Space of modular forms of level 6728, weight 2, and trivial character

• Label: 6728.2.a
• Level: $6728$
• Weight: $2$
• Character orbit: 6728.a
• Rep. character: $\chi_{6728}(1,\cdot)$
• Character field: $\Q$
• Dimension: $203$
• Newform subspaces: $32$
• Sturm bound: $1740$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 6728 = 2^{3} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6728.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(1740\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6728))\).

	Total	New	Old
Modular forms	930	203	727
Cusp forms	811	203	608
Eisenstein series	119	0	119

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

\(2\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(45\)
\(+\)	\(-\)	$-$	\(56\)
\(-\)	\(+\)	$-$	\(53\)
\(-\)	\(-\)	$+$	\(49\)
Plus space		\(+\)	\(94\)
Minus space		\(-\)	\(109\)

Trace form

\( 203 q + 4 q^{7} + 209 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6728))\) 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	29
6728.2.a.a	$6728$	$2$	6728.a	1.a	$1$	$1$	$1$	$53.723$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
6728.2.a.b	$6728$	$2$	6728.a	1.a	$1$	$1$	$1$	$53.723$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}-2q^{9}-3q^{11}+\cdots\)
6728.2.a.c	$6728$	$2$	6728.a	1.a	$1$	$1$	$1$	$53.723$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+2q^{7}-2q^{9}+3q^{11}+\cdots\)
6728.2.a.d	$6728$	$2$	6728.a	1.a	$1$	$1$	$1$	$53.723$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
6728.2.a.e	$6728$	$2$	6728.a	1.a	$1$	$2$	$2$	$53.723$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(5\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2\beta q^{3}+(2+\beta )q^{5}+(-2-\beta )q^{7}+\cdots\)
6728.2.a.f	$6728$	$2$	6728.a	1.a	$1$	$2$	$2$	$53.723$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1+2\beta )q^{5}-4q^{7}+\cdots\)
6728.2.a.g	$6728$	$2$	6728.a	1.a	$1$	$2$	$2$	$53.723$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(5\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+(2+\beta )q^{5}+(-2-\beta )q^{7}+\cdots\)
6728.2.a.h	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{3}+(2\beta _{1}-\beta _{2})q^{5}+\cdots\)
6728.2.a.i	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(9\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
6728.2.a.j	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	3.3.568.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{2})q^{5}+(2+\beta _{2})q^{9}+\cdots\)
6728.2.a.k	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$2$	\(0\)	\(-1\)	\(-7\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-3+\beta _{1}-\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
6728.2.a.l	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	3.3.3221.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{1}q^{5}-q^{7}+(3+\beta _{1}+\beta _{2})q^{9}+\cdots\)
6728.2.a.m	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(2\beta _{1}-\beta _{2})q^{5}+3q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
6728.2.a.n	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-7\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-3+\beta _{1}-\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
6728.2.a.o	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	3.3.3221.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{1}q^{5}-q^{7}+(3+\beta _{1}+\beta _{2})q^{9}+\cdots\)
6728.2.a.p	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(2\beta _{1}-\beta _{2})q^{5}+3q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
6728.2.a.q	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(1\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-\beta _{2}q^{5}+3q^{7}+(-2\beta _{1}+\cdots)q^{9}+\cdots\)
6728.2.a.r	$6728$	$2$	6728.a	1.a	$1$	$3$	$3$	$53.723$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(3\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{2})q^{3}+(2\beta _{1}-\beta _{2})q^{5}+\cdots\)
6728.2.a.s	$6728$	$2$	6728.a	1.a	$1$	$4$	$4$	$53.723$	4.4.8468.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(2\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}-\beta _{3}q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
6728.2.a.t	$6728$	$2$	6728.a	1.a	$1$	$4$	$4$	$53.723$	4.4.8468.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}-\beta _{3}q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
6728.2.a.u	$6728$	$2$	6728.a	1.a	$1$	$6$	$6$	$53.723$	6.6.5173625.1	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
6728.2.a.v	$6728$	$2$	6728.a	1.a	$1$	$6$	$6$	$53.723$	6.6.50158625.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{2})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
6728.2.a.w	$6728$	$2$	6728.a	1.a	$1$	$6$	$6$	$53.723$	6.6.50158625.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{2})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
6728.2.a.x	$6728$	$2$	6728.a	1.a	$1$	$6$	$6$	$53.723$	6.6.5173625.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{5}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
6728.2.a.y	$6728$	$2$	6728.a	1.a	$1$	$12$	$12$	$53.723$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{7}+\beta _{9}+\beta _{11})q^{5}+\cdots\)
6728.2.a.z	$6728$	$2$	6728.a	1.a	$1$	$12$	$12$	$53.723$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(-1\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{10}q^{7}+(-\beta _{2}+\cdots)q^{9}+\cdots\)
6728.2.a.ba	$6728$	$2$	6728.a	1.a	$1$	$12$	$12$	$53.723$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(-4\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{7}+\beta _{9}+\beta _{11})q^{5}+\cdots\)
6728.2.a.bb	$6728$	$2$	6728.a	1.a	$1$	$12$	$12$	$53.723$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(4\)	\(-4\)	\(-1\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{10}q^{7}+(-\beta _{2}+\cdots)q^{9}+\cdots\)
6728.2.a.bc	$6728$	$2$	6728.a	1.a	$1$	$16$	$16$	$53.723$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(4\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{5}q^{5}+\beta _{9}q^{7}+(2-\beta _{3}+\cdots)q^{9}+\cdots\)
6728.2.a.bd	$6728$	$2$	6728.a	1.a	$1$	$16$	$16$	$53.723$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(4\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{5}q^{5}+\beta _{9}q^{7}+(2-\beta _{3}+\cdots)q^{9}+\cdots\)
6728.2.a.be	$6728$	$2$	6728.a	1.a	$1$	$24$	$24$	$53.723$		None	✓	$1$	$1$	\(0\)	\(-4\)	\(8\)	\(-2\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
6728.2.a.bf	$6728$	$2$	6728.a	1.a	$1$	$24$	$24$	$53.723$		None	✓	$1$	$0$	\(0\)	\(4\)	\(8\)	\(-2\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6728))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6728)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(841))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1682))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3364))\)\(^{\oplus 2}\)