Space of modular forms of level 475, weight 6, and trivial character

• Label: 475.6.a
• Level: $475$
• Weight: $6$
• Character orbit: 475.a
• Rep. character: $\chi_{475}(1,\cdot)$
• Character field: $\Q$
• Dimension: $142$
• Newform subspaces: $15$
• Sturm bound: $300$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(300\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(475))\).

	Total	New	Old
Modular forms	256	142	114
Cusp forms	244	142	102
Eisenstein series	12	0	12

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

\(5\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(33\)
\(+\)	\(-\)	$-$	\(35\)
\(-\)	\(+\)	$-$	\(39\)
\(-\)	\(-\)	$+$	\(35\)
Plus space		\(+\)	\(68\)
Minus space		\(-\)	\(74\)

Trace form

\( 142 q + 2 q^{2} - 2 q^{3} + 2234 q^{4} + 122 q^{6} + 233 q^{7} - 120 q^{8} + 11438 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(475))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	19
475.6.a.a	$475$	$6$	475.a	1.a	$1$	$1$	$1$	$76.182$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(1\)	\(0\)	\(167\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}-28q^{4}+2q^{6}+167q^{7}+\cdots\)
475.6.a.b	$475$	$6$	475.a	1.a	$1$	$1$	$1$	$76.182$	\(\Q\)	None	✓	$1$	$0$	\(6\)	\(-4\)	\(0\)	\(-248\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}-4q^{3}+4q^{4}-24q^{6}-248q^{7}+\cdots\)
475.6.a.c	$475$	$6$	475.a	1.a	$1$	$1$	$1$	$76.182$	\(\Q\)	None	✓	$1$	$0$	\(7\)	\(11\)	\(0\)	\(197\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{2}+11q^{3}+17q^{4}+77q^{6}+197q^{7}+\cdots\)
475.6.a.d	$475$	$6$	475.a	1.a	$1$	$2$	$2$	$76.182$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$1$	\(7\)	\(7\)	\(0\)	\(-72\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(4-\beta )q^{2}+(2+3\beta )q^{3}+(28-7\beta )q^{4}+\cdots\)
475.6.a.e	$475$	$6$	475.a	1.a	$1$	$4$	$4$	$76.182$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-9\)	\(-6\)	\(0\)	\(190\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1}-\beta _{2})q^{2}+(-3-3\beta _{2}+\cdots)q^{3}+\cdots\)
475.6.a.f	$475$	$6$	475.a	1.a	$1$	$5$	$5$	$76.182$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(16\)	\(0\)	\(312\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(4-\beta _{1}-\beta _{4})q^{3}+(15+\cdots)q^{4}+\cdots\)
475.6.a.g	$475$	$6$	475.a	1.a	$1$	$6$	$6$	$76.182$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(20\)	\(0\)	\(80\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3-\beta _{4})q^{3}+(9-2\beta _{1}+\cdots)q^{4}+\cdots\)
475.6.a.h	$475$	$6$	475.a	1.a	$1$	$9$	$9$	$76.182$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(-38\)	\(0\)	\(-80\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-4-\beta _{3})q^{3}+(23+\cdots)q^{4}+\cdots\)
475.6.a.i	$475$	$6$	475.a	1.a	$1$	$9$	$9$	$76.182$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(-4\)	\(-9\)	\(0\)	\(-313\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(18+\beta _{2}+\cdots)q^{4}+\cdots\)
475.6.a.j	$475$	$6$	475.a	1.a	$1$	$15$	$15$	$76.182$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-13\)	\(-18\)	\(0\)	\(-196\)		$-$	$-$	$+$	$2^{3}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{4})q^{3}+(2^{4}+\cdots)q^{4}+\cdots\)
475.6.a.k	$475$	$6$	475.a	1.a	$1$	$15$	$15$	$76.182$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-11\)	\(-18\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{4})q^{3}+(2^{4}+\cdots)q^{4}+\cdots\)
475.6.a.l	$475$	$6$	475.a	1.a	$1$	$15$	$15$	$76.182$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(11\)	\(18\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{4})q^{3}+(2^{4}-\beta _{1}+\cdots)q^{4}+\cdots\)
475.6.a.m	$475$	$6$	475.a	1.a	$1$	$15$	$15$	$76.182$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(13\)	\(18\)	\(0\)	\(196\)		$+$	$-$	$-$	$2^{3}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{4})q^{3}+(2^{4}-2\beta _{1}+\cdots)q^{4}+\cdots\)
475.6.a.n	$475$	$6$	475.a	1.a	$1$	$20$	$20$	$76.182$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{13}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{11}q^{3}+(13+\beta _{2})q^{4}+(-2+\cdots)q^{6}+\cdots\)
475.6.a.o	$475$	$6$	475.a	1.a	$1$	$24$	$24$	$76.182$		None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(475))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(475)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 2}\)