Space of modular forms of level 576, weight 6, and trivial character

• Label: 576.6.a
• Level: $576$
• Weight: $6$
• Character orbit: 576.a
• Rep. character: $\chi_{576}(1,\cdot)$
• Character field: $\Q$
• Dimension: $49$
• Newform subspaces: $42$
• Sturm bound: $576$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	576.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 42 \)
Sturm bound:			\(576\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(576))\).

	Total	New	Old
Modular forms	504	51	453
Cusp forms	456	49	407
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(124\)	\(9\)	\(115\)		\(112\)	\(9\)	\(103\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(128\)	\(16\)	\(112\)		\(116\)	\(15\)	\(101\)		\(12\)	\(1\)	\(11\)
\(-\)	\(+\)	\(-\)		\(128\)	\(11\)	\(117\)		\(116\)	\(11\)	\(105\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(124\)	\(15\)	\(109\)		\(112\)	\(14\)	\(98\)		\(12\)	\(1\)	\(11\)
Plus space		\(+\)		\(248\)	\(24\)	\(224\)		\(224\)	\(23\)	\(201\)		\(24\)	\(1\)	\(23\)
Minus space		\(-\)		\(256\)	\(27\)	\(229\)		\(232\)	\(26\)	\(206\)		\(24\)	\(1\)	\(23\)

Trace form

49 * q - 2 * q^5 + 234 * q^13 + 406 * q^17 + 25007 * q^25 + 4070 * q^29 - 8494 * q^37 + 9134 * q^41 + 103241 * q^49 - 63586 * q^53 + 2074 * q^61 + 10092 * q^65 + 5034 * q^73 - 161568 * q^77 + 154468 * q^85 + 41438 * q^89 - 51774 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(576))\) 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
576.6.a.a	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				18.6.a.a	$1$	$1$	\(0\)	\(0\)	\(-96\)	\(-148\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-96q^{5}-148q^{7}+384q^{11}+334q^{13}+\cdots\)
576.6.a.b	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				18.6.a.a	$1$	$0$	\(0\)	\(0\)	\(-96\)	\(148\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-96q^{5}+148q^{7}-384q^{11}+334q^{13}+\cdots\)
576.6.a.c	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				96.6.a.a	$1$	$1$	\(0\)	\(0\)	\(-86\)	\(-180\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-86q^{5}-180q^{7}+684q^{11}-222q^{13}+\cdots\)
576.6.a.d	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				96.6.a.a	$1$	$1$	\(0\)	\(0\)	\(-86\)	\(180\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-86q^{5}+180q^{7}-684q^{11}-222q^{13}+\cdots\)
576.6.a.e	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.6.a.b	$2$	$1$	\(0\)	\(0\)	\(-82\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-82q^{5}+1194q^{13}-2242q^{17}+\cdots\)
576.6.a.f	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				288.6.a.a	$2$	$0$	\(0\)	\(0\)	\(-76\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-76q^{5}-1194q^{13}-808q^{17}+2651q^{25}+\cdots\)
576.6.a.g	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				8.6.a.a	$1$	$0$	\(0\)	\(0\)	\(-74\)	\(-24\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-74q^{5}-24q^{7}+124q^{11}-478q^{13}+\cdots\)
576.6.a.h	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				8.6.a.a	$1$	$1$	\(0\)	\(0\)	\(-74\)	\(24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-74q^{5}+24q^{7}-124q^{11}-478q^{13}+\cdots\)
576.6.a.i	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				6.6.a.a	$1$	$1$	\(0\)	\(0\)	\(-66\)	\(-176\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-66q^{5}-176q^{7}+60q^{11}+658q^{13}+\cdots\)
576.6.a.j	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				6.6.a.a	$1$	$0$	\(0\)	\(0\)	\(-66\)	\(176\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-66q^{5}+176q^{7}-60q^{11}+658q^{13}+\cdots\)
576.6.a.k	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				24.6.a.a	$1$	$0$	\(0\)	\(0\)	\(-34\)	\(-240\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-34q^{5}-240q^{7}-124q^{11}-46q^{13}+\cdots\)
576.6.a.l	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				24.6.a.a	$1$	$1$	\(0\)	\(0\)	\(-34\)	\(240\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-34q^{5}+240q^{7}+124q^{11}-46q^{13}+\cdots\)
576.6.a.m	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				72.6.a.c	$1$	$0$	\(0\)	\(0\)	\(-16\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{5}-12q^{7}+448q^{11}+206q^{13}+\cdots\)
576.6.a.n	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				72.6.a.c	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{5}+12q^{7}-448q^{11}+206q^{13}+\cdots\)
576.6.a.o	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				96.6.a.b	$1$	$0$	\(0\)	\(0\)	\(-14\)	\(-100\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}-10^{2}q^{7}-220q^{11}+818q^{13}+\cdots\)
576.6.a.p	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				96.6.a.b	$1$	$0$	\(0\)	\(0\)	\(-14\)	\(100\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}+10^{2}q^{7}+220q^{11}+818q^{13}+\cdots\)
576.6.a.q	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				36.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-236\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-236q^{7}-1202q^{13}-1432q^{19}+\cdots\)
576.6.a.r	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				36.6.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(236\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+236q^{7}-1202q^{13}+1432q^{19}+\cdots\)
576.6.a.s	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				3.6.a.a	$1$	$0$	\(0\)	\(0\)	\(6\)	\(-40\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}-40q^{7}-564q^{11}-638q^{13}+\cdots\)
576.6.a.t	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				3.6.a.a	$1$	$1$	\(0\)	\(0\)	\(6\)	\(40\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}+40q^{7}+564q^{11}-638q^{13}+\cdots\)
576.6.a.u	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				32.6.a.a	$1$	$0$	\(0\)	\(0\)	\(14\)	\(-208\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{5}-208q^{7}-536q^{11}-694q^{13}+\cdots\)
576.6.a.v	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				32.6.a.a	$1$	$0$	\(0\)	\(0\)	\(14\)	\(208\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{5}+208q^{7}+536q^{11}-694q^{13}+\cdots\)
576.6.a.w	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				72.6.a.c	$1$	$0$	\(0\)	\(0\)	\(16\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{5}-12q^{7}-448q^{11}+206q^{13}+\cdots\)
576.6.a.x	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				72.6.a.c	$1$	$1$	\(0\)	\(0\)	\(16\)	\(12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{5}+12q^{7}+448q^{11}+206q^{13}+\cdots\)
576.6.a.y	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				96.6.a.c	$1$	$1$	\(0\)	\(0\)	\(26\)	\(-36\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+26q^{5}-6^{2}q^{7}-180q^{11}-318q^{13}+\cdots\)
576.6.a.z	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				96.6.a.c	$1$	$1$	\(0\)	\(0\)	\(26\)	\(36\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+26q^{5}+6^{2}q^{7}+180q^{11}-318q^{13}+\cdots\)
576.6.a.ba	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				24.6.a.c	$1$	$1$	\(0\)	\(0\)	\(38\)	\(-120\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+38q^{5}-120q^{7}-524q^{11}+962q^{13}+\cdots\)
576.6.a.bb	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				24.6.a.c	$1$	$0$	\(0\)	\(0\)	\(38\)	\(120\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+38q^{5}+120q^{7}+524q^{11}+962q^{13}+\cdots\)
576.6.a.bc	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				4.6.a.a	$1$	$0$	\(0\)	\(0\)	\(54\)	\(-88\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+54q^{5}-88q^{7}+540q^{11}+418q^{13}+\cdots\)
576.6.a.bd	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				4.6.a.a	$1$	$1$	\(0\)	\(0\)	\(54\)	\(88\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+54q^{5}+88q^{7}-540q^{11}+418q^{13}+\cdots\)
576.6.a.be	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				288.6.a.a	$2$	$0$	\(0\)	\(0\)	\(76\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+76q^{5}-1194q^{13}+808q^{17}+2651q^{25}+\cdots\)
576.6.a.bf	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				24.6.a.b	$1$	$1$	\(0\)	\(0\)	\(94\)	\(-144\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+94q^{5}-12^{2}q^{7}+380q^{11}-814q^{13}+\cdots\)
576.6.a.bg	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				24.6.a.b	$1$	$0$	\(0\)	\(0\)	\(94\)	\(144\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+94q^{5}+12^{2}q^{7}-380q^{11}-814q^{13}+\cdots\)
576.6.a.bh	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				18.6.a.a	$1$	$1$	\(0\)	\(0\)	\(96\)	\(-148\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+96q^{5}-148q^{7}-384q^{11}+334q^{13}+\cdots\)
576.6.a.bi	$576$	$6$	576.a	1.a	$1$	$1$	$1$	$92.381$	\(\Q\)	None	✓				18.6.a.a	$1$	$0$	\(0\)	\(0\)	\(96\)	\(148\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+96q^{5}+148q^{7}+384q^{11}+334q^{13}+\cdots\)
576.6.a.bj	$576$	$6$	576.a	1.a	$1$	$2$	$2$	$92.381$	\(\Q(\sqrt{15}) \)	None	✓				288.6.a.m	$2$	$0$	\(0\)	\(0\)	\(-88\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-44q^{5}+\beta q^{7}+4\beta q^{11}+726q^{13}+\cdots\)
576.6.a.bk	$576$	$6$	576.a	1.a	$1$	$2$	$2$	$92.381$	\(\Q(\sqrt{181}) \)	None	✓				288.6.a.p	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+2\beta q^{7}-2^{5}q^{11}-10q^{13}+\cdots\)
576.6.a.bl	$576$	$6$	576.a	1.a	$1$	$2$	$2$	$92.381$	\(\Q(\sqrt{181}) \)	None	✓				288.6.a.p	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-2\beta q^{7}+2^{5}q^{11}-10q^{13}+\cdots\)
576.6.a.bm	$576$	$6$	576.a	1.a	$1$	$2$	$2$	$92.381$	\(\Q(\sqrt{31}) \)	None	✓				96.6.a.g	$1$	$0$	\(0\)	\(0\)	\(36\)	\(-120\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(18+\beta )q^{5}+(-60-\beta )q^{7}+(-10^{2}+\cdots)q^{11}+\cdots\)
576.6.a.bn	$576$	$6$	576.a	1.a	$1$	$2$	$2$	$92.381$	\(\Q(\sqrt{31}) \)	None	✓				96.6.a.g	$1$	$0$	\(0\)	\(0\)	\(36\)	\(120\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(18+\beta )q^{5}+(60+\beta )q^{7}+(10^{2}-6\beta )q^{11}+\cdots\)
576.6.a.bo	$576$	$6$	576.a	1.a	$1$	$2$	$2$	$92.381$	\(\Q(\sqrt{15}) \)	None	✓				288.6.a.m	$2$	$0$	\(0\)	\(0\)	\(88\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+44q^{5}-\beta q^{7}+4\beta q^{11}+726q^{13}+\cdots\)
576.6.a.bp	$576$	$6$	576.a	1.a	$1$	$2$	$2$	$92.381$	\(\Q(\sqrt{3}) \)	None	✓		✓		32.6.a.d	$2$	$1$	\(0\)	\(0\)	\(92\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+46q^{5}+2\beta q^{7}-\beta q^{11}+42q^{13}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(576))\) into lower level spaces

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