Properties

Label 225.4.f
Level $225$
Weight $4$
Character orbit 225.f
Rep. character $\chi_{225}(107,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $36$
Newform subspaces $4$
Sturm bound $120$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(225, [\chi])\).

Total New Old
Modular forms 204 36 168
Cusp forms 156 36 120
Eisenstein series 48 0 48

Trace form

\( 36 q - 24 q^{7} + O(q^{10}) \) \( 36 q - 24 q^{7} + 108 q^{13} + 336 q^{16} - 1056 q^{22} + 576 q^{28} + 1776 q^{31} - 828 q^{37} + 96 q^{43} - 1344 q^{46} + 312 q^{52} + 3864 q^{58} + 4128 q^{61} - 1632 q^{67} - 3972 q^{73} - 10560 q^{76} + 7848 q^{82} - 7968 q^{88} + 2736 q^{91} - 2772 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
225.4.f.a $4$ $13.275$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-36\) \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-9+9\zeta_{8}^{2})q^{7}+\cdots\)
225.4.f.b $4$ $13.275$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(36\) \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(9-9\zeta_{8}^{2})q^{7}+\cdots\)
225.4.f.c $12$ $13.275$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-24\) \(q-\beta _{6}q^{2}+(-6\beta _{1}+\beta _{5})q^{4}+(-2-2\beta _{1}+\cdots)q^{7}+\cdots\)
225.4.f.d $16$ $13.275$ 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{13}q^{2}+(\beta _{9}+\beta _{15})q^{4}+(7\beta _{3}+2\beta _{14}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)