Space of modular forms of level 240, weight 8, and trivial character

• Label: 240.8.a
• Level: $240$
• Weight: $8$
• Character orbit: 240.a
• Rep. character: $\chi_{240}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $21$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(240))\).

	Total	New	Old
Modular forms	348	28	320
Cusp forms	324	28	296
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
\(+\)	\(+\)	\(+\)	\(+\)		\(46\)	\(3\)	\(43\)		\(43\)	\(3\)	\(40\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(41\)	\(3\)	\(38\)		\(38\)	\(3\)	\(35\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(43\)	\(4\)	\(39\)		\(40\)	\(4\)	\(36\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(44\)	\(4\)	\(40\)		\(41\)	\(4\)	\(37\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(43\)	\(3\)	\(40\)		\(40\)	\(3\)	\(37\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(44\)	\(4\)	\(40\)		\(41\)	\(4\)	\(37\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(42\)	\(4\)	\(38\)		\(39\)	\(4\)	\(35\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(45\)	\(3\)	\(42\)		\(42\)	\(3\)	\(39\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(176\)	\(15\)	\(161\)		\(164\)	\(15\)	\(149\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(172\)	\(13\)	\(159\)		\(160\)	\(13\)	\(147\)		\(12\)	\(0\)	\(12\)

Trace form

28 * q + 54 * q^3 + 640 * q^7 + 20412 * q^9 + 2408 * q^11 - 6750 * q^15 + 11136 * q^19 - 143416 * q^23 + 437500 * q^25 + 39366 * q^27 + 103376 * q^29 - 241536 * q^31 - 831152 * q^37 - 789156 * q^39 - 254344 * q^41 + 1272216 * q^43 - 1252248 * q^47 + 4846044 * q^49 - 748332 * q^51 + 445440 * q^53 - 1331000 * q^55 - 1551096 * q^57 - 3525608 * q^59 + 4790040 * q^61 + 466560 * q^63 + 83000 * q^65 - 1670696 * q^67 - 2395224 * q^69 + 12408576 * q^71 - 1267064 * q^73 + 843750 * q^75 + 15928320 * q^77 + 11613296 * q^79 + 14880348 * q^81 - 15261432 * q^83 - 3767000 * q^85 - 7902036 * q^87 - 15994392 * q^89 - 1463136 * q^91 + 16203576 * q^97 + 1755432 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(240))\) 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
240.8.a.a	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				30.8.a.f	$1$	$1$	\(0\)	\(-27\)	\(-125\)	\(-1604\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-5^{3}q^{5}-1604q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.b	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				60.8.a.c	$1$	$1$	\(0\)	\(-27\)	\(-125\)	\(-92\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-5^{3}q^{5}-92q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.c	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				15.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(-125\)	\(420\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-5^{3}q^{5}+420q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.d	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				120.8.a.a	$1$	$0$	\(0\)	\(-27\)	\(-125\)	\(540\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-5^{3}q^{5}+540q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.e	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				30.8.a.c	$1$	$0$	\(0\)	\(-27\)	\(125\)	\(232\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+5^{3}q^{5}+232q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.f	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				120.8.a.b	$1$	$1$	\(0\)	\(-27\)	\(125\)	\(776\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+5^{3}q^{5}+776q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.g	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				60.8.a.d	$1$	$0$	\(0\)	\(-27\)	\(125\)	\(1408\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+5^{3}q^{5}+1408q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.h	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				15.8.a.b	$1$	$0$	\(0\)	\(27\)	\(-125\)	\(-1380\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-5^{3}q^{5}-1380q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.i	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				60.8.a.a	$1$	$0$	\(0\)	\(27\)	\(-125\)	\(-1028\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-5^{3}q^{5}-1028q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.j	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				30.8.a.d	$1$	$0$	\(0\)	\(27\)	\(-125\)	\(988\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-5^{3}q^{5}+988q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.k	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				30.8.a.a	$1$	$0$	\(0\)	\(27\)	\(-125\)	\(1084\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-5^{3}q^{5}+1084q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.l	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				30.8.a.e	$1$	$1$	\(0\)	\(27\)	\(125\)	\(-512\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+5^{3}q^{5}-2^{9}q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.m	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				30.8.a.b	$1$	$1$	\(0\)	\(27\)	\(125\)	\(-416\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+5^{3}q^{5}-416q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.n	$240$	$8$	240.a	1.a	$1$	$1$	$1$	$74.972$	\(\Q\)	None	✓				60.8.a.b	$1$	$1$	\(0\)	\(27\)	\(125\)	\(832\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+5^{3}q^{5}+832q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.o	$240$	$8$	240.a	1.a	$1$	$2$	$2$	$74.972$	\(\Q(\sqrt{10761}) \)	None	✓		✓		120.8.a.g	$1$	$0$	\(0\)	\(-54\)	\(-250\)	\(896\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-5^{3}q^{5}+(448-\beta )q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.p	$240$	$8$	240.a	1.a	$1$	$2$	$2$	$74.972$	\(\Q(\sqrt{601}) \)	None	✓		✓		15.8.a.c	$1$	$0$	\(0\)	\(-54\)	\(250\)	\(-1304\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+5^{3}q^{5}+(-652-7\beta )q^{7}+\cdots\)
240.8.a.q	$240$	$8$	240.a	1.a	$1$	$2$	$2$	$74.972$	\(\Q(\sqrt{7849}) \)	None	✓		✓		120.8.a.h	$1$	$1$	\(0\)	\(-54\)	\(250\)	\(-952\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+5^{3}q^{5}+(-476-\beta )q^{7}+\cdots\)
240.8.a.r	$240$	$8$	240.a	1.a	$1$	$2$	$2$	$74.972$	\(\Q(\sqrt{114}) \)	None	✓				120.8.a.d	$1$	$1$	\(0\)	\(54\)	\(-250\)	\(-328\)		$+$	$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-5^{3}q^{5}+(-164+\beta )q^{7}+\cdots\)
240.8.a.s	$240$	$8$	240.a	1.a	$1$	$2$	$2$	$74.972$	\(\Q(\sqrt{106}) \)	None	✓				120.8.a.c	$1$	$1$	\(0\)	\(54\)	\(-250\)	\(824\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-5^{3}q^{5}+(412+7\beta )q^{7}+3^{6}q^{9}+\cdots\)
240.8.a.t	$240$	$8$	240.a	1.a	$1$	$2$	$2$	$74.972$	\(\Q(\sqrt{6946}) \)	None	✓				120.8.a.f	$1$	$0$	\(0\)	\(54\)	\(250\)	\(-448\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+5^{3}q^{5}+(-224+\beta )q^{7}+\cdots\)
240.8.a.u	$240$	$8$	240.a	1.a	$1$	$2$	$2$	$74.972$	\(\Q(\sqrt{106}) \)	None	✓				120.8.a.e	$1$	$0$	\(0\)	\(54\)	\(250\)	\(704\)		$+$	$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+5^{3}q^{5}+(352+\beta )q^{7}+3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(240))\) into lower level spaces

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