Space of modular forms of level 1050, weight 6, and Character 1050.g

• Label: 1050.6.g
• Level: $1050$
• Weight: $6$
• Character orbit: 1050.g
• Rep. character: $\chi_{1050}(799,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $29$
• Sturm bound: $1440$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(1440\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(1050, [\chi])\).

	Total	New	Old
Modular forms	1224	92	1132
Cusp forms	1176	92	1084
Eisenstein series	48	0	48

Trace form

92 * q - 1472 * q^4 - 7452 * q^9 + 23552 * q^16 + 7024 * q^19 + 1764 * q^21 + 19104 * q^26 + 8680 * q^29 - 9680 * q^31 + 17056 * q^34 + 119232 * q^36 - 22320 * q^39 - 44376 * q^41 + 62304 * q^46 - 220892 * q^49 + 20808 * q^51 + 121072 * q^59 - 132504 * q^61 - 376832 * q^64 + 54720 * q^66 - 140976 * q^69 - 160816 * q^71 + 54176 * q^74 - 112384 * q^76 - 336048 * q^79 + 603612 * q^81 - 28224 * q^84 - 123840 * q^86 - 343672 * q^89 + 66248 * q^91 + 34240 * q^94

Decomposition of \(S_{6}^{\mathrm{new}}(1050, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1050.6.g.a	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}+9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.b	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					42.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}+9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.c	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}+9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.d	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}-9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.e	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}-9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.f	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}-9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.g	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}-9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.h	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					42.6.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}-9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.i	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					42.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}+9 i q^{3}-16 q^{4}-36 q^{6}+\cdots\)
1050.6.g.j	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}+9 i q^{3}-16 q^{4}+36 q^{6}+\cdots\)
1050.6.g.k	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}-9 i q^{3}-16 q^{4}+36 q^{6}+\cdots\)
1050.6.g.l	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					42.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}-9 i q^{3}-16 q^{4}+36 q^{6}+\cdots\)
1050.6.g.m	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					42.6.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}+9 i q^{3}-16 q^{4}+36 q^{6}+\cdots\)
1050.6.g.n	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}+9 i q^{3}-16 q^{4}+36 q^{6}+\cdots\)
1050.6.g.o	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					42.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 i q^{2}+9 i q^{3}-16 q^{4}+36 q^{6}+\cdots\)
1050.6.g.p	$1050$	$6$	1050.g	5.b	$2$	$2$	$2$	$168.403$	\(\Q(\sqrt{-1}) \)	None					210.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}-9 i q^{3}-16 q^{4}+36 q^{6}+\cdots\)
1050.6.g.q	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\Q(i, \sqrt{274})\)	None		✓			1050.6.a.w	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
1050.6.g.r	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					210.6.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
1050.6.g.s	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\Q(i, \sqrt{1066})\)	None					210.6.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
1050.6.g.t	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\Q(i, \sqrt{2059})\)	None		✓			1050.6.a.v	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
1050.6.g.u	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\Q(i, \sqrt{499})\)	None		✓			1050.6.a.q	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}-9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
1050.6.g.v	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\Q(i, \sqrt{46})\)	None		✓			1050.6.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
1050.6.g.w	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					210.6.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
1050.6.g.x	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					210.6.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
1050.6.g.y	$1050$	$6$	1050.g	5.b	$2$	$4$	$4$	$168.403$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					210.6.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}-9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
1050.6.g.z	$1050$	$6$	1050.g	5.b	$2$	$6$	$6$	$168.403$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			1050.6.a.bh	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
1050.6.g.ba	$1050$	$6$	1050.g	5.b	$2$	$6$	$6$	$168.403$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓			1050.6.a.bg	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
1050.6.g.bb	$1050$	$6$	1050.g	5.b	$2$	$6$	$6$	$168.403$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓			1050.6.a.bf	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 5^{2}\cdot 17^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}-9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
1050.6.g.bc	$1050$	$6$	1050.g	5.b	$2$	$6$	$6$	$168.403$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			1050.6.a.be	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(1050, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)