Space of modular forms of level 700, weight 4, and trivial character

• Label: 700.4.a
• Level: $700$
• Weight: $4$
• Character orbit: 700.a
• Rep. character: $\chi_{700}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $20$
• Sturm bound: $480$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(480\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(700))\).

	Total	New	Old
Modular forms	378	28	350
Cusp forms	342	28	314
Eisenstein series	36	0	36

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
\(+\)	\(+\)	\(+\)	\(+\)		\(51\)	\(0\)	\(51\)		\(45\)	\(0\)	\(45\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(45\)	\(0\)	\(45\)		\(39\)	\(0\)	\(39\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(45\)	\(0\)	\(45\)		\(39\)	\(0\)	\(39\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(51\)	\(0\)	\(51\)		\(45\)	\(0\)	\(45\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(48\)	\(7\)	\(41\)		\(45\)	\(7\)	\(38\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(45\)	\(7\)	\(38\)		\(42\)	\(7\)	\(35\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(45\)	\(7\)	\(38\)		\(42\)	\(7\)	\(35\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(48\)	\(7\)	\(41\)		\(45\)	\(7\)	\(38\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(192\)	\(14\)	\(178\)		\(174\)	\(14\)	\(160\)		\(18\)	\(0\)	\(18\)
Minus space			\(-\)		\(186\)	\(14\)	\(172\)		\(168\)	\(14\)	\(154\)		\(18\)	\(0\)	\(18\)

Trace form

28 * q + 2 * q^3 + 228 * q^9 - 88 * q^11 + 78 * q^13 - 20 * q^17 + 46 * q^19 - 98 * q^21 + 112 * q^23 - 100 * q^27 - 84 * q^29 - 220 * q^31 - 792 * q^33 + 92 * q^37 + 328 * q^39 + 160 * q^41 - 532 * q^43 + 644 * q^47 + 1372 * q^49 + 800 * q^51 + 536 * q^53 + 1212 * q^57 + 946 * q^59 + 86 * q^61 + 476 * q^63 - 856 * q^67 - 3384 * q^69 - 1472 * q^71 + 2104 * q^73 + 420 * q^77 + 1584 * q^79 - 1008 * q^81 - 306 * q^83 - 2444 * q^87 - 4252 * q^89 - 350 * q^91 + 1904 * q^93 - 1268 * q^97 - 3052 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(700))\) 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
700.4.a.a	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.a.f	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+7q^{7}+54q^{9}+55q^{11}+69q^{13}+\cdots\)
700.4.a.b	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.a.e	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-7q^{7}+37q^{9}+28q^{11}-82q^{13}+\cdots\)
700.4.a.c	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.e.b	$1$	$1$	\(0\)	\(-7\)	\(0\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+7q^{7}+22q^{9}-7q^{11}-3q^{13}+\cdots\)
700.4.a.d	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.e.a	$1$	$0$	\(0\)	\(-5\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-7q^{7}-2q^{9}-65q^{11}+13q^{13}+\cdots\)
700.4.a.e	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				28.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-7q^{7}-11q^{9}-12q^{11}+82q^{13}+\cdots\)
700.4.a.f	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓	✓			700.4.a.f	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+7q^{7}-26q^{9}-37q^{11}+38q^{13}+\cdots\)
700.4.a.g	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.a.d	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+7q^{7}-26q^{9}-7q^{11}+23q^{13}+\cdots\)
700.4.a.h	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓	✓			700.4.a.f	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-7q^{7}-26q^{9}-37q^{11}-38q^{13}+\cdots\)
700.4.a.i	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.a.c	$1$	$0$	\(0\)	\(4\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+7q^{7}-11q^{9}+68q^{11}-22q^{13}+\cdots\)
700.4.a.j	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.a.b	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-7q^{7}-2q^{9}-15q^{11}+13q^{13}+\cdots\)
700.4.a.k	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.a.a	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-7q^{7}-2q^{9}+15q^{11}-17q^{13}+\cdots\)
700.4.a.l	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.e.a	$1$	$1$	\(0\)	\(5\)	\(0\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+7q^{7}-2q^{9}-65q^{11}-13q^{13}+\cdots\)
700.4.a.m	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				140.4.e.b	$1$	$0$	\(0\)	\(7\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-7q^{7}+22q^{9}-7q^{11}+3q^{13}+\cdots\)
700.4.a.n	$700$	$4$	700.a	1.a	$1$	$1$	$1$	$41.301$	\(\Q\)	None	✓				28.4.a.a	$1$	$0$	\(0\)	\(10\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}+7q^{7}+73q^{9}-40q^{11}+\cdots\)
700.4.a.o	$700$	$4$	700.a	1.a	$1$	$2$	$2$	$41.301$	\(\Q(\sqrt{61}) \)	None	✓	✓	✓		700.4.a.o	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+7q^{7}+(35+2\beta )q^{9}+\cdots\)
700.4.a.p	$700$	$4$	700.a	1.a	$1$	$2$	$2$	$41.301$	\(\Q(\sqrt{6}) \)	None	✓				140.4.e.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-7q^{7}-21q^{9}+(24+14\beta )q^{11}+\cdots\)
700.4.a.q	$700$	$4$	700.a	1.a	$1$	$2$	$2$	$41.301$	\(\Q(\sqrt{6}) \)	None	✓				140.4.e.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+7q^{7}-21q^{9}+(24-14\beta )q^{11}+\cdots\)
700.4.a.r	$700$	$4$	700.a	1.a	$1$	$2$	$2$	$41.301$	\(\Q(\sqrt{61}) \)	None	✓	✓			700.4.a.o	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-14\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-7q^{7}+(35+2\beta )q^{9}+(24+\cdots)q^{11}+\cdots\)
700.4.a.s	$700$	$4$	700.a	1.a	$1$	$3$	$3$	$41.301$	3.3.15629.1	None	✓	✓	✓		700.4.a.s	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(21\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{3}+7q^{7}+(13-\beta _{1}-2\beta _{2})q^{9}+\cdots\)
700.4.a.t	$700$	$4$	700.a	1.a	$1$	$3$	$3$	$41.301$	3.3.15629.1	None	✓	✓	✓		700.4.a.s	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-21\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{2})q^{3}-7q^{7}+(13-\beta _{1}-2\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(700))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(700)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)