Space of modular forms of level 4400, weight 2, and Character 4400.b

• Label: 4400.2.b
• Level: $4400$
• Weight: $2$
• Character orbit: 4400.b
• Rep. character: $\chi_{4400}(4049,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $29$
• Sturm bound: $1440$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(1440\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4400, [\chi])\).

	Total	New	Old
Modular forms	756	90	666
Cusp forms	684	90	594
Eisenstein series	72	0	72

Trace form

90 * q - 90 * q^9 + 6 * q^11 - 16 * q^19 + 16 * q^21 - 12 * q^29 + 4 * q^31 + 4 * q^41 - 90 * q^49 + 16 * q^51 - 52 * q^59 - 52 * q^61 + 16 * q^69 - 12 * q^71 - 40 * q^79 + 42 * q^81 - 4 * q^89 + 64 * q^91 - 18 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(4400, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4400.2.b.a	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					440.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}-i q^{7}-6 q^{9}+q^{11}+6 i q^{13}+\cdots\)
4400.2.b.b	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					88.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}-2 i q^{7}-6 q^{9}+q^{11}+\cdots\)
4400.2.b.c	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					220.2.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-q^{9}-q^{11}-2\beta q^{17}-4 q^{19}+\cdots\)
4400.2.b.d	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					550.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-q^{9}-q^{11}-3 i q^{13}-4 i q^{17}+\cdots\)
4400.2.b.e	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					550.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-4 i q^{7}-q^{9}+q^{11}+5 i q^{13}+\cdots\)
4400.2.b.f	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					220.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-2\beta q^{7}-q^{9}+q^{11}-2\beta q^{13}+\cdots\)
4400.2.b.g	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					110.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-5 i q^{7}+2 q^{9}-q^{11}-2 i q^{13}+\cdots\)
4400.2.b.h	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					11.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-2 i q^{7}+2 q^{9}-q^{11}+4 i q^{13}+\cdots\)
4400.2.b.i	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					110.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+3 i q^{7}+2 q^{9}-q^{11}-6 i q^{13}+\cdots\)
4400.2.b.j	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					110.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+i q^{7}+2 q^{9}+q^{11}-2 i q^{13}+\cdots\)
4400.2.b.k	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					44.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-2 i q^{7}+2 q^{9}+q^{11}+4 i q^{13}+\cdots\)
4400.2.b.l	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					440.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+3 q^{9}-q^{11}-2\beta q^{13}+2\beta q^{17}+\cdots\)
4400.2.b.m	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					440.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+3 q^{9}+q^{11}-8 q^{19}+4\beta q^{23}+\cdots\)
4400.2.b.n	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					55.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 q^{9}+q^{11}-\beta q^{13}+3\beta q^{17}+\cdots\)
4400.2.b.o	$4400$	$2$	4400.b	5.b	$2$	$2$	$2$	$35.134$	\(\Q(\sqrt{-1}) \)	None					440.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{7}+3 q^{9}+q^{11}-3\beta q^{13}+\cdots\)
4400.2.b.p	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{33})\)	None					110.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{1}q^{7}+(-6+\beta _{3})q^{9}+q^{11}+\cdots\)
4400.2.b.q	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(\zeta_{8})\)	None					55.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{2} q^{3}-\beta_1 q^{7}-5 q^{9}-q^{11}+\cdots\)
4400.2.b.r	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{13})\)	None					1100.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}+2\beta _{2})q^{7}+(-4+\cdots)q^{9}+\cdots\)
4400.2.b.s	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{21})\)	None					1100.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+3\beta _{2})q^{7}+(-3+\beta _{3})q^{9}+\cdots\)
4400.2.b.t	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{17})\)	None					440.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{1}q^{7}+(-2+\beta _{3})q^{9}-q^{11}+\cdots\)
4400.2.b.u	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{17})\)	None					440.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(-2+\beta _{3})q^{9}+\cdots\)
4400.2.b.v	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{17})\)	None					88.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+2\beta _{1}q^{7}+(-2+\beta _{3})q^{9}+\cdots\)
4400.2.b.w	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{17})\)	None					440.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{2})q^{7}+(-2+\beta _{3})q^{9}+\cdots\)
4400.2.b.x	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{5})\)	None					275.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{3})q^{3}+(3\beta _{1}+2\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots\)
4400.2.b.y	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{13})\)	None					275.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+3\beta _{2})q^{7}+(-1+\beta _{3})q^{9}+\cdots\)
4400.2.b.z	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{5})\)	None					2200.2.a.p	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{3})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
4400.2.b.ba	$4400$	$2$	4400.b	5.b	$2$	$4$	$4$	$35.134$	\(\Q(i, \sqrt{5})\)	None					2200.2.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
4400.2.b.bb	$4400$	$2$	4400.b	5.b	$2$	$6$	$6$	$35.134$	6.0.96668224.1	None					2200.2.a.u	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{2}-\beta _{4})q^{7}+(-2+\beta _{3}+\cdots)q^{9}+\cdots\)
4400.2.b.bc	$4400$	$2$	4400.b	5.b	$2$	$6$	$6$	$35.134$	6.0.44836416.1	None					2200.2.a.t	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{3})q^{3}+(-\beta _{3}-\beta _{5})q^{7}+(-2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4400, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(110, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(220, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(275, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(440, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(550, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(880, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2200, [\chi])\)\(^{\oplus 2}\)