Space of modular forms of level 432, weight 4, and trivial character

• Label: 432.4.a
• Level: $432$
• Weight: $4$
• Character orbit: 432.a
• Rep. character: $\chi_{432}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $19$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(432))\).

	Total	New	Old
Modular forms	234	24	210
Cusp forms	198	24	174
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
\(+\)	\(+\)	\(+\)		\(60\)	\(7\)	\(53\)		\(51\)	\(7\)	\(44\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)		\(57\)	\(5\)	\(52\)		\(48\)	\(5\)	\(43\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)		\(57\)	\(6\)	\(51\)		\(48\)	\(6\)	\(42\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)		\(60\)	\(6\)	\(54\)		\(51\)	\(6\)	\(45\)		\(9\)	\(0\)	\(9\)
Plus space		\(+\)		\(120\)	\(13\)	\(107\)		\(102\)	\(13\)	\(89\)		\(18\)	\(0\)	\(18\)
Minus space		\(-\)		\(114\)	\(11\)	\(103\)		\(96\)	\(11\)	\(85\)		\(18\)	\(0\)	\(18\)

Trace form

\( 24 q - 18 q^{7} - 102 q^{19} + 516 q^{25} - 108 q^{31} + 528 q^{37} + 120 q^{43} + 1560 q^{49} + 540 q^{55} - 168 q^{61} + 1290 q^{67} + 384 q^{73} + 2070 q^{79} - 2544 q^{85} + 2118 q^{91} - 24 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(432))\) 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
432.4.a.a	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				27.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-15\)	\(25\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-15q^{5}+5^{2}q^{7}-15q^{11}+20q^{13}+\cdots\)
432.4.a.b	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				54.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-12q^{5}+7q^{7}+60q^{11}-79q^{13}+\cdots\)
432.4.a.c	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				108.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-9\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{5}+q^{7}+63q^{11}-28q^{13}-72q^{17}+\cdots\)
432.4.a.d	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				216.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-3q^{7}+28q^{11}-11q^{13}+\cdots\)
432.4.a.e	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				54.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-29\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-29q^{7}-57q^{11}+20q^{13}+\cdots\)
432.4.a.f	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				216.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+9q^{7}-17q^{11}-44q^{13}-56q^{17}+\cdots\)
432.4.a.g	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				108.4.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-17\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-17q^{7}+89q^{13}-107q^{19}-5^{3}q^{25}+\cdots\)
432.4.a.h	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				108.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(37\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+37q^{7}-19q^{13}+163q^{19}-5^{3}q^{25}+\cdots\)
432.4.a.i	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				216.4.a.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(9\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+9q^{7}+17q^{11}-44q^{13}+56q^{17}+\cdots\)
432.4.a.j	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				54.4.a.b	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-29\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-29q^{7}+57q^{11}+20q^{13}+\cdots\)
432.4.a.k	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				216.4.a.a	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-3q^{7}-28q^{11}-11q^{13}+\cdots\)
432.4.a.l	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				108.4.a.a	$1$	$1$	\(0\)	\(0\)	\(9\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{5}+q^{7}-63q^{11}-28q^{13}+72q^{17}+\cdots\)
432.4.a.m	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				54.4.a.a	$1$	$1$	\(0\)	\(0\)	\(12\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}+7q^{7}-60q^{11}-79q^{13}+\cdots\)
432.4.a.n	$432$	$4$	432.a	1.a	$1$	$1$	$1$	$25.489$	\(\Q\)	None	✓				27.4.a.a	$1$	$0$	\(0\)	\(0\)	\(15\)	\(25\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+15q^{5}+5^{2}q^{7}+15q^{11}+20q^{13}+\cdots\)
432.4.a.o	$432$	$4$	432.a	1.a	$1$	$2$	$2$	$25.489$	\(\Q(\sqrt{33}) \)	None	✓				216.4.a.e	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(-24\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{5}+(-12-\beta )q^{7}+q^{11}+\cdots\)
432.4.a.p	$432$	$4$	432.a	1.a	$1$	$2$	$2$	$25.489$	\(\Q(\sqrt{5}) \)	None	✓				216.4.a.f	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(6\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{5}+(3-2\beta )q^{7}+(26-3\beta )q^{11}+\cdots\)
432.4.a.q	$432$	$4$	432.a	1.a	$1$	$2$	$2$	$25.489$	\(\Q(\sqrt{2}) \)	None	✓		✓		27.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-22\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-11q^{7}+\beta q^{11}+29q^{13}+\cdots\)
432.4.a.r	$432$	$4$	432.a	1.a	$1$	$2$	$2$	$25.489$	\(\Q(\sqrt{5}) \)	None	✓				216.4.a.f	$1$	$0$	\(0\)	\(0\)	\(4\)	\(6\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(3-2\beta )q^{7}+(-26+3\beta )q^{11}+\cdots\)
432.4.a.s	$432$	$4$	432.a	1.a	$1$	$2$	$2$	$25.489$	\(\Q(\sqrt{33}) \)	None	✓		✓		216.4.a.e	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-24\)		$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(-12-\beta )q^{7}-q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(432))\) into lower level spaces

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