Space of modular forms of level 2730, weight 2, and trivial character

• Label: 2730.2.a
• Level: $2730$
• Weight: $2$
• Character orbit: 2730.a
• Rep. character: $\chi_{2730}(1,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $38$
• Sturm bound: $1344$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 2730 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2730.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(1344\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(11\), \(17\), \(19\), \(23\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2730))\).

	Total	New	Old
Modular forms	688	47	641
Cusp forms	657	47	610
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(5\)	\(7\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(-\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	$+$	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space					\(+\)	\(16\)
Minus space					\(-\)	\(31\)

Trace form

\( 47 q - q^{2} - q^{3} + 47 q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + 47 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2730))\) 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	5	7	13
2730.2.a.a	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
2730.2.a.b	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
2730.2.a.c	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
2730.2.a.d	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.e	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.f	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.g	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(1\)	\(-1\)		$+$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
2730.2.a.h	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.i	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.j	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(1\)	\(1\)		$+$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.k	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(-1\)		$+$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.l	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(-1\)		$+$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.m	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(1\)		$+$	$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.n	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-1\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.o	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(1\)		$+$	$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.p	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(1\)		$+$	$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.q	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.r	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.s	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.t	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.u	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.v	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.w	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.x	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(1\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.y	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
2730.2.a.z	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.ba	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.bb	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(1\)	\(-1\)		$-$	$-$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
2730.2.a.bc	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(1\)	\(-1\)		$-$	$-$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
2730.2.a.bd	$2730$	$2$	2730.a	1.a	$1$	$1$	$1$	$21.799$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.be	$2730$	$2$	2730.a	1.a	$1$	$2$	$2$	$21.799$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(2\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
2730.2.a.bf	$2730$	$2$	2730.a	1.a	$1$	$2$	$2$	$21.799$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(2\)	\(2\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.bg	$2730$	$2$	2730.a	1.a	$1$	$2$	$2$	$21.799$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-2\)	\(2\)	\(2\)	\(-2\)		$+$	$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2730.2.a.bh	$2730$	$2$	2730.a	1.a	$1$	$2$	$2$	$21.799$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-2\)	\(2\)	\(2\)	\(2\)		$+$	$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.bi	$2730$	$2$	2730.a	1.a	$1$	$2$	$2$	$21.799$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(2\)	\(-2\)	\(-2\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.bj	$2730$	$2$	2730.a	1.a	$1$	$2$	$2$	$21.799$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(2\)	\(2\)		$-$	$+$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
2730.2.a.bk	$2730$	$2$	2730.a	1.a	$1$	$2$	$2$	$21.799$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
2730.2.a.bl	$2730$	$2$	2730.a	1.a	$1$	$3$	$3$	$21.799$	3.3.892.1	None	✓	$1$	$0$	\(3\)	\(3\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2730))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2730)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(455))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(546))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(910))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1365))\)\(^{\oplus 2}\)