Space of modular forms of level 867, weight 4, and trivial character

• Label: 867.4.a
• Level: $867$
• Weight: $4$
• Character orbit: 867.a
• Rep. character: $\chi_{867}(1,\cdot)$
• Character field: $\Q$
• Dimension: $136$
• Newform subspaces: $23$
• Sturm bound: $408$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(408\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(867))\).

	Total	New	Old
Modular forms	324	136	188
Cusp forms	288	136	152
Eisenstein series	36	0	36

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

\(3\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(35\)
\(+\)	\(-\)	\(-\)	\(33\)
\(-\)	\(+\)	\(-\)	\(29\)
\(-\)	\(-\)	\(+\)	\(39\)
Plus space		\(+\)	\(74\)
Minus space		\(-\)	\(62\)

Trace form

136 * q - 4 * q^2 + 564 * q^4 - 12 * q^6 - 8 * q^7 - 48 * q^8 + 1224 * q^9 - 100 * q^10 - 16 * q^11 - 28 * q^13 + 288 * q^14 + 84 * q^15 + 2348 * q^16 - 36 * q^18 + 192 * q^19 - 32 * q^20 + 84 * q^21 + 284 * q^22 - 144 * q^24 + 3020 * q^25 + 204 * q^26 + 720 * q^28 - 432 * q^29 + 408 * q^30 + 48 * q^31 + 312 * q^32 + 36 * q^33 - 232 * q^35 + 5076 * q^36 - 192 * q^37 + 932 * q^38 + 24 * q^39 - 176 * q^40 - 1440 * q^41 - 252 * q^42 + 600 * q^43 - 1056 * q^44 - 820 * q^46 + 80 * q^47 + 96 * q^48 + 6720 * q^49 - 160 * q^50 - 688 * q^52 + 744 * q^53 - 108 * q^54 - 40 * q^55 + 1056 * q^56 - 312 * q^57 - 400 * q^58 - 216 * q^59 + 204 * q^60 - 64 * q^61 + 248 * q^62 - 72 * q^63 + 11552 * q^64 + 656 * q^65 + 1032 * q^66 + 1656 * q^67 - 240 * q^69 - 12 * q^70 + 2128 * q^71 - 432 * q^72 + 1048 * q^73 + 96 * q^74 + 1104 * q^75 + 532 * q^76 + 1096 * q^77 - 1620 * q^78 - 1768 * q^79 + 1968 * q^80 + 11016 * q^81 + 1076 * q^82 - 3416 * q^83 + 624 * q^84 - 124 * q^86 + 516 * q^87 + 184 * q^88 + 384 * q^89 - 900 * q^90 + 600 * q^91 + 352 * q^92 + 1836 * q^93 - 4492 * q^94 + 3024 * q^95 + 936 * q^96 - 3160 * q^97 - 3056 * q^98 - 144 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(867))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	17
867.4.a.a	$867$	$4$	867.a	1.a	$1$	$1$	$1$	$51.155$	\(\Q\)	None	✓	✓			867.4.a.a	$1$	$1$	\(-3\)	\(-3\)	\(9\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-3q^{3}+q^{4}+9q^{5}+9q^{6}+\cdots\)
867.4.a.b	$867$	$4$	867.a	1.a	$1$	$1$	$1$	$51.155$	\(\Q\)	None	✓	✓			867.4.a.a	$1$	$1$	\(-3\)	\(3\)	\(-9\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}-9q^{5}-9q^{6}+\cdots\)
867.4.a.c	$867$	$4$	867.a	1.a	$1$	$1$	$1$	$51.155$	\(\Q\)	None	✓				51.4.a.b	$1$	$0$	\(-1\)	\(-3\)	\(20\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-7q^{4}+20q^{5}+3q^{6}+\cdots\)
867.4.a.d	$867$	$4$	867.a	1.a	$1$	$1$	$1$	$51.155$	\(\Q\)	None	✓				51.4.a.a	$1$	$1$	\(-1\)	\(3\)	\(-16\)	\(-34\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}-7q^{4}-2^{4}q^{5}-3q^{6}+\cdots\)
867.4.a.e	$867$	$4$	867.a	1.a	$1$	$1$	$1$	$51.155$	\(\Q\)	None	✓				51.4.a.c	$1$	$1$	\(1\)	\(3\)	\(10\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}+10q^{5}+3q^{6}+\cdots\)
867.4.a.f	$867$	$4$	867.a	1.a	$1$	$1$	$1$	$51.155$	\(\Q\)	None	✓	✓			867.4.a.f	$1$	$1$	\(4\)	\(-3\)	\(2\)	\(31\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-3q^{3}+8q^{4}+2q^{5}-12q^{6}+\cdots\)
867.4.a.g	$867$	$4$	867.a	1.a	$1$	$1$	$1$	$51.155$	\(\Q\)	None	✓	✓			867.4.a.f	$1$	$1$	\(4\)	\(3\)	\(-2\)	\(-31\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+3q^{3}+8q^{4}-2q^{5}+12q^{6}+\cdots\)
867.4.a.h	$867$	$4$	867.a	1.a	$1$	$2$	$2$	$51.155$	\(\Q(\sqrt{13}) \)	None	✓	✓			867.4.a.h	$1$	$1$	\(-4\)	\(-6\)	\(-20\)	\(-6\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}-3q^{3}+(9+4\beta )q^{4}+\cdots\)
867.4.a.i	$867$	$4$	867.a	1.a	$1$	$2$	$2$	$51.155$	\(\Q(\sqrt{13}) \)	None	✓	✓			867.4.a.h	$1$	$1$	\(-4\)	\(6\)	\(20\)	\(6\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+3q^{3}+(9+4\beta )q^{4}+\cdots\)
867.4.a.j	$867$	$4$	867.a	1.a	$1$	$2$	$2$	$51.155$	\(\Q(\sqrt{2}) \)	None	✓				51.4.a.d	$1$	$1$	\(0\)	\(6\)	\(-6\)	\(8\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+3q^{3}+10q^{4}+(-3-4\beta )q^{5}+\cdots\)
867.4.a.k	$867$	$4$	867.a	1.a	$1$	$3$	$3$	$51.155$	3.3.5912.1	None	✓				51.4.a.e	$1$	$0$	\(5\)	\(-9\)	\(-8\)	\(8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}-3q^{3}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
867.4.a.l	$867$	$4$	867.a	1.a	$1$	$4$	$4$	$51.155$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			867.4.a.l	$1$	$0$	\(1\)	\(-12\)	\(-9\)	\(-35\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
867.4.a.m	$867$	$4$	867.a	1.a	$1$	$4$	$4$	$51.155$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			867.4.a.l	$1$	$0$	\(1\)	\(12\)	\(9\)	\(35\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
867.4.a.n	$867$	$4$	867.a	1.a	$1$	$4$	$4$	$51.155$	4.4.2873868.1	None	✓				51.4.d.a	$1$	$0$	\(2\)	\(-12\)	\(12\)	\(14\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
867.4.a.o	$867$	$4$	867.a	1.a	$1$	$4$	$4$	$51.155$	4.4.2873868.1	None	✓				51.4.d.a	$1$	$1$	\(2\)	\(12\)	\(-12\)	\(-14\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
867.4.a.p	$867$	$4$	867.a	1.a	$1$	$8$	$8$	$51.155$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				51.4.e.a	$1$	$0$	\(-4\)	\(-24\)	\(24\)	\(28\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(4-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
867.4.a.q	$867$	$4$	867.a	1.a	$1$	$8$	$8$	$51.155$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				51.4.e.a	$1$	$1$	\(-4\)	\(24\)	\(-24\)	\(-28\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(4-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
867.4.a.r	$867$	$4$	867.a	1.a	$1$	$9$	$9$	$51.155$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			867.4.a.r	$1$	$1$	\(-6\)	\(-27\)	\(-6\)	\(33\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(2-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
867.4.a.s	$867$	$4$	867.a	1.a	$1$	$9$	$9$	$51.155$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		867.4.a.r	$1$	$1$	\(-6\)	\(27\)	\(6\)	\(-33\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(2-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
867.4.a.t	$867$	$4$	867.a	1.a	$1$	$15$	$15$	$51.155$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓		867.4.a.t	$1$	$0$	\(6\)	\(-45\)	\(-6\)	\(-33\)		$+$	$+$	$+$	$17^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
867.4.a.u	$867$	$4$	867.a	1.a	$1$	$15$	$15$	$51.155$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓			867.4.a.t	$1$	$0$	\(6\)	\(45\)	\(6\)	\(33\)		$-$	$-$	$+$	$17^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
867.4.a.v	$867$	$4$	867.a	1.a	$1$	$20$	$20$	$51.155$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		51.4.h.a	$1$	$1$	\(0\)	\(-60\)	\(-32\)	\(-56\)		$+$	$-$	$-$	$2^{5}\cdot 17^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
867.4.a.w	$867$	$4$	867.a	1.a	$1$	$20$	$20$	$51.155$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		51.4.h.a	$1$	$0$	\(0\)	\(60\)	\(32\)	\(56\)		$-$	$-$	$+$	$2^{5}\cdot 17^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}+(2-\beta _{16}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(867))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(867)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 2}\)