Space of modular forms of level 1728, weight 2, and trivial character

• Label: 1728.2.a
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.a
• Rep. character: $\chi_{1728}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $30$
• Sturm bound: $576$
• Trace bound: $11$

Defining parameters

Level:	$N$	$=$	$1728 = 2^{6} \cdot 3^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1728.a (trivial)
Character field:			$\Q$
Newform subspaces:			$30$
Sturm bound:			$576$
Trace bound:			$11$
Distinguishing $T_p$:			$5$, $7$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(1728))$.

	Total	New	Old
Modular forms	324	32	292
Cusp forms	253	32	221
Eisenstein series	71	0	71

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
$+$	$+$	$+$	$7$
$+$	$-$	$-$	$9$
$-$	$+$	$-$	$9$
$-$	$-$	$+$	$7$
Plus space		$+$	$14$
Minus space		$-$	$18$

Trace form

$32 q - 8 q^{13} + 32 q^{25} - 8 q^{37} + 32 q^{49} + 8 q^{61} + 16 q^{85} + 16 q^{97}+O(q^{100})$

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(1728))$ 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
1728.2.a.a	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				216.2.a.a	$1$	$0$	$0$	$0$	$-4$	$-3$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-4q^{5}-3q^{7}-4q^{11}-q^{13}-4q^{17}+\cdots$
1728.2.a.b	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				216.2.a.a	$1$	$0$	$0$	$0$	$-4$	$3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-4q^{5}+3q^{7}+4q^{11}-q^{13}-4q^{17}+\cdots$
1728.2.a.c	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				54.2.a.a	$1$	$1$	$0$	$0$	$-3$	$-1$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-3q^{5}-q^{7}+3q^{11}+4q^{13}-2q^{19}+\cdots$
1728.2.a.d	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				54.2.a.a	$1$	$1$	$0$	$0$	$-3$	$1$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-3q^{5}+q^{7}-3q^{11}+4q^{13}+2q^{19}+\cdots$
1728.2.a.e	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.a	$1$	$1$	$0$	$0$	$-2$	$-3$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{5}-3q^{7}+6q^{11}+3q^{13}-2q^{17}+\cdots$
1728.2.a.f	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.b	$1$	$1$	$0$	$0$	$-2$	$-1$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{5}-q^{7}+2q^{11}-q^{13}+6q^{17}+\cdots$
1728.2.a.g	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.b	$1$	$0$	$0$	$0$	$-2$	$1$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{5}+q^{7}-2q^{11}-q^{13}+6q^{17}+\cdots$
1728.2.a.h	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.a	$1$	$0$	$0$	$0$	$-2$	$3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{5}+3q^{7}-6q^{11}+3q^{13}-2q^{17}+\cdots$
1728.2.a.i	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				216.2.a.b	$1$	$0$	$0$	$0$	$-1$	$-3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-3q^{7}-5q^{11}-4q^{13}+8q^{17}+\cdots$
1728.2.a.j	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.e	$1$	$0$	$0$	$0$	$-1$	$-3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-3q^{7}+3q^{11}-4q^{17}+6q^{19}+\cdots$
1728.2.a.k	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.e	$1$	$1$	$0$	$0$	$-1$	$3$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}+3q^{7}-3q^{11}-4q^{17}-6q^{19}+\cdots$
1728.2.a.l	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				216.2.a.b	$1$	$0$	$0$	$0$	$-1$	$3$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{5}+3q^{7}+5q^{11}-4q^{13}+8q^{17}+\cdots$
1728.2.a.m	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	$\Q(\sqrt{-3})$	✓				108.2.a.a	$2$	$0$	$0$	$0$	$0$	$-5$		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	$q-5q^{7}+7q^{13}-q^{19}-5q^{25}+4q^{31}+\cdots$
1728.2.a.n	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	$\Q(\sqrt{-3})$	✓				27.2.a.a	$2$	$1$	$0$	$0$	$0$	$-1$		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	$q-q^{7}-5q^{13}+7q^{19}-5q^{25}-4q^{31}+\cdots$
1728.2.a.o	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	$\Q(\sqrt{-3})$	✓				27.2.a.a	$2$	$1$	$0$	$0$	$0$	$1$		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	$q+q^{7}-5q^{13}-7q^{19}-5q^{25}+4q^{31}+\cdots$
1728.2.a.p	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	$\Q(\sqrt{-3})$	✓				108.2.a.a	$2$	$0$	$0$	$0$	$0$	$5$		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	$q+5q^{7}+7q^{13}+q^{19}-5q^{25}-4q^{31}+\cdots$
1728.2.a.q	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.e	$1$	$1$	$0$	$0$	$1$	$-3$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{5}-3q^{7}-3q^{11}+4q^{17}+6q^{19}+\cdots$
1728.2.a.r	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				216.2.a.b	$1$	$1$	$0$	$0$	$1$	$-3$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{5}-3q^{7}+5q^{11}-4q^{13}-8q^{17}+\cdots$
1728.2.a.s	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				216.2.a.b	$1$	$1$	$0$	$0$	$1$	$3$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{5}+3q^{7}-5q^{11}-4q^{13}-8q^{17}+\cdots$
1728.2.a.t	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.e	$1$	$0$	$0$	$0$	$1$	$3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{5}+3q^{7}+3q^{11}+4q^{17}-6q^{19}+\cdots$
1728.2.a.u	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.a	$1$	$1$	$0$	$0$	$2$	$-3$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{5}-3q^{7}-6q^{11}+3q^{13}+2q^{17}+\cdots$
1728.2.a.v	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.b	$1$	$1$	$0$	$0$	$2$	$-1$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{5}-q^{7}-2q^{11}-q^{13}-6q^{17}+\cdots$
1728.2.a.w	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.b	$1$	$0$	$0$	$0$	$2$	$1$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{5}+q^{7}+2q^{11}-q^{13}-6q^{17}+\cdots$
1728.2.a.x	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				864.2.a.a	$1$	$0$	$0$	$0$	$2$	$3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{5}+3q^{7}+6q^{11}+3q^{13}+2q^{17}+\cdots$
1728.2.a.y	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				54.2.a.a	$1$	$0$	$0$	$0$	$3$	$-1$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+3q^{5}-q^{7}-3q^{11}+4q^{13}-2q^{19}+\cdots$
1728.2.a.z	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				54.2.a.a	$1$	$0$	$0$	$0$	$3$	$1$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+3q^{5}+q^{7}+3q^{11}+4q^{13}+2q^{19}+\cdots$
1728.2.a.ba	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				216.2.a.a	$1$	$0$	$0$	$0$	$4$	$-3$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+4q^{5}-3q^{7}+4q^{11}-q^{13}+4q^{17}+\cdots$
1728.2.a.bb	$1728$	$2$	1728.a	1.a	$1$	$1$	$1$	$13.798$	$\Q$	None	✓				216.2.a.a	$1$	$0$	$0$	$0$	$4$	$3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+4q^{5}+3q^{7}-4q^{11}-q^{13}+4q^{17}+\cdots$
1728.2.a.bc	$1728$	$2$	1728.a	1.a	$1$	$2$	$2$	$13.798$	$\Q(\sqrt{13})$	None	✓		✓		864.2.a.m	$2$	$1$	$0$	$0$	$0$	$0$		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-\beta q^{5}+\beta q^{7}-q^{11}-4q^{13}+2\beta q^{19}+\cdots$
1728.2.a.bd	$1728$	$2$	1728.a	1.a	$1$	$2$	$2$	$13.798$	$\Q(\sqrt{13})$	None	✓		✓		864.2.a.m	$2$	$0$	$0$	$0$	$0$	$0$		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta q^{5}-\beta q^{7}+q^{11}-4q^{13}-2\beta q^{19}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(1728))$ into lower level spaces

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