We demonstrate a high-capacity and ultra-compact silicon photonic transmitter directly coupled to a 7-channel multicore fiber (MCF). The integrated transmitter contains seven GeSi electro-absorption modulators (EAMs) connected with the perfectly vertical grating coupler (PVGC) array. A high port density for spatial-division multiplexing (SDM) is thus enabled by the out-of-plane coupling while not requiring angled polishing of the MCF. Relying on four-level pulse amplitude modulation (PAM-4) and digital signal processing (DSP), each of the EAMs can support a data rate of 160 Gbit/s in the back-to-back system. The obtained bit-error-rate (BER) is below the low-latency 7% hard-decision forward-error-correct (HD-FEC) coding limit. The integrated transmitter has a total data-rate-capacity of 1.12 Tbit/s over the seven spatial channels and a compact footprint of 1.8 mm × 0.45mm. BER performance of the optimum spatial channel is evaluated at different wavelengths, revealing its potential for possible wavelength-division multiplexing (WDM) by the on-chip wavelength (de)multiplexers in the future.