Space of modular forms of level 28, weight 4, and trivial character

• Label: 28.4.a
• Level: $28$
• Weight: $4$
• Character orbit: 28.a
• Rep. character: $\chi_{28}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $16$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(16\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(28))\).

	Total	New	Old
Modular forms	15	2	13
Cusp forms	9	2	7
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
\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(8\)	\(1\)	\(7\)		\(5\)	\(1\)	\(4\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(7\)	\(1\)	\(6\)		\(4\)	\(1\)	\(3\)		\(3\)	\(0\)	\(3\)

Trace form

2 * q - 6 * q^3 - 2 * q^5 + 62 * q^9 - 52 * q^11 - 94 * q^13 + 104 * q^15 - 88 * q^17 + 94 * q^19 + 98 * q^21 + 152 * q^23 - 150 * q^25 - 612 * q^27 + 184 * q^29 + 140 * q^31 + 352 * q^33 + 98 * q^35 + 136 * q^37 - 208 * q^39 - 240 * q^41 + 244 * q^43 - 650 * q^45 - 204 * q^47 + 98 * q^49 + 460 * q^51 + 108 * q^53 + 248 * q^55 + 12 * q^57 + 814 * q^59 - 466 * q^61 - 588 * q^63 - 396 * q^65 - 336 * q^67 + 1504 * q^69 - 1288 * q^71 - 708 * q^73 + 254 * q^75 + 196 * q^77 - 1144 * q^79 + 2318 * q^81 + 1046 * q^83 + 284 * q^85 + 1604 * q^87 - 1788 * q^89 - 490 * q^91 - 2968 * q^93 + 200 * q^95 + 1800 * q^97 - 2788 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(28))\) 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
28.4.a.a	$28$	$4$	28.a	1.a	$1$	$1$	$1$	$1.652$	\(\Q\)	None	✓	✓	✓	✓	28.4.a.a	$1$	$1$	\(0\)	\(-10\)	\(-8\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}-8q^{5}-7q^{7}+73q^{9}-40q^{11}+\cdots\)
28.4.a.b	$28$	$4$	28.a	1.a	$1$	$1$	$1$	$1.652$	\(\Q\)	None	✓	✓	✓	✓	28.4.a.b	$1$	$0$	\(0\)	\(4\)	\(6\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+6q^{5}+7q^{7}-11q^{9}-12q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(28))\) into lower level spaces

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