Space of modular forms of level 4788, weight 2, and trivial character

• Label: 4788.2.a
• Level: $4788$
• Weight: $2$
• Character orbit: 4788.a
• Rep. character: $\chi_{4788}(1,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $20$
• Sturm bound: $1920$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(1920\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4788))\).

	Total	New	Old
Modular forms	984	46	938
Cusp forms	937	46	891
Eisenstein series	47	0	47

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

\(2\)	\(3\)	\(7\)	\(19\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(7\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(6\)
Plus space				\(+\)	\(20\)
Minus space				\(-\)	\(26\)

Trace form

\( 46 q + 8 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4788))\) 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	19
4788.2.a.a	$4788$	$2$	4788.a	1.a	$1$	$1$	$1$	$38.232$	\(\Q\)	None	✓				1596.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-6q^{11}+2q^{13}+2q^{17}+\cdots\)
4788.2.a.b	$4788$	$2$	4788.a	1.a	$1$	$1$	$1$	$38.232$	\(\Q\)	None	✓				1596.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-2q^{11}+2q^{13}-q^{19}-2q^{23}+\cdots\)
4788.2.a.c	$4788$	$2$	4788.a	1.a	$1$	$1$	$1$	$38.232$	\(\Q\)	None	✓				1596.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+2q^{11}+6q^{13}+8q^{17}+q^{19}+\cdots\)
4788.2.a.d	$4788$	$2$	4788.a	1.a	$1$	$1$	$1$	$38.232$	\(\Q\)	None	✓				1596.2.a.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}+2q^{11}+2q^{13}-2q^{17}+\cdots\)
4788.2.a.e	$4788$	$2$	4788.a	1.a	$1$	$1$	$1$	$38.232$	\(\Q\)	None	✓				532.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}-4q^{11}+4q^{13}-6q^{17}+\cdots\)
4788.2.a.f	$4788$	$2$	4788.a	1.a	$1$	$1$	$1$	$38.232$	\(\Q\)	None	✓				1596.2.a.d	$1$	$1$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}+2q^{11}-6q^{13}-2q^{17}+\cdots\)
4788.2.a.g	$4788$	$2$	4788.a	1.a	$1$	$2$	$2$	$38.232$	\(\Q(\sqrt{21}) \)	None	✓				532.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+q^{7}+(2-\beta )q^{11}-q^{13}+(-2+\cdots)q^{17}+\cdots\)
4788.2.a.h	$4788$	$2$	4788.a	1.a	$1$	$2$	$2$	$38.232$	\(\Q(\sqrt{2}) \)	None	✓				1596.2.a.h	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+q^{7}-2q^{11}-2\beta q^{13}-3\beta q^{17}+\cdots\)
4788.2.a.i	$4788$	$2$	4788.a	1.a	$1$	$2$	$2$	$38.232$	\(\Q(\sqrt{3}) \)	None	✓	✓			4788.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{7}-\beta q^{11}-4q^{13}-2\beta q^{17}+q^{19}+\cdots\)
4788.2.a.j	$4788$	$2$	4788.a	1.a	$1$	$2$	$2$	$38.232$	\(\Q(\sqrt{3}) \)	None	✓				1596.2.a.i	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-q^{7}-2\beta q^{11}-4q^{13}+\cdots\)
4788.2.a.k	$4788$	$2$	4788.a	1.a	$1$	$2$	$2$	$38.232$	\(\Q(\sqrt{7}) \)	None	✓				1596.2.a.f	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-q^{7}-2\beta q^{11}+(-3+\beta )q^{17}+\cdots\)
4788.2.a.l	$4788$	$2$	4788.a	1.a	$1$	$2$	$2$	$38.232$	\(\Q(\sqrt{5}) \)	None	✓				532.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}+(2-5\beta )q^{11}+(1-2\beta )q^{13}+\cdots\)
4788.2.a.m	$4788$	$2$	4788.a	1.a	$1$	$2$	$2$	$38.232$	\(\Q(\sqrt{5}) \)	None	✓				1596.2.a.g	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+q^{7}+2q^{11}+2\beta q^{13}+\cdots\)
4788.2.a.n	$4788$	$2$	4788.a	1.a	$1$	$2$	$2$	$38.232$	\(\Q(\sqrt{5}) \)	None	✓				532.2.a.d	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+2\beta )q^{5}-q^{7}+(1+\beta )q^{11}+(-3+\cdots)q^{13}+\cdots\)
4788.2.a.o	$4788$	$2$	4788.a	1.a	$1$	$3$	$3$	$38.232$	3.3.733.1	None	✓		✓		532.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{5}-q^{7}+(1+\beta _{2})q^{11}+\cdots\)
4788.2.a.p	$4788$	$2$	4788.a	1.a	$1$	$3$	$3$	$38.232$	3.3.1016.1	None	✓		✓		1596.2.a.j	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+q^{7}-2q^{11}+(1+\beta _{1})q^{13}+\cdots\)
4788.2.a.q	$4788$	$2$	4788.a	1.a	$1$	$4$	$4$	$38.232$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓	✓	✓	4788.2.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-q^{7}+(-\beta _{1}-\beta _{3})q^{11}+(-1+\cdots)q^{13}+\cdots\)
4788.2.a.r	$4788$	$2$	4788.a	1.a	$1$	$4$	$4$	$38.232$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓	✓	✓	4788.2.a.r	$2$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+q^{7}+(-\beta _{1}+\beta _{2})q^{11}+(-3+\cdots)q^{13}+\cdots\)
4788.2.a.s	$4788$	$2$	4788.a	1.a	$1$	$4$	$4$	$38.232$	4.4.32448.1	None	✓	✓	✓		4788.2.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+q^{7}+(\beta _{1}+\beta _{2})q^{11}+(3-\beta _{3})q^{13}+\cdots\)
4788.2.a.t	$4788$	$2$	4788.a	1.a	$1$	$6$	$6$	$38.232$	6.6.56310016.1	None	✓	✓	✓	✓	4788.2.a.t	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{5}-q^{7}+\beta _{5}q^{11}+(1-\beta _{3})q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4788))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4788)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(266))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(399))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(532))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(684))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(798))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1197))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1596))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2394))\)\(^{\oplus 2}\)