Space of modular forms of level 1404, weight 2, and trivial character

• Label: 1404.2.a
• Level: $1404$
• Weight: $2$
• Character orbit: 1404.a
• Rep. character: $\chi_{1404}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $10$
• Sturm bound: $504$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1404 = 2^{2} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1404.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(504\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1404))\).

	Total	New	Old
Modular forms	270	16	254
Cusp forms	235	16	219
Eisenstein series	35	0	35

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

\(2\)	\(3\)	\(13\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(8\)

Trace form

\( 16 q - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1404))\) 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	3	13
1404.2.a.a	$1404$	$2$	1404.a	1.a	$1$	$1$	$1$	$11.211$	\(\Q\)	None	✓	✓			1404.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+2q^{7}+q^{13}+2q^{19}-6q^{23}+\cdots\)
1404.2.a.b	$1404$	$2$	1404.a	1.a	$1$	$1$	$1$	$11.211$	\(\Q\)	None	✓	✓			1404.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+4q^{11}-q^{13}-6q^{17}+\cdots\)
1404.2.a.c	$1404$	$2$	1404.a	1.a	$1$	$1$	$1$	$11.211$	\(\Q\)	None	✓	✓			1404.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-4q^{11}-q^{13}+6q^{17}+\cdots\)
1404.2.a.d	$1404$	$2$	1404.a	1.a	$1$	$1$	$1$	$11.211$	\(\Q\)	None	✓	✓			1404.2.a.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+2q^{7}+q^{13}+2q^{19}+6q^{23}+\cdots\)
1404.2.a.e	$1404$	$2$	1404.a	1.a	$1$	$2$	$2$	$11.211$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓		1404.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+q^{7}+(-2+\beta )q^{11}+\cdots\)
1404.2.a.f	$1404$	$2$	1404.a	1.a	$1$	$2$	$2$	$11.211$	\(\Q(\sqrt{13}) \)	None	✓	✓			1404.2.a.f	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+(-1+2\beta )q^{7}+(-3-\beta )q^{11}+\cdots\)
1404.2.a.g	$1404$	$2$	1404.a	1.a	$1$	$2$	$2$	$11.211$	\(\Q(\sqrt{7}) \)	None	✓	✓			1404.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2q^{7}+2\beta q^{11}-q^{13}+6q^{19}+\cdots\)
1404.2.a.h	$1404$	$2$	1404.a	1.a	$1$	$2$	$2$	$11.211$	\(\Q(\sqrt{7}) \)	None	✓	✓			1404.2.a.h	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2q^{7}-2\beta q^{11}+q^{13}-2\beta q^{17}+\cdots\)
1404.2.a.i	$1404$	$2$	1404.a	1.a	$1$	$2$	$2$	$11.211$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓		1404.2.a.f	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-1+2\beta )q^{7}+(3+\beta )q^{11}+\cdots\)
1404.2.a.j	$1404$	$2$	1404.a	1.a	$1$	$2$	$2$	$11.211$	\(\Q(\sqrt{13}) \)	None	✓	✓			1404.2.a.e	$1$	$0$	\(0\)	\(0\)	\(3\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+q^{7}+(2-\beta )q^{11}-q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1404))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1404)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(351))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(702))\)\(^{\oplus 2}\)