Space of modular forms of level 3640, weight 2, and trivial character

• Label: 3640.2.a
• Level: $3640$
• Weight: $2$
• Character orbit: 3640.a
• Rep. character: $\chi_{3640}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $27$
• Sturm bound: $1344$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(1344\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3640))\).

	Total	New	Old
Modular forms	688	72	616
Cusp forms	657	72	585
Eisenstein series	31	0	31

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

\(2\)	\(5\)	\(7\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(4\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(4\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(7\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(5\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(6\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(5\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(3\)
Plus space				\(+\)	\(32\)
Minus space				\(-\)	\(40\)

Trace form

\( 72 q + 8 q^{3} + 48 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	7	13
3640.2.a.a	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
3640.2.a.b	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}-2q^{9}+5q^{11}-q^{13}+\cdots\)
3640.2.a.c	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-q^{7}-2q^{9}-3q^{11}-q^{13}+\cdots\)
3640.2.a.d	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-3q^{9}+2q^{11}+q^{13}+\cdots\)
3640.2.a.e	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{7}-3q^{9}-q^{13}+6q^{17}+\cdots\)
3640.2.a.f	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-3q^{9}-2q^{11}-q^{13}+\cdots\)
3640.2.a.g	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
3640.2.a.h	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+q^{7}+q^{9}+6q^{11}-q^{13}+\cdots\)
3640.2.a.i	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(1\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3640.2.a.j	$3640$	$2$	3640.a	1.a	$1$	$1$	$1$	$29.066$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-q^{7}+q^{9}-q^{13}+2q^{15}+\cdots\)
3640.2.a.k	$3640$	$2$	3640.a	1.a	$1$	$2$	$2$	$29.066$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}-q^{7}+(-2+\beta )q^{9}-q^{13}+\cdots\)
3640.2.a.l	$3640$	$2$	3640.a	1.a	$1$	$2$	$2$	$29.066$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}+q^{7}+\beta q^{9}+(-2+2\beta )q^{11}+\cdots\)
3640.2.a.m	$3640$	$2$	3640.a	1.a	$1$	$2$	$2$	$29.066$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}-3q^{9}+\beta q^{11}+q^{13}+\cdots\)
3640.2.a.n	$3640$	$2$	3640.a	1.a	$1$	$2$	$2$	$29.066$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+q^{7}+(-2+\beta )q^{9}-4\beta q^{11}+\cdots\)
3640.2.a.o	$3640$	$2$	3640.a	1.a	$1$	$3$	$3$	$29.066$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(-3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}-q^{5}-q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
3640.2.a.p	$3640$	$2$	3640.a	1.a	$1$	$3$	$3$	$29.066$	3.3.1101.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(3\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+q^{7}+(4-\beta _{1}+\beta _{2})q^{9}+\cdots\)
3640.2.a.q	$3640$	$2$	3640.a	1.a	$1$	$3$	$3$	$29.066$	3.3.1573.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(3\)	\(-3\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}-q^{7}+(2+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3640.2.a.r	$3640$	$2$	3640.a	1.a	$1$	$4$	$4$	$29.066$	4.4.13676.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(-4\)		$+$	$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}-q^{7}+(3+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
3640.2.a.s	$3640$	$2$	3640.a	1.a	$1$	$4$	$4$	$29.066$	4.4.2225.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(-4\)		$+$	$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}-q^{7}+(\beta _{1}+\beta _{2})q^{9}+\cdots\)
3640.2.a.t	$3640$	$2$	3640.a	1.a	$1$	$4$	$4$	$29.066$	4.4.24197.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(4\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+q^{7}+\beta _{2}q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
3640.2.a.u	$3640$	$2$	3640.a	1.a	$1$	$4$	$4$	$29.066$	4.4.24197.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+q^{5}-q^{7}+(4+\beta _{1})q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
3640.2.a.v	$3640$	$2$	3640.a	1.a	$1$	$4$	$4$	$29.066$	4.4.2225.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(-4\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}-q^{7}+(\beta _{1}+\beta _{2})q^{9}+\cdots\)
3640.2.a.w	$3640$	$2$	3640.a	1.a	$1$	$4$	$4$	$29.066$	4.4.23724.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+q^{7}+(1+\beta _{3})q^{9}+(1+\cdots)q^{11}+\cdots\)
3640.2.a.x	$3640$	$2$	3640.a	1.a	$1$	$4$	$4$	$29.066$	4.4.4913.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-4\)	\(-4\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-q^{5}-q^{7}+(1-\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
3640.2.a.y	$3640$	$2$	3640.a	1.a	$1$	$5$	$5$	$29.066$	5.5.1194649.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(5\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+q^{7}+(\beta _{1}+\beta _{2})q^{9}+\cdots\)
3640.2.a.z	$3640$	$2$	3640.a	1.a	$1$	$5$	$5$	$29.066$	5.5.2112217.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(5\)	\(5\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+q^{7}+(1+\beta _{2}-\beta _{3}+\cdots)q^{9}+\cdots\)
3640.2.a.ba	$3640$	$2$	3640.a	1.a	$1$	$7$	$7$	$29.066$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(4\)	\(7\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+q^{5}+q^{7}+(2-\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3640))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3640)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(364))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(455))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(728))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(910))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1820))\)\(^{\oplus 2}\)