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

• Label: 3600.3.e
• Level: $3600$
• Weight: $3$
• Character orbit: 3600.e
• Rep. character: $\chi_{3600}(3151,\cdot)$
• Character field: $\Q$
• Dimension: $95$
• Newform subspaces: $36$
• Sturm bound: $2160$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(2160\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	1512	95	1417
Cusp forms	1368	95	1273
Eisenstein series	144	0	144

Trace form

\( 95 q - 10 q^{13} + 30 q^{17} + 102 q^{29} + 46 q^{37} - 18 q^{41} - 481 q^{49} + 6 q^{53} + 46 q^{61} - 154 q^{73} - 240 q^{77} - 18 q^{89} + 134 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3600.3.e.a	$3600$	$3$	3600.e	4.b	$2$	$1$	$1$	$98.093$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				720.3.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-24q^{13}-30q^{17}+40q^{29}-24q^{37}+\cdots\)
3600.3.e.b	$3600$	$3$	3600.e	4.b	$2$	$1$	$1$	$98.093$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				720.3.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-24q^{13}+30q^{17}-40q^{29}-24q^{37}+\cdots\)
3600.3.e.c	$3600$	$3$	3600.e	4.b	$2$	$1$	$1$	$98.093$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				16.3.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-10q^{13}-30q^{17}-42q^{29}+70q^{37}+\cdots\)
3600.3.e.d	$3600$	$3$	3600.e	4.b	$2$	$1$	$1$	$98.093$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				720.3.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+24q^{13}-30q^{17}-40q^{29}+24q^{37}+\cdots\)
3600.3.e.e	$3600$	$3$	3600.e	4.b	$2$	$1$	$1$	$98.093$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				720.3.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+24q^{13}+30q^{17}+40q^{29}+24q^{37}+\cdots\)
3600.3.e.f	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					240.3.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+2\beta q^{11}-24 q^{13}-24 q^{17}+\cdots\)
3600.3.e.g	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			3600.3.e.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3\beta q^{7}-23 q^{13}-21\beta q^{19}-11\beta q^{31}+\cdots\)
3600.3.e.h	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					144.3.g.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{7}-22 q^{13}+2\beta q^{19}-3\beta q^{31}+\cdots\)
3600.3.e.i	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-15}) \)	None					400.3.b.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{7}-5\beta q^{11}-20q^{13}+15q^{17}+\cdots\)
3600.3.e.j	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					400.3.b.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{7}+7\beta q^{11}-16 q^{13}-3 q^{17}+\cdots\)
3600.3.e.k	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					1200.3.e.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta q^{7}+2\beta q^{11}-13 q^{13}+18 q^{17}+\cdots\)
3600.3.e.l	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					1200.3.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}-2\beta q^{11}-11 q^{13}-6 q^{17}+\cdots\)
3600.3.e.m	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					240.3.j.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+\beta q^{11}-6 q^{13}-18 q^{17}+\cdots\)
3600.3.e.n	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			3600.3.e.n	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{7}-q^{13}-\beta q^{19}-7\beta q^{31}+\cdots\)
3600.3.e.o	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-5}) \)	\(\Q(\sqrt{-5}) \)					80.3.h.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{7}+\beta q^{23}-22q^{29}+62q^{41}+\cdots\)
3600.3.e.p	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			3600.3.e.n	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{7}+q^{13}+\beta q^{19}+7\beta q^{31}+\cdots\)
3600.3.e.q	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					240.3.j.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}-\beta q^{11}+6 q^{13}+18 q^{17}+\cdots\)
3600.3.e.r	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					1200.3.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+2\beta q^{11}+11 q^{13}+6 q^{17}+\cdots\)
3600.3.e.s	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					1200.3.e.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta q^{7}-2\beta q^{11}+13 q^{13}-18 q^{17}+\cdots\)
3600.3.e.t	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					48.3.g.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+3\beta q^{11}+14 q^{13}-6 q^{17}+\cdots\)
3600.3.e.u	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					400.3.b.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{7}-7\beta q^{11}+16 q^{13}+3 q^{17}+\cdots\)
3600.3.e.v	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-15}) \)	None					400.3.b.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{7}+5\beta q^{11}+20q^{13}-15q^{17}+\cdots\)
3600.3.e.w	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			3600.3.e.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3\beta q^{7}+23 q^{13}+21\beta q^{19}+11\beta q^{31}+\cdots\)
3600.3.e.x	$3600$	$3$	3600.e	4.b	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-3}) \)	None					240.3.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}-2\beta q^{11}+24 q^{13}+24 q^{17}+\cdots\)
3600.3.e.y	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{-10})\)	None		✓			3600.3.e.y	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{2}q^{7}-\beta _{1}q^{11}-13q^{13}-\beta _{3}q^{17}+\cdots\)
3600.3.e.z	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					240.3.e.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{7}+\beta _{1}q^{11}+(-4-\beta _{3})q^{13}+\cdots\)
3600.3.e.ba	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					720.3.e.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 3\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{7}+\beta _{2}q^{11}-4q^{13}-\beta _{3}q^{17}+\cdots\)
3600.3.e.bb	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					80.3.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{7}+2\beta _{2}q^{11}+(-4+\beta _{3})q^{13}+\cdots\)
3600.3.e.bc	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{11})\)	None					1200.3.e.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{7}+\beta _{1}q^{11}-q^{13}+(-12+\cdots)q^{17}+\cdots\)
3600.3.e.bd	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					80.3.h.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{7}+\beta _{3}q^{11}-\beta _{1}q^{13}+2\beta _{1}q^{17}+\cdots\)
3600.3.e.be	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					240.3.j.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{7}-\beta _{3}q^{11}-\beta _{1}q^{13}+\beta _{1}q^{17}+\cdots\)
3600.3.e.bf	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{11})\)	None					1200.3.e.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{7}-\beta _{1}q^{11}+q^{13}+(12+\cdots)q^{17}+\cdots\)
3600.3.e.bg	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					720.3.e.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{7}+\beta _{2}q^{11}+8q^{13}-\beta _{3}q^{17}+\cdots\)
3600.3.e.bh	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					240.3.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{7}+(\beta _{1}-\beta _{2})q^{11}+(8+\beta _{3})q^{13}+\cdots\)
3600.3.e.bi	$3600$	$3$	3600.e	4.b	$2$	$4$	$4$	$98.093$	\(\Q(\sqrt{-3}, \sqrt{-10})\)	None		✓			3600.3.e.y	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{2}q^{7}+\beta _{1}q^{11}+13q^{13}+\beta _{3}q^{17}+\cdots\)
3600.3.e.bj	$3600$	$3$	3600.e	4.b	$2$	$8$	$8$	$98.093$	8.0.3317760000.8	None			✓		720.3.j.h	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{15}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{7}+\beta _{1}q^{11}+\beta _{4}q^{13}+\beta _{2}q^{17}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(3600, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(1200, [\chi])\)\(^{\oplus 2}\)