Properties

Label 450.2.p
Level $450$
Weight $2$
Character orbit 450.p
Rep. character $\chi_{450}(257,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $72$
Newform subspaces $8$
Sturm bound $180$
Trace bound $14$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.p (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 45 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 8 \)
Sturm bound: \(180\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

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

Total New Old
Modular forms 408 72 336
Cusp forms 312 72 240
Eisenstein series 96 0 96

Trace form

\( 72 q - 4 q^{3} + 16 q^{6} + O(q^{10}) \) \( 72 q - 4 q^{3} + 16 q^{6} + 48 q^{11} + 4 q^{12} + 36 q^{16} + 8 q^{18} + 16 q^{21} + 24 q^{23} + 8 q^{27} - 16 q^{33} - 32 q^{36} + 24 q^{37} - 36 q^{38} - 72 q^{41} - 44 q^{42} + 48 q^{46} - 48 q^{47} - 8 q^{48} - 88 q^{51} - 24 q^{56} - 52 q^{57} - 12 q^{58} + 24 q^{61} + 80 q^{63} - 44 q^{66} + 12 q^{67} + 36 q^{68} + 16 q^{72} + 48 q^{77} + 24 q^{78} - 112 q^{81} + 48 q^{82} - 60 q^{83} - 36 q^{86} + 56 q^{87} - 96 q^{91} + 24 q^{92} - 52 q^{93} + 8 q^{96} + 36 q^{97} + O(q^{100}) \)

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.p.a 450.p 45.l $8$ $3.593$ \(\Q(\zeta_{24})\) None \(0\) \(-4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{7}q^{2}+(-1-\zeta_{24}^{2}+\zeta_{24}^{4}+\cdots)q^{3}+\cdots\)
450.2.p.b 450.p 45.l $8$ $3.593$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{7}q^{2}+(2\zeta_{24}-\zeta_{24}^{5})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
450.2.p.c 450.p 45.l $8$ $3.593$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{7}q^{2}+(\zeta_{24}^{3}-2\zeta_{24}^{7})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
450.2.p.d 450.p 45.l $8$ $3.593$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{7}q^{2}+(-2\zeta_{24}+\zeta_{24}^{5})q^{3}+\cdots\)
450.2.p.e 450.p 45.l $8$ $3.593$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{7}q^{2}+(-\zeta_{24}^{3}+2\zeta_{24}^{7})q^{3}+\cdots\)
450.2.p.f 450.p 45.l $8$ $3.593$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{7}q^{2}+(\zeta_{24}-2\zeta_{24}^{5})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
450.2.p.g 450.p 45.l $8$ $3.593$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{7}q^{2}+(-\zeta_{24}-\zeta_{24}^{5})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
450.2.p.h 450.p 45.l $16$ $3.593$ 16.0.\(\cdots\).9 None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{12}]$ \(q+(\beta _{7}+\beta _{15})q^{2}+(\beta _{1}+\beta _{3}+\beta _{4}+\beta _{10}+\cdots)q^{3}+\cdots\)

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

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