Properties

Label 1200.2.h
Level $1200$
Weight $2$
Character orbit 1200.h
Rep. character $\chi_{1200}(1151,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $14$
Sturm bound $480$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1200, [\chi])\).

Total New Old
Modular forms 276 38 238
Cusp forms 204 38 166
Eisenstein series 72 0 72

Trace form

\( 38q - 6q^{9} + O(q^{10}) \) \( 38q - 6q^{9} - 4q^{13} - 12q^{21} + 28q^{37} - 22q^{49} + 36q^{57} - 20q^{61} + 48q^{69} + 44q^{73} - 6q^{81} - 12q^{93} - 52q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1200, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1200.2.h.a \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(0\) \(0\) \(q+(-1-\zeta_{6})q^{3}+3\zeta_{6}q^{9}-3q^{11}+\cdots\)
1200.2.h.b \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(0\) \(0\) \(q+(-1-\zeta_{6})q^{3}+3\zeta_{6}q^{9}+3q^{11}+\cdots\)
1200.2.h.c \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{6}q^{3}-3\zeta_{6}q^{7}-3q^{9}-7q^{13}+\cdots\)
1200.2.h.d \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{6}q^{3}-\zeta_{6}q^{7}-3q^{9}-5q^{13}-5\zeta_{6}q^{19}+\cdots\)
1200.2.h.e \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{6}q^{3}-2\zeta_{6}q^{7}-3q^{9}+2q^{13}+\cdots\)
1200.2.h.f \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{6}q^{3}-\zeta_{6}q^{7}-3q^{9}+5q^{13}+5\zeta_{6}q^{19}+\cdots\)
1200.2.h.g \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{6}q^{3}-3\zeta_{6}q^{7}-3q^{9}+7q^{13}+\cdots\)
1200.2.h.h \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(0\) \(q+(1+\zeta_{6})q^{3}+3\zeta_{6}q^{9}-3q^{11}-2q^{13}+\cdots\)
1200.2.h.i \(2\) \(9.582\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(0\) \(q+(2-\zeta_{6})q^{3}+(3-3\zeta_{6})q^{9}+3q^{11}+\cdots\)
1200.2.h.j \(4\) \(9.582\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(-4\) \(0\) \(0\) \(q+(-1+\beta _{1})q^{3}+(-1-2\beta _{1})q^{9}+\beta _{3}q^{11}+\cdots\)
1200.2.h.k \(4\) \(9.582\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{7}+\beta _{2}q^{9}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
1200.2.h.l \(4\) \(9.582\) \(\Q(\sqrt{-2}, \sqrt{-5})\) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
1200.2.h.m \(4\) \(9.582\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}q^{3}-\zeta_{12}^{2}q^{7}+3q^{9}-2\zeta_{12}q^{11}+\cdots\)
1200.2.h.n \(4\) \(9.582\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(4\) \(0\) \(0\) \(q+(1-\beta _{1})q^{3}+(-1-2\beta _{1})q^{9}-\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1200, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)