Space of modular forms of level 720, weight 6, and trivial character

• Label: 720.6.a
• Level: $720$
• Weight: $6$
• Character orbit: 720.a
• Rep. character: $\chi_{720}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $36$
• Sturm bound: $864$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(864\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(720))\).

	Total	New	Old
Modular forms	744	50	694
Cusp forms	696	50	646
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.

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(-\)	$+$	\(7\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(-\)	$-$	\(8\)
Plus space			\(+\)	\(24\)
Minus space			\(-\)	\(26\)

Trace form

\( 50 q + 26 q^{7} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(720))\) 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	5
720.6.a.a	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-25\)	\(-192\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}-192q^{7}-148q^{11}+286q^{13}+\cdots\)
720.6.a.b	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-25\)	\(-128\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}-2^{7}q^{7}-308q^{11}-1058q^{13}+\cdots\)
720.6.a.c	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-25\)	\(-98\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}-98q^{7}-354q^{11}+404q^{13}+\cdots\)
720.6.a.d	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-25\)	\(-56\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}-56q^{7}+156q^{11}+350q^{13}+\cdots\)
720.6.a.e	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-25\)	\(-32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}-2^{5}q^{7}+12q^{11}-154q^{13}+\cdots\)
720.6.a.f	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-25\)	\(16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+2^{4}q^{7}-564q^{11}-370q^{13}+\cdots\)
720.6.a.g	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-25\)	\(80\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+80q^{7}+684q^{11}-978q^{13}+\cdots\)
720.6.a.h	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-25\)	\(108\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+108q^{7}-604q^{11}-306q^{13}+\cdots\)
720.6.a.i	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-25\)	\(160\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+160q^{7}-596q^{11}-122q^{13}+\cdots\)
720.6.a.j	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-25\)	\(172\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+172q^{7}+132q^{11}-946q^{13}+\cdots\)
720.6.a.k	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(25\)	\(-242\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}-242q^{7}+656q^{11}-206q^{13}+\cdots\)
720.6.a.l	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(25\)	\(-218\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}-218q^{7}-480q^{11}-622q^{13}+\cdots\)
720.6.a.m	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(25\)	\(-164\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}-164q^{7}+720q^{11}+698q^{13}+\cdots\)
720.6.a.n	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(25\)	\(-108\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}-108q^{7}-8q^{11}+162q^{13}+\cdots\)
720.6.a.o	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(25\)	\(-98\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}-98q^{7}+354q^{11}+404q^{13}+\cdots\)
720.6.a.p	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(25\)	\(-44\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}-44q^{7}+6^{3}q^{11}+770q^{13}+\cdots\)
720.6.a.q	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(25\)	\(-12\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}-12q^{7}+112q^{11}-974q^{13}+\cdots\)
720.6.a.r	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(25\)	\(22\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+22q^{7}-768q^{11}-46q^{13}+\cdots\)
720.6.a.s	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(25\)	\(28\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+28q^{7}-208q^{11}-422q^{13}+\cdots\)
720.6.a.t	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(25\)	\(62\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+62q^{7}-12^{2}q^{11}-654q^{13}+\cdots\)
720.6.a.u	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(25\)	\(100\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+10^{2}q^{7}-136q^{11}+82q^{13}+\cdots\)
720.6.a.v	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(25\)	\(118\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+118q^{7}+192q^{11}+1106q^{13}+\cdots\)
720.6.a.w	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(25\)	\(132\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+132q^{7}+472q^{11}-686q^{13}+\cdots\)
720.6.a.x	$720$	$6$	720.a	1.a	$1$	$1$	$1$	$115.476$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(25\)	\(244\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+244q^{7}-12^{2}q^{11}+50q^{13}+\cdots\)
720.6.a.y	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{145}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-50\)	\(-80\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(-40-5\beta )q^{7}+(20^{2}-5\beta )q^{11}+\cdots\)
720.6.a.z	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{129}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-50\)	\(-52\)		$+$	$-$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(-26-\beta )q^{7}+(280-2\beta )q^{11}+\cdots\)
720.6.a.ba	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{2161}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-50\)	\(8\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(4+\beta )q^{7}+(244+2\beta )q^{11}+\cdots\)
720.6.a.bb	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{3289}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-50\)	\(80\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(40-\beta )q^{7}+(-140+\beta )q^{11}+\cdots\)
720.6.a.bc	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-50\)	\(80\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(40-\beta )q^{7}+(-120-5\beta )q^{11}+\cdots\)
720.6.a.bd	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{409}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-50\)	\(112\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(56-\beta )q^{7}+(124+2\beta )q^{11}+\cdots\)
720.6.a.be	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{145}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(50\)	\(-80\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(-40-5\beta )q^{7}+(-20^{2}+\cdots)q^{11}+\cdots\)
720.6.a.bf	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{1489}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(50\)	\(-16\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(-8-\beta )q^{7}+(2^{5}-4\beta )q^{11}+\cdots\)
720.6.a.bg	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(50\)	\(80\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(40-\beta )q^{7}+(120+5\beta )q^{11}+\cdots\)
720.6.a.bh	$720$	$6$	720.a	1.a	$1$	$2$	$2$	$115.476$	\(\Q(\sqrt{3289}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(50\)	\(80\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(40-\beta )q^{7}+(140-\beta )q^{11}+\cdots\)
720.6.a.bi	$720$	$6$	720.a	1.a	$1$	$3$	$3$	$115.476$	3.3.2521041.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-75\)	\(-18\)		$+$	$+$	$+$	$+$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(-6-\beta _{1})q^{7}+(186+\beta _{1}+\cdots)q^{11}+\cdots\)
720.6.a.bj	$720$	$6$	720.a	1.a	$1$	$3$	$3$	$115.476$	3.3.2521041.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(75\)	\(-18\)		$+$	$+$	$-$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(-6-\beta _{1})q^{7}+(-186-\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(720))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(720)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 20}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 18}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 15}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)