Space of modular forms of level 8, weight 22, and trivial character

• Label: 8.22.a
• Level: $8$
• Weight: $22$
• Character orbit: 8.a
• Rep. character: $\chi_{8}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $2$
• Sturm bound: $22$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(22\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(8))\).

	Total	New	Old
Modular forms	23	5	18
Cusp forms	19	5	14
Eisenstein series	4	0	4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	Dim
\(+\)	\(2\)
\(-\)	\(3\)

Trace form

5 * q - 8668 * q^3 - 22003634 * q^5 + 740760072 * q^7 + 29917044817 * q^9 + 13471294636 * q^11 - 356632250986 * q^13 - 1862971179880 * q^15 + 1203867602330 * q^17 + 55090597707316 * q^19 + 63483640028832 * q^21 - 215967104297192 * q^23 + 186686289431851 * q^25 + 3014062843435496 * q^27 + 3202555900321062 * q^29 + 2556850700943904 * q^31 - 6758392137812368 * q^33 + 7975251928887216 * q^35 - 26458703887352466 * q^37 + 86979314693434424 * q^39 - 229981070418310926 * q^41 + 231195720126249164 * q^43 - 859998337885344938 * q^45 + 305620128011018544 * q^47 - 1397370909615771011 * q^49 + 5114029246370682760 * q^51 - 4194423668594372354 * q^53 + 7292989855958396744 * q^55 - 10021125333234603248 * q^57 + 10009049496232751836 * q^59 - 13732181343142521274 * q^61 + 36014708061348485352 * q^63 - 23512551689317845212 * q^65 + 881069099479908964 * q^67 - 26426991957433856032 * q^69 - 31691319343619281336 * q^71 - 31008308284406859502 * q^73 - 26746902752653120100 * q^75 + 139184164190911890912 * q^77 - 217016023518521947312 * q^79 + 520915702457008912765 * q^81 - 317800628207406564748 * q^83 + 367067042098068889852 * q^85 - 384429083625446912136 * q^87 + 370148514466841267106 * q^89 - 887770236365977512720 * q^91 + 1575602859006642094720 * q^93 - 822320774641135950856 * q^95 + 459335324412563312746 * q^97 + 817207776598242539644 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(8))\) 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
8.22.a.a	$8$	$22$	8.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{358549}) \)	None	✓	✓	✓	✓	8.22.a.a	$1$	$1$	\(0\)	\(-105432\)	\(2108140\)	\(444771792\)		$+$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-52716-\beta )q^{3}+(1054070+20\beta )q^{5}+\cdots\)
8.22.a.b	$8$	$22$	8.a	1.a	$1$	$3$	$3$	$22.358$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	8.22.a.b	$1$	$0$	\(0\)	\(96764\)	\(-24111774\)	\(295988280\)		$-$	$-$	$2^{21}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(32255-\beta _{1})q^{3}+(-8037261+9\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(8))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(8)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)