Space of modular forms of level 2100, weight 2, and trivial character

• Label: 2100.2.a
• Level: $2100$
• Weight: $2$
• Character orbit: 2100.a
• Rep. character: $\chi_{2100}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $18$
• Sturm bound: $960$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(960\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2100))\).

	Total	New	Old
Modular forms	516	18	498
Cusp forms	445	18	427
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.

\(2\)	\(3\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(30\)	\(0\)	\(30\)		\(25\)	\(0\)	\(25\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(33\)	\(0\)	\(33\)		\(27\)	\(0\)	\(27\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(35\)	\(0\)	\(35\)		\(29\)	\(0\)	\(29\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(34\)	\(0\)	\(34\)		\(28\)	\(0\)	\(28\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(36\)	\(0\)	\(36\)		\(30\)	\(0\)	\(30\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(33\)	\(0\)	\(33\)		\(27\)	\(0\)	\(27\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(31\)	\(0\)	\(31\)		\(25\)	\(0\)	\(25\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(32\)	\(0\)	\(32\)		\(26\)	\(0\)	\(26\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(33\)	\(3\)	\(30\)		\(30\)	\(3\)	\(27\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(30\)	\(2\)	\(28\)		\(27\)	\(2\)	\(25\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(31\)	\(1\)	\(30\)		\(28\)	\(1\)	\(27\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(32\)	\(3\)	\(29\)		\(29\)	\(3\)	\(26\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(30\)	\(2\)	\(28\)		\(27\)	\(2\)	\(25\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(33\)	\(3\)	\(30\)		\(30\)	\(3\)	\(27\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(32\)	\(3\)	\(29\)		\(29\)	\(3\)	\(26\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(31\)	\(1\)	\(30\)		\(28\)	\(1\)	\(27\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(250\)	\(6\)	\(244\)		\(215\)	\(6\)	\(209\)		\(35\)	\(0\)	\(35\)
Minus space				\(-\)		\(266\)	\(12\)	\(254\)		\(230\)	\(12\)	\(218\)		\(36\)	\(0\)	\(36\)

Trace form

18 * q + 18 * q^9 - 4 * q^11 - 4 * q^13 - 12 * q^17 - 2 * q^21 + 4 * q^23 + 28 * q^29 - 8 * q^31 - 12 * q^37 - 8 * q^39 + 28 * q^41 + 8 * q^43 - 8 * q^47 + 18 * q^49 - 4 * q^51 + 12 * q^53 + 8 * q^57 + 32 * q^59 + 44 * q^61 + 16 * q^67 + 4 * q^71 + 20 * q^73 + 8 * q^77 + 16 * q^79 + 18 * q^81 + 32 * q^83 - 8 * q^87 + 36 * q^89 + 16 * q^93 + 20 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2100))\) 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	3	5	7
2100.2.a.a	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				84.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
2100.2.a.b	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓			2100.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-3q^{11}+4q^{13}+\cdots\)
2100.2.a.c	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓	✓	✓	2100.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-q^{11}-2q^{13}+6q^{19}+\cdots\)
2100.2.a.d	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				420.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+6q^{11}+4q^{13}+\cdots\)
2100.2.a.e	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				420.2.k.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
2100.2.a.f	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓			2100.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-q^{11}+2q^{13}-8q^{17}+\cdots\)
2100.2.a.g	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓			2100.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+q^{11}+4q^{13}+2q^{17}+\cdots\)
2100.2.a.h	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				420.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}-4q^{13}+\cdots\)
2100.2.a.i	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				420.2.k.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
2100.2.a.j	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				420.2.k.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
2100.2.a.k	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				420.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}-4q^{13}+\cdots\)
2100.2.a.l	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓			2100.2.a.f	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-q^{11}-2q^{13}+8q^{17}+\cdots\)
2100.2.a.m	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓			2100.2.a.g	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+q^{11}-4q^{13}-2q^{17}+\cdots\)
2100.2.a.n	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				420.2.k.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
2100.2.a.o	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓	✓	✓	2100.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-3q^{11}-4q^{13}+\cdots\)
2100.2.a.p	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓			2100.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-q^{11}+2q^{13}+6q^{19}+\cdots\)
2100.2.a.q	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				420.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+2q^{11}-4q^{13}+\cdots\)
2100.2.a.r	$2100$	$2$	2100.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓				84.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+2q^{11}+6q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2100))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2100)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 2}\)