Space of modular forms of level 4900, weight 2, and Character 4900.e

• Label: 4900.2.e
• Level: $4900$
• Weight: $2$
• Character orbit: 4900.e
• Rep. character: $\chi_{4900}(2549,\cdot)$
• Character field: $\Q$
• Dimension: $62$
• Newform subspaces: $21$
• Sturm bound: $1680$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(1680\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(3\), \(11\), \(19\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4900, [\chi])\).

	Total	New	Old
Modular forms	912	62	850
Cusp forms	768	62	706
Eisenstein series	144	0	144

Trace form

\( 62 q - 62 q^{9} + 4 q^{11} - 12 q^{29} - 20 q^{31} - 8 q^{39} + 16 q^{41} - 8 q^{51} + 4 q^{59} + 8 q^{61} + 12 q^{69} + 28 q^{71} + 40 q^{79} + 54 q^{81} + 16 q^{89} - 44 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4900, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4900.2.e.a	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					140.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}-6 q^{9}-5 q^{11}-3 i q^{13}+\cdots\)
4900.2.e.b	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					140.2.i.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}-6 q^{9}-2 q^{11}+6 i q^{13}+\cdots\)
4900.2.e.c	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					140.2.i.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}-6 q^{9}-2 q^{11}+6 i q^{13}+\cdots\)
4900.2.e.d	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None		✓			4900.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-q^{9}-q^{11}+2 i q^{13}+4 i q^{17}+\cdots\)
4900.2.e.e	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None		✓			4900.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-q^{9}-q^{11}+2 i q^{13}+4 i q^{17}+\cdots\)
4900.2.e.f	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					20.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-q^{9}-\beta q^{13}-3\beta q^{17}-4 q^{19}+\cdots\)
4900.2.e.g	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					700.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-q^{9}+3 q^{11}+4 i q^{13}+\cdots\)
4900.2.e.h	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					28.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}-3 q^{11}+2 i q^{13}+\cdots\)
4900.2.e.i	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					28.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}-3 q^{11}+2 i q^{13}+\cdots\)
4900.2.e.j	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					980.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}-q^{11}-5 i q^{13}-i q^{17}+\cdots\)
4900.2.e.k	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					980.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}-q^{11}-5 i q^{13}-i q^{17}+\cdots\)
4900.2.e.l	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					140.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}+3 q^{11}-i q^{13}+3 i q^{17}+\cdots\)
4900.2.e.m	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					140.2.i.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}+6 q^{11}+2 i q^{13}+\cdots\)
4900.2.e.n	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					140.2.i.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}+6 q^{11}+2 i q^{13}+\cdots\)
4900.2.e.o	$4900$	$2$	4900.e	5.b	$2$	$2$	$2$	$39.127$	\(\Q(\sqrt{-1}) \)	None					700.2.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 q^{9}-5 q^{11}-6 i q^{13}-4 i q^{17}+\cdots\)
4900.2.e.p	$4900$	$2$	4900.e	5.b	$2$	$4$	$4$	$39.127$	\(\Q(\zeta_{8})\)	None					196.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta_{2} q^{3}-5 q^{9}+4 q^{11}+3\beta_{2} q^{13}+\cdots\)
4900.2.e.q	$4900$	$2$	4900.e	5.b	$2$	$4$	$4$	$39.127$	\(\Q(\zeta_{8})\)	None					980.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{3}-2\beta_{3} q^{9}+(2\beta_{3}-1)q^{11}+\cdots\)
4900.2.e.r	$4900$	$2$	4900.e	5.b	$2$	$4$	$4$	$39.127$	\(\Q(\zeta_{8})\)	None					980.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{3}-2\beta_{3} q^{9}+(2\beta_{3}-1)q^{11}+\cdots\)
4900.2.e.s	$4900$	$2$	4900.e	5.b	$2$	$6$	$6$	$39.127$	6.0.4227136.2	None					700.2.i.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-2-\beta _{4}-\beta _{5})q^{9}+(1+\beta _{5})q^{11}+\cdots\)
4900.2.e.t	$4900$	$2$	4900.e	5.b	$2$	$6$	$6$	$39.127$	6.0.4227136.2	None					700.2.i.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-2-\beta _{4}-\beta _{5})q^{9}+(1+\beta _{5})q^{11}+\cdots\)
4900.2.e.u	$4900$	$2$	4900.e	5.b	$2$	$8$	$8$	$39.127$	8.0.40960000.1	None		✓	✓		4900.2.a.bg	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{5}-\beta _{7})q^{3}+\beta _{2}q^{9}+(-1+2\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4900, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4900, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(490, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(700, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(980, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1225, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2450, [\chi])\)\(^{\oplus 2}\)