Space of modular forms of level 448, weight 2, and trivial character

• Label: 448.2.a
• Level: $448$
• Weight: $2$
• Character orbit: 448.a
• Rep. character: $\chi_{448}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $10$
• Sturm bound: $128$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(128\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(448))\).

	Total	New	Old
Modular forms	76	12	64
Cusp forms	53	12	41
Eisenstein series	23	0	23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(7\)

Trace form

\( 12 q + 12 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(448))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	7
448.2.a.a	$448$	$2$	448.a	1.a	$1$	$1$	$1$	$3.577$	\(\Q\)	None	✓				14.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{7}+q^{9}+4q^{13}+6q^{17}+\cdots\)
448.2.a.b	$448$	$2$	448.a	1.a	$1$	$1$	$1$	$3.577$	\(\Q\)	None	✓				224.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
448.2.a.c	$448$	$2$	448.a	1.a	$1$	$1$	$1$	$3.577$	\(\Q\)	None	✓				56.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+4q^{5}+q^{7}+q^{9}-8q^{15}+\cdots\)
448.2.a.d	$448$	$2$	448.a	1.a	$1$	$1$	$1$	$3.577$	\(\Q\)	None	✓				56.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-3q^{9}+4q^{11}-2q^{13}+\cdots\)
448.2.a.e	$448$	$2$	448.a	1.a	$1$	$1$	$1$	$3.577$	\(\Q\)	None	✓				56.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-3q^{9}-4q^{11}-2q^{13}+\cdots\)
448.2.a.f	$448$	$2$	448.a	1.a	$1$	$1$	$1$	$3.577$	\(\Q\)	None	✓				224.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
448.2.a.g	$448$	$2$	448.a	1.a	$1$	$1$	$1$	$3.577$	\(\Q\)	None	✓				14.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{7}+q^{9}+4q^{13}+6q^{17}+\cdots\)
448.2.a.h	$448$	$2$	448.a	1.a	$1$	$1$	$1$	$3.577$	\(\Q\)	None	✓				56.2.a.b	$1$	$0$	\(0\)	\(2\)	\(4\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{5}-q^{7}+q^{9}+8q^{15}+\cdots\)
448.2.a.i	$448$	$2$	448.a	1.a	$1$	$2$	$2$	$3.577$	\(\Q(\sqrt{5}) \)	None	✓		✓		224.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-1+\beta )q^{5}-q^{7}+\cdots\)
448.2.a.j	$448$	$2$	448.a	1.a	$1$	$2$	$2$	$3.577$	\(\Q(\sqrt{5}) \)	None	✓		✓		224.2.a.c	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1+\beta )q^{5}+q^{7}+(3+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(448))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(448)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)