Space of modular forms of level 1400, weight 2, and trivial character

• Label: 1400.2.a
• Level: $1400$
• Weight: $2$
• Character orbit: 1400.a
• Rep. character: $\chi_{1400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $20$
• Sturm bound: $480$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(480\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1400))\).

	Total	New	Old
Modular forms	264	28	236
Cusp forms	217	28	189
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(11\)
Minus space			\(-\)	\(17\)

Trace form

\( 28 q + 2 q^{3} + 20 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\) 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	5	7
1400.2.a.a	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓				56.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{7}+q^{9}+2q^{17}-2q^{19}+\cdots\)
1400.2.a.b	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓	✓			1400.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{7}+q^{9}+5q^{11}-8q^{17}+\cdots\)
1400.2.a.c	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓	✓			1400.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
1400.2.a.d	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓				280.2.g.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}-q^{11}+q^{13}+3q^{17}+\cdots\)
1400.2.a.e	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓	✓			1400.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}-q^{11}+6q^{13}+\cdots\)
1400.2.a.f	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓	✓			1400.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}+q^{11}+2q^{13}+4q^{17}+\cdots\)
1400.2.a.g	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓				56.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
1400.2.a.h	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓	✓			1400.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}+q^{11}-2q^{13}-4q^{17}+\cdots\)
1400.2.a.i	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓	✓			1400.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}-q^{11}-6q^{13}+\cdots\)
1400.2.a.j	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓				280.2.g.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}-q^{11}-q^{13}-3q^{17}+\cdots\)
1400.2.a.k	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓				280.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}-2q^{9}-5q^{11}-q^{13}+\cdots\)
1400.2.a.l	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓	✓			1400.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
1400.2.a.m	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓	✓			1400.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{7}+q^{9}+5q^{11}+8q^{17}+\cdots\)
1400.2.a.n	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	\(\Q\)	None	✓				280.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{7}+6q^{9}-5q^{11}+5q^{13}+\cdots\)
1400.2.a.o	$1400$	$2$	1400.a	1.a	$1$	$2$	$2$	$11.179$	\(\Q(\sqrt{17}) \)	None	✓	✓			1400.2.a.o	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
1400.2.a.p	$1400$	$2$	1400.a	1.a	$1$	$2$	$2$	$11.179$	\(\Q(\sqrt{17}) \)	None	✓				280.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
1400.2.a.q	$1400$	$2$	1400.a	1.a	$1$	$2$	$2$	$11.179$	\(\Q(\sqrt{17}) \)	None	✓	✓			1400.2.a.o	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
1400.2.a.r	$1400$	$2$	1400.a	1.a	$1$	$2$	$2$	$11.179$	\(\Q(\sqrt{33}) \)	None	✓		✓		280.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{7}+(5+\beta )q^{9}+(4-\beta )q^{11}+\cdots\)
1400.2.a.s	$1400$	$2$	1400.a	1.a	$1$	$3$	$3$	$11.179$	3.3.568.1	None	✓		✓		280.2.g.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
1400.2.a.t	$1400$	$2$	1400.a	1.a	$1$	$3$	$3$	$11.179$	3.3.568.1	None	✓		✓		280.2.g.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1400)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 2}\)