Space of modular forms of level 85, weight 4, and trivial character

• Label: 85.4.a
• Level: $85$
• Weight: $4$
• Character orbit: 85.a
• Rep. character: $\chi_{85}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $7$
• Sturm bound: $36$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(36\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(85))\).

	Total	New	Old
Modular forms	30	16	14
Cusp forms	26	16	10
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
\(+\)	\(+\)	\(+\)		\(8\)	\(3\)	\(5\)		\(7\)	\(3\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(7\)	\(4\)	\(3\)		\(6\)	\(4\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(8\)	\(6\)	\(2\)		\(7\)	\(6\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(16\)	\(9\)	\(7\)		\(14\)	\(9\)	\(5\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(14\)	\(7\)	\(7\)		\(12\)	\(7\)	\(5\)		\(2\)	\(0\)	\(2\)

Trace form

16 * q + 4 * q^2 + 44 * q^4 + 10 * q^5 + 68 * q^6 - 12 * q^7 - 48 * q^8 + 116 * q^9 + 20 * q^10 - 12 * q^11 - 204 * q^12 + 8 * q^13 - 236 * q^14 + 396 * q^16 + 68 * q^17 - 80 * q^18 - 444 * q^19 + 120 * q^20 - 192 * q^21 - 208 * q^22 + 100 * q^23 + 944 * q^24 + 400 * q^25 - 540 * q^26 + 576 * q^27 - 548 * q^28 + 556 * q^29 - 200 * q^30 + 136 * q^31 - 740 * q^32 - 152 * q^33 - 260 * q^35 - 488 * q^36 - 788 * q^37 + 1224 * q^38 - 1056 * q^39 + 240 * q^40 + 392 * q^41 + 2020 * q^42 - 1120 * q^43 + 636 * q^44 - 390 * q^45 - 1160 * q^46 - 560 * q^47 - 2304 * q^48 + 2080 * q^49 + 100 * q^50 + 1100 * q^52 - 1440 * q^53 + 1900 * q^54 - 1928 * q^56 + 976 * q^57 - 1292 * q^58 - 876 * q^59 + 980 * q^60 - 92 * q^61 - 1736 * q^62 + 2252 * q^63 + 3372 * q^64 + 260 * q^65 - 656 * q^66 + 1976 * q^67 + 816 * q^68 - 184 * q^69 + 180 * q^70 - 632 * q^71 - 5732 * q^72 + 648 * q^73 + 1004 * q^74 - 996 * q^76 + 424 * q^77 - 2340 * q^78 + 408 * q^79 - 320 * q^80 + 2208 * q^81 + 3036 * q^82 + 80 * q^83 - 4216 * q^84 + 170 * q^85 + 4008 * q^86 - 2072 * q^87 - 1244 * q^88 + 4084 * q^89 + 3080 * q^90 - 1152 * q^91 + 432 * q^92 + 2144 * q^93 + 892 * q^94 + 1520 * q^95 + 1660 * q^96 - 1528 * q^97 - 2656 * q^98 - 564 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(85))\) 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
85.4.a.a	$85$	$4$	85.a	1.a	$1$	$1$	$1$	$5.015$	\(\Q\)	None	✓	✓			85.4.a.a	$1$	$1$	\(3\)	\(-7\)	\(5\)	\(-22\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-7q^{3}+q^{4}+5q^{5}-21q^{6}+\cdots\)
85.4.a.b	$85$	$4$	85.a	1.a	$1$	$1$	$1$	$5.015$	\(\Q\)	None	✓	✓			85.4.a.b	$1$	$1$	\(3\)	\(-5\)	\(-5\)	\(-22\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-5q^{3}+q^{4}-5q^{5}-15q^{6}+\cdots\)
85.4.a.c	$85$	$4$	85.a	1.a	$1$	$1$	$1$	$5.015$	\(\Q\)	None	✓	✓			85.4.a.c	$1$	$0$	\(3\)	\(10\)	\(5\)	\(-22\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+10q^{3}+q^{4}+5q^{5}+30q^{6}+\cdots\)
85.4.a.d	$85$	$4$	85.a	1.a	$1$	$2$	$2$	$5.015$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		85.4.a.d	$1$	$1$	\(-4\)	\(-2\)	\(10\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}+(-1+\beta )q^{3}+(-1+\cdots)q^{4}+\cdots\)
85.4.a.e	$85$	$4$	85.a	1.a	$1$	$3$	$3$	$5.015$	3.3.1304.1	None	✓	✓	✓		85.4.a.e	$1$	$1$	\(-6\)	\(-4\)	\(-15\)	\(8\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+\cdots\)
85.4.a.f	$85$	$4$	85.a	1.a	$1$	$3$	$3$	$5.015$	3.3.568.1	None	✓	✓	✓	✓	85.4.a.f	$1$	$0$	\(3\)	\(9\)	\(-15\)	\(34\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{2})q^{2}+(4-3\beta _{1})q^{3}+(-1+\cdots)q^{4}+\cdots\)
85.4.a.g	$85$	$4$	85.a	1.a	$1$	$5$	$5$	$5.015$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		85.4.a.g	$1$	$0$	\(2\)	\(-1\)	\(25\)	\(10\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}-\beta _{2}q^{3}+(6-2\beta _{1}+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(85))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(85)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)