Space of modular forms of level 975, weight 2, and Character 975.bc

• Label: 975.2.bc
• Level: $975$
• Weight: $2$
• Character orbit: 975.bc
• Rep. character: $\chi_{975}(751,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $90$
• Newform subspaces: $14$
• Sturm bound: $280$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.bc (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(280\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(975, [\chi])\).

	Total	New	Old
Modular forms	304	90	214
Cusp forms	256	90	166
Eisenstein series	48	0	48

Trace form

\( 90 q + q^{3} + 46 q^{4} - 3 q^{7} - 45 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(975, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
975.2.bc.a	$975$	$2$	975.bc	13.e	$6$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.bc.a	$2$	$1$	\(-3\)	\(-1\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.bc.b	$975$	$2$	975.bc	13.e	$6$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.bc.b	$2$	$0$	\(-3\)	\(1\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.bc.c	$975$	$2$	975.bc	13.e	$6$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None					39.2.j.a	$2$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(2-\zeta_{6})q^{7}+\cdots\)
975.2.bc.d	$975$	$2$	975.bc	13.e	$6$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.bc.d	$2$	$0$	\(0\)	\(-1\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(4-2\zeta_{6})q^{7}+\cdots\)
975.2.bc.e	$975$	$2$	975.bc	13.e	$6$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.bc.d	$2$	$0$	\(0\)	\(1\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(-4+2\zeta_{6})q^{7}+\cdots\)
975.2.bc.f	$975$	$2$	975.bc	13.e	$6$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.bc.b	$2$	$0$	\(3\)	\(-1\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.bc.g	$975$	$2$	975.bc	13.e	$6$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.bc.a	$2$	$0$	\(3\)	\(1\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.bc.h	$975$	$2$	975.bc	13.e	$6$	$4$	$2$	$7.785$	\(\Q(\zeta_{12})\)	None					195.2.bb.a	$2$	$0$	\(6\)	\(2\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{2}q^{3}+\cdots\)
975.2.bc.i	$975$	$2$	975.bc	13.e	$6$	$8$	$4$	$7.785$	8.0.56070144.2	None					195.2.bb.c	$2$	$0$	\(-6\)	\(4\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{2}+\beta _{5}+\beta _{6})q^{2}+\beta _{6}q^{3}+\cdots\)
975.2.bc.j	$975$	$2$	975.bc	13.e	$6$	$8$	$4$	$7.785$	8.0.191102976.5	None					195.2.bb.b	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-12\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{5}-\beta _{6})q^{2}-\beta _{1}q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
975.2.bc.k	$975$	$2$	975.bc	13.e	$6$	$12$	$6$	$7.785$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			975.2.bc.k	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+\beta _{6}q^{3}+(1-\beta _{2}+\beta _{6}+\cdots)q^{4}+\cdots\)
975.2.bc.l	$975$	$2$	975.bc	13.e	$6$	$12$	$6$	$7.785$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			975.2.bc.k	$2$	$0$	\(0\)	\(6\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}+(1+\beta _{6})q^{3}+(-\beta _{6}-\beta _{10}+\cdots)q^{4}+\cdots\)
975.2.bc.m	$975$	$2$	975.bc	13.e	$6$	$16$	$8$	$7.785$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					195.2.v.a	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{5})q^{2}+(-1-\beta _{3})q^{3}+(-\beta _{3}+\cdots)q^{4}+\cdots\)
975.2.bc.n	$975$	$2$	975.bc	13.e	$6$	$16$	$8$	$7.785$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					195.2.v.a	$2$	$0$	\(0\)	\(8\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{2}-\beta _{3}q^{3}+(1-\beta _{2}+\beta _{3}+\beta _{13}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(975, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(975, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 2}\)