Space of modular forms of level 440, weight 2, and trivial character

• Label: 440.2.a
• Level: $440$
• Weight: $2$
• Character orbit: 440.a
• Rep. character: $\chi_{440}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $7$
• Sturm bound: $144$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(144\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(440))\).

	Total	New	Old
Modular forms	80	10	70
Cusp forms	65	10	55
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(13\)	\(2\)	\(11\)		\(11\)	\(2\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(14\)	\(2\)	\(12\)		\(12\)	\(2\)	\(10\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(10\)	\(2\)	\(8\)		\(8\)	\(2\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(11\)	\(1\)	\(10\)		\(9\)	\(1\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(34\)	\(2\)	\(32\)		\(27\)	\(2\)	\(25\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(46\)	\(8\)	\(38\)		\(38\)	\(8\)	\(30\)		\(8\)	\(0\)	\(8\)

Trace form

10 * q + 4 * q^3 - 2 * q^5 + 8 * q^7 + 6 * q^9 + 8 * q^13 + 8 * q^19 + 16 * q^21 - 4 * q^23 + 10 * q^25 + 16 * q^27 + 12 * q^29 + 28 * q^31 - 4 * q^33 - 4 * q^37 - 16 * q^39 - 12 * q^41 + 16 * q^43 + 6 * q^45 - 28 * q^47 - 2 * q^49 - 16 * q^51 - 20 * q^53 - 16 * q^57 - 8 * q^59 - 4 * q^61 + 8 * q^63 - 4 * q^67 - 24 * q^69 + 12 * q^71 + 4 * q^75 - 8 * q^77 - 16 * q^79 - 6 * q^81 - 40 * q^87 + 8 * q^91 - 16 * q^93 - 8 * q^95 - 12 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(440))\) 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	5	11
440.2.a.a	$440$	$2$	440.a	1.a	$1$	$1$	$1$	$3.513$	\(\Q\)	None	✓	✓	✓	✓	440.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{9}-q^{11}-8q^{19}+\cdots\)
440.2.a.b	$440$	$2$	440.a	1.a	$1$	$1$	$1$	$3.513$	\(\Q\)	None	✓	✓	✓	✓	440.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{9}+q^{11}-4q^{13}+\cdots\)
440.2.a.c	$440$	$2$	440.a	1.a	$1$	$1$	$1$	$3.513$	\(\Q\)	None	✓	✓			440.2.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}-3q^{9}-q^{11}+6q^{13}+\cdots\)
440.2.a.d	$440$	$2$	440.a	1.a	$1$	$1$	$1$	$3.513$	\(\Q\)	None	✓	✓			440.2.a.d	$1$	$0$	\(0\)	\(3\)	\(1\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+q^{7}+6q^{9}-q^{11}-6q^{13}+\cdots\)
440.2.a.e	$440$	$2$	440.a	1.a	$1$	$2$	$2$	$3.513$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	440.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}-\beta q^{7}+(1+\beta )q^{9}+q^{11}+\cdots\)
440.2.a.f	$440$	$2$	440.a	1.a	$1$	$2$	$2$	$3.513$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	440.2.a.f	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+(2-\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
440.2.a.g	$440$	$2$	440.a	1.a	$1$	$2$	$2$	$3.513$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	440.2.a.g	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+(2+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(440))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(440)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 2}\)