Space of modular forms of level 3600, weight 2, and Character 3600.w

• Label: 3600.2.w
• Level: $3600$
• Weight: $2$
• Character orbit: 3600.w
• Rep. character: $\chi_{3600}(593,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $72$
• Newform subspaces: $13$
• Sturm bound: $1440$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.w (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(1440\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3600, [\chi])\).

	Total	New	Old
Modular forms	1584	72	1512
Cusp forms	1296	72	1224
Eisenstein series	288	0	288

Trace form

\( 72 q - 32 q^{31} - 8 q^{37} - 32 q^{43} - 64 q^{61} - 48 q^{67} + 16 q^{73} - 48 q^{91} + 16 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3600, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3600.2.w.a	$3600$	$2$	3600.w	15.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None					450.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3\beta_1-3)q^{7}-\beta_{2} q^{11}+(3\beta_1+3)q^{13}+\cdots\)
3600.2.w.b	$3600$	$2$	3600.w	15.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None					45.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2\beta_1-2)q^{7}-2\beta_{2} q^{11}+(-\beta_1-1)q^{13}+\cdots\)
3600.2.w.c	$3600$	$2$	3600.w	15.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None					1800.2.s.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_1-1)q^{7}+\beta_{2} q^{11}+(-3\beta_1-3)q^{13}+\cdots\)
3600.2.w.d	$3600$	$2$	3600.w	15.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None					360.2.s.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-4\beta_{2} q^{11}+(3\beta_1-3)q^{13}+4\beta_1 q^{19}+\cdots\)
3600.2.w.e	$3600$	$2$	3600.w	15.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None					1800.2.s.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_1+1)q^{7}+\beta_{2} q^{11}+(3\beta_1+3)q^{13}+\cdots\)
3600.2.w.f	$3600$	$2$	3600.w	15.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None					180.2.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2\beta_1+2)q^{7}+2\beta_{2} q^{11}+(-3\beta_1-3)q^{13}+\cdots\)
3600.2.w.g	$3600$	$2$	3600.w	15.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None					90.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2\beta_1+2)q^{7}+2\beta_{2} q^{11}+(3\beta_1+3)q^{13}+\cdots\)
3600.2.w.h	$3600$	$2$	3600.w	15.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None					450.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3\beta_1+3)q^{7}+\beta_{2} q^{11}+(-3\beta_1-3)q^{13}+\cdots\)
3600.2.w.i	$3600$	$2$	3600.w	15.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None					1800.2.s.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{3}+2\beta_{2}-2)q^{7}+\beta_{6} q^{11}+\cdots\)
3600.2.w.j	$3600$	$2$	3600.w	15.e	$4$	$8$	$4$	$28.746$	\(\Q(i, \sqrt{2}, \sqrt{5})\)	None					360.2.s.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{1}+\beta _{4})q^{7}+(-\beta _{3}-\beta _{6}+\cdots)q^{11}+\cdots\)
3600.2.w.k	$3600$	$2$	3600.w	15.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None					900.2.j.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{3} q^{7}+\beta_{5} q^{11}+\beta_1 q^{13}-\beta_{6} q^{17}+\cdots\)
3600.2.w.l	$3600$	$2$	3600.w	15.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None					225.2.f.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{3} q^{7}+\beta_{7} q^{11}+3\beta_1 q^{13}+\beta_{4} q^{17}+\cdots\)
3600.2.w.m	$3600$	$2$	3600.w	15.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None					1800.2.s.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2\beta_{2}-\beta_1+2)q^{7}-\beta_{6} q^{11}+\beta_{3} q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1200, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1800, [\chi])\)\(^{\oplus 2}\)