Space of modular forms of level 6084, weight 2, and trivial character

• Label: 6084.2.a
• Level: $6084$
• Weight: $2$
• Character orbit: 6084.a
• Rep. character: $\chi_{6084}(1,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $32$
• Sturm bound: $2184$
• Trace bound: $43$

Defining parameters

Level:	\( N \)	\(=\)	\( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6084.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(2184\)
Trace bound:			\(43\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6084))\).

	Total	New	Old
Modular forms	1176	64	1112
Cusp forms	1009	64	945
Eisenstein series	167	0	167

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(147\)	\(0\)	\(147\)		\(120\)	\(0\)	\(120\)		\(27\)	\(0\)	\(27\)
\(+\)	\(+\)	\(-\)	\(-\)		\(153\)	\(0\)	\(153\)		\(125\)	\(0\)	\(125\)		\(28\)	\(0\)	\(28\)
\(+\)	\(-\)	\(+\)	\(-\)		\(154\)	\(0\)	\(154\)		\(126\)	\(0\)	\(126\)		\(28\)	\(0\)	\(28\)
\(+\)	\(-\)	\(-\)	\(+\)		\(148\)	\(0\)	\(148\)		\(120\)	\(0\)	\(120\)		\(28\)	\(0\)	\(28\)
\(-\)	\(+\)	\(+\)	\(-\)		\(147\)	\(15\)	\(132\)		\(133\)	\(15\)	\(118\)		\(14\)	\(0\)	\(14\)
\(-\)	\(+\)	\(-\)	\(+\)		\(141\)	\(10\)	\(131\)		\(127\)	\(10\)	\(117\)		\(14\)	\(0\)	\(14\)
\(-\)	\(-\)	\(+\)	\(+\)		\(140\)	\(18\)	\(122\)		\(126\)	\(18\)	\(108\)		\(14\)	\(0\)	\(14\)
\(-\)	\(-\)	\(-\)	\(-\)		\(146\)	\(21\)	\(125\)		\(132\)	\(21\)	\(111\)		\(14\)	\(0\)	\(14\)
Plus space			\(+\)		\(576\)	\(28\)	\(548\)		\(493\)	\(28\)	\(465\)		\(83\)	\(0\)	\(83\)
Minus space			\(-\)		\(600\)	\(36\)	\(564\)		\(516\)	\(36\)	\(480\)		\(84\)	\(0\)	\(84\)

Trace form

64 * q - 2 * q^5 - 2 * q^7 - 6 * q^11 - 2 * q^19 + 10 * q^23 + 50 * q^25 - 10 * q^29 - 6 * q^31 + 12 * q^35 - 8 * q^37 - 10 * q^41 + 6 * q^43 - 6 * q^47 + 46 * q^49 + 2 * q^53 + 16 * q^55 - 6 * q^59 - 2 * q^61 + 30 * q^67 + 38 * q^71 + 24 * q^73 + 26 * q^79 + 6 * q^83 - 4 * q^85 - 10 * q^89 - 14 * q^95 - 32 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6084))\) 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
6084.2.a.a	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				468.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-4q^{7}+4q^{11}-8q^{23}+11q^{25}+\cdots\)
6084.2.a.b	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				156.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+2q^{7}-4q^{11}-2q^{17}+2q^{19}+\cdots\)
6084.2.a.c	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				52.2.e.b	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+4q^{7}-3q^{17}-2q^{19}+6q^{23}+\cdots\)
6084.2.a.d	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				156.2.b.b	$1$	$2$	\(0\)	\(0\)	\(-2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}-6q^{11}-2q^{17}-8q^{23}+\cdots\)
6084.2.a.e	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				156.2.i.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-2q^{11}+4q^{17}+4q^{19}+\cdots\)
6084.2.a.f	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				52.2.e.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}+5q^{11}-3q^{17}-3q^{19}+\cdots\)
6084.2.a.g	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				468.2.l.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-5q^{7}-8q^{19}-5q^{25}-11q^{31}+\cdots\)
6084.2.a.h	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				156.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+6q^{17}-2q^{19}-5q^{25}+6q^{29}+\cdots\)
6084.2.a.i	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				36.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{7}-8q^{19}-5q^{25}+4q^{31}+10q^{37}+\cdots\)
6084.2.a.j	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				468.2.l.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+5q^{7}+8q^{19}-5q^{25}+11q^{31}+\cdots\)
6084.2.a.k	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				52.2.e.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}-5q^{11}-3q^{17}+3q^{19}+\cdots\)
6084.2.a.l	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				156.2.i.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}+2q^{11}+4q^{17}-4q^{19}+\cdots\)
6084.2.a.m	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				52.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}-2q^{11}-6q^{17}+6q^{19}+\cdots\)
6084.2.a.n	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				156.2.b.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{7}+6q^{11}-2q^{17}-8q^{23}+\cdots\)
6084.2.a.o	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				52.2.e.b	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-4q^{7}-3q^{17}+2q^{19}+6q^{23}+\cdots\)
6084.2.a.p	$6084$	$2$	6084.a	1.a	$1$	$1$	$1$	$48.581$	\(\Q\)	None	✓				468.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-4q^{7}-4q^{11}+8q^{23}+11q^{25}+\cdots\)
6084.2.a.q	$6084$	$2$	6084.a	1.a	$1$	$2$	$2$	$48.581$	\(\Q(\sqrt{7}) \)	None	✓				468.2.l.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2q^{7}+2\beta q^{11}+3\beta q^{17}+\cdots\)
6084.2.a.r	$6084$	$2$	6084.a	1.a	$1$	$2$	$2$	$48.581$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				468.2.b.b	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$N(\mathrm{U}(1))$	\(q-\beta q^{7}+\beta q^{19}-5q^{25}+3\beta q^{31}+\cdots\)
6084.2.a.s	$6084$	$2$	6084.a	1.a	$1$	$2$	$2$	$48.581$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				468.2.t.b	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+\beta q^{7}-2\beta q^{19}-5q^{25}+\beta q^{31}+\cdots\)
6084.2.a.t	$6084$	$2$	6084.a	1.a	$1$	$2$	$2$	$48.581$	\(\Q(\sqrt{3}) \)	None	✓				52.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{7}+3\beta q^{11}+3q^{17}+3\beta q^{19}+\cdots\)
6084.2.a.u	$6084$	$2$	6084.a	1.a	$1$	$2$	$2$	$48.581$	\(\Q(\sqrt{3}) \)	None	✓				156.2.q.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+2\beta q^{7}+2\beta q^{11}+3q^{17}+\cdots\)
6084.2.a.v	$6084$	$2$	6084.a	1.a	$1$	$2$	$2$	$48.581$	\(\Q(\sqrt{3}) \)	None	✓				156.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-\beta q^{11}+6q^{17}+2\beta q^{19}+\cdots\)
6084.2.a.w	$6084$	$2$	6084.a	1.a	$1$	$2$	$2$	$48.581$	\(\Q(\sqrt{7}) \)	None	✓				468.2.l.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+2q^{7}+2\beta q^{11}-3\beta q^{17}+\cdots\)
6084.2.a.x	$6084$	$2$	6084.a	1.a	$1$	$3$	$3$	$48.581$	\(\Q(\zeta_{14})^+\)	None	✓				676.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{5}+(-2+2\beta _{1}-3\beta _{2})q^{7}+\cdots\)
6084.2.a.y	$6084$	$2$	6084.a	1.a	$1$	$3$	$3$	$48.581$	\(\Q(\zeta_{14})^+\)	None	✓				2028.2.a.i	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{5}+(-1-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
6084.2.a.z	$6084$	$2$	6084.a	1.a	$1$	$3$	$3$	$48.581$	\(\Q(\zeta_{14})^+\)	None	✓				2028.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{5}+(-2-\beta _{1}-\beta _{2})q^{7}+\cdots\)
6084.2.a.ba	$6084$	$2$	6084.a	1.a	$1$	$3$	$3$	$48.581$	\(\Q(\zeta_{14})^+\)	None	✓				2028.2.a.k	$1$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta _{1}+\beta _{2})q^{5}+(3-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
6084.2.a.bb	$6084$	$2$	6084.a	1.a	$1$	$3$	$3$	$48.581$	\(\Q(\zeta_{14})^+\)	None	✓				2028.2.a.i	$1$	$1$	\(0\)	\(0\)	\(2\)	\(6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(3-\beta _{1}+2\beta _{2})q^{7}+(-2+\cdots)q^{11}+\cdots\)
6084.2.a.bc	$6084$	$2$	6084.a	1.a	$1$	$3$	$3$	$48.581$	\(\Q(\zeta_{14})^+\)	None	✓				676.2.a.g	$1$	$0$	\(0\)	\(0\)	\(8\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{5}+(-\beta _{1}-2\beta _{2})q^{7}+(3+\cdots)q^{11}+\cdots\)
6084.2.a.bd	$6084$	$2$	6084.a	1.a	$1$	$4$	$4$	$48.581$	\(\Q(\sqrt{3}, \sqrt{43})\)	None	✓		✓		156.2.q.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(-\beta _{1}+\beta _{2})q^{7}-2\beta _{2}q^{11}+\cdots\)
6084.2.a.be	$6084$	$2$	6084.a	1.a	$1$	$6$	$6$	$48.581$	6.6.25969216.1	None	✓	✓	✓		6084.2.a.be	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{5}+(-1-\beta _{2})q^{7}+(\beta _{1}-\beta _{4}+\cdots)q^{11}+\cdots\)
6084.2.a.bf	$6084$	$2$	6084.a	1.a	$1$	$6$	$6$	$48.581$	6.6.25969216.1	None	✓	✓	✓		6084.2.a.be	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{5}+(1+\beta _{2})q^{7}+(\beta _{1}-\beta _{4})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6084))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6084)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1014))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2028))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3042))\)\(^{\oplus 2}\)