Space of modular forms of level 722, weight 4, and trivial character

• Label: 722.4.a
• Level: $722$
• Weight: $4$
• Character orbit: 722.a
• Rep. character: $\chi_{722}(1,\cdot)$
• Character field: $\Q$
• Dimension: $85$
• Newform subspaces: $21$
• Sturm bound: $380$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(380\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(722))\).

	Total	New	Old
Modular forms	305	85	220
Cusp forms	265	85	180
Eisenstein series	40	0	40

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(19\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(80\)	\(22\)	\(58\)		\(70\)	\(22\)	\(48\)		\(10\)	\(0\)	\(10\)
\(+\)	\(-\)	\(-\)		\(73\)	\(20\)	\(53\)		\(63\)	\(20\)	\(43\)		\(10\)	\(0\)	\(10\)
\(-\)	\(+\)	\(-\)		\(75\)	\(18\)	\(57\)		\(65\)	\(18\)	\(47\)		\(10\)	\(0\)	\(10\)
\(-\)	\(-\)	\(+\)		\(77\)	\(25\)	\(52\)		\(67\)	\(25\)	\(42\)		\(10\)	\(0\)	\(10\)
Plus space		\(+\)		\(157\)	\(47\)	\(110\)		\(137\)	\(47\)	\(90\)		\(20\)	\(0\)	\(20\)
Minus space		\(-\)		\(148\)	\(38\)	\(110\)		\(128\)	\(38\)	\(90\)		\(20\)	\(0\)	\(20\)

Trace form

85 * q + 2 * q^2 - 8 * q^3 + 340 * q^4 + 16 * q^5 - 20 * q^6 - 12 * q^7 + 8 * q^8 + 775 * q^9 + 20 * q^10 - 78 * q^11 - 32 * q^12 + 22 * q^13 + 88 * q^14 + 304 * q^15 + 1360 * q^16 - 2 * q^17 - 22 * q^18 + 64 * q^20 + 48 * q^21 + 168 * q^22 + 168 * q^23 - 80 * q^24 + 1909 * q^25 - 112 * q^26 - 368 * q^27 - 48 * q^28 + 226 * q^29 - 8 * q^30 - 228 * q^31 + 32 * q^32 - 28 * q^33 + 196 * q^34 - 1136 * q^35 + 3100 * q^36 - 346 * q^37 + 168 * q^39 + 80 * q^40 + 1098 * q^41 + 968 * q^42 - 486 * q^43 - 312 * q^44 + 1244 * q^45 - 600 * q^46 - 168 * q^47 - 128 * q^48 + 2945 * q^49 - 18 * q^50 + 1924 * q^51 + 88 * q^52 + 1254 * q^53 - 80 * q^54 + 732 * q^55 + 352 * q^56 - 248 * q^58 + 416 * q^59 + 1216 * q^60 + 232 * q^61 - 392 * q^62 - 680 * q^63 + 5440 * q^64 - 272 * q^65 + 336 * q^66 + 572 * q^67 - 8 * q^68 - 2020 * q^69 - 264 * q^70 - 1248 * q^71 - 88 * q^72 - 458 * q^73 - 496 * q^74 + 84 * q^75 + 1020 * q^77 - 2120 * q^78 - 756 * q^79 + 256 * q^80 + 8069 * q^81 - 2068 * q^82 - 1298 * q^83 + 192 * q^84 - 3100 * q^85 - 896 * q^86 - 2232 * q^87 + 672 * q^88 + 1706 * q^89 + 2588 * q^90 + 688 * q^91 + 672 * q^92 + 2288 * q^93 + 1536 * q^94 - 320 * q^96 - 854 * q^97 + 3170 * q^98 - 434 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	19
722.4.a.a	$722$	$4$	722.a	1.a	$1$	$1$	$1$	$42.599$	\(\Q\)	None	✓				38.4.c.a	$1$	$1$	\(-2\)	\(-5\)	\(3\)	\(-32\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-5q^{3}+4q^{4}+3q^{5}+10q^{6}+\cdots\)
722.4.a.b	$722$	$4$	722.a	1.a	$1$	$1$	$1$	$42.599$	\(\Q\)	None	✓				38.4.c.b	$1$	$1$	\(-2\)	\(5\)	\(-12\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+5q^{3}+4q^{4}-12q^{5}-10q^{6}+\cdots\)
722.4.a.c	$722$	$4$	722.a	1.a	$1$	$1$	$1$	$42.599$	\(\Q\)	None	✓				38.4.c.b	$1$	$1$	\(2\)	\(-5\)	\(-12\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-5q^{3}+4q^{4}-12q^{5}-10q^{6}+\cdots\)
722.4.a.d	$722$	$4$	722.a	1.a	$1$	$1$	$1$	$42.599$	\(\Q\)	None	✓				38.4.a.a	$1$	$0$	\(2\)	\(2\)	\(-9\)	\(-31\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+4q^{4}-9q^{5}+4q^{6}+\cdots\)
722.4.a.e	$722$	$4$	722.a	1.a	$1$	$1$	$1$	$42.599$	\(\Q\)	None	✓				38.4.c.a	$1$	$1$	\(2\)	\(5\)	\(3\)	\(-32\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+5q^{3}+4q^{4}+3q^{5}+10q^{6}+\cdots\)
722.4.a.f	$722$	$4$	722.a	1.a	$1$	$2$	$2$	$42.599$	\(\Q(\sqrt{73}) \)	None	✓				38.4.a.c	$1$	$1$	\(-4\)	\(-9\)	\(-9\)	\(-18\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-4-\beta )q^{3}+4q^{4}+(-3+\cdots)q^{5}+\cdots\)
722.4.a.g	$722$	$4$	722.a	1.a	$1$	$2$	$2$	$42.599$	\(\Q(\sqrt{73}) \)	None	✓	✓			722.4.a.g	$1$	$0$	\(-4\)	\(5\)	\(13\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(3-\beta )q^{3}+4q^{4}+(8-3\beta )q^{5}+\cdots\)
722.4.a.h	$722$	$4$	722.a	1.a	$1$	$2$	$2$	$42.599$	\(\Q(\sqrt{73}) \)	None	✓	✓			722.4.a.g	$1$	$1$	\(4\)	\(-5\)	\(13\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-2-\beta )q^{3}+4q^{4}+(5+3\beta )q^{5}+\cdots\)
722.4.a.i	$722$	$4$	722.a	1.a	$1$	$2$	$2$	$42.599$	\(\Q(\sqrt{177}) \)	None	✓				38.4.a.b	$1$	$0$	\(4\)	\(-1\)	\(10\)	\(57\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-\beta q^{3}+4q^{4}+(6-2\beta )q^{5}+\cdots\)
722.4.a.j	$722$	$4$	722.a	1.a	$1$	$3$	$3$	$42.599$	3.3.253788.1	None	✓				38.4.c.c	$1$	$0$	\(-6\)	\(5\)	\(1\)	\(26\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
722.4.a.k	$722$	$4$	722.a	1.a	$1$	$3$	$3$	$42.599$	3.3.253788.1	None	✓				38.4.c.c	$1$	$0$	\(6\)	\(-5\)	\(1\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
722.4.a.l	$722$	$4$	722.a	1.a	$1$	$4$	$4$	$42.599$	\(\Q(\sqrt{462 -38 \sqrt{5}})\)	None	✓	✓			722.4.a.l	$1$	$1$	\(-8\)	\(-3\)	\(27\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(8+2\beta _{2}+\cdots)q^{5}+\cdots\)
722.4.a.m	$722$	$4$	722.a	1.a	$1$	$4$	$4$	$42.599$	\(\Q(\sqrt{462 -38 \sqrt{5}})\)	None	✓	✓			722.4.a.l	$1$	$0$	\(8\)	\(3\)	\(27\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(8+2\beta _{2}+\cdots)q^{5}+\cdots\)
722.4.a.n	$722$	$4$	722.a	1.a	$1$	$6$	$6$	$42.599$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			722.4.a.n	$1$	$1$	\(-12\)	\(-7\)	\(22\)	\(-18\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-2-\beta _{1}-\beta _{2})q^{3}+4q^{4}+\cdots\)
722.4.a.o	$722$	$4$	722.a	1.a	$1$	$6$	$6$	$42.599$	6.6.6719782761.1	None	✓				38.4.e.a	$1$	$1$	\(-12\)	\(9\)	\(-27\)	\(-21\)		$+$	$-$	$-$	$19$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(-4-\beta _{3}+\cdots)q^{5}+\cdots\)
722.4.a.p	$722$	$4$	722.a	1.a	$1$	$6$	$6$	$42.599$	6.6.6719782761.1	None	✓				38.4.e.a	$1$	$1$	\(12\)	\(-9\)	\(-27\)	\(-21\)		$-$	$+$	$-$	$19$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-1+\beta _{3})q^{3}+4q^{4}+(-5+\cdots)q^{5}+\cdots\)
722.4.a.q	$722$	$4$	722.a	1.a	$1$	$6$	$6$	$42.599$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			722.4.a.n	$1$	$0$	\(12\)	\(7\)	\(22\)	\(-18\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+(2+\beta _{1}+\beta _{2})q^{3}+4q^{4}+(4+\cdots)q^{5}+\cdots\)
722.4.a.r	$722$	$4$	722.a	1.a	$1$	$8$	$8$	$42.599$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			722.4.a.r	$1$	$0$	\(-16\)	\(10\)	\(-18\)	\(-14\)		$+$	$+$	$+$	$19$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1-\beta _{3})q^{3}+4q^{4}+(-1+\beta _{2}+\cdots)q^{5}+\cdots\)
722.4.a.s	$722$	$4$	722.a	1.a	$1$	$8$	$8$	$42.599$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		722.4.a.r	$1$	$1$	\(16\)	\(-10\)	\(-18\)	\(-14\)		$-$	$+$	$-$	$19$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-1+\beta _{3})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots\)
722.4.a.t	$722$	$4$	722.a	1.a	$1$	$9$	$9$	$42.599$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		38.4.e.b	$1$	$0$	\(-18\)	\(-9\)	\(3\)	\(33\)		$+$	$+$	$+$	$3\cdot 19^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+\beta _{3})q^{3}+4q^{4}-\beta _{6}q^{5}+\cdots\)
722.4.a.u	$722$	$4$	722.a	1.a	$1$	$9$	$9$	$42.599$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		38.4.e.b	$1$	$0$	\(18\)	\(9\)	\(3\)	\(33\)		$-$	$-$	$+$	$3\cdot 19^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1-\beta _{3})q^{3}+4q^{4}-\beta _{6}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(722))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(722)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 2}\)