Space of modular forms of level 9300, weight 2, and Character 9300.g

• Label: 9300.2.g
• Level: $9300$
• Weight: $2$
• Character orbit: 9300.g
• Rep. character: $\chi_{9300}(3349,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $21$
• Sturm bound: $3840$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(3840\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(9300, [\chi])\).

	Total	New	Old
Modular forms	1956	92	1864
Cusp forms	1884	92	1792
Eisenstein series	72	0	72

Trace form

\( 92 q - 92 q^{9} - 24 q^{11} + 16 q^{19} - 24 q^{29} + 8 q^{41} - 100 q^{49} + 24 q^{51} + 24 q^{59} + 32 q^{61} - 24 q^{69} - 16 q^{71} + 92 q^{81} - 56 q^{89} + 48 q^{91} + 24 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(9300, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
9300.2.g.a	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None					1860.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 i q^{7}-q^{9}-4 q^{11}-4 i q^{13}+\cdots\)
9300.2.g.b	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None					1860.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+2 i q^{7}-q^{9}-4 q^{11}-4 i q^{13}+\cdots\)
9300.2.g.c	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None		✓			9300.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-q^{9}-3 q^{11}-4 i q^{13}-i q^{17}+\cdots\)
9300.2.g.d	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None					372.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+i q^{7}-q^{9}-6 i q^{13}+8 i q^{17}+\cdots\)
9300.2.g.e	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None					372.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+4 i q^{7}-q^{9}-2 i q^{13}+\cdots\)
9300.2.g.f	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None		✓			9300.2.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+4 i q^{7}-q^{9}-4 i q^{13}+\cdots\)
9300.2.g.g	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None					372.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+i q^{7}-q^{9}+2 i q^{13}+q^{19}+\cdots\)
9300.2.g.h	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None					1860.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}-q^{9}-2 i q^{13}+4 i q^{17}+\cdots\)
9300.2.g.i	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None					372.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+5 i q^{7}-q^{9}+2 q^{11}-4 i q^{13}+\cdots\)
9300.2.g.j	$9300$	$2$	9300.g	5.b	$2$	$2$	$2$	$74.261$	\(\Q(\sqrt{-1}) \)	None		✓			9300.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+4 i q^{7}-q^{9}+3 q^{11}+2 i q^{13}+\cdots\)
9300.2.g.k	$9300$	$2$	9300.g	5.b	$2$	$4$	$4$	$74.261$	\(\Q(\zeta_{12})\)	None					1860.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{3}+(-\beta_{2}+\beta_1)q^{7}-q^{9}+\cdots\)
9300.2.g.l	$9300$	$2$	9300.g	5.b	$2$	$4$	$4$	$74.261$	\(\Q(i, \sqrt{6})\)	None					1860.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{2}q^{7}-q^{9}+(2\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots\)
9300.2.g.m	$9300$	$2$	9300.g	5.b	$2$	$4$	$4$	$74.261$	\(\Q(i, \sqrt{33})\)	None		✓			9300.2.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-q^{9}+(1-\beta _{3})q^{11}+(-2\beta _{1}+\cdots)q^{17}+\cdots\)
9300.2.g.n	$9300$	$2$	9300.g	5.b	$2$	$4$	$4$	$74.261$	\(\Q(i, \sqrt{17})\)	None					372.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+(2-2\beta _{3})q^{11}+\cdots\)
9300.2.g.o	$9300$	$2$	9300.g	5.b	$2$	$6$	$6$	$74.261$	6.0.2611456.1	None		✓			9300.2.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(\beta _{4}-\beta _{5})q^{7}-q^{9}+(-2+\cdots)q^{11}+\cdots\)
9300.2.g.p	$9300$	$2$	9300.g	5.b	$2$	$6$	$6$	$74.261$	6.0.5089536.1	None					1860.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(\beta _{2}+2\beta _{4}-\beta _{5})q^{7}-q^{9}+\cdots\)
9300.2.g.q	$9300$	$2$	9300.g	5.b	$2$	$6$	$6$	$74.261$	6.0.2611456.1	None					1860.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(\beta _{3}-\beta _{4})q^{7}-q^{9}-2\beta _{2}q^{11}+\cdots\)
9300.2.g.r	$9300$	$2$	9300.g	5.b	$2$	$6$	$6$	$74.261$	6.0.932935936.2	None					1860.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+2q^{11}+\beta _{1}q^{13}+\cdots\)
9300.2.g.s	$9300$	$2$	9300.g	5.b	$2$	$8$	$8$	$74.261$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					1860.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(\beta _{1}-\beta _{4})q^{7}-q^{9}+(-\beta _{3}+\cdots)q^{11}+\cdots\)
9300.2.g.t	$9300$	$2$	9300.g	5.b	$2$	$12$	$12$	$74.261$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			9300.2.a.y	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(\beta _{4}+\beta _{6})q^{7}-q^{9}+(-1+\cdots)q^{11}+\cdots\)
9300.2.g.u	$9300$	$2$	9300.g	5.b	$2$	$12$	$12$	$74.261$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			9300.2.a.z	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{7}q^{3}+(-\beta _{7}+\beta _{8})q^{7}-q^{9}+\beta _{4}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(9300, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(9300, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(155, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(310, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(465, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(620, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(775, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(930, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1550, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1860, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2325, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(3100, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(4650, [\chi])\)\(^{\oplus 2}\)