Space of modular forms of level 784, weight 2, and trivial character

• Label: 784.2.a
• Level: $784$
• Weight: $2$
• Character orbit: 784.a
• Rep. character: $\chi_{784}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $14$
• Sturm bound: $224$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(224\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(784))\).

	Total	New	Old
Modular forms	136	23	113
Cusp forms	89	18	71
Eisenstein series	47	5	42

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
\(+\)	\(+\)	\(+\)		\(32\)	\(4\)	\(28\)		\(21\)	\(4\)	\(17\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(36\)	\(6\)	\(30\)		\(24\)	\(6\)	\(18\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(36\)	\(7\)	\(29\)		\(24\)	\(5\)	\(19\)		\(12\)	\(2\)	\(10\)
\(-\)	\(-\)	\(+\)		\(32\)	\(6\)	\(26\)		\(20\)	\(3\)	\(17\)		\(12\)	\(3\)	\(9\)
Plus space		\(+\)		\(64\)	\(10\)	\(54\)		\(41\)	\(7\)	\(34\)		\(23\)	\(3\)	\(20\)
Minus space		\(-\)		\(72\)	\(13\)	\(59\)		\(48\)	\(11\)	\(37\)		\(24\)	\(2\)	\(22\)

Trace form

18 * q + 2 * q^5 + 16 * q^9 - 6 * q^11 + 2 * q^13 - 2 * q^15 + 2 * q^17 + 8 * q^19 + 6 * q^23 + 4 * q^25 - 4 * q^29 + 8 * q^31 + 10 * q^37 + 20 * q^39 - 6 * q^41 + 16 * q^43 + 10 * q^45 - 24 * q^47 + 14 * q^51 + 2 * q^53 - 8 * q^55 - 6 * q^57 - 6 * q^61 - 12 * q^65 - 30 * q^67 - 16 * q^69 + 8 * q^71 + 2 * q^73 + 32 * q^75 + 6 * q^79 + 2 * q^81 + 8 * q^83 - 10 * q^85 + 16 * q^87 + 2 * q^89 - 42 * q^93 + 66 * q^95 + 18 * q^97 + 20 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(784))\) 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
784.2.a.a	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				56.2.i.a	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}+6q^{9}+q^{11}+2q^{13}+\cdots\)
784.2.a.b	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				14.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+4q^{13}-6q^{17}+2q^{19}+\cdots\)
784.2.a.c	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				56.2.i.b	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{9}-3q^{11}+6q^{13}+\cdots\)
784.2.a.d	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				28.2.e.a	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
784.2.a.e	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				56.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-3q^{9}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
784.2.a.f	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}-4q^{11}-8q^{23}-5q^{25}+2q^{29}+\cdots\)
784.2.a.g	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				28.2.e.a	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
784.2.a.h	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				56.2.i.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{9}-3q^{11}-6q^{13}+\cdots\)
784.2.a.i	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				56.2.a.b	$1$	$0$	\(0\)	\(2\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{5}+q^{9}+8q^{15}+2q^{17}+\cdots\)
784.2.a.j	$784$	$2$	784.a	1.a	$1$	$1$	$1$	$6.260$	\(\Q\)	None	✓				56.2.i.a	$1$	$0$	\(0\)	\(3\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+6q^{9}+q^{11}-2q^{13}+\cdots\)
784.2.a.k	$784$	$2$	784.a	1.a	$1$	$2$	$2$	$6.260$	\(\Q(\sqrt{2}) \)	None	✓		✓		392.2.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{5}-q^{9}-6q^{11}+4\beta q^{13}+\cdots\)
784.2.a.l	$784$	$2$	784.a	1.a	$1$	$2$	$2$	$6.260$	\(\Q(\sqrt{2}) \)	None	✓				98.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}-q^{9}+2q^{11}+4q^{15}+\cdots\)
784.2.a.m	$784$	$2$	784.a	1.a	$1$	$2$	$2$	$6.260$	\(\Q(\sqrt{2}) \)	None	✓				196.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}+3\beta q^{13}+\cdots\)
784.2.a.n	$784$	$2$	784.a	1.a	$1$	$2$	$2$	$6.260$	\(\Q(\sqrt{2}) \)	None	✓		✓		392.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{5}+5q^{9}+4q^{11}+\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(784))\) into lower level spaces

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