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

• Label: 784.2.m
• Level: $784$
• Weight: $2$
• Character orbit: 784.m
• Rep. character: $\chi_{784}(197,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $154$
• Newform subspaces: $12$
• Sturm bound: $224$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.m (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 16 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 12 \)
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	240	174	66
Cusp forms	208	154	54
Eisenstein series	32	20	12

Trace form

154 * q + 2 * q^2 + 2 * q^3 + 2 * q^5 + 8 * q^6 - 16 * q^8 - 8 * q^10 - 2 * q^11 + 2 * q^13 - 12 * q^15 + 16 * q^16 + 4 * q^17 + 22 * q^18 + 10 * q^19 + 24 * q^20 - 28 * q^22 + 24 * q^24 + 8 * q^26 - 16 * q^27 - 18 * q^29 + 48 * q^30 + 16 * q^31 - 8 * q^32 + 4 * q^33 - 16 * q^34 + 12 * q^36 - 6 * q^37 + 12 * q^38 - 12 * q^40 + 10 * q^43 + 32 * q^44 - 2 * q^45 + 12 * q^46 - 16 * q^47 - 60 * q^48 - 38 * q^50 - 56 * q^51 - 12 * q^52 + 10 * q^53 - 32 * q^54 - 24 * q^58 + 30 * q^59 - 28 * q^60 - 14 * q^61 - 52 * q^62 + 24 * q^64 + 12 * q^65 + 24 * q^66 + 10 * q^67 - 20 * q^69 - 100 * q^72 + 20 * q^74 + 18 * q^75 - 44 * q^76 + 32 * q^78 + 8 * q^79 - 20 * q^80 - 70 * q^81 + 36 * q^82 + 42 * q^83 + 16 * q^85 + 36 * q^88 + 60 * q^90 + 44 * q^92 - 4 * q^93 - 28 * q^94 - 52 * q^95 + 24 * q^96 + 4 * q^97 - 74 * 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.m.a	$784$	$2$	784.m	16.e	$4$	$2$	$1$	$6.260$	\(\Q(\sqrt{-1}) \)	None					112.2.m.b	$2$	$0$	\(-2\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{2}-2 i q^{4}+(-2 i-2)q^{5}+\cdots\)
784.2.m.b	$784$	$2$	784.m	16.e	$4$	$2$	$1$	$6.260$	\(\Q(\sqrt{-1}) \)	None					16.2.e.a	$2$	$0$	\(-2\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+(-i+1)q^{3}+2 i q^{4}+\cdots\)
784.2.m.c	$784$	$2$	784.m	16.e	$4$	$2$	$1$	$6.260$	\(\Q(\sqrt{-1}) \)	None					112.2.m.a	$2$	$0$	\(-2\)	\(4\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{2}+(-2 i+2)q^{3}-2 i q^{4}+\cdots\)
784.2.m.d	$784$	$2$	784.m	16.e	$4$	$4$	$2$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.w.a	$2$	$1$	\(-2\)	\(-2\)	\(-6\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{3}-\beta_{2})q^{2}+\beta_{3} q^{3}+(-\beta_{3}-\beta_1)q^{4}+\cdots\)
784.2.m.e	$784$	$2$	784.m	16.e	$4$	$4$	$2$	$6.260$	\(\Q(\zeta_{12})\)	None					112.2.w.a	$2$	$0$	\(-2\)	\(2\)	\(6\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_1-1)q^{2}+(\beta_{2}-\beta_1+1)q^{3}+\cdots\)
784.2.m.f	$784$	$2$	784.m	16.e	$4$	$4$	$2$	$6.260$	\(\Q(i, \sqrt{7})\)	\(\Q(\sqrt{-7}) \)		✓			784.2.m.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}-2\beta _{3})q^{8}+\cdots\)
784.2.m.g	$784$	$2$	784.m	16.e	$4$	$8$	$4$	$6.260$	8.0.214798336.3	None					112.2.m.c	$2$	$0$	\(2\)	\(0\)	\(4\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{2}+(\beta _{1}+\beta _{4})q^{3}+(1+\beta _{4}-\beta _{6}+\cdots)q^{4}+\cdots\)
784.2.m.h	$784$	$2$	784.m	16.e	$4$	$12$	$6$	$6.260$	12.0.\(\cdots\).1	None					112.2.m.d	$2$	$0$	\(2\)	\(-4\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+(\beta _{2}-\beta _{10})q^{3}+\beta _{1}q^{4}+(-1+\cdots)q^{5}+\cdots\)
784.2.m.i	$784$	$2$	784.m	16.e	$4$	$20$	$10$	$6.260$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		✓			784.2.m.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{2}+\beta _{9}q^{3}-\beta _{17}q^{4}+\beta _{2}q^{5}+\cdots\)
784.2.m.j	$784$	$2$	784.m	16.e	$4$	$24$	$12$	$6.260$		None					112.2.w.c	$2$	$0$	\(4\)	\(0\)	\(-4\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
784.2.m.k	$784$	$2$	784.m	16.e	$4$	$24$	$12$	$6.260$		None					112.2.w.c	$2$	$0$	\(4\)	\(0\)	\(4\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
784.2.m.l	$784$	$2$	784.m	16.e	$4$	$48$	$24$	$6.260$		None		✓	✓		784.2.m.l	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

\( S_{2}^{\mathrm{old}}(784, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)