Space of modular forms of level 1584, weight 4, and trivial character

• Label: 1584.4.a
• Level: $1584$
• Weight: $4$
• Character orbit: 1584.a
• Rep. character: $\chi_{1584}(1,\cdot)$
• Character field: $\Q$
• Dimension: $75$
• Newform subspaces: $44$
• Sturm bound: $1152$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1584.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 44 \)
Sturm bound:			\(1152\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1584))\).

	Total	New	Old
Modular forms	888	75	813
Cusp forms	840	75	765
Eisenstein series	48	0	48

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

\(2\)	\(3\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(7\)
\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(-\)	\(+\)	$-$	\(10\)
\(+\)	\(-\)	\(-\)	$+$	\(13\)
\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(-\)	$+$	\(8\)
\(-\)	\(-\)	\(+\)	$+$	\(11\)
\(-\)	\(-\)	\(-\)	$-$	\(11\)
Plus space			\(+\)	\(39\)
Minus space			\(-\)	\(36\)

Trace form

\( 75 q + 2 q^{5} - 36 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1584))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	11
1584.4.a.a	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-22\)	\(20\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-22q^{5}+20q^{7}+11q^{11}+22q^{13}+\cdots\)
1584.4.a.b	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-14\)	\(8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}+8q^{7}-11q^{11}-50q^{13}+\cdots\)
1584.4.a.c	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(0\)	\(-12\)	\(-26\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-12q^{5}-26q^{7}+11q^{11}-34q^{13}+\cdots\)
1584.4.a.d	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-12\)	\(-22\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-12q^{5}-22q^{7}+11q^{11}-48q^{13}+\cdots\)
1584.4.a.e	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-16\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{5}-2^{4}q^{7}+11q^{11}+10q^{13}+\cdots\)
1584.4.a.f	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{5}-8q^{7}-11q^{11}+18q^{13}+\cdots\)
1584.4.a.g	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{5}-2q^{7}-11q^{11}+38q^{17}+\cdots\)
1584.4.a.h	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(22\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}+22q^{7}+11q^{11}-54q^{13}+\cdots\)
1584.4.a.i	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{7}+11q^{11}+80q^{13}-30q^{17}+\cdots\)
1584.4.a.j	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-11q^{11}-88q^{13}+66q^{17}+\cdots\)
1584.4.a.k	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(3\)	\(10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+10q^{7}+11q^{11}-2^{4}q^{13}+\cdots\)
1584.4.a.l	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(4\)	\(26\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+26q^{7}+11q^{11}-2^{5}q^{13}+\cdots\)
1584.4.a.m	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(6\)	\(8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}+8q^{7}-11q^{11}-30q^{13}+\cdots\)
1584.4.a.n	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(14\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}+14q^{7}+11q^{11}+6q^{13}+\cdots\)
1584.4.a.o	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(7\)	\(6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{5}+6q^{7}-11q^{11}-40q^{13}+\cdots\)
1584.4.a.p	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(7\)	\(26\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{5}+26q^{7}-11q^{11}+52q^{13}+\cdots\)
1584.4.a.q	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(8\)	\(22\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{5}+22q^{7}-11q^{11}-54q^{13}+\cdots\)
1584.4.a.r	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(12\)	\(-26\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}-26q^{7}-11q^{11}-34q^{13}+\cdots\)
1584.4.a.s	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(12\)	\(-14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}-14q^{7}+11q^{11}+56q^{13}+\cdots\)
1584.4.a.t	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(14\)	\(32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14q^{5}+2^{5}q^{7}-11q^{11}-38q^{13}+\cdots\)
1584.4.a.u	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(18\)	\(28\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{5}+28q^{7}+11q^{11}-18q^{13}+\cdots\)
1584.4.a.v	$1584$	$4$	1584.a	1.a	$1$	$1$	$1$	$93.459$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(19\)	\(-14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+19q^{5}-14q^{7}+11q^{11}-72q^{13}+\cdots\)
1584.4.a.w	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-20\)	\(16\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-10+\beta )q^{5}+(8-5\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.x	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(-2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-8-2\beta )q^{5}+(-1-\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.y	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{31}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(24\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-6+\beta )q^{5}+(12-\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.z	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-11\)	\(-10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{5}+(-2-6\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.ba	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{5}+(1+3\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bb	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{70}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-36\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{5}+(-18+\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bc	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-20\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+2\beta )q^{5}+(-10-\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bd	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{185}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(6\)	\(-22\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{5}+(-11+\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.be	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{137}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(6\)	\(-16\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{5}+(-8-2\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bf	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(-10\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{5}+(-5-\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bg	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(56\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3+4\beta )q^{5}+(28-\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bh	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{70}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-36\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(-18-\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bi	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{31}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(12\)	\(24\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(6+\beta )q^{5}+(12+\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bj	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(14\)	\(-24\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{5}+(-12-2\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bk	$1584$	$4$	1584.a	1.a	$1$	$2$	$2$	$93.459$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(20\)	\(16\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(10+\beta )q^{5}+(8+5\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bl	$1584$	$4$	1584.a	1.a	$1$	$3$	$3$	$93.459$	3.3.11109.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(-24\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{5}+(-8+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1584.4.a.bm	$1584$	$4$	1584.a	1.a	$1$	$3$	$3$	$93.459$	3.3.4364.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(-10\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(-3+\beta _{2})q^{7}-11q^{11}+\cdots\)
1584.4.a.bn	$1584$	$4$	1584.a	1.a	$1$	$3$	$3$	$93.459$	3.3.123209.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-28\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-10-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1584.4.a.bo	$1584$	$4$	1584.a	1.a	$1$	$3$	$3$	$93.459$	3.3.142161.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(12\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(4-\beta _{2})q^{7}-11q^{11}+\cdots\)
1584.4.a.bp	$1584$	$4$	1584.a	1.a	$1$	$3$	$3$	$93.459$	3.3.4364.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-10\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(-3+\beta _{2})q^{7}+11q^{11}+\cdots\)
1584.4.a.bq	$1584$	$4$	1584.a	1.a	$1$	$4$	$4$	$93.459$	4.4.8611212.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-8\)		$+$	$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{5}+(-2-\beta _{2})q^{7}+11q^{11}+\cdots\)
1584.4.a.br	$1584$	$4$	1584.a	1.a	$1$	$4$	$4$	$93.459$	4.4.8611212.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-8\)		$+$	$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{5}+(-2-\beta _{2})q^{7}-11q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1584))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1584)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(528))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(792))\)\(^{\oplus 2}\)