Space of modular forms of level 1700, weight 2, and Character 1700.m

• Label: 1700.2.m
• Level: $1700$
• Weight: $2$
• Character orbit: 1700.m
• Rep. character: $\chi_{1700}(149,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $56$
• Newform subspaces: $6$
• Sturm bound: $540$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1700.m (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 85 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(540\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1700, [\chi])\).

	Total	New	Old
Modular forms	576	56	520
Cusp forms	504	56	448
Eisenstein series	72	0	72

Trace form

\( 56 q - 12 q^{11} - 40 q^{21} + 4 q^{29} - 24 q^{31} - 24 q^{39} + 4 q^{41} + 12 q^{51} - 24 q^{61} + 8 q^{69} + 68 q^{71} + 24 q^{79} - 56 q^{81} + 56 q^{89} + 80 q^{91} + 32 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1700, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1700.2.m.a	$1700$	$2$	1700.m	85.j	$4$	$4$	$2$	$13.575$	\(\Q(i, \sqrt{13})\)	None					68.2.e.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2})q^{3}+(1-\beta _{1}+\beta _{3})q^{7}+\cdots\)
1700.2.m.b	$1700$	$2$	1700.m	85.j	$4$	$4$	$2$	$13.575$	\(\Q(i, \sqrt{13})\)	None					68.2.e.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{1}+\beta _{3})q^{3}+(-1+\beta _{2})q^{7}+\cdots\)
1700.2.m.c	$1700$	$2$	1700.m	85.j	$4$	$12$	$6$	$13.575$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					340.2.o.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{5}-\beta _{8}+\beta _{9})q^{3}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1700.2.m.d	$1700$	$2$	1700.m	85.j	$4$	$12$	$6$	$13.575$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			1700.2.o.e	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+\beta _{11}q^{7}+(-\beta _{9}-\beta _{10}+\beta _{11})q^{9}+\cdots\)
1700.2.m.e	$1700$	$2$	1700.m	85.j	$4$	$12$	$6$	$13.575$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			1700.2.o.e	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}-\beta _{9}q^{7}+(\beta _{9}+\beta _{10}-\beta _{11})q^{9}+\cdots\)
1700.2.m.f	$1700$	$2$	1700.m	85.j	$4$	$12$	$6$	$13.575$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					340.2.o.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{5}+\beta _{7}-\beta _{10})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1700, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1700, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(340, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(425, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(850, [\chi])\)\(^{\oplus 2}\)