Space of modular forms of level 70, weight 6, and trivial character

• Label: 70.6.a
• Level: $70$
• Weight: $6$
• Character orbit: 70.a
• Rep. character: $\chi_{70}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	70.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(72\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(70))\).

	Total	New	Old
Modular forms	64	10	54
Cusp forms	56	10	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.

\(2\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(8\)	\(1\)	\(7\)		\(7\)	\(1\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(8\)	\(2\)	\(6\)		\(7\)	\(2\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(8\)	\(1\)	\(7\)		\(7\)	\(1\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(31\)	\(4\)	\(27\)		\(27\)	\(4\)	\(23\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(33\)	\(6\)	\(27\)		\(29\)	\(6\)	\(23\)		\(4\)	\(0\)	\(4\)

Trace form

10 * q + 8 * q^2 - 44 * q^3 + 160 * q^4 + 16 * q^6 + 128 * q^8 + 986 * q^9 + 1248 * q^11 - 704 * q^12 - 808 * q^13 - 1100 * q^15 + 2560 * q^16 - 1980 * q^17 + 5800 * q^18 + 1220 * q^19 - 980 * q^21 - 1856 * q^22 - 6024 * q^23 + 256 * q^24 + 6250 * q^25 - 2528 * q^26 + 15400 * q^27 - 3412 * q^29 + 3600 * q^30 + 9928 * q^31 + 2048 * q^32 - 6208 * q^33 - 17072 * q^34 + 2450 * q^35 + 15776 * q^36 - 15508 * q^37 + 17040 * q^38 - 24504 * q^39 + 28412 * q^41 + 7056 * q^42 - 21408 * q^43 + 19968 * q^44 - 11600 * q^45 - 32096 * q^46 - 84296 * q^47 - 11264 * q^48 + 24010 * q^49 + 5000 * q^50 - 4344 * q^51 - 12928 * q^52 + 46228 * q^53 + 61024 * q^54 - 27800 * q^55 - 12000 * q^57 + 19344 * q^58 - 39324 * q^59 - 17600 * q^60 - 62800 * q^61 - 66400 * q^62 - 46256 * q^63 + 40960 * q^64 + 77500 * q^65 + 66240 * q^66 + 151304 * q^67 - 31680 * q^68 - 160624 * q^69 + 9800 * q^70 - 137080 * q^71 + 92800 * q^72 - 52436 * q^73 - 47888 * q^74 - 27500 * q^75 + 19520 * q^76 + 47432 * q^77 - 99296 * q^78 - 17240 * q^79 + 121610 * q^81 + 34096 * q^82 + 302668 * q^83 - 15680 * q^84 + 46600 * q^85 + 2368 * q^86 + 216600 * q^87 - 29696 * q^88 + 216996 * q^89 - 142400 * q^90 - 86828 * q^91 - 96384 * q^92 - 336208 * q^93 - 29280 * q^94 - 109500 * q^95 + 4096 * q^96 + 108340 * q^97 + 19208 * q^98 + 627680 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(70))\) 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	7
70.6.a.a	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	✓	✓	✓	70.6.a.a	$1$	$1$	\(-4\)	\(-23\)	\(25\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-23q^{3}+2^{4}q^{4}+5^{2}q^{5}+92q^{6}+\cdots\)
70.6.a.b	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	✓	✓	✓	70.6.a.b	$1$	$0$	\(-4\)	\(-9\)	\(25\)	\(-49\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+5^{2}q^{5}+6^{2}q^{6}+\cdots\)
70.6.a.c	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	✓	✓	✓	70.6.a.c	$1$	$0$	\(-4\)	\(-3\)	\(-25\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-3q^{3}+2^{4}q^{4}-5^{2}q^{5}+12q^{6}+\cdots\)
70.6.a.d	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	✓	✓	✓	70.6.a.d	$1$	$1$	\(-4\)	\(11\)	\(-25\)	\(-49\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+11q^{3}+2^{4}q^{4}-5^{2}q^{5}-44q^{6}+\cdots\)
70.6.a.e	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	✓	✓	✓	70.6.a.e	$1$	$1$	\(4\)	\(-17\)	\(25\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-17q^{3}+2^{4}q^{4}+5^{2}q^{5}-68q^{6}+\cdots\)
70.6.a.f	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	✓	✓	✓	70.6.a.f	$1$	$1$	\(4\)	\(-11\)	\(-25\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-11q^{3}+2^{4}q^{4}-5^{2}q^{5}-44q^{6}+\cdots\)
70.6.a.g	$70$	$6$	70.a	1.a	$1$	$2$	$2$	$11.227$	\(\Q(\sqrt{3369}) \)	None	✓	✓	✓	✓	70.6.a.g	$1$	$0$	\(8\)	\(3\)	\(-50\)	\(-98\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+(2-\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
70.6.a.h	$70$	$6$	70.a	1.a	$1$	$2$	$2$	$11.227$	\(\Q(\sqrt{1129}) \)	None	✓	✓	✓	✓	70.6.a.h	$1$	$0$	\(8\)	\(5\)	\(50\)	\(98\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+(3-\beta )q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(70))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(70)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)