Properties

Label 450.3.b
Level $450$
Weight $3$
Character orbit 450.b
Rep. character $\chi_{450}(449,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $3$
Sturm bound $270$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(270\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(450, [\chi])\).

Total New Old
Modular forms 204 12 192
Cusp forms 156 12 144
Eisenstein series 48 0 48

Trace form

\( 12 q + 24 q^{4} + O(q^{10}) \) \( 12 q + 24 q^{4} + 48 q^{16} + 72 q^{19} + 168 q^{31} + 24 q^{34} + 288 q^{46} + 36 q^{49} - 240 q^{61} + 96 q^{64} + 144 q^{76} - 528 q^{79} - 504 q^{91} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
450.3.b.a 450.b 15.d $4$ $12.262$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{3}q^{2}+2q^{4}+11\zeta_{8}q^{7}+2\zeta_{8}^{3}q^{8}+\cdots\)
450.3.b.b 450.b 15.d $4$ $12.262$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{3}q^{2}+2q^{4}+2\zeta_{8}q^{7}+2\zeta_{8}^{3}q^{8}+\cdots\)
450.3.b.c 450.b 15.d $4$ $12.262$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}^{3}q^{2}+2q^{4}+\zeta_{8}q^{7}-2\zeta_{8}^{3}q^{8}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)