Space of modular forms of level 2016, weight 4, and trivial character

• Label: 2016.4.a
• Level: $2016$
• Weight: $4$
• Character orbit: 2016.a
• Rep. character: $\chi_{2016}(1,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $34$
• Sturm bound: $1536$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2016.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(1536\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2016))\).

	Total	New	Old
Modular forms	1184	90	1094
Cusp forms	1120	90	1030
Eisenstein series	64	0	64

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
\(+\)	\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(+\)	\(-\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	$-$	\(13\)
\(+\)	\(-\)	\(-\)	$+$	\(15\)
\(-\)	\(+\)	\(+\)	$-$	\(9\)
\(-\)	\(+\)	\(-\)	$+$	\(9\)
\(-\)	\(-\)	\(+\)	$+$	\(14\)
\(-\)	\(-\)	\(-\)	$-$	\(12\)
Plus space			\(+\)	\(47\)
Minus space			\(-\)	\(43\)

Trace form

\( 90 q - 4 q^{5} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2016))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
2016.4.a.a	$2016$	$4$	2016.a	1.a	$1$	$1$	$1$	$118.948$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}-7q^{7}-4q^{11}-46q^{13}+82q^{17}+\cdots\)
2016.4.a.b	$2016$	$4$	2016.a	1.a	$1$	$1$	$1$	$118.948$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}+7q^{7}+4q^{11}-46q^{13}+82q^{17}+\cdots\)
2016.4.a.c	$2016$	$4$	2016.a	1.a	$1$	$1$	$1$	$118.948$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{7}+20q^{11}-20q^{13}+50q^{17}+\cdots\)
2016.4.a.d	$2016$	$4$	2016.a	1.a	$1$	$1$	$1$	$118.948$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{7}-20q^{11}-20q^{13}+50q^{17}+\cdots\)
2016.4.a.e	$2016$	$4$	2016.a	1.a	$1$	$1$	$1$	$118.948$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(18\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{5}-7q^{7}-44q^{11}+58q^{13}+\cdots\)
2016.4.a.f	$2016$	$4$	2016.a	1.a	$1$	$1$	$1$	$118.948$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(18\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{5}+7q^{7}+44q^{11}+58q^{13}+\cdots\)
2016.4.a.g	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{43}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(-14\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-8+\beta )q^{5}-7q^{7}+(2+\beta )q^{11}+\cdots\)
2016.4.a.h	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{43}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-16\)	\(14\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-8+\beta )q^{5}+7q^{7}+(-2-\beta )q^{11}+\cdots\)
2016.4.a.i	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{137}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-14\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{5}-7q^{7}+(11+\beta )q^{11}+\cdots\)
2016.4.a.j	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{137}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(14\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{5}+7q^{7}+(-11-\beta )q^{11}+\cdots\)
2016.4.a.k	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{11}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(-14\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{5}-7q^{7}+(22-7\beta )q^{11}+\cdots\)
2016.4.a.l	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{11}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(14\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{5}+7q^{7}+(-22+7\beta )q^{11}+\cdots\)
2016.4.a.m	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-14\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}-7q^{7}+(-24+\beta )q^{11}+\cdots\)
2016.4.a.n	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(14\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+7q^{7}+(24-\beta )q^{11}+(-26+\cdots)q^{13}+\cdots\)
2016.4.a.o	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(-14\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{5}-7q^{7}+(-22-2\beta )q^{11}+\cdots\)
2016.4.a.p	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(6\)	\(14\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{5}+7q^{7}+(22+2\beta )q^{11}+\cdots\)
2016.4.a.q	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(-14\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{5}-7q^{7}+(-11-11\beta )q^{11}+\cdots\)
2016.4.a.r	$2016$	$4$	2016.a	1.a	$1$	$2$	$2$	$118.948$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(14\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{5}+7q^{7}+(11+11\beta )q^{11}+\cdots\)
2016.4.a.s	$2016$	$4$	2016.a	1.a	$1$	$3$	$3$	$118.948$	3.3.2981.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(-21\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{5}-7q^{7}+(10+5\beta _{1}+\cdots)q^{11}+\cdots\)
2016.4.a.t	$2016$	$4$	2016.a	1.a	$1$	$3$	$3$	$118.948$	3.3.2981.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(21\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{5}+7q^{7}+(-10-5\beta _{1}+\cdots)q^{11}+\cdots\)
2016.4.a.u	$2016$	$4$	2016.a	1.a	$1$	$3$	$3$	$118.948$	3.3.37341.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(-21\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{5}-7q^{7}+(2^{4}+\beta _{1}-\beta _{2})q^{11}+\cdots\)
2016.4.a.v	$2016$	$4$	2016.a	1.a	$1$	$3$	$3$	$118.948$	3.3.37341.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(21\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{5}+7q^{7}+(-2^{4}-\beta _{1}+\cdots)q^{11}+\cdots\)
2016.4.a.w	$2016$	$4$	2016.a	1.a	$1$	$3$	$3$	$118.948$	3.3.621.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(6\)	\(-21\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2+2\beta _{1}+\beta _{2})q^{5}-7q^{7}+(-3\beta _{1}+\cdots)q^{11}+\cdots\)
2016.4.a.x	$2016$	$4$	2016.a	1.a	$1$	$3$	$3$	$118.948$	3.3.621.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(21\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2+2\beta _{1}+\beta _{2})q^{5}+7q^{7}+(3\beta _{1}+7\beta _{2})q^{11}+\cdots\)
2016.4.a.y	$2016$	$4$	2016.a	1.a	$1$	$3$	$3$	$118.948$	3.3.22700.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-21\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{2})q^{5}-7q^{7}+(-1+\beta _{1}+2\beta _{2})q^{11}+\cdots\)
2016.4.a.z	$2016$	$4$	2016.a	1.a	$1$	$3$	$3$	$118.948$	3.3.22700.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(10\)	\(21\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{2})q^{5}+7q^{7}+(1-\beta _{1}-2\beta _{2})q^{11}+\cdots\)
2016.4.a.ba	$2016$	$4$	2016.a	1.a	$1$	$4$	$4$	$118.948$	4.4.777569.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-28\)		$-$	$+$	$+$	$-$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{5}-7q^{7}+(6+\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
2016.4.a.bb	$2016$	$4$	2016.a	1.a	$1$	$4$	$4$	$118.948$	4.4.777569.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(28\)		$+$	$+$	$-$	$-$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{5}+7q^{7}+(-6-\beta _{1}+\cdots)q^{11}+\cdots\)
2016.4.a.bc	$2016$	$4$	2016.a	1.a	$1$	$4$	$4$	$118.948$	4.4.777569.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(-28\)		$+$	$+$	$+$	$+$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}-7q^{7}+(-6-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
2016.4.a.bd	$2016$	$4$	2016.a	1.a	$1$	$4$	$4$	$118.948$	4.4.777569.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(28\)		$-$	$+$	$-$	$+$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+7q^{7}+(6+\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
2016.4.a.be	$2016$	$4$	2016.a	1.a	$1$	$5$	$5$	$118.948$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-35\)		$-$	$+$	$+$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}-7q^{7}+(-9-\beta _{3})q^{11}+(5+\cdots)q^{13}+\cdots\)
2016.4.a.bf	$2016$	$4$	2016.a	1.a	$1$	$5$	$5$	$118.948$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-35\)		$+$	$+$	$+$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}-7q^{7}+(9+\beta _{3})q^{11}+(5-\beta _{1}+\cdots)q^{13}+\cdots\)
2016.4.a.bg	$2016$	$4$	2016.a	1.a	$1$	$5$	$5$	$118.948$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(35\)		$+$	$+$	$-$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+7q^{7}+(-9-\beta _{3})q^{11}+(5+\cdots)q^{13}+\cdots\)
2016.4.a.bh	$2016$	$4$	2016.a	1.a	$1$	$5$	$5$	$118.948$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(35\)		$-$	$+$	$-$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+7q^{7}+(9+\beta _{3})q^{11}+(5-\beta _{1}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2016))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2016)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1008))\)\(^{\oplus 2}\)