Space of modular forms of level 9900, weight 2, and Character 9900.c

• Label: 9900.2.c
• Level: $9900$
• Weight: $2$
• Character orbit: 9900.c
• Rep. character: $\chi_{9900}(5149,\cdot)$
• Character field: $\Q$
• Dimension: $74$
• Newform subspaces: $26$
• Sturm bound: $4320$
• Trace bound: $59$

Defining parameters

Level:	\( N \)	\(=\)	\( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9900.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(4320\)
Trace bound:			\(59\)
Distinguishing \(T_p\):			\(7\), \(13\), \(17\), \(29\), \(41\), \(59\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(9900, [\chi])\).

	Total	New	Old
Modular forms	2232	74	2158
Cusp forms	2088	74	2014
Eisenstein series	144	0	144

Trace form

\( 74 q + 2 q^{11} + 4 q^{19} + 20 q^{29} - 6 q^{31} - 12 q^{41} - 78 q^{49} + 6 q^{59} - 22 q^{71} - 8 q^{79} + 18 q^{89} - 28 q^{91}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(9900, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
9900.2.c.a	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					3300.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{7}-q^{11}-4 i q^{13}+5 i q^{17}+\cdots\)
9900.2.c.b	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None		✓			9900.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}-q^{11}+3 i q^{13}-8 i q^{17}+\cdots\)
9900.2.c.c	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					660.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{11}+2\beta q^{13}+\beta q^{17}-2 q^{19}+\cdots\)
9900.2.c.d	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					660.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-q^{11}-\beta q^{13}-2 q^{19}+8 q^{31}+\cdots\)
9900.2.c.e	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					220.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{11}-2\beta q^{17}+4 q^{19}-3\beta q^{23}+\cdots\)
9900.2.c.f	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					132.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-q^{11}-\beta q^{13}+2\beta q^{17}+\cdots\)
9900.2.c.g	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					44.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}+q^{11}+4 i q^{13}-6 i q^{17}+\cdots\)
9900.2.c.h	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None		✓			9900.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}+q^{11}+3 i q^{13}+8 i q^{17}+\cdots\)
9900.2.c.i	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					3300.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}+q^{11}-4 i q^{13}+3 i q^{17}+\cdots\)
9900.2.c.j	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					660.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+q^{11}-2\beta q^{13}-3\beta q^{17}+\cdots\)
9900.2.c.k	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					132.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+q^{11}-3\beta q^{13}+2\beta q^{17}+\cdots\)
9900.2.c.l	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					660.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+q^{11}+\beta q^{13}+4\beta q^{17}+\cdots\)
9900.2.c.m	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					220.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+q^{11}-2\beta q^{13}+4 q^{19}+\cdots\)
9900.2.c.n	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					3300.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}+q^{11}+i q^{13}-2 i q^{17}+\cdots\)
9900.2.c.o	$9900$	$2$	9900.c	5.b	$2$	$2$	$2$	$79.052$	\(\Q(\sqrt{-1}) \)	None					3300.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}+q^{11}-4 i q^{13}+i q^{17}+\cdots\)
9900.2.c.p	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{7})\)	None					1980.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{7}-q^{11}+(-\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
9900.2.c.q	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{13})\)	None					660.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{7}-q^{11}+\beta _{1}q^{13}+(-\beta _{1}+\beta _{2}+\cdots)q^{17}+\cdots\)
9900.2.c.r	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{13})\)	None					1100.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+3\beta _{2})q^{7}-q^{11}+(2\beta _{1}+3\beta _{2}+\cdots)q^{13}+\cdots\)
9900.2.c.s	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{6})\)	None		✓			9900.2.a.bm	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{7}-q^{11}+2\beta _{1}q^{13}+(-3\beta _{1}+\cdots)q^{17}+\cdots\)
9900.2.c.t	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{145})\)	None					3300.2.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{2}q^{7}-q^{11}-\beta _{1}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
9900.2.c.u	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(\zeta_{12})\)	None					1980.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{2} q^{7}-q^{11}+(-\beta_{2}-2\beta_1)q^{13}+\cdots\)
9900.2.c.v	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{13})\)	None					660.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{7}+q^{11}+(\beta _{1}+2\beta _{2})q^{13}+(\beta _{1}+\cdots)q^{17}+\cdots\)
9900.2.c.w	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{7})\)	None					1980.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{7}+q^{11}+(-\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
9900.2.c.x	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{21})\)	None					1100.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+3\beta _{2})q^{7}+q^{11}-\beta _{2}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
9900.2.c.y	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(i, \sqrt{6})\)	None		✓			9900.2.a.bm	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{7}+q^{11}+2\beta _{1}q^{13}+(3\beta _{1}+\cdots)q^{17}+\cdots\)
9900.2.c.z	$9900$	$2$	9900.c	5.b	$2$	$4$	$4$	$79.052$	\(\Q(\zeta_{12})\)	None					1980.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{2} q^{7}+q^{11}+(-\beta_{2}-2\beta_1)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(9900, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(9900, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(110, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(165, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(220, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(275, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(330, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(495, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(550, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(660, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(825, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(990, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1650, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1980, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2475, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(3300, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(4950, [\chi])\)\(^{\oplus 2}\)