Space of modular forms of level 70, weight 8, and trivial character

• Label: 70.8.a
• Level: $70$
• Weight: $8$
• Character orbit: 70.a
• Rep. character: $\chi_{70}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $8$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	70.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(70))\).

	Total	New	Old
Modular forms	88	14	74
Cusp forms	80	14	66
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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(1\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(6\)

Trace form

14 * q - 16 * q^2 + 52 * q^3 + 896 * q^4 - 1376 * q^6 - 1024 * q^8 + 11294 * q^9 + 1360 * q^11 + 3328 * q^12 + 19144 * q^13 - 3500 * q^15 + 57344 * q^16 - 42580 * q^17 + 13360 * q^18 + 49540 * q^19 - 56252 * q^21 + 98432 * q^22 - 43368 * q^23 - 88064 * q^24 + 218750 * q^25 + 145088 * q^26 + 13960 * q^27 + 36900 * q^29 - 108000 * q^30 - 657592 * q^31 - 65536 * q^32 + 941984 * q^33 - 96928 * q^34 - 85750 * q^35 + 722816 * q^36 + 131236 * q^37 + 638240 * q^38 + 603912 * q^39 - 350380 * q^41 - 296352 * q^42 - 989616 * q^43 + 87040 * q^44 + 1402000 * q^45 + 1556672 * q^46 - 2168488 * q^47 + 212992 * q^48 + 1647086 * q^49 - 250000 * q^50 + 4192680 * q^51 + 1225216 * q^52 - 293924 * q^53 - 4981184 * q^54 - 811000 * q^55 - 2796480 * q^57 - 3844512 * q^58 - 1042940 * q^59 - 224000 * q^60 - 2117744 * q^61 + 5872960 * q^62 + 4450768 * q^63 + 3670016 * q^64 + 642500 * q^65 + 5698176 * q^66 - 5810888 * q^67 - 2725120 * q^68 - 7587088 * q^69 - 686000 * q^70 + 361112 * q^71 + 855040 * q^72 - 7778812 * q^73 + 9188512 * q^74 + 812500 * q^75 + 3170560 * q^76 - 3652264 * q^77 - 5083712 * q^78 - 6320104 * q^79 - 3972370 * q^81 + 2230688 * q^82 - 5462164 * q^83 - 3600128 * q^84 + 179000 * q^85 - 7249536 * q^86 - 16119720 * q^87 + 6299648 * q^88 - 9191156 * q^89 + 10096000 * q^90 + 755972 * q^91 - 2775552 * q^92 + 32323664 * q^93 - 1415232 * q^94 - 18047500 * q^95 - 5636096 * q^96 - 23006020 * q^97 - 1882384 * q^98 - 1263088 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(70))\) 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
70.8.a.a	$70$	$8$	70.a	1.a	$1$	$1$	$1$	$21.867$	\(\Q\)	None	✓	✓	✓	✓	70.8.a.a	$1$	$1$	\(8\)	\(-93\)	\(125\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-93q^{3}+2^{6}q^{4}+5^{3}q^{5}-744q^{6}+\cdots\)
70.8.a.b	$70$	$8$	70.a	1.a	$1$	$1$	$1$	$21.867$	\(\Q\)	None	✓	✓	✓	✓	70.8.a.b	$1$	$1$	\(8\)	\(9\)	\(-125\)	\(-343\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+9q^{3}+2^{6}q^{4}-5^{3}q^{5}+72q^{6}+\cdots\)
70.8.a.c	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{9241}) \)	None	✓	✓	✓	✓	70.8.a.c	$1$	$1$	\(-16\)	\(11\)	\(-250\)	\(686\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(6-\beta )q^{3}+2^{6}q^{4}-5^{3}q^{5}+\cdots\)
70.8.a.d	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{11761}) \)	None	✓	✓	✓	✓	70.8.a.d	$1$	$0$	\(-16\)	\(25\)	\(-250\)	\(-686\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(13-\beta )q^{3}+2^{6}q^{4}-5^{3}q^{5}+\cdots\)
70.8.a.e	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{18481}) \)	None	✓	✓	✓	✓	70.8.a.e	$1$	$0$	\(-16\)	\(31\)	\(250\)	\(686\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(2^{4}-\beta )q^{3}+2^{6}q^{4}+5^{3}q^{5}+\cdots\)
70.8.a.f	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{1401}) \)	None	✓	✓	✓	✓	70.8.a.f	$1$	$1$	\(-16\)	\(45\)	\(250\)	\(-686\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(23-\beta )q^{3}+2^{6}q^{4}+5^{3}q^{5}+\cdots\)
70.8.a.g	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{8761}) \)	None	✓	✓	✓	✓	70.8.a.g	$1$	$0$	\(16\)	\(-5\)	\(-250\)	\(686\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-2-\beta )q^{3}+2^{6}q^{4}-5^{3}q^{5}+\cdots\)
70.8.a.h	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{12121}) \)	None	✓	✓	✓	✓	70.8.a.h	$1$	$0$	\(16\)	\(29\)	\(250\)	\(-686\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+(15-\beta )q^{3}+2^{6}q^{4}+5^{3}q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(70))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(70)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)