Space of modular forms of level 1470, weight 2, and trivial character

• Label: 1470.2.a
• Level: $1470$
• Weight: $2$
• Character orbit: 1470.a
• Rep. character: $\chi_{1470}(1,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $22$
• Sturm bound: $672$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1470.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(672\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(19\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1470))\).

	Total	New	Old
Modular forms	368	26	342
Cusp forms	305	26	279
Eisenstein series	63	0	63

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
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(3\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(3\)
Plus space				\(+\)	\(6\)
Minus space				\(-\)	\(20\)

Trace form

\( 26 q - 2 q^{3} + 26 q^{4} + 26 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1470))\) 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
1470.2.a.a	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.b	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.c	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.d	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.e	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.f	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.g	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.h	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.i	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.j	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.k	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.l	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.m	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.n	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.o	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
1470.2.a.p	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
1470.2.a.q	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
1470.2.a.r	$1470$	$2$	1470.a	1.a	$1$	$1$	$1$	$11.738$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)
1470.2.a.s	$1470$	$2$	1470.a	1.a	$1$	$2$	$2$	$11.738$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(-2\)	\(2\)	\(0\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.t	$1470$	$2$	1470.a	1.a	$1$	$2$	$2$	$11.738$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(2\)	\(-2\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.u	$1470$	$2$	1470.a	1.a	$1$	$2$	$2$	$11.738$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(-2\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.v	$1470$	$2$	1470.a	1.a	$1$	$2$	$2$	$11.738$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1470))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1470)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(735))\)\(^{\oplus 2}\)