Space of modular forms of level 8001, weight 2, and trivial character

• Label: 8001.2.a
• Level: $8001$
• Weight: $2$
• Character orbit: 8001.a
• Rep. character: $\chi_{8001}(1,\cdot)$
• Character field: $\Q$
• Dimension: $314$
• Newform subspaces: $27$
• Sturm bound: $2048$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(2048\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8001))\).

	Total	New	Old
Modular forms	1032	314	718
Cusp forms	1017	314	703
Eisenstein series	15	0	15

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

\(3\)	\(7\)	\(127\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(124\)	\(32\)	\(92\)		\(123\)	\(32\)	\(91\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(132\)	\(30\)	\(102\)		\(130\)	\(30\)	\(100\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(134\)	\(40\)	\(94\)		\(132\)	\(40\)	\(92\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(126\)	\(22\)	\(104\)		\(124\)	\(22\)	\(102\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(134\)	\(47\)	\(87\)		\(132\)	\(47\)	\(85\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(126\)	\(48\)	\(78\)		\(124\)	\(48\)	\(76\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(124\)	\(43\)	\(81\)		\(122\)	\(43\)	\(79\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(132\)	\(52\)	\(80\)		\(130\)	\(52\)	\(78\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(500\)	\(145\)	\(355\)		\(493\)	\(145\)	\(348\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(532\)	\(169\)	\(363\)		\(524\)	\(169\)	\(355\)		\(8\)	\(0\)	\(8\)

Trace form

314 * q + 320 * q^4 - 4 * q^5 + 4 * q^10 + 16 * q^11 - 8 * q^13 + 332 * q^16 - 4 * q^17 + 4 * q^19 + 4 * q^20 - 28 * q^22 - 4 * q^23 + 290 * q^25 - 40 * q^26 + 12 * q^31 - 30 * q^32 - 26 * q^34 + 8 * q^35 - 16 * q^37 + 6 * q^38 - 32 * q^40 - 8 * q^41 - 8 * q^43 + 38 * q^44 + 16 * q^46 + 36 * q^47 + 314 * q^49 + 32 * q^50 - 30 * q^52 + 16 * q^53 - 8 * q^55 + 20 * q^58 + 20 * q^59 + 12 * q^61 - 62 * q^62 + 332 * q^64 - 36 * q^65 - 8 * q^67 + 72 * q^68 + 28 * q^70 + 52 * q^71 - 64 * q^73 - 38 * q^74 - 24 * q^76 + 16 * q^79 - 20 * q^80 - 24 * q^82 + 16 * q^83 + 28 * q^85 + 24 * q^86 - 10 * q^88 - 32 * q^89 - 4 * q^91 - 52 * q^92 + 40 * q^94 - 44 * q^95 - 108 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\) 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	7	127
8001.2.a.a	$8001$	$2$	8001.a	1.a	$1$	$1$	$1$	$63.888$	\(\Q\)	None	✓				2667.2.a.e	$1$	$2$	\(-2\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-3q^{5}-q^{7}+6q^{10}+\cdots\)
8001.2.a.b	$8001$	$2$	8001.a	1.a	$1$	$1$	$1$	$63.888$	\(\Q\)	None	✓	✓			8001.2.a.b	$1$	$2$	\(-2\)	\(0\)	\(-1\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{5}-q^{7}+2q^{10}+\cdots\)
8001.2.a.c	$8001$	$2$	8001.a	1.a	$1$	$1$	$1$	$63.888$	\(\Q\)	None	✓				2667.2.a.d	$1$	$2$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}+q^{7}-6q^{11}-q^{13}+\cdots\)
8001.2.a.d	$8001$	$2$	8001.a	1.a	$1$	$1$	$1$	$63.888$	\(\Q\)	None	✓				2667.2.a.b	$1$	$1$	\(1\)	\(0\)	\(-4\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-4q^{5}-q^{7}-3q^{8}-4q^{10}+\cdots\)
8001.2.a.e	$8001$	$2$	8001.a	1.a	$1$	$1$	$1$	$63.888$	\(\Q\)	None	✓				2667.2.a.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{7}-3q^{8}+2q^{13}+q^{14}+\cdots\)
8001.2.a.f	$8001$	$2$	8001.a	1.a	$1$	$1$	$1$	$63.888$	\(\Q\)	None	✓				2667.2.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{7}+2q^{13}-2q^{14}+\cdots\)
8001.2.a.g	$8001$	$2$	8001.a	1.a	$1$	$1$	$1$	$63.888$	\(\Q\)	None	✓	✓			8001.2.a.b	$1$	$0$	\(2\)	\(0\)	\(1\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{5}-q^{7}+2q^{10}+\cdots\)
8001.2.a.h	$8001$	$2$	8001.a	1.a	$1$	$2$	$2$	$63.888$	\(\Q(\sqrt{17}) \)	None	✓				2667.2.a.i	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{4}+(-1-\beta )q^{5}-q^{7}+\cdots\)
8001.2.a.i	$8001$	$2$	8001.a	1.a	$1$	$2$	$2$	$63.888$	\(\Q(\sqrt{17}) \)	None	✓				2667.2.a.f	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+(1+\beta )q^{5}-q^{7}+(2-2\beta )q^{11}+\cdots\)
8001.2.a.j	$8001$	$2$	8001.a	1.a	$1$	$2$	$2$	$63.888$	\(\Q(\sqrt{2}) \)	None	✓				2667.2.a.g	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-\beta q^{5}+q^{7}-2\beta q^{8}-2q^{10}+\cdots\)
8001.2.a.k	$8001$	$2$	8001.a	1.a	$1$	$2$	$2$	$63.888$	\(\Q(\sqrt{6}) \)	None	✓				2667.2.a.h	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+4q^{4}+\beta q^{5}+q^{7}+2\beta q^{8}+\cdots\)
8001.2.a.l	$8001$	$2$	8001.a	1.a	$1$	$7$	$7$	$63.888$	7.7.118870813.1	None	✓				2667.2.a.j	$1$	$1$	\(2\)	\(0\)	\(8\)	\(7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
8001.2.a.m	$8001$	$2$	8001.a	1.a	$1$	$11$	$11$	$63.888$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				2667.2.a.k	$1$	$1$	\(2\)	\(0\)	\(-1\)	\(-11\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}-q^{7}+\cdots\)
8001.2.a.n	$8001$	$2$	8001.a	1.a	$1$	$12$	$12$	$63.888$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				889.2.a.a	$1$	$0$	\(7\)	\(0\)	\(7\)	\(12\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{6}+\beta _{7})q^{4}+\cdots\)
8001.2.a.o	$8001$	$2$	8001.a	1.a	$1$	$13$	$13$	$63.888$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓				2667.2.a.l	$1$	$1$	\(-4\)	\(0\)	\(-12\)	\(13\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{10}+\cdots)q^{5}+\cdots\)
8001.2.a.p	$8001$	$2$	8001.a	1.a	$1$	$14$	$14$	$63.888$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				2667.2.a.m	$1$	$0$	\(5\)	\(0\)	\(4\)	\(-14\)		$-$	$+$	$+$	$-$	$5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}-q^{7}+\cdots\)
8001.2.a.q	$8001$	$2$	8001.a	1.a	$1$	$15$	$15$	$63.888$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓				889.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-7\)	\(-15\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{8}q^{5}-q^{7}+\cdots\)
8001.2.a.r	$8001$	$2$	8001.a	1.a	$1$	$16$	$16$	$63.888$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		2667.2.a.o	$1$	$1$	\(-5\)	\(0\)	\(1\)	\(-16\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{13}q^{5}-q^{7}+\cdots\)
8001.2.a.s	$8001$	$2$	8001.a	1.a	$1$	$16$	$16$	$63.888$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓				2667.2.a.n	$1$	$0$	\(-4\)	\(0\)	\(-5\)	\(-16\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{7}q^{5}-q^{7}+\cdots\)
8001.2.a.t	$8001$	$2$	8001.a	1.a	$1$	$16$	$16$	$63.888$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓				889.2.a.c	$1$	$0$	\(2\)	\(0\)	\(9\)	\(-16\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{11})q^{5}+\cdots\)
8001.2.a.u	$8001$	$2$	8001.a	1.a	$1$	$18$	$18$	$63.888$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓				2667.2.a.p	$1$	$0$	\(6\)	\(0\)	\(10\)	\(18\)		$-$	$-$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{12})q^{5}+\cdots\)
8001.2.a.v	$8001$	$2$	8001.a	1.a	$1$	$19$	$19$	$63.888$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓		✓		2667.2.a.q	$1$	$0$	\(-4\)	\(0\)	\(-5\)	\(19\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{10}q^{5}+q^{7}+\cdots\)
8001.2.a.w	$8001$	$2$	8001.a	1.a	$1$	$20$	$20$	$63.888$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		889.2.a.d	$1$	$1$	\(-8\)	\(0\)	\(-3\)	\(20\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{16}q^{5}+q^{7}+\cdots\)
8001.2.a.x	$8001$	$2$	8001.a	1.a	$1$	$22$	$22$	$63.888$		None	✓	✓	✓	✓	8001.2.a.x	$2$	$1$	\(0\)	\(0\)	\(0\)	\(22\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
8001.2.a.y	$8001$	$2$	8001.a	1.a	$1$	$28$	$28$	$63.888$		None	✓	✓	✓		8001.2.a.y	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
8001.2.a.z	$8001$	$2$	8001.a	1.a	$1$	$32$	$32$	$63.888$		None	✓	✓	✓	✓	8001.2.a.z	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-32\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
8001.2.a.ba	$8001$	$2$	8001.a	1.a	$1$	$40$	$40$	$63.888$		None	✓	✓	✓	✓	8001.2.a.ba	$2$	$0$	\(0\)	\(0\)	\(0\)	\(40\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8001))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8001)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(127))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(381))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(889))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1143))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2667))\)\(^{\oplus 2}\)