The optimal standard local converse theorem for Sp_{2N}, SO_{2N}, SO_{2N+1}, and G2 requires twisting up to roughly half the dual group's standard representation dimension, except for an improved SO_{2N} bound for odd N.
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On sharpness in Local Converse Theorems for classical groups and $G_2$
The optimal standard local converse theorem for Sp_{2N}, SO_{2N}, SO_{2N+1}, and G2 requires twisting up to roughly half the dual group's standard representation dimension, except for an improved SO_{2N} bound for odd N.