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Properties

Label 600.2.w
Level $600$
Weight $2$
Character orbit 600.w
Rep. character $\chi_{600}(293,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $136$
Newform subspaces $11$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.w (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 120 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 11 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

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

Total New Old
Modular forms 264 152 112
Cusp forms 216 136 80
Eisenstein series 48 16 32

Trace form

\( 136 q - 4 q^{6} + 8 q^{7} + O(q^{10}) \) \( 136 q - 4 q^{6} + 8 q^{7} + 8 q^{12} - 16 q^{16} + 20 q^{18} + 20 q^{22} - 28 q^{28} + 16 q^{31} + 16 q^{33} - 20 q^{36} + 32 q^{42} - 72 q^{46} - 44 q^{48} - 8 q^{52} + 16 q^{57} - 44 q^{58} - 24 q^{63} - 116 q^{66} - 32 q^{72} + 8 q^{73} + 56 q^{76} - 64 q^{78} + 24 q^{81} - 64 q^{82} - 64 q^{87} + 116 q^{88} + 92 q^{96} - 8 q^{97} + O(q^{100}) \)

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.w.a 600.w 120.w $4$ $4.791$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-30}) \) \(-4\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1+\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+\cdots\)
600.2.w.b 600.w 120.w $4$ $4.791$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-6}) \) \(-4\) \(0\) \(0\) \(4\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1+\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+\cdots\)
600.2.w.c 600.w 120.w $4$ $4.791$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(1-\zeta_{12}+\zeta_{12}^{3})q^{3}+\cdots\)
600.2.w.d 600.w 120.w $4$ $4.791$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
600.2.w.e 600.w 120.w $4$ $4.791$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
600.2.w.f 600.w 120.w $4$ $4.791$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(-1+\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
600.2.w.g 600.w 120.w $4$ $4.791$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-30}) \) \(4\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(1-\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
600.2.w.h 600.w 120.w $4$ $4.791$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-6}) \) \(4\) \(0\) \(0\) \(4\) $\mathrm{U}(1)[D_{4}]$ \(q+(1-\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
600.2.w.i 600.w 120.w $8$ $4.791$ 8.0.3317760000.5 \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{2}+(\beta _{4}+\beta _{6})q^{3}+\beta _{2}q^{4}+(2+\cdots)q^{6}+\cdots\)
600.2.w.j 600.w 120.w $32$ $4.791$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
600.2.w.k 600.w 120.w $64$ $4.791$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

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

\( S_{2}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)