Space of modular forms of level 1078, weight 6, and trivial character

• Label: 1078.6.a
• Level: $1078$
• Weight: $6$
• Character orbit: 1078.a
• Rep. character: $\chi_{1078}(1,\cdot)$
• Character field: $\Q$
• Dimension: $169$
• Newform subspaces: $30$
• Sturm bound: $1008$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(1008\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1078))\).

	Total	New	Old
Modular forms	856	169	687
Cusp forms	824	169	655
Eisenstein series	32	0	32

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

\(2\)	\(7\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(20\)
\(+\)	\(+\)	\(-\)	$-$	\(20\)
\(+\)	\(-\)	\(+\)	$-$	\(22\)
\(+\)	\(-\)	\(-\)	$+$	\(22\)
\(-\)	\(+\)	\(+\)	$-$	\(22\)
\(-\)	\(+\)	\(-\)	$+$	\(18\)
\(-\)	\(-\)	\(+\)	$+$	\(21\)
\(-\)	\(-\)	\(-\)	$-$	\(24\)
Plus space			\(+\)	\(81\)
Minus space			\(-\)	\(88\)

Trace form

\( 169 q + 4 q^{2} + 16 q^{3} + 2704 q^{4} + 86 q^{5} - 80 q^{6} + 64 q^{8} + 12761 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1078))\) 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	7	11
1078.6.a.a	$1078$	$6$	1078.a	1.a	$1$	$1$	$1$	$172.894$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(-1\)	\(51\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-q^{3}+2^{4}q^{4}+51q^{5}+4q^{6}+\cdots\)
1078.6.a.b	$1078$	$6$	1078.a	1.a	$1$	$1$	$1$	$172.894$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(21\)	\(-81\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+21q^{3}+2^{4}q^{4}-3^{4}q^{5}-84q^{6}+\cdots\)
1078.6.a.c	$1078$	$6$	1078.a	1.a	$1$	$1$	$1$	$172.894$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(30\)	\(-42\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+30q^{3}+2^{4}q^{4}-42q^{5}-120q^{6}+\cdots\)
1078.6.a.d	$1078$	$6$	1078.a	1.a	$1$	$1$	$1$	$172.894$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(-5\)	\(49\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-5q^{3}+2^{4}q^{4}+7^{2}q^{5}-20q^{6}+\cdots\)
1078.6.a.e	$1078$	$6$	1078.a	1.a	$1$	$1$	$1$	$172.894$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(20\)	\(-6\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+20q^{3}+2^{4}q^{4}-6q^{5}+80q^{6}+\cdots\)
1078.6.a.f	$1078$	$6$	1078.a	1.a	$1$	$1$	$1$	$172.894$	\(\Q\)	None	✓	$1$	$0$	\(4\)	\(29\)	\(31\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+29q^{3}+2^{4}q^{4}+31q^{5}+116q^{6}+\cdots\)
1078.6.a.g	$1078$	$6$	1078.a	1.a	$1$	$2$	$2$	$172.894$	\(\Q(\sqrt{113}) \)	None	✓	$1$	$0$	\(-8\)	\(-15\)	\(47\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-6-3\beta )q^{3}+2^{4}q^{4}+(26+\cdots)q^{5}+\cdots\)
1078.6.a.h	$1078$	$6$	1078.a	1.a	$1$	$2$	$2$	$172.894$	\(\Q(\sqrt{793}) \)	None	✓	$1$	$1$	\(8\)	\(-29\)	\(13\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-14-\beta )q^{3}+2^{4}q^{4}+(4+\cdots)q^{5}+\cdots\)
1078.6.a.i	$1078$	$6$	1078.a	1.a	$1$	$2$	$2$	$172.894$	\(\Q(\sqrt{337}) \)	None	✓	$1$	$0$	\(8\)	\(-7\)	\(63\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-3-\beta )q^{3}+2^{4}q^{4}+(29+\cdots)q^{5}+\cdots\)
1078.6.a.j	$1078$	$6$	1078.a	1.a	$1$	$3$	$3$	$172.894$	3.3.682584.1	None	✓	$1$	$1$	\(-12\)	\(-7\)	\(137\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-2-\beta _{1}-\beta _{2})q^{3}+2^{4}q^{4}+\cdots\)
1078.6.a.k	$1078$	$6$	1078.a	1.a	$1$	$3$	$3$	$172.894$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(12\)	\(15\)	\(-119\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+(5-\beta _{1})q^{3}+2^{4}q^{4}+(-40+\cdots)q^{5}+\cdots\)
1078.6.a.l	$1078$	$6$	1078.a	1.a	$1$	$4$	$4$	$172.894$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-16\)	\(-25\)	\(-25\)	\(0\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-6-\beta _{1})q^{3}+2^{4}q^{4}+(-6+\cdots)q^{5}+\cdots\)
1078.6.a.m	$1078$	$6$	1078.a	1.a	$1$	$4$	$4$	$172.894$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-16\)	\(15\)	\(-7\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-4q^{2}+(4-\beta _{1})q^{3}+2^{4}q^{4}+(-1+\cdots)q^{5}+\cdots\)
1078.6.a.n	$1078$	$6$	1078.a	1.a	$1$	$5$	$5$	$172.894$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(20\)	\(-25\)	\(-25\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-5+\beta _{1})q^{3}+2^{4}q^{4}+(-5+\cdots)q^{5}+\cdots\)
1078.6.a.o	$1078$	$6$	1078.a	1.a	$1$	$6$	$6$	$172.894$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(-24\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1078.6.a.p	$1078$	$6$	1078.a	1.a	$1$	$6$	$6$	$172.894$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(-24\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}-\beta _{2}q^{5}-4\beta _{1}q^{6}+\cdots\)
1078.6.a.q	$1078$	$6$	1078.a	1.a	$1$	$6$	$6$	$172.894$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(24\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{3}\cdot 11$	$\mathrm{SU}(2)$	\(q+4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+\beta _{3}q^{5}+4\beta _{1}q^{6}+\cdots\)
1078.6.a.r	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-32\)	\(-14\)	\(-50\)	\(0\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-2+\beta _{1})q^{3}+2^{4}q^{4}+(-6+\cdots)q^{5}+\cdots\)
1078.6.a.s	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-32\)	\(-4\)	\(-100\)	\(0\)		$+$	$+$	$+$	$+$	$2^{5}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-1+\beta _{1})q^{3}+2^{4}q^{4}+(-12+\cdots)q^{5}+\cdots\)
1078.6.a.t	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-32\)	\(4\)	\(100\)	\(0\)		$+$	$-$	$+$	$-$	$2^{5}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(1-\beta _{1})q^{3}+2^{4}q^{4}+(12+\beta _{1}+\cdots)q^{5}+\cdots\)
1078.6.a.u	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-32\)	\(14\)	\(50\)	\(0\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(2-\beta _{1})q^{3}+2^{4}q^{4}+(6-\beta _{1}+\cdots)q^{5}+\cdots\)
1078.6.a.v	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(32\)	\(-32\)	\(-100\)	\(0\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-4+\beta _{1})q^{3}+2^{4}q^{4}+(-13+\cdots)q^{5}+\cdots\)
1078.6.a.w	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(32\)	\(-22\)	\(-50\)	\(0\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-3+\beta _{1})q^{3}+2^{4}q^{4}+(-6+\cdots)q^{5}+\cdots\)
1078.6.a.x	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(32\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{6}\cdot 11$	$\mathrm{SU}(2)$	\(q+4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}-\beta _{2}q^{5}+4\beta _{1}q^{6}+\cdots\)
1078.6.a.y	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(32\)	\(22\)	\(50\)	\(0\)		$-$	$+$	$+$	$-$	$2^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+(3-\beta _{1})q^{3}+2^{4}q^{4}+(6+\beta _{5}+\cdots)q^{5}+\cdots\)
1078.6.a.z	$1078$	$6$	1078.a	1.a	$1$	$8$	$8$	$172.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(32\)	\(32\)	\(100\)	\(0\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+4q^{2}+(4-\beta _{1})q^{3}+2^{4}q^{4}+(13-\beta _{4}+\cdots)q^{5}+\cdots\)
1078.6.a.ba	$1078$	$6$	1078.a	1.a	$1$	$10$	$10$	$172.894$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$1$	\(40\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{7}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}-\beta _{5}q^{3}+2^{4}q^{4}+(2\beta _{5}-\beta _{6}+\cdots)q^{5}+\cdots\)
1078.6.a.bb	$1078$	$6$	1078.a	1.a	$1$	$12$	$12$	$172.894$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$1$	\(-48\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$2^{8}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q-4q^{2}+\beta _{6}q^{3}+2^{4}q^{4}-\beta _{8}q^{5}-4\beta _{6}q^{6}+\cdots\)
1078.6.a.bc	$1078$	$6$	1078.a	1.a	$1$	$12$	$12$	$172.894$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$0$	\(-48\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q-4q^{2}+\beta _{6}q^{3}+2^{4}q^{4}+(-\beta _{7}+\beta _{9}+\cdots)q^{5}+\cdots\)
1078.6.a.bd	$1078$	$6$	1078.a	1.a	$1$	$14$	$14$	$172.894$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$2$	$0$	\(56\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{9}\cdot 3^{6}\cdot 7^{8}$	$\mathrm{SU}(2)$	\(q+4q^{2}+\beta _{3}q^{3}+2^{4}q^{4}+(\beta _{5}+\beta _{9}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1078))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1078)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(539))\)\(^{\oplus 2}\)