Space of modular forms of level 6200, weight 2, and trivial character

• Label: 6200.2.a
• Level: $6200$
• Weight: $2$
• Character orbit: 6200.a
• Rep. character: $\chi_{6200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $142$
• Newform subspaces: $32$
• Sturm bound: $1920$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 6200 = 2^{3} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(1920\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6200))\).

	Total	New	Old
Modular forms	984	142	842
Cusp forms	937	142	795
Eisenstein series	47	0	47

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(5\)	\(31\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(15\)
\(+\)	\(+\)	\(-\)	\(-\)	\(21\)
\(+\)	\(-\)	\(+\)	\(-\)	\(20\)
\(+\)	\(-\)	\(-\)	\(+\)	\(16\)
\(-\)	\(+\)	\(+\)	\(-\)	\(19\)
\(-\)	\(+\)	\(-\)	\(+\)	\(13\)
\(-\)	\(-\)	\(+\)	\(+\)	\(17\)
\(-\)	\(-\)	\(-\)	\(-\)	\(21\)
Plus space			\(+\)	\(61\)
Minus space			\(-\)	\(81\)

Trace form

\( 142 q + 2 q^{3} + 130 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6200))\) 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	5	31
6200.2.a.a	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				1240.2.a.g	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{7}+6q^{9}-2q^{11}+2q^{13}+\cdots\)
6200.2.a.b	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				1240.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-4q^{13}+4q^{19}-6q^{23}+\cdots\)
6200.2.a.c	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				1240.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
6200.2.a.d	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓	✓			6200.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}-5q^{11}-5q^{13}+2q^{17}+\cdots\)
6200.2.a.e	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				1240.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-4q^{11}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
6200.2.a.f	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				1240.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+2q^{11}-2q^{13}-8q^{19}+2q^{23}+\cdots\)
6200.2.a.g	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓	✓			6200.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}-5q^{11}+5q^{13}-2q^{17}+\cdots\)
6200.2.a.h	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				248.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}-3q^{9}+2q^{11}+4q^{13}+q^{19}+\cdots\)
6200.2.a.i	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				1240.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}-2q^{11}+2q^{13}+\cdots\)
6200.2.a.j	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				1240.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+2q^{13}-3q^{17}+q^{19}+\cdots\)
6200.2.a.k	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				248.2.a.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+2q^{11}-4q^{13}-6q^{17}+\cdots\)
6200.2.a.l	$6200$	$2$	6200.a	1.a	$1$	$1$	$1$	$49.507$	\(\Q\)	None	✓				248.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
6200.2.a.m	$6200$	$2$	6200.a	1.a	$1$	$2$	$2$	$49.507$	\(\Q(\sqrt{33}) \)	None	✓				248.2.a.d	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-\beta q^{7}+q^{9}-2q^{11}-2\beta q^{13}+\cdots\)
6200.2.a.n	$6200$	$2$	6200.a	1.a	$1$	$2$	$2$	$49.507$	\(\Q(\sqrt{2}) \)	None	✓				1240.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2q^{7}-q^{9}+(2-\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
6200.2.a.o	$6200$	$2$	6200.a	1.a	$1$	$2$	$2$	$49.507$	\(\Q(\sqrt{17}) \)	None	✓				1240.2.a.h	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{9}+(-2+2\beta )q^{11}+\cdots\)
6200.2.a.p	$6200$	$2$	6200.a	1.a	$1$	$3$	$3$	$49.507$	3.3.316.1	None	✓				248.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-1-2\beta _{1}+\cdots)q^{7}+\cdots\)
6200.2.a.q	$6200$	$2$	6200.a	1.a	$1$	$3$	$3$	$49.507$	3.3.148.1	None	✓				1240.2.a.j	$1$	$1$	\(0\)	\(3\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(1-\beta _{1}-\beta _{2})q^{7}+(1+\cdots)q^{9}+\cdots\)
6200.2.a.r	$6200$	$2$	6200.a	1.a	$1$	$4$	$4$	$49.507$	4.4.112820.1	None	✓				1240.2.a.k	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{3})q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
6200.2.a.s	$6200$	$2$	6200.a	1.a	$1$	$5$	$5$	$49.507$	5.5.288633.1	None	✓	✓	✓		6200.2.a.s	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{3}-\beta _{4})q^{7}+\beta _{2}q^{9}+(-1+\cdots)q^{11}+\cdots\)
6200.2.a.t	$6200$	$2$	6200.a	1.a	$1$	$5$	$5$	$49.507$	5.5.288633.1	None	✓	✓			6200.2.a.s	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{3}+\beta _{4})q^{7}+\beta _{2}q^{9}+\cdots\)
6200.2.a.u	$6200$	$2$	6200.a	1.a	$1$	$6$	$6$	$49.507$	6.6.112647633.1	None	✓	✓	✓		6200.2.a.u	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{2}q^{7}+(\beta _{2}-\beta _{3})q^{9}+(-1+\cdots)q^{11}+\cdots\)
6200.2.a.v	$6200$	$2$	6200.a	1.a	$1$	$6$	$6$	$49.507$	6.6.112647633.1	None	✓	✓			6200.2.a.u	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{2}q^{7}+(\beta _{2}-\beta _{3})q^{9}+(-1+\cdots)q^{11}+\cdots\)
6200.2.a.w	$6200$	$2$	6200.a	1.a	$1$	$6$	$6$	$49.507$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				1240.2.a.m	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{7}+(4-\beta _{3}+\beta _{4})q^{9}+\cdots\)
6200.2.a.x	$6200$	$2$	6200.a	1.a	$1$	$6$	$6$	$49.507$	6.6.473125168.1	None	✓				1240.2.a.l	$1$	$0$	\(0\)	\(3\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{2}-\beta _{4})q^{7}+(\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
6200.2.a.y	$6200$	$2$	6200.a	1.a	$1$	$8$	$8$	$49.507$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		6200.2.a.y	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{4}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
6200.2.a.z	$6200$	$2$	6200.a	1.a	$1$	$8$	$8$	$49.507$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			6200.2.a.y	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{4}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
6200.2.a.ba	$6200$	$2$	6200.a	1.a	$1$	$10$	$10$	$49.507$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			6200.2.a.ba	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{8}q^{7}+(1+\beta _{2})q^{9}+\beta _{3}q^{11}+\cdots\)
6200.2.a.bb	$6200$	$2$	6200.a	1.a	$1$	$10$	$10$	$49.507$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		6200.2.a.ba	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{8}q^{7}+(1+\beta _{2})q^{9}+\beta _{3}q^{11}+\cdots\)
6200.2.a.bc	$6200$	$2$	6200.a	1.a	$1$	$11$	$11$	$49.507$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1240.2.d.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-8\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
6200.2.a.bd	$6200$	$2$	6200.a	1.a	$1$	$11$	$11$	$49.507$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1240.2.d.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{9}q^{7}+(1+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
6200.2.a.be	$6200$	$2$	6200.a	1.a	$1$	$11$	$11$	$49.507$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1240.2.d.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{9}q^{7}+(1+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
6200.2.a.bf	$6200$	$2$	6200.a	1.a	$1$	$11$	$11$	$49.507$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1240.2.d.a	$1$	$0$	\(0\)	\(4\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6200)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(248))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(775))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1550))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3100))\)\(^{\oplus 2}\)