Space of modular forms of level 6900, weight 2, and trivial character

• Label: 6900.2.a
• Level: $6900$
• Weight: $2$
• Character orbit: 6900.a
• Rep. character: $\chi_{6900}(1,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $30$
• Sturm bound: $2880$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(2880\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6900))\).

	Total	New	Old
Modular forms	1476	68	1408
Cusp forms	1405	68	1337
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.

\(2\)	\(3\)	\(5\)	\(23\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(11\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(8\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(10\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(10\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(6\)
Plus space				\(+\)	\(28\)
Minus space				\(-\)	\(40\)

Trace form

\( 68 q - 4 q^{7} + 68 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6900))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	23
6900.2.a.a	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}+q^{9}-5q^{11}-q^{13}+\cdots\)
6900.2.a.b	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+4q^{13}+3q^{17}+\cdots\)
6900.2.a.c	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
6900.2.a.d	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}+q^{9}+q^{11}-q^{13}-5q^{19}+\cdots\)
6900.2.a.e	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}-2q^{13}-6q^{17}+\cdots\)
6900.2.a.f	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}+q^{9}+q^{11}+q^{13}-5q^{19}+\cdots\)
6900.2.a.g	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+6q^{13}-2q^{17}+6q^{19}+\cdots\)
6900.2.a.h	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}+q^{9}-5q^{11}+q^{13}+\cdots\)
6900.2.a.i	$6900$	$2$	6900.a	1.a	$1$	$1$	$1$	$55.097$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+5q^{7}+q^{9}-4q^{13}+3q^{17}+\cdots\)
6900.2.a.j	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{15}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-6\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}+q^{9}+(1-\beta )q^{11}+(-1+\cdots)q^{13}+\cdots\)
6900.2.a.k	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-5\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2-\beta )q^{7}+q^{9}+\beta q^{11}+\cdots\)
6900.2.a.l	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-5\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2-\beta )q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
6900.2.a.m	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{7}+q^{9}+4\beta q^{11}-4\beta q^{13}+\cdots\)
6900.2.a.n	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+(-1+\beta )q^{11}+q^{13}+\cdots\)
6900.2.a.o	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
6900.2.a.p	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{7}+q^{9}-4q^{13}+\cdots\)
6900.2.a.q	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
6900.2.a.r	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+(-2-\beta )q^{11}-\beta q^{13}+\cdots\)
6900.2.a.s	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+2\beta )q^{7}+q^{9}+(-1+3\beta )q^{11}+\cdots\)
6900.2.a.t	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+(-1+\beta )q^{11}-q^{13}+\cdots\)
6900.2.a.u	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{7}+q^{9}+2\beta q^{11}-2q^{13}+\cdots\)
6900.2.a.v	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(5\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(3-\beta )q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
6900.2.a.w	$6900$	$2$	6900.a	1.a	$1$	$2$	$2$	$55.097$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(3-\beta )q^{7}+q^{9}+(3-\beta )q^{11}+\cdots\)
6900.2.a.x	$6900$	$2$	6900.a	1.a	$1$	$3$	$3$	$55.097$	3.3.3144.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta _{1})q^{7}+q^{9}+(1+\beta _{2})q^{11}+\cdots\)
6900.2.a.y	$6900$	$2$	6900.a	1.a	$1$	$3$	$3$	$55.097$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-\beta _{2}q^{11}+(1-\beta _{1}+\cdots)q^{13}+\cdots\)
6900.2.a.z	$6900$	$2$	6900.a	1.a	$1$	$3$	$3$	$55.097$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-\beta _{2}q^{11}+(-1+\beta _{1}+\cdots)q^{13}+\cdots\)
6900.2.a.ba	$6900$	$2$	6900.a	1.a	$1$	$4$	$4$	$55.097$	4.4.175557.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1}-\beta _{3})q^{7}+q^{9}-\beta _{1}q^{11}+\cdots\)
6900.2.a.bb	$6900$	$2$	6900.a	1.a	$1$	$4$	$4$	$55.097$	4.4.175557.1	None	✓	$1$	$1$	\(0\)	\(4\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{1}+\beta _{3})q^{7}+q^{9}-\beta _{1}q^{11}+\cdots\)
6900.2.a.bc	$6900$	$2$	6900.a	1.a	$1$	$7$	$7$	$55.097$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-7\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{7}+q^{9}+(\beta _{1}-\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)
6900.2.a.bd	$6900$	$2$	6900.a	1.a	$1$	$7$	$7$	$55.097$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(7\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{7}+q^{9}+(\beta _{1}-\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6900))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6900)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(575))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1725))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3450))\)\(^{\oplus 2}\)