Space of modular forms of level 1776, weight 2, and trivial character

• Label: 1776.2.a
• Level: $1776$
• Weight: $2$
• Character orbit: 1776.a
• Rep. character: $\chi_{1776}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $21$
• Sturm bound: $608$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1776.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(608\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1776))\).

	Total	New	Old
Modular forms	316	36	280
Cusp forms	293	36	257
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(37\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(33\)	\(5\)	\(28\)		\(31\)	\(5\)	\(26\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(45\)	\(4\)	\(41\)		\(42\)	\(4\)	\(38\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(46\)	\(5\)	\(41\)		\(43\)	\(5\)	\(38\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(34\)	\(4\)	\(30\)		\(31\)	\(4\)	\(27\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(38\)	\(6\)	\(32\)		\(35\)	\(6\)	\(29\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(42\)	\(3\)	\(39\)		\(39\)	\(3\)	\(36\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(41\)	\(2\)	\(39\)		\(38\)	\(2\)	\(36\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(37\)	\(7\)	\(30\)		\(34\)	\(7\)	\(27\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(150\)	\(14\)	\(136\)		\(139\)	\(14\)	\(125\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(166\)	\(22\)	\(144\)		\(154\)	\(22\)	\(132\)		\(12\)	\(0\)	\(12\)

Trace form

36 * q + 36 * q^9 + 4 * q^15 - 4 * q^19 - 24 * q^23 + 44 * q^25 + 20 * q^31 + 8 * q^33 + 8 * q^39 + 4 * q^43 + 44 * q^49 - 4 * q^51 + 16 * q^53 - 16 * q^55 + 8 * q^59 - 16 * q^69 + 32 * q^71 + 8 * q^73 + 16 * q^75 + 16 * q^77 + 4 * q^79 + 36 * q^81 + 16 * q^83 - 16 * q^85 - 12 * q^87 + 16 * q^89 + 32 * q^91 + 16 * q^95 + 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1776))\) 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	37
1776.2.a.a	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				888.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)
1776.2.a.b	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				444.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}+q^{9}+4q^{11}+\cdots\)
1776.2.a.c	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				222.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-3q^{11}-q^{13}-3q^{17}+\cdots\)
1776.2.a.d	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				888.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{9}+2q^{13}-4q^{15}+\cdots\)
1776.2.a.e	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				222.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{7}+q^{9}+q^{11}-3q^{13}+\cdots\)
1776.2.a.f	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				222.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-3q^{7}+q^{9}-5q^{11}+\cdots\)
1776.2.a.g	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				888.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+q^{7}+q^{9}+3q^{11}-5q^{13}+\cdots\)
1776.2.a.h	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				222.2.a.d	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}+q^{9}-q^{11}+q^{13}-3q^{17}+\cdots\)
1776.2.a.i	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				444.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-4q^{11}-2q^{13}-6q^{19}+\cdots\)
1776.2.a.j	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				888.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-6q^{13}-4q^{17}-4q^{19}+\cdots\)
1776.2.a.k	$1776$	$2$	1776.a	1.a	$1$	$1$	$1$	$14.181$	\(\Q\)	None	✓				222.2.a.b	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
1776.2.a.l	$1776$	$2$	1776.a	1.a	$1$	$2$	$2$	$14.181$	\(\Q(\sqrt{2}) \)	None	✓				888.2.a.g	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2+\beta )q^{5}+(2+2\beta )q^{7}+q^{9}+\cdots\)
1776.2.a.m	$1776$	$2$	1776.a	1.a	$1$	$2$	$2$	$14.181$	\(\Q(\sqrt{6}) \)	None	✓		✓		444.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}-2q^{7}+q^{9}+(2-2\beta )q^{13}+\cdots\)
1776.2.a.n	$1776$	$2$	1776.a	1.a	$1$	$2$	$2$	$14.181$	\(\Q(\sqrt{41}) \)	None	✓		✓		888.2.a.h	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta )q^{7}+q^{9}+(-1-\beta )q^{11}+\cdots\)
1776.2.a.o	$1776$	$2$	1776.a	1.a	$1$	$2$	$2$	$14.181$	\(\Q(\sqrt{3}) \)	None	✓				444.2.a.c	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta )q^{5}+2\beta q^{7}+q^{9}+\cdots\)
1776.2.a.p	$1776$	$2$	1776.a	1.a	$1$	$2$	$2$	$14.181$	\(\Q(\sqrt{5}) \)	None	✓		✓		888.2.a.f	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-8\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}-4q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
1776.2.a.q	$1776$	$2$	1776.a	1.a	$1$	$2$	$2$	$14.181$	\(\Q(\sqrt{3}) \)	None	✓				888.2.a.e	$1$	$0$	\(0\)	\(2\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}+2q^{7}+q^{9}+(2+2\beta )q^{11}+\cdots\)
1776.2.a.r	$1776$	$2$	1776.a	1.a	$1$	$3$	$3$	$14.181$	3.3.568.1	None	✓		✓		888.2.a.j	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(\beta _{1}+\beta _{2})q^{5}-\beta _{2}q^{7}+q^{9}+\cdots\)
1776.2.a.s	$1776$	$2$	1776.a	1.a	$1$	$3$	$3$	$14.181$	3.3.316.1	None	✓		✓		888.2.a.i	$1$	$0$	\(0\)	\(3\)	\(2\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1}-\beta _{2})q^{5}+(1+2\beta _{1}+\cdots)q^{7}+\cdots\)
1776.2.a.t	$1776$	$2$	1776.a	1.a	$1$	$3$	$3$	$14.181$	3.3.148.1	None	✓		✓		111.2.a.a	$1$	$0$	\(0\)	\(3\)	\(4\)	\(4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{2})q^{5}+(2-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1776.2.a.u	$1776$	$2$	1776.a	1.a	$1$	$4$	$4$	$14.181$	4.4.6224.1	None	✓		✓		111.2.a.b	$1$	$0$	\(0\)	\(-4\)	\(-2\)	\(-4\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta _{3})q^{5}+(-2+\beta _{2}+\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1776))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1776)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(296))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(444))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(592))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(888))\)\(^{\oplus 2}\)