Note on Bernoulli numbers associated to some Dirichlet character of prime conductor

We fix an integer ({n geq 1}) and a divisor m of n such that n/m is odd. Let p be a prime number of the form ({p=2nell+1}) for some odd prime number ({ell}) with ({ell nmid m}). Let ({S=pB_{1,2mell}}) be the p times of the generalised Bernoulli number associated to an odd Dirichlet character of conductor p and order ({2mell}), which is an algebraic integer of the ({2mell})th cyclotomic field. It is known that ({S neq 0}). More strongly, we show that when ({ell}) is sufficiently large, the trace of ({zeta^{-1}S}) to the ({2m})th cyclotomic field does not vanish for any({ell})th root ({zeta}) of unity. We also show a related result on indivisibility of relative class numbers.