Space of modular forms of level 3600, weight 3, and Character 3600.c

• Label: 3600.3.c
• Level: $3600$
• Weight: $3$
• Character orbit: 3600.c
• Rep. character: $\chi_{3600}(449,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $12$
• Sturm bound: $2160$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(2160\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(3600, [\chi])\).

	Total	New	Old
Modular forms	1512	72	1440
Cusp forms	1368	72	1296
Eisenstein series	144	0	144

Trace form

\( 72 q + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(3600, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3600.3.c.a	$3600$	$3$	3600.c	15.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}q^{7}-\zeta_{8}^{2}q^{11}-17\zeta_{8}q^{13}-7\zeta_{8}^{3}q^{17}+\cdots\)
3600.3.c.b	$3600$	$3$	3600.c	15.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\zeta_{8}q^{7}-4\zeta_{8}^{2}q^{11}-4\zeta_{8}q^{13}+\cdots\)
3600.3.c.c	$3600$	$3$	3600.c	15.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-6\zeta_{8}q^{7}+4\zeta_{8}^{2}q^{11}+4\zeta_{8}q^{13}+\cdots\)
3600.3.c.d	$3600$	$3$	3600.c	15.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}q^{7}+\zeta_{8}^{2}q^{11}+7\zeta_{8}q^{13}-\zeta_{8}^{3}q^{17}+\cdots\)
3600.3.c.e	$3600$	$3$	3600.c	15.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\zeta_{8}q^{7}-13\zeta_{8}^{2}q^{11}+\zeta_{8}q^{13}+\cdots\)
3600.3.c.f	$3600$	$3$	3600.c	15.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+11\zeta_{8}q^{7}+\zeta_{8}^{2}q^{11}-7\zeta_{8}q^{13}+\cdots\)
3600.3.c.g	$3600$	$3$	3600.c	15.d	$2$	$8$	$8$	$98.093$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3\beta _{1}-\beta _{6})q^{7}+(-\beta _{2}-2\beta _{5})q^{11}+\cdots\)
3600.3.c.h	$3600$	$3$	3600.c	15.d	$2$	$8$	$8$	$98.093$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{7}+(\beta _{2}-\beta _{4})q^{11}+(-\beta _{1}-\beta _{5}+\cdots)q^{13}+\cdots\)
3600.3.c.i	$3600$	$3$	3600.c	15.d	$2$	$8$	$8$	$98.093$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}-\beta _{3})q^{7}+(-\beta _{2}-2\beta _{4})q^{11}+\cdots\)
3600.3.c.j	$3600$	$3$	3600.c	15.d	$2$	$8$	$8$	$98.093$	8.0.8540717056.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{4})q^{7}+\beta _{2}q^{11}+(5\beta _{1}+2\beta _{4}+\cdots)q^{13}+\cdots\)
3600.3.c.k	$3600$	$3$	3600.c	15.d	$2$	$8$	$8$	$98.093$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}-\beta _{3})q^{7}+(-\beta _{2}-\beta _{5})q^{11}+\cdots\)
3600.3.c.l	$3600$	$3$	3600.c	15.d	$2$	$8$	$8$	$98.093$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(4\beta _{1}-\beta _{5})q^{7}+(-\beta _{2}-3\beta _{4})q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(3600, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(3600, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 20}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(1800, [\chi])\)\(^{\oplus 2}\)