Space of modular forms of level 784, weight 4, and trivial character

• Label: 784.4.a
• Level: $784$
• Weight: $4$
• Character orbit: 784.a
• Rep. character: $\chi_{784}(1,\cdot)$
• Character field: $\Q$
• Dimension: $59$
• Newform subspaces: $34$
• Sturm bound: $448$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(448\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(784))\).

	Total	New	Old
Modular forms	360	64	296
Cusp forms	312	59	253
Eisenstein series	48	5	43

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

\(2\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(92\)	\(16\)	\(76\)		\(80\)	\(16\)	\(64\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(88\)	\(15\)	\(73\)		\(76\)	\(15\)	\(61\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(88\)	\(15\)	\(73\)		\(76\)	\(13\)	\(63\)		\(12\)	\(2\)	\(10\)
\(-\)	\(-\)	\(+\)		\(92\)	\(18\)	\(74\)		\(80\)	\(15\)	\(65\)		\(12\)	\(3\)	\(9\)
Plus space		\(+\)		\(184\)	\(34\)	\(150\)		\(160\)	\(31\)	\(129\)		\(24\)	\(3\)	\(21\)
Minus space		\(-\)		\(176\)	\(30\)	\(146\)		\(152\)	\(28\)	\(124\)		\(24\)	\(2\)	\(22\)

Trace form

59 * q - 4 * q^3 + 477 * q^9 + 22 * q^11 + 24 * q^13 - 58 * q^15 + 52 * q^17 + 20 * q^19 - 194 * q^23 + 1299 * q^25 - 424 * q^27 - 62 * q^29 + 104 * q^31 + 208 * q^33 - 168 * q^37 - 100 * q^39 + 100 * q^41 - 116 * q^43 - 112 * q^45 - 264 * q^47 + 1318 * q^51 + 528 * q^53 + 1168 * q^55 + 554 * q^57 + 292 * q^59 + 752 * q^61 + 392 * q^65 + 662 * q^67 + 304 * q^69 + 400 * q^71 - 612 * q^73 - 2236 * q^75 - 810 * q^79 + 2227 * q^81 - 876 * q^83 + 1442 * q^85 - 776 * q^87 - 564 * q^89 + 1334 * q^93 - 1478 * q^95 - 428 * q^97 - 384 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\) 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	7
784.4.a.a	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				28.4.a.a	$1$	$0$	\(0\)	\(-10\)	\(8\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}+8q^{5}+73q^{9}+40q^{11}+\cdots\)
784.4.a.b	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				7.4.c.a	$1$	$1$	\(0\)	\(-7\)	\(7\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+7q^{5}+22q^{9}+5q^{11}-14q^{13}+\cdots\)
784.4.a.c	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				14.4.c.a	$1$	$0$	\(0\)	\(-5\)	\(9\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+9q^{5}-2q^{9}+57q^{11}+70q^{13}+\cdots\)
784.4.a.d	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				392.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(-12\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-12q^{5}-11q^{9}-12q^{11}+\cdots\)
784.4.a.e	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				8.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+2q^{5}-11q^{9}+44q^{11}-22q^{13}+\cdots\)
784.4.a.f	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				196.4.a.a	$1$	$0$	\(0\)	\(-4\)	\(20\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+20q^{5}-11q^{9}-44q^{11}+\cdots\)
784.4.a.g	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				7.4.a.a	$1$	$0$	\(0\)	\(-2\)	\(-16\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2^{4}q^{5}-23q^{9}+8q^{11}-28q^{13}+\cdots\)
784.4.a.h	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				14.4.a.b	$1$	$0$	\(0\)	\(-2\)	\(12\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+12q^{5}-23q^{9}-48q^{11}+\cdots\)
784.4.a.i	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				56.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(16\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2^{4}q^{5}-23q^{9}-24q^{11}+\cdots\)
784.4.a.j	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				14.4.c.b	$1$	$0$	\(0\)	\(-1\)	\(-7\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-7q^{5}-26q^{9}-35q^{11}-66q^{13}+\cdots\)
784.4.a.k	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3^{3}q^{9}+68q^{11}+40q^{23}-5^{3}q^{25}+\cdots\)
784.4.a.l	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				14.4.c.b	$1$	$1$	\(0\)	\(1\)	\(7\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+7q^{5}-26q^{9}-35q^{11}+66q^{13}+\cdots\)
784.4.a.m	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				196.4.a.a	$1$	$0$	\(0\)	\(4\)	\(-20\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-20q^{5}-11q^{9}-44q^{11}+\cdots\)
784.4.a.n	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				28.4.a.b	$1$	$0$	\(0\)	\(4\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-6q^{5}-11q^{9}+12q^{11}+82q^{13}+\cdots\)
784.4.a.o	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				392.4.a.b	$1$	$1$	\(0\)	\(4\)	\(12\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+12q^{5}-11q^{9}-12q^{11}+\cdots\)
784.4.a.p	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				14.4.c.a	$1$	$1$	\(0\)	\(5\)	\(-9\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-9q^{5}-2q^{9}+57q^{11}-70q^{13}+\cdots\)
784.4.a.q	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				56.4.a.b	$1$	$1$	\(0\)	\(6\)	\(-8\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}-8q^{5}+9q^{9}-56q^{11}+28q^{13}+\cdots\)
784.4.a.r	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				7.4.c.a	$1$	$0$	\(0\)	\(7\)	\(-7\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-7q^{5}+22q^{9}+5q^{11}+14q^{13}+\cdots\)
784.4.a.s	$784$	$4$	784.a	1.a	$1$	$1$	$1$	$46.257$	\(\Q\)	None	✓				14.4.a.a	$1$	$0$	\(0\)	\(8\)	\(14\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+14q^{5}+37q^{9}+28q^{11}+\cdots\)
784.4.a.t	$784$	$4$	784.a	1.a	$1$	$2$	$2$	$46.257$	\(\Q(\sqrt{57}) \)	None	✓				56.4.a.c	$1$	$1$	\(0\)	\(-2\)	\(-22\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-11-\beta )q^{5}+(31+\cdots)q^{9}+\cdots\)
784.4.a.u	$784$	$4$	784.a	1.a	$1$	$2$	$2$	$46.257$	\(\Q(\sqrt{37}) \)	None	✓				28.4.e.a	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-7-2\beta )q^{5}+10q^{9}+(-2^{4}+\cdots)q^{11}+\cdots\)
784.4.a.v	$784$	$4$	784.a	1.a	$1$	$2$	$2$	$46.257$	\(\Q(\sqrt{2}) \)	None	✓				392.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+10\beta q^{5}-5^{2}q^{9}-54q^{11}+\cdots\)
784.4.a.w	$784$	$4$	784.a	1.a	$1$	$2$	$2$	$46.257$	\(\Q(\sqrt{2}) \)	None	✓				196.4.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}-5^{2}q^{9}+26q^{11}+\cdots\)
784.4.a.x	$784$	$4$	784.a	1.a	$1$	$2$	$2$	$46.257$	\(\Q(\sqrt{2}) \)	None	✓				392.4.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{5}+5q^{9}+4q^{11}+\beta q^{13}+\cdots\)
784.4.a.y	$784$	$4$	784.a	1.a	$1$	$2$	$2$	$46.257$	\(\Q(\sqrt{2}) \)	None	✓				98.4.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5\beta q^{3}-14\beta q^{5}+23q^{9}+14q^{11}+\cdots\)
784.4.a.z	$784$	$4$	784.a	1.a	$1$	$2$	$2$	$46.257$	\(\Q(\sqrt{22}) \)	None	✓				98.4.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}+61q^{9}-20q^{11}+7\beta q^{13}+\cdots\)
784.4.a.ba	$784$	$4$	784.a	1.a	$1$	$2$	$2$	$46.257$	\(\Q(\sqrt{37}) \)	None	✓				28.4.e.a	$1$	$0$	\(0\)	\(0\)	\(14\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(7-2\beta )q^{5}+10q^{9}+(-2^{4}+\cdots)q^{11}+\cdots\)
784.4.a.bb	$784$	$4$	784.a	1.a	$1$	$3$	$3$	$46.257$	3.3.1929.1	None	✓				56.4.i.b	$1$	$1$	\(0\)	\(-7\)	\(3\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
784.4.a.bc	$784$	$4$	784.a	1.a	$1$	$3$	$3$	$46.257$	3.3.1929.1	None	✓				56.4.i.a	$1$	$0$	\(0\)	\(-1\)	\(13\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(4+\beta _{1})q^{5}+(15+3\beta _{1}-2\beta _{2})q^{9}+\cdots\)
784.4.a.bd	$784$	$4$	784.a	1.a	$1$	$3$	$3$	$46.257$	3.3.1929.1	None	✓				56.4.i.a	$1$	$1$	\(0\)	\(1\)	\(-13\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-4-\beta _{1})q^{5}+(15+3\beta _{1}+\cdots)q^{9}+\cdots\)
784.4.a.be	$784$	$4$	784.a	1.a	$1$	$3$	$3$	$46.257$	3.3.1929.1	None	✓				56.4.i.b	$1$	$0$	\(0\)	\(7\)	\(-3\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
784.4.a.bf	$784$	$4$	784.a	1.a	$1$	$4$	$4$	$46.257$	\(\Q(\sqrt{2}, \sqrt{65})\)	None	✓		✓		49.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2\cdot 7$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{3}q^{5}+(10-3\beta _{1})q^{9}+(-5^{2}+\cdots)q^{11}+\cdots\)
784.4.a.bg	$784$	$4$	784.a	1.a	$1$	$4$	$4$	$46.257$	\(\Q(\sqrt{2}, \sqrt{65})\)	None	✓				392.4.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2\beta _{1}+\beta _{3})q^{3}+(3\beta _{1}+\beta _{3})q^{5}+(10+\cdots)q^{9}+\cdots\)
784.4.a.bh	$784$	$4$	784.a	1.a	$1$	$4$	$4$	$46.257$	\(\Q(\sqrt{2}, \sqrt{113})\)	None	✓				392.4.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2\beta _{1}+\beta _{3})q^{3}+(9\beta _{1}-\beta _{3})q^{5}+(34+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(784))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(784)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)