Space of modular forms of level 1248, weight 2, and Character 1248.bc

• Label: 1248.2.bc
• Level: $1248$
• Weight: $2$
• Character orbit: 1248.bc
• Rep. character: $\chi_{1248}(31,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $56$
• Newform subspaces: $20$
• Sturm bound: $448$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.bc (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 52 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(448\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1248, [\chi])\).

	Total	New	Old
Modular forms	480	56	424
Cusp forms	416	56	360
Eisenstein series	64	0	64

Trace form

\( 56 q + 8 q^{5} - 56 q^{9} + 16 q^{21} - 8 q^{37} + 56 q^{41} - 8 q^{45} - 48 q^{53} + 16 q^{57} + 16 q^{61} - 8 q^{65} - 40 q^{73} + 56 q^{81} + 16 q^{85} + 24 q^{89} - 16 q^{93} + 24 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1248, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1248.2.bc.a	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.a	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{3}+(-2 i-2)q^{5}+(-2 i-2)q^{7}+\cdots\)
1248.2.bc.b	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.b	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{3}+(-2 i-2)q^{5}+(-i-1)q^{7}+\cdots\)
1248.2.bc.c	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{3}+(-2 i-2)q^{5}+(i+1)q^{7}+\cdots\)
1248.2.bc.d	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.a	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{3}+(-2 i-2)q^{5}+(2 i+2)q^{7}+\cdots\)
1248.2.bc.e	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.e	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{3}+(-i-1)q^{5}-q^{9}+(-2 i-2)q^{11}+\cdots\)
1248.2.bc.f	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.e	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{3}+(-i-1)q^{5}-q^{9}+(2 i+2)q^{11}+\cdots\)
1248.2.bc.g	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{3}+(-3 i-3)q^{7}-q^{9}+(4 i+4)q^{11}+\cdots\)
1248.2.bc.h	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{3}-q^{9}+(-i-1)q^{11}+(-2 i-3)q^{13}+\cdots\)
1248.2.bc.i	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{3}-q^{9}+(i+1)q^{11}+(-2 i-3)q^{13}+\cdots\)
1248.2.bc.j	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{3}+(3 i+3)q^{7}-q^{9}+(-4 i-4)q^{11}+\cdots\)
1248.2.bc.k	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.k	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{3}+(2 i+2)q^{5}+(-3 i-3)q^{7}+\cdots\)
1248.2.bc.l	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.l	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{3}+(2 i+2)q^{5}+(-2 i-2)q^{7}+\cdots\)
1248.2.bc.m	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.l	$2$	$0$	\(0\)	\(0\)	\(4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{3}+(2 i+2)q^{5}+(2 i+2)q^{7}+\cdots\)
1248.2.bc.n	$1248$	$2$	1248.bc	52.f	$4$	$2$	$1$	$9.965$	\(\Q(\sqrt{-1}) \)	None		✓			1248.2.bc.k	$2$	$0$	\(0\)	\(0\)	\(4\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{3}+(2 i+2)q^{5}+(3 i+3)q^{7}+\cdots\)
1248.2.bc.o	$1248$	$2$	1248.bc	52.f	$4$	$4$	$2$	$9.965$	\(\Q(i, \sqrt{17})\)	None		✓			1248.2.bc.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(-1+\beta _{2})q^{7}-q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
1248.2.bc.p	$1248$	$2$	1248.bc	52.f	$4$	$4$	$2$	$9.965$	\(\Q(i, \sqrt{17})\)	None		✓			1248.2.bc.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(1-\beta _{2})q^{7}-q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
1248.2.bc.q	$1248$	$2$	1248.bc	52.f	$4$	$4$	$2$	$9.965$	\(\Q(\zeta_{12})\)	None		✓			1248.2.bc.q	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_1 q^{3}+(-\beta_{2}+\beta_1+1)q^{5}+\cdots\)
1248.2.bc.r	$1248$	$2$	1248.bc	52.f	$4$	$4$	$2$	$9.965$	\(\Q(\zeta_{12})\)	None		✓			1248.2.bc.q	$2$	$0$	\(0\)	\(0\)	\(4\)	\(4\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_1 q^{3}+(-\beta_{2}+\beta_1+1)q^{5}+\cdots\)
1248.2.bc.s	$1248$	$2$	1248.bc	52.f	$4$	$6$	$3$	$9.965$	6.0.3182656.1	None		✓			1248.2.bc.s	$2$	$0$	\(0\)	\(0\)	\(2\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}-\beta _{5}q^{5}+(-1+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
1248.2.bc.t	$1248$	$2$	1248.bc	52.f	$4$	$6$	$3$	$9.965$	6.0.3182656.1	None		✓			1248.2.bc.s	$2$	$0$	\(0\)	\(0\)	\(2\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}-\beta _{5}q^{5}+(1-\beta _{2}-\beta _{3}+\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1248, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1248, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(208, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(416, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(624, [\chi])\)\(^{\oplus 2}\)