tokens: lighthouse effect. Comparing the neutrino modulation frequency with the GW frequency, we found that the neutrino peak frequency is m /2 times larger than the GW peak frequency, where m represents the deformation mode of the PNS (around the rotation axis) induced by the low- T /| W | instability. In the non-rotating model, the neutrino modulation frequency, as previously identified, is almost the same as the SASI frequency, and it is half of the GW frequency. We discussed the detectability of these neutrino modulations by IceCube and HK at a source distance of 10 kpc. If we observe our rotating models along the equatorial direction, the characteristic peak in the power spectra for the event rates at IceCube would be above the noise level. The characteristic peak due to the SASI in our non-rotating model observed along either polar or equatorial directions is also above the noise level. This analysis indicates that the joint observation of the GW and neutrino signals would be useful to extract information on growing multiple modes of non-axisymmetric instabilities such as the low- T /| W | instability and the spiral SASI activity, which is otherwise inaccessible by electromagnetic-wave observations. Finally, we shall discuss several major limitations of our work. Even when the pre-collapse core is a slow rotator (Heger et al. 2005 ) but with a sufficiently weak initial magnetic fields, the magnetic fields may be amplified to a dynamically relevant strength by the magnetorotational instability (MRI, e.g. Akiyama et al. 2003 ; Obergaulinger et al. 2009 ; Masada, Takiwaki Kotake 2015 ; Rembiasz et al. 2016 ). Although the role of the MRI on the post-bounce dynamics remains still unclear, the magnetic fields should play a key role in the angular momentum transport especially in the vicinity of the (differentially rotating) PNS surface. Possible impacts on the growth of the low- T /| W | instability with magnetic fields remain to be clarified. If progenitors have both strong initial magnetic fields ( G) and rapid rotation, the magnetic fields are amplified due to the field wrapping to dynamically relevant strength shortly after bounce, leading to the formation of magnetohydrodynamic (MHD) jets. This would also make the post-bounce dynamics such as the shock evolution and the post-shock flow very different comparing to those of the non-magnetized models. Such difference should also affect the angular momentum transport, not to mention the onset criteria of the low- T /| W | instability. Although there are a few recent 3D MHD core-collapse simulations, the development of the low- T /| W | instability has been only reported in Scheidegger et al. ( 2010 ) and suggested in Kuroda et al. ( 2020 ) (see, however, Mösta et al. 2014 ). Also, the convective dynamo may amplify the magnetic fields and modify the dynamics in the late phase of the core collapse (e.g. Raynaud et al. 2020 ). To answer these questions, one apparently need to perform 3D GR-MHD
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