Space of modular forms of level 405, weight 4, and trivial character

• Label: 405.4.a
• Level: $405$
• Weight: $4$
• Character orbit: 405.a
• Rep. character: $\chi_{405}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $14$
• Sturm bound: $216$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(216\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(405))\).

	Total	New	Old
Modular forms	174	48	126
Cusp forms	150	48	102
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.

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(13\)
\(+\)	\(-\)	\(-\)	\(11\)
\(-\)	\(+\)	\(-\)	\(11\)
\(-\)	\(-\)	\(+\)	\(13\)
Plus space		\(+\)	\(26\)
Minus space		\(-\)	\(22\)

Trace form

48 * q + 192 * q^4 + 24 * q^7 - 48 * q^13 + 408 * q^16 + 60 * q^19 + 144 * q^22 + 1200 * q^25 + 888 * q^28 + 240 * q^31 + 36 * q^34 - 336 * q^37 + 180 * q^40 - 1020 * q^43 + 2268 * q^46 + 2268 * q^49 + 1068 * q^52 + 3852 * q^58 - 1236 * q^61 + 48 * q^64 - 4836 * q^67 - 360 * q^70 + 2004 * q^73 - 876 * q^76 - 624 * q^79 - 3492 * q^82 + 720 * q^85 + 6264 * q^88 - 3624 * q^91 - 5796 * q^94 - 3468 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\) 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
405.4.a.a	$405$	$4$	405.a	1.a	$1$	$1$	$1$	$23.896$	\(\Q\)	None	✓	✓			405.4.a.a	$1$	$1$	\(-5\)	\(0\)	\(5\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}+5q^{5}+9q^{7}-45q^{8}+\cdots\)
405.4.a.b	$405$	$4$	405.a	1.a	$1$	$1$	$1$	$23.896$	\(\Q\)	None	✓	✓			405.4.a.a	$1$	$0$	\(5\)	\(0\)	\(-5\)	\(9\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+17q^{4}-5q^{5}+9q^{7}+45q^{8}+\cdots\)
405.4.a.c	$405$	$4$	405.a	1.a	$1$	$2$	$2$	$23.896$	\(\Q(\sqrt{3}) \)	None	✓	✓			405.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(-10\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-4-2\beta )q^{4}-5q^{5}+\cdots\)
405.4.a.d	$405$	$4$	405.a	1.a	$1$	$2$	$2$	$23.896$	\(\Q(\sqrt{33}) \)	None	✓				45.4.e.a	$1$	$1$	\(-1\)	\(0\)	\(10\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+\beta q^{4}+5q^{5}+(5-\beta )q^{7}+(-8+\cdots)q^{8}+\cdots\)
405.4.a.e	$405$	$4$	405.a	1.a	$1$	$2$	$2$	$23.896$	\(\Q(\sqrt{33}) \)	None	✓				45.4.e.a	$1$	$1$	\(1\)	\(0\)	\(-10\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+\beta q^{4}-5q^{5}+(5-\beta )q^{7}+(8+\cdots)q^{8}+\cdots\)
405.4.a.f	$405$	$4$	405.a	1.a	$1$	$2$	$2$	$23.896$	\(\Q(\sqrt{3}) \)	None	✓	✓			405.4.a.c	$1$	$1$	\(2\)	\(0\)	\(10\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-4+2\beta )q^{4}+5q^{5}+\cdots\)
405.4.a.g	$405$	$4$	405.a	1.a	$1$	$3$	$3$	$23.896$	3.3.7032.1	None	✓	✓			405.4.a.g	$1$	$1$	\(-1\)	\(0\)	\(15\)	\(-25\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)
405.4.a.h	$405$	$4$	405.a	1.a	$1$	$3$	$3$	$23.896$	3.3.2292.1	None	✓				45.4.e.b	$1$	$1$	\(-1\)	\(0\)	\(15\)	\(-43\)		$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(3-\beta _{1}-\beta _{2})q^{4}+5q^{5}+\cdots\)
405.4.a.i	$405$	$4$	405.a	1.a	$1$	$3$	$3$	$23.896$	3.3.7032.1	None	✓	✓			405.4.a.g	$1$	$0$	\(1\)	\(0\)	\(-15\)	\(-25\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2-\beta _{1}+\beta _{2})q^{4}-5q^{5}+\cdots\)
405.4.a.j	$405$	$4$	405.a	1.a	$1$	$3$	$3$	$23.896$	3.3.2292.1	None	✓				45.4.e.b	$1$	$1$	\(1\)	\(0\)	\(-15\)	\(-43\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(3-\beta _{1}-\beta _{2})q^{4}-5q^{5}+\cdots\)
405.4.a.k	$405$	$4$	405.a	1.a	$1$	$6$	$6$	$23.896$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		405.4.a.k	$1$	$1$	\(-4\)	\(0\)	\(-30\)	\(40\)		$-$	$+$	$-$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(6-\beta _{1}+\beta _{3})q^{4}+\cdots\)
405.4.a.l	$405$	$4$	405.a	1.a	$1$	$6$	$6$	$23.896$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			405.4.a.k	$1$	$0$	\(4\)	\(0\)	\(30\)	\(40\)		$-$	$-$	$+$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(6-\beta _{1}+\beta _{3})q^{4}+5q^{5}+\cdots\)
405.4.a.m	$405$	$4$	405.a	1.a	$1$	$7$	$7$	$23.896$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		45.4.e.c	$1$	$0$	\(-2\)	\(0\)	\(-35\)	\(22\)		$+$	$+$	$+$	$2\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(5+\beta _{2})q^{4}-5q^{5}+(3-\beta _{5}+\cdots)q^{7}+\cdots\)
405.4.a.n	$405$	$4$	405.a	1.a	$1$	$7$	$7$	$23.896$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		45.4.e.c	$1$	$0$	\(2\)	\(0\)	\(35\)	\(22\)		$-$	$-$	$+$	$2\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+5q^{5}+(3-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(405))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(405)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)