Spectrum shape measurements in nuclear $\beta$ decay can be used to test physics beyond the Standard Model with results being complementary to high-energy collider experiments. In particular, Beyond Standard Model sensitivity of the weak interaction is expressed through the so-called Fierz interference term. Additionally, the $\beta$ spectrum shape is a useful tool to probe Standard Model effects, among which the most prominent is \textit{weak magnetism}, a higher-order recoil correction induced by nuclear pion exchange. To study effects in the $\beta$ spectrum shape at a precision level competitive with the LHC, a new spectrometer was designed and built. It consists of a 3D low-pressure gas electron tracker and a plastic scintillator used for triggering the data acquisition and recording the $\beta$ particle energy. In this Letter, the results from $\beta$ spectrum shape measurements on the allowed Gamow-Teller transition $^{114}\text{In} \rightarrow ^{114}\text{Sn}$ are presented, including a first extraction of the weak magnetism form factor in the high nuclear mass range and a new estimate of the $90\%$ confidence interval for the Fierz interference term.