Space of modular forms of level 432, weight 8, and trivial character

• Label: 432.8.a
• Level: $432$
• Weight: $8$
• Character orbit: 432.a
• Rep. character: $\chi_{432}(1,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $26$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(432))\).

	Total	New	Old
Modular forms	522	56	466
Cusp forms	486	56	430
Eisenstein series	36	0	36

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

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(132\)	\(15\)	\(117\)		\(123\)	\(15\)	\(108\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)		\(129\)	\(13\)	\(116\)		\(120\)	\(13\)	\(107\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)		\(129\)	\(14\)	\(115\)		\(120\)	\(14\)	\(106\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)		\(132\)	\(14\)	\(118\)		\(123\)	\(14\)	\(109\)		\(9\)	\(0\)	\(9\)
Plus space		\(+\)		\(264\)	\(29\)	\(235\)		\(246\)	\(29\)	\(217\)		\(18\)	\(0\)	\(18\)
Minus space		\(-\)		\(258\)	\(27\)	\(231\)		\(240\)	\(27\)	\(213\)		\(18\)	\(0\)	\(18\)

Trace form

56 * q + 1006 * q^7 + 20714 * q^19 + 864756 * q^25 + 296084 * q^31 + 331088 * q^37 + 568632 * q^43 + 6335672 * q^49 + 355100 * q^55 + 1850040 * q^61 - 8142502 * q^67 - 3091200 * q^73 - 17719818 * q^79 + 7289936 * q^85 - 29608810 * q^91 + 6577864 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(432))\) 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	3
432.8.a.a	$432$	$8$	432.a	1.a	$1$	$1$	$1$	$134.950$	\(\Q\)	None	✓				54.8.a.c	$1$	$0$	\(0\)	\(0\)	\(-312\)	\(-323\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-312q^{5}-323q^{7}+3720q^{11}-14179q^{13}+\cdots\)
432.8.a.b	$432$	$8$	432.a	1.a	$1$	$1$	$1$	$134.950$	\(\Q\)	None	✓				54.8.a.a	$1$	$1$	\(0\)	\(0\)	\(-120\)	\(-377\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-120q^{5}-377q^{7}+600q^{11}+5369q^{13}+\cdots\)
432.8.a.c	$432$	$8$	432.a	1.a	$1$	$1$	$1$	$134.950$	\(\Q\)	None	✓				54.8.a.b	$1$	$0$	\(0\)	\(0\)	\(-105\)	\(937\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-105q^{5}+937q^{7}+5943q^{11}+68q^{13}+\cdots\)
432.8.a.d	$432$	$8$	432.a	1.a	$1$	$1$	$1$	$134.950$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				27.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-1763\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-1763q^{7}+12605q^{13}-14357q^{19}+\cdots\)
432.8.a.e	$432$	$8$	432.a	1.a	$1$	$1$	$1$	$134.950$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				108.8.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(1255\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+1255q^{7}+2009q^{13}-43091q^{19}+\cdots\)
432.8.a.f	$432$	$8$	432.a	1.a	$1$	$1$	$1$	$134.950$	\(\Q\)	None	✓				54.8.a.b	$1$	$1$	\(0\)	\(0\)	\(105\)	\(937\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+105q^{5}+937q^{7}-5943q^{11}+68q^{13}+\cdots\)
432.8.a.g	$432$	$8$	432.a	1.a	$1$	$1$	$1$	$134.950$	\(\Q\)	None	✓				54.8.a.a	$1$	$1$	\(0\)	\(0\)	\(120\)	\(-377\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+120q^{5}-377q^{7}-600q^{11}+5369q^{13}+\cdots\)
432.8.a.h	$432$	$8$	432.a	1.a	$1$	$1$	$1$	$134.950$	\(\Q\)	None	✓				54.8.a.c	$1$	$0$	\(0\)	\(0\)	\(312\)	\(-323\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+312q^{5}-323q^{7}-3720q^{11}-14179q^{13}+\cdots\)
432.8.a.i	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{1289}) \)	None	✓				108.8.a.b	$1$	$1$	\(0\)	\(0\)	\(-324\)	\(980\)		$-$	$+$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-162-\beta )q^{5}+(490+3\beta )q^{7}+(405+\cdots)q^{11}+\cdots\)
432.8.a.j	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{65}) \)	None	✓				27.8.a.b	$1$	$1$	\(0\)	\(0\)	\(-180\)	\(-700\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-90-\beta )q^{5}+(-350-45\beta )q^{7}+\cdots\)
432.8.a.k	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{329}) \)	None	✓				54.8.a.g	$1$	$1$	\(0\)	\(0\)	\(-48\)	\(-880\)		$-$	$+$	$-$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-24-\beta )q^{5}+(-440-\beta )q^{7}+(-3615+\cdots)q^{11}+\cdots\)
432.8.a.l	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{195}) \)	None	✓				108.8.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-3106\)		$-$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-1553q^{7}-13\beta q^{11}-799q^{13}+\cdots\)
432.8.a.m	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{30}) \)	None	✓				108.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(26\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+13q^{7}+41\beta q^{11}-691q^{13}+\cdots\)
432.8.a.n	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{3}) \)	None	✓				27.8.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(1118\)		$-$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+17\beta q^{5}+559q^{7}+227\beta q^{11}-8671q^{13}+\cdots\)
432.8.a.o	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{42}) \)	None	✓				27.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2522\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+5\beta q^{5}+1261q^{7}-19\beta q^{11}+9581q^{13}+\cdots\)
432.8.a.p	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{329}) \)	None	✓				54.8.a.g	$1$	$0$	\(0\)	\(0\)	\(48\)	\(-880\)		$-$	$-$	$+$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(24+\beta )q^{5}+(-440-\beta )q^{7}+(3615+\cdots)q^{11}+\cdots\)
432.8.a.q	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{65}) \)	None	✓				27.8.a.b	$1$	$0$	\(0\)	\(0\)	\(180\)	\(-700\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(90-\beta )q^{5}+(-350+45\beta )q^{7}+(-5445+\cdots)q^{11}+\cdots\)
432.8.a.r	$432$	$8$	432.a	1.a	$1$	$2$	$2$	$134.950$	\(\Q(\sqrt{1289}) \)	None	✓				108.8.a.b	$1$	$0$	\(0\)	\(0\)	\(324\)	\(980\)		$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(162-\beta )q^{5}+(490-3\beta )q^{7}+(-405+\cdots)q^{11}+\cdots\)
432.8.a.s	$432$	$8$	432.a	1.a	$1$	$3$	$3$	$134.950$	3.3.2239860.1	None	✓				216.8.a.a	$1$	$1$	\(0\)	\(0\)	\(-36\)	\(-321\)		$+$	$-$	$-$	$2^{5}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-12+\beta _{1})q^{5}+(-107-\beta _{2})q^{7}+\cdots\)
432.8.a.t	$432$	$8$	432.a	1.a	$1$	$3$	$3$	$134.950$	3.3.1573064.1	None	✓				216.8.a.b	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(489\)		$+$	$-$	$-$	$2^{5}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{5}+(163-\beta _{2})q^{7}+(-185+\cdots)q^{11}+\cdots\)
432.8.a.u	$432$	$8$	432.a	1.a	$1$	$3$	$3$	$134.950$	3.3.1573064.1	None	✓				216.8.a.b	$1$	$0$	\(0\)	\(0\)	\(9\)	\(489\)		$+$	$+$	$+$	$2^{5}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{5}+(163-\beta _{2})q^{7}+(185+\cdots)q^{11}+\cdots\)
432.8.a.v	$432$	$8$	432.a	1.a	$1$	$3$	$3$	$134.950$	3.3.2239860.1	None	✓				216.8.a.a	$1$	$1$	\(0\)	\(0\)	\(36\)	\(-321\)		$+$	$-$	$-$	$2^{5}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(12-\beta _{1})q^{5}+(-107-\beta _{2})q^{7}+(460+\cdots)q^{11}+\cdots\)
432.8.a.w	$432$	$8$	432.a	1.a	$1$	$4$	$4$	$134.950$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		216.8.a.e	$1$	$1$	\(0\)	\(0\)	\(-188\)	\(180\)		$+$	$-$	$-$	$2^{9}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+(-47-\beta _{2})q^{5}+(45+\beta _{1})q^{7}+(365+\cdots)q^{11}+\cdots\)
432.8.a.x	$432$	$8$	432.a	1.a	$1$	$4$	$4$	$134.950$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				216.8.a.f	$1$	$0$	\(0\)	\(0\)	\(-104\)	\(492\)		$+$	$+$	$+$	$2^{10}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+(-26-\beta _{1})q^{5}+(123-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
432.8.a.y	$432$	$8$	432.a	1.a	$1$	$4$	$4$	$134.950$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				216.8.a.f	$1$	$0$	\(0\)	\(0\)	\(104\)	\(492\)		$+$	$+$	$+$	$2^{10}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+(26+\beta _{1})q^{5}+(123-\beta _{1}-\beta _{2})q^{7}+\cdots\)
432.8.a.z	$432$	$8$	432.a	1.a	$1$	$4$	$4$	$134.950$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				216.8.a.e	$1$	$0$	\(0\)	\(0\)	\(188\)	\(180\)		$+$	$+$	$+$	$2^{9}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+(47+\beta _{2})q^{5}+(45+\beta _{1})q^{7}+(-365+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(432))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(432)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 15}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 2}\)