Space of modular forms of level 726, weight 2, and trivial character

• Label: 726.2.a
• Level: $726$
• Weight: $2$
• Character orbit: 726.a
• Rep. character: $\chi_{726}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $13$
• Sturm bound: $264$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	726.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(264\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(726))\).

	Total	New	Old
Modular forms	156	17	139
Cusp forms	109	17	92
Eisenstein series	47	0	47

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

\(2\)	\(3\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(13\)

Trace form

17 * q - q^2 - q^3 + 17 * q^4 + 2 * q^5 + q^6 + 4 * q^7 - q^8 + 17 * q^9 + 2 * q^10 - q^12 + 6 * q^13 + 8 * q^14 + 6 * q^15 + 17 * q^16 + 6 * q^17 - q^18 + 2 * q^20 - 4 * q^21 + q^24 + 15 * q^25 + 2 * q^26 - q^27 + 4 * q^28 - 22 * q^29 + 6 * q^30 + 4 * q^31 - q^32 - 2 * q^34 + 17 * q^36 + 10 * q^37 - 4 * q^38 - 6 * q^39 + 2 * q^40 - 2 * q^41 + 4 * q^42 - 16 * q^43 + 2 * q^45 + 8 * q^46 + 32 * q^47 - q^48 + 29 * q^49 - 15 * q^50 + 10 * q^51 + 6 * q^52 + 2 * q^53 + q^54 + 8 * q^56 + 8 * q^57 - 2 * q^58 - 4 * q^59 + 6 * q^60 + 14 * q^61 + 16 * q^62 + 4 * q^63 + 17 * q^64 + 28 * q^65 + 28 * q^67 + 6 * q^68 + 8 * q^69 + 8 * q^70 + 8 * q^71 - q^72 + 10 * q^73 - 14 * q^74 + q^75 - 14 * q^78 - 20 * q^79 + 2 * q^80 + 17 * q^81 + 22 * q^82 + 4 * q^83 - 4 * q^84 - 12 * q^85 + 4 * q^86 - 10 * q^87 - 2 * q^89 + 2 * q^90 - 72 * q^91 - 72 * q^93 + 8 * q^94 - 8 * q^95 + q^96 - 62 * q^97 - 9 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(726))\) 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	3	11
726.2.a.a	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓	✓			726.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(-1\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
726.2.a.b	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓	✓	✓	✓	726.2.a.b	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
726.2.a.c	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓				66.2.a.b	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+4q^{7}+\cdots\)
726.2.a.d	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓				66.2.a.c	$1$	$1$	\(-1\)	\(1\)	\(-4\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-4q^{5}-q^{6}+2q^{7}+\cdots\)
726.2.a.e	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓	✓			726.2.a.e	$1$	$1$	\(-1\)	\(1\)	\(-1\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)
726.2.a.f	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓	✓	✓	✓	726.2.a.a	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)
726.2.a.g	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓	✓			726.2.a.b	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
726.2.a.h	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓	✓			726.2.a.e	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
726.2.a.i	$726$	$2$	726.a	1.a	$1$	$1$	$1$	$5.797$	\(\Q\)	None	✓				66.2.a.a	$1$	$0$	\(1\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
726.2.a.j	$726$	$2$	726.a	1.a	$1$	$2$	$2$	$5.797$	\(\Q(\sqrt{5}) \)	None	✓		✓		66.2.e.a	$1$	$0$	\(-2\)	\(-2\)	\(-1\)	\(-7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(1-3\beta )q^{5}+q^{6}+\cdots\)
726.2.a.k	$726$	$2$	726.a	1.a	$1$	$2$	$2$	$5.797$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	66.2.e.b	$1$	$0$	\(-2\)	\(2\)	\(5\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+(3-\beta )q^{5}-q^{6}+\cdots\)
726.2.a.l	$726$	$2$	726.a	1.a	$1$	$2$	$2$	$5.797$	\(\Q(\sqrt{5}) \)	None	✓		✓		66.2.e.a	$1$	$0$	\(2\)	\(-2\)	\(-1\)	\(7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(1-3\beta )q^{5}-q^{6}+\cdots\)
726.2.a.m	$726$	$2$	726.a	1.a	$1$	$2$	$2$	$5.797$	\(\Q(\sqrt{5}) \)	None	✓		✓		66.2.e.b	$1$	$0$	\(2\)	\(2\)	\(5\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(3-\beta )q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(726))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(726)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 2}\)