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Properties

Label 600.3.u
Level $600$
Weight $3$
Character orbit 600.u
Rep. character $\chi_{600}(193,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $36$
Newform subspaces $8$
Sturm bound $360$
Trace bound $21$

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Defining parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.u (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 8 \)
Sturm bound: \(360\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(600, [\chi])\).

Total New Old
Modular forms 528 36 492
Cusp forms 432 36 396
Eisenstein series 96 0 96

Trace form

\( 36 q + 8 q^{7} + O(q^{10}) \) \( 36 q + 8 q^{7} - 32 q^{11} - 44 q^{13} - 28 q^{17} + 96 q^{23} - 32 q^{31} - 72 q^{33} - 52 q^{37} - 64 q^{41} + 96 q^{43} + 32 q^{47} + 12 q^{53} + 96 q^{57} + 240 q^{61} + 24 q^{63} + 224 q^{67} - 64 q^{71} + 100 q^{73} - 240 q^{77} - 324 q^{81} - 608 q^{83} - 360 q^{87} + 448 q^{91} - 144 q^{93} + 356 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(600, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
600.3.u.a 600.u 5.c $4$ $16.349$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(-4+4\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
600.3.u.b 600.u 5.c $4$ $16.349$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{1}q^{3}+(-4+4\beta _{2}+3\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
600.3.u.c 600.u 5.c $4$ $16.349$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(-2+2\beta _{2}+\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
600.3.u.d 600.u 5.c $4$ $16.349$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(2-2\beta _{2}+\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
600.3.u.e 600.u 5.c $4$ $16.349$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{1}q^{3}+(3-3\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
600.3.u.f 600.u 5.c $4$ $16.349$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(16\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(4-4\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
600.3.u.g 600.u 5.c $4$ $16.349$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(16\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(4-4\beta _{2}-3\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
600.3.u.h 600.u 5.c $8$ $16.349$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{1}q^{3}+(-\beta _{3}+\beta _{6})q^{7}-3\beta _{2}q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(600, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)