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Properties

Label 450.3.k
Level $450$
Weight $3$
Character orbit 450.k
Rep. character $\chi_{450}(149,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $72$
Newform subspaces $3$
Sturm bound $270$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.k (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 45 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(270\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(7\)

Dimensions

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

Total New Old
Modular forms 384 72 312
Cusp forms 336 72 264
Eisenstein series 48 0 48

Trace form

\( 72 q - 72 q^{4} + 16 q^{6} - 20 q^{9} + O(q^{10}) \) \( 72 q - 72 q^{4} + 16 q^{6} - 20 q^{9} - 72 q^{11} + 72 q^{14} - 144 q^{16} - 104 q^{21} - 16 q^{24} - 72 q^{29} + 60 q^{31} + 128 q^{36} + 416 q^{39} + 288 q^{41} + 48 q^{46} + 204 q^{49} - 228 q^{51} - 160 q^{54} - 144 q^{56} - 288 q^{59} - 96 q^{61} + 576 q^{64} - 144 q^{66} - 312 q^{69} - 288 q^{74} - 144 q^{79} + 908 q^{81} - 136 q^{84} - 216 q^{86} - 336 q^{91} + 168 q^{94} - 32 q^{96} - 624 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.k.a 450.k 45.h $8$ $12.262$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{24}^{4}-\zeta_{24}^{7})q^{2}+(-2\zeta_{24}+\zeta_{24}^{3}+\cdots)q^{3}+\cdots\)
450.3.k.b 450.k 45.h $32$ $12.262$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
450.3.k.c 450.k 45.h $32$ $12.262$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

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

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