Space of modular forms of level 1456, weight 2, and trivial character

• Label: 1456.2.a
• Level: $1456$
• Weight: $2$
• Character orbit: 1456.a
• Rep. character: $\chi_{1456}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $22$
• Sturm bound: $448$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(448\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1456))\).

	Total	New	Old
Modular forms	236	36	200
Cusp forms	213	36	177
Eisenstein series	23	0	23

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

\(2\)	\(7\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(5\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(15\)
Minus space			\(-\)	\(21\)

Trace form

\( 36 q + 2 q^{7} + 44 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1456))\) 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	7	13
1456.2.a.a	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				182.2.a.e	$1$	$1$	\(0\)	\(-3\)	\(-4\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{5}+q^{7}+6q^{9}-q^{11}+\cdots\)
1456.2.a.b	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				182.2.a.b	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+6q^{9}+5q^{11}-q^{13}+\cdots\)
1456.2.a.c	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				728.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(3\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{5}-q^{7}+q^{9}-q^{13}-6q^{15}+\cdots\)
1456.2.a.d	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				182.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
1456.2.a.e	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				182.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{7}-2q^{9}+q^{11}+q^{13}+\cdots\)
1456.2.a.f	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				364.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-q^{7}-3q^{9}+2q^{11}-q^{13}+\cdots\)
1456.2.a.g	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				91.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+q^{7}-3q^{9}+6q^{11}-q^{13}+\cdots\)
1456.2.a.h	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				728.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{7}-3q^{9}-2q^{11}+q^{13}+\cdots\)
1456.2.a.i	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				182.2.a.c	$1$	$1$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}-3q^{9}-4q^{11}-q^{13}+\cdots\)
1456.2.a.j	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				728.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}-3q^{11}-q^{13}+\cdots\)
1456.2.a.k	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				91.2.a.b	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{5}-q^{7}+q^{9}+q^{13}-6q^{15}+\cdots\)
1456.2.a.l	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				728.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
1456.2.a.m	$1456$	$2$	1456.a	1.a	$1$	$1$	$1$	$11.626$	\(\Q\)	None	✓				364.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}+q^{13}+\cdots\)
1456.2.a.n	$1456$	$2$	1456.a	1.a	$1$	$2$	$2$	$11.626$	\(\Q(\sqrt{3}) \)	None	✓		✓		364.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+\beta q^{5}-q^{7}+(1-2\beta )q^{9}+\cdots\)
1456.2.a.o	$1456$	$2$	1456.a	1.a	$1$	$2$	$2$	$11.626$	\(\Q(\sqrt{17}) \)	None	✓		✓		728.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{5}+q^{7}+(1+\beta )q^{9}+\cdots\)
1456.2.a.p	$1456$	$2$	1456.a	1.a	$1$	$2$	$2$	$11.626$	\(\Q(\sqrt{6}) \)	None	✓		✓		364.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1-\beta )q^{5}+q^{7}+3q^{9}+\cdots\)
1456.2.a.q	$1456$	$2$	1456.a	1.a	$1$	$2$	$2$	$11.626$	\(\Q(\sqrt{2}) \)	None	✓		✓		91.2.a.c	$1$	$0$	\(0\)	\(0\)	\(6\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(3+\beta )q^{5}-q^{7}-q^{9}+3\beta q^{11}+\cdots\)
1456.2.a.r	$1456$	$2$	1456.a	1.a	$1$	$2$	$2$	$11.626$	\(\Q(\sqrt{3}) \)	None	✓				728.2.a.f	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+\beta q^{5}+q^{7}+(1+2\beta )q^{9}+\cdots\)
1456.2.a.s	$1456$	$2$	1456.a	1.a	$1$	$2$	$2$	$11.626$	\(\Q(\sqrt{2}) \)	None	✓				728.2.a.e	$1$	$0$	\(0\)	\(4\)	\(-2\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(-1-\beta )q^{5}-q^{7}+(3+\cdots)q^{9}+\cdots\)
1456.2.a.t	$1456$	$2$	1456.a	1.a	$1$	$3$	$3$	$11.626$	3.3.316.1	None	✓		✓		91.2.a.d	$1$	$0$	\(0\)	\(2\)	\(2\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{2})q^{3}+(1-\beta _{1})q^{5}+q^{7}+\cdots\)
1456.2.a.u	$1456$	$2$	1456.a	1.a	$1$	$4$	$4$	$11.626$	4.4.183064.1	None	✓		✓		728.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}-q^{7}+(2+\beta _{2})q^{9}+\cdots\)
1456.2.a.v	$1456$	$2$	1456.a	1.a	$1$	$4$	$4$	$11.626$	4.4.64268.1	None	✓		✓		728.2.a.i	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{2})q^{5}+q^{7}+(3+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1456))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1456)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(364))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(728))\)\(^{\oplus 2}\)