Space of modular forms of level 28, weight 10, and trivial character

• Label: 28.10.a
• Level: $28$
• Weight: $10$
• Character orbit: 28.a
• Rep. character: $\chi_{28}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $40$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(40\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(28))\).

	Total	New	Old
Modular forms	39	4	35
Cusp forms	33	4	29
Eisenstein series	6	0	6

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
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(2\)

Trace form

4 * q - 294 * q^3 + 3150 * q^5 + 7712 * q^9 + 11796 * q^11 - 62006 * q^13 - 135592 * q^15 - 78540 * q^17 + 566174 * q^19 - 369754 * q^21 - 241224 * q^23 + 5716992 * q^25 - 5935860 * q^27 - 4650852 * q^29 - 3677156 * q^31 + 13692784 * q^33 + 100842 * q^35 - 20278756 * q^37 + 35792768 * q^39 - 22213044 * q^41 - 23098756 * q^43 + 50202502 * q^45 + 34123908 * q^47 + 23059204 * q^49 - 163474196 * q^51 - 51210552 * q^53 + 30454648 * q^55 - 4482636 * q^57 + 254130030 * q^59 - 49895426 * q^61 + 88135908 * q^63 - 281225148 * q^65 + 9282168 * q^67 + 388112368 * q^69 - 232237320 * q^71 - 12708528 * q^73 - 87156706 * q^75 - 271264980 * q^77 - 381768008 * q^79 + 590930888 * q^81 + 676068582 * q^83 - 337114532 * q^85 - 250815628 * q^87 - 925489656 * q^89 - 738398738 * q^91 + 1218203672 * q^93 + 1975826616 * q^95 + 2994959604 * q^97 - 2672796508 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(28))\) 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
28.10.a.a	$28$	$10$	28.a	1.a	$1$	$2$	$2$	$14.421$	\(\Q(\sqrt{4561}) \)	None	✓	✓	✓	✓	28.10.a.a	$1$	$1$	\(0\)	\(-224\)	\(1596\)	\(4802\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-112-\beta )q^{3}+(798+9\beta )q^{5}+7^{4}q^{7}+\cdots\)
28.10.a.b	$28$	$10$	28.a	1.a	$1$	$2$	$2$	$14.421$	\(\Q(\sqrt{11209}) \)	None	✓	✓	✓	✓	28.10.a.b	$1$	$0$	\(0\)	\(-70\)	\(1554\)	\(-4802\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-35-\beta )q^{3}+(777-19\beta )q^{5}-7^{4}q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(28))\) into lower level spaces

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