I need the dimensions of the Atkin-Lehner eigenspace for the paper I'm writing.

As is well known, the cuspidal space $S_{k}(\Gamma_{0}(N))$ can be decomposed by Atkin Lehner involution. For example, when $N=21=3\times7$, we have the decomposition $$ S_{k}(\Gamma_{0}(N))=S^{(++)} \oplus S^{(+-)} \oplus S^{(-+)} \oplus S^{(--)} $$ where $S^{(++)}$, $S^{(+-)}$, $S^{(-+)}$, and $S^{(--)}$ are the subspaces of $S_{k}(\Gamma_{0}(N))$ for which pairs of eigenvalues for $W_3$, $W_7$ are (+1,+1), (+1,-1), (-1,+1), (-1,-1) respectively.

In addition, dimensions of each summand spaces are 0,0,1,0. This is from Table 5 of Antwerp IV(https://wstein.org/Tables/antwerp/table5/).

For k=2, the dimension of each Atkin-Lehner eigenspace has already been given by Table 5 of Antwerp IV and David Kohel.

And for any weight k, I found the following documentation: https://trac.sagemath.org/ticket/9455. However, using this, it is difficult for me to find the dimensions of the eigenspaces.

So, is there any idea or data?

I need the dimensions of the Atkin-Lehner eigenspace for the paper I'm writing.

As is well known, the cuspidal space $S_{k}(\Gamma_{0}(N))$ can be decomposed by Atkin Lehner involution. For example, when $N=21=3\times7$, we have the decomposition $$ S_{k}(\Gamma_{0}(N))=S^{(++)} \oplus S^{(+-)} \oplus S^{(-+)} \oplus S^{(--)} $$ where $S^{(++)}$, $S^{(+-)}$, $S^{(-+)}$, and $S^{(--)}$ are the subspaces of $S_{k}(\Gamma_{0}(N))$ for which pairs of eigenvalues for $W_3$, $W_7$ are (+1,+1), (+1,-1), (-1,+1), (-1,-1) respectively.

In addition, dimensions of each summand spaces are 0,0,1,0. This is from Table 5 of Antwerp IV.

For k=2, the dimension of each Atkin-Lehner eigenspace has already been given by Table 5 of Antwerp IV and David Kohel.

And for any weight k, I found the following documentation. However, using this, it is difficult for me to find the dimensions of the eigenspaces.

So, is there any idea or data?

I need the dimensions of the Atkin-Lehner eigenspace for the paper I'm writing.

As is well known, the cuspidal space $S_{k}(\Gamma_{0}(N))$ can be decomposed by Atkin Lehner involution. For example, when $N=21=3\times7$, we have the decomposition $$ S_{k}(\Gamma_{0}(N))=S^{(++)} \oplus S^{(+-)} \oplus S^{(-+)} \oplus S^{(--)} $$ where $S^{(++)}$, $S^{(+-)}$, $S^{(-+)}$, and $S^{(--)}$ are the subspace of $S_{k}(\Gamma_{0}(N))$ for which pairs of eigenvalues for $W_3$, $W_7$ are (+1,+1), (+1,-1), (-1,+1), (-1,-1) respectively.

In addition, dimensions of each summand spaces are 0,0,1,0. This is from Table 5 of Antwerp IV(https://wstein.org/Tables/antwerp/table5/).

For k=2, the dimension of each Atkin-Lehner eigenspace has already been given by Table 5 of Antwerp IV and David Kohel.

And for any weight k, I found the following documentation: https://trac.sagemath.org/ticket/9455. However, using this, it is difficult for me to find the dimensions of the eigenspaces.

So, is there any idea or data?

I need the dimensions of the Atkin-Lehner eigenspace for the paper I'm writing.

As is well known, the cuspidal space $S_{k}(\Gamma_{0}(N))$ can be decomposed by Atkin Lehner involution. For example, when $N=21=3\times7$, we have the decomposition $$ S_{k}(\Gamma_{0}(N))=S^{(++)} \oplus S^{(+-)} \oplus S^{(-+)} \oplus S^{(--)} $$ where $S^{(++)}$, $S^{(+-)}$, $S^{(-+)}$, and $S^{(--)}$ are the subspaces of $S_{k}(\Gamma_{0}(N))$ for which pairs of eigenvalues for $W_3$, $W_7$ are (+1,+1), (+1,-1), (-1,+1), (-1,-1) respectively.

In addition, dimensions of each summand spaces are 0,0,1,0. This is from Table 5 of Antwerp IV(https://wstein.org/Tables/antwerp/table5/).

For k=2, the dimension of each Atkin-Lehner eigenspace has already been given by Table 5 of Antwerp IV and David Kohel.

And for any weight k, I found the following documentation: https://trac.sagemath.org/ticket/9455. However, using this, it is difficult for me to find the dimensions of the eigenspaces.

So, is there any idea or data?