Space of modular forms of level 448, weight 7, and Character 448.c

• Label: 448.7.c
• Level: $448$
• Weight: $7$
• Character orbit: 448.c
• Rep. character: $\chi_{448}(321,\cdot)$
• Character field: $\Q$
• Dimension: $94$
• Newform subspaces: $12$
• Sturm bound: $448$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	448.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(448\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(448, [\chi])\).

	Total	New	Old
Modular forms	396	98	298
Cusp forms	372	94	278
Eisenstein series	24	4	20

Trace form

\( 94 q - 21874 q^{9} - 20896 q^{21} - 268754 q^{25} + 33204 q^{29} + 107284 q^{37} - 2 q^{49} + 221844 q^{53} + 2912 q^{57} + 62496 q^{65} - 560108 q^{77} + 3430942 q^{81} - 3153024 q^{85} + 2990848 q^{93}+O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(448, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
448.7.c.a	$448$	$7$	448.c	7.b	$2$	$1$	$1$	$103.064$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-343\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-7^{3}q^{7}+3^{6}q^{9}-1962q^{11}-22734q^{23}+\cdots\)
448.7.c.b	$448$	$7$	448.c	7.b	$2$	$1$	$1$	$103.064$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(343\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+7^{3}q^{7}+3^{6}q^{9}+1962q^{11}+22734q^{23}+\cdots\)
448.7.c.c	$448$	$7$	448.c	7.b	$2$	$2$	$2$	$103.064$	\(\Q(\sqrt{-510}) \)	None					7.7.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-266\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+\beta q^{5}+(-133-7\beta )q^{7}-1311q^{9}+\cdots\)
448.7.c.d	$448$	$7$	448.c	7.b	$2$	$2$	$2$	$103.064$	\(\Q(\sqrt{-510}) \)	None					7.7.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(266\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-\beta q^{5}+(133-7\beta )q^{7}-1311q^{9}+\cdots\)
448.7.c.e	$448$	$7$	448.c	7.b	$2$	$4$	$4$	$103.064$	4.0.211968.1	None					14.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-308\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+5\beta _{2}q^{5}+(-77-14\beta _{1}+\cdots)q^{7}+\cdots\)
448.7.c.f	$448$	$7$	448.c	7.b	$2$	$4$	$4$	$103.064$	4.0.903168.1	None					28.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$2^{7}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{1}q^{5}+(-7+\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
448.7.c.g	$448$	$7$	448.c	7.b	$2$	$4$	$4$	$103.064$	4.0.903168.1	None					28.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(28\)	$2^{7}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-\beta _{1}q^{5}+(7+\beta _{1}-2\beta _{2}+\beta _{3})q^{7}+\cdots\)
448.7.c.h	$448$	$7$	448.c	7.b	$2$	$4$	$4$	$103.064$	4.0.211968.1	None					14.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(308\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-5\beta _{2}q^{5}+(77-14\beta _{1}+7\beta _{2}+\cdots)q^{7}+\cdots\)
448.7.c.i	$448$	$7$	448.c	7.b	$2$	$12$	$12$	$103.064$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					56.7.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-564\)	$2^{51}\cdot 7^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+\beta _{8}q^{5}+(-47-\beta _{4}+\beta _{5}+\cdots)q^{7}+\cdots\)
448.7.c.j	$448$	$7$	448.c	7.b	$2$	$12$	$12$	$103.064$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					56.7.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(564\)	$2^{51}\cdot 7^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}-\beta _{8}q^{5}+(47-\beta _{4}-\beta _{5})q^{7}+\cdots\)
448.7.c.k	$448$	$7$	448.c	7.b	$2$	$24$	$24$	$103.064$		None					224.7.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
448.7.c.l	$448$	$7$	448.c	7.b	$2$	$24$	$24$	$103.064$		None					224.7.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

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