Space of modular forms of level 10, weight 18, and trivial character

• Label: 10.18.a
• Level: $10$
• Weight: $18$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $4$
• Sturm bound: $27$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(27\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	27	7	20
Cusp forms	23	7	16
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.

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(2\)	\(4\)		\(5\)	\(2\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(13\)	\(3\)	\(10\)		\(11\)	\(3\)	\(8\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(14\)	\(4\)	\(10\)		\(12\)	\(4\)	\(8\)		\(2\)	\(0\)	\(2\)

Trace form

7 * q - 256 * q^2 - 4964 * q^3 + 458752 * q^4 - 390625 * q^5 + 2628608 * q^6 + 49640552 * q^7 - 16777216 * q^8 + 446713291 * q^9 - 100000000 * q^10 - 302591916 * q^11 - 325320704 * q^12 - 5259218734 * q^13 + 9048365056 * q^14 - 10782812500 * q^15 + 30064771072 * q^16 - 1140224898 * q^17 + 55132074752 * q^18 + 84274529420 * q^19 - 25600000000 * q^20 + 87503214224 * q^21 - 445042363392 * q^22 - 82347597864 * q^23 + 172268453888 * q^24 + 1068115234375 * q^25 + 476233773568 * q^26 + 3526468806520 * q^27 + 3253243215872 * q^28 + 107401405170 * q^29 - 3760400000000 * q^30 + 3844292097344 * q^31 - 1099511627776 * q^32 - 24080865991248 * q^33 - 7144639271424 * q^34 - 7349815625000 * q^35 + 29275802238976 * q^36 - 90600510737158 * q^37 + 25686817745920 * q^38 + 119903106480872 * q^39 - 6553600000000 * q^40 - 42297288282186 * q^41 + 9278596538368 * q^42 + 48708157195796 * q^43 - 19830663806976 * q^44 - 184168223828125 * q^45 - 21657337608192 * q^46 - 468908924842368 * q^47 - 21320217657344 * q^48 + 595144774851519 * q^49 - 39062500000000 * q^50 + 537128820158904 * q^51 - 344668158951424 * q^52 + 771768939695466 * q^53 + 2314789782886400 * q^54 - 1097828067187500 * q^55 + 592993652310016 * q^56 - 4894114504687120 * q^57 + 1130080958784000 * q^58 + 3305979545258340 * q^59 - 706662400000000 * q^60 - 1143358756633726 * q^61 - 6675522216353792 * q^62 + 7955317090432616 * q^63 + 1970324836974592 * q^64 - 2740256724218750 * q^65 - 989472716795904 * q^66 - 12767118471683908 * q^67 - 74725778915328 * q^68 + 18058617414221232 * q^69 - 693552800000000 * q^70 - 1493803906388136 * q^71 + 3613135650947072 * q^72 - 13995582151311034 * q^73 - 14043218385135104 * q^74 - 757446289062500 * q^75 + 5523015560069120 * q^76 - 12354227646083136 * q^77 + 4600556316166144 * q^78 + 20916431337421040 * q^79 - 1677721600000000 * q^80 + 17039696628198367 * q^81 + 3612164220016128 * q^82 + 22441018762155756 * q^83 + 5734610647384064 * q^84 + 12972901867968750 * q^85 - 6648223406492672 * q^86 + 142187591241889800 * q^87 - 29166296327258112 * q^88 - 83338386922064970 * q^89 + 18995934700000000 * q^90 - 123790374889787696 * q^91 - 5396732173615104 * q^92 - 202446187448100448 * q^93 + 5543183562141696 * q^94 + 105178423867187500 * q^95 + 11289785394003968 * q^96 + 363067859693827022 * q^97 - 198820889786956032 * q^98 - 442209053189154108 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\) 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	5
10.18.a.a	$10$	$18$	10.a	1.a	$1$	$1$	$1$	$18.322$	\(\Q\)	None	✓	✓	✓	✓	10.18.a.a	$1$	$1$	\(256\)	\(-14976\)	\(390625\)	\(14808668\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}-14976q^{3}+2^{16}q^{4}+5^{8}q^{5}+\cdots\)
10.18.a.b	$10$	$18$	10.a	1.a	$1$	$2$	$2$	$18.322$	\(\Q(\sqrt{36061}) \)	None	✓	✓	✓	✓	10.18.a.b	$1$	$1$	\(-512\)	\(-6308\)	\(-781250\)	\(6543844\)		$+$	$+$	$+$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+(-3154-\beta )q^{3}+2^{16}q^{4}+\cdots\)
10.18.a.c	$10$	$18$	10.a	1.a	$1$	$2$	$2$	$18.322$	\(\Q(\sqrt{83281}) \)	None	✓	✓	✓	✓	10.18.a.c	$1$	$0$	\(-512\)	\(-1308\)	\(781250\)	\(603844\)		$+$	$-$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+(-654-\beta )q^{3}+2^{16}q^{4}+\cdots\)
10.18.a.d	$10$	$18$	10.a	1.a	$1$	$2$	$2$	$18.322$	\(\Q(\sqrt{2941}) \)	None	✓	✓	✓	✓	10.18.a.d	$1$	$0$	\(512\)	\(17628\)	\(-781250\)	\(27684196\)		$-$	$+$	$-$	$2^{5}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+(8814-\beta )q^{3}+2^{16}q^{4}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)