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Properties

Label 450.2.f
Level $450$
Weight $2$
Character orbit 450.f
Rep. character $\chi_{450}(107,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $12$
Newform subspaces $3$
Sturm bound $180$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(180\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(450, [\chi])\).

Total New Old
Modular forms 228 12 216
Cusp forms 132 12 120
Eisenstein series 96 0 96

Trace form

\( 12 q - 8 q^{7} + O(q^{10}) \) \( 12 q - 8 q^{7} + 12 q^{13} - 12 q^{16} - 8 q^{22} - 8 q^{28} + 48 q^{31} + 12 q^{37} - 32 q^{46} - 12 q^{52} - 4 q^{58} - 80 q^{61} + 16 q^{67} - 4 q^{73} - 16 q^{76} + 28 q^{82} - 8 q^{88} + 96 q^{91} + 12 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
450.2.f.a 450.f 15.e $4$ $3.593$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-3+3\zeta_{8}^{2})q^{7}+\cdots\)
450.2.f.b 450.f 15.e $4$ $3.593$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-2+2\zeta_{8}^{2})q^{7}+\cdots\)
450.2.f.c 450.f 15.e $4$ $3.593$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(3-3\zeta_{8}^{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

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