Properties

Label 1200.3.c
Level $1200$
Weight $3$
Character orbit 1200.c
Rep. character $\chi_{1200}(449,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $13$
Sturm bound $720$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(720\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1200, [\chi])\).

Total New Old
Modular forms 516 74 442
Cusp forms 444 70 374
Eisenstein series 72 4 68

Trace form

\( 70 q + 2 q^{9} + O(q^{10}) \) \( 70 q + 2 q^{9} - 36 q^{19} - 4 q^{21} - 60 q^{31} - 116 q^{39} - 322 q^{49} - 208 q^{51} - 132 q^{61} - 128 q^{69} - 180 q^{79} + 142 q^{81} + 104 q^{91} - 96 q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1200, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1200.3.c.a 1200.c 15.d $2$ $32.698$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) 75.3.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3iq^{3}+11iq^{7}-9q^{9}-iq^{13}+\cdots\)
1200.3.c.b 1200.c 15.d $2$ $32.698$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) 300.3.g.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3iq^{3}+13iq^{7}-9q^{9}-23iq^{13}+\cdots\)
1200.3.c.c 1200.c 15.d $2$ $32.698$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) 12.3.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3iq^{3}+2iq^{7}-9q^{9}-22iq^{13}+\cdots\)
1200.3.c.d 1200.c 15.d $4$ $32.698$ \(\Q(i, \sqrt{11})\) None 75.3.c.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(-3+\beta _{3})q^{9}+(-1-2\beta _{3})q^{11}+\cdots\)
1200.3.c.e 1200.c 15.d $4$ $32.698$ \(\Q(i, \sqrt{5})\) None 60.3.g.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{1}q^{7}+(1+2\beta _{2}+\cdots)q^{9}+\cdots\)
1200.3.c.f 1200.c 15.d $4$ $32.698$ \(\Q(i, \sqrt{5})\) None 15.3.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{3})q^{3}-3\beta _{1}q^{7}+(1+2\beta _{2}+\cdots)q^{9}+\cdots\)
1200.3.c.g 1200.c 15.d $4$ $32.698$ \(\Q(\zeta_{8})\) None 600.3.l.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}q^{7}+(7-2\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1200.3.c.h 1200.c 15.d $4$ $32.698$ \(\Q(\zeta_{8})\) None 150.3.d.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\zeta_{8}+\zeta_{8}^{3})q^{3}+7\zeta_{8}q^{7}+(7-2\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1200.3.c.i 1200.c 15.d $4$ $32.698$ \(\Q(\zeta_{8})\) None 24.3.e.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\zeta_{8}-\zeta_{8}^{2})q^{3}-3\zeta_{8}q^{7}+(7+\zeta_{8}^{3})q^{9}+\cdots\)
1200.3.c.j 1200.c 15.d $4$ $32.698$ \(\Q(i, \sqrt{35})\) None 300.3.g.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-8\beta _{2}q^{7}+(9+\beta _{3})q^{9}+(3+\cdots)q^{11}+\cdots\)
1200.3.c.k 1200.c 15.d $8$ $32.698$ 8.0.40960000.1 None 30.3.d.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+(-2\beta _{1}-2\beta _{2}+\beta _{7})q^{7}+\cdots\)
1200.3.c.l 1200.c 15.d $12$ $32.698$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 600.3.l.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{6}-\beta _{10})q^{3}+(-2\beta _{6}-\beta _{7}+\beta _{8}+\cdots)q^{7}+\cdots\)
1200.3.c.m 1200.c 15.d $16$ $32.698$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 120.3.l.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{7}+(-3+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1200, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)