Space of modular forms of level 405, weight 2, and trivial character

• Label: 405.2.a
• Level: $405$
• Weight: $2$
• Character orbit: 405.a
• Rep. character: $\chi_{405}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $10$
• Sturm bound: $108$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(108\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(405))\).

	Total	New	Old
Modular forms	66	16	50
Cusp forms	43	16	27
Eisenstein series	23	0	23

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

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(13\)	\(3\)	\(10\)		\(8\)	\(3\)	\(5\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(17\)	\(5\)	\(12\)		\(11\)	\(5\)	\(6\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(19\)	\(5\)	\(14\)		\(13\)	\(5\)	\(8\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(17\)	\(3\)	\(14\)		\(11\)	\(3\)	\(8\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(30\)	\(6\)	\(24\)		\(19\)	\(6\)	\(13\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(36\)	\(10\)	\(26\)		\(24\)	\(10\)	\(14\)		\(12\)	\(0\)	\(12\)

Trace form

16 * q + 16 * q^4 - 4 * q^7 - 4 * q^13 + 40 * q^16 - 16 * q^19 + 12 * q^22 + 16 * q^25 + 8 * q^28 - 16 * q^31 - 24 * q^34 - 4 * q^37 - 12 * q^40 - 28 * q^43 - 52 * q^52 - 12 * q^58 + 8 * q^61 + 16 * q^64 + 56 * q^67 - 12 * q^70 - 16 * q^73 - 64 * q^76 + 32 * q^79 - 48 * q^82 - 12 * q^85 + 36 * q^88 + 16 * q^91 - 24 * q^94 - 16 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(405))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
405.2.a.a	$405$	$2$	405.a	1.a	$1$	$1$	$1$	$3.234$	\(\Q\)	None	✓	✓			405.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{5}+2q^{10}-5q^{11}+\cdots\)
405.2.a.b	$405$	$2$	405.a	1.a	$1$	$1$	$1$	$3.234$	\(\Q\)	None	✓				45.2.e.a	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}-3q^{7}+3q^{8}-q^{10}+\cdots\)
405.2.a.c	$405$	$2$	405.a	1.a	$1$	$1$	$1$	$3.234$	\(\Q\)	None	✓	✓			405.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{5}+2q^{7}-3q^{11}-4q^{13}+\cdots\)
405.2.a.d	$405$	$2$	405.a	1.a	$1$	$1$	$1$	$3.234$	\(\Q\)	None	✓	✓			405.2.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}+2q^{7}+3q^{11}-4q^{13}+\cdots\)
405.2.a.e	$405$	$2$	405.a	1.a	$1$	$1$	$1$	$3.234$	\(\Q\)	None	✓				45.2.e.a	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{5}-3q^{7}-3q^{8}-q^{10}+\cdots\)
405.2.a.f	$405$	$2$	405.a	1.a	$1$	$1$	$1$	$3.234$	\(\Q\)	None	✓	✓			405.2.a.a	$1$	$0$	\(2\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{5}+2q^{10}+5q^{11}+\cdots\)
405.2.a.g	$405$	$2$	405.a	1.a	$1$	$2$	$2$	$3.234$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		405.2.a.g	$1$	$1$	\(-2\)	\(0\)	\(2\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(2-2\beta )q^{4}+q^{5}+(-3+\cdots)q^{7}+\cdots\)
405.2.a.h	$405$	$2$	405.a	1.a	$1$	$2$	$2$	$3.234$	\(\Q(\sqrt{3}) \)	None	✓	✓			405.2.a.g	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(2+2\beta )q^{4}-q^{5}+(-3+\cdots)q^{7}+\cdots\)
405.2.a.i	$405$	$2$	405.a	1.a	$1$	$3$	$3$	$3.234$	3.3.564.1	None	✓		✓		45.2.e.b	$1$	$0$	\(-1\)	\(0\)	\(-3\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
405.2.a.j	$405$	$2$	405.a	1.a	$1$	$3$	$3$	$3.234$	3.3.564.1	None	✓		✓		45.2.e.b	$1$	$0$	\(1\)	\(0\)	\(3\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(405))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(405)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)