Space of modular forms of level 405, weight 4, and Character 405.e

• Label: 405.4.e
• Level: $405$
• Weight: $4$
• Character orbit: 405.e
• Rep. character: $\chi_{405}(136,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $96$
• Newform subspaces: $24$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(216\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(405, [\chi])\).

	Total	New	Old
Modular forms	348	96	252
Cusp forms	300	96	204
Eisenstein series	48	0	48

Trace form

96 * q - 192 * q^4 - 60 * q^7 + 120 * q^13 - 768 * q^16 - 960 * q^19 + 72 * q^22 - 1200 * q^25 + 1920 * q^28 - 600 * q^31 - 1350 * q^34 - 1680 * q^37 - 450 * q^40 + 156 * q^43 - 1188 * q^46 - 756 * q^49 + 3756 * q^52 + 1188 * q^58 + 84 * q^61 + 1644 * q^64 - 1464 * q^67 - 360 * q^70 - 5064 * q^73 + 9750 * q^76 + 3036 * q^79 + 6984 * q^82 + 720 * q^85 + 864 * q^88 - 14088 * q^91 - 3924 * q^94 + 2532 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(405, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
405.4.e.a	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None		✓			405.4.a.a	$2$	$0$	\(-5\)	\(0\)	\(5\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\zeta_{6})q^{2}-17\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.b	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					45.4.a.a	$2$	$0$	\(-5\)	\(0\)	\(5\)	\(30\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\zeta_{6})q^{2}-17\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.c	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					5.4.a.a	$2$	$0$	\(-4\)	\(0\)	\(-5\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{2}-8\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.d	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					15.4.a.b	$2$	$0$	\(-3\)	\(0\)	\(5\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{2}-\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.e	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					135.4.a.a	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}+4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.f	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					135.4.a.b	$2$	$0$	\(-1\)	\(0\)	\(-5\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+7\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.g	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					15.4.a.a	$2$	$0$	\(-1\)	\(0\)	\(-5\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+7\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.h	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					135.4.a.b	$2$	$0$	\(1\)	\(0\)	\(5\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+7\zeta_{6}q^{4}+5\zeta_{6}q^{5}+(6+\cdots)q^{7}+\cdots\)
405.4.e.i	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					15.4.a.a	$2$	$0$	\(1\)	\(0\)	\(5\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+7\zeta_{6}q^{4}+5\zeta_{6}q^{5}+(24+\cdots)q^{7}+\cdots\)
405.4.e.j	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					135.4.a.a	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}+4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.k	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					15.4.a.b	$2$	$0$	\(3\)	\(0\)	\(-5\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\zeta_{6})q^{2}-\zeta_{6}q^{4}-5\zeta_{6}q^{5}+(-20+\cdots)q^{7}+\cdots\)
405.4.e.l	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					5.4.a.a	$2$	$0$	\(4\)	\(0\)	\(5\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{2}-8\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.m	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None		✓			405.4.a.a	$2$	$0$	\(5\)	\(0\)	\(-5\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\zeta_{6})q^{2}-17\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.n	$405$	$4$	405.e	9.c	$3$	$2$	$1$	$23.896$	\(\Q(\sqrt{-3}) \)	None					45.4.a.a	$2$	$0$	\(5\)	\(0\)	\(-5\)	\(30\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\zeta_{6})q^{2}-17\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.o	$405$	$4$	405.e	9.c	$3$	$4$	$2$	$23.896$	\(\Q(\zeta_{12})\)	None		✓			405.4.a.c	$2$	$0$	\(-2\)	\(0\)	\(-10\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta_{2}-\beta_1)q^{2}+(2\beta_{3}-2\beta_{2}-4\beta_1+4)q^{4}+\cdots\)
405.4.e.p	$405$	$4$	405.e	9.c	$3$	$4$	$2$	$23.896$	\(\Q(\zeta_{12})\)	None		✓			405.4.a.c	$2$	$0$	\(2\)	\(0\)	\(10\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta_{2}+\beta_1)q^{2}+(2\beta_{3}-2\beta_{2}-4\beta_1+4)q^{4}+\cdots\)
405.4.e.q	$405$	$4$	405.e	9.c	$3$	$6$	$3$	$23.896$	6.0.84779568.3	None					135.4.a.e	$2$	$0$	\(-5\)	\(0\)	\(-15\)	\(4\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-2\beta _{3}+\beta _{5})q^{2}+(-7+7\beta _{3}+\cdots)q^{4}+\cdots\)
405.4.e.r	$405$	$4$	405.e	9.c	$3$	$6$	$3$	$23.896$	6.0.95327307.1	None					135.4.a.f	$2$	$0$	\(-1\)	\(0\)	\(15\)	\(-44\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{2}+(-\beta _{1}-7\beta _{3}-\beta _{4}+\beta _{5})q^{4}+\cdots\)
405.4.e.s	$405$	$4$	405.e	9.c	$3$	$6$	$3$	$23.896$	6.0.148347072.2	None		✓			405.4.a.g	$2$	$0$	\(-1\)	\(0\)	\(15\)	\(25\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{3})q^{2}+(-2-\beta _{1}-2\beta _{2}+\cdots)q^{4}+\cdots\)
405.4.e.t	$405$	$4$	405.e	9.c	$3$	$6$	$3$	$23.896$	6.0.95327307.1	None					135.4.a.f	$2$	$0$	\(1\)	\(0\)	\(-15\)	\(-44\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{4})q^{2}+(-7-\beta _{2}+7\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
405.4.e.u	$405$	$4$	405.e	9.c	$3$	$6$	$3$	$23.896$	6.0.148347072.2	None		✓			405.4.a.g	$2$	$0$	\(1\)	\(0\)	\(-15\)	\(25\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(\beta _{1}+2\beta _{2}-\beta _{3}+\beta _{4})q^{4}+\cdots\)
405.4.e.v	$405$	$4$	405.e	9.c	$3$	$6$	$3$	$23.896$	6.0.84779568.3	None					135.4.a.e	$2$	$0$	\(5\)	\(0\)	\(15\)	\(4\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\beta _{3}+\beta _{5})q^{2}+(-3\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
405.4.e.w	$405$	$4$	405.e	9.c	$3$	$12$	$6$	$23.896$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			405.4.a.k	$2$	$0$	\(-4\)	\(0\)	\(-30\)	\(-40\)	$2^{2}\cdot 3^{9}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{2}-\beta _{3})q^{2}+(-6+6\beta _{2}+\beta _{5}+\cdots)q^{4}+\cdots\)
405.4.e.x	$405$	$4$	405.e	9.c	$3$	$12$	$6$	$23.896$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			405.4.a.k	$2$	$0$	\(4\)	\(0\)	\(30\)	\(-40\)	$2^{2}\cdot 3^{9}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{2}-\beta _{9})q^{2}+(-6\beta _{2}-\beta _{3}-\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(405, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(405, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)