The space of Newforms of weight 4 on $\Gamma_1(4)$ decomposes as \[ S_{ 4 }^{new}(\Gamma_1(4)) = \bigoplus_{\chi \bmod 4 } S_{ 4 }^{new}(\Gamma_0(4), \chi) \] where the direct sum is over all Dirichlet characters mod \(4\). If $\chi$ and $\chi'$ are in the same Galois orbit, then $S_{ 4 }^{new}(\Gamma_0(4), \chi)$ and $S_{ 4 }^{new}(\Gamma_0(4), \chi')$ are Galois conjugate, so in particular they have the same dimension. Note that this dimension is automatically zero if the parity of the character is not the same as the parity of the weight.

The table below gives the dimensions of the spaces of newforms for \( \Gamma_0(4) \) of weight 4 and characters in each Galois orbit, with links to each space.

Dimension of $S_{ 4 }^{new}(\Gamma_0(4), \chi)$ |
Parity | $S_{ 4 }(\chi_{ 4 }(n,\cdot)):=S^{new}_{ 4 }(\Gamma_0(4),\chi_{ 4 }(n,\cdot))$ for characters $\chi$ grouped by Galois orbit |
---|---|---|

0 | even | \(S_{4}(\chi_{4}(1, \cdot)) \) |

0 | odd | \(S_{4}(\chi_{4}(3, \cdot)) \) |