Properties

Label 450.6.f
Level $450$
Weight $6$
Character orbit 450.f
Rep. character $\chi_{450}(107,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $60$
Newform subspaces $7$
Sturm bound $540$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 7 \)
Sturm bound: \(540\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(450, [\chi])\).

Total New Old
Modular forms 948 60 888
Cusp forms 852 60 792
Eisenstein series 96 0 96

Trace form

\( 60q + 152q^{7} + O(q^{10}) \) \( 60q + 152q^{7} + 2892q^{13} - 15360q^{16} - 928q^{22} + 2432q^{28} + 240q^{31} - 29988q^{37} + 13440q^{43} + 83840q^{46} - 46272q^{52} + 24016q^{58} + 160400q^{61} - 281584q^{67} + 267196q^{73} - 78080q^{76} - 180592q^{82} - 14848q^{88} - 652320q^{91} + 154572q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
450.6.f.a \(4\) \(72.173\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-132\) \(q+(-2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{2}+2^{4}\zeta_{8}q^{4}+\cdots\)
450.6.f.b \(4\) \(72.173\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(88\) \(q+(2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{2}-2^{4}\zeta_{8}q^{4}+(22+\cdots)q^{7}+\cdots\)
450.6.f.c \(4\) \(72.173\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(132\) \(q+(-2\zeta_{8}^{2}+2\zeta_{8}^{3})q^{2}-2^{4}\zeta_{8}q^{4}+\cdots\)
450.6.f.d \(4\) \(72.173\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(208\) \(q+(-2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{2}+2^{4}\zeta_{8}q^{4}+\cdots\)
450.6.f.e \(12\) \(72.173\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(-144\) \(q+(2\beta _{3}+2\beta _{4})q^{2}+2^{4}\beta _{2}q^{4}+(-12+\cdots)q^{7}+\cdots\)
450.6.f.f \(16\) \(72.173\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-528\) \(q+(-2\beta _{1}+2\beta _{5})q^{2}-2^{4}\beta _{4}q^{4}+(-33+\cdots)q^{7}+\cdots\)
450.6.f.g \(16\) \(72.173\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(528\) \(q+(2\beta _{1}+2\beta _{5})q^{2}+2^{4}\beta _{4}q^{4}+(33-33\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)