Space of modular forms of level 49, weight 12, and Character 49.c

• Label: 49.12.c
• Level: $49$
• Weight: $12$
• Character orbit: 49.c
• Rep. character: $\chi_{49}(18,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $70$
• Newform subspaces: $10$
• Sturm bound: $56$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	49.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(56\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(49, [\chi])\).

	Total	New	Old
Modular forms	110	78	32
Cusp forms	94	70	24
Eisenstein series	16	8	8

Trace form

\( 70 q - 45 q^{2} + 244 q^{3} - 37897 q^{4} + 8782 q^{5} - 38140 q^{6} - 115098 q^{8} - 1926445 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(49, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
49.12.c.a	$49$	$12$	49.c	7.c	$3$	$2$	$1$	$37.649$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-7}) \)		$4$	$0$	\(-67\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-67\zeta_{6}q^{2}+(-2441+2441\zeta_{6})q^{4}+\cdots\)
49.12.c.b	$49$	$12$	49.c	7.c	$3$	$2$	$1$	$37.649$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(24\)	\(-252\)	\(-4830\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+24\zeta_{6}q^{2}+(-252+252\zeta_{6})q^{3}+\cdots\)
49.12.c.c	$49$	$12$	49.c	7.c	$3$	$2$	$1$	$37.649$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(24\)	\(252\)	\(4830\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+24\zeta_{6}q^{2}+(252-252\zeta_{6})q^{3}+(1472+\cdots)q^{4}+\cdots\)
49.12.c.d	$49$	$12$	49.c	7.c	$3$	$4$	$2$	$37.649$	\(\Q(\sqrt{-3}, \sqrt{-1123})\)	None		$2$	$0$	\(54\)	\(-120\)	\(13500\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3^{3}+3^{3}\beta _{1}+\beta _{3})q^{2}+(60\beta _{1}-6\beta _{2}+\cdots)q^{3}+\cdots\)
49.12.c.e	$49$	$12$	49.c	7.c	$3$	$4$	$2$	$37.649$	\(\Q(\sqrt{-3}, \sqrt{-1123})\)	None		$2$	$0$	\(54\)	\(120\)	\(-13500\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3^{3}+3^{3}\beta _{1}+\beta _{3})q^{2}+(-60\beta _{1}+6\beta _{2}+\cdots)q^{3}+\cdots\)
49.12.c.f	$49$	$12$	49.c	7.c	$3$	$6$	$3$	$37.649$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(-77\)	\(-140\)	\(5026\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-26+\beta _{2}-26\beta _{3}-\beta _{5})q^{2}+(-11\beta _{1}+\cdots)q^{3}+\cdots\)
49.12.c.g	$49$	$12$	49.c	7.c	$3$	$6$	$3$	$37.649$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(-77\)	\(140\)	\(-5026\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-26+\beta _{2}-26\beta _{3}-\beta _{5})q^{2}+(11\beta _{1}+\cdots)q^{3}+\cdots\)
49.12.c.h	$49$	$12$	49.c	7.c	$3$	$8$	$4$	$37.649$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(108\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3^{3}\beta _{1}-\beta _{3})q^{2}+(-\beta _{4}+\beta _{5})q^{3}+\cdots\)
49.12.c.i	$49$	$12$	49.c	7.c	$3$	$12$	$6$	$37.649$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(22\)	\(244\)	\(8782\)	\(0\)	$2^{20}\cdot 3^{3}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-\beta _{1}-4\beta _{2})q^{2}+(-\beta _{1}+40\beta _{2}+\cdots)q^{3}+\cdots\)
49.12.c.j	$49$	$12$	49.c	7.c	$3$	$24$	$12$	$37.649$		None		$4$	$0$	\(-110\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{12}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(49, [\chi]) \cong \) \(S_{12}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)