Properties

Label 600.2.d
Level $600$
Weight $2$
Character orbit 600.d
Rep. character $\chi_{600}(349,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $8$
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.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(240\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(37\)

Dimensions

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

Total New Old
Modular forms 132 36 96
Cusp forms 108 36 72
Eisenstein series 24 0 24

Trace form

\( 36 q - 4 q^{4} + 4 q^{6} + 36 q^{9} + O(q^{10}) \) \( 36 q - 4 q^{4} + 4 q^{6} + 36 q^{9} + 28 q^{14} + 4 q^{16} + 4 q^{24} + 36 q^{26} + 40 q^{31} - 32 q^{34} - 4 q^{36} + 16 q^{39} + 8 q^{41} - 32 q^{46} - 36 q^{49} + 4 q^{54} - 8 q^{56} + 20 q^{64} - 32 q^{66} - 64 q^{71} - 88 q^{74} - 16 q^{76} - 24 q^{79} + 36 q^{81} - 8 q^{84} - 100 q^{86} + 40 q^{89} - 32 q^{94} + 4 q^{96} + O(q^{100}) \)

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.d.a 600.d 40.f $2$ $4.791$ \(\Q(\sqrt{-1}) \) None \(-2\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+i)q^{2}-q^{3}-2iq^{4}+(1-i)q^{6}+\cdots\)
600.2.d.b 600.d 40.f $2$ $4.791$ \(\Q(\sqrt{-1}) \) None \(-2\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+i)q^{2}+q^{3}-2iq^{4}+(-1+\cdots)q^{6}+\cdots\)
600.2.d.c 600.d 40.f $2$ $4.791$ \(\Q(\sqrt{-1}) \) None \(2\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+i)q^{2}-q^{3}+2iq^{4}+(-1-i)q^{6}+\cdots\)
600.2.d.d 600.d 40.f $2$ $4.791$ \(\Q(\sqrt{-1}) \) None \(2\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+i)q^{2}+q^{3}+2iq^{4}+(1+i)q^{6}+\cdots\)
600.2.d.e 600.d 40.f $6$ $4.791$ 6.0.399424.1 None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)
600.2.d.f 600.d 40.f $6$ $4.791$ 6.0.399424.1 None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+q^{3}-\beta _{1}q^{4}-\beta _{2}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)
600.2.d.g 600.d 40.f $8$ $4.791$ 8.0.214798336.3 None \(-2\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{6}q^{2}-q^{3}+(-1-\beta _{1}+\beta _{3}+\beta _{6}+\cdots)q^{4}+\cdots\)
600.2.d.h 600.d 40.f $8$ $4.791$ 8.0.214798336.3 None \(2\) \(8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{2}+q^{3}+(-1-\beta _{1}+\beta _{3}+\beta _{6}+\cdots)q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)