Space of modular forms of level 2, weight 8, and trivial character

• Label: 2.8.a
• Level: $2$
• Weight: $8$
• Character orbit: 2.a
• Rep. character: $\chi_{2}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $2$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 2 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	2.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(2\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(2))\).

	Total	New	Old
Modular forms	3	1	2
Cusp forms	1	1	0
Eisenstein series	2	0	2

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

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

q - 8 * q^2 + 12 * q^3 + 64 * q^4 - 210 * q^5 - 96 * q^6 + 1016 * q^7 - 512 * q^8 - 2043 * q^9 + 1680 * q^10 + 1092 * q^11 + 768 * q^12 + 1382 * q^13 - 8128 * q^14 - 2520 * q^15 + 4096 * q^16 + 14706 * q^17 + 16344 * q^18 - 39940 * q^19 - 13440 * q^20 + 12192 * q^21 - 8736 * q^22 + 68712 * q^23 - 6144 * q^24 - 34025 * q^25 - 11056 * q^26 - 50760 * q^27 + 65024 * q^28 - 102570 * q^29 + 20160 * q^30 + 227552 * q^31 - 32768 * q^32 + 13104 * q^33 - 117648 * q^34 - 213360 * q^35 - 130752 * q^36 + 160526 * q^37 + 319520 * q^38 + 16584 * q^39 + 107520 * q^40 + 10842 * q^41 - 97536 * q^42 - 630748 * q^43 + 69888 * q^44 + 429030 * q^45 - 549696 * q^46 + 472656 * q^47 + 49152 * q^48 + 208713 * q^49 + 272200 * q^50 + 176472 * q^51 + 88448 * q^52 - 1494018 * q^53 + 406080 * q^54 - 229320 * q^55 - 520192 * q^56 - 479280 * q^57 + 820560 * q^58 + 2640660 * q^59 - 161280 * q^60 + 827702 * q^61 - 1820416 * q^62 - 2075688 * q^63 + 262144 * q^64 - 290220 * q^65 - 104832 * q^66 - 126004 * q^67 + 941184 * q^68 + 824544 * q^69 + 1706880 * q^70 - 1414728 * q^71 + 1046016 * q^72 + 980282 * q^73 - 1284208 * q^74 - 408300 * q^75 - 2556160 * q^76 + 1109472 * q^77 - 132672 * q^78 - 3566800 * q^79 - 860160 * q^80 + 3858921 * q^81 - 86736 * q^82 + 5672892 * q^83 + 780288 * q^84 - 3088260 * q^85 + 5045984 * q^86 - 1230840 * q^87 - 559104 * q^88 - 11951190 * q^89 - 3432240 * q^90 + 1404112 * q^91 + 4397568 * q^92 + 2730624 * q^93 - 3781248 * q^94 + 8387400 * q^95 - 393216 * q^96 + 8682146 * q^97 - 1669704 * q^98 - 2230956 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\) 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
2.8.a.a	$2$	$8$	2.a	1.a	$1$	$1$	$1$	$0.625$	\(\Q\)	None	✓	✓	✓	✓	2.8.a.a	$1$	$0$	\(-8\)	\(12\)	\(-210\)	\(1016\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+12q^{3}+2^{6}q^{4}-210q^{5}+\cdots\)