Space of modular forms of level 448, weight 6, and trivial character

• Label: 448.6.a
• Level: $448$
• Weight: $6$
• Character orbit: 448.a
• Rep. character: $\chi_{448}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $32$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(448))\).

	Total	New	Old
Modular forms	332	60	272
Cusp forms	308	60	248
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
\(+\)	\(+\)	\(+\)		\(81\)	\(15\)	\(66\)		\(75\)	\(15\)	\(60\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(85\)	\(16\)	\(69\)		\(79\)	\(16\)	\(63\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(83\)	\(15\)	\(68\)		\(77\)	\(15\)	\(62\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(83\)	\(14\)	\(69\)		\(77\)	\(14\)	\(63\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(164\)	\(29\)	\(135\)		\(152\)	\(29\)	\(123\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(168\)	\(31\)	\(137\)		\(156\)	\(31\)	\(125\)		\(12\)	\(0\)	\(12\)

Trace form

60 * q + 4860 * q^9 - 808 * q^17 + 43732 * q^25 - 4072 * q^29 - 11344 * q^33 + 10648 * q^37 + 28168 * q^41 + 72048 * q^45 + 144060 * q^49 - 52984 * q^53 + 61616 * q^57 - 96160 * q^61 - 7680 * q^65 - 240432 * q^69 - 10072 * q^73 + 7448 * q^77 + 436076 * q^81 + 264800 * q^85 - 70232 * q^89 - 728064 * q^93 - 160808 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\) 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
448.6.a.a	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				56.6.a.b	$1$	$0$	\(0\)	\(-30\)	\(-32\)	\(49\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-30q^{3}-2^{5}q^{5}+7^{2}q^{7}+657q^{9}+\cdots\)
448.6.a.b	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				28.6.a.b	$1$	$1$	\(0\)	\(-26\)	\(-16\)	\(-49\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-26q^{3}-2^{4}q^{5}-7^{2}q^{7}+433q^{9}+\cdots\)
448.6.a.c	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				7.6.a.a	$1$	$1$	\(0\)	\(-14\)	\(56\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-14q^{3}+56q^{5}+7^{2}q^{7}-47q^{9}+\cdots\)
448.6.a.d	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				224.6.a.a	$1$	$0$	\(0\)	\(-14\)	\(64\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{3}+2^{6}q^{5}-7^{2}q^{7}-47q^{9}+\cdots\)
448.6.a.e	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				14.6.a.a	$1$	$0$	\(0\)	\(-10\)	\(-84\)	\(49\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}-84q^{5}+7^{2}q^{7}-143q^{9}+\cdots\)
448.6.a.f	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				14.6.a.b	$1$	$1$	\(0\)	\(-8\)	\(-10\)	\(-49\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-10q^{5}-7^{2}q^{7}-179q^{9}+\cdots\)
448.6.a.g	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				56.6.a.a	$1$	$0$	\(0\)	\(-6\)	\(-4\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-4q^{5}-7^{2}q^{7}-207q^{9}-240q^{11}+\cdots\)
448.6.a.h	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				28.6.a.a	$1$	$0$	\(0\)	\(-2\)	\(96\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+96q^{5}-7^{2}q^{7}-239q^{9}+\cdots\)
448.6.a.i	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				28.6.a.a	$1$	$0$	\(0\)	\(2\)	\(96\)	\(49\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+96q^{5}+7^{2}q^{7}-239q^{9}+\cdots\)
448.6.a.j	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				56.6.a.a	$1$	$0$	\(0\)	\(6\)	\(-4\)	\(49\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}-4q^{5}+7^{2}q^{7}-207q^{9}+240q^{11}+\cdots\)
448.6.a.k	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				14.6.a.b	$1$	$1$	\(0\)	\(8\)	\(-10\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-10q^{5}+7^{2}q^{7}-179q^{9}+\cdots\)
448.6.a.l	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				14.6.a.a	$1$	$0$	\(0\)	\(10\)	\(-84\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}-84q^{5}-7^{2}q^{7}-143q^{9}+\cdots\)
448.6.a.m	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				7.6.a.a	$1$	$1$	\(0\)	\(14\)	\(56\)	\(-49\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14q^{3}+56q^{5}-7^{2}q^{7}-47q^{9}+\cdots\)
448.6.a.n	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				224.6.a.a	$1$	$1$	\(0\)	\(14\)	\(64\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14q^{3}+2^{6}q^{5}+7^{2}q^{7}-47q^{9}+\cdots\)
448.6.a.o	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				28.6.a.b	$1$	$1$	\(0\)	\(26\)	\(-16\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+26q^{3}-2^{4}q^{5}+7^{2}q^{7}+433q^{9}+\cdots\)
448.6.a.p	$448$	$6$	448.a	1.a	$1$	$1$	$1$	$71.852$	\(\Q\)	None	✓				56.6.a.b	$1$	$0$	\(0\)	\(30\)	\(-32\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+30q^{3}-2^{5}q^{5}-7^{2}q^{7}+657q^{9}+\cdots\)
448.6.a.q	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{177}) \)	None	✓				56.6.a.c	$1$	$0$	\(0\)	\(-26\)	\(62\)	\(-98\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-13-\beta )q^{3}+(31+5\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.r	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{193}) \)	None	✓				56.6.a.d	$1$	$1$	\(0\)	\(-14\)	\(-42\)	\(98\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(-21-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.s	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{61}) \)	None	✓				224.6.a.c	$1$	$1$	\(0\)	\(-14\)	\(-34\)	\(98\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(-17+7\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.t	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{345}) \)	None	✓				56.6.a.e	$1$	$1$	\(0\)	\(-6\)	\(-82\)	\(98\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}+(-41+3\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.u	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{57}) \)	None	✓				7.6.a.b	$1$	$0$	\(0\)	\(-6\)	\(18\)	\(-98\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-3\beta )q^{3}+(9-5\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.v	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{345}) \)	None	✓				56.6.a.e	$1$	$1$	\(0\)	\(6\)	\(-82\)	\(-98\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(-41+3\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.w	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{57}) \)	None	✓				7.6.a.b	$1$	$0$	\(0\)	\(6\)	\(18\)	\(98\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+3\beta )q^{3}+(9-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.x	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{193}) \)	None	✓				56.6.a.d	$1$	$1$	\(0\)	\(14\)	\(-42\)	\(-98\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(-21+5\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.y	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{61}) \)	None	✓				224.6.a.c	$1$	$0$	\(0\)	\(14\)	\(-34\)	\(-98\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(-17-7\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.z	$448$	$6$	448.a	1.a	$1$	$2$	$2$	$71.852$	\(\Q(\sqrt{177}) \)	None	✓				56.6.a.c	$1$	$0$	\(0\)	\(26\)	\(62\)	\(98\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(13-\beta )q^{3}+(31-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.ba	$448$	$6$	448.a	1.a	$1$	$3$	$3$	$71.852$	3.3.367637.1	None	✓				224.6.a.e	$1$	$1$	\(0\)	\(-8\)	\(14\)	\(-147\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(5+\beta _{2})q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.bb	$448$	$6$	448.a	1.a	$1$	$3$	$3$	$71.852$	3.3.367637.1	None	✓				224.6.a.e	$1$	$0$	\(0\)	\(8\)	\(14\)	\(147\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(5+\beta _{2})q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.bc	$448$	$6$	448.a	1.a	$1$	$4$	$4$	$71.852$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		224.6.a.g	$1$	$1$	\(0\)	\(-18\)	\(30\)	\(196\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}+(7+\beta _{1}+\beta _{2})q^{5}+\cdots\)
448.6.a.bd	$448$	$6$	448.a	1.a	$1$	$4$	$4$	$71.852$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		224.6.a.g	$1$	$0$	\(0\)	\(18\)	\(30\)	\(-196\)		$-$	$+$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{3}+(7+\beta _{1}+\beta _{2})q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.be	$448$	$6$	448.a	1.a	$1$	$5$	$5$	$71.852$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		224.6.a.i	$1$	$1$	\(0\)	\(-10\)	\(-36\)	\(-245\)		$+$	$+$	$+$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(-7+\beta _{3})q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.bf	$448$	$6$	448.a	1.a	$1$	$5$	$5$	$71.852$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		224.6.a.i	$1$	$0$	\(0\)	\(10\)	\(-36\)	\(245\)		$+$	$-$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(-7+\beta _{3})q^{5}+7^{2}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(448))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(448)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 7}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)