Space of modular forms of level 4800, weight 2, and Character 4800.k

• Label: 4800.2.k
• Level: $4800$
• Weight: $2$
• Character orbit: 4800.k
• Rep. character: $\chi_{4800}(2401,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $19$
• Sturm bound: $1920$
• Trace bound: $57$

Defining parameters

Level:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(1920\)
Trace bound:			\(57\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(23\), \(73\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4800, [\chi])\).

	Total	New	Old
Modular forms	1032	76	956
Cusp forms	888	76	812
Eisenstein series	144	0	144

Trace form

\( 76 q - 76 q^{9} - 24 q^{17} + 24 q^{41} + 92 q^{49} - 16 q^{57} - 40 q^{73} + 76 q^{81} - 24 q^{89} + 40 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4800, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4800.2.k.a	$4800$	$2$	4800.k	8.b	$2$	$2$	$2$	$38.328$	\(\Q(\sqrt{-1}) \)	None					960.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}-4 q^{7}-q^{9}+4 i q^{11}-2 i q^{13}+\cdots\)
4800.2.k.b	$4800$	$2$	4800.k	8.b	$2$	$2$	$2$	$38.328$	\(\Q(\sqrt{-1}) \)	None					960.2.d.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-2 q^{7}-q^{9}+2 i q^{11}-2 i q^{13}+\cdots\)
4800.2.k.c	$4800$	$2$	4800.k	8.b	$2$	$2$	$2$	$38.328$	\(\Q(\sqrt{-1}) \)	None					960.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}-2 q^{7}-q^{9}+2 i q^{11}-2 i q^{13}+\cdots\)
4800.2.k.d	$4800$	$2$	4800.k	8.b	$2$	$2$	$2$	$38.328$	\(\Q(\sqrt{-1}) \)	None					960.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-q^{9}-6 i q^{13}+2 i q^{19}+\cdots\)
4800.2.k.e	$4800$	$2$	4800.k	8.b	$2$	$2$	$2$	$38.328$	\(\Q(\sqrt{-1}) \)	None					960.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}-q^{9}-6 i q^{13}-2 i q^{19}+\cdots\)
4800.2.k.f	$4800$	$2$	4800.k	8.b	$2$	$2$	$2$	$38.328$	\(\Q(\sqrt{-1}) \)	None					960.2.d.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{7}-q^{9}+2 i q^{11}+2 i q^{13}+\cdots\)
4800.2.k.g	$4800$	$2$	4800.k	8.b	$2$	$2$	$2$	$38.328$	\(\Q(\sqrt{-1}) \)	None					960.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{7}-q^{9}-2 i q^{11}-2 i q^{13}+\cdots\)
4800.2.k.h	$4800$	$2$	4800.k	8.b	$2$	$2$	$2$	$38.328$	\(\Q(\sqrt{-1}) \)	None					960.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+4 q^{7}-q^{9}+4 i q^{11}+2 i q^{13}+\cdots\)
4800.2.k.i	$4800$	$2$	4800.k	8.b	$2$	$4$	$4$	$38.328$	\(\Q(\zeta_{12})\)	None					960.2.k.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{3}-2 q^{7}-q^{9}+(\beta_{2}-2\beta_1)q^{11}+\cdots\)
4800.2.k.j	$4800$	$2$	4800.k	8.b	$2$	$4$	$4$	$38.328$	\(\Q(\zeta_{12})\)	None					192.2.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}-\beta_{3} q^{7}-q^{9}-6 q^{17}+\cdots\)
4800.2.k.k	$4800$	$2$	4800.k	8.b	$2$	$4$	$4$	$38.328$	\(\Q(i, \sqrt{19})\)	None		✓			4800.2.k.k	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{7}-q^{9}-2\beta _{1}q^{11}+\cdots\)
4800.2.k.l	$4800$	$2$	4800.k	8.b	$2$	$4$	$4$	$38.328$	\(\Q(\zeta_{12})\)	None		✓			4800.2.k.l	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{3}-\beta_{3} q^{7}-q^{9}+\beta_{2} q^{13}+\cdots\)
4800.2.k.m	$4800$	$2$	4800.k	8.b	$2$	$4$	$4$	$38.328$	\(\Q(\zeta_{12})\)	None		✓			4800.2.k.l	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}-\beta_{3} q^{7}-q^{9}+\beta_{2} q^{13}+\cdots\)
4800.2.k.n	$4800$	$2$	4800.k	8.b	$2$	$4$	$4$	$38.328$	\(\Q(i, \sqrt{19})\)	None		✓			4800.2.k.k	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{2}q^{7}-q^{9}-2\beta _{1}q^{11}+\cdots\)
4800.2.k.o	$4800$	$2$	4800.k	8.b	$2$	$4$	$4$	$38.328$	\(\Q(\zeta_{12})\)	None					960.2.k.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}+2 q^{7}-q^{9}+(-\beta_{2}+2\beta_1)q^{11}+\cdots\)
4800.2.k.p	$4800$	$2$	4800.k	8.b	$2$	$8$	$8$	$38.328$	8.0.49787136.1	None		✓			4800.2.k.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{7}q^{7}-q^{9}+(-\beta _{2}-\beta _{6}+\cdots)q^{11}+\cdots\)
4800.2.k.q	$4800$	$2$	4800.k	8.b	$2$	$8$	$8$	$38.328$	8.0.49787136.1	None					960.2.d.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-\beta _{6}q^{7}-q^{9}+(-\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)
4800.2.k.r	$4800$	$2$	4800.k	8.b	$2$	$8$	$8$	$38.328$	8.0.49787136.1	None					960.2.d.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-\beta _{6}q^{7}-q^{9}+(-\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)
4800.2.k.s	$4800$	$2$	4800.k	8.b	$2$	$8$	$8$	$38.328$	8.0.49787136.1	None		✓			4800.2.k.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{7}q^{7}-q^{9}+(-\beta _{2}-\beta _{6}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4800, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4800, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(960, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1600, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2400, [\chi])\)\(^{\oplus 2}\)