Space of modular forms of level 3900, weight 2, and Character 3900.by

• Label: 3900.2.by
• Level: $3900$
• Weight: $2$
• Character orbit: 3900.by
• Rep. character: $\chi_{3900}(1849,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $80$
• Newform subspaces: $11$
• Sturm bound: $1680$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.by (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(1680\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3900, [\chi])\).

	Total	New	Old
Modular forms	1752	80	1672
Cusp forms	1608	80	1528
Eisenstein series	144	0	144

Trace form

\( 80 q + 40 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3900, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3900.2.by.a	$3900$	$2$	3900.by	65.n	$6$	$4$	$2$	$31.142$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+3\zeta_{12}q^{7}+(1+\cdots)q^{9}+\cdots\)
3900.2.by.b	$3900$	$2$	3900.by	65.n	$6$	$4$	$2$	$31.142$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{9}+(-3+3\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
3900.2.by.c	$3900$	$2$	3900.by	65.n	$6$	$4$	$2$	$31.142$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}q^{7}+(1+\cdots)q^{9}+\cdots\)
3900.2.by.d	$3900$	$2$	3900.by	65.n	$6$	$4$	$2$	$31.142$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{9}+(2-2\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
3900.2.by.e	$3900$	$2$	3900.by	65.n	$6$	$4$	$2$	$31.142$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}q^{7}+(1-\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
3900.2.by.f	$3900$	$2$	3900.by	65.n	$6$	$4$	$2$	$31.142$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}-\zeta_{12}^{3})q^{3}+4\zeta_{12}q^{7}+(1-\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
3900.2.by.g	$3900$	$2$	3900.by	65.n	$6$	$8$	$4$	$31.142$	8.0.3317760000.2	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{3}+(-\beta _{1}+\beta _{3}-\beta _{5})q^{7}+\beta _{2}q^{9}+\cdots\)
3900.2.by.h	$3900$	$2$	3900.by	65.n	$6$	$8$	$4$	$31.142$	8.0.49787136.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{4}q^{3}-\beta _{6}q^{7}+(1+\beta _{3})q^{9}-2\beta _{3}q^{11}+\cdots\)
3900.2.by.i	$3900$	$2$	3900.by	65.n	$6$	$12$	$6$	$31.142$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+\beta _{7}q^{7}+\beta _{4}q^{9}+(-\beta _{3}-\beta _{7}+\cdots)q^{13}+\cdots\)
3900.2.by.j	$3900$	$2$	3900.by	65.n	$6$	$12$	$6$	$31.142$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{3}+\beta _{11}q^{7}+(1+\beta _{6})q^{9}+2\beta _{5}q^{11}+\cdots\)
3900.2.by.k	$3900$	$2$	3900.by	65.n	$6$	$16$	$8$	$31.142$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{12}q^{3}+(\beta _{4}+\beta _{11})q^{7}-\beta _{3}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3900, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3900, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(390, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(780, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(975, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1300, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1950, [\chi])\)\(^{\oplus 2}\)