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

• Label: 975.2.o
• Level: $975$
• Weight: $2$
• Character orbit: 975.o
• Rep. character: $\chi_{975}(476,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $164$
• Newform subspaces: $17$
• Sturm bound: $280$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.o (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(280\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\), \(37\)

Dimensions

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

	Total	New	Old
Modular forms	304	188	116
Cusp forms	256	164	92
Eisenstein series	48	24	24

Trace form

164 * q + 4 * q^3 - 16 * q^6 + 12 * q^7 + 4 * q^9 + 8 * q^13 - 132 * q^16 - 12 * q^18 + 24 * q^19 + 12 * q^21 + 16 * q^22 - 28 * q^24 + 4 * q^27 - 36 * q^28 - 48 * q^31 + 20 * q^33 - 8 * q^34 - 36 * q^37 - 32 * q^39 + 8 * q^46 - 4 * q^48 - 12 * q^52 + 68 * q^54 + 40 * q^57 - 32 * q^58 - 64 * q^61 - 4 * q^63 + 52 * q^67 - 108 * q^72 + 60 * q^73 + 84 * q^76 + 16 * q^78 + 24 * q^79 - 52 * q^81 - 52 * q^84 - 32 * q^87 + 8 * q^91 + 88 * q^93 - 16 * q^94 + 44 * q^96 - 52 * q^97 - 64 * q^99

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.o.a	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(\zeta_{12})\)	None		✓			975.2.o.a	$4$	$0$	\(0\)	\(-6\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-2\zeta_{12}-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+\cdots\)
975.2.o.b	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(i, \sqrt{11})\)	None		✓			975.2.o.b	$4$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}+2\beta _{2}q^{4}+(-3-\beta _{3})q^{9}+\cdots\)
975.2.o.c	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(i, \sqrt{6})\)	None		✓			975.2.o.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-3q^{6}+(-1+\cdots)q^{7}+\cdots\)
975.2.o.d	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(i, \sqrt{6})\)	None		✓			975.2.o.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-3q^{6}+(1+\cdots)q^{7}+\cdots\)
975.2.o.e	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		✓			975.2.o.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2\zeta_{12}-\zeta_{12}^{3})q^{3}-2\zeta_{12}^{3}q^{4}+(-2+\cdots)q^{7}+\cdots\)
975.2.o.f	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(i, \sqrt{6})\)	None		✓			975.2.o.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{1}q^{3}+\beta _{2}q^{4}+3\beta _{2}q^{6}+\cdots\)
975.2.o.g	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(i, \sqrt{6})\)	None		✓			975.2.o.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{1}q^{3}+\beta _{2}q^{4}+3\beta _{2}q^{6}+\cdots\)
975.2.o.h	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		✓			975.2.o.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2\zeta_{12}+\zeta_{12}^{3})q^{3}-2\zeta_{12}^{3}q^{4}+\cdots\)
975.2.o.i	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(i, \sqrt{11})\)	None		✓			975.2.o.b	$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+2\beta _{2}q^{4}+(-3-\beta _{3})q^{9}+\cdots\)
975.2.o.j	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(\zeta_{8})\)	None					39.2.f.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(1+\zeta_{8}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)
975.2.o.k	$975$	$2$	975.o	39.f	$4$	$4$	$2$	$7.785$	\(\Q(\zeta_{12})\)	None		✓			975.2.o.a	$4$	$0$	\(0\)	\(6\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-2\zeta_{12}-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+\cdots\)
975.2.o.l	$975$	$2$	975.o	39.f	$4$	$8$	$4$	$7.785$	8.0.619810816.2	None		✓			975.2.o.l	$2$	$1$	\(-6\)	\(-2\)	\(0\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{3}+\beta _{4}-\beta _{7})q^{2}+(-\beta _{1}+\cdots)q^{3}+\cdots\)
975.2.o.m	$975$	$2$	975.o	39.f	$4$	$8$	$4$	$7.785$	8.0.619810816.2	None		✓			975.2.o.l	$2$	$0$	\(-6\)	\(2\)	\(0\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{3}+\beta _{4}-\beta _{7})q^{2}+(-\beta _{6}+\cdots)q^{3}+\cdots\)
975.2.o.n	$975$	$2$	975.o	39.f	$4$	$8$	$4$	$7.785$	8.0.619810816.2	None		✓			975.2.o.l	$2$	$0$	\(6\)	\(-2\)	\(0\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{3}-\beta _{4}+\beta _{7})q^{2}+(\beta _{6}+\beta _{7})q^{3}+\cdots\)
975.2.o.o	$975$	$2$	975.o	39.f	$4$	$8$	$4$	$7.785$	8.0.619810816.2	None		✓			975.2.o.l	$2$	$0$	\(6\)	\(2\)	\(0\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{3}-\beta _{4}+\beta _{7})q^{2}+(\beta _{1}+\beta _{5}+\cdots)q^{3}+\cdots\)
975.2.o.p	$975$	$2$	975.o	39.f	$4$	$40$	$20$	$7.785$		None					195.2.o.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)		$\mathrm{SU}(2)[C_{4}]$
975.2.o.q	$975$	$2$	975.o	39.f	$4$	$48$	$24$	$7.785$		None			✓		195.2.n.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

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