Space of modular forms of level 21, weight 10, and trivial character

• Label: 21.10.a
• Level: $21$
• Weight: $10$
• Character orbit: 21.a
• Rep. character: $\chi_{21}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $26$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(26\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(21))\).

	Total	New	Old
Modular forms	26	8	18
Cusp forms	22	8	14
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
\(+\)	\(+\)	\(+\)		\(6\)	\(2\)	\(4\)		\(5\)	\(2\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(12\)	\(3\)	\(9\)		\(10\)	\(3\)	\(7\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(14\)	\(5\)	\(9\)		\(12\)	\(5\)	\(7\)		\(2\)	\(0\)	\(2\)

Trace form

8 * q + 2 * q^2 + 2050 * q^4 + 4552 * q^5 - 6156 * q^6 - 4802 * q^7 + 9390 * q^8 + 52488 * q^9 + 16108 * q^10 - 78140 * q^11 + 115992 * q^12 + 279176 * q^13 + 24010 * q^14 - 3564 * q^15 + 214546 * q^16 - 378912 * q^17 + 13122 * q^18 + 586184 * q^19 + 1411100 * q^20 - 388962 * q^21 + 2396416 * q^22 + 725724 * q^23 - 1495908 * q^24 - 2577248 * q^25 + 957428 * q^26 - 4057690 * q^28 + 8576440 * q^29 + 3227688 * q^30 - 8356584 * q^31 - 45299498 * q^32 + 5409504 * q^33 + 28069404 * q^34 - 6204184 * q^35 + 13450050 * q^36 - 37232136 * q^37 - 30025936 * q^38 - 14262480 * q^39 + 52787556 * q^40 + 13951376 * q^41 - 6223392 * q^42 + 50242720 * q^43 - 42073024 * q^44 + 29865672 * q^45 - 98920632 * q^46 + 62344392 * q^47 + 92730096 * q^48 + 46118408 * q^49 + 73994302 * q^50 + 23031540 * q^51 + 231982372 * q^52 - 217936680 * q^53 - 40389516 * q^54 + 130559768 * q^55 + 63083874 * q^56 + 5276664 * q^57 - 443463044 * q^58 + 69912216 * q^59 - 29254608 * q^60 - 85792600 * q^61 - 308107200 * q^62 - 31505922 * q^63 - 56923702 * q^64 + 246963088 * q^65 + 35066520 * q^66 + 179326184 * q^67 - 853640772 * q^68 + 547698672 * q^69 + 291106844 * q^70 + 151882212 * q^71 + 61607790 * q^72 + 781362720 * q^73 - 758102628 * q^74 - 638703792 * q^75 - 219853280 * q^76 + 467349848 * q^77 - 408004128 * q^78 + 75820360 * q^79 - 225047332 * q^80 + 344373768 * q^81 - 1050294052 * q^82 - 167217024 * q^83 - 298722816 * q^84 - 2054101752 * q^85 + 3203864920 * q^86 - 472571496 * q^87 + 1195436784 * q^88 + 473386912 * q^89 + 105684588 * q^90 - 720175148 * q^91 + 3191894664 * q^92 + 89676072 * q^93 + 130819488 * q^94 + 1028816560 * q^95 - 374635692 * q^96 + 2488240800 * q^97 + 11529602 * q^98 - 512676540 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(21))\) 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
21.10.a.a	$21$	$10$	21.a	1.a	$1$	$1$	$1$	$10.816$	\(\Q\)	None	✓	✓	✓	✓	21.10.a.a	$1$	$1$	\(-24\)	\(81\)	\(-144\)	\(2401\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-24q^{2}+3^{4}q^{3}+2^{6}q^{4}-12^{2}q^{5}+\cdots\)
21.10.a.b	$21$	$10$	21.a	1.a	$1$	$2$	$2$	$10.816$	\(\Q(\sqrt{2353}) \)	None	✓	✓	✓	✓	21.10.a.b	$1$	$1$	\(9\)	\(-162\)	\(1170\)	\(-4802\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{2}-3^{4}q^{3}+(101-9\beta )q^{4}+\cdots\)
21.10.a.c	$21$	$10$	21.a	1.a	$1$	$2$	$2$	$10.816$	\(\Q(\sqrt{345}) \)	None	✓	✓	✓	✓	21.10.a.c	$1$	$0$	\(30\)	\(-162\)	\(1128\)	\(4802\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(15-\beta )q^{2}-3^{4}q^{3}+(58-30\beta )q^{4}+\cdots\)
21.10.a.d	$21$	$10$	21.a	1.a	$1$	$3$	$3$	$10.816$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	21.10.a.d	$1$	$0$	\(-13\)	\(243\)	\(2398\)	\(-7203\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{2}+3^{4}q^{3}+(555+7\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(21)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)