Space of modular forms of level 5, weight 10, and trivial character

• Label: 5.10.a
• Level: $5$
• Weight: $10$
• Character orbit: 5.a
• Rep. character: $\chi_{5}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $5$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	5.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(5\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(5))\).

	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
\(+\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(3\)	\(2\)	\(1\)		\(2\)	\(2\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

3 * q - 18 * q^2 + 146 * q^3 + 596 * q^4 + 625 * q^5 - 4424 * q^6 + 5942 * q^7 - 12600 * q^8 - 4181 * q^9 - 1250 * q^10 - 22224 * q^11 + 227152 * q^12 - 914 * q^13 - 474288 * q^14 + 233750 * q^15 - 144112 * q^16 + 907662 * q^17 - 1008394 * q^18 - 1305260 * q^19 + 932500 * q^20 + 601116 * q^21 + 4083944 * q^22 - 1581774 * q^23 - 3754080 * q^24 + 1171875 * q^25 - 2375964 * q^26 + 313100 * q^27 + 3305504 * q^28 + 529410 * q^29 - 3905000 * q^30 - 1751884 * q^31 - 4067808 * q^32 + 717232 * q^33 + 19920572 * q^34 - 1588750 * q^35 + 14797508 * q^36 - 36053298 * q^37 + 3821400 * q^38 + 33625748 * q^39 - 17475000 * q^40 - 15901014 * q^41 - 61126176 * q^42 + 46492906 * q^43 - 5121168 * q^44 + 5745625 * q^45 + 48651696 * q^46 + 22853022 * q^47 - 38239424 * q^48 - 9204929 * q^49 - 7031250 * q^50 - 54632204 * q^51 + 139262232 * q^52 + 703446 * q^53 - 72617360 * q^54 + 43870000 * q^55 - 10570560 * q^56 + 87911000 * q^57 - 231081500 * q^58 - 140181180 * q^59 + 78130000 * q^60 - 228831074 * q^61 + 331039104 * q^62 + 198326886 * q^63 - 213219264 * q^64 + 144346250 * q^65 + 430656992 * q^66 - 45604738 * q^67 - 265631256 * q^68 - 153977772 * q^69 - 254010000 * q^70 - 197098404 * q^71 - 139728600 * q^72 + 533029126 * q^73 + 639347892 * q^74 + 57031250 * q^75 + 354443280 * q^76 - 996146736 * q^77 - 794572208 * q^78 + 101918360 * q^79 - 299990000 * q^80 - 486443657 * q^81 - 268068116 * q^82 + 1664055066 * q^83 + 1378576512 * q^84 - 51263750 * q^85 - 161213784 * q^86 - 523405300 * q^87 - 368023200 * q^88 + 810150030 * q^89 - 697116250 * q^90 + 189838876 * q^91 - 700560288 * q^92 - 31047288 * q^93 - 1063675648 * q^94 + 445137500 * q^95 + 1404784256 * q^96 + 618891222 * q^97 - 621046626 * q^98 - 1654739152 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\) 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
5.10.a.a	$5$	$10$	5.a	1.a	$1$	$1$	$1$	$2.575$	\(\Q\)	None	✓	✓	✓	✓	5.10.a.a	$1$	$1$	\(-8\)	\(-114\)	\(-625\)	\(4242\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-114q^{3}-448q^{4}-5^{4}q^{5}+\cdots\)
5.10.a.b	$5$	$10$	5.a	1.a	$1$	$2$	$2$	$2.575$	\(\Q(\sqrt{1009}) \)	None	✓	✓	✓	✓	5.10.a.b	$1$	$0$	\(-10\)	\(260\)	\(1250\)	\(1700\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{2}+(130+2\beta )q^{3}+(522+\cdots)q^{4}+\cdots\)