Space of modular forms of level 42, weight 8, and trivial character

• Label: 42.8.a
• Level: $42$
• Weight: $8$
• Character orbit: 42.a
• Rep. character: $\chi_{42}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $64$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(64\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(42))\).

	Total	New	Old
Modular forms	60	6	54
Cusp forms	52	6	46
Eisenstein series	8	0	8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(7\)	\(0\)	\(7\)		\(6\)	\(0\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(8\)	\(1\)	\(7\)		\(7\)	\(1\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(31\)	\(4\)	\(27\)		\(27\)	\(4\)	\(23\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(29\)	\(2\)	\(27\)		\(25\)	\(2\)	\(23\)		\(4\)	\(0\)	\(4\)

Trace form

6 * q - 16 * q^2 + 384 * q^4 + 220 * q^5 - 1024 * q^8 + 4374 * q^9 + 6240 * q^10 - 5272 * q^11 + 5580 * q^13 + 27432 * q^15 + 24576 * q^16 + 22940 * q^17 - 11664 * q^18 + 73680 * q^19 + 14080 * q^20 - 18522 * q^21 - 45504 * q^22 - 116224 * q^23 + 9258 * q^25 + 287456 * q^26 + 446692 * q^29 - 29376 * q^30 - 464160 * q^31 - 65536 * q^32 - 380592 * q^33 + 533856 * q^34 + 117992 * q^35 + 279936 * q^36 + 133956 * q^37 - 317312 * q^38 + 505224 * q^39 + 399360 * q^40 - 976788 * q^41 - 148176 * q^42 - 1306200 * q^43 - 337408 * q^44 + 160380 * q^45 + 62976 * q^46 - 2065104 * q^47 + 705894 * q^49 + 974736 * q^50 - 30672 * q^51 + 357120 * q^52 - 2016540 * q^53 - 1490352 * q^55 - 2461752 * q^57 - 2872416 * q^58 - 1129472 * q^59 + 1755648 * q^60 - 2307492 * q^61 + 3870464 * q^62 + 1572864 * q^64 + 5564184 * q^65 - 867456 * q^66 + 7014984 * q^67 + 1468160 * q^68 - 598320 * q^69 - 3358656 * q^70 + 4247600 * q^71 - 746496 * q^72 - 527268 * q^73 - 2345312 * q^74 + 3762720 * q^75 + 4715520 * q^76 + 7677712 * q^77 - 3174336 * q^78 + 1256016 * q^79 + 901120 * q^80 + 3188646 * q^81 + 5080032 * q^82 + 545360 * q^83 - 1185408 * q^84 + 15411864 * q^85 - 14661056 * q^86 - 543456 * q^87 - 2912256 * q^88 - 2961924 * q^89 + 4548960 * q^90 - 6873720 * q^91 - 7438336 * q^92 + 12985056 * q^93 + 521088 * q^94 - 18982912 * q^95 - 23944452 * q^97 - 1882384 * q^98 - 3843288 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(42))\) 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	3	7
42.8.a.a	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	✓	✓	✓	42.8.a.a	$1$	$0$	\(-8\)	\(-27\)	\(-410\)	\(-343\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}-410q^{5}+\cdots\)
42.8.a.b	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	✓	✓	✓	42.8.a.b	$1$	$1$	\(-8\)	\(-27\)	\(-18\)	\(343\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}-18q^{5}+6^{3}q^{6}+\cdots\)
42.8.a.c	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	✓	✓	✓	42.8.a.c	$1$	$1$	\(-8\)	\(27\)	\(-122\)	\(-343\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-122q^{5}+\cdots\)
42.8.a.d	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	✓	✓	✓	42.8.a.d	$1$	$0$	\(-8\)	\(27\)	\(270\)	\(343\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+270q^{5}+\cdots\)
42.8.a.e	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	✓	✓	✓	42.8.a.e	$1$	$0$	\(8\)	\(-27\)	\(30\)	\(343\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+30q^{5}-6^{3}q^{6}+\cdots\)
42.8.a.f	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	✓	✓	✓	42.8.a.f	$1$	$0$	\(8\)	\(27\)	\(470\)	\(-343\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+470q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(42)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)