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Properties

Label 600.2.y
Level $600$
Weight $2$
Character orbit 600.y
Rep. character $\chi_{600}(121,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $64$
Newform subspaces $6$
Sturm bound $240$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.y (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 6 \)
Sturm bound: \(240\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(600, [\chi])\).

Total New Old
Modular forms 512 64 448
Cusp forms 448 64 384
Eisenstein series 64 0 64

Trace form

\( 64 q + 2 q^{5} - 4 q^{7} - 16 q^{9} + O(q^{10}) \) \( 64 q + 2 q^{5} - 4 q^{7} - 16 q^{9} + 4 q^{13} - 2 q^{15} + 4 q^{17} + 12 q^{19} - 4 q^{21} - 36 q^{23} + 4 q^{25} - 4 q^{29} + 6 q^{31} + 12 q^{33} + 4 q^{35} - 14 q^{37} - 4 q^{41} + 64 q^{43} + 2 q^{45} + 24 q^{47} + 108 q^{49} - 48 q^{51} + 18 q^{53} + 52 q^{55} - 8 q^{57} + 24 q^{59} - 16 q^{61} - 4 q^{63} - 26 q^{65} + 8 q^{67} - 8 q^{69} + 24 q^{71} + 40 q^{73} - 8 q^{75} - 4 q^{79} - 16 q^{81} - 52 q^{83} + 38 q^{85} - 40 q^{87} + 6 q^{89} + 8 q^{91} - 96 q^{93} - 120 q^{95} - 38 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(600, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
600.2.y.a 600.y 25.d $4$ $4.791$ \(\Q(\zeta_{10})\) None \(0\) \(-1\) \(-5\) \(12\) $\mathrm{SU}(2)[C_{5}]$ \(q-\zeta_{10}^{3}q^{3}+(-2\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{5}+\cdots\)
600.2.y.b 600.y 25.d $4$ $4.791$ \(\Q(\zeta_{10})\) None \(0\) \(1\) \(-5\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{10}^{3}q^{3}+(-2\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{5}+\cdots\)
600.2.y.c 600.y 25.d $12$ $4.791$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-3\) \(4\) \(-8\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{5}q^{3}-\beta _{4}q^{5}+(-2+\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots\)
600.2.y.d 600.y 25.d $12$ $4.791$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(3\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{5}q^{3}+(\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots\)
600.2.y.e 600.y 25.d $16$ $4.791$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(-4\) \(1\) \(2\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\beta _{7}+\beta _{8}+\beta _{11})q^{3}-\beta _{13}q^{5}+\cdots\)
600.2.y.f 600.y 25.d $16$ $4.791$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(4\) \(7\) \(-10\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{3}q^{3}+\beta _{10}q^{5}+(-\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)