Space of modular forms of level 3750, weight 2, and Character 3750.c

• Label: 3750.2.c
• Level: $3750$
• Weight: $2$
• Character orbit: 3750.c
• Rep. character: $\chi_{3750}(1249,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $11$
• Sturm bound: $1500$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(1500\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3750, [\chi])\).

	Total	New	Old
Modular forms	810	80	730
Cusp forms	690	80	610
Eisenstein series	120	0	120

Trace form

80 * q - 80 * q^4 - 80 * q^9 + 80 * q^16 + 20 * q^19 - 20 * q^21 + 20 * q^26 - 20 * q^29 - 20 * q^31 - 20 * q^34 + 80 * q^36 + 20 * q^39 + 20 * q^41 - 60 * q^49 - 80 * q^64 - 20 * q^74 - 20 * q^76 + 20 * q^79 + 80 * q^81 + 20 * q^84 - 20 * q^89 - 40 * q^91

Decomposition of \(S_{2}^{\mathrm{new}}(3750, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3750.2.c.a	$3750$	$2$	3750.c	5.b	$2$	$4$	$4$	$29.944$	\(\Q(i, \sqrt{5})\)	None		✓			3750.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-\beta _{3}q^{3}-q^{4}-q^{6}+3\beta _{1}q^{7}+\cdots\)
3750.2.c.b	$3750$	$2$	3750.c	5.b	$2$	$4$	$4$	$29.944$	\(\Q(i, \sqrt{5})\)	None					150.2.g.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-\beta _{3}q^{3}-q^{4}-q^{6}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
3750.2.c.c	$3750$	$2$	3750.c	5.b	$2$	$4$	$4$	$29.944$	\(\Q(i, \sqrt{5})\)	None					150.2.g.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-\beta _{3}q^{3}-q^{4}+q^{6}-2\beta _{3}q^{7}+\cdots\)
3750.2.c.d	$3750$	$2$	3750.c	5.b	$2$	$4$	$4$	$29.944$	\(\Q(i, \sqrt{5})\)	None		✓			3750.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+\beta _{3}q^{3}-q^{4}+q^{6}+\beta _{1}q^{7}+\cdots\)
3750.2.c.e	$3750$	$2$	3750.c	5.b	$2$	$8$	$8$	$29.944$	\(\Q(\zeta_{20})\)	None					150.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{2}-\beta_1 q^{3}-q^{4}-q^{6}+(-\beta_{7}-\beta_{2}+\beta_1)q^{7}+\cdots\)
3750.2.c.f	$3750$	$2$	3750.c	5.b	$2$	$8$	$8$	$29.944$	8.0.324000000.1	None		✓			3750.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}-\beta _{5}q^{3}-q^{4}-q^{6}+(\beta _{1}-\beta _{5}+\cdots)q^{7}+\cdots\)
3750.2.c.g	$3750$	$2$	3750.c	5.b	$2$	$8$	$8$	$29.944$	8.0.324000000.1	None		✓			3750.2.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}-\beta _{5}q^{3}-q^{4}-q^{6}+(\beta _{1}+2\beta _{3}+\cdots)q^{7}+\cdots\)
3750.2.c.h	$3750$	$2$	3750.c	5.b	$2$	$8$	$8$	$29.944$	8.0.\(\cdots\).17	None					150.2.g.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+\beta _{3}q^{3}-q^{4}-q^{6}+\beta _{1}q^{7}+\cdots\)
3750.2.c.i	$3750$	$2$	3750.c	5.b	$2$	$8$	$8$	$29.944$	8.0.324000000.1	None		✓			3750.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}-\beta _{5}q^{3}-q^{4}+q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
3750.2.c.j	$3750$	$2$	3750.c	5.b	$2$	$8$	$8$	$29.944$	8.0.324000000.1	None		✓			3750.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}-\beta _{5}q^{3}-q^{4}+q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
3750.2.c.k	$3750$	$2$	3750.c	5.b	$2$	$16$	$16$	$29.944$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None			✓		150.2.h.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{2}+\beta _{6}q^{3}-q^{4}+q^{6}+(-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3750, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3750, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(250, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(375, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(625, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(750, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1250, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1875, [\chi])\)\(^{\oplus 2}\)