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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$29$	$3$	$26$		$26$	$3$	$23$		$3$	$0$	$3$
$+$	$+$	$-$	$-$		$34$	$3$	$31$		$31$	$3$	$28$		$3$	$0$	$3$
$+$	$-$	$+$	$-$		$32$	$2$	$30$		$29$	$2$	$27$		$3$	$0$	$3$
$+$	$-$	$-$	$+$		$31$	$2$	$29$		$28$	$2$	$26$		$3$	$0$	$3$
$-$	$+$	$+$	$-$		$32$	$3$	$29$		$29$	$3$	$26$		$3$	$0$	$3$
$-$	$+$	$-$	$+$		$31$	$2$	$29$		$28$	$2$	$26$		$3$	$0$	$3$
$-$	$-$	$+$	$+$		$33$	$2$	$31$		$30$	$2$	$28$		$3$	$0$	$3$
$-$	$-$	$-$	$-$		$30$	$3$	$27$		$27$	$3$	$24$		$3$	$0$	$3$
Plus space			$+$		$124$	$9$	$115$		$112$	$9$	$103$		$12$	$0$	$12$
Minus space			$-$		$128$	$11$	$117$		$116$	$11$	$105$		$12$	$0$	$12$

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}$