Space of modular forms of level 5700, weight 2, and trivial character

• Label: 5700.2.a
• Level: $5700$
• Weight: $2$
• Character orbit: 5700.a
• Rep. character: $\chi_{5700}(1,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $30$
• Sturm bound: $2400$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5700.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(2400\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5700))\).

	Total	New	Old
Modular forms	1236	56	1180
Cusp forms	1165	56	1109
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.

\(2\)	\(3\)	\(5\)	\(19\)	Fricke	Dim.
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(5\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(9\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(5\)
Plus space				\(+\)	\(23\)
Minus space				\(-\)	\(33\)

Trace form

\( 56 q - 2 q^{7} + 56 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5700))\) 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	19
5700.2.a.a	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}+4q^{13}-6q^{17}+\cdots\)
5700.2.a.b	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+2q^{17}-q^{19}+q^{21}+\cdots\)
5700.2.a.c	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}-4q^{17}+\cdots\)
5700.2.a.d	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}+6q^{17}+\cdots\)
5700.2.a.e	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+4q^{13}+q^{19}-q^{21}+\cdots\)
5700.2.a.f	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}+4q^{13}+2q^{17}+\cdots\)
5700.2.a.g	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
5700.2.a.h	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
5700.2.a.i	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(5\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
5700.2.a.j	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-5\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-5q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
5700.2.a.k	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
5700.2.a.l	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-5q^{11}+6q^{13}+\cdots\)
5700.2.a.m	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-4q^{11}-6q^{17}+\cdots\)
5700.2.a.n	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-4q^{11}+4q^{17}+\cdots\)
5700.2.a.o	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-4q^{13}+q^{19}-q^{21}+\cdots\)
5700.2.a.p	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+2q^{11}-2q^{13}-6q^{17}+\cdots\)
5700.2.a.q	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-2q^{17}-q^{19}+q^{21}+\cdots\)
5700.2.a.r	$5700$	$2$	5700.a	1.a	$1$	$1$	$1$	$45.515$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}+4q^{11}+2q^{17}+\cdots\)
5700.2.a.s	$5700$	$2$	5700.a	1.a	$1$	$2$	$2$	$45.515$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-8\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}+q^{9}+(1+\beta )q^{11}+2\beta q^{13}+\cdots\)
5700.2.a.t	$5700$	$2$	5700.a	1.a	$1$	$2$	$2$	$45.515$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
5700.2.a.u	$5700$	$2$	5700.a	1.a	$1$	$2$	$2$	$45.515$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{7}+q^{9}+(-1+\beta )q^{11}+\cdots\)
5700.2.a.v	$5700$	$2$	5700.a	1.a	$1$	$2$	$2$	$45.515$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(8\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}+q^{9}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)
5700.2.a.w	$5700$	$2$	5700.a	1.a	$1$	$3$	$3$	$45.515$	3.3.564.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-4\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta _{2})q^{7}+q^{9}+(2+\beta _{1}+\cdots)q^{11}+\cdots\)
5700.2.a.x	$5700$	$2$	5700.a	1.a	$1$	$3$	$3$	$45.515$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{2}q^{7}+q^{9}+(-1-\beta _{1})q^{11}+\cdots\)
5700.2.a.y	$5700$	$2$	5700.a	1.a	$1$	$3$	$3$	$45.515$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{2}q^{7}+q^{9}+(-1-\beta _{1})q^{11}+\cdots\)
5700.2.a.z	$5700$	$2$	5700.a	1.a	$1$	$3$	$3$	$45.515$	3.3.1524.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{7}+q^{9}+(-1-\beta _{2})q^{11}+\cdots\)
5700.2.a.ba	$5700$	$2$	5700.a	1.a	$1$	$4$	$4$	$45.515$	\(\Q(\sqrt{6}, \sqrt{22})\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-2\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{2})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
5700.2.a.bb	$5700$	$2$	5700.a	1.a	$1$	$4$	$4$	$45.515$	\(\Q(\sqrt{6}, \sqrt{22})\)	None	✓	$1$	$0$	\(0\)	\(4\)	\(0\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{2})q^{7}+q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
5700.2.a.bc	$5700$	$2$	5700.a	1.a	$1$	$5$	$5$	$45.515$	5.5.13090800.1	None	✓	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{7}+q^{9}+(1-\beta _{3})q^{11}+\cdots\)
5700.2.a.bd	$5700$	$2$	5700.a	1.a	$1$	$5$	$5$	$45.515$	5.5.13090800.1	None	✓	$1$	$0$	\(0\)	\(5\)	\(0\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{7}+q^{9}+(1-\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5700))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5700)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1425))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\)\(^{\oplus 2}\)