Space of modular forms of level 756, weight 3, and Character 756.bk

• Label: 756.3.bk
• Level: $756$
• Weight: $3$
• Character orbit: 756.bk
• Rep. character: $\chi_{756}(53,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $42$
• Newform subspaces: $7$
• Sturm bound: $432$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.bk (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(432\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(756, [\chi])\).

	Total	New	Old
Modular forms	612	42	570
Cusp forms	540	42	498
Eisenstein series	72	0	72

Trace form

\( 42 q - 5 q^{7} - 20 q^{13} + 45 q^{19} + 115 q^{25} - 35 q^{31} - 30 q^{37} + 78 q^{43} - 33 q^{49} - 232 q^{55} - 47 q^{61} + 110 q^{67} + 113 q^{73} + 82 q^{79} + 164 q^{85} + 280 q^{91} + 470 q^{97}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(756, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
756.3.bk.a	$756$	$3$	756.bk	21.h	$6$	$2$	$1$	$20.600$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			756.3.bk.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-3+8\zeta_{6})q^{7}-q^{13}+(-26+26\zeta_{6})q^{19}+\cdots\)
756.3.bk.b	$756$	$3$	756.bk	21.h	$6$	$2$	$1$	$20.600$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			756.3.bk.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(5-8\zeta_{6})q^{7}+23q^{13}+(-26+26\zeta_{6})q^{19}+\cdots\)
756.3.bk.c	$756$	$3$	756.bk	21.h	$6$	$2$	$1$	$20.600$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			756.3.bk.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(11\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(8-5\zeta_{6})q^{7}-22q^{13}+(37-37\zeta_{6})q^{19}+\cdots\)
756.3.bk.d	$756$	$3$	756.bk	21.h	$6$	$4$	$2$	$20.600$	\(\Q(\sqrt{-3}, \sqrt{-5})\)	None		✓			756.3.bk.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}-7q^{7}+(\beta _{1}-\beta _{3})q^{11}+3q^{13}+\cdots\)
756.3.bk.e	$756$	$3$	756.bk	21.h	$6$	$4$	$2$	$20.600$	\(\Q(\zeta_{12})\)	None		✓			756.3.bk.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+2\beta_1 q^{5}+(7\beta_{2}-7)q^{7}+(\beta_{3}-\beta_1)q^{11}+\cdots\)
756.3.bk.f	$756$	$3$	756.bk	21.h	$6$	$12$	$6$	$20.600$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			756.3.bk.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2^{6}\cdot 3^{7}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{4}q^{5}-\beta _{6}q^{7}+(-\beta _{5}+\beta _{10})q^{11}+\cdots\)
756.3.bk.g	$756$	$3$	756.bk	21.h	$6$	$16$	$8$	$20.600$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		756.3.bk.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(24\)	$3^{11}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{7}-\beta _{9})q^{5}+(\beta _{3}-2\beta _{4})q^{7}+(-\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(756, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)