Space of modular forms of level 850, weight 4, and trivial character

• Label: 850.4.a
• Level: $850$
• Weight: $4$
• Character orbit: 850.a
• Rep. character: $\chi_{850}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $23$
• Sturm bound: $540$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	850.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(540\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(850))\).

	Total	New	Old
Modular forms	418	76	342
Cusp forms	394	76	318
Eisenstein series	24	0	24

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

\(2\)	\(5\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(56\)	\(8\)	\(48\)		\(53\)	\(8\)	\(45\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(50\)	\(10\)	\(40\)		\(47\)	\(10\)	\(37\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(49\)	\(10\)	\(39\)		\(46\)	\(10\)	\(36\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(53\)	\(10\)	\(43\)		\(50\)	\(10\)	\(40\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(52\)	\(7\)	\(45\)		\(49\)	\(7\)	\(42\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(52\)	\(11\)	\(41\)		\(49\)	\(11\)	\(38\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(52\)	\(12\)	\(40\)		\(49\)	\(12\)	\(37\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(54\)	\(8\)	\(46\)		\(51\)	\(8\)	\(43\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(213\)	\(41\)	\(172\)		\(201\)	\(41\)	\(160\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(205\)	\(35\)	\(170\)		\(193\)	\(35\)	\(158\)		\(12\)	\(0\)	\(12\)

Trace form

76 * q - 14 * q^3 + 304 * q^4 - 20 * q^6 - 16 * q^7 + 776 * q^9 - 50 * q^11 - 56 * q^12 - 148 * q^13 + 40 * q^14 + 1216 * q^16 + 34 * q^17 + 40 * q^18 + 184 * q^19 + 232 * q^21 + 212 * q^22 + 236 * q^23 - 80 * q^24 + 152 * q^26 + 28 * q^27 - 64 * q^28 + 62 * q^29 - 168 * q^31 - 316 * q^33 - 68 * q^34 + 3104 * q^36 - 314 * q^37 - 80 * q^38 - 500 * q^39 + 4 * q^41 - 432 * q^42 + 612 * q^43 - 200 * q^44 - 256 * q^46 - 280 * q^47 - 224 * q^48 + 1812 * q^49 - 102 * q^51 - 592 * q^52 + 132 * q^53 + 472 * q^54 + 160 * q^56 - 152 * q^57 - 468 * q^58 + 1764 * q^59 - 1478 * q^61 - 152 * q^62 + 3136 * q^63 + 4864 * q^64 + 72 * q^66 + 1780 * q^67 + 136 * q^68 + 5176 * q^69 + 4476 * q^71 + 160 * q^72 + 1912 * q^73 - 20 * q^74 + 736 * q^76 + 864 * q^77 + 2136 * q^78 + 1492 * q^79 + 8280 * q^81 + 1576 * q^82 - 516 * q^83 + 928 * q^84 + 232 * q^86 + 1744 * q^87 + 848 * q^88 + 5200 * q^89 - 2168 * q^91 + 944 * q^92 + 1688 * q^93 - 528 * q^94 - 320 * q^96 + 1076 * q^97 - 8202 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(850))\) 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	17
850.4.a.a	$850$	$4$	850.a	1.a	$1$	$1$	$1$	$50.152$	\(\Q\)	None	✓				170.4.a.b	$1$	$1$	\(-2\)	\(-7\)	\(0\)	\(10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-7q^{3}+4q^{4}+14q^{6}+10q^{7}+\cdots\)
850.4.a.b	$850$	$4$	850.a	1.a	$1$	$1$	$1$	$50.152$	\(\Q\)	None	✓				170.4.a.a	$1$	$1$	\(2\)	\(-4\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{3}+4q^{4}-8q^{6}+4q^{7}+\cdots\)
850.4.a.c	$850$	$4$	850.a	1.a	$1$	$1$	$1$	$50.152$	\(\Q\)	None	✓				34.4.a.b	$1$	$0$	\(2\)	\(2\)	\(0\)	\(-24\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+4q^{4}+4q^{6}-24q^{7}+\cdots\)
850.4.a.d	$850$	$4$	850.a	1.a	$1$	$1$	$1$	$50.152$	\(\Q\)	None	✓				34.4.a.a	$1$	$1$	\(2\)	\(2\)	\(0\)	\(10\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+4q^{4}+4q^{6}+10q^{7}+\cdots\)
850.4.a.e	$850$	$4$	850.a	1.a	$1$	$2$	$2$	$50.152$	\(\Q(\sqrt{13}) \)	None	✓				34.4.a.c	$1$	$0$	\(-4\)	\(-6\)	\(0\)	\(6\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-3-\beta )q^{3}+4q^{4}+(6+2\beta )q^{6}+\cdots\)
850.4.a.f	$850$	$4$	850.a	1.a	$1$	$2$	$2$	$50.152$	\(\Q(\sqrt{19}) \)	None	✓				170.4.a.f	$1$	$1$	\(-4\)	\(-2\)	\(0\)	\(-46\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+\beta )q^{3}+4q^{4}+(2-2\beta )q^{6}+\cdots\)
850.4.a.g	$850$	$4$	850.a	1.a	$1$	$2$	$2$	$50.152$	\(\Q(\sqrt{6}) \)	None	✓				170.4.a.e	$1$	$1$	\(-4\)	\(4\)	\(0\)	\(28\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(2+\beta )q^{3}+4q^{4}+(-4-2\beta )q^{6}+\cdots\)
850.4.a.h	$850$	$4$	850.a	1.a	$1$	$2$	$2$	$50.152$	\(\Q(\sqrt{145}) \)	None	✓				170.4.a.d	$1$	$1$	\(4\)	\(-3\)	\(0\)	\(-28\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-1-\beta )q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
850.4.a.i	$850$	$4$	850.a	1.a	$1$	$2$	$2$	$50.152$	\(\Q(\sqrt{43}) \)	None	✓				170.4.a.c	$1$	$0$	\(4\)	\(6\)	\(0\)	\(34\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(3+\beta )q^{3}+4q^{4}+(6+2\beta )q^{6}+\cdots\)
850.4.a.j	$850$	$4$	850.a	1.a	$1$	$3$	$3$	$50.152$	3.3.62013.1	None	✓				170.4.a.h	$1$	$0$	\(-6\)	\(1\)	\(0\)	\(-20\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}-\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+(-7+\cdots)q^{7}+\cdots\)
850.4.a.k	$850$	$4$	850.a	1.a	$1$	$3$	$3$	$50.152$	3.3.43476.1	None	✓	✓			850.4.a.k	$1$	$0$	\(-6\)	\(1\)	\(0\)	\(9\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}+(4+\cdots)q^{7}+\cdots\)
850.4.a.l	$850$	$4$	850.a	1.a	$1$	$3$	$3$	$50.152$	3.3.3732.1	None	✓	✓			850.4.a.l	$1$	$0$	\(-6\)	\(7\)	\(0\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(2+\beta _{2})q^{3}+4q^{4}+(-4-2\beta _{2})q^{6}+\cdots\)
850.4.a.m	$850$	$4$	850.a	1.a	$1$	$3$	$3$	$50.152$	3.3.3732.1	None	✓	✓			850.4.a.l	$1$	$1$	\(6\)	\(-7\)	\(0\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-2-\beta _{2})q^{3}+4q^{4}+(-4+\cdots)q^{6}+\cdots\)
850.4.a.n	$850$	$4$	850.a	1.a	$1$	$3$	$3$	$50.152$	3.3.701288.1	None	✓				170.4.a.g	$1$	$0$	\(6\)	\(-7\)	\(0\)	\(10\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-2-\beta _{1})q^{3}+4q^{4}+(-4+\cdots)q^{6}+\cdots\)
850.4.a.o	$850$	$4$	850.a	1.a	$1$	$3$	$3$	$50.152$	3.3.43476.1	None	✓	✓	✓		850.4.a.k	$1$	$1$	\(6\)	\(-1\)	\(0\)	\(-9\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}+(-4+\cdots)q^{7}+\cdots\)
850.4.a.p	$850$	$4$	850.a	1.a	$1$	$5$	$5$	$50.152$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			850.4.a.p	$1$	$1$	\(-10\)	\(-5\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+(2+2\beta _{1}+\cdots)q^{6}+\cdots\)
850.4.a.q	$850$	$4$	850.a	1.a	$1$	$5$	$5$	$50.152$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		850.4.a.q	$1$	$1$	\(-10\)	\(1\)	\(0\)	\(-53\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}+(-11+\cdots)q^{7}+\cdots\)
850.4.a.r	$850$	$4$	850.a	1.a	$1$	$5$	$5$	$50.152$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				170.4.c.a	$1$	$1$	\(-10\)	\(5\)	\(0\)	\(26\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1+\beta _{2})q^{3}+4q^{4}+(-2-2\beta _{2}+\cdots)q^{6}+\cdots\)
850.4.a.s	$850$	$4$	850.a	1.a	$1$	$5$	$5$	$50.152$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		170.4.c.a	$1$	$1$	\(10\)	\(-5\)	\(0\)	\(-26\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-1-\beta _{2})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
850.4.a.t	$850$	$4$	850.a	1.a	$1$	$5$	$5$	$50.152$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			850.4.a.q	$1$	$0$	\(10\)	\(-1\)	\(0\)	\(53\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}+(11+\cdots)q^{7}+\cdots\)
850.4.a.u	$850$	$4$	850.a	1.a	$1$	$5$	$5$	$50.152$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		850.4.a.p	$1$	$0$	\(10\)	\(5\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(2+2\beta _{1}+\cdots)q^{6}+\cdots\)
850.4.a.v	$850$	$4$	850.a	1.a	$1$	$7$	$7$	$50.152$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		170.4.c.b	$1$	$0$	\(-14\)	\(-1\)	\(0\)	\(26\)		$+$	$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-2q^{2}-\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+(4+\cdots)q^{7}+\cdots\)
850.4.a.w	$850$	$4$	850.a	1.a	$1$	$7$	$7$	$50.152$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		170.4.c.b	$1$	$0$	\(14\)	\(1\)	\(0\)	\(-26\)		$-$	$-$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+(-4+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(850))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(850)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(425))\)\(^{\oplus 2}\)