 Prime-power old block at a bad prime

ID: prime-power-old-block-at-a-bad-prime

p

divides the primitive level

M

N = M p^{r}

, put

e_{j} = p^{jk /2} f (p^{j} z)

. Then

U_{p} e_{0} = a_{p} e_{0}

and

U_{p} e_{j} = p^{k /2} e_{j - 1}

for

j \geq 1

. The normalized Fricke involution acts by

e_{j} \mapsto ε e_{r - j}

, where

W_{M} f = ε f

. These two operators act irreducibly: a nonzero invariant subspace contains an eigenvector of

U_{p}

a_{p}

or zero. Either it contains

e_{0}

immediately or applying the reversal and a power of

U_{p}

produces

e_{0}

, using

a_{p}^{2} \neq = p^{k}

. Reversal then produces

e_{r}

, and its successive images produce every basis vector.

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