Space of modular forms of level 1600, weight 2, and trivial character

• Label: 1600.2.a
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.a
• Rep. character: $\chi_{1600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $30$
• Sturm bound: $480$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(480\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1600))\).

	Total	New	Old
Modular forms	276	41	235
Cusp forms	205	35	170
Eisenstein series	71	6	65

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

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(66\)	\(9\)	\(57\)		\(49\)	\(8\)	\(41\)		\(17\)	\(1\)	\(16\)
\(+\)	\(-\)	\(-\)		\(70\)	\(12\)	\(58\)		\(52\)	\(10\)	\(42\)		\(18\)	\(2\)	\(16\)
\(-\)	\(+\)	\(-\)		\(72\)	\(10\)	\(62\)		\(54\)	\(9\)	\(45\)		\(18\)	\(1\)	\(17\)
\(-\)	\(-\)	\(+\)		\(68\)	\(10\)	\(58\)		\(50\)	\(8\)	\(42\)		\(18\)	\(2\)	\(16\)
Plus space		\(+\)		\(134\)	\(19\)	\(115\)		\(99\)	\(16\)	\(83\)		\(35\)	\(3\)	\(32\)
Minus space		\(-\)		\(142\)	\(22\)	\(120\)		\(106\)	\(19\)	\(87\)		\(36\)	\(3\)	\(33\)

Trace form

35 * q + 31 * q^9 - 10 * q^13 - 2 * q^17 - 8 * q^21 + 6 * q^29 + 16 * q^33 - 2 * q^37 - 2 * q^41 + 19 * q^49 + 30 * q^53 + 16 * q^57 - 18 * q^61 + 24 * q^69 + 6 * q^73 + 16 * q^77 + 3 * q^81 - 10 * q^89 + 32 * q^93 - 18 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\) 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
1600.2.a.a	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				200.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{7}+6q^{9}-q^{11}+4q^{13}+\cdots\)
1600.2.a.b	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				200.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{7}+6q^{9}+q^{11}-4q^{13}+\cdots\)
1600.2.a.c	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				20.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}+2q^{13}+6q^{17}+\cdots\)
1600.2.a.d	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				40.2.c.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
1600.2.a.e	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				160.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1600.2.a.f	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				40.2.c.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
1600.2.a.g	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				800.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}-5q^{11}+5q^{17}+\cdots\)
1600.2.a.h	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				800.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}+5q^{11}-5q^{17}+\cdots\)
1600.2.a.i	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				50.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}-3q^{11}-4q^{13}+\cdots\)
1600.2.a.j	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				50.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}+3q^{11}+4q^{13}+\cdots\)
1600.2.a.k	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				40.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-3q^{9}+4q^{11}-2q^{13}-2q^{17}+\cdots\)
1600.2.a.l	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.2.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}-4q^{13}+8q^{17}-10q^{29}+\cdots\)
1600.2.a.m	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.2.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+4q^{13}-8q^{17}-10q^{29}+\cdots\)
1600.2.a.n	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+6q^{13}-2q^{17}+10q^{29}+\cdots\)
1600.2.a.o	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				40.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-3q^{9}-4q^{11}-2q^{13}-2q^{17}+\cdots\)
1600.2.a.p	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				50.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}-3q^{11}+4q^{13}+\cdots\)
1600.2.a.q	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				50.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}+3q^{11}-4q^{13}+\cdots\)
1600.2.a.r	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				800.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}-5q^{11}-5q^{17}+\cdots\)
1600.2.a.s	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				800.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}+5q^{11}+5q^{17}+\cdots\)
1600.2.a.t	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				160.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
1600.2.a.u	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				40.2.c.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
1600.2.a.v	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				40.2.c.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
1600.2.a.w	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				20.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{7}+q^{9}+2q^{13}+6q^{17}+\cdots\)
1600.2.a.x	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				200.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{7}+6q^{9}-q^{11}-4q^{13}+\cdots\)
1600.2.a.y	$1600$	$2$	1600.a	1.a	$1$	$1$	$1$	$12.776$	\(\Q\)	None	✓				200.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{7}+6q^{9}+q^{11}+4q^{13}+\cdots\)
1600.2.a.z	$1600$	$2$	1600.a	1.a	$1$	$2$	$2$	$12.776$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.2.c.b	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(6\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(-1-\beta )q^{3}+(3-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
1600.2.a.ba	$1600$	$2$	1600.a	1.a	$1$	$2$	$2$	$12.776$	\(\Q(\sqrt{5}) \)	None	✓		✓		800.2.a.k	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2\beta q^{7}+2q^{9}-\beta q^{11}-4q^{13}+\cdots\)
1600.2.a.bb	$1600$	$2$	1600.a	1.a	$1$	$2$	$2$	$12.776$	\(\Q(\sqrt{5}) \)	None	✓				800.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2\beta q^{7}+2q^{9}+\beta q^{11}+4q^{13}+\cdots\)
1600.2.a.bc	$1600$	$2$	1600.a	1.a	$1$	$2$	$2$	$12.776$	\(\Q(\sqrt{2}) \)	None	✓				160.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{7}+5q^{9}+2\beta q^{11}-2q^{13}+\cdots\)
1600.2.a.bd	$1600$	$2$	1600.a	1.a	$1$	$2$	$2$	$12.776$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.2.c.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(1+\beta )q^{3}+(-3+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1600)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 2}\)