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

• Label: 784.2.x
• Level: $784$
• Weight: $2$
• Character orbit: 784.x
• Rep. character: $\chi_{784}(165,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $304$
• Newform subspaces: $16$
• Sturm bound: $224$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.x (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 112 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(224\)
Trace bound:			\(5\)
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 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 8 * q^6 - 4 * q^8 - 4 * q^10 + 10 * q^11 + 2 * q^12 + 8 * q^13 - 32 * q^15 - 18 * q^16 + 4 * q^17 - 44 * q^18 + 2 * q^19 - 8 * q^20 + 16 * q^22 - 18 * q^24 + 2 * q^26 + 20 * q^27 - 48 * q^29 - 34 * q^30 - 20 * q^31 - 8 * q^32 + 4 * q^33 + 32 * q^34 + 80 * q^36 + 18 * q^37 - 18 * q^38 - 50 * q^40 + 12 * q^44 + 28 * q^45 - 24 * q^46 + 44 * q^47 - 40 * q^48 - 76 * q^50 + 6 * q^51 - 4 * q^52 - 14 * q^53 - 82 * q^54 + 24 * q^58 + 18 * q^59 - 26 * q^60 + 2 * q^61 - 28 * q^62 - 132 * q^64 + 4 * q^65 - 2 * q^66 - 50 * q^67 - 24 * q^68 + 20 * q^69 + 62 * q^72 + 52 * q^74 + 24 * q^75 + 68 * q^76 - 204 * q^78 + 4 * q^79 + 52 * q^80 + 92 * q^81 + 18 * q^82 - 32 * q^83 - 36 * q^85 - 18 * q^86 + 44 * q^88 + 20 * q^90 + 76 * q^92 + 26 * q^93 + 54 * q^94 - 20 * q^95 + 128 * q^96 + 16 * q^97 - 72 * 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.x.a	$784$	$2$	784.x	112.w	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.w.a	$2$	$0$	\(-2\)	\(4\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(1+\zeta_{12}+\cdots)q^{3}+\cdots\)
784.2.x.b	$784$	$2$	784.x	112.w	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.m.a	$4$	$0$	\(2\)	\(-4\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12}-\zeta_{12}^{2})q^{2}+(-2\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
784.2.x.c	$784$	$2$	784.x	112.w	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					16.2.e.a	$4$	$0$	\(2\)	\(-2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1+\zeta_{12}-\zeta_{12}^{2})q^{2}+(-\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
784.2.x.d	$784$	$2$	784.x	112.w	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.m.b	$4$	$0$	\(2\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12}-\zeta_{12}^{2})q^{2}+(-2\zeta_{12}+2\zeta_{12}^{3})q^{4}+\cdots\)
784.2.x.e	$784$	$2$	784.x	112.w	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.m.b	$4$	$0$	\(2\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12}-\zeta_{12}^{2})q^{2}+(-2\zeta_{12}+2\zeta_{12}^{3})q^{4}+\cdots\)
784.2.x.f	$784$	$2$	784.x	112.w	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					16.2.e.a	$4$	$0$	\(2\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1+\zeta_{12}-\zeta_{12}^{2})q^{2}+(\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
784.2.x.g	$784$	$2$	784.x	112.w	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.m.a	$4$	$0$	\(2\)	\(4\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12}-\zeta_{12}^{2})q^{2}+(2\zeta_{12}+2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
784.2.x.h	$784$	$2$	784.x	112.w	$12$	$4$	$1$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.w.a	$2$	$0$	\(4\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1+\zeta_{12}^{3})q^{2}+(-1+\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots\)
784.2.x.i	$784$	$2$	784.x	112.w	$12$	$8$	$2$	$6.260$	\(\Q(i, \sqrt{3}, \sqrt{7})\)	\(\Q(\sqrt{-7}) \)		✓			784.2.m.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+\beta _{1}q^{2}+(-1-\beta _{2}+\beta _{4}+\beta _{6})q^{4}+\cdots\)
784.2.x.j	$784$	$2$	784.x	112.w	$12$	$16$	$4$	$6.260$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					112.2.m.c	$4$	$0$	\(-2\)	\(0\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{9}+\beta _{12})q^{2}+(\beta _{8}-\beta _{13})q^{3}+(-1+\cdots)q^{4}+\cdots\)
784.2.x.k	$784$	$2$	784.x	112.w	$12$	$16$	$4$	$6.260$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					112.2.m.c	$4$	$0$	\(-2\)	\(0\)	\(4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{9}+\beta _{12})q^{2}+(-\beta _{8}+\beta _{13})q^{3}+\cdots\)
784.2.x.l	$784$	$2$	784.x	112.w	$12$	$24$	$6$	$6.260$		None					112.2.m.d	$4$	$0$	\(-2\)	\(-4\)	\(-4\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
784.2.x.m	$784$	$2$	784.x	112.w	$12$	$24$	$6$	$6.260$		None					112.2.m.d	$4$	$0$	\(-2\)	\(4\)	\(4\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
784.2.x.n	$784$	$2$	784.x	112.w	$12$	$40$	$10$	$6.260$		None		✓			784.2.m.i	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
784.2.x.o	$784$	$2$	784.x	112.w	$12$	$48$	$12$	$6.260$		None					112.2.w.c	$4$	$0$	\(-4\)	\(0\)	\(-4\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$
784.2.x.p	$784$	$2$	784.x	112.w	$12$	$96$	$24$	$6.260$		None		✓	✓		784.2.m.l	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$

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

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