Space of modular forms of level 6300, weight 2, and trivial character

• Label: 6300.2.a
• Level: $6300$
• Weight: $2$
• Character orbit: 6300.a
• Rep. character: $\chi_{6300}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $38$
• Sturm bound: $2880$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(2880\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6300))\).

	Total	New	Old
Modular forms	1512	48	1464
Cusp forms	1369	48	1321
Eisenstein series	143	0	143

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	Dim
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(6\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(7\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(7\)
Plus space				\(+\)	\(24\)
Minus space				\(-\)	\(24\)

Trace form

\( 48 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6300))\) 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	7
6300.2.a.a	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-6q^{11}+4q^{13}+6q^{17}+2q^{19}+\cdots\)
6300.2.a.b	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-4q^{11}-2q^{13}+2q^{17}-2q^{19}+\cdots\)
6300.2.a.c	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{11}-q^{13}-5q^{17}-8q^{19}+\cdots\)
6300.2.a.d	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{11}+q^{13}-3q^{17}+2q^{19}+\cdots\)
6300.2.a.e	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{11}+4q^{13}+2q^{19}-3q^{23}+\cdots\)
6300.2.a.f	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-q^{11}-4q^{13}+2q^{17}-4q^{19}+\cdots\)
6300.2.a.g	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-4q^{13}+4q^{17}+4q^{19}+8q^{23}+\cdots\)
6300.2.a.h	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+4q^{13}-6q^{17}+2q^{19}-6q^{23}+\cdots\)
6300.2.a.i	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+4q^{13}+6q^{17}+2q^{19}+6q^{23}+\cdots\)
6300.2.a.j	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+q^{11}-2q^{13}-8q^{17}-2q^{19}+\cdots\)
6300.2.a.k	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+q^{11}-2q^{13}+6q^{19}-q^{23}+\cdots\)
6300.2.a.l	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+2q^{11}-4q^{13}+2q^{17}+2q^{19}+\cdots\)
6300.2.a.m	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+3q^{11}+4q^{13}-6q^{17}-4q^{19}+\cdots\)
6300.2.a.n	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+4q^{11}+6q^{13}+2q^{17}+6q^{19}+\cdots\)
6300.2.a.o	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+5q^{11}+6q^{13}+4q^{17}-6q^{19}+\cdots\)
6300.2.a.p	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+6q^{11}-2q^{13}-4q^{19}-6q^{23}+\cdots\)
6300.2.a.q	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-4q^{11}+2q^{17}-6q^{19}+6q^{23}+\cdots\)
6300.2.a.r	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-4q^{11}+2q^{13}-2q^{17}-2q^{19}+\cdots\)
6300.2.a.s	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{11}-4q^{13}+2q^{19}+3q^{23}+\cdots\)
6300.2.a.t	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{11}+q^{13}+5q^{17}-8q^{19}+\cdots\)
6300.2.a.u	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-2q^{11}-4q^{13}+2q^{17}-2q^{19}+\cdots\)
6300.2.a.v	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-2q^{11}-4q^{13}+6q^{17}+6q^{19}+\cdots\)
6300.2.a.w	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-2q^{11}+6q^{13}-4q^{17}-4q^{19}+\cdots\)
6300.2.a.x	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-q^{11}+4q^{13}-2q^{17}-4q^{19}+\cdots\)
6300.2.a.y	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{13}-4q^{17}+4q^{19}-8q^{23}+\cdots\)
6300.2.a.z	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+q^{11}+2q^{13}+6q^{19}+q^{23}+\cdots\)
6300.2.a.ba	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+q^{11}+2q^{13}+8q^{17}-2q^{19}+\cdots\)
6300.2.a.bb	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+3q^{11}-4q^{13}+6q^{17}-4q^{19}+\cdots\)
6300.2.a.bc	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{11}-6q^{13}-2q^{17}+6q^{19}+\cdots\)
6300.2.a.bd	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{11}-2q^{17}-6q^{19}-6q^{23}+\cdots\)
6300.2.a.be	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+5q^{11}-6q^{13}-4q^{17}-6q^{19}+\cdots\)
6300.2.a.bf	$6300$	$2$	6300.a	1.a	$1$	$1$	$1$	$50.306$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+5q^{11}+3q^{13}-q^{17}+6q^{19}+\cdots\)
6300.2.a.bg	$6300$	$2$	6300.a	1.a	$1$	$2$	$2$	$50.306$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-q^{7}+\beta q^{11}-2q^{13}+2q^{19}-\beta q^{23}+\cdots\)
6300.2.a.bh	$6300$	$2$	6300.a	1.a	$1$	$2$	$2$	$50.306$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+\beta q^{11}+2\beta q^{17}-3\beta q^{23}+\cdots\)
6300.2.a.bi	$6300$	$2$	6300.a	1.a	$1$	$2$	$2$	$50.306$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+\beta q^{11}-2\beta q^{17}+3\beta q^{23}+\cdots\)
6300.2.a.bj	$6300$	$2$	6300.a	1.a	$1$	$2$	$2$	$50.306$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+q^{7}+\beta q^{11}+2q^{13}+2q^{19}+\beta q^{23}+\cdots\)
6300.2.a.bk	$6300$	$2$	6300.a	1.a	$1$	$4$	$4$	$50.306$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-q^{7}+(-\beta _{1}-\beta _{3})q^{11}+(-1+\beta _{2}+\cdots)q^{13}+\cdots\)
6300.2.a.bl	$6300$	$2$	6300.a	1.a	$1$	$4$	$4$	$50.306$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+q^{7}+(-\beta _{1}-\beta _{3})q^{11}+(1-\beta _{2})q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6300))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6300)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(630))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3150))\)\(^{\oplus 2}\)