Space of modular forms of level 4050, weight 2, and Character 4050.c

• Label: 4050.2.c
• Level: $4050$
• Weight: $2$
• Character orbit: 4050.c
• Rep. character: $\chi_{4050}(649,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $28$
• Sturm bound: $1620$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(1620\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(19\), \(29\), \(41\), \(71\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4050, [\chi])\).

	Total	New	Old
Modular forms	882	72	810
Cusp forms	738	72	666
Eisenstein series	144	0	144

Trace form

\( 72 q - 72 q^{4} + 72 q^{16} - 24 q^{31} - 12 q^{34} + 24 q^{46} - 144 q^{49} + 36 q^{61} - 72 q^{64} - 24 q^{79} + 24 q^{94}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4050, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4050.2.c.a	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					90.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+i q^{7}-i q^{8}-6 q^{11}+\cdots\)
4050.2.c.b	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None		✓			4050.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+i q^{7}+i q^{8}-6 q^{11}+\cdots\)
4050.2.c.c	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					18.2.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+2 i q^{7}-i q^{8}-3 q^{11}+\cdots\)
4050.2.c.d	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					90.2.e.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+4 i q^{7}+i q^{8}-3 q^{11}+\cdots\)
4050.2.c.e	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					450.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+4 i q^{7}+i q^{8}-3 q^{11}+\cdots\)
4050.2.c.f	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					810.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+5 i q^{7}-i q^{8}+i q^{13}+\cdots\)
4050.2.c.g	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					162.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+4 i q^{7}-i q^{8}-i q^{13}+\cdots\)
4050.2.c.h	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					450.2.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+2 i q^{7}-i q^{8}+4 i q^{13}+\cdots\)
4050.2.c.i	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					810.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+i q^{7}-i q^{8}+5 i q^{13}+\cdots\)
4050.2.c.j	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None		✓			4050.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+i q^{7}-i q^{8}-4 i q^{13}+\cdots\)
4050.2.c.k	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					810.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+i q^{7}+i q^{8}+5 i q^{13}+\cdots\)
4050.2.c.l	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None		✓			4050.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+i q^{7}+i q^{8}-4 i q^{13}+\cdots\)
4050.2.c.m	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					450.2.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+2 i q^{7}+i q^{8}+4 i q^{13}+\cdots\)
4050.2.c.n	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					162.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+4 i q^{7}+i q^{8}-i q^{13}+\cdots\)
4050.2.c.o	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					810.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+5 i q^{7}+i q^{8}+i q^{13}+\cdots\)
4050.2.c.p	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					90.2.e.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+4 i q^{7}-i q^{8}+3 q^{11}+\cdots\)
4050.2.c.q	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					450.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+4 i q^{7}-i q^{8}+3 q^{11}+\cdots\)
4050.2.c.r	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					18.2.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+2 i q^{7}+i q^{8}+3 q^{11}+\cdots\)
4050.2.c.s	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None		✓			4050.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{4}+i q^{7}-i q^{8}+6 q^{11}+\cdots\)
4050.2.c.t	$4050$	$2$	4050.c	5.b	$2$	$2$	$2$	$32.339$	\(\Q(\sqrt{-1}) \)	None					90.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-q^{4}+i q^{7}+i q^{8}+6 q^{11}+\cdots\)
4050.2.c.u	$4050$	$2$	4050.c	5.b	$2$	$4$	$4$	$32.339$	\(\Q(\zeta_{12})\)	None		✓			4050.2.a.bn	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{2}-q^{4}+(-2\beta_{2}+\beta_1)q^{7}+\cdots\)
4050.2.c.v	$4050$	$2$	4050.c	5.b	$2$	$4$	$4$	$32.339$	\(\Q(i, \sqrt{33})\)	None					90.2.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-q^{4}+\beta _{1}q^{7}-\beta _{2}q^{8}+(-2+\cdots)q^{11}+\cdots\)
4050.2.c.w	$4050$	$2$	4050.c	5.b	$2$	$4$	$4$	$32.339$	\(\Q(i, \sqrt{6})\)	None					450.2.e.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-q^{4}+(2\beta _{1}-\beta _{2})q^{7}-\beta _{1}q^{8}+\cdots\)
4050.2.c.x	$4050$	$2$	4050.c	5.b	$2$	$4$	$4$	$32.339$	\(\Q(\zeta_{12})\)	None					810.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{2}-q^{4}+(\beta_{2}+2\beta_1)q^{7}+\cdots\)
4050.2.c.y	$4050$	$2$	4050.c	5.b	$2$	$4$	$4$	$32.339$	\(\Q(i, \sqrt{6})\)	None					450.2.e.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-q^{4}+(2\beta _{1}-\beta _{2})q^{7}+\beta _{1}q^{8}+\cdots\)
4050.2.c.z	$4050$	$2$	4050.c	5.b	$2$	$4$	$4$	$32.339$	\(\Q(\zeta_{12})\)	None					810.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{2}-q^{4}+(\beta_{2}+2\beta_1)q^{7}+\cdots\)
4050.2.c.ba	$4050$	$2$	4050.c	5.b	$2$	$4$	$4$	$32.339$	\(\Q(i, \sqrt{33})\)	None					90.2.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-q^{4}+\beta _{1}q^{7}+\beta _{2}q^{8}+(2+\cdots)q^{11}+\cdots\)
4050.2.c.bb	$4050$	$2$	4050.c	5.b	$2$	$4$	$4$	$32.339$	\(\Q(\zeta_{12})\)	None		✓			4050.2.a.bn	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{2}-q^{4}+(-2\beta_{2}+\beta_1)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4050, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4050, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(810, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1350, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2025, [\chi])\)\(^{\oplus 2}\)