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

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

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(1728))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs		$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	2	3	$q$-expansion
1728.2.a.a	$1$	$13.798$	$\Q$	None	$0$	$0$	$-4$	$-3$	$+$	$-$	$q-4q^{5}-3q^{7}-4q^{11}-q^{13}-4q^{17}+\cdots$
1728.2.a.b	$1$	$13.798$	$\Q$	None	$0$	$0$	$-4$	$3$	$-$	$+$	$q-4q^{5}+3q^{7}+4q^{11}-q^{13}-4q^{17}+\cdots$
1728.2.a.c	$1$	$13.798$	$\Q$	None	$0$	$0$	$-3$	$-1$	$+$	$+$	$q-3q^{5}-q^{7}+3q^{11}+4q^{13}-2q^{19}+\cdots$
1728.2.a.d	$1$	$13.798$	$\Q$	None	$0$	$0$	$-3$	$1$	$-$	$-$	$q-3q^{5}+q^{7}-3q^{11}+4q^{13}+2q^{19}+\cdots$
1728.2.a.e	$1$	$13.798$	$\Q$	None	$0$	$0$	$-2$	$-3$	$-$	$-$	$q-2q^{5}-3q^{7}+6q^{11}+3q^{13}-2q^{17}+\cdots$
1728.2.a.f	$1$	$13.798$	$\Q$	None	$0$	$0$	$-2$	$-1$	$+$	$+$	$q-2q^{5}-q^{7}+2q^{11}-q^{13}+6q^{17}+\cdots$
1728.2.a.g	$1$	$13.798$	$\Q$	None	$0$	$0$	$-2$	$1$	$+$	$-$	$q-2q^{5}+q^{7}-2q^{11}-q^{13}+6q^{17}+\cdots$
1728.2.a.h	$1$	$13.798$	$\Q$	None	$0$	$0$	$-2$	$3$	$-$	$+$	$q-2q^{5}+3q^{7}-6q^{11}+3q^{13}-2q^{17}+\cdots$
1728.2.a.i	$1$	$13.798$	$\Q$	None	$0$	$0$	$-1$	$-3$	$-$	$+$	$q-q^{5}-3q^{7}-5q^{11}-4q^{13}+8q^{17}+\cdots$
1728.2.a.j	$1$	$13.798$	$\Q$	None	$0$	$0$	$-1$	$-3$	$-$	$+$	$q-q^{5}-3q^{7}+3q^{11}-4q^{17}+6q^{19}+\cdots$
1728.2.a.k	$1$	$13.798$	$\Q$	None	$0$	$0$	$-1$	$3$	$-$	$-$	$q-q^{5}+3q^{7}-3q^{11}-4q^{17}-6q^{19}+\cdots$
1728.2.a.l	$1$	$13.798$	$\Q$	None	$0$	$0$	$-1$	$3$	$+$	$-$	$q-q^{5}+3q^{7}+5q^{11}-4q^{13}+8q^{17}+\cdots$
1728.2.a.m	$1$	$13.798$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$-5$	$-$	$+$	$q-5q^{7}+7q^{13}-q^{19}-5q^{25}+4q^{31}+\cdots$
1728.2.a.n	$1$	$13.798$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$-1$	$+$	$+$	$q-q^{7}-5q^{13}+7q^{19}-5q^{25}-4q^{31}+\cdots$
1728.2.a.o	$1$	$13.798$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$1$	$-$	$-$	$q+q^{7}-5q^{13}-7q^{19}-5q^{25}+4q^{31}+\cdots$
1728.2.a.p	$1$	$13.798$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$5$	$+$	$-$	$q+5q^{7}+7q^{13}+q^{19}-5q^{25}-4q^{31}+\cdots$
1728.2.a.q	$1$	$13.798$	$\Q$	None	$0$	$0$	$1$	$-3$	$-$	$-$	$q+q^{5}-3q^{7}-3q^{11}+4q^{17}+6q^{19}+\cdots$
1728.2.a.r	$1$	$13.798$	$\Q$	None	$0$	$0$	$1$	$-3$	$-$	$-$	$q+q^{5}-3q^{7}+5q^{11}-4q^{13}-8q^{17}+\cdots$
1728.2.a.s	$1$	$13.798$	$\Q$	None	$0$	$0$	$1$	$3$	$+$	$+$	$q+q^{5}+3q^{7}-5q^{11}-4q^{13}-8q^{17}+\cdots$
1728.2.a.t	$1$	$13.798$	$\Q$	None	$0$	$0$	$1$	$3$	$-$	$+$	$q+q^{5}+3q^{7}+3q^{11}+4q^{17}-6q^{19}+\cdots$
1728.2.a.u	$1$	$13.798$	$\Q$	None	$0$	$0$	$2$	$-3$	$-$	$-$	$q+2q^{5}-3q^{7}-6q^{11}+3q^{13}+2q^{17}+\cdots$
1728.2.a.v	$1$	$13.798$	$\Q$	None	$0$	$0$	$2$	$-1$	$+$	$+$	$q+2q^{5}-q^{7}-2q^{11}-q^{13}-6q^{17}+\cdots$
1728.2.a.w	$1$	$13.798$	$\Q$	None	$0$	$0$	$2$	$1$	$+$	$-$	$q+2q^{5}+q^{7}+2q^{11}-q^{13}-6q^{17}+\cdots$
1728.2.a.x	$1$	$13.798$	$\Q$	None	$0$	$0$	$2$	$3$	$-$	$+$	$q+2q^{5}+3q^{7}+6q^{11}+3q^{13}+2q^{17}+\cdots$
1728.2.a.y	$1$	$13.798$	$\Q$	None	$0$	$0$	$3$	$-1$	$+$	$-$	$q+3q^{5}-q^{7}-3q^{11}+4q^{13}-2q^{19}+\cdots$
1728.2.a.z	$1$	$13.798$	$\Q$	None	$0$	$0$	$3$	$1$	$-$	$+$	$q+3q^{5}+q^{7}+3q^{11}+4q^{13}+2q^{19}+\cdots$
1728.2.a.ba	$1$	$13.798$	$\Q$	None	$0$	$0$	$4$	$-3$	$+$	$-$	$q+4q^{5}-3q^{7}+4q^{11}-q^{13}+4q^{17}+\cdots$
1728.2.a.bb	$1$	$13.798$	$\Q$	None	$0$	$0$	$4$	$3$	$-$	$+$	$q+4q^{5}+3q^{7}-4q^{11}-q^{13}+4q^{17}+\cdots$
1728.2.a.bc	$2$	$13.798$	$\Q(\sqrt{13}) $	None	$0$	$0$	$0$	$0$	$+$	$+$	$q-\beta q^{5}+\beta q^{7}-q^{11}-4q^{13}+2\beta q^{19}+\cdots$
1728.2.a.bd	$2$	$13.798$	$\Q(\sqrt{13}) $	None	$0$	$0$	$0$	$0$	$+$	$-$	$q-\beta q^{5}-\beta q^{7}+q^{11}-4q^{13}-2\beta q^{19}+\cdots$

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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 \(T_p\):			\(5\), \(7\), \(11\)

\(2\)	\(3\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(9\)
\(-\)	\(+\)	\(-\)	\(9\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(18\)

Introduction

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Modular forms

Varieties

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Representations

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Properties

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1728))\) into newform subspaces

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

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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	$11$

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs		$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	2	3	$q$-expansion
1728.2.a.a	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-4\)	\(-3\)	\(+\)	\(-\)	\(q-4q^{5}-3q^{7}-4q^{11}-q^{13}-4q^{17}+\cdots\)
1728.2.a.b	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-4\)	\(3\)	\(-\)	\(+\)	\(q-4q^{5}+3q^{7}+4q^{11}-q^{13}-4q^{17}+\cdots\)
1728.2.a.c	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-3\)	\(-1\)	\(+\)	\(+\)	\(q-3q^{5}-q^{7}+3q^{11}+4q^{13}-2q^{19}+\cdots\)
1728.2.a.d	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-3\)	\(1\)	\(-\)	\(-\)	\(q-3q^{5}+q^{7}-3q^{11}+4q^{13}+2q^{19}+\cdots\)
1728.2.a.e	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-2\)	\(-3\)	\(-\)	\(-\)	\(q-2q^{5}-3q^{7}+6q^{11}+3q^{13}-2q^{17}+\cdots\)
1728.2.a.f	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-2\)	\(-1\)	\(+\)	\(+\)	\(q-2q^{5}-q^{7}+2q^{11}-q^{13}+6q^{17}+\cdots\)
1728.2.a.g	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-2\)	\(1\)	\(+\)	\(-\)	\(q-2q^{5}+q^{7}-2q^{11}-q^{13}+6q^{17}+\cdots\)
1728.2.a.h	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-2\)	\(3\)	\(-\)	\(+\)	\(q-2q^{5}+3q^{7}-6q^{11}+3q^{13}-2q^{17}+\cdots\)
1728.2.a.i	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-1\)	\(-3\)	\(-\)	\(+\)	\(q-q^{5}-3q^{7}-5q^{11}-4q^{13}+8q^{17}+\cdots\)
1728.2.a.j	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-1\)	\(-3\)	\(-\)	\(+\)	\(q-q^{5}-3q^{7}+3q^{11}-4q^{17}+6q^{19}+\cdots\)
1728.2.a.k	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-1\)	\(3\)	\(-\)	\(-\)	\(q-q^{5}+3q^{7}-3q^{11}-4q^{17}-6q^{19}+\cdots\)
1728.2.a.l	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(-1\)	\(3\)	\(+\)	\(-\)	\(q-q^{5}+3q^{7}+5q^{11}-4q^{13}+8q^{17}+\cdots\)
1728.2.a.m	\(1\)	\(13.798\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(-5\)	\(-\)	\(+\)	\(q-5q^{7}+7q^{13}-q^{19}-5q^{25}+4q^{31}+\cdots\)
1728.2.a.n	\(1\)	\(13.798\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(-1\)	\(+\)	\(+\)	\(q-q^{7}-5q^{13}+7q^{19}-5q^{25}-4q^{31}+\cdots\)
1728.2.a.o	\(1\)	\(13.798\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(1\)	\(-\)	\(-\)	\(q+q^{7}-5q^{13}-7q^{19}-5q^{25}+4q^{31}+\cdots\)
1728.2.a.p	\(1\)	\(13.798\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(5\)	\(+\)	\(-\)	\(q+5q^{7}+7q^{13}+q^{19}-5q^{25}-4q^{31}+\cdots\)
1728.2.a.q	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(1\)	\(-3\)	\(-\)	\(-\)	\(q+q^{5}-3q^{7}-3q^{11}+4q^{17}+6q^{19}+\cdots\)
1728.2.a.r	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(1\)	\(-3\)	\(-\)	\(-\)	\(q+q^{5}-3q^{7}+5q^{11}-4q^{13}-8q^{17}+\cdots\)
1728.2.a.s	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(1\)	\(3\)	\(+\)	\(+\)	\(q+q^{5}+3q^{7}-5q^{11}-4q^{13}-8q^{17}+\cdots\)
1728.2.a.t	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(1\)	\(3\)	\(-\)	\(+\)	\(q+q^{5}+3q^{7}+3q^{11}+4q^{17}-6q^{19}+\cdots\)
1728.2.a.u	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(2\)	\(-3\)	\(-\)	\(-\)	\(q+2q^{5}-3q^{7}-6q^{11}+3q^{13}+2q^{17}+\cdots\)
1728.2.a.v	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(2\)	\(-1\)	\(+\)	\(+\)	\(q+2q^{5}-q^{7}-2q^{11}-q^{13}-6q^{17}+\cdots\)
1728.2.a.w	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(2\)	\(1\)	\(+\)	\(-\)	\(q+2q^{5}+q^{7}+2q^{11}-q^{13}-6q^{17}+\cdots\)
1728.2.a.x	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(2\)	\(3\)	\(-\)	\(+\)	\(q+2q^{5}+3q^{7}+6q^{11}+3q^{13}+2q^{17}+\cdots\)
1728.2.a.y	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(3\)	\(-1\)	\(+\)	\(-\)	\(q+3q^{5}-q^{7}-3q^{11}+4q^{13}-2q^{19}+\cdots\)
1728.2.a.z	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(3\)	\(1\)	\(-\)	\(+\)	\(q+3q^{5}+q^{7}+3q^{11}+4q^{13}+2q^{19}+\cdots\)
1728.2.a.ba	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(4\)	\(-3\)	\(+\)	\(-\)	\(q+4q^{5}-3q^{7}+4q^{11}-q^{13}+4q^{17}+\cdots\)
1728.2.a.bb	\(1\)	\(13.798\)	\(\Q\)	None	\(0\)	\(0\)	\(4\)	\(3\)	\(-\)	\(+\)	\(q+4q^{5}+3q^{7}-4q^{11}-q^{13}+4q^{17}+\cdots\)
1728.2.a.bc	\(2\)	\(13.798\)	\(\Q(\sqrt{13}) \)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(+\)	\(+\)	\(q-\beta q^{5}+\beta q^{7}-q^{11}-4q^{13}+2\beta q^{19}+\cdots\)
1728.2.a.bd	\(2\)	\(13.798\)	\(\Q(\sqrt{13}) \)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(+\)	\(-\)	\(q-\beta q^{5}-\beta q^{7}+q^{11}-4q^{13}-2\beta q^{19}+\cdots\)