Space of modular forms of level 5, weight 8, and trivial character

• Label: 5.8.a
• Level: $5$
• Weight: $8$
• Character orbit: 5.a
• Rep. character: $\chi_{5}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $4$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	5.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(4\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(5))\).

	Total	New	Old
Modular forms	5	3	2
Cusp forms	3	3	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.

\(5\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(3\)	\(2\)	\(1\)		\(2\)	\(2\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

3 * q + 6 * q^2 - 28 * q^3 + 164 * q^4 - 125 * q^5 - 344 * q^6 - 1744 * q^7 + 2280 * q^8 + 5671 * q^9 - 4250 * q^10 + 4716 * q^11 - 26624 * q^12 + 7402 * q^13 + 30528 * q^14 - 8500 * q^15 - 112 * q^16 - 39594 * q^17 + 29582 * q^18 + 12820 * q^19 - 3500 * q^20 + 9816 * q^21 + 103832 * q^22 - 111168 * q^23 - 171840 * q^24 + 46875 * q^25 - 111084 * q^26 + 305720 * q^27 + 53648 * q^28 - 153870 * q^29 + 211000 * q^30 + 149936 * q^31 + 100896 * q^32 - 449216 * q^33 - 281812 * q^34 - 193000 * q^35 - 211852 * q^36 - 12894 * q^37 + 702120 * q^38 + 589352 * q^39 - 75000 * q^40 + 732246 * q^41 - 1649808 * q^42 - 75908 * q^43 + 1445808 * q^44 - 679625 * q^45 - 1485024 * q^46 - 502344 * q^47 + 1964032 * q^48 + 713779 * q^49 + 93750 * q^50 + 1754536 * q^51 - 1415784 * q^52 - 3672078 * q^53 - 288080 * q^54 - 546500 * q^55 - 431520 * q^56 + 2021840 * q^57 + 3386980 * q^58 - 2457540 * q^59 + 2512000 * q^60 + 1333066 * q^61 + 4839072 * q^62 - 1831968 * q^63 - 5032896 * q^64 + 40250 * q^65 - 6448768 * q^66 + 3719156 * q^67 - 5744952 * q^68 - 1367928 * q^69 + 1938000 * q^70 + 5725176 * q^71 + 1178760 * q^72 + 301582 * q^73 + 1109508 * q^74 - 437500 * q^75 - 876240 * q^76 + 2894832 * q^77 + 7266064 * q^78 + 4332080 * q^79 - 5102000 * q^80 - 7423037 * q^81 - 12540308 * q^82 - 3652188 * q^83 + 4967808 * q^84 + 1885750 * q^85 + 17787096 * q^86 - 20541640 * q^87 + 10712160 * q^88 + 8036190 * q^89 - 4107250 * q^90 - 11701424 * q^91 - 6316944 * q^92 + 14003664 * q^93 + 11417168 * q^94 - 8087500 * q^95 + 4714496 * q^96 - 272394 * q^97 - 38814042 * q^98 + 2910812 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(5))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5
5.8.a.a	$5$	$8$	5.a	1.a	$1$	$1$	$1$	$1.562$	\(\Q\)	None	✓	✓	✓	✓	5.8.a.a	$1$	$1$	\(-14\)	\(-48\)	\(125\)	\(-1644\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{2}-48q^{3}+68q^{4}+5^{3}q^{5}+\cdots\)
5.8.a.b	$5$	$8$	5.a	1.a	$1$	$2$	$2$	$1.562$	\(\Q(\sqrt{19}) \)	None	✓	✓	✓	✓	5.8.a.b	$1$	$0$	\(20\)	\(20\)	\(-250\)	\(-100\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(10+\beta )q^{2}+(10-8\beta )q^{3}+(48+20\beta )q^{4}+\cdots\)