Space of modular forms of level 2800, weight 2, and trivial character

• Label: 2800.2.a
• Level: $2800$
• Weight: $2$
• Character orbit: 2800.a
• Rep. character: $\chi_{2800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $57$
• Newform subspaces: $44$
• Sturm bound: $960$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 44 \)
Sturm bound:			\(960\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2800))\).

	Total	New	Old
Modular forms	516	57	459
Cusp forms	445	57	388
Eisenstein series	71	0	71

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

\(2\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(7\)
\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(-\)	\(+\)	$-$	\(7\)
\(+\)	\(-\)	\(-\)	$+$	\(7\)
\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(-\)	$-$	\(9\)
Plus space			\(+\)	\(26\)
Minus space			\(-\)	\(31\)

Trace form

\( 57 q - q^{7} + 53 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\) 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	5	7
2800.2.a.a	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+6q^{9}-3q^{11}+q^{13}+\cdots\)
2800.2.a.b	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+6q^{9}+5q^{11}+6q^{13}+\cdots\)
2800.2.a.c	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{7}+6q^{9}+5q^{11}+5q^{13}+\cdots\)
2800.2.a.d	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{7}+q^{9}-5q^{11}+8q^{17}+\cdots\)
2800.2.a.e	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}-3q^{11}+4q^{13}+\cdots\)
2800.2.a.f	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
2800.2.a.g	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}+4q^{13}-6q^{17}+\cdots\)
2800.2.a.h	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
2800.2.a.i	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}+5q^{11}-q^{13}+\cdots\)
2800.2.a.j	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}+q^{11}-6q^{13}+\cdots\)
2800.2.a.k	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}+q^{11}-q^{13}-3q^{17}+\cdots\)
2800.2.a.l	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
2800.2.a.m	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}-4q^{11}+6q^{13}-2q^{17}+\cdots\)
2800.2.a.n	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}-q^{11}-2q^{13}-4q^{17}+\cdots\)
2800.2.a.o	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}+4q^{13}+4q^{17}-4q^{19}+\cdots\)
2800.2.a.p	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
2800.2.a.q	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}+5q^{11}-6q^{13}+4q^{17}+\cdots\)
2800.2.a.r	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}-q^{11}+2q^{13}+4q^{17}+\cdots\)
2800.2.a.s	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}-4q^{13}-4q^{17}-4q^{19}+\cdots\)
2800.2.a.t	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}+5q^{11}+6q^{13}-4q^{17}+\cdots\)
2800.2.a.u	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+q^{11}+q^{13}+3q^{17}+\cdots\)
2800.2.a.v	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+q^{11}+6q^{13}+\cdots\)
2800.2.a.w	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+3q^{11}-q^{13}+\cdots\)
2800.2.a.x	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
2800.2.a.y	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}-2q^{9}-3q^{11}+q^{13}+\cdots\)
2800.2.a.z	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}-2q^{9}+3q^{11}-5q^{13}+\cdots\)
2800.2.a.ba	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{7}+q^{9}-3q^{11}-4q^{13}+\cdots\)
2800.2.a.bb	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
2800.2.a.bc	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{7}+q^{9}-5q^{11}-8q^{17}+\cdots\)
2800.2.a.bd	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{7}+q^{9}+2q^{17}+2q^{19}+\cdots\)
2800.2.a.be	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{7}+6q^{9}+5q^{11}+3q^{13}+\cdots\)
2800.2.a.bf	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{7}+6q^{9}-3q^{11}-q^{13}+\cdots\)
2800.2.a.bg	$2800$	$2$	2800.a	1.a	$1$	$1$	$1$	$22.358$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{7}+6q^{9}+5q^{11}-6q^{13}+\cdots\)
2800.2.a.bh	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+q^{7}+(3+2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
2800.2.a.bi	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bj	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}+(3-2\beta )q^{11}+\cdots\)
2800.2.a.bk	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{7}+(5+\beta )q^{9}+(-4+\beta )q^{11}+\cdots\)
2800.2.a.bl	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{7}+3q^{9}-2\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bm	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{7}+3q^{9}+2\beta q^{11}+(2+\beta )q^{13}+\cdots\)
2800.2.a.bn	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bo	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+(3-2\beta )q^{11}+\cdots\)
2800.2.a.bp	$2800$	$2$	2800.a	1.a	$1$	$2$	$2$	$22.358$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{7}+(3+2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
2800.2.a.bq	$2800$	$2$	2800.a	1.a	$1$	$3$	$3$	$22.358$	3.3.568.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
2800.2.a.br	$2800$	$2$	2800.a	1.a	$1$	$3$	$3$	$22.358$	3.3.568.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 2}\)