Space of modular forms of level 28, weight 6, and trivial character

• Label: 28.6.a
• Level: $28$
• Weight: $6$
• Character orbit: 28.a
• Rep. character: $\chi_{28}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $24$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$28 = 2^{2} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	28.a (trivial)
Character field:			$\Q$
Newform subspaces:			$2$
Sturm bound:			$24$
Trace bound:			$3$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{6}(\Gamma_0(28))$.

	Total	New	Old
Modular forms	23	2	21
Cusp forms	17	2	15
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
$+$	$+$	$+$		$5$	$0$	$5$		$3$	$0$	$3$		$2$	$0$	$2$
$+$	$-$	$-$		$7$	$0$	$7$		$5$	$0$	$5$		$2$	$0$	$2$
$-$	$+$	$-$		$5$	$1$	$4$		$4$	$1$	$3$		$1$	$0$	$1$
$-$	$-$	$+$		$6$	$1$	$5$		$5$	$1$	$4$		$1$	$0$	$1$
Plus space		$+$		$11$	$1$	$10$		$8$	$1$	$7$		$3$	$0$	$3$
Minus space		$-$		$12$	$1$	$11$		$9$	$1$	$8$		$3$	$0$	$3$

Trace form

2 * q + 24 * q^3 - 80 * q^5 + 194 * q^9 - 712 * q^11 + 1256 * q^13 + 608 * q^15 - 964 * q^17 - 2792 * q^19 - 1372 * q^21 + 3440 * q^23 + 3222 * q^25 + 5904 * q^27 - 7628 * q^29 - 1456 * q^31 + 1648 * q^33 - 5488 * q^35 - 22268 * q^37 + 16640 * q^39 + 25356 * q^41 - 10088 * q^43 + 29872 * q^45 - 24624 * q^47 + 4802 * q^49 - 60176 * q^51 + 14124 * q^53 + 69248 * q^55 - 69960 * q^57 + 35944 * q^59 + 6560 * q^61 - 32928 * q^63 - 43968 * q^65 + 51240 * q^67 + 86752 * q^69 + 25088 * q^71 - 32364 * q^73 - 86776 * q^75 - 35672 * q^77 + 17648 * q^79 + 79370 * q^81 + 63704 * q^83 - 155872 * q^85 - 75856 * q^87 - 25404 * q^89 - 5488 * q^91 + 136976 * q^93 - 34144 * q^95 + 124284 * q^97 + 175544 * q^99

Decomposition of $S_{6}^{\mathrm{new}}(\Gamma_0(28))$ 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
28.6.a.a	$28$	$6$	28.a	1.a	$1$	$1$	$1$	$4.491$	$\Q$	None	✓	✓	✓	✓	28.6.a.a	$1$	$1$	$0$	$-2$	$-96$	$49$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}-96q^{5}+7^{2}q^{7}-239q^{9}+\cdots$
28.6.a.b	$28$	$6$	28.a	1.a	$1$	$1$	$1$	$4.491$	$\Q$	None	✓	✓	✓	✓	28.6.a.b	$1$	$0$	$0$	$26$	$16$	$-49$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+26q^{3}+2^{4}q^{5}-7^{2}q^{7}+433q^{9}+\cdots$

Decomposition of $S_{6}^{\mathrm{old}}(\Gamma_0(28))$ into lower level spaces

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