Space of modular forms of level 2100, weight 4, and trivial character

• Label: 2100.4.a
• Level: $2100$
• Weight: $4$
• Character orbit: 2100.a
• Rep. character: $\chi_{2100}(1,\cdot)$
• Character field: $\Q$
• Dimension: $58$
• Newform subspaces: $29$
• Sturm bound: $1920$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2100.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(1920\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2100))\).

	Total	New	Old
Modular forms	1476	58	1418
Cusp forms	1404	58	1346
Eisenstein series	72	0	72

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
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(96\)	\(0\)	\(96\)		\(90\)	\(0\)	\(90\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(93\)	\(0\)	\(93\)		\(87\)	\(0\)	\(87\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(91\)	\(0\)	\(91\)		\(85\)	\(0\)	\(85\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(92\)	\(0\)	\(92\)		\(86\)	\(0\)	\(86\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(90\)	\(0\)	\(90\)		\(84\)	\(0\)	\(84\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(93\)	\(0\)	\(93\)		\(87\)	\(0\)	\(87\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(95\)	\(0\)	\(95\)		\(89\)	\(0\)	\(89\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(94\)	\(0\)	\(94\)		\(88\)	\(0\)	\(88\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(90\)	\(6\)	\(84\)		\(87\)	\(6\)	\(81\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(93\)	\(7\)	\(86\)		\(90\)	\(7\)	\(83\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(92\)	\(9\)	\(83\)		\(89\)	\(9\)	\(80\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(91\)	\(7\)	\(84\)		\(88\)	\(7\)	\(81\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(93\)	\(7\)	\(86\)		\(90\)	\(7\)	\(83\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(90\)	\(6\)	\(84\)		\(87\)	\(6\)	\(81\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(91\)	\(7\)	\(84\)		\(88\)	\(7\)	\(81\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(92\)	\(9\)	\(83\)		\(89\)	\(9\)	\(80\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(746\)	\(32\)	\(714\)		\(710\)	\(32\)	\(678\)		\(36\)	\(0\)	\(36\)
Minus space				\(-\)		\(730\)	\(26\)	\(704\)		\(694\)	\(26\)	\(668\)		\(36\)	\(0\)	\(36\)

Trace form

58 * q + 522 * q^9 + 40 * q^11 - 84 * q^13 - 36 * q^17 + 64 * q^19 + 42 * q^21 - 352 * q^23 - 628 * q^29 + 176 * q^31 + 288 * q^33 - 532 * q^37 + 24 * q^39 + 212 * q^41 + 248 * q^43 + 656 * q^47 + 2842 * q^49 + 24 * q^51 - 924 * q^53 - 960 * q^57 - 1712 * q^59 + 1028 * q^61 - 1336 * q^67 + 864 * q^69 - 112 * q^71 - 1076 * q^73 - 224 * q^77 - 384 * q^79 + 4698 * q^81 - 1712 * q^83 - 1392 * q^87 + 4164 * q^89 + 784 * q^91 + 168 * q^93 - 2228 * q^97 + 360 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2100))\) 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
2100.4.a.a	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				420.4.a.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-7q^{7}+9q^{9}-6^{2}q^{11}+34q^{13}+\cdots\)
2100.4.a.b	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				420.4.a.e	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-7q^{7}+9q^{9}-2^{4}q^{11}+14q^{13}+\cdots\)
2100.4.a.c	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				420.4.a.f	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-7q^{7}+9q^{9}+6^{2}q^{11}-38q^{13}+\cdots\)
2100.4.a.d	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				420.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}-44q^{11}+42q^{13}+\cdots\)
2100.4.a.e	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓	✓			2100.4.a.e	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}-26q^{11}-39q^{13}+\cdots\)
2100.4.a.f	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓	✓			2100.4.a.f	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}-6q^{11}-79q^{13}+\cdots\)
2100.4.a.g	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				84.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}+4q^{11}-54q^{13}+\cdots\)
2100.4.a.h	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				420.4.k.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}+54q^{11}-14q^{13}+\cdots\)
2100.4.a.i	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				420.4.a.b	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}-6^{2}q^{11}+34q^{13}+\cdots\)
2100.4.a.j	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓	✓			2100.4.a.e	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}-26q^{11}+39q^{13}+\cdots\)
2100.4.a.k	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓	✓			2100.4.a.f	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}-6q^{11}+79q^{13}+\cdots\)
2100.4.a.l	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				84.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}+6^{2}q^{11}-62q^{13}+\cdots\)
2100.4.a.m	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				420.4.k.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}+54q^{11}+14q^{13}+\cdots\)
2100.4.a.n	$2100$	$4$	2100.a	1.a	$1$	$1$	$1$	$123.904$	\(\Q\)	None	✓				420.4.a.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(7\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+7q^{7}+9q^{9}+2^{5}q^{11}-42q^{13}+\cdots\)
2100.4.a.o	$2100$	$4$	2100.a	1.a	$1$	$2$	$2$	$123.904$	\(\Q(\sqrt{421}) \)	None	✓				420.4.a.i	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(14\)		$-$	$+$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}+(-6-\beta )q^{11}+\cdots\)
2100.4.a.p	$2100$	$4$	2100.a	1.a	$1$	$2$	$2$	$123.904$	\(\Q(\sqrt{29}) \)	None	✓	✓			2100.4.a.p	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(14\)		$-$	$+$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}+(9+2\beta )q^{11}+\cdots\)
2100.4.a.q	$2100$	$4$	2100.a	1.a	$1$	$2$	$2$	$123.904$	\(\Q(\sqrt{109}) \)	None	✓	✓			2100.4.a.q	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(14\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}+(3^{3}-2\beta )q^{11}+\cdots\)
2100.4.a.r	$2100$	$4$	2100.a	1.a	$1$	$2$	$2$	$123.904$	\(\Q(\sqrt{109}) \)	None	✓				420.4.a.g	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-14\)		$-$	$-$	$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}+(6+\beta )q^{11}+\cdots\)
2100.4.a.s	$2100$	$4$	2100.a	1.a	$1$	$2$	$2$	$123.904$	\(\Q(\sqrt{29}) \)	None	✓	✓			2100.4.a.p	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-14\)		$-$	$-$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}+(9+2\beta )q^{11}+\cdots\)
2100.4.a.t	$2100$	$4$	2100.a	1.a	$1$	$2$	$2$	$123.904$	\(\Q(\sqrt{109}) \)	None	✓	✓			2100.4.a.q	$1$	$1$	\(0\)	\(6\)	\(0\)	\(-14\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}+(3^{3}-2\beta )q^{11}+\cdots\)
2100.4.a.u	$2100$	$4$	2100.a	1.a	$1$	$2$	$2$	$123.904$	\(\Q(\sqrt{130}) \)	None	✓				420.4.a.h	$1$	$1$	\(0\)	\(6\)	\(0\)	\(14\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+7q^{7}+9q^{9}-8q^{11}+(-2+\cdots)q^{13}+\cdots\)
2100.4.a.v	$2100$	$4$	2100.a	1.a	$1$	$3$	$3$	$123.904$	3.3.683389.1	None	✓	✓			2100.4.a.v	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(-21\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}-7q^{7}+9q^{9}+(-3-2\beta _{1}+\cdots)q^{11}+\cdots\)
2100.4.a.w	$2100$	$4$	2100.a	1.a	$1$	$3$	$3$	$123.904$	3.3.698565.1	None	✓	✓	✓		2100.4.a.w	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(-21\)		$-$	$+$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-3q^{3}-7q^{7}+9q^{9}+(2-\beta _{1}+\beta _{2})q^{11}+\cdots\)
2100.4.a.x	$2100$	$4$	2100.a	1.a	$1$	$3$	$3$	$123.904$	3.3.4281.1	None	✓		✓		420.4.k.b	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(21\)		$-$	$+$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}+(-18-\beta _{1}+\cdots)q^{11}+\cdots\)
2100.4.a.y	$2100$	$4$	2100.a	1.a	$1$	$3$	$3$	$123.904$	3.3.4281.1	None	✓		✓		420.4.k.b	$1$	$1$	\(0\)	\(9\)	\(0\)	\(-21\)		$-$	$-$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}+(-18-\beta _{1}+\cdots)q^{11}+\cdots\)
2100.4.a.z	$2100$	$4$	2100.a	1.a	$1$	$3$	$3$	$123.904$	3.3.683389.1	None	✓	✓	✓		2100.4.a.v	$1$	$1$	\(0\)	\(9\)	\(0\)	\(21\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+7q^{7}+9q^{9}+(-3-2\beta _{1}+\cdots)q^{11}+\cdots\)
2100.4.a.ba	$2100$	$4$	2100.a	1.a	$1$	$3$	$3$	$123.904$	3.3.698565.1	None	✓	✓			2100.4.a.w	$1$	$0$	\(0\)	\(9\)	\(0\)	\(21\)		$-$	$-$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+3q^{3}+7q^{7}+9q^{9}+(2-\beta _{1}+\beta _{2})q^{11}+\cdots\)
2100.4.a.bb	$2100$	$4$	2100.a	1.a	$1$	$6$	$6$	$123.904$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		420.4.k.c	$1$	$0$	\(0\)	\(-18\)	\(0\)	\(-42\)		$-$	$+$	$-$	$+$	$+$	$2^{9}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-7q^{7}+9q^{9}-\beta _{3}q^{11}+(-4+\cdots)q^{13}+\cdots\)
2100.4.a.bc	$2100$	$4$	2100.a	1.a	$1$	$6$	$6$	$123.904$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		420.4.k.c	$1$	$0$	\(0\)	\(18\)	\(0\)	\(42\)		$-$	$-$	$-$	$-$	$+$	$2^{9}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+7q^{7}+9q^{9}-\beta _{3}q^{11}+(4+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2100))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2100)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 24}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 2}\)