Space of modular forms of level 576, weight 8, and trivial character

• Label: 576.8.a
• Level: $576$
• Weight: $8$
• Character orbit: 576.a
• Rep. character: $\chi_{576}(1,\cdot)$
• Character field: $\Q$
• Dimension: $69$
• Newform subspaces: $46$
• Sturm bound: $768$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	576.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 46 \)
Sturm bound:			\(768\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(576))\).

	Total	New	Old
Modular forms	696	71	625
Cusp forms	648	69	579
Eisenstein series	48	2	46

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	Dim
\(+\)	\(+\)	$+$	\(15\)
\(+\)	\(-\)	$-$	\(20\)
\(-\)	\(+\)	$-$	\(13\)
\(-\)	\(-\)	$+$	\(21\)
Plus space		\(+\)	\(36\)
Minus space		\(-\)	\(33\)

Trace form

\( 69 q - 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(576))\) 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
576.8.a.a	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(-556\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-556q^{5}-8898q^{13}+5816q^{17}+\cdots\)
576.8.a.b	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-530\)	\(-120\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-530q^{5}-120q^{7}+7196q^{11}+9626q^{13}+\cdots\)
576.8.a.c	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-530\)	\(120\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-530q^{5}+120q^{7}-7196q^{11}+9626q^{13}+\cdots\)
576.8.a.d	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-378\)	\(-832\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-378q^{5}-832q^{7}-2484q^{11}-14870q^{13}+\cdots\)
576.8.a.e	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-378\)	\(832\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-378q^{5}+832q^{7}+2484q^{11}-14870q^{13}+\cdots\)
576.8.a.f	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-210\)	\(-1016\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-210q^{5}-1016q^{7}-1092q^{11}+\cdots\)
576.8.a.g	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-210\)	\(1016\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-210q^{5}+1016q^{7}+1092q^{11}+\cdots\)
576.8.a.h	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-114\)	\(-1576\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-114q^{5}-1576q^{7}+7332q^{11}+\cdots\)
576.8.a.i	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-114\)	\(1576\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-114q^{5}+1576q^{7}-7332q^{11}+\cdots\)
576.8.a.j	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-82\)	\(-456\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-82q^{5}-456q^{7}-2524q^{11}+10778q^{13}+\cdots\)
576.8.a.k	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-82\)	\(456\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-82q^{5}+456q^{7}+2524q^{11}+10778q^{13}+\cdots\)
576.8.a.l	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-70\)	\(-92\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-70q^{5}-92q^{7}-3124q^{11}-1174q^{13}+\cdots\)
576.8.a.m	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-70\)	\(92\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-70q^{5}+92q^{7}+3124q^{11}-1174q^{13}+\cdots\)
576.8.a.n	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-58\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-58q^{5}+8898q^{13}+40094q^{17}+\cdots\)
576.8.a.o	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-26\)	\(-1056\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-26q^{5}-1056q^{7}-6412q^{11}-5206q^{13}+\cdots\)
576.8.a.p	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-26\)	\(1056\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-26q^{5}+1056q^{7}+6412q^{11}-5206q^{13}+\cdots\)
576.8.a.q	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-508\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-508q^{7}+14614q^{13}+57448q^{19}+\cdots\)
576.8.a.r	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(508\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+508q^{7}+14614q^{13}-57448q^{19}+\cdots\)
576.8.a.s	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(110\)	\(-504\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+110q^{5}-504q^{7}-3812q^{11}-9574q^{13}+\cdots\)
576.8.a.t	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(110\)	\(504\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+110q^{5}+504q^{7}+3812q^{11}-9574q^{13}+\cdots\)
576.8.a.u	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(270\)	\(-1112\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+270q^{5}-1112q^{7}+5724q^{11}+\cdots\)
576.8.a.v	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(270\)	\(1112\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+270q^{5}+1112q^{7}-5724q^{11}+\cdots\)
576.8.a.w	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(390\)	\(-64\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+390q^{5}-2^{6}q^{7}-948q^{11}+5098q^{13}+\cdots\)
576.8.a.x	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(390\)	\(64\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+390q^{5}+2^{6}q^{7}+948q^{11}+5098q^{13}+\cdots\)
576.8.a.y	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(430\)	\(-1224\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+430q^{5}-1224q^{7}-3164q^{11}+\cdots\)
576.8.a.z	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(430\)	\(1224\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+430q^{5}+1224q^{7}+3164q^{11}+\cdots\)
576.8.a.ba	$576$	$8$	576.a	1.a	$1$	$1$	$1$	$179.934$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(556\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+556q^{5}-8898q^{13}-5816q^{17}+\cdots\)
576.8.a.bb	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{435}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-280\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-140q^{5}+\beta q^{7}+4\beta q^{11}+2238q^{13}+\cdots\)
576.8.a.bc	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-224\)	\(-840\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-112+\beta )q^{5}+(-420+4\beta )q^{7}+\cdots\)
576.8.a.bd	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-224\)	\(840\)		$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-112+\beta )q^{5}+(420-4\beta )q^{7}+(320+\cdots)q^{11}+\cdots\)
576.8.a.be	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-180\)	\(-1248\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-90+8\beta )q^{5}+(-624+14\beta )q^{7}+\cdots\)
576.8.a.bf	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-180\)	\(1248\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-90+8\beta )q^{5}+(624-14\beta )q^{7}+\cdots\)
576.8.a.bg	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-28\)	\(-936\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-14+3\beta )q^{5}+(-468-7\beta )q^{7}+\cdots\)
576.8.a.bh	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-28\)	\(936\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-14+3\beta )q^{5}+(468+7\beta )q^{7}+(-1692+\cdots)q^{11}+\cdots\)
576.8.a.bi	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{10}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-520\)		$-$	$+$	$-$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-260q^{7}-20\beta q^{11}-6890q^{13}+\cdots\)
576.8.a.bj	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{10}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(520\)		$+$	$+$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+260q^{7}+20\beta q^{11}-6890q^{13}+\cdots\)
576.8.a.bk	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{15}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(140\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+70q^{5}+2\beta q^{7}-13\beta q^{11}-13758q^{13}+\cdots\)
576.8.a.bl	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{235}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(180\)	\(-1032\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(90+\beta )q^{5}+(-516+5\beta )q^{7}+(-1420+\cdots)q^{11}+\cdots\)
576.8.a.bm	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{235}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(180\)	\(1032\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(90+\beta )q^{5}+(516-5\beta )q^{7}+(1420+\cdots)q^{11}+\cdots\)
576.8.a.bn	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(196\)	\(-504\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(98+\beta )q^{5}+(-252-3\beta )q^{7}+(-828+\cdots)q^{11}+\cdots\)
576.8.a.bo	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(196\)	\(504\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(98+\beta )q^{5}+(252+3\beta )q^{7}+(828+\cdots)q^{11}+\cdots\)
576.8.a.bp	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(224\)	\(-840\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(112+\beta )q^{5}+(-420-4\beta )q^{7}+(320+\cdots)q^{11}+\cdots\)
576.8.a.bq	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(224\)	\(840\)		$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(112+\beta )q^{5}+(420+4\beta )q^{7}+(-320+\cdots)q^{11}+\cdots\)
576.8.a.br	$576$	$8$	576.a	1.a	$1$	$2$	$2$	$179.934$	\(\Q(\sqrt{435}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(280\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+140q^{5}+\beta q^{7}-4\beta q^{11}+2238q^{13}+\cdots\)
576.8.a.bs	$576$	$8$	576.a	1.a	$1$	$4$	$4$	$179.934$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+\beta _{3}q^{7}+(-160-\beta _{1})q^{11}+\cdots\)
576.8.a.bt	$576$	$8$	576.a	1.a	$1$	$4$	$4$	$179.934$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}-\beta _{3}q^{7}+(160+\beta _{1})q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(576))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(576)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 18}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 14}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 15}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 7}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)