Space of modular forms of level 448, weight 8, and trivial character

• Label: 448.8.a
• Level: $448$
• Weight: $8$
• Character orbit: 448.a
• Rep. character: $\chi_{448}(1,\cdot)$
• Character field: $\Q$
• Dimension: $84$
• Newform subspaces: $32$
• Sturm bound: $512$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(512\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(448))\).

	Total	New	Old
Modular forms	460	84	376
Cusp forms	436	84	352
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
\(+\)	\(+\)	\(+\)		\(117\)	\(21\)	\(96\)		\(111\)	\(21\)	\(90\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(113\)	\(20\)	\(93\)		\(107\)	\(20\)	\(87\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(115\)	\(21\)	\(94\)		\(109\)	\(21\)	\(88\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(115\)	\(22\)	\(93\)		\(109\)	\(22\)	\(87\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(232\)	\(43\)	\(189\)		\(220\)	\(43\)	\(177\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(228\)	\(41\)	\(187\)		\(216\)	\(41\)	\(175\)		\(12\)	\(0\)	\(12\)

Trace form

84 * q + 61236 * q^9 - 5816 * q^17 + 1280252 * q^25 + 51688 * q^29 + 198032 * q^33 - 415576 * q^37 - 319848 * q^41 - 2210928 * q^45 + 9882516 * q^49 + 8639800 * q^53 + 1372688 * q^57 + 4559776 * q^61 + 666624 * q^65 - 5832528 * q^69 + 2685304 * q^73 + 5957224 * q^77 + 58155428 * q^81 - 15068000 * q^85 + 4781048 * q^89 - 17418240 * q^93 - 3818296 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(448))\) 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
448.8.a.a	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				14.8.a.a	$1$	$0$	\(0\)	\(-82\)	\(-448\)	\(343\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-82q^{3}-448q^{5}+7^{3}q^{7}+4537q^{9}+\cdots\)
448.8.a.b	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				14.8.a.b	$1$	$0$	\(0\)	\(-66\)	\(400\)	\(343\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-66q^{3}+20^{2}q^{5}+7^{3}q^{7}+2169q^{9}+\cdots\)
448.8.a.c	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				56.8.a.b	$1$	$0$	\(0\)	\(-46\)	\(160\)	\(-343\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-46q^{3}+160q^{5}-7^{3}q^{7}-71q^{9}+\cdots\)
448.8.a.d	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				7.8.a.a	$1$	$1$	\(0\)	\(-42\)	\(84\)	\(-343\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-42q^{3}+84q^{5}-7^{3}q^{7}-423q^{9}+\cdots\)
448.8.a.e	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				56.8.a.a	$1$	$0$	\(0\)	\(-18\)	\(-160\)	\(343\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-18q^{3}-160q^{5}+7^{3}q^{7}-1863q^{9}+\cdots\)
448.8.a.f	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				56.8.a.a	$1$	$0$	\(0\)	\(18\)	\(-160\)	\(-343\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{3}-160q^{5}-7^{3}q^{7}-1863q^{9}+\cdots\)
448.8.a.g	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				7.8.a.a	$1$	$1$	\(0\)	\(42\)	\(84\)	\(343\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+42q^{3}+84q^{5}+7^{3}q^{7}-423q^{9}+\cdots\)
448.8.a.h	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				56.8.a.b	$1$	$0$	\(0\)	\(46\)	\(160\)	\(343\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+46q^{3}+160q^{5}+7^{3}q^{7}-71q^{9}+\cdots\)
448.8.a.i	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				14.8.a.b	$1$	$0$	\(0\)	\(66\)	\(400\)	\(-343\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+66q^{3}+20^{2}q^{5}-7^{3}q^{7}+2169q^{9}+\cdots\)
448.8.a.j	$448$	$8$	448.a	1.a	$1$	$1$	$1$	$139.948$	\(\Q\)	None	✓				14.8.a.a	$1$	$0$	\(0\)	\(82\)	\(-448\)	\(-343\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+82q^{3}-448q^{5}-7^{3}q^{7}+4537q^{9}+\cdots\)
448.8.a.k	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{865}) \)	None	✓				7.8.a.b	$1$	$0$	\(0\)	\(-94\)	\(-330\)	\(-686\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-47-\beta )q^{3}+(-165-5\beta )q^{5}+\cdots\)
448.8.a.l	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{1969}) \)	None	✓				14.8.a.c	$1$	$1$	\(0\)	\(-70\)	\(-126\)	\(686\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-35-\beta )q^{3}+(-63+9\beta )q^{5}+7^{3}q^{7}+\cdots\)
448.8.a.m	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{249}) \)	None	✓				56.8.a.c	$1$	$1$	\(0\)	\(-42\)	\(-14\)	\(-686\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-21-3\beta )q^{3}+(-7-11\beta )q^{5}+\cdots\)
448.8.a.n	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{3529}) \)	None	✓				28.8.a.a	$1$	$1$	\(0\)	\(-14\)	\(-42\)	\(-686\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(-21+3\beta )q^{5}-7^{3}q^{7}+\cdots\)
448.8.a.o	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{1009}) \)	None	✓				28.8.a.b	$1$	$0$	\(0\)	\(-14\)	\(294\)	\(-686\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(147+11\beta )q^{5}-7^{3}q^{7}+\cdots\)
448.8.a.p	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{3529}) \)	None	✓				28.8.a.a	$1$	$1$	\(0\)	\(14\)	\(-42\)	\(686\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(-21-3\beta )q^{5}+7^{3}q^{7}+\cdots\)
448.8.a.q	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{1009}) \)	None	✓				28.8.a.b	$1$	$0$	\(0\)	\(14\)	\(294\)	\(686\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(147-11\beta )q^{5}+7^{3}q^{7}+\cdots\)
448.8.a.r	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{249}) \)	None	✓				56.8.a.c	$1$	$1$	\(0\)	\(42\)	\(-14\)	\(686\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(21-3\beta )q^{3}+(-7+11\beta )q^{5}+7^{3}q^{7}+\cdots\)
448.8.a.s	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{1969}) \)	None	✓				14.8.a.c	$1$	$1$	\(0\)	\(70\)	\(-126\)	\(-686\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(35-\beta )q^{3}+(-63-9\beta )q^{5}-7^{3}q^{7}+\cdots\)
448.8.a.t	$448$	$8$	448.a	1.a	$1$	$2$	$2$	$139.948$	\(\Q(\sqrt{865}) \)	None	✓				7.8.a.b	$1$	$0$	\(0\)	\(94\)	\(-330\)	\(686\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(47-\beta )q^{3}+(-165+5\beta )q^{5}+7^{3}q^{7}+\cdots\)
448.8.a.u	$448$	$8$	448.a	1.a	$1$	$3$	$3$	$139.948$	3.3.3109313.1	None	✓				56.8.a.e	$1$	$1$	\(0\)	\(-28\)	\(-138\)	\(1029\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-9+\beta _{1})q^{3}+(-46+\beta _{1}+\beta _{2})q^{5}+\cdots\)
448.8.a.v	$448$	$8$	448.a	1.a	$1$	$3$	$3$	$139.948$	3.3.294792.1	None	✓				56.8.a.d	$1$	$0$	\(0\)	\(-12\)	\(598\)	\(-1029\)		$+$	$+$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}+(199+\beta _{1}+\beta _{2})q^{5}+\cdots\)
448.8.a.w	$448$	$8$	448.a	1.a	$1$	$3$	$3$	$139.948$	3.3.294792.1	None	✓				56.8.a.d	$1$	$0$	\(0\)	\(12\)	\(598\)	\(1029\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{3}+(199+\beta _{1}+\beta _{2})q^{5}+\cdots\)
448.8.a.x	$448$	$8$	448.a	1.a	$1$	$3$	$3$	$139.948$	3.3.3109313.1	None	✓				56.8.a.e	$1$	$1$	\(0\)	\(28\)	\(-138\)	\(-1029\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(9-\beta _{1})q^{3}+(-46+\beta _{1}+\beta _{2})q^{5}+\cdots\)
448.8.a.y	$448$	$8$	448.a	1.a	$1$	$4$	$4$	$139.948$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				224.8.a.a	$1$	$1$	\(0\)	\(-70\)	\(70\)	\(1372\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-18+\beta _{1})q^{3}+(19-3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
448.8.a.z	$448$	$8$	448.a	1.a	$1$	$4$	$4$	$139.948$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				224.8.a.a	$1$	$0$	\(0\)	\(70\)	\(70\)	\(-1372\)		$+$	$+$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(18-\beta _{1})q^{3}+(19-3\beta _{1}-\beta _{2})q^{5}+\cdots\)
448.8.a.ba	$448$	$8$	448.a	1.a	$1$	$5$	$5$	$139.948$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				224.8.a.c	$1$	$1$	\(0\)	\(-54\)	\(-84\)	\(-1715\)		$-$	$+$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{1})q^{3}+(-17-2\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
448.8.a.bb	$448$	$8$	448.a	1.a	$1$	$5$	$5$	$139.948$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				224.8.a.c	$1$	$0$	\(0\)	\(54\)	\(-84\)	\(1715\)		$-$	$-$	$+$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(11+\beta _{1})q^{3}+(-17-2\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
448.8.a.bc	$448$	$8$	448.a	1.a	$1$	$6$	$6$	$139.948$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		224.8.a.e	$1$	$0$	\(0\)	\(-16\)	\(-180\)	\(-2058\)		$+$	$+$	$+$	$2^{18}\cdot 23$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+(-30-\beta _{2})q^{5}-7^{3}q^{7}+\cdots\)
448.8.a.bd	$448$	$8$	448.a	1.a	$1$	$6$	$6$	$139.948$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		224.8.a.f	$1$	$1$	\(0\)	\(0\)	\(-84\)	\(-2058\)		$-$	$+$	$-$	$2^{18}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-14+2\beta _{1}+\beta _{2})q^{5}-7^{3}q^{7}+\cdots\)
448.8.a.be	$448$	$8$	448.a	1.a	$1$	$6$	$6$	$139.948$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		224.8.a.f	$1$	$0$	\(0\)	\(0\)	\(-84\)	\(2058\)		$-$	$-$	$+$	$2^{18}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-14+2\beta _{1}+\beta _{2})q^{5}+7^{3}q^{7}+\cdots\)
448.8.a.bf	$448$	$8$	448.a	1.a	$1$	$6$	$6$	$139.948$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		224.8.a.e	$1$	$1$	\(0\)	\(16\)	\(-180\)	\(2058\)		$+$	$-$	$-$	$2^{18}\cdot 23$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{3}+(-30-\beta _{2})q^{5}+7^{3}q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(448))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(448)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 7}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)