Space of modular forms of level 1764, weight 4, and trivial character

• Label: 1764.4.a
• Level: $1764$
• Weight: $4$
• Character orbit: 1764.a
• Rep. character: $\chi_{1764}(1,\cdot)$
• Character field: $\Q$
• Dimension: $51$
• Newform subspaces: $29$
• Sturm bound: $1344$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(1344\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1764))\).

	Total	New	Old
Modular forms	1056	51	1005
Cusp forms	960	51	909
Eisenstein series	96	0	96

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

\(2\)	\(3\)	\(7\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)	\(16\)
\(-\)	\(-\)	\(-\)	\(-\)	\(15\)
Plus space			\(+\)	\(28\)
Minus space			\(-\)	\(23\)

Trace form

\( 51 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\) 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	7
1764.4.a.a	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				196.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-20\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-20q^{5}-44q^{11}+44q^{13}+72q^{17}+\cdots\)
1764.4.a.b	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				12.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-18\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-18q^{5}-6^{2}q^{11}+10q^{13}+18q^{17}+\cdots\)
1764.4.a.c	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				28.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}+40q^{11}+12q^{13}-58q^{17}+\cdots\)
1764.4.a.d	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				588.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+20q^{11}-4q^{13}-24q^{17}+\cdots\)
1764.4.a.e	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				252.4.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-89q^{13}+163q^{19}-5^{3}q^{25}+19q^{31}+\cdots\)
1764.4.a.f	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				252.4.k.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-19q^{13}+107q^{19}-5^{3}q^{25}-17^{2}q^{31}+\cdots\)
1764.4.a.g	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				252.4.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+19q^{13}-107q^{19}-5^{3}q^{25}+17^{2}q^{31}+\cdots\)
1764.4.a.h	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				252.4.k.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+89q^{13}-163q^{19}-5^{3}q^{25}-19q^{31}+\cdots\)
1764.4.a.i	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				588.4.a.b	$1$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+20q^{11}+4q^{13}+24q^{17}+\cdots\)
1764.4.a.j	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				84.4.a.a	$1$	$1$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}-6^{2}q^{11}-62q^{13}+114q^{17}+\cdots\)
1764.4.a.k	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				28.4.a.b	$1$	$1$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}+12q^{11}+82q^{13}-30q^{17}+\cdots\)
1764.4.a.l	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				84.4.a.b	$1$	$1$	\(0\)	\(0\)	\(14\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{5}-4q^{11}-54q^{13}-14q^{17}+\cdots\)
1764.4.a.m	$1764$	$4$	1764.a	1.a	$1$	$1$	$1$	$104.079$	\(\Q\)	None	✓				196.4.a.a	$1$	$1$	\(0\)	\(0\)	\(20\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+20q^{5}-44q^{11}-44q^{13}-72q^{17}+\cdots\)
1764.4.a.n	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{37}) \)	None	✓				28.4.e.a	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-7-2\beta )q^{5}+(-2^{4}+7\beta )q^{11}+\cdots\)
1764.4.a.o	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{193}) \)	None	✓				84.4.i.a	$1$	$0$	\(0\)	\(0\)	\(-11\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{5}+(-1+7\beta )q^{11}+5\beta q^{13}+\cdots\)
1764.4.a.p	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{57}) \)	None	✓				84.4.i.b	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{5}+(26+\beta )q^{11}+(-33+\cdots)q^{13}+\cdots\)
1764.4.a.q	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{2}) \)	None	✓	✓			1764.4.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7\beta q^{5}-28q^{11}-3\beta q^{13}-35\beta q^{17}+\cdots\)
1764.4.a.r	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{385}) \)	None	✓				252.4.k.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+\beta q^{11}-54q^{13}-2\beta q^{17}+\cdots\)
1764.4.a.s	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{7}) \)	None	✓				252.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-\beta q^{11}-26q^{13}-5\beta q^{17}+\cdots\)
1764.4.a.t	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{7}) \)	None	✓				252.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+13\beta q^{11}+30q^{13}+23\beta q^{17}+\cdots\)
1764.4.a.u	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{385}) \)	None	✓				252.4.k.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-\beta q^{11}+54q^{13}-2\beta q^{17}+\cdots\)
1764.4.a.v	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{2}) \)	None	✓				196.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}+26q^{11}-24\beta q^{13}-73\beta q^{17}+\cdots\)
1764.4.a.w	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{2}) \)	None	✓	✓			1764.4.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7\beta q^{5}+28q^{11}+3\beta q^{13}-35\beta q^{17}+\cdots\)
1764.4.a.x	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{57}) \)	None	✓				84.4.i.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(26+\beta )q^{11}+(33+5\beta )q^{13}+\cdots\)
1764.4.a.y	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{193}) \)	None	✓				84.4.i.a	$1$	$1$	\(0\)	\(0\)	\(11\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(6-\beta )q^{5}+(6-7\beta )q^{11}+(-5+5\beta )q^{13}+\cdots\)
1764.4.a.z	$1764$	$4$	1764.a	1.a	$1$	$2$	$2$	$104.079$	\(\Q(\sqrt{37}) \)	None	✓				28.4.e.a	$1$	$0$	\(0\)	\(0\)	\(14\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(7+2\beta )q^{5}+(-2^{4}+7\beta )q^{11}+(-14+\cdots)q^{13}+\cdots\)
1764.4.a.ba	$1764$	$4$	1764.a	1.a	$1$	$4$	$4$	$104.079$	4.4.136768.1	None	✓				588.4.a.j	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(-3\beta _{1}+\beta _{3})q^{11}-3\beta _{3}q^{13}+\cdots\)
1764.4.a.bb	$1764$	$4$	1764.a	1.a	$1$	$4$	$4$	$104.079$	\(\Q(\sqrt{7}, \sqrt{109})\)	None	✓	✓	✓		1764.4.a.bb	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{8}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+\beta _{3}q^{11}-\beta _{2}q^{13}+9\beta _{1}q^{17}+\cdots\)
1764.4.a.bc	$1764$	$4$	1764.a	1.a	$1$	$4$	$4$	$104.079$	4.4.136768.1	None	✓				588.4.a.j	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(-3\beta _{1}+\beta _{3})q^{11}+3\beta _{3}q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1764))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1764)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(882))\)\(^{\oplus 2}\)