Space of modular forms of level 784, weight 2, and Character 784.w

• Label: 784.2.w
• Level: $784$
• Weight: $2$
• Character orbit: 784.w
• Rep. character: $\chi_{784}(19,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $304$
• Newform subspaces: $7$
• Sturm bound: $224$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.w (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 112 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(224\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(784, [\chi])\).

	Total	New	Old
Modular forms	480	336	144
Cusp forms	416	304	112
Eisenstein series	64	32	32

Trace form

304 * q + 2 * q^2 + 6 * q^3 + 6 * q^4 + 6 * q^5 - 28 * q^8 + 24 * q^10 - 6 * q^11 + 6 * q^12 - 18 * q^16 + 12 * q^17 + 16 * q^18 + 6 * q^19 + 20 * q^23 + 6 * q^24 + 6 * q^26 + 16 * q^29 + 38 * q^30 + 12 * q^32 + 12 * q^33 - 80 * q^36 - 14 * q^37 + 6 * q^38 + 4 * q^39 + 66 * q^40 - 32 * q^43 - 24 * q^44 - 12 * q^45 + 8 * q^46 - 52 * q^50 - 2 * q^51 - 84 * q^52 - 14 * q^53 - 42 * q^54 - 8 * q^58 - 42 * q^59 - 122 * q^60 + 6 * q^61 - 36 * q^64 + 4 * q^65 - 126 * q^66 - 26 * q^67 - 24 * q^68 + 32 * q^71 + 22 * q^72 - 104 * q^74 - 24 * q^75 + 20 * q^78 - 12 * q^80 + 92 * q^81 - 42 * q^82 + 4 * q^85 + 102 * q^86 + 12 * q^87 - 40 * q^88 - 164 * q^92 + 26 * q^93 + 42 * q^94 - 36 * q^96 - 48 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(784, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
784.2.w.a	$784$	$2$	784.w	112.v	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.j.c	$4$	$1$	\(-2\)	\(-6\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-1+\zeta_{12}+\cdots)q^{3}+\cdots\)
784.2.w.b	$784$	$2$	784.w	112.v	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.j.c	$4$	$0$	\(-2\)	\(6\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(1-\zeta_{12}+\cdots)q^{3}+\cdots\)
784.2.w.c	$784$	$2$	784.w	112.v	$12$	$8$	$2$	$6.260$	8.0.49787136.1	\(\Q(\sqrt{-7}) \)					112.2.j.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+\beta _{1}q^{2}+(-1-\beta _{2}+\beta _{4}+\beta _{6})q^{4}+\cdots\)
784.2.w.d	$784$	$2$	784.w	112.v	$12$	$8$	$2$	$6.260$	\(\Q(\zeta_{24})\)	None					112.2.j.a	$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\beta_{2}+\beta_1+1)q^{2}+(-2\beta_{4}+2\beta_1)q^{4}+\cdots\)
784.2.w.e	$784$	$2$	784.w	112.v	$12$	$32$	$8$	$6.260$		None					112.2.j.d	$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
784.2.w.f	$784$	$2$	784.w	112.v	$12$	$56$	$14$	$6.260$		None					112.2.v.a	$4$	$0$	\(-2\)	\(6\)	\(6\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
784.2.w.g	$784$	$2$	784.w	112.v	$12$	$192$	$48$	$6.260$		None		✓	✓		784.2.j.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(784, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)