Space of modular forms of level 1400, weight 2, and Character 1400.q

• Label: 1400.2.q
• Level: $1400$
• Weight: $2$
• Character orbit: 1400.q
• Rep. character: $\chi_{1400}(401,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $76$
• Newform subspaces: $15$
• Sturm bound: $480$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.q (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(480\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1400, [\chi])\).

	Total	New	Old
Modular forms	528	76	452
Cusp forms	432	76	356
Eisenstein series	96	0	96

Trace form

76 * q - 2 * q^3 - 4 * q^7 - 36 * q^9 + 2 * q^11 - 8 * q^13 + 2 * q^17 + 10 * q^19 + 18 * q^21 - 10 * q^23 + 4 * q^27 + 16 * q^29 + 10 * q^31 - 6 * q^33 - 6 * q^37 + 4 * q^39 + 24 * q^41 - 32 * q^43 - 6 * q^47 - 20 * q^49 + 22 * q^51 - 6 * q^53 - 36 * q^57 + 18 * q^59 + 2 * q^61 + 40 * q^63 + 14 * q^67 - 28 * q^69 + 26 * q^73 + 34 * q^77 + 30 * q^79 - 22 * q^81 + 72 * q^83 + 24 * q^87 - 10 * q^89 + 16 * q^91 + 46 * q^93 + 8 * q^97 - 112 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(1400, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1400.2.q.a	$1400$	$2$	1400.q	7.c	$3$	$2$	$1$	$11.179$	\(\Q(\sqrt{-3}) \)	None					280.2.q.c	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}+(-3+2\zeta_{6})q^{7}+\cdots\)
1400.2.q.b	$1400$	$2$	1400.q	7.c	$3$	$2$	$1$	$11.179$	\(\Q(\sqrt{-3}) \)	None		✓			1400.2.q.b	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}+(2-3\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots\)
1400.2.q.c	$1400$	$2$	1400.q	7.c	$3$	$2$	$1$	$11.179$	\(\Q(\sqrt{-3}) \)	None					280.2.q.b	$2$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+(1-3\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots\)
1400.2.q.d	$1400$	$2$	1400.q	7.c	$3$	$2$	$1$	$11.179$	\(\Q(\sqrt{-3}) \)	None					56.2.i.b	$2$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+(1+2\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots\)
1400.2.q.e	$1400$	$2$	1400.q	7.c	$3$	$2$	$1$	$11.179$	\(\Q(\sqrt{-3}) \)	None					280.2.q.a	$2$	$0$	\(0\)	\(1\)	\(0\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+(3-\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots\)
1400.2.q.f	$1400$	$2$	1400.q	7.c	$3$	$2$	$1$	$11.179$	\(\Q(\sqrt{-3}) \)	None		✓			1400.2.q.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}+(-2+3\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots\)
1400.2.q.g	$1400$	$2$	1400.q	7.c	$3$	$2$	$1$	$11.179$	\(\Q(\sqrt{-3}) \)	None					56.2.i.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\zeta_{6})q^{3}+(-3+2\zeta_{6})q^{7}-6\zeta_{6}q^{9}+\cdots\)
1400.2.q.h	$1400$	$2$	1400.q	7.c	$3$	$4$	$2$	$11.179$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					280.2.q.d	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1400.2.q.i	$1400$	$2$	1400.q	7.c	$3$	$6$	$3$	$11.179$	\(\Q(\zeta_{18})\)	None		✓			1400.2.q.i	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{18}+\zeta_{18}^{3}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{3}+\cdots\)
1400.2.q.j	$1400$	$2$	1400.q	7.c	$3$	$6$	$3$	$11.179$	6.0.11337408.1	None					280.2.q.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{3}+\beta _{4}-\beta _{5})q^{3}+(-1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
1400.2.q.k	$1400$	$2$	1400.q	7.c	$3$	$6$	$3$	$11.179$	\(\Q(\zeta_{18})\)	None		✓			1400.2.q.i	$2$	$0$	\(0\)	\(3\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{4})q^{3}+(2\zeta_{18}+\cdots)q^{7}+\cdots\)
1400.2.q.l	$1400$	$2$	1400.q	7.c	$3$	$8$	$4$	$11.179$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			1400.2.q.l	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{5})q^{3}+(-\beta _{1}-\beta _{3}+\beta _{4})q^{7}+\cdots\)
1400.2.q.m	$1400$	$2$	1400.q	7.c	$3$	$8$	$4$	$11.179$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			1400.2.q.l	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{5})q^{3}+(\beta _{1}+\beta _{3}-\beta _{4})q^{7}+\cdots\)
1400.2.q.n	$1400$	$2$	1400.q	7.c	$3$	$12$	$6$	$11.179$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					280.2.bg.a	$2$	$0$	\(0\)	\(-1\)	\(0\)	\(-3\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{3})q^{3}-\beta _{4}q^{7}+(\beta _{6}-\beta _{10}+\cdots)q^{9}+\cdots\)
1400.2.q.o	$1400$	$2$	1400.q	7.c	$3$	$12$	$6$	$11.179$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					280.2.bg.a	$2$	$0$	\(0\)	\(1\)	\(0\)	\(3\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{4}q^{7}+(\beta _{6}-\beta _{10}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1400, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(700, [\chi])\)\(^{\oplus 2}\)