Space of modular forms of level 240, weight 12, and trivial character

• Label: 240.12.a
• Level: $240$
• Weight: $12$
• Character orbit: 240.a
• Rep. character: $\chi_{240}(1,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $22$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(240))\).

	Total	New	Old
Modular forms	540	44	496
Cusp forms	516	44	472
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
\(+\)	\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(23\)
Minus space			\(-\)	\(21\)

Trace form

44 * q + 486 * q^3 - 169920 * q^7 + 2598156 * q^9 - 1081688 * q^11 - 1518750 * q^15 + 18693312 * q^19 + 13479496 * q^23 + 429687500 * q^25 + 28697814 * q^27 - 155346416 * q^29 + 58859328 * q^31 + 1045524112 * q^37 - 65305764 * q^39 + 144100888 * q^41 - 678976936 * q^43 - 5716727832 * q^47 + 12017580140 * q^49 + 56110644 * q^51 - 4572817920 * q^53 - 4026275000 * q^55 - 1131254424 * q^57 + 30929182040 * q^59 + 6445845496 * q^61 - 10033606080 * q^63 - 5624825000 * q^65 + 8101989656 * q^67 - 8096707512 * q^69 - 77537467008 * q^71 + 16957280744 * q^73 + 4746093750 * q^75 - 102161410560 * q^77 - 78118469456 * q^79 + 153418513644 * q^81 + 29851077192 * q^83 + 42238925000 * q^85 - 59810510484 * q^87 + 29701819848 * q^89 + 56948460832 * q^91 + 66428319064 * q^97 - 63872594712 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(240))\) 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
240.12.a.a	$240$	$12$	240.a	1.a	$1$	$1$	$1$	$184.402$	\(\Q\)	None	✓				30.12.a.f	$1$	$1$	\(0\)	\(-243\)	\(-3125\)	\(-10556\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}-5^{5}q^{5}-10556q^{7}+3^{10}q^{9}+\cdots\)
240.12.a.b	$240$	$12$	240.a	1.a	$1$	$1$	$1$	$184.402$	\(\Q\)	None	✓				30.12.a.c	$1$	$0$	\(0\)	\(-243\)	\(3125\)	\(-56672\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+5^{5}q^{5}-56672q^{7}+3^{10}q^{9}+\cdots\)
240.12.a.c	$240$	$12$	240.a	1.a	$1$	$1$	$1$	$184.402$	\(\Q\)	None	✓				30.12.a.d	$1$	$0$	\(0\)	\(243\)	\(-3125\)	\(-29348\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}-5^{5}q^{5}-29348q^{7}+3^{10}q^{9}+\cdots\)
240.12.a.d	$240$	$12$	240.a	1.a	$1$	$1$	$1$	$184.402$	\(\Q\)	None	✓				30.12.a.a	$1$	$0$	\(0\)	\(243\)	\(-3125\)	\(22876\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}-5^{5}q^{5}+22876q^{7}+3^{10}q^{9}+\cdots\)
240.12.a.e	$240$	$12$	240.a	1.a	$1$	$1$	$1$	$184.402$	\(\Q\)	None	✓				15.12.a.a	$1$	$1$	\(0\)	\(243\)	\(3125\)	\(-27984\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+5^{5}q^{5}-27984q^{7}+3^{10}q^{9}+\cdots\)
240.12.a.f	$240$	$12$	240.a	1.a	$1$	$1$	$1$	$184.402$	\(\Q\)	None	✓				30.12.a.e	$1$	$1$	\(0\)	\(243\)	\(3125\)	\(5152\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+5^{5}q^{5}+5152q^{7}+3^{10}q^{9}+\cdots\)
240.12.a.g	$240$	$12$	240.a	1.a	$1$	$1$	$1$	$184.402$	\(\Q\)	None	✓				30.12.a.b	$1$	$1$	\(0\)	\(243\)	\(3125\)	\(57376\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+5^{5}q^{5}+57376q^{7}+3^{10}q^{9}+\cdots\)
240.12.a.h	$240$	$12$	240.a	1.a	$1$	$2$	$2$	$184.402$	\(\Q(\sqrt{1119913}) \)	None	✓				60.12.a.c	$1$	$1$	\(0\)	\(-486\)	\(-6250\)	\(-36496\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}-5^{5}q^{5}+(-18248-\beta )q^{7}+\cdots\)
240.12.a.i	$240$	$12$	240.a	1.a	$1$	$2$	$2$	$184.402$	\(\Q(\sqrt{534073}) \)	None	✓				120.12.a.a	$1$	$0$	\(0\)	\(-486\)	\(-6250\)	\(-11440\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}-5^{5}q^{5}+(-5720-\beta )q^{7}+\cdots\)
240.12.a.j	$240$	$12$	240.a	1.a	$1$	$2$	$2$	$184.402$	\(\Q(\sqrt{1609}) \)	None	✓				15.12.a.b	$1$	$1$	\(0\)	\(-486\)	\(-6250\)	\(10864\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}-5^{5}q^{5}+(5432-319\beta )q^{7}+\cdots\)
240.12.a.k	$240$	$12$	240.a	1.a	$1$	$2$	$2$	$184.402$	\(\Q(\sqrt{1009}) \)	None	✓				120.12.a.b	$1$	$1$	\(0\)	\(-486\)	\(6250\)	\(19448\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+5^{5}q^{5}+(9724-83\beta )q^{7}+\cdots\)
240.12.a.l	$240$	$12$	240.a	1.a	$1$	$2$	$2$	$184.402$	\(\Q(\sqrt{193}) \)	None	✓				60.12.a.d	$1$	$0$	\(0\)	\(-486\)	\(6250\)	\(44504\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+5^{5}q^{5}+(22252-29\beta )q^{7}+\cdots\)
240.12.a.m	$240$	$12$	240.a	1.a	$1$	$2$	$2$	$184.402$	\(\Q(\sqrt{1801}) \)	None	✓				15.12.a.c	$1$	$0$	\(0\)	\(486\)	\(-6250\)	\(-7784\)		$-$	$-$	$+$	$+$	$2^{8}$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}-5^{5}q^{5}+(-3892-14\beta )q^{7}+\cdots\)
240.12.a.n	$240$	$12$	240.a	1.a	$1$	$2$	$2$	$184.402$	\(\Q(\sqrt{26929}) \)	None	✓				60.12.a.a	$1$	$0$	\(0\)	\(486\)	\(-6250\)	\(11816\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}-5^{5}q^{5}+(5908-\beta )q^{7}+3^{10}q^{9}+\cdots\)
240.12.a.o	$240$	$12$	240.a	1.a	$1$	$2$	$2$	$184.402$	\(\Q(\sqrt{25489}) \)	None	✓		✓		60.12.a.b	$1$	$1$	\(0\)	\(486\)	\(6250\)	\(-65584\)		$-$	$-$	$-$	$-$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+5^{5}q^{5}+(-32792-\beta )q^{7}+\cdots\)
240.12.a.p	$240$	$12$	240.a	1.a	$1$	$3$	$3$	$184.402$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		120.12.a.g	$1$	$0$	\(0\)	\(-729\)	\(-9375\)	\(5148\)		$+$	$+$	$+$	$+$	$2^{10}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}-5^{5}q^{5}+(1716+\beta _{1}+3\beta _{2})q^{7}+\cdots\)
240.12.a.q	$240$	$12$	240.a	1.a	$1$	$3$	$3$	$184.402$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		120.12.a.h	$1$	$1$	\(0\)	\(-729\)	\(9375\)	\(-64368\)		$+$	$+$	$-$	$-$	$2^{10}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+5^{5}q^{5}+(-21456-\beta _{1}+\cdots)q^{7}+\cdots\)
240.12.a.r	$240$	$12$	240.a	1.a	$1$	$3$	$3$	$184.402$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		15.12.a.d	$1$	$0$	\(0\)	\(-729\)	\(9375\)	\(14608\)		$-$	$+$	$-$	$+$	$2^{11}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+5^{5}q^{5}+(4868+\beta _{1}+4\beta _{2})q^{7}+\cdots\)
240.12.a.s	$240$	$12$	240.a	1.a	$1$	$3$	$3$	$184.402$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				120.12.a.d	$1$	$1$	\(0\)	\(729\)	\(-9375\)	\(-49148\)		$+$	$-$	$+$	$-$	$2^{11}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}-5^{5}q^{5}+(-16383-\beta _{1}+\cdots)q^{7}+\cdots\)
240.12.a.t	$240$	$12$	240.a	1.a	$1$	$3$	$3$	$184.402$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				120.12.a.c	$1$	$1$	\(0\)	\(729\)	\(-9375\)	\(9108\)		$+$	$-$	$+$	$-$	$2^{11}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}-5^{5}q^{5}+(3036-3\beta _{1}+\beta _{2})q^{7}+\cdots\)
240.12.a.u	$240$	$12$	240.a	1.a	$1$	$3$	$3$	$184.402$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				120.12.a.f	$1$	$0$	\(0\)	\(729\)	\(9375\)	\(-34848\)		$+$	$-$	$-$	$+$	$2^{11}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+5^{5}q^{5}+(-11616-\beta _{1}+\cdots)q^{7}+\cdots\)
240.12.a.v	$240$	$12$	240.a	1.a	$1$	$3$	$3$	$184.402$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				120.12.a.e	$1$	$0$	\(0\)	\(729\)	\(9375\)	\(23408\)		$+$	$-$	$-$	$+$	$2^{11}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+5^{5}q^{5}+(7803-\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(240))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(240)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 20}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)