Space of modular forms of level 448, weight 4, and trivial character

• Label: 448.4.a
• Level: $448$
• Weight: $4$
• Character orbit: 448.a
• Rep. character: $\chi_{448}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $24$
• Sturm bound: $256$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(256\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(448))\).

	Total	New	Old
Modular forms	204	36	168
Cusp forms	180	36	144
Eisenstein series	24	0	24

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(53\)	\(9\)	\(44\)		\(47\)	\(9\)	\(38\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(49\)	\(8\)	\(41\)		\(43\)	\(8\)	\(35\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(51\)	\(9\)	\(42\)		\(45\)	\(9\)	\(36\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(51\)	\(10\)	\(41\)		\(45\)	\(10\)	\(35\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(104\)	\(19\)	\(85\)		\(92\)	\(19\)	\(73\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(100\)	\(17\)	\(83\)		\(88\)	\(17\)	\(71\)		\(12\)	\(0\)	\(12\)

Trace form

36 * q + 324 * q^9 + 104 * q^17 + 812 * q^25 + 200 * q^29 + 464 * q^33 + 8 * q^37 - 392 * q^41 - 1968 * q^45 + 1764 * q^49 - 1192 * q^53 - 688 * q^57 + 1824 * q^61 - 1536 * q^65 + 2544 * q^69 - 296 * q^73 - 952 * q^77 + 1076 * q^81 - 480 * q^85 + 88 * q^89 + 9216 * q^93 - 1816 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(448))\) 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
448.4.a.a	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				28.4.a.a	$1$	$0$	\(0\)	\(-10\)	\(8\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}+8q^{5}+7q^{7}+73q^{9}-40q^{11}+\cdots\)
448.4.a.b	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				14.4.a.a	$1$	$0$	\(0\)	\(-8\)	\(14\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+14q^{5}-7q^{7}+37q^{9}+28q^{11}+\cdots\)
448.4.a.c	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				56.4.a.b	$1$	$0$	\(0\)	\(-6\)	\(-8\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-8q^{5}-7q^{7}+9q^{9}-56q^{11}+\cdots\)
448.4.a.d	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				28.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(-6\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-6q^{5}+7q^{7}-11q^{9}+12q^{11}+\cdots\)
448.4.a.e	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				7.4.a.a	$1$	$0$	\(0\)	\(-2\)	\(-16\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2^{4}q^{5}+7q^{7}-23q^{9}-8q^{11}+\cdots\)
448.4.a.f	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				224.4.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-7q^{7}-23q^{9}+20q^{11}+20q^{13}+\cdots\)
448.4.a.g	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				14.4.a.b	$1$	$1$	\(0\)	\(-2\)	\(12\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+12q^{5}-7q^{7}-23q^{9}+48q^{11}+\cdots\)
448.4.a.h	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				56.4.a.a	$1$	$0$	\(0\)	\(-2\)	\(16\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2^{4}q^{5}+7q^{7}-23q^{9}+24q^{11}+\cdots\)
448.4.a.i	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				7.4.a.a	$1$	$0$	\(0\)	\(2\)	\(-16\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2^{4}q^{5}-7q^{7}-23q^{9}+8q^{11}+\cdots\)
448.4.a.j	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				224.4.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+7q^{7}-23q^{9}-20q^{11}+20q^{13}+\cdots\)
448.4.a.k	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				14.4.a.b	$1$	$1$	\(0\)	\(2\)	\(12\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+12q^{5}+7q^{7}-23q^{9}-48q^{11}+\cdots\)
448.4.a.l	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				56.4.a.a	$1$	$0$	\(0\)	\(2\)	\(16\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2^{4}q^{5}-7q^{7}-23q^{9}-24q^{11}+\cdots\)
448.4.a.m	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				28.4.a.b	$1$	$1$	\(0\)	\(4\)	\(-6\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-6q^{5}-7q^{7}-11q^{9}-12q^{11}+\cdots\)
448.4.a.n	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				56.4.a.b	$1$	$0$	\(0\)	\(6\)	\(-8\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}-8q^{5}+7q^{7}+9q^{9}+56q^{11}+\cdots\)
448.4.a.o	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				14.4.a.a	$1$	$0$	\(0\)	\(8\)	\(14\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+14q^{5}+7q^{7}+37q^{9}-28q^{11}+\cdots\)
448.4.a.p	$448$	$4$	448.a	1.a	$1$	$1$	$1$	$26.433$	\(\Q\)	None	✓				28.4.a.a	$1$	$0$	\(0\)	\(10\)	\(8\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}+8q^{5}-7q^{7}+73q^{9}+40q^{11}+\cdots\)
448.4.a.q	$448$	$4$	448.a	1.a	$1$	$2$	$2$	$26.433$	\(\Q(\sqrt{37}) \)	None	✓				224.4.a.c	$1$	$1$	\(0\)	\(-6\)	\(6\)	\(-14\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}+(3-\beta )q^{5}-7q^{7}+(19+\cdots)q^{9}+\cdots\)
448.4.a.r	$448$	$4$	448.a	1.a	$1$	$2$	$2$	$26.433$	\(\Q(\sqrt{57}) \)	None	✓				56.4.a.c	$1$	$1$	\(0\)	\(-2\)	\(-22\)	\(-14\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-11-\beta )q^{5}-7q^{7}+\cdots\)
448.4.a.s	$448$	$4$	448.a	1.a	$1$	$2$	$2$	$26.433$	\(\Q(\sqrt{57}) \)	None	✓				56.4.a.c	$1$	$1$	\(0\)	\(2\)	\(-22\)	\(14\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-11-\beta )q^{5}+7q^{7}+\cdots\)
448.4.a.t	$448$	$4$	448.a	1.a	$1$	$2$	$2$	$26.433$	\(\Q(\sqrt{37}) \)	None	✓				224.4.a.c	$1$	$0$	\(0\)	\(6\)	\(6\)	\(14\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(3-\beta )q^{5}+7q^{7}+(19+\cdots)q^{9}+\cdots\)
448.4.a.u	$448$	$4$	448.a	1.a	$1$	$3$	$3$	$26.433$	3.3.2981.1	None	✓		✓		224.4.a.e	$1$	$1$	\(0\)	\(-8\)	\(-10\)	\(21\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(-3-\beta _{2})q^{5}+7q^{7}+\cdots\)
448.4.a.v	$448$	$4$	448.a	1.a	$1$	$3$	$3$	$26.433$	3.3.621.1	None	✓		✓		224.4.a.f	$1$	$1$	\(0\)	\(0\)	\(6\)	\(-21\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}-7q^{7}+\cdots\)
448.4.a.w	$448$	$4$	448.a	1.a	$1$	$3$	$3$	$26.433$	3.3.621.1	None	✓		✓		224.4.a.f	$1$	$0$	\(0\)	\(0\)	\(6\)	\(21\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}+7q^{7}+\cdots\)
448.4.a.x	$448$	$4$	448.a	1.a	$1$	$3$	$3$	$26.433$	3.3.2981.1	None	✓		✓		224.4.a.e	$1$	$0$	\(0\)	\(8\)	\(-10\)	\(-21\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(-3-\beta _{2})q^{5}-7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(448))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(448)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)