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

• Label: 3600.2.f
• Level: $3600$
• Weight: $2$
• Character orbit: 3600.f
• Rep. character: $\chi_{3600}(2449,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $22$
• Sturm bound: $1440$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(1440\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	792	46	746
Cusp forms	648	44	604
Eisenstein series	144	2	142

Trace form

\( 44 q + 4 q^{11} - 4 q^{19} - 12 q^{29} + 16 q^{31} - 8 q^{41} - 20 q^{49} - 16 q^{61} - 56 q^{71} - 32 q^{79} + 24 q^{89}+O(q^{100}) \)

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.f.a	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					90.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-6 q^{11}-2\beta q^{13}+3\beta q^{17}+\cdots\)
3600.2.f.b	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					600.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{7}-6 q^{11}+3 i q^{13}+2 i q^{17}+\cdots\)
3600.2.f.c	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					120.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 q^{11}-3\beta q^{13}+3\beta q^{17}-4 q^{19}+\cdots\)
3600.2.f.d	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					1800.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}-4 q^{11}-i q^{13}-4 i q^{17}+\cdots\)
3600.2.f.e	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					15.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 q^{11}+\beta q^{13}-\beta q^{17}+4 q^{19}+\cdots\)
3600.2.f.f	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					50.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}-3 q^{11}-4 i q^{13}-3 i q^{17}+\cdots\)
3600.2.f.g	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					360.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-2 q^{11}+2\beta q^{13}+\beta q^{17}+\cdots\)
3600.2.f.h	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					900.2.a.d	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+i q^{7}+7 i q^{13}-7 q^{19}-11 q^{31}+\cdots\)
3600.2.f.i	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					30.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-\beta q^{13}-3\beta q^{17}-4 q^{19}+\cdots\)
3600.2.f.j	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					20.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+\beta q^{13}-3\beta q^{17}-4 q^{19}+\cdots\)
3600.2.f.k	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					225.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{7}+\beta q^{13}-q^{19}+7 q^{31}+2\beta q^{37}+\cdots\)
3600.2.f.l	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					120.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-3\beta q^{13}-\beta q^{17}+4 q^{19}+\cdots\)
3600.2.f.m	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					36.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2\beta q^{7}-\beta q^{13}+8 q^{19}+4 q^{31}+\cdots\)
3600.2.f.n	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					200.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}+q^{11}+4 i q^{13}+5 i q^{17}+\cdots\)
3600.2.f.o	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					600.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}+2 q^{11}-3 i q^{13}+6 i q^{17}+\cdots\)
3600.2.f.p	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					75.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}+2 q^{11}-i q^{13}-2 i q^{17}+\cdots\)
3600.2.f.q	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					360.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+2 q^{11}+2\beta q^{13}-\beta q^{17}+\cdots\)
3600.2.f.r	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					24.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 q^{11}-\beta q^{13}+\beta q^{17}-4 q^{19}+\cdots\)
3600.2.f.s	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					1800.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}+4 q^{11}-i q^{13}+4 i q^{17}+\cdots\)
3600.2.f.t	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					40.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+4 q^{11}+\beta q^{13}-\beta q^{17}+\cdots\)
3600.2.f.u	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					90.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+6 q^{11}-2\beta q^{13}-3\beta q^{17}+\cdots\)
3600.2.f.v	$3600$	$2$	3600.f	5.b	$2$	$2$	$2$	$28.746$	\(\Q(\sqrt{-1}) \)	None					300.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}+6 q^{11}-5 i q^{13}+6 i q^{17}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(3600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1200, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1800, [\chi])\)\(^{\oplus 2}\)