Space of modular forms of level 85, weight 6, and trivial character

• Label: 85.6.a
• Level: $85$
• Weight: $6$
• Character orbit: 85.a
• Rep. character: $\chi_{85}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $4$
• Sturm bound: $54$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(54\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(85))\).

	Total	New	Old
Modular forms	46	28	18
Cusp forms	42	28	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.

\(5\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(11\)	\(8\)	\(3\)		\(10\)	\(8\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(12\)	\(7\)	\(5\)		\(11\)	\(7\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(12\)	\(8\)	\(4\)		\(11\)	\(8\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(11\)	\(5\)	\(6\)		\(10\)	\(5\)	\(5\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(22\)	\(13\)	\(9\)		\(20\)	\(13\)	\(7\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(24\)	\(15\)	\(9\)		\(22\)	\(15\)	\(7\)		\(2\)	\(0\)	\(2\)

Trace form

28 * q + 4 * q^2 - 36 * q^3 + 556 * q^4 - 50 * q^5 - 148 * q^6 - 384 * q^7 + 624 * q^8 + 2972 * q^9 - 200 * q^10 + 1188 * q^11 - 156 * q^12 - 300 * q^13 - 1196 * q^14 + 7468 * q^16 - 1156 * q^17 - 320 * q^18 - 2976 * q^19 - 2400 * q^20 + 2616 * q^21 + 2088 * q^22 - 10172 * q^23 - 11536 * q^24 + 17500 * q^25 - 1812 * q^26 - 11988 * q^27 - 3652 * q^28 + 2128 * q^29 + 4000 * q^30 + 10304 * q^31 + 3772 * q^32 + 18340 * q^33 - 2900 * q^35 + 41800 * q^36 - 13272 * q^37 - 13944 * q^38 + 34524 * q^39 - 9600 * q^40 - 39268 * q^41 - 34988 * q^42 + 9684 * q^43 - 3636 * q^44 + 5550 * q^45 - 23432 * q^46 + 41428 * q^47 + 31008 * q^48 + 11988 * q^49 + 2500 * q^50 - 99828 * q^52 + 82920 * q^53 - 3380 * q^54 + 96568 * q^56 + 10480 * q^57 - 145492 * q^58 - 62064 * q^59 + 7700 * q^60 - 107236 * q^61 + 21208 * q^62 - 231604 * q^63 + 218188 * q^64 + 29600 * q^65 + 146176 * q^66 - 115068 * q^67 - 55488 * q^68 - 166816 * q^69 + 69900 * q^70 - 121076 * q^71 + 314428 * q^72 + 240180 * q^73 - 63772 * q^74 - 22500 * q^75 - 359092 * q^76 + 145852 * q^77 - 125268 * q^78 + 78792 * q^79 - 154400 * q^80 + 172740 * q^81 + 291372 * q^82 - 24700 * q^83 + 411176 * q^84 - 14450 * q^85 - 551976 * q^86 + 214528 * q^87 - 326268 * q^88 - 54776 * q^89 - 9700 * q^90 - 242868 * q^91 - 760944 * q^92 - 377524 * q^93 + 561476 * q^94 - 144400 * q^95 - 230660 * q^96 + 67696 * q^97 + 566480 * q^98 + 505152 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(85))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	17
85.6.a.a	$85$	$6$	85.a	1.a	$1$	$5$	$5$	$13.633$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	85.6.a.a	$1$	$1$	\(-7\)	\(-36\)	\(125\)	\(-204\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-8+\beta _{1}-\beta _{3})q^{3}+\cdots\)
85.6.a.b	$85$	$6$	85.a	1.a	$1$	$7$	$7$	$13.633$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	85.6.a.b	$1$	$0$	\(9\)	\(18\)	\(-175\)	\(50\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(3+\beta _{2})q^{3}+(20+2\beta _{1}+\cdots)q^{4}+\cdots\)
85.6.a.c	$85$	$6$	85.a	1.a	$1$	$8$	$8$	$13.633$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	85.6.a.c	$1$	$1$	\(-3\)	\(-36\)	\(-200\)	\(-184\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-4-\beta _{3})q^{3}+(23+\beta _{1}+\cdots)q^{4}+\cdots\)
85.6.a.d	$85$	$6$	85.a	1.a	$1$	$8$	$8$	$13.633$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	85.6.a.d	$1$	$0$	\(5\)	\(18\)	\(200\)	\(-46\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{3})q^{3}+(24-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(85))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(85)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)