pole> of order at most $k$.
Consequently the <linear map> $f\mapsto\omega_f$ injects $M_{2k}$ into the <meromorphic> sections of $K^{\otimes k}$ with poles bounded by
$$
D=k[\infty]+\lfloor k/2\rfloor[i]+\lfloor2k/3\rfloor[\rho].
$$
This section space is $H^0(X(1),K^{\otimes k}(D))$. It is finite-dimensional on a <compact Riemann surface> by the <Riemann-Roch theorem>. Concretely choose one nonzero <meromorphic> section of the line bundle; dividing by it identifies these sections with a <Riemann-Roch space> with a fixed finite <divisor> bound. This proves the requested finite-dimensionality by compact-surface theory.
In fact the injection is an isomorphism. If a descended coefficient has order at least $-\lfloor k(e-1)/e\rfloor$, its pulled-back exponent $ed+k(e-1)$ is nonnegative; the <modular cusp> <pole> bound likewise gives a <holomorphic> <Fourier expansion of a modular form>. These pullbacks therefore recover <modular forms>. Since $X(1)$ is the <Riemann sphere>, $\deg K=-2$, and a line bundle of degree $d$ on the sphere has $\max(0,d+1)$ sections. Hence
$$
\boxed{\dim M_{2k}=\max\left(0,1-k+\left\lfloor\frac{k}{2}\right\rfloor+\left\lfloor\frac{2k}{3}\right\rfloor\right)\quad(k\geq0).}
$$
For negative $k$, the valence formula excludes a nonzero <holomorphic> <modular form> because all its orders would be nonnegative but their sum would be $k/6<0$; the space is zero. The mechanism behind finite-dimensionality is <modular forms as meromorphic differentials on X(1)>, with explicit elliptic and <modular cusp> <pole> bounds.
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