Space of modular forms of level 3456, weight 2, and trivial character

• Label: 3456.2.a
• Level: $3456$
• Weight: $2$
• Character orbit: 3456.a
• Rep. character: $\chi_{3456}(1,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $40$
• Sturm bound: $1152$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 40 \)
Sturm bound:			\(1152\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3456))\).

	Total	New	Old
Modular forms	624	64	560
Cusp forms	529	64	465
Eisenstein series	95	0	95

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

\(2\)	\(3\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(14\)
\(+\)	\(-\)	$-$	\(18\)
\(-\)	\(+\)	$-$	\(18\)
\(-\)	\(-\)	$+$	\(14\)
Plus space		\(+\)	\(28\)
Minus space		\(-\)	\(36\)

Trace form

\( 64 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3456))\) 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
3456.2.a.a	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(0\)	\(-3\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-5q^{7}-q^{11}-6q^{13}-6q^{17}+\cdots\)
3456.2.a.b	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-5q^{7}-q^{11}+6q^{13}+6q^{17}+\cdots\)
3456.2.a.c	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+5q^{7}+q^{11}-6q^{13}-6q^{17}+\cdots\)
3456.2.a.d	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+5q^{7}+q^{11}+6q^{13}+6q^{17}+\cdots\)
3456.2.a.e	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}+q^{11}-2q^{13}+2q^{17}+\cdots\)
3456.2.a.f	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}+q^{11}+2q^{13}-2q^{17}+\cdots\)
3456.2.a.g	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{7}-q^{11}-2q^{13}+2q^{17}+\cdots\)
3456.2.a.h	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{7}-q^{11}+2q^{13}-2q^{17}+\cdots\)
3456.2.a.i	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}-q^{11}-2q^{13}-2q^{17}+\cdots\)
3456.2.a.j	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}-q^{11}+2q^{13}+2q^{17}+\cdots\)
3456.2.a.k	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}+q^{11}-2q^{13}-2q^{17}+\cdots\)
3456.2.a.l	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}+q^{11}+2q^{13}+2q^{17}+\cdots\)
3456.2.a.m	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-5q^{7}+q^{11}-6q^{13}+6q^{17}+\cdots\)
3456.2.a.n	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-5q^{7}+q^{11}+6q^{13}-6q^{17}+\cdots\)
3456.2.a.o	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+5q^{7}-q^{11}-6q^{13}+6q^{17}+\cdots\)
3456.2.a.p	$3456$	$2$	3456.a	1.a	$1$	$1$	$1$	$27.596$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+5q^{7}-q^{11}+6q^{13}-6q^{17}+\cdots\)
3456.2.a.q	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(-1-\beta )q^{7}+(1+2\beta )q^{11}+\cdots\)
3456.2.a.r	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(-1+2\beta )q^{7}+(1-\beta )q^{11}+\cdots\)
3456.2.a.s	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(-1+2\beta )q^{7}+(1-\beta )q^{11}+\cdots\)
3456.2.a.t	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(-1-\beta )q^{7}+(1+2\beta )q^{11}+\cdots\)
3456.2.a.u	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}-q^{7}+(3+\beta )q^{11}+(-1+\cdots)q^{13}+\cdots\)
3456.2.a.v	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}-q^{7}+(3+\beta )q^{11}+(1+\cdots)q^{13}+\cdots\)
3456.2.a.w	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+q^{7}+(-3-\beta )q^{11}+\cdots\)
3456.2.a.x	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+q^{7}+(-3-\beta )q^{11}+\cdots\)
3456.2.a.y	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(1+\beta )q^{7}+(-1-2\beta )q^{11}+\cdots\)
3456.2.a.z	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(1-2\beta )q^{7}+(-1+\beta )q^{11}+\cdots\)
3456.2.a.ba	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(1-2\beta )q^{7}+(-1+\beta )q^{11}+\cdots\)
3456.2.a.bb	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(1+\beta )q^{7}+(-1-2\beta )q^{11}+\cdots\)
3456.2.a.bc	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-q^{7}+(-3+\beta )q^{11}+(-1+\cdots)q^{13}+\cdots\)
3456.2.a.bd	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-q^{7}+(-3+\beta )q^{11}+(1+\cdots)q^{13}+\cdots\)
3456.2.a.be	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-1+\beta )q^{7}+(-1+2\beta )q^{11}+\cdots\)
3456.2.a.bf	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-1-2\beta )q^{7}+(-1-\beta )q^{11}+\cdots\)
3456.2.a.bg	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-1-2\beta )q^{7}+(-1-\beta )q^{11}+\cdots\)
3456.2.a.bh	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-1+\beta )q^{7}+(-1+2\beta )q^{11}+\cdots\)
3456.2.a.bi	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(1-\beta )q^{7}+(1-2\beta )q^{11}+\cdots\)
3456.2.a.bj	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(1+2\beta )q^{7}+(1+\beta )q^{11}+\cdots\)
3456.2.a.bk	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(1+2\beta )q^{7}+(1+\beta )q^{11}+\cdots\)
3456.2.a.bl	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(1-\beta )q^{7}+(1-2\beta )q^{11}+\cdots\)
3456.2.a.bm	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+q^{7}+(3-\beta )q^{11}+(-1+\cdots)q^{13}+\cdots\)
3456.2.a.bn	$3456$	$2$	3456.a	1.a	$1$	$2$	$2$	$27.596$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+q^{7}+(3-\beta )q^{11}+(1-2\beta )q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3456))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3456)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(864))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1152))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1728))\)\(^{\oplus 2}\)