Space of modular forms of level 240, weight 6, and trivial character

• Label: 240.6.a
• Level: $240$
• Weight: $6$
• Character orbit: 240.a
• Rep. character: $\chi_{240}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $17$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(240))\).

	Total	New	Old
Modular forms	252	20	232
Cusp forms	228	20	208
Eisenstein series	24	0	24

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
\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(9\)
Minus space			\(-\)	\(11\)

Trace form

20 * q - 18 * q^3 + 320 * q^7 + 1620 * q^9 - 1208 * q^11 + 450 * q^15 - 4416 * q^19 + 1672 * q^23 + 12500 * q^25 - 1458 * q^27 - 8144 * q^29 - 2496 * q^31 + 21296 * q^37 - 21636 * q^39 + 9640 * q^41 + 19128 * q^43 - 5016 * q^47 + 20628 * q^49 + 5220 * q^51 + 84480 * q^53 + 24200 * q^55 + 12888 * q^57 + 104504 * q^59 - 49176 * q^61 + 25920 * q^63 - 42200 * q^65 - 115912 * q^67 + 11160 * q^69 + 11136 * q^71 + 52568 * q^73 - 11250 * q^75 - 99840 * q^77 + 166576 * q^79 + 131220 * q^81 + 118824 * q^83 + 66200 * q^85 + 90828 * q^87 + 61944 * q^89 - 60000 * q^91 + 77352 * q^97 - 97848 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(240))\) 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	5
240.6.a.a	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				30.6.a.a	$1$	$0$	\(0\)	\(-9\)	\(-25\)	\(-164\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-5^{2}q^{5}-164q^{7}+3^{4}q^{9}+\cdots\)
240.6.a.b	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				15.6.a.b	$1$	$0$	\(0\)	\(-9\)	\(-25\)	\(-12\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-5^{2}q^{5}-12q^{7}+3^{4}q^{9}-112q^{11}+\cdots\)
240.6.a.c	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				120.6.a.e	$1$	$1$	\(0\)	\(-9\)	\(-25\)	\(28\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-5^{2}q^{5}+28q^{7}+3^{4}q^{9}+208q^{11}+\cdots\)
240.6.a.d	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				60.6.a.c	$1$	$0$	\(0\)	\(-9\)	\(-25\)	\(244\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-5^{2}q^{5}+244q^{7}+3^{4}q^{9}+\cdots\)
240.6.a.e	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				60.6.a.d	$1$	$1$	\(0\)	\(-9\)	\(25\)	\(-56\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+5^{2}q^{5}-56q^{7}+3^{4}q^{9}-156q^{11}+\cdots\)
240.6.a.f	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				30.6.a.b	$1$	$1$	\(0\)	\(-9\)	\(25\)	\(-32\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+5^{2}q^{5}-2^{5}q^{7}+3^{4}q^{9}-12q^{11}+\cdots\)
240.6.a.g	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				120.6.a.f	$1$	$0$	\(0\)	\(-9\)	\(25\)	\(160\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+5^{2}q^{5}+160q^{7}+3^{4}q^{9}+\cdots\)
240.6.a.h	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				120.6.a.b	$1$	$0$	\(0\)	\(9\)	\(-25\)	\(-108\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-5^{2}q^{5}-108q^{7}+3^{4}q^{9}+\cdots\)
240.6.a.i	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				60.6.a.a	$1$	$1$	\(0\)	\(9\)	\(-25\)	\(-44\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-5^{2}q^{5}-44q^{7}+3^{4}q^{9}-6^{3}q^{11}+\cdots\)
240.6.a.j	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				120.6.a.a	$1$	$0$	\(0\)	\(9\)	\(-25\)	\(100\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-5^{2}q^{5}+10^{2}q^{7}+3^{4}q^{9}+\cdots\)
240.6.a.k	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				15.6.a.a	$1$	$1$	\(0\)	\(9\)	\(-25\)	\(132\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-5^{2}q^{5}+132q^{7}+3^{4}q^{9}+\cdots\)
240.6.a.l	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				120.6.a.d	$1$	$1$	\(0\)	\(9\)	\(25\)	\(-128\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+5^{2}q^{5}-2^{7}q^{7}+3^{4}q^{9}+308q^{11}+\cdots\)
240.6.a.m	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				60.6.a.b	$1$	$0$	\(0\)	\(9\)	\(25\)	\(16\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+5^{2}q^{5}+2^{4}q^{7}+3^{4}q^{9}+564q^{11}+\cdots\)
240.6.a.n	$240$	$6$	240.a	1.a	$1$	$1$	$1$	$38.492$	\(\Q\)	None	✓				120.6.a.c	$1$	$1$	\(0\)	\(9\)	\(25\)	\(80\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+5^{2}q^{5}+80q^{7}+3^{4}q^{9}-684q^{11}+\cdots\)
240.6.a.o	$240$	$6$	240.a	1.a	$1$	$2$	$2$	$38.492$	\(\Q(\sqrt{1489}) \)	None	✓		✓		120.6.a.g	$1$	$1$	\(0\)	\(-18\)	\(-50\)	\(-16\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-9q^{3}-5^{2}q^{5}+(-8-\beta )q^{7}+3^{4}q^{9}+\cdots\)
240.6.a.p	$240$	$6$	240.a	1.a	$1$	$2$	$2$	$38.492$	\(\Q(\sqrt{2161}) \)	None	✓		✓		120.6.a.h	$1$	$0$	\(0\)	\(-18\)	\(50\)	\(8\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-9q^{3}+5^{2}q^{5}+(4+\beta )q^{7}+3^{4}q^{9}+\cdots\)
240.6.a.q	$240$	$6$	240.a	1.a	$1$	$2$	$2$	$38.492$	\(\Q(\sqrt{409}) \)	None	✓		✓		15.6.a.c	$1$	$0$	\(0\)	\(18\)	\(50\)	\(112\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+9q^{3}+5^{2}q^{5}+(56-\beta )q^{7}+3^{4}q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(240))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(240)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)