Space of modular forms of level 1200, weight 2, and Character 1200.v

• Label: 1200.2.v
• Level: $1200$
• Weight: $2$
• Character orbit: 1200.v
• Rep. character: $\chi_{1200}(257,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $68$
• Newform subspaces: $13$
• Sturm bound: $480$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.v (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 13 \)
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	552	76	476
Cusp forms	408	68	340
Eisenstein series	144	8	136

Trace form

\( 68 q - 2 q^{3} - 4 q^{7} + O(q^{10}) \)

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

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.2.v.a	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(\zeta_{8})\)	None		$4$	$1$	\(0\)	\(-4\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(-2+2\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
1200.2.v.b	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(-1+\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
1200.2.v.c	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8}+\zeta_{8}^{2})q^{3}+(3+3\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
1200.2.v.d	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\zeta_{8}+\zeta_{8}^{3})q^{3}+(-1-\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
1200.2.v.e	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{3}q^{3}+2\beta _{1}q^{7}-3\beta _{2}q^{9}-4\beta _{3}q^{13}+\cdots\)
1200.2.v.f	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{3}q^{3}+\beta _{1}q^{7}-3\beta _{2}q^{9}+3\beta _{3}q^{13}+\cdots\)
1200.2.v.g	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-15}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{3}q^{3}-3\beta _{2}q^{9}+4\beta _{1}q^{17}-4\beta _{2}q^{19}+\cdots\)
1200.2.v.h	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta _{3}q^{3}+3\beta _{1}q^{7}-3\beta _{2}q^{9}-\beta _{3}q^{13}+\cdots\)
1200.2.v.i	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{3})q^{3}+(-1+\beta _{2})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
1200.2.v.j	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\zeta_{8}^{2})q^{3}+(-1-\zeta_{8}+\zeta_{8}^{2}+\zeta_{8}^{3})q^{7}+\cdots\)
1200.2.v.k	$1200$	$2$	1200.v	15.e	$4$	$4$	$2$	$9.582$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(2-2\zeta_{8}^{2})q^{7}+\cdots\)
1200.2.v.l	$1200$	$2$	1200.v	15.e	$4$	$8$	$4$	$9.582$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{24}^{3}q^{3}-2\zeta_{24}q^{7}+(2\zeta_{24}^{2}-\zeta_{24}^{6}+\cdots)q^{9}+\cdots\)
1200.2.v.m	$1200$	$2$	1200.v	15.e	$4$	$16$	$8$	$9.582$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{9}q^{3}+(-\beta _{1}-\beta _{2}-\beta _{5})q^{7}+(-\beta _{3}+\cdots)q^{9}+\cdots\)

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

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