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Properties

Label 600.3.l
Level $600$
Weight $3$
Character orbit 600.l
Rep. character $\chi_{600}(401,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $7$
Sturm bound $360$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.l (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(360\)
Trace bound: \(7\)
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 38 226
Cusp forms 216 38 178
Eisenstein series 48 0 48

Trace form

\( 38q + 2q^{3} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 38q + 2q^{3} - 4q^{7} + 2q^{9} - 12q^{13} + 4q^{19} - 20q^{21} + 26q^{27} - 68q^{31} + 80q^{33} + 4q^{37} + 4q^{39} + 164q^{43} + 250q^{49} + 196q^{51} - 76q^{57} + 260q^{61} - 172q^{63} - 156q^{67} + 112q^{69} - 196q^{73} - 196q^{79} - 310q^{81} + 56q^{87} - 120q^{91} + 412q^{93} + 44q^{97} - 36q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.l.a \(2\) \(16.349\) \(\Q(\sqrt{-2}) \) None \(0\) \(-2\) \(0\) \(2\) \(q+(-1+\beta )q^{3}+q^{7}+(-7-2\beta )q^{9}+\cdots\)
600.3.l.b \(2\) \(16.349\) \(\Q(\sqrt{-2}) \) None \(0\) \(-2\) \(0\) \(12\) \(q+(-1+\beta )q^{3}+6q^{7}+(-7-2\beta )q^{9}+\cdots\)
600.3.l.c \(2\) \(16.349\) \(\Q(\sqrt{-2}) \) None \(0\) \(2\) \(0\) \(-2\) \(q+(1-\beta )q^{3}-q^{7}+(-7-2\beta )q^{9}-3\beta q^{11}+\cdots\)
600.3.l.d \(6\) \(16.349\) 6.0.574198272.1 None \(0\) \(-4\) \(0\) \(10\) \(q+(-1-\beta _{2})q^{3}+(2+\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
600.3.l.e \(6\) \(16.349\) 6.0.574198272.1 None \(0\) \(4\) \(0\) \(-10\) \(q+(1+\beta _{2})q^{3}+(-2-\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
600.3.l.f \(8\) \(16.349\) 8.0.\(\cdots\).5 None \(0\) \(4\) \(0\) \(-16\) \(q+\beta _{2}q^{3}+(-1+\beta _{3}+\beta _{4}+\beta _{5})q^{7}+\cdots\)
600.3.l.g \(12\) \(16.349\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{3}-\beta _{5}q^{7}+(-1-\beta _{9})q^{9}+(-\beta _{7}+\cdots)q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)