Space of modular forms of level 1210, weight 2, and Character 1210.b

• Label: 1210.2.b
• Level: $1210$
• Weight: $2$
• Character orbit: 1210.b
• Rep. character: $\chi_{1210}(969,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $13$
• Sturm bound: $396$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1210.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(396\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1210, [\chi])\).

	Total	New	Old
Modular forms	222	54	168
Cusp forms	174	54	120
Eisenstein series	48	0	48

Trace form

54 * q - 54 * q^4 + 4 * q^6 - 54 * q^9 - 4 * q^10 + 4 * q^14 + 4 * q^15 + 54 * q^16 - 16 * q^19 - 4 * q^24 - 12 * q^25 - 4 * q^26 - 4 * q^29 + 4 * q^30 + 24 * q^31 + 8 * q^34 + 8 * q^35 + 54 * q^36 - 16 * q^39 + 4 * q^40 + 36 * q^41 + 32 * q^45 + 12 * q^46 - 82 * q^49 - 24 * q^50 - 48 * q^51 - 16 * q^54 - 4 * q^56 - 4 * q^59 - 4 * q^60 + 36 * q^61 - 54 * q^64 - 16 * q^65 + 48 * q^69 - 24 * q^70 - 24 * q^75 + 16 * q^76 + 16 * q^79 + 54 * q^81 - 8 * q^85 - 8 * q^86 + 12 * q^89 + 20 * q^90 - 8 * q^91 + 20 * q^94 + 4 * q^96

Decomposition of \(S_{2}^{\mathrm{new}}(1210, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1210.2.b.a	$1210$	$2$	1210.b	5.b	$2$	$2$	$2$	$9.662$	\(\Q(\sqrt{-1}) \)	None					110.2.b.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+(i-2)q^{5}+\cdots\)
1210.2.b.b	$1210$	$2$	1210.b	5.b	$2$	$2$	$2$	$9.662$	\(\Q(\sqrt{-1}) \)	None		✓			1210.2.b.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+3 i q^{3}-q^{4}+(-2 i-1)q^{5}+\cdots\)
1210.2.b.c	$1210$	$2$	1210.b	5.b	$2$	$2$	$2$	$9.662$	\(\Q(\sqrt{-1}) \)	None		✓			1210.2.b.b	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+3 i q^{3}-q^{4}+(-2 i-1)q^{5}+\cdots\)
1210.2.b.d	$1210$	$2$	1210.b	5.b	$2$	$2$	$2$	$9.662$	\(\Q(\sqrt{-1}) \)	None					110.2.b.b	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+2 i q^{3}-q^{4}+(2 i+1)q^{5}+\cdots\)
1210.2.b.e	$1210$	$2$	1210.b	5.b	$2$	$2$	$2$	$9.662$	\(\Q(\sqrt{-1}) \)	None					110.2.b.c	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+3 i q^{3}-q^{4}+(-i+2)q^{5}+\cdots\)
1210.2.b.f	$1210$	$2$	1210.b	5.b	$2$	$4$	$4$	$9.662$	\(\Q(i, \sqrt{5})\)	None					110.2.j.a	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+2\beta _{1}q^{3}-q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1210.2.b.g	$1210$	$2$	1210.b	5.b	$2$	$4$	$4$	$9.662$	\(\Q(i, \sqrt{5})\)	None					110.2.j.a	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+2\beta _{1}q^{3}-q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1210.2.b.h	$1210$	$2$	1210.b	5.b	$2$	$4$	$4$	$9.662$	\(\Q(i, \sqrt{5})\)	None		✓			1210.2.b.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}-q^{4}-\beta _{2}q^{5}-\beta _{3}q^{6}+\cdots\)
1210.2.b.i	$1210$	$2$	1210.b	5.b	$2$	$4$	$4$	$9.662$	\(\Q(\zeta_{12})\)	None		✓			1210.2.b.i	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{3}q^{2}+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
1210.2.b.j	$1210$	$2$	1210.b	5.b	$2$	$4$	$4$	$9.662$	\(\Q(\zeta_{12})\)	None		✓			1210.2.b.i	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}^{3}q^{2}+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
1210.2.b.k	$1210$	$2$	1210.b	5.b	$2$	$8$	$8$	$9.662$	8.0.\(\cdots\).1	None					110.2.j.b	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(1+\cdots)q^{5}+\cdots\)
1210.2.b.l	$1210$	$2$	1210.b	5.b	$2$	$8$	$8$	$9.662$	8.0.\(\cdots\).1	None					110.2.j.b	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
1210.2.b.m	$1210$	$2$	1210.b	5.b	$2$	$8$	$8$	$9.662$	8.0.303595776.1	None		✓			1210.2.b.m	$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}+(\beta _{2}-\beta _{4}+\beta _{6})q^{3}-q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1210, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1210, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(110, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(605, [\chi])\)\(^{\oplus 2}\)