Space of modular forms of level 405, weight 3, and Character 405.g

• Label: 405.3.g
• Level: $405$
• Weight: $3$
• Character orbit: 405.g
• Rep. character: $\chi_{405}(82,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $88$
• Newform subspaces: $8$
• Sturm bound: $162$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(162\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(405, [\chi])\).

	Total	New	Old
Modular forms	240	104	136
Cusp forms	192	88	104
Eisenstein series	48	16	32

Trace form

88 * q + 4 * q^7 - 8 * q^10 - 44 * q^13 - 248 * q^16 - 28 * q^22 + 16 * q^25 - 88 * q^28 + 8 * q^31 - 20 * q^37 + 84 * q^40 + 28 * q^43 + 200 * q^46 - 140 * q^52 + 224 * q^55 - 36 * q^58 - 184 * q^61 + 160 * q^67 + 132 * q^70 + 448 * q^73 - 432 * q^76 + 560 * q^82 + 604 * q^85 + 324 * q^88 + 56 * q^91 - 392 * q^97

Decomposition of \(S_{3}^{\mathrm{new}}(405, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
405.3.g.a	$405$	$3$	405.g	5.c	$4$	$4$	$2$	$11.035$	\(\Q(\zeta_{12})\)	None		✓			405.3.g.a	$2$	$0$	\(-2\)	\(0\)	\(10\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(-1+2\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
405.3.g.b	$405$	$3$	405.g	5.c	$4$	$4$	$2$	$11.035$	\(\Q(\zeta_{12})\)	None		✓			405.3.g.a	$2$	$0$	\(2\)	\(0\)	\(-10\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(1+\cdots)q^{4}+\cdots\)
405.3.g.c	$405$	$3$	405.g	5.c	$4$	$8$	$4$	$11.035$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			405.3.g.c	$2$	$0$	\(-2\)	\(0\)	\(-2\)	\(26\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{6}q^{2}+(-\beta _{1}-4\beta _{2}-\beta _{5}-\beta _{6}+\cdots)q^{4}+\cdots\)
405.3.g.d	$405$	$3$	405.g	5.c	$4$	$8$	$4$	$11.035$	8.0.49787136.1	None		✓			405.3.g.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-52\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta _{2}-\beta _{7})q^{2}+(-2\beta _{1}-\beta _{5}-\beta _{6}+\cdots)q^{4}+\cdots\)
405.3.g.e	$405$	$3$	405.g	5.c	$4$	$8$	$4$	$11.035$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			405.3.g.c	$2$	$0$	\(2\)	\(0\)	\(2\)	\(26\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}+(-\beta _{1}-4\beta _{2}-\beta _{5}-\beta _{6}+\cdots)q^{4}+\cdots\)
405.3.g.f	$405$	$3$	405.g	5.c	$4$	$16$	$8$	$11.035$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓			405.3.g.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(40\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{9}q^{2}+(-3\beta _{2}-\beta _{5}-\beta _{6})q^{4}+(\beta _{9}+\cdots)q^{5}+\cdots\)
405.3.g.g	$405$	$3$	405.g	5.c	$4$	$20$	$10$	$11.035$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None					45.3.k.a	$2$	$0$	\(-2\)	\(0\)	\(-2\)	\(2\)	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{2}+(2\beta _{5}+\beta _{9})q^{4}+\beta _{7}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
405.3.g.h	$405$	$3$	405.g	5.c	$4$	$20$	$10$	$11.035$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None					45.3.k.a	$2$	$0$	\(2\)	\(0\)	\(2\)	\(2\)	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{2}+(-2\beta _{5}-\beta _{9})q^{4}-\beta _{12}q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(405, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(405, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)