Space of modular forms of level 722, weight 2, and Character 722.e

• Label: 722.2.e
• Level: $722$
• Weight: $2$
• Character orbit: 722.e
• Rep. character: $\chi_{722}(99,\cdot)$
• Character field: $\Q(\zeta_{9})$
• Dimension: $174$
• Newform subspaces: $19$
• Sturm bound: $190$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.e (of order \(9\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{9})\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(190\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(722, [\chi])\).

	Total	New	Old
Modular forms	690	174	516
Cusp forms	450	174	276
Eisenstein series	240	0	240

Trace form

174 * q + 3 * q^3 + 3 * q^6 + 12 * q^7 + 3 * q^8 + 3 * q^9 + 12 * q^11 - 12 * q^13 - 12 * q^14 + 6 * q^15 + 12 * q^17 + 6 * q^18 - 24 * q^20 - 24 * q^21 + 12 * q^23 + 3 * q^24 - 6 * q^26 - 3 * q^27 + 6 * q^28 + 18 * q^29 - 6 * q^31 - 3 * q^33 + 12 * q^34 + 12 * q^35 + 3 * q^36 + 12 * q^37 - 24 * q^39 - 3 * q^41 + 12 * q^42 + 6 * q^43 + 24 * q^45 - 30 * q^47 - 6 * q^48 - 45 * q^49 - 3 * q^50 - 21 * q^51 + 6 * q^52 - 24 * q^53 - 9 * q^54 - 18 * q^55 - 12 * q^56 - 48 * q^58 + 3 * q^59 + 6 * q^60 - 6 * q^61 - 18 * q^62 - 12 * q^63 - 87 * q^64 - 12 * q^65 - 3 * q^66 + 9 * q^67 + 9 * q^68 + 6 * q^69 + 12 * q^70 + 18 * q^71 - 6 * q^72 + 30 * q^73 + 18 * q^74 - 12 * q^77 + 18 * q^78 - 6 * q^79 + 33 * q^81 - 3 * q^82 + 36 * q^83 - 6 * q^84 + 24 * q^85 - 12 * q^86 + 24 * q^87 + 6 * q^88 - 12 * q^90 + 12 * q^91 - 6 * q^92 - 6 * q^93 + 12 * q^94 - 3 * q^97 - 21 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(722, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
722.2.e.a	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.e.a	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(-1-\zeta_{18}^{2}+\cdots)q^{3}+\cdots\)
722.2.e.b	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.e.a	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+(-1+\zeta_{18}^{3}-\zeta_{18}^{5})q^{3}+\cdots\)
722.2.e.c	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.a.b	$6$	$0$	\(0\)	\(0\)	\(0\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.d	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.a.b	$6$	$0$	\(0\)	\(0\)	\(0\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.e	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.a.a	$6$	$0$	\(0\)	\(0\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.f	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.a.a	$6$	$0$	\(0\)	\(0\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.g	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None		✓			722.2.a.a	$6$	$0$	\(0\)	\(0\)	\(0\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}-3\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.h	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None		✓			722.2.a.a	$6$	$0$	\(0\)	\(0\)	\(0\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+3\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.i	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.c.a	$6$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.j	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.c.a	$6$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.k	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.e.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(\zeta_{18}^{2}+\zeta_{18}^{3}+\cdots)q^{3}+\cdots\)
722.2.e.l	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.e.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(1-\zeta_{18}^{3}+\zeta_{18}^{5})q^{3}+\cdots\)
722.2.e.m	$722$	$2$	722.e	19.e	$9$	$6$	$1$	$5.765$	\(\Q(\zeta_{18})\)	None					38.2.e.a	$2$	$0$	\(0\)	\(6\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+(1+\zeta_{18}^{2})q^{3}+\cdots\)
722.2.e.n	$722$	$2$	722.e	19.e	$9$	$12$	$2$	$5.765$	12.0.\(\cdots\).3	None					38.2.c.b	$6$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+\beta _{10}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+\beta _{2}q^{4}+\cdots\)
722.2.e.o	$722$	$2$	722.e	19.e	$9$	$12$	$2$	$5.765$	12.0.\(\cdots\).3	None					38.2.c.b	$6$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q-\beta _{10}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+\beta _{2}q^{4}+\cdots\)
722.2.e.p	$722$	$2$	722.e	19.e	$9$	$12$	$2$	$5.765$	12.0.\(\cdots\).1	None		✓			722.2.a.h	$6$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q-\beta _{9}q^{2}-2\beta _{7}q^{3}-\beta _{11}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
722.2.e.q	$722$	$2$	722.e	19.e	$9$	$12$	$2$	$5.765$	12.0.\(\cdots\).1	None		✓			722.2.a.h	$6$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\beta _{3}-\beta _{9})q^{2}-2\beta _{1}q^{3}+(-\beta _{5}+\beta _{11})q^{4}+\cdots\)
722.2.e.r	$722$	$2$	722.e	19.e	$9$	$24$	$4$	$5.765$		None		✓			722.2.a.m	$6$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$\mathrm{SU}(2)[C_{9}]$
722.2.e.s	$722$	$2$	722.e	19.e	$9$	$24$	$4$	$5.765$		None		✓			722.2.a.m	$6$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$\mathrm{SU}(2)[C_{9}]$

Decomposition of \(S_{2}^{\mathrm{old}}(722, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(722, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(361, [\chi])\)\(^{\oplus 2}\)