Space of modular forms of level 56, weight 10, and trivial character

• Label: 56.10.a
• Level: $56$
• Weight: $10$
• Character orbit: 56.a
• Rep. character: $\chi_{56}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $4$
• Sturm bound: $80$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(80\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(56))\).

	Total	New	Old
Modular forms	76	14	62
Cusp forms	68	14	54
Eisenstein series	8	0	8

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	Dim
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(3\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(8\)

Trace form

14 * q + 146 * q^3 - 122 * q^5 + 146586 * q^9 - 138196 * q^11 + 223242 * q^13 - 268328 * q^15 + 67120 * q^17 - 393930 * q^19 + 388962 * q^21 + 1046216 * q^23 + 1979326 * q^25 + 5797628 * q^27 + 2181488 * q^29 + 1334572 * q^31 - 719712 * q^33 - 9003750 * q^35 + 22501632 * q^37 + 17008288 * q^39 + 15210552 * q^41 - 16288908 * q^43 + 37718910 * q^45 + 197234964 * q^47 + 80707214 * q^49 - 65701604 * q^51 + 89103044 * q^53 + 203189624 * q^55 - 160526532 * q^57 - 182192874 * q^59 - 352452250 * q^61 - 127915676 * q^63 + 671642356 * q^65 - 274357376 * q^67 - 578909648 * q^69 - 1773704 * q^71 - 161286956 * q^73 + 1102830118 * q^75 + 140612164 * q^77 - 739810712 * q^79 + 923328234 * q^81 + 86165534 * q^83 - 1625460148 * q^85 - 1616152540 * q^87 - 763078436 * q^89 - 239874306 * q^91 - 537246520 * q^93 - 1058290440 * q^95 - 251757456 * q^97 - 127984260 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(56))\) 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
56.10.a.a	$56$	$10$	56.a	1.a	$1$	$3$	$3$	$28.842$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	56.10.a.a	$1$	$1$	\(0\)	\(-92\)	\(274\)	\(-7203\)		$+$	$+$	$+$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-31-\beta _{1})q^{3}+(92+\beta _{1}-\beta _{2})q^{5}+\cdots\)
56.10.a.b	$56$	$10$	56.a	1.a	$1$	$3$	$3$	$28.842$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	56.10.a.b	$1$	$1$	\(0\)	\(84\)	\(-2958\)	\(7203\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(28-\beta _{1})q^{3}+(-986+5\beta _{1}-\beta _{2})q^{5}+\cdots\)
56.10.a.c	$56$	$10$	56.a	1.a	$1$	$4$	$4$	$28.842$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	56.10.a.c	$1$	$0$	\(0\)	\(70\)	\(1022\)	\(9604\)		$+$	$-$	$-$	$2^{11}\cdot 3$	$\mathrm{SU}(2)$	\(q+(17-\beta _{1})q^{3}+(255-\beta _{1}-\beta _{2})q^{5}+\cdots\)
56.10.a.d	$56$	$10$	56.a	1.a	$1$	$4$	$4$	$28.842$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	56.10.a.d	$1$	$0$	\(0\)	\(84\)	\(1540\)	\(-9604\)		$-$	$+$	$-$	$2^{13}\cdot 3^{3}\cdot 17$	$\mathrm{SU}(2)$	\(q+(21-\beta _{1})q^{3}+(385-2\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(56))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(56)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)