Space of modular forms of level 728, weight 2, and trivial character

• Label: 728.2.a
• Level: $728$
• Weight: $2$
• Character orbit: 728.a
• Rep. character: $\chi_{728}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $9$
• Sturm bound: $224$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	728.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(224\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(728))\).

	Total	New	Old
Modular forms	120	18	102
Cusp forms	105	18	87
Eisenstein series	15	0	15

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

\(2\)	\(7\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(11\)	\(2\)	\(9\)		\(10\)	\(2\)	\(8\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(17\)	\(2\)	\(15\)		\(15\)	\(2\)	\(13\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(16\)	\(2\)	\(14\)		\(14\)	\(2\)	\(12\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(16\)	\(2\)	\(14\)		\(14\)	\(2\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(19\)	\(4\)	\(15\)		\(17\)	\(4\)	\(13\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(11\)	\(1\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(14\)	\(1\)	\(13\)		\(12\)	\(1\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(14\)	\(4\)	\(10\)		\(12\)	\(4\)	\(8\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(54\)	\(6\)	\(48\)		\(47\)	\(6\)	\(41\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(66\)	\(12\)	\(54\)		\(58\)	\(12\)	\(46\)		\(8\)	\(0\)	\(8\)

Trace form

18 * q - 4 * q^3 + 30 * q^9 + 8 * q^11 + 24 * q^15 + 12 * q^17 - 12 * q^19 - 4 * q^21 + 10 * q^25 + 8 * q^27 + 24 * q^33 - 12 * q^37 + 4 * q^41 - 12 * q^43 + 24 * q^45 + 8 * q^47 + 18 * q^49 - 32 * q^51 + 16 * q^53 - 16 * q^55 + 8 * q^57 - 36 * q^59 - 16 * q^61 - 8 * q^65 - 8 * q^67 + 8 * q^69 + 12 * q^73 - 4 * q^75 - 16 * q^79 + 50 * q^81 + 12 * q^83 + 8 * q^85 - 48 * q^87 - 4 * q^89 + 6 * q^91 + 24 * q^93 - 52 * q^95 + 20 * q^97 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(728))\) 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	13
728.2.a.a	$728$	$2$	728.a	1.a	$1$	$1$	$1$	$5.813$	\(\Q\)	None	✓	✓	✓	✓	728.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}-q^{13}+\cdots\)
728.2.a.b	$728$	$2$	728.a	1.a	$1$	$1$	$1$	$5.813$	\(\Q\)	None	✓	✓			728.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}+3q^{11}-q^{13}+\cdots\)
728.2.a.c	$728$	$2$	728.a	1.a	$1$	$1$	$1$	$5.813$	\(\Q\)	None	✓	✓	✓	✓	728.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-3q^{9}+2q^{11}+q^{13}+\cdots\)
728.2.a.d	$728$	$2$	728.a	1.a	$1$	$1$	$1$	$5.813$	\(\Q\)	None	✓	✓			728.2.a.d	$1$	$0$	\(0\)	\(2\)	\(3\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{5}+q^{7}+q^{9}-q^{13}+6q^{15}+\cdots\)
728.2.a.e	$728$	$2$	728.a	1.a	$1$	$2$	$2$	$5.813$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	728.2.a.e	$1$	$1$	\(0\)	\(-4\)	\(-2\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(-1+\beta )q^{5}+q^{7}+\cdots\)
728.2.a.f	$728$	$2$	728.a	1.a	$1$	$2$	$2$	$5.813$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	728.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-\beta q^{5}-q^{7}+(1-2\beta )q^{9}+\cdots\)
728.2.a.g	$728$	$2$	728.a	1.a	$1$	$2$	$2$	$5.813$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	728.2.a.g	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}-q^{7}+(1+\beta )q^{9}+\cdots\)
728.2.a.h	$728$	$2$	728.a	1.a	$1$	$4$	$4$	$5.813$	4.4.183064.1	None	✓	✓	✓	✓	728.2.a.h	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{5}+q^{7}+(2+\beta _{2})q^{9}+\cdots\)
728.2.a.i	$728$	$2$	728.a	1.a	$1$	$4$	$4$	$5.813$	4.4.64268.1	None	✓	✓	✓	✓	728.2.a.i	$1$	$0$	\(0\)	\(1\)	\(2\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{5}-q^{7}+(3+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(728))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(728)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(364))\)\(^{\oplus 2}\)