Space of modular forms of level 3744, weight 2, and trivial character

• Label: 3744.2.a
• Level: $3744$
• Weight: $2$
• Character orbit: 3744.a
• Rep. character: $\chi_{3744}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $32$
• Sturm bound: $1344$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(1344\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3744))\).

	Total	New	Old
Modular forms	704	60	644
Cusp forms	641	60	581
Eisenstein series	63	0	63

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

\(2\)	\(3\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(10\)
\(+\)	\(-\)	\(-\)	\(+\)	\(7\)
\(-\)	\(+\)	\(+\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)	\(11\)
Plus space			\(+\)	\(27\)
Minus space			\(-\)	\(33\)

Trace form

\( 60 q - 8 q^{17} + 44 q^{25} - 16 q^{29} + 24 q^{41} + 44 q^{49} + 56 q^{53} + 8 q^{61} + 8 q^{73} - 40 q^{77} - 80 q^{85} + 40 q^{89} + 72 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3744))\) 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	3	13
3744.2.a.a	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}-2q^{11}-q^{13}-6q^{17}+\cdots\)
3744.2.a.b	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓	✓			3744.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{11}+q^{13}+4q^{19}-4q^{23}+\cdots\)
3744.2.a.c	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓	✓			3744.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{11}+q^{13}-4q^{19}+4q^{23}+\cdots\)
3744.2.a.d	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{7}+2q^{11}-q^{13}-6q^{17}+\cdots\)
3744.2.a.e	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				416.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}+2q^{11}+q^{13}+3q^{17}+\cdots\)
3744.2.a.f	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				416.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}-2q^{11}+q^{13}+3q^{17}+\cdots\)
3744.2.a.g	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{13}-2q^{17}+2q^{19}+8q^{23}+\cdots\)
3744.2.a.h	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+4q^{11}+q^{13}+6q^{17}-6q^{19}+\cdots\)
3744.2.a.i	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-4q^{11}+q^{13}+6q^{17}+6q^{19}+\cdots\)
3744.2.a.j	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+q^{13}-2q^{17}-2q^{19}-8q^{23}+\cdots\)
3744.2.a.k	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-6q^{11}-q^{13}+2q^{17}+\cdots\)
3744.2.a.l	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{11}+q^{13}+6q^{17}-8q^{19}+\cdots\)
3744.2.a.m	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓	✓			3744.2.a.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{11}+q^{13}-4q^{19}-4q^{23}+\cdots\)
3744.2.a.n	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓	✓			3744.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{11}+q^{13}+4q^{19}+4q^{23}+\cdots\)
3744.2.a.o	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{11}+q^{13}+6q^{17}+8q^{19}+\cdots\)
3744.2.a.p	$3744$	$2$	3744.a	1.a	$1$	$1$	$1$	$29.896$	\(\Q\)	None	✓				1248.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}+6q^{11}-q^{13}+2q^{17}+\cdots\)
3744.2.a.q	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{5}) \)	None	✓				416.2.a.d	$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{5}-\beta q^{7}-2\beta q^{11}-q^{13}+3q^{17}+\cdots\)
3744.2.a.r	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{5}) \)	None	✓				1248.2.a.l	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+(-1-\beta )q^{7}+2q^{11}+\cdots\)
3744.2.a.s	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{5}) \)	None	✓				1248.2.a.l	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+(1+\beta )q^{7}-2q^{11}+\cdots\)
3744.2.a.t	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{2}) \)	None	✓	✓			3744.2.a.t	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{7}-\beta q^{11}-q^{13}+\beta q^{19}-2\beta q^{23}+\cdots\)
3744.2.a.u	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{2}) \)	None	✓	✓			3744.2.a.t	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{7}+\beta q^{11}-q^{13}+\beta q^{19}+2\beta q^{23}+\cdots\)
3744.2.a.v	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{5}) \)	None	✓				1248.2.a.k	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-6\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-3+\beta )q^{7}-2\beta q^{11}+\cdots\)
3744.2.a.w	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{5}) \)	None	✓				1248.2.a.k	$1$	$0$	\(0\)	\(0\)	\(2\)	\(6\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(3-\beta )q^{7}+2\beta q^{11}-q^{13}+\cdots\)
3744.2.a.x	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{17}) \)	None	✓				416.2.a.c	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-1-\beta )q^{7}+2q^{11}+\cdots\)
3744.2.a.y	$3744$	$2$	3744.a	1.a	$1$	$2$	$2$	$29.896$	\(\Q(\sqrt{17}) \)	None	✓				416.2.a.c	$1$	$0$	\(0\)	\(0\)	\(3\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(1+\beta )q^{7}-2q^{11}-q^{13}+\cdots\)
3744.2.a.z	$3744$	$2$	3744.a	1.a	$1$	$3$	$3$	$29.896$	3.3.148.1	None	✓		✓		1248.2.a.o	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{5}-\beta _{2}q^{7}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
3744.2.a.ba	$3744$	$2$	3744.a	1.a	$1$	$3$	$3$	$29.896$	3.3.148.1	None	✓				1248.2.a.o	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{5}+\beta _{2}q^{7}+(1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
3744.2.a.bb	$3744$	$2$	3744.a	1.a	$1$	$4$	$4$	$29.896$	\(\Q(\zeta_{20})^+\)	None	✓	✓	✓		3744.2.a.bb	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{5}-\beta _{2}q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
3744.2.a.bc	$3744$	$2$	3744.a	1.a	$1$	$4$	$4$	$29.896$	\(\Q(\zeta_{24})^+\)	None	✓	✓	✓		3744.2.a.bc	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+\beta _{3}q^{7}+(-2-\beta _{2})q^{11}+\cdots\)
3744.2.a.bd	$3744$	$2$	3744.a	1.a	$1$	$4$	$4$	$29.896$	\(\Q(\zeta_{24})^+\)	None	✓	✓	✓		3744.2.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-\beta _{3}q^{7}+(2+\beta _{2})q^{11}-q^{13}+\cdots\)
3744.2.a.be	$3744$	$2$	3744.a	1.a	$1$	$4$	$4$	$29.896$	4.4.13448.1	None	✓		✓		416.2.a.f	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{5}+(\beta _{1}-\beta _{2})q^{7}+\beta _{1}q^{11}+q^{13}+\cdots\)
3744.2.a.bf	$3744$	$2$	3744.a	1.a	$1$	$4$	$4$	$29.896$	\(\Q(\zeta_{20})^+\)	None	✓	✓	✓		3744.2.a.bb	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{5}-\beta _{2}q^{7}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3744))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3744)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(416))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(624))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(936))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1248))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1872))\)\(^{\oplus 2}\)