Space of modular forms of level 700, weight 2, and trivial character

• Label: 700.2.a
• Level: $700$
• Weight: $2$
• Character orbit: 700.a
• Rep. character: $\chi_{700}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $10$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(700))\).

	Total	New	Old
Modular forms	138	10	128
Cusp forms	103	10	93
Eisenstein series	35	0	35

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(15\)	\(0\)	\(15\)		\(10\)	\(0\)	\(10\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(21\)	\(0\)	\(21\)		\(15\)	\(0\)	\(15\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(21\)	\(0\)	\(21\)		\(15\)	\(0\)	\(15\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(15\)	\(0\)	\(15\)		\(9\)	\(0\)	\(9\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(15\)	\(2\)	\(13\)		\(12\)	\(2\)	\(10\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(18\)	\(2\)	\(16\)		\(15\)	\(2\)	\(13\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(18\)	\(3\)	\(15\)		\(15\)	\(3\)	\(12\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(15\)	\(3\)	\(12\)		\(12\)	\(3\)	\(9\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(66\)	\(5\)	\(61\)		\(49\)	\(5\)	\(44\)		\(17\)	\(0\)	\(17\)
Minus space			\(-\)		\(72\)	\(5\)	\(67\)		\(54\)	\(5\)	\(49\)		\(18\)	\(0\)	\(18\)

Trace form

10 * q - 4 * q^3 + 6 * q^9 + 4 * q^13 + 4 * q^17 - 8 * q^19 - 4 * q^27 - 4 * q^29 + 4 * q^31 + 12 * q^33 + 8 * q^37 + 12 * q^39 - 24 * q^41 - 12 * q^43 + 4 * q^47 + 10 * q^49 - 36 * q^51 - 4 * q^53 - 20 * q^57 - 52 * q^59 - 24 * q^61 + 8 * q^63 - 20 * q^67 + 36 * q^69 + 36 * q^71 - 16 * q^73 - 8 * q^77 - 16 * q^79 + 42 * q^81 + 16 * q^83 + 36 * q^87 + 8 * q^89 - 8 * q^91 + 4 * q^93 - 4 * q^97 + 36 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(700))\) 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
700.2.a.a	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓				140.2.e.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+6q^{9}+3q^{11}-q^{13}+\cdots\)
700.2.a.b	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓				140.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{7}+6q^{9}-5q^{11}+3q^{13}+\cdots\)
700.2.a.c	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓	✓			700.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}+3q^{11}-4q^{13}+\cdots\)
700.2.a.d	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓				140.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
700.2.a.e	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓	✓			700.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}-5q^{11}+6q^{13}-4q^{17}+\cdots\)
700.2.a.f	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓				140.2.e.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}-4q^{13}-4q^{17}+4q^{19}+\cdots\)
700.2.a.g	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓	✓			700.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}-5q^{11}-6q^{13}+4q^{17}+\cdots\)
700.2.a.h	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓				140.2.e.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}+4q^{13}+4q^{17}+4q^{19}+\cdots\)
700.2.a.i	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓	✓			700.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
700.2.a.j	$700$	$2$	700.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓				140.2.e.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{7}+6q^{9}+3q^{11}+q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(700))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(700)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)