Space of modular forms of level 448, weight 5, and Character 448.c

• Label: 448.5.c
• Level: $448$
• Weight: $5$
• Character orbit: 448.c
• Rep. character: $\chi_{448}(321,\cdot)$
• Character field: $\Q$
• Dimension: $62$
• Newform subspaces: $10$
• Sturm bound: $320$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	448.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(320\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(448, [\chi])\).

	Total	New	Old
Modular forms	268	66	202
Cusp forms	244	62	182
Eisenstein series	24	4	20

Trace form

\( 62 q - 1570 q^{9} - 448 q^{21} - 6754 q^{25} - 860 q^{29} - 2332 q^{37} - 2 q^{49} + 484 q^{53} + 320 q^{57} + 2496 q^{65} + 12004 q^{77} + 43966 q^{81} + 36096 q^{85} + 39424 q^{93}+O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(448, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
448.5.c.a	$448$	$5$	448.c	7.b	$2$	$1$	$1$	$46.310$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-49\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-7^{2}q^{7}+3^{4}q^{9}-206q^{11}+734q^{23}+\cdots\)
448.5.c.b	$448$	$5$	448.c	7.b	$2$	$1$	$1$	$46.310$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(49\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+7^{2}q^{7}+3^{4}q^{9}+206q^{11}-734q^{23}+\cdots\)
448.5.c.c	$448$	$5$	448.c	7.b	$2$	$2$	$2$	$46.310$	\(\Q(\sqrt{-3}) \)	None					28.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}-3\beta q^{5}+(7\beta-7)q^{7}+33 q^{9}+\cdots\)
448.5.c.d	$448$	$5$	448.c	7.b	$2$	$2$	$2$	$46.310$	\(\Q(\sqrt{-3}) \)	None					28.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}+3\beta q^{5}+(7\beta+7)q^{7}+33 q^{9}+\cdots\)
448.5.c.e	$448$	$5$	448.c	7.b	$2$	$4$	$4$	$46.310$	4.0.1308672.3	None					14.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-76\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-19+\beta _{1}+\cdots)q^{7}+\cdots\)
448.5.c.f	$448$	$5$	448.c	7.b	$2$	$4$	$4$	$46.310$	4.0.1308672.3	None					14.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(76\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(19+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
448.5.c.g	$448$	$5$	448.c	7.b	$2$	$8$	$8$	$46.310$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					56.5.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-56\)	$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-7+\beta _{5})q^{7}+(-31+\cdots)q^{9}+\cdots\)
448.5.c.h	$448$	$5$	448.c	7.b	$2$	$8$	$8$	$46.310$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					56.5.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(56\)	$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(7+\beta _{4})q^{7}+(-31+\cdots)q^{9}+\cdots\)
448.5.c.i	$448$	$5$	448.c	7.b	$2$	$16$	$16$	$46.310$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					224.5.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{54}\cdot 7$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}-\beta _{7}q^{5}+(\beta _{5}-\beta _{8})q^{7}+(-31+\cdots)q^{9}+\cdots\)
448.5.c.j	$448$	$5$	448.c	7.b	$2$	$16$	$16$	$46.310$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					224.5.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{54}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{5}q^{5}+(\beta _{1}+\beta _{7})q^{7}+(-23+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(448, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)