Space of modular forms of level 1050, weight 2, and Character 1050.bc

• Label: 1050.2.bc
• Level: $1050$
• Weight: $2$
• Character orbit: 1050.bc
• Rep. character: $\chi_{1050}(157,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $96$
• Newform subspaces: $8$
• Sturm bound: $480$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.bc (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 35 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(480\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1050, [\chi])\).

	Total	New	Old
Modular forms	1056	96	960
Cusp forms	864	96	768
Eisenstein series	192	0	192

Trace form

96 * q + 12 * q^7 - 16 * q^11 + 48 * q^16 + 8 * q^21 - 8 * q^22 + 8 * q^23 + 48 * q^26 + 12 * q^28 + 96 * q^31 - 12 * q^33 - 96 * q^36 + 16 * q^37 + 48 * q^38 - 4 * q^42 + 48 * q^43 + 16 * q^46 + 24 * q^47 - 32 * q^51 - 16 * q^53 - 32 * q^56 + 16 * q^57 - 36 * q^58 + 24 * q^61 + 48 * q^67 + 128 * q^71 + 24 * q^73 + 32 * q^78 + 48 * q^81 - 48 * q^82 + 32 * q^86 - 60 * q^87 - 4 * q^88 + 8 * q^91 - 16 * q^92 - 8 * q^93 - 16 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(1050, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1050.2.bc.a	$1050$	$2$	1050.bc	35.k	$12$	$8$	$2$	$8.384$	\(\Q(\zeta_{24})\)	None		✓			1050.2.bc.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q-\zeta_{24}^{7}q^{2}+(-\zeta_{24}^{3}+\zeta_{24}^{7})q^{3}+\cdots\)
1050.2.bc.b	$1050$	$2$	1050.bc	35.k	$12$	$8$	$2$	$8.384$	\(\Q(\zeta_{24})\)	None		✓			1050.2.bc.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{24}^{3}-\zeta_{24}^{7})q^{2}+\zeta_{24}^{7}q^{3}+(\zeta_{24}^{2}+\cdots)q^{4}+\cdots\)
1050.2.bc.c	$1050$	$2$	1050.bc	35.k	$12$	$8$	$2$	$8.384$	\(\Q(\zeta_{24})\)	None		✓			1050.2.bc.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q-\zeta_{24}^{7}q^{2}+(\zeta_{24}^{3}-\zeta_{24}^{7})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
1050.2.bc.d	$1050$	$2$	1050.bc	35.k	$12$	$8$	$2$	$8.384$	\(\Q(\zeta_{24})\)	None		✓			1050.2.bc.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q-\zeta_{24}q^{2}-\zeta_{24}^{5}q^{3}+\zeta_{24}^{2}q^{4}+\zeta_{24}^{6}q^{6}+\cdots\)
1050.2.bc.e	$1050$	$2$	1050.bc	35.k	$12$	$16$	$4$	$8.384$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			1050.2.bc.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{2}q^{2}-\beta _{11}q^{3}+(\beta _{8}-\beta _{9})q^{4}-\beta _{9}q^{6}+\cdots\)
1050.2.bc.f	$1050$	$2$	1050.bc	35.k	$12$	$16$	$4$	$8.384$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			1050.2.bc.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{11}q^{2}+\beta _{2}q^{3}-\beta _{8}q^{4}+\beta _{9}q^{6}+\cdots\)
1050.2.bc.g	$1050$	$2$	1050.bc	35.k	$12$	$16$	$4$	$8.384$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					210.2.u.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q-\beta _{6}q^{2}-\beta _{12}q^{3}+(-\beta _{5}+\beta _{13})q^{4}+\cdots\)
1050.2.bc.h	$1050$	$2$	1050.bc	35.k	$12$	$16$	$4$	$8.384$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					210.2.u.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{15}q^{2}+\beta _{2}q^{3}+(\beta _{5}-\beta _{13})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)