Properties

Label 600.3.c
Level $600$
Weight $3$
Character orbit 600.c
Rep. character $\chi_{600}(449,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $4$
Sturm bound $360$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(360\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\)

Dimensions

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

Total New Old
Modular forms 264 36 228
Cusp forms 216 36 180
Eisenstein series 48 0 48

Trace form

\( 36q - 16q^{9} + O(q^{10}) \) \( 36q - 16q^{9} - 8q^{21} + 56q^{31} + 176q^{39} - 36q^{49} + 84q^{51} - 48q^{61} - 416q^{69} - 344q^{79} + 136q^{81} + 320q^{91} + 500q^{99} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
600.3.c.a \(4\) \(16.349\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{8}-\zeta_{8}^{2})q^{3}-3\zeta_{8}q^{7}+(7+\zeta_{8}^{3})q^{9}+\cdots\)
600.3.c.b \(4\) \(16.349\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}q^{7}+(7-2\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
600.3.c.c \(12\) \(16.349\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{6}+\beta _{7})q^{3}+(-2\beta _{6}-\beta _{7}-\beta _{8}+\cdots)q^{7}+\cdots\)
600.3.c.d \(16\) \(16.349\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{7}+(-3+\cdots)q^{9}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)