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

• Label: 975.2.i
• Level: $975$
• Weight: $2$
• Character orbit: 975.i
• Rep. character: $\chi_{975}(451,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $86$
• 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.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 17 \)
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	86	218
Cusp forms	256	86	170
Eisenstein series	48	0	48

Trace form

86 * q + 2 * q^2 - q^3 - 40 * q^4 + q^7 - 12 * q^8 - 43 * q^9 - 2 * q^11 - 4 * q^12 - 5 * q^13 + 32 * q^14 - 38 * q^16 - 10 * q^17 - 4 * q^18 + 2 * q^19 - 14 * q^21 - 8 * q^22 + 10 * q^23 - 12 * q^24 - 18 * q^26 + 2 * q^27 + 18 * q^28 + 12 * q^29 + 34 * q^31 - 6 * q^32 - 2 * q^33 - 108 * q^34 - 40 * q^36 + 4 * q^37 - 40 * q^38 + 22 * q^41 - 8 * q^42 + 9 * q^43 + 64 * q^44 - 8 * q^46 + 36 * q^47 + 12 * q^48 - 34 * q^49 - 32 * q^51 + 2 * q^52 + 60 * q^53 - 32 * q^56 + 24 * q^57 - 18 * q^58 + 2 * q^59 + 7 * q^61 - 16 * q^62 + q^63 + 132 * q^64 - 32 * q^66 - 3 * q^67 - 54 * q^68 + 10 * q^69 + 54 * q^71 + 6 * q^72 - 50 * q^73 - 46 * q^74 + 56 * q^76 + 36 * q^77 + 12 * q^78 - 74 * q^79 - 43 * q^81 + 10 * q^82 + 56 * q^83 + 22 * q^84 + 48 * q^86 - 30 * q^87 - 28 * q^88 - 12 * q^89 - 7 * q^91 - 136 * q^92 + 3 * q^93 + 4 * q^94 + 48 * q^96 + 11 * q^97 + 94 * q^98 + 4 * 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.i.a	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.i.a	$2$	$0$	\(-2\)	\(-1\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
975.2.i.b	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.i.b	$2$	$0$	\(-1\)	\(1\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.c	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.i.c	$2$	$0$	\(-1\)	\(1\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.d	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None					195.2.i.b	$2$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{4}-\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
975.2.i.e	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.i.c	$2$	$0$	\(1\)	\(-1\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.f	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None					39.2.e.a	$2$	$0$	\(1\)	\(-1\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.g	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.i.b	$2$	$0$	\(1\)	\(-1\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.h	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None		✓			975.2.i.a	$2$	$0$	\(2\)	\(1\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
975.2.i.i	$975$	$2$	975.i	13.c	$3$	$2$	$1$	$7.785$	\(\Q(\sqrt{-3}) \)	None					195.2.i.a	$2$	$0$	\(2\)	\(1\)	\(0\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
975.2.i.j	$975$	$2$	975.i	13.c	$3$	$4$	$2$	$7.785$	\(\Q(\zeta_{12})\)	None					195.2.i.c	$2$	$0$	\(-2\)	\(2\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta_{2}-\beta_1)q^{2}+\beta_1 q^{3}+(2\beta_{3}-2\beta_{2}+2\beta_1-2)q^{4}+\cdots\)
975.2.i.k	$975$	$2$	975.i	13.c	$3$	$4$	$2$	$7.785$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None					39.2.e.b	$2$	$0$	\(1\)	\(2\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
975.2.i.l	$975$	$2$	975.i	13.c	$3$	$6$	$3$	$7.785$	6.0.1714608.1	None					195.2.i.d	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{3}-\beta _{5})q^{2}+(-1-\beta _{4})q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
975.2.i.m	$975$	$2$	975.i	13.c	$3$	$6$	$3$	$7.785$	6.0.591408.1	None					195.2.i.e	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{5}q^{2}+(-1+\beta _{4})q^{3}+(\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
975.2.i.n	$975$	$2$	975.i	13.c	$3$	$12$	$6$	$7.785$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			975.2.i.n	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{3})q^{2}+(-1-\beta _{6})q^{3}+(\beta _{6}+\cdots)q^{4}+\cdots\)
975.2.i.o	$975$	$2$	975.i	13.c	$3$	$12$	$6$	$7.785$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					195.2.ba.a	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{5}q^{2}-\beta _{3}q^{3}+(-1+\beta _{3}-\beta _{6}+\cdots)q^{4}+\cdots\)
975.2.i.p	$975$	$2$	975.i	13.c	$3$	$12$	$6$	$7.785$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			975.2.i.n	$2$	$0$	\(0\)	\(6\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{3})q^{2}+(1+\beta _{6})q^{3}+(\beta _{6}-\beta _{9}+\cdots)q^{4}+\cdots\)
975.2.i.q	$975$	$2$	975.i	13.c	$3$	$12$	$6$	$7.785$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					195.2.ba.a	$2$	$0$	\(0\)	\(6\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{5}q^{2}+\beta _{3}q^{3}+(-1+\beta _{3}-\beta _{6}+\cdots)q^{4}+\cdots\)

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

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