Space of modular forms of level 1350, weight 4, and Character 1350.c

• Label: 1350.4.c
• Level: $1350$
• Weight: $4$
• Character orbit: 1350.c
• Rep. character: $\chi_{1350}(649,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $28$
• Sturm bound: $1080$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(1080\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1350, [\chi])\).

	Total	New	Old
Modular forms	846	72	774
Cusp forms	774	72	702
Eisenstein series	72	0	72

Trace form

\( 72 q - 288 q^{4} + 1152 q^{16} + 24 q^{19} + 564 q^{31} + 912 q^{34} + 288 q^{46} - 6324 q^{49} - 576 q^{61} - 4608 q^{64} - 96 q^{76} + 3396 q^{79} - 7572 q^{91} + 6192 q^{94}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1350, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1350.4.c.a	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					54.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+7 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.b	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					54.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+29 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.c	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+34 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.d	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+4 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.e	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None		✓			1350.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+23 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.f	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+13 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.g	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+8 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.h	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None		✓			1350.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+19 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.i	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+22 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.j	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+14 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.k	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+14 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.l	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+22 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.m	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None		✓			1350.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+19 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.n	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+8 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.o	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+13 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.p	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None		✓			1350.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+23 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.q	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+4 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.r	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					270.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-4 q^{4}+34 i q^{7}-8 i q^{8}+\cdots\)
1350.4.c.s	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					54.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+29 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.t	$1350$	$4$	1350.c	5.b	$2$	$2$	$2$	$79.653$	\(\Q(\sqrt{-1}) \)	None					54.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}-4 q^{4}+7 i q^{7}+8 i q^{8}+\cdots\)
1350.4.c.u	$1350$	$4$	1350.c	5.b	$2$	$4$	$4$	$79.653$	\(\Q(i, \sqrt{401})\)	None					270.4.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}-4q^{4}-\beta _{2}q^{7}+8\beta _{1}q^{8}+\cdots\)
1350.4.c.v	$1350$	$4$	1350.c	5.b	$2$	$4$	$4$	$79.653$	\(\Q(i, \sqrt{21})\)	None		✓			1350.4.a.be	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-4q^{4}+(-5\beta _{1}+\beta _{2})q^{7}+\cdots\)
1350.4.c.w	$1350$	$4$	1350.c	5.b	$2$	$4$	$4$	$79.653$	\(\Q(i, \sqrt{6})\)	None		✓			1350.4.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-4q^{4}+(-10\beta _{1}-\beta _{2})q^{7}+\cdots\)
1350.4.c.x	$1350$	$4$	1350.c	5.b	$2$	$4$	$4$	$79.653$	\(\Q(i, \sqrt{209})\)	None		✓			1350.4.a.bd	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-4q^{4}+(-7\beta _{1}-\beta _{2})q^{7}+\cdots\)
1350.4.c.y	$1350$	$4$	1350.c	5.b	$2$	$4$	$4$	$79.653$	\(\Q(i, \sqrt{209})\)	None		✓			1350.4.a.bd	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}-4q^{4}+(-7\beta _{1}-\beta _{2})q^{7}+\cdots\)
1350.4.c.z	$1350$	$4$	1350.c	5.b	$2$	$4$	$4$	$79.653$	\(\Q(i, \sqrt{6})\)	None		✓			1350.4.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}-4q^{4}+(-10\beta _{1}-\beta _{2})q^{7}+\cdots\)
1350.4.c.ba	$1350$	$4$	1350.c	5.b	$2$	$4$	$4$	$79.653$	\(\Q(i, \sqrt{21})\)	None		✓			1350.4.a.be	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}-4q^{4}+(-5\beta _{1}+\beta _{2})q^{7}+\cdots\)
1350.4.c.bb	$1350$	$4$	1350.c	5.b	$2$	$4$	$4$	$79.653$	\(\Q(i, \sqrt{401})\)	None					270.4.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-4q^{4}-\beta _{2}q^{7}-8\beta _{1}q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1350, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1350, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 2}\)