Space of modular forms of level 28, weight 8, and trivial character

• Label: 28.8.a
• Level: $28$
• Weight: $8$
• Character orbit: 28.a
• Rep. character: $\chi_{28}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $32$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(32\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(28))\).

	Total	New	Old
Modular forms	31	4	27
Cusp forms	25	4	21
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(8\)	\(2\)	\(6\)		\(7\)	\(2\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(16\)	\(2\)	\(14\)		\(13\)	\(2\)	\(11\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(15\)	\(2\)	\(13\)		\(12\)	\(2\)	\(10\)		\(3\)	\(0\)	\(3\)

Trace form

4 * q - 252 * q^5 + 524 * q^9 + 3936 * q^11 - 4340 * q^13 - 3376 * q^15 - 13440 * q^17 + 23408 * q^19 - 9604 * q^21 + 23280 * q^23 + 39300 * q^25 - 105840 * q^27 + 26688 * q^29 - 15680 * q^31 - 5600 * q^33 + 115248 * q^35 + 240320 * q^37 - 538096 * q^39 - 438480 * q^41 + 729440 * q^43 - 216860 * q^45 + 651840 * q^47 + 470596 * q^49 + 1158400 * q^51 - 1807080 * q^53 - 4260032 * q^55 + 2409480 * q^57 - 1027152 * q^59 + 1919428 * q^61 + 1728720 * q^63 + 7277784 * q^65 - 2365440 * q^67 - 12886832 * q^69 + 5659584 * q^71 - 6865320 * q^73 + 8976464 * q^75 + 3745560 * q^77 + 10421632 * q^79 - 12079924 * q^81 - 28388640 * q^83 + 19841704 * q^85 - 4449760 * q^87 + 10770984 * q^89 + 9604000 * q^91 + 28469840 * q^93 - 25102320 * q^95 - 44079840 * q^97 + 23566784 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(28))\) 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
28.8.a.a	$28$	$8$	28.a	1.a	$1$	$2$	$2$	$8.747$	\(\Q(\sqrt{3529}) \)	None	✓	✓	✓	✓	28.8.a.a	$1$	$0$	\(0\)	\(-14\)	\(42\)	\(686\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(21-3\beta )q^{5}+7^{3}q^{7}+\cdots\)
28.8.a.b	$28$	$8$	28.a	1.a	$1$	$2$	$2$	$8.747$	\(\Q(\sqrt{1009}) \)	None	✓	✓	✓	✓	28.8.a.b	$1$	$1$	\(0\)	\(14\)	\(-294\)	\(-686\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(-147+11\beta )q^{5}-7^{3}q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(28))\) into lower level spaces

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