Space of modular forms of level 51, weight 6, and trivial character

• Label: 51.6.a
• Level: $51$
• Weight: $6$
• Character orbit: 51.a
• Rep. character: $\chi_{51}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $4$
• Sturm bound: $36$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	51.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(36\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(51))\).

	Total	New	Old
Modular forms	32	14	18
Cusp forms	28	14	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.

\(3\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(10\)	\(5\)	\(5\)		\(9\)	\(5\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(8\)	\(4\)	\(4\)		\(7\)	\(4\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(8\)	\(2\)	\(6\)		\(7\)	\(2\)	\(5\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(14\)	\(5\)	\(9\)		\(12\)	\(5\)	\(7\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(18\)	\(9\)	\(9\)		\(16\)	\(9\)	\(7\)		\(2\)	\(0\)	\(2\)

Trace form

14 * q + 4 * q^2 - 18 * q^3 + 300 * q^4 - 72 * q^6 - 156 * q^7 - 720 * q^8 + 1134 * q^9 + 596 * q^10 + 1168 * q^11 - 72 * q^12 - 774 * q^13 + 2352 * q^14 + 594 * q^15 + 8244 * q^16 + 324 * q^18 + 478 * q^19 - 5152 * q^20 + 2484 * q^21 - 4620 * q^22 - 7512 * q^23 - 3456 * q^24 + 8208 * q^25 + 17268 * q^26 - 1458 * q^27 - 9680 * q^28 - 2472 * q^29 - 1368 * q^30 - 6048 * q^31 - 13656 * q^32 + 14886 * q^33 - 4624 * q^34 - 8892 * q^35 + 24300 * q^36 + 1020 * q^37 - 20484 * q^38 - 22644 * q^39 + 8400 * q^40 - 25680 * q^41 - 31644 * q^42 - 5334 * q^43 - 46560 * q^44 - 41116 * q^46 + 75884 * q^47 - 6912 * q^48 + 77022 * q^49 - 97152 * q^50 - 10404 * q^51 - 82112 * q^52 + 70612 * q^53 - 5832 * q^54 + 48366 * q^55 + 194688 * q^56 + 19224 * q^57 - 176640 * q^58 - 57648 * q^59 + 70596 * q^60 - 102596 * q^61 + 101560 * q^62 - 12636 * q^63 + 168500 * q^64 + 187384 * q^65 - 103500 * q^66 + 101280 * q^67 - 29538 * q^69 + 349928 * q^70 - 51544 * q^71 - 58320 * q^72 + 136352 * q^73 - 228672 * q^74 + 17874 * q^75 - 275424 * q^76 + 164052 * q^77 + 45900 * q^78 - 104260 * q^79 - 156432 * q^80 + 91854 * q^81 + 109404 * q^82 + 35740 * q^83 + 138564 * q^84 - 63002 * q^85 + 153900 * q^86 - 5400 * q^87 - 210456 * q^88 + 141444 * q^89 + 48276 * q^90 - 332044 * q^91 - 812416 * q^92 - 106092 * q^93 + 209784 * q^94 + 90576 * q^95 + 121176 * q^96 + 42160 * q^97 - 588532 * q^98 + 94608 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(51))\) 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	17
51.6.a.a	$51$	$6$	51.a	1.a	$1$	$2$	$2$	$8.180$	\(\Q(\sqrt{145}) \)	None	✓	✓	✓	✓	51.6.a.a	$1$	$1$	\(-7\)	\(18\)	\(-113\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+9q^{3}+(13+7\beta )q^{4}+\cdots\)
51.6.a.b	$51$	$6$	51.a	1.a	$1$	$3$	$3$	$8.180$	3.3.76361.1	None	✓	✓	✓	✓	51.6.a.b	$1$	$1$	\(5\)	\(-27\)	\(-37\)	\(-176\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}-9q^{3}+(12+\beta _{1}+2\beta _{2})q^{4}+\cdots\)
51.6.a.c	$51$	$6$	51.a	1.a	$1$	$4$	$4$	$8.180$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	51.6.a.c	$1$	$0$	\(5\)	\(36\)	\(146\)	\(60\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+9q^{3}+(3^{3}+4\beta _{2}+\beta _{3})q^{4}+\cdots\)
51.6.a.d	$51$	$6$	51.a	1.a	$1$	$5$	$5$	$8.180$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	51.6.a.d	$1$	$0$	\(1\)	\(-45\)	\(4\)	\(-40\)		$+$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-9q^{3}+(24+\beta _{1}+2\beta _{2}+\beta _{4})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(51))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(51)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)