Properties

Label 600.2.bc
Level $600$
Weight $2$
Character orbit 600.bc
Rep. character $\chi_{600}(169,\cdot)$
Character field $\Q(\zeta_{10})$
Dimension $56$
Newform subspaces $3$
Sturm bound $240$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bc (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 3 \)
Sturm bound: \(240\)
Trace bound: \(5\)
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 56 456
Cusp forms 448 56 392
Eisenstein series 64 0 64

Trace form

\( 56 q - 4 q^{5} + 14 q^{9} + O(q^{10}) \) \( 56 q - 4 q^{5} + 14 q^{9} - 2 q^{15} + 12 q^{19} + 4 q^{21} + 60 q^{23} - 6 q^{25} + 16 q^{29} - 6 q^{31} - 4 q^{35} + 20 q^{37} - 16 q^{41} + 4 q^{45} + 40 q^{47} - 12 q^{49} + 48 q^{51} + 60 q^{53} + 32 q^{55} + 24 q^{59} - 4 q^{61} + 16 q^{65} - 8 q^{69} - 24 q^{71} - 40 q^{73} - 8 q^{75} - 4 q^{79} - 14 q^{81} - 60 q^{83} - 48 q^{85} - 60 q^{87} - 24 q^{89} - 8 q^{91} - 88 q^{95} - 30 q^{97} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
600.2.bc.a 600.bc 25.e $8$ $4.791$ \(\Q(\zeta_{20})\) None 600.2.bc.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\zeta_{20}q^{3}+(\zeta_{20}^{2}+\zeta_{20}^{3}+\zeta_{20}^{5}+\cdots)q^{5}+\cdots\)
600.2.bc.b 600.bc 25.e $24$ $4.791$ None 600.2.bc.b \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{10}]$
600.2.bc.c 600.bc 25.e $24$ $4.791$ None 600.2.bc.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$

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

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