Space of modular forms of level 21, weight 18, and trivial character

• Label: 21.18.a
• Level: $21$
• Weight: $18$
• Character orbit: 21.a
• Rep. character: $\chi_{21}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(21))\).

	Total	New	Old
Modular forms	48	16	32
Cusp forms	44	16	28
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.

\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(11\)	\(4\)	\(7\)		\(10\)	\(4\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(12\)	\(4\)	\(8\)		\(11\)	\(4\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(13\)	\(5\)	\(8\)		\(12\)	\(5\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(12\)	\(3\)	\(9\)		\(11\)	\(3\)	\(8\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(23\)	\(7\)	\(16\)		\(21\)	\(7\)	\(14\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(25\)	\(9\)	\(16\)		\(23\)	\(9\)	\(14\)		\(2\)	\(0\)	\(2\)

Trace form

16 * q - 1054 * q^2 + 1081858 * q^4 - 1672808 * q^5 - 1758348 * q^6 - 11529602 * q^7 - 237216402 * q^8 + 688747536 * q^9 + 1686014508 * q^10 + 721831288 * q^11 - 910194408 * q^12 + 6865298144 * q^13 + 1164489802 * q^14 - 20126156184 * q^15 + 22424901394 * q^16 + 112134797496 * q^17 - 45371243934 * q^18 + 16504751024 * q^19 - 163266777700 * q^20 - 75645718722 * q^21 - 238369736640 * q^22 + 1183839164376 * q^23 - 882414635364 * q^24 + 998438952832 * q^25 + 4626183149876 * q^26 - 2479475498906 * q^28 - 4193233111088 * q^29 + 7368770427048 * q^30 + 15375119829824 * q^31 - 37033193813738 * q^32 + 9589452046632 * q^33 + 24815642260956 * q^34 + 2587219629596 * q^35 + 46570439487618 * q^36 - 8662143490672 * q^37 - 94139941159504 * q^38 - 12707967202704 * q^39 + 150373860544356 * q^40 + 44395847461352 * q^41 - 19365303992832 * q^42 - 180096775065328 * q^43 + 95402374593920 * q^44 - 72008899262568 * q^45 + 404595620944968 * q^46 + 169949830773552 * q^47 + 269907402989808 * q^48 + 531726889113616 * q^49 + 693360438222622 * q^50 - 379692021878136 * q^51 - 1277204425585756 * q^52 + 724604198807232 * q^53 - 75691115776908 * q^54 - 136570940296752 * q^55 + 777055893445986 * q^56 + 1164777242454144 * q^57 + 2444250987740412 * q^58 + 1578318222475920 * q^59 + 549364395122352 * q^60 + 3261547717573904 * q^61 + 751154652800064 * q^62 - 496311560535042 * q^63 + 4547012918358346 * q^64 - 5591285973385712 * q^65 - 314943866335080 * q^66 - 2193572728443904 * q^67 - 2939982830206020 * q^68 - 11296634492761992 * q^69 - 4316721935795556 * q^70 + 14380322803142856 * q^71 - 10211388273517842 * q^72 - 24652443269289520 * q^73 - 15083387415808356 * q^74 - 2762092998829152 * q^75 + 8444379928901024 * q^76 + 8089166365042784 * q^77 + 32161736737167840 * q^78 - 70635650892690304 * q^79 - 94286064332576932 * q^80 + 29648323021629456 * q^81 + 92364391328021532 * q^82 + 65576027375396064 * q^83 - 14872553466494976 * q^84 + 93831464072660208 * q^85 - 139789476022480232 * q^86 - 42455923734408192 * q^87 - 69329575662011088 * q^88 + 210861415379622088 * q^89 + 72577396127828268 * q^90 - 41234333258331724 * q^91 - 157075001592615288 * q^92 - 69655031147206512 * q^93 + 196555696028343840 * q^94 - 326281746398240480 * q^95 + 77078556553885524 * q^96 - 214729255992423280 * q^97 - 35027508820359454 * q^98 + 31072470063606648 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(21))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7
21.18.a.a	$21$	$18$	21.a	1.a	$1$	$3$	$3$	$38.477$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	21.18.a.a	$1$	$1$	\(-408\)	\(19683\)	\(-1463514\)	\(17294403\)		$-$	$-$	$+$	$2^{3}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+(-136+\beta _{1})q^{2}+3^{8}q^{3}+(9984+\cdots)q^{4}+\cdots\)
21.18.a.b	$21$	$18$	21.a	1.a	$1$	$4$	$4$	$38.477$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	21.18.a.b	$1$	$1$	\(-375\)	\(-26244\)	\(-154140\)	\(-23059204\)		$+$	$+$	$+$	$2^{3}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-94+\beta _{1})q^{2}-3^{8}q^{3}+(78600+\cdots)q^{4}+\cdots\)
21.18.a.c	$21$	$18$	21.a	1.a	$1$	$4$	$4$	$38.477$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	21.18.a.c	$1$	$0$	\(-18\)	\(-26244\)	\(851508\)	\(23059204\)		$+$	$-$	$-$	$2^{8}\cdot 3^{2}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{2}-3^{8}q^{3}+(73952+59\beta _{1}+\cdots)q^{4}+\cdots\)
21.18.a.d	$21$	$18$	21.a	1.a	$1$	$5$	$5$	$38.477$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	21.18.a.d	$1$	$0$	\(-253\)	\(32805\)	\(-906662\)	\(-28824005\)		$-$	$+$	$-$	$2^{8}\cdot 3^{5}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-51+\beta _{1})q^{2}+3^{8}q^{3}+(88394+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(21)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)