Space of modular forms of level 4005, weight 2, and trivial character

• Label: 4005.2.a
• Level: $4005$
• Weight: $2$
• Character orbit: 4005.a
• Rep. character: $\chi_{4005}(1,\cdot)$
• Character field: $\Q$
• Dimension: $148$
• Newform subspaces: $24$
• Sturm bound: $1080$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(1080\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4005))\).

	Total	New	Old
Modular forms	548	148	400
Cusp forms	533	148	385
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.

\(3\)	\(5\)	\(89\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(13\)
\(+\)	\(+\)	\(-\)	$-$	\(17\)
\(+\)	\(-\)	\(+\)	$-$	\(17\)
\(+\)	\(-\)	\(-\)	$+$	\(13\)
\(-\)	\(+\)	\(+\)	$-$	\(23\)
\(-\)	\(+\)	\(-\)	$+$	\(21\)
\(-\)	\(-\)	\(+\)	$+$	\(21\)
\(-\)	\(-\)	\(-\)	$-$	\(23\)
Plus space			\(+\)	\(68\)
Minus space			\(-\)	\(80\)

Trace form

\( 148 q - 4 q^{2} + 152 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	5	89
4005.2.a.a	$4005$	$2$	4005.a	1.a	$1$	$1$	$1$	$31.980$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}+4q^{7}+3q^{8}-q^{10}+\cdots\)
4005.2.a.b	$4005$	$2$	4005.a	1.a	$1$	$1$	$1$	$31.980$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{5}-4q^{7}-2q^{11}+4q^{13}+\cdots\)
4005.2.a.c	$4005$	$2$	4005.a	1.a	$1$	$1$	$1$	$31.980$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}-4q^{7}+2q^{11}+4q^{13}+\cdots\)
4005.2.a.d	$4005$	$2$	4005.a	1.a	$1$	$1$	$1$	$31.980$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{5}-3q^{8}-q^{10}-4q^{11}+\cdots\)
4005.2.a.e	$4005$	$2$	4005.a	1.a	$1$	$2$	$2$	$31.980$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}-q^{5}-\beta q^{7}+\cdots\)
4005.2.a.f	$4005$	$2$	4005.a	1.a	$1$	$2$	$2$	$31.980$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}-q^{5}+(-1+\beta )q^{7}-\beta q^{8}+\cdots\)
4005.2.a.g	$4005$	$2$	4005.a	1.a	$1$	$2$	$2$	$31.980$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}-q^{5}+(4+\beta )q^{8}+\cdots\)
4005.2.a.h	$4005$	$2$	4005.a	1.a	$1$	$2$	$2$	$31.980$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}+(1+\beta )q^{7}-3q^{8}+\cdots\)
4005.2.a.i	$4005$	$2$	4005.a	1.a	$1$	$3$	$3$	$31.980$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}-q^{5}+(1+\beta _{2})q^{7}+\cdots\)
4005.2.a.j	$4005$	$2$	4005.a	1.a	$1$	$3$	$3$	$31.980$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-3\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots\)
4005.2.a.k	$4005$	$2$	4005.a	1.a	$1$	$4$	$4$	$31.980$	4.4.8069.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-4\)	\(12\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-q^{5}+3q^{7}+\cdots\)
4005.2.a.l	$4005$	$2$	4005.a	1.a	$1$	$4$	$4$	$31.980$	4.4.725.1	None	✓	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{2}+(\beta _{1}+\beta _{3})q^{4}+\cdots\)
4005.2.a.m	$4005$	$2$	4005.a	1.a	$1$	$4$	$4$	$31.980$	4.4.2777.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1-\beta _{3})q^{4}+q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
4005.2.a.n	$4005$	$2$	4005.a	1.a	$1$	$6$	$6$	$31.980$	6.6.10407557.1	None	✓	$1$	$0$	\(4\)	\(0\)	\(6\)	\(1\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(2+\beta _{2}+\beta _{3})q^{4}+q^{5}+\cdots\)
4005.2.a.o	$4005$	$2$	4005.a	1.a	$1$	$7$	$7$	$31.980$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(-7\)	\(-16\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{3})q^{4}-q^{5}+\cdots\)
4005.2.a.p	$4005$	$2$	4005.a	1.a	$1$	$8$	$8$	$31.980$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(8\)	\(-6\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{1}-\beta _{3}-\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
4005.2.a.q	$4005$	$2$	4005.a	1.a	$1$	$9$	$9$	$31.980$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(9\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
4005.2.a.r	$4005$	$2$	4005.a	1.a	$1$	$10$	$10$	$31.980$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(-6\)	\(0\)	\(-10\)	\(7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
4005.2.a.s	$4005$	$2$	4005.a	1.a	$1$	$10$	$10$	$31.980$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-10\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-q^{5}+\beta _{6}q^{7}+\cdots\)
4005.2.a.t	$4005$	$2$	4005.a	1.a	$1$	$10$	$10$	$31.980$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(9\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+q^{5}+(1-\beta _{4}+\cdots)q^{7}+\cdots\)
4005.2.a.u	$4005$	$2$	4005.a	1.a	$1$	$12$	$12$	$31.980$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(12\)	\(-8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
4005.2.a.v	$4005$	$2$	4005.a	1.a	$1$	$12$	$12$	$31.980$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(3\)	\(0\)	\(-12\)	\(-8\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
4005.2.a.w	$4005$	$2$	4005.a	1.a	$1$	$17$	$17$	$31.980$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(-5\)	\(0\)	\(-17\)	\(12\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-q^{5}+(1+\beta _{8}+\cdots)q^{7}+\cdots\)
4005.2.a.x	$4005$	$2$	4005.a	1.a	$1$	$17$	$17$	$31.980$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(0\)	\(17\)	\(12\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+q^{5}+(1+\beta _{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4005))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4005)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(89))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(267))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(445))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(801))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1335))\)\(^{\oplus 2}\)