Space of modular forms of level 784, weight 5, and Character 784.d

• Label: 784.5.d
• Level: $784$
• Weight: $5$
• Character orbit: 784.d
• Rep. character: $\chi_{784}(687,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $13$
• Sturm bound: $560$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	784.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(560\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(784, [\chi])\).

	Total	New	Old
Modular forms	472	82	390
Cusp forms	424	82	342
Eisenstein series	48	0	48

Trace form

\( 82 q + 36 q^{5} - 2382 q^{9} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(784, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
784.5.d.a	$784$	$5$	784.d	4.b	$2$	$2$	$2$	$81.042$	\(\Q(\sqrt{-3}) \)	None					16.5.c.a	$2$	$0$	\(0\)	\(0\)	\(-36\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}-18q^{5}-111q^{9}-9\zeta_{6}q^{11}+\cdots\)
784.5.d.b	$784$	$5$	784.d	4.b	$2$	$2$	$2$	$81.042$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓	✓			784.5.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-17\beta q^{5}+3^{4}q^{9}-239\beta q^{13}-79\beta q^{17}+\cdots\)
784.5.d.c	$784$	$5$	784.d	4.b	$2$	$2$	$2$	$81.042$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓	✓			784.5.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+31\beta q^{5}+3^{4}q^{9}+\beta q^{13}+401\beta q^{17}+\cdots\)
784.5.d.d	$784$	$5$	784.d	4.b	$2$	$4$	$4$	$81.042$	\(\Q(\sqrt{2}, \sqrt{-91})\)	None		✓			784.5.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-4\beta _{1}q^{5}-101q^{9}+3\beta _{2}q^{11}+\cdots\)
784.5.d.e	$784$	$5$	784.d	4.b	$2$	$4$	$4$	$81.042$	\(\Q(\sqrt{-21}, \sqrt{127})\)	None		✓			784.5.d.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}-3q^{9}-\beta _{3}q^{11}+\cdots\)
784.5.d.f	$784$	$5$	784.d	4.b	$2$	$4$	$4$	$81.042$	\(\Q(\sqrt{-7}, \sqrt{13})\)	None					112.5.d.a	$2$	$0$	\(0\)	\(0\)	\(36\)	\(0\)	$2^{6}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(9-\beta _{2})q^{5}+(-17+2\beta _{2}+\cdots)q^{9}+\cdots\)
784.5.d.g	$784$	$5$	784.d	4.b	$2$	$6$	$6$	$81.042$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					112.5.r.c	$2$	$0$	\(0\)	\(0\)	\(-18\)	\(0\)	$2^{7}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-3-\beta _{2})q^{5}+(-47+\beta _{2}+\cdots)q^{9}+\cdots\)
784.5.d.h	$784$	$5$	784.d	4.b	$2$	$6$	$6$	$81.042$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					112.5.r.c	$2$	$0$	\(0\)	\(0\)	\(18\)	\(0\)	$2^{7}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(3+\beta _{2})q^{5}+(-47+\beta _{2}+\cdots)q^{9}+\cdots\)
784.5.d.i	$784$	$5$	784.d	4.b	$2$	$8$	$8$	$81.042$	8.0.\(\cdots\).16	None		✓			784.5.d.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{24}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{5}q^{5}+(-3^{4}-\beta _{3})q^{9}+\cdots\)
784.5.d.j	$784$	$5$	784.d	4.b	$2$	$8$	$8$	$81.042$	8.0.\(\cdots\).1	None					112.5.d.b	$2$	$0$	\(0\)	\(0\)	\(36\)	\(0\)	$2^{22}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(4+\beta _{2})q^{5}+(10-\beta _{7})q^{9}+\cdots\)
784.5.d.k	$784$	$5$	784.d	4.b	$2$	$10$	$10$	$81.042$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					112.5.r.a	$2$	$0$	\(0\)	\(0\)	\(-18\)	\(0\)	$2^{23}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(-2+\beta _{1})q^{5}+(-15-\beta _{1}+\cdots)q^{9}+\cdots\)
784.5.d.l	$784$	$5$	784.d	4.b	$2$	$10$	$10$	$81.042$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					112.5.r.a	$2$	$0$	\(0\)	\(0\)	\(18\)	\(0\)	$2^{23}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(2-\beta _{1})q^{5}+(-15-\beta _{1}+\cdots)q^{9}+\cdots\)
784.5.d.m	$784$	$5$	784.d	4.b	$2$	$16$	$16$	$81.042$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓		784.5.d.m	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{44}\cdot 3^{2}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{9}+\beta _{10})q^{5}+(-6^{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(784, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)