Space of modular forms of level 470, weight 2, and trivial character

• Label: 470.2.a
• Level: $470$
• Weight: $2$
• Character orbit: 470.a
• Rep. character: $\chi_{470}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $10$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 470 = 2 \cdot 5 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	470.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(470))\).

	Total	New	Old
Modular forms	76	17	59
Cusp forms	69	17	52
Eisenstein series	7	0	7

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

\(2\)	\(5\)	\(47\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(13\)

Trace form

\( 17 q + q^{2} + 4 q^{3} + 17 q^{4} - q^{5} + 8 q^{7} + q^{8} + 17 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(470))\) 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	47
470.2.a.a	$470$	$2$	470.a	1.a	$1$	$1$	$1$	$3.753$	\(\Q\)	None	✓	✓	✓	✓	470.2.a.a	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
470.2.a.b	$470$	$2$	470.a	1.a	$1$	$1$	$1$	$3.753$	\(\Q\)	None	✓	✓	✓	✓	470.2.a.b	$1$	$1$	\(-1\)	\(1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
470.2.a.c	$470$	$2$	470.a	1.a	$1$	$1$	$1$	$3.753$	\(\Q\)	None	✓	✓			470.2.a.c	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
470.2.a.d	$470$	$2$	470.a	1.a	$1$	$1$	$1$	$3.753$	\(\Q\)	None	✓	✓	✓	✓	470.2.a.d	$1$	$1$	\(1\)	\(-3\)	\(1\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}+q^{5}-3q^{6}-3q^{7}+\cdots\)
470.2.a.e	$470$	$2$	470.a	1.a	$1$	$1$	$1$	$3.753$	\(\Q\)	None	✓	✓	✓	✓	470.2.a.e	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-3q^{7}+\cdots\)
470.2.a.f	$470$	$2$	470.a	1.a	$1$	$1$	$1$	$3.753$	\(\Q\)	None	✓	✓			470.2.a.f	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+5q^{7}+\cdots\)
470.2.a.g	$470$	$2$	470.a	1.a	$1$	$2$	$2$	$3.753$	\(\Q(\sqrt{21}) \)	None	✓	✓	✓		470.2.a.g	$1$	$0$	\(-2\)	\(1\)	\(2\)	\(8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
470.2.a.h	$470$	$2$	470.a	1.a	$1$	$3$	$3$	$3.753$	3.3.837.1	None	✓	✓	✓	✓	470.2.a.h	$1$	$0$	\(-3\)	\(0\)	\(-3\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
470.2.a.i	$470$	$2$	470.a	1.a	$1$	$3$	$3$	$3.753$	3.3.229.1	None	✓	✓	✓	✓	470.2.a.i	$1$	$0$	\(3\)	\(2\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(1+\beta _{2})q^{6}+\cdots\)
470.2.a.j	$470$	$2$	470.a	1.a	$1$	$3$	$3$	$3.753$	3.3.1373.1	None	✓	✓	✓		470.2.a.j	$1$	$0$	\(3\)	\(3\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(470))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(470)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(47))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(94))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(235))\)\(^{\oplus 2}\)