Properties

Label 1200.2.o
Level $1200$
Weight $2$
Character orbit 1200.o
Rep. character $\chi_{1200}(1199,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $9$
Sturm bound $480$
Trace bound $21$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.o (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 60 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(480\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

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

Total New Old
Modular forms 276 36 240
Cusp forms 204 36 168
Eisenstein series 72 0 72

Trace form

\( 36 q + O(q^{10}) \) \( 36 q - 24 q^{21} + 60 q^{49} + 48 q^{61} + 72 q^{81} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1200.2.o.a 1200.o 60.h $4$ $9.582$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{7}-3q^{9}-\zeta_{12}^{3}q^{11}+\cdots\)
1200.2.o.b 1200.o 60.h $4$ $9.582$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{7}-3q^{9}+\zeta_{12}^{3}q^{11}+\cdots\)
1200.2.o.c 1200.o 60.h $4$ $9.582$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+(-3+3\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
1200.2.o.d 1200.o 60.h $4$ $9.582$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+(-3+3\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
1200.2.o.e 1200.o 60.h $4$ $9.582$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+(\beta _{1}-\beta _{3})q^{7}-3\beta _{2}q^{9}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
1200.2.o.f 1200.o 60.h $4$ $9.582$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+(\beta _{1}-\beta _{3})q^{7}-3\beta _{2}q^{9}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
1200.2.o.g 1200.o 60.h $4$ $9.582$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\zeta_{12}^{3}q^{3}+3\zeta_{12}^{3}q^{7}+3q^{9}+7\zeta_{12}q^{13}+\cdots\)
1200.2.o.h 1200.o 60.h $4$ $9.582$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{12}q^{3}+\zeta_{12}q^{7}+3q^{9}-\zeta_{12}^{3}q^{13}+\cdots\)
1200.2.o.i 1200.o 60.h $4$ $9.582$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{12}q^{3}+2\zeta_{12}q^{7}+3q^{9}-\zeta_{12}^{3}q^{13}+\cdots\)

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

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