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Properties

Label 450.2.l
Level $450$
Weight $2$
Character orbit 450.l
Rep. character $\chi_{450}(19,\cdot)$
Character field $\Q(\zeta_{10})$
Dimension $48$
Newform subspaces $4$
Sturm bound $180$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.l (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 4 \)
Sturm bound: \(180\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

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

Total New Old
Modular forms 392 48 344
Cusp forms 328 48 280
Eisenstein series 64 0 64

Trace form

\( 48 q + 12 q^{4} + 6 q^{5} + O(q^{10}) \) \( 48 q + 12 q^{4} + 6 q^{5} + 2 q^{10} - 8 q^{11} + 4 q^{14} - 12 q^{16} + 10 q^{17} - 14 q^{19} + 4 q^{20} - 20 q^{22} + 10 q^{23} + 44 q^{25} + 20 q^{26} - 10 q^{28} + 22 q^{29} + 18 q^{31} - 10 q^{34} + 14 q^{35} - 2 q^{40} - 30 q^{41} - 2 q^{44} + 16 q^{46} + 70 q^{47} - 32 q^{49} + 8 q^{50} - 2 q^{55} - 4 q^{56} + 30 q^{61} - 60 q^{62} + 12 q^{64} - 62 q^{65} - 70 q^{67} - 52 q^{70} + 42 q^{71} - 44 q^{74} - 16 q^{76} - 120 q^{77} - 40 q^{79} - 4 q^{80} - 90 q^{83} - 74 q^{85} + 2 q^{86} + 10 q^{88} - 56 q^{89} + 36 q^{91} + 10 q^{92} + 8 q^{94} - 40 q^{97} - 40 q^{98} + O(q^{100}) \)

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.l.a 450.l 25.e $8$ $3.593$ \(\Q(\zeta_{20})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\zeta_{20}q^{2}+\zeta_{20}^{2}q^{4}+(\zeta_{20}+\zeta_{20}^{2}+\cdots)q^{5}+\cdots\)
450.2.l.b 450.l 25.e $8$ $3.593$ \(\Q(\zeta_{20})\) None \(0\) \(0\) \(10\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\zeta_{20}q^{2}+\zeta_{20}^{2}q^{4}+(1-\zeta_{20}-\zeta_{20}^{3}+\cdots)q^{5}+\cdots\)
450.2.l.c 450.l 25.e $16$ $3.593$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q-\beta _{8}q^{2}-\beta _{10}q^{4}+(-1-\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)
450.2.l.d 450.l 25.e $16$ $3.593$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q-\beta _{11}q^{2}+(1+\beta _{3}+\beta _{4}+\beta _{5})q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)