Space of modular forms of level 3700, weight 2, and Character 3700.d

• Label: 3700.2.d
• Level: $3700$
• Weight: $2$
• Character orbit: 3700.d
• Rep. character: $\chi_{3700}(149,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $11$
• Sturm bound: $1140$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(1140\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3700, [\chi])\).

	Total	New	Old
Modular forms	588	54	534
Cusp forms	552	54	498
Eisenstein series	36	0	36

Trace form

54 * q - 54 * q^9 - 4 * q^11 - 4 * q^19 + 20 * q^21 + 4 * q^29 - 8 * q^31 - 12 * q^39 + 8 * q^41 - 90 * q^49 + 8 * q^51 + 32 * q^59 + 8 * q^61 - 36 * q^69 + 28 * q^71 + 40 * q^79 + 38 * q^81 + 8 * q^89 - 56 * q^91 + 28 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(3700, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3700.2.d.a	$3700$	$2$	3700.d	5.b	$2$	$2$	$2$	$29.545$	\(\Q(\sqrt{-1}) \)	None					740.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+3 i q^{7}-6 q^{9}+5 q^{11}+\cdots\)
3700.2.d.b	$3700$	$2$	3700.d	5.b	$2$	$2$	$2$	$29.545$	\(\Q(\sqrt{-1}) \)	None		✓			3700.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}+2 i q^{7}-q^{9}+6 q^{11}+\cdots\)
3700.2.d.c	$3700$	$2$	3700.d	5.b	$2$	$2$	$2$	$29.545$	\(\Q(\sqrt{-1}) \)	None					740.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+i q^{7}+2 q^{9}-3 q^{11}+6 i q^{13}+\cdots\)
3700.2.d.d	$3700$	$2$	3700.d	5.b	$2$	$2$	$2$	$29.545$	\(\Q(\sqrt{-1}) \)	None					740.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+i q^{7}+2 q^{9}-3 q^{11}-4 i q^{13}+\cdots\)
3700.2.d.e	$3700$	$2$	3700.d	5.b	$2$	$2$	$2$	$29.545$	\(\Q(\sqrt{-1}) \)	None					148.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-3 i q^{7}+2 q^{9}+5 q^{11}+\cdots\)
3700.2.d.f	$3700$	$2$	3700.d	5.b	$2$	$4$	$4$	$29.545$	\(\Q(i, \sqrt{17})\)	None					148.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{1}q^{7}+(-2+\beta _{3})q^{9}+(1+\cdots)q^{11}+\cdots\)
3700.2.d.g	$3700$	$2$	3700.d	5.b	$2$	$4$	$4$	$29.545$	\(\Q(\zeta_{12})\)	None					740.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{2}+\beta_1)q^{3}+(\beta_{2}+3\beta_1)q^{7}+\cdots\)
3700.2.d.h	$3700$	$2$	3700.d	5.b	$2$	$6$	$6$	$29.545$	6.0.350464.1	None					740.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(2\beta _{4}+\beta _{5})q^{7}+(\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
3700.2.d.i	$3700$	$2$	3700.d	5.b	$2$	$8$	$8$	$29.545$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					740.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{4})q^{3}+(-\beta _{1}-\beta _{4})q^{7}+(-3+\cdots)q^{9}+\cdots\)
3700.2.d.j	$3700$	$2$	3700.d	5.b	$2$	$10$	$10$	$29.545$	10.0.\(\cdots\).1	None		✓			3700.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{3})q^{7}+(\beta _{4}+\beta _{5})q^{9}+\cdots\)
3700.2.d.k	$3700$	$2$	3700.d	5.b	$2$	$12$	$12$	$29.545$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		3700.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-\beta _{8}q^{7}+(-1+\beta _{3}-\beta _{6}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3700, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3700, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(185, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(370, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(740, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(925, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1850, [\chi])\)\(^{\oplus 2}\)