Space of modular forms of level 2028, weight 4, and trivial character

• Label: 2028.4.a
• Level: $2028$
• Weight: $4$
• Character orbit: 2028.a
• Rep. character: $\chi_{2028}(1,\cdot)$
• Character field: $\Q$
• Dimension: $77$
• Newform subspaces: $19$
• Sturm bound: $1456$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2028.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(1456\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2028))\).

	Total	New	Old
Modular forms	1134	77	1057
Cusp forms	1050	77	973
Eisenstein series	84	0	84

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
\(-\)	\(+\)	\(+\)	\(-\)	\(20\)
\(-\)	\(+\)	\(-\)	\(+\)	\(19\)
\(-\)	\(-\)	\(+\)	\(+\)	\(21\)
\(-\)	\(-\)	\(-\)	\(-\)	\(17\)
Plus space			\(+\)	\(40\)
Minus space			\(-\)	\(37\)

Trace form

\( 77 q - 3 q^{3} + 2 q^{5} + 12 q^{7} + 693 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2028))\) 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
2028.4.a.a	$2028$	$4$	2028.a	1.a	$1$	$1$	$1$	$119.656$	\(\Q\)	None	✓				156.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(6\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{5}+4q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
2028.4.a.b	$2028$	$4$	2028.a	1.a	$1$	$1$	$1$	$119.656$	\(\Q\)	None	✓				156.4.a.b	$1$	$0$	\(0\)	\(3\)	\(2\)	\(32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}+2^{5}q^{7}+9q^{9}+68q^{11}+\cdots\)
2028.4.a.c	$2028$	$4$	2028.a	1.a	$1$	$1$	$1$	$119.656$	\(\Q\)	None	✓				12.4.a.a	$1$	$0$	\(0\)	\(3\)	\(18\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+18q^{5}-8q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
2028.4.a.d	$2028$	$4$	2028.a	1.a	$1$	$2$	$2$	$119.656$	\(\Q(\sqrt{22}) \)	None	✓				156.4.a.c	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(-8\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta q^{5}+(-4-3\beta )q^{7}+9q^{9}+\cdots\)
2028.4.a.e	$2028$	$4$	2028.a	1.a	$1$	$2$	$2$	$119.656$	\(\Q(\sqrt{10}) \)	None	✓				156.4.a.d	$1$	$0$	\(0\)	\(6\)	\(-24\)	\(-8\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-12+\beta )q^{5}+(-4-3\beta )q^{7}+\cdots\)
2028.4.a.f	$2028$	$4$	2028.a	1.a	$1$	$2$	$2$	$119.656$	\(\Q(\sqrt{3}) \)	None	✓				156.4.q.b	$1$	$1$	\(0\)	\(6\)	\(-24\)	\(-6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-12+3\beta )q^{5}+(-3+\beta )q^{7}+\cdots\)
2028.4.a.g	$2028$	$4$	2028.a	1.a	$1$	$2$	$2$	$119.656$	\(\Q(\sqrt{3}) \)	None	✓				156.4.b.a	$2$	$1$	\(0\)	\(6\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+2\beta q^{5}-3\beta q^{7}+9q^{9}-5\beta q^{11}+\cdots\)
2028.4.a.h	$2028$	$4$	2028.a	1.a	$1$	$2$	$2$	$119.656$	\(\Q(\sqrt{3}) \)	None	✓				156.4.q.a	$2$	$1$	\(0\)	\(6\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2\beta q^{5}-11\beta q^{7}+9q^{9}+22\beta q^{11}+\cdots\)
2028.4.a.i	$2028$	$4$	2028.a	1.a	$1$	$2$	$2$	$119.656$	\(\Q(\sqrt{3}) \)	None	✓				156.4.q.b	$1$	$1$	\(0\)	\(6\)	\(24\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(12+3\beta )q^{5}+(3+\beta )q^{7}+9q^{9}+\cdots\)
2028.4.a.j	$2028$	$4$	2028.a	1.a	$1$	$4$	$4$	$119.656$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				156.4.i.b	$1$	$1$	\(0\)	\(-12\)	\(-3\)	\(15\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-1+\beta _{2})q^{5}+(4-\beta _{1})q^{7}+\cdots\)
2028.4.a.k	$2028$	$4$	2028.a	1.a	$1$	$4$	$4$	$119.656$	4.4.47664588.1	None	✓				156.4.b.b	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta _{1}q^{5}-\beta _{2}q^{7}+9q^{9}+(3\beta _{1}+\cdots)q^{11}+\cdots\)
2028.4.a.l	$2028$	$4$	2028.a	1.a	$1$	$4$	$4$	$119.656$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				156.4.i.b	$1$	$1$	\(0\)	\(-12\)	\(3\)	\(-15\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1-\beta _{2})q^{5}+(-4+\beta _{1})q^{7}+\cdots\)
2028.4.a.m	$2028$	$4$	2028.a	1.a	$1$	$4$	$4$	$119.656$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				156.4.i.a	$1$	$0$	\(0\)	\(12\)	\(-7\)	\(-11\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2+\beta _{2})q^{5}+(-3+\beta _{1}+\cdots)q^{7}+\cdots\)
2028.4.a.n	$2028$	$4$	2028.a	1.a	$1$	$4$	$4$	$119.656$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				156.4.i.a	$1$	$0$	\(0\)	\(12\)	\(7\)	\(11\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(2-\beta _{2})q^{5}+(3-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
2028.4.a.o	$2028$	$4$	2028.a	1.a	$1$	$6$	$6$	$119.656$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				156.4.q.c	$2$	$0$	\(0\)	\(-18\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(\beta _{1}-\beta _{2})q^{5}+(\beta _{2}+\beta _{4})q^{7}+\cdots\)
2028.4.a.p	$2028$	$4$	2028.a	1.a	$1$	$9$	$9$	$119.656$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		2028.4.a.p	$1$	$1$	\(0\)	\(-27\)	\(-3\)	\(-47\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-\beta _{1}q^{5}+(-6-\beta _{2}+\beta _{7})q^{7}+\cdots\)
2028.4.a.q	$2028$	$4$	2028.a	1.a	$1$	$9$	$9$	$119.656$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		2028.4.a.p	$1$	$0$	\(0\)	\(-27\)	\(3\)	\(47\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta _{1}q^{5}+(6+\beta _{2}-\beta _{7})q^{7}+\cdots\)
2028.4.a.r	$2028$	$4$	2028.a	1.a	$1$	$9$	$9$	$119.656$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		2028.4.a.r	$1$	$1$	\(0\)	\(27\)	\(-11\)	\(-37\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-\beta _{4})q^{5}+(-4-\beta _{5}+\cdots)q^{7}+\cdots\)
2028.4.a.s	$2028$	$4$	2028.a	1.a	$1$	$9$	$9$	$119.656$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		2028.4.a.r	$1$	$0$	\(0\)	\(27\)	\(11\)	\(37\)		$-$	$-$	$+$	$+$	$2^{3}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(1+\beta _{4})q^{5}+(4+\beta _{5})q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2028))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2028)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1014))\)\(^{\oplus 2}\)