Space of modular forms of level 3042, weight 2, and trivial character

• Label: 3042.2.a
• Level: $3042$
• Weight: $2$
• Character orbit: 3042.a
• Rep. character: $\chi_{3042}(1,\cdot)$
• Character field: $\Q$
• Dimension: $65$
• Newform subspaces: $35$
• Sturm bound: $1092$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 35 \)
Sturm bound:			\(1092\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3042))\).

	Total	New	Old
Modular forms	602	65	537
Cusp forms	491	65	426
Eisenstein series	111	0	111

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
\(+\)	\(+\)	\(+\)	\(+\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(11\)
\(+\)	\(-\)	\(-\)	\(+\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(12\)
Plus space			\(+\)	\(29\)
Minus space			\(-\)	\(36\)

Trace form

65 * q - q^2 + 65 * q^4 - 2 * q^5 - q^8 - 4 * q^10 - 4 * q^14 + 65 * q^16 - 6 * q^17 + 12 * q^19 - 2 * q^20 - 6 * q^22 - 8 * q^23 + 69 * q^25 + 8 * q^29 + 8 * q^31 - q^32 + 2 * q^34 + 12 * q^35 - 6 * q^37 + 10 * q^38 - 4 * q^40 - 10 * q^41 + 14 * q^43 + 12 * q^46 + 24 * q^47 + 85 * q^49 - 7 * q^50 - 4 * q^53 + 16 * q^55 - 4 * q^56 - 6 * q^58 - 12 * q^59 - 12 * q^61 + 16 * q^62 + 65 * q^64 + 8 * q^67 - 6 * q^68 + 20 * q^70 - 16 * q^71 - 22 * q^73 + 20 * q^74 + 12 * q^76 - 12 * q^77 + 4 * q^79 - 2 * q^80 + 6 * q^82 + 24 * q^83 - 16 * q^85 - 8 * q^86 - 6 * q^88 + 14 * q^89 - 8 * q^92 + 16 * q^94 - 20 * q^95 + 6 * q^97 - 9 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3042))\) 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
3042.2.a.a	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				26.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(-3\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-3q^{5}+q^{7}-q^{8}+3q^{10}+\cdots\)
3042.2.a.b	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				234.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+2q^{7}-q^{8}+2q^{10}+\cdots\)
3042.2.a.c	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				78.2.b.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+2q^{7}-q^{8}+2q^{10}+\cdots\)
3042.2.a.d	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				78.2.e.b	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+2q^{7}-q^{8}+q^{10}+\cdots\)
3042.2.a.e	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				26.2.c.a	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+4q^{7}-q^{8}-q^{10}+\cdots\)
3042.2.a.f	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				78.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-4q^{7}-q^{8}-2q^{10}+\cdots\)
3042.2.a.g	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				26.2.b.a	$1$	$1$	\(-1\)	\(0\)	\(3\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+3q^{5}-3q^{7}-q^{8}-3q^{10}+\cdots\)
3042.2.a.h	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				78.2.e.a	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+3q^{5}-2q^{7}-q^{8}-3q^{10}+\cdots\)
3042.2.a.i	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				78.2.e.a	$1$	$1$	\(1\)	\(0\)	\(-3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{5}+2q^{7}+q^{8}-3q^{10}+\cdots\)
3042.2.a.j	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				26.2.b.a	$1$	$0$	\(1\)	\(0\)	\(-3\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{5}+3q^{7}+q^{8}-3q^{10}+\cdots\)
3042.2.a.k	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				26.2.c.a	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
3042.2.a.l	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				26.2.a.b	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
3042.2.a.m	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				78.2.e.b	$1$	$1$	\(1\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-2q^{7}+q^{8}+q^{10}+\cdots\)
3042.2.a.n	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				78.2.b.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots\)
3042.2.a.o	$3042$	$2$	3042.a	1.a	$1$	$1$	$1$	$24.290$	\(\Q\)	None	✓				234.2.a.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+2q^{7}+q^{8}+2q^{10}+\cdots\)
3042.2.a.p	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{3}) \)	None	✓				78.2.i.a	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-2+\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
3042.2.a.q	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{10}) \)	None	✓				234.2.h.d	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+\beta )q^{5}+\beta q^{7}-q^{8}+\cdots\)
3042.2.a.r	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{10}) \)	None	✓				234.2.h.d	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+\beta )q^{5}-\beta q^{7}-q^{8}+\cdots\)
3042.2.a.s	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{3}) \)	None	✓				78.2.i.b	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{5}+(-3+\beta )q^{7}-q^{8}+\cdots\)
3042.2.a.t	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{3}) \)	None	✓				234.2.l.b	$2$	$0$	\(-2\)	\(0\)	\(6\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+3q^{5}-q^{8}-3q^{10}+q^{16}+\cdots\)
3042.2.a.u	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{3}) \)	None	✓				234.2.l.b	$2$	$1$	\(2\)	\(0\)	\(-6\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{5}+q^{8}-3q^{10}+q^{16}+\cdots\)
3042.2.a.v	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{3}) \)	None	✓				78.2.i.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}+(3+\beta )q^{7}+q^{8}+\cdots\)
3042.2.a.w	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{10}) \)	None	✓				234.2.h.d	$1$	$0$	\(2\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta )q^{5}+\beta q^{7}+q^{8}+\cdots\)
3042.2.a.x	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{10}) \)	None	✓				234.2.h.d	$1$	$0$	\(2\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta )q^{5}-\beta q^{7}+q^{8}+\cdots\)
3042.2.a.y	$3042$	$2$	3042.a	1.a	$1$	$2$	$2$	$24.290$	\(\Q(\sqrt{3}) \)	None	✓				78.2.i.a	$1$	$0$	\(2\)	\(0\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(2+\beta )q^{5}+(1+\beta )q^{7}+\cdots\)
3042.2.a.z	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓				338.2.a.g	$1$	$0$	\(-3\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2\beta _{1}q^{5}+(-2\beta _{1}+2\beta _{2})q^{7}+\cdots\)
3042.2.a.ba	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓				1014.2.a.l	$1$	$0$	\(-3\)	\(0\)	\(-1\)	\(9\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1-3\beta _{1}+\beta _{2})q^{5}+(4+\cdots)q^{7}+\cdots\)
3042.2.a.bb	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		3042.2.a.bb	$1$	$1$	\(-3\)	\(0\)	\(1\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{5}+(-2\beta _{1}-\beta _{2})q^{7}+\cdots\)
3042.2.a.bc	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		3042.2.a.bb	$1$	$0$	\(-3\)	\(0\)	\(1\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{5}+(2\beta _{1}+\beta _{2})q^{7}+\cdots\)
3042.2.a.bd	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓		✓		1014.2.a.m	$1$	$1$	\(-3\)	\(0\)	\(3\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta _{1}+\beta _{2})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
3042.2.a.be	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓		✓		1014.2.a.m	$1$	$1$	\(3\)	\(0\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta _{1}-\beta _{2})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
3042.2.a.bf	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		3042.2.a.bb	$1$	$0$	\(3\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{1}q^{5}+(-2\beta _{1}-\beta _{2})q^{7}+\cdots\)
3042.2.a.bg	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		3042.2.a.bb	$1$	$1$	\(3\)	\(0\)	\(-1\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{1}q^{5}+(2\beta _{1}+\beta _{2})q^{7}+\cdots\)
3042.2.a.bh	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓				1014.2.a.l	$1$	$0$	\(3\)	\(0\)	\(1\)	\(-9\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-\beta _{1}-2\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
3042.2.a.bi	$3042$	$2$	3042.a	1.a	$1$	$3$	$3$	$24.290$	\(\Q(\zeta_{14})^+\)	None	✓				338.2.a.g	$1$	$0$	\(3\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2\beta _{1}q^{5}+(2\beta _{1}-2\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3042))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3042)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1014))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\)\(^{\oplus 2}\)