Space of modular forms of level 1250, weight 2, and Character 1250.b

• Label: 1250.2.b
• Level: $1250$
• Weight: $2$
• Character orbit: 1250.b
• Rep. character: $\chi_{1250}(1249,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $6$
• Sturm bound: $375$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1250 = 2 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1250.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(375\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1250, [\chi])\).

	Total	New	Old
Modular forms	218	40	178
Cusp forms	158	40	118
Eisenstein series	60	0	60

Trace form

40 * q - 40 * q^4 - 40 * q^9 + 40 * q^16 - 20 * q^19 + 20 * q^21 - 10 * q^26 + 10 * q^29 + 20 * q^31 + 10 * q^34 + 40 * q^36 - 20 * q^39 - 10 * q^41 - 60 * q^49 + 30 * q^51 - 30 * q^54 - 30 * q^59 + 10 * q^61 - 40 * q^64 + 30 * q^66 + 40 * q^69 - 40 * q^71 + 10 * q^74 + 20 * q^76 - 20 * q^79 - 20 * q^84 + 30 * q^86 + 50 * q^89 + 40 * q^91 + 10 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(1250, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1250.2.b.a	$1250$	$2$	1250.b	5.b	$2$	$4$	$4$	$9.981$	\(\Q(i, \sqrt{5})\)	None		✓			1250.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-\beta _{3}q^{3}-q^{4}-q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1250.2.b.b	$1250$	$2$	1250.b	5.b	$2$	$4$	$4$	$9.981$	\(\Q(i, \sqrt{5})\)	None					50.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(\beta _{1}-\beta _{3})q^{3}-q^{4}+(1-\beta _{2}+\cdots)q^{6}+\cdots\)
1250.2.b.c	$1250$	$2$	1250.b	5.b	$2$	$8$	$8$	$9.981$	\(\Q(\zeta_{20})\)	None					50.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{2}+(-\beta_{2}-\beta_1)q^{3}-q^{4}+\cdots\)
1250.2.b.d	$1250$	$2$	1250.b	5.b	$2$	$8$	$8$	$9.981$	8.0.\(\cdots\).2	None		✓			1250.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{2}+\beta _{1}q^{3}-q^{4}-\beta _{6}q^{6}+(2\beta _{4}+\cdots)q^{7}+\cdots\)
1250.2.b.e	$1250$	$2$	1250.b	5.b	$2$	$8$	$8$	$9.981$	8.0.\(\cdots\).15	None					50.2.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+\beta _{1}q^{3}-q^{4}+(1+\beta _{4}-\beta _{7})q^{6}+\cdots\)
1250.2.b.f	$1250$	$2$	1250.b	5.b	$2$	$8$	$8$	$9.981$	8.0.324000000.1	None		✓			1250.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}+(-\beta _{3}-\beta _{7})q^{3}-q^{4}-\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1250, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1250, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(250, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(625, [\chi])\)\(^{\oplus 2}\)