Space of modular forms of level 1078, weight 2, and trivial character

• Label: 1078.2.a
• Level: $1078$
• Weight: $2$
• Character orbit: 1078.a
• Rep. character: $\chi_{1078}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $24$
• Sturm bound: $336$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(336\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(3\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1078))\).

	Total	New	Old
Modular forms	184	35	149
Cusp forms	153	35	118
Eisenstein series	31	0	31

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
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(14\)
Minus space			\(-\)	\(21\)

Trace form

35 * q - q^2 + 35 * q^4 - 2 * q^5 + 4 * q^6 - q^8 + 39 * q^9 - 6 * q^10 - q^11 + 2 * q^13 + 16 * q^15 + 35 * q^16 + 6 * q^17 + 3 * q^18 + 12 * q^19 - 2 * q^20 - q^22 + 8 * q^23 + 4 * q^24 + 37 * q^25 - 2 * q^26 + 24 * q^27 + 10 * q^29 + 24 * q^30 + 16 * q^31 - q^32 - 2 * q^34 + 39 * q^36 - 54 * q^37 + 8 * q^38 - 32 * q^39 - 6 * q^40 - 18 * q^41 - 20 * q^43 - q^44 - 34 * q^45 + 16 * q^46 - 16 * q^47 - 23 * q^50 + 2 * q^52 - 22 * q^53 + 16 * q^54 - 6 * q^55 - 40 * q^57 - 30 * q^58 - 8 * q^59 + 16 * q^60 + 18 * q^61 - 8 * q^62 + 35 * q^64 - 20 * q^65 + 4 * q^66 - 28 * q^67 + 6 * q^68 + 16 * q^71 + 3 * q^72 - 2 * q^73 - 22 * q^74 + 24 * q^75 + 12 * q^76 + 24 * q^78 - 56 * q^79 - 2 * q^80 + 43 * q^81 - 10 * q^82 + 4 * q^83 - 36 * q^85 + 12 * q^86 - 24 * q^87 - q^88 - 18 * q^89 - 6 * q^90 + 8 * q^92 - 8 * q^93 + 8 * q^94 + 4 * q^96 - 18 * q^97 - 13 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\) 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	7	11
1078.2.a.a	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.e.d	$1$	$1$	\(-1\)	\(-3\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}+q^{4}-4q^{5}+3q^{6}-q^{8}+\cdots\)
1078.2.a.b	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.a.b	$1$	$1$	\(-1\)	\(-2\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}-2q^{5}+2q^{6}-q^{8}+\cdots\)
1078.2.a.c	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.e.c	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}-2q^{9}+\cdots\)
1078.2.a.d	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{5}-q^{8}-3q^{9}-4q^{10}+\cdots\)
1078.2.a.e	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.e.c	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}-2q^{9}+\cdots\)
1078.2.a.f	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.e.d	$1$	$0$	\(-1\)	\(3\)	\(4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}+q^{4}+4q^{5}-3q^{6}-q^{8}+\cdots\)
1078.2.a.g	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.e.a	$1$	$1$	\(1\)	\(-3\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}-2q^{5}-3q^{6}+q^{8}+\cdots\)
1078.2.a.h	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓	✓			1078.2.a.h	$1$	$0$	\(1\)	\(-2\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-2q^{5}-2q^{6}+q^{8}+\cdots\)
1078.2.a.i	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.e.b	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}-2q^{9}+\cdots\)
1078.2.a.j	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.a.c	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}+q^{8}-3q^{9}-2q^{10}+\cdots\)
1078.2.a.k	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.e.b	$1$	$0$	\(1\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}-2q^{9}+\cdots\)
1078.2.a.l	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓	✓			1078.2.a.h	$1$	$0$	\(1\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{5}+2q^{6}+q^{8}+\cdots\)
1078.2.a.m	$1078$	$2$	1078.a	1.a	$1$	$1$	$1$	$8.608$	\(\Q\)	None	✓				154.2.e.a	$1$	$0$	\(1\)	\(3\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}+q^{4}+2q^{5}+3q^{6}+q^{8}+\cdots\)
1078.2.a.n	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{7}) \)	None	✓				154.2.e.f	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}+(-1-\beta )q^{5}-\beta q^{6}+\cdots\)
1078.2.a.o	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{2}) \)	None	✓	✓			1078.2.a.o	$2$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{5}-q^{8}-3q^{9}-\beta q^{10}+\cdots\)
1078.2.a.p	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{2}) \)	None	✓	✓			1078.2.a.p	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2\beta q^{5}-q^{8}-3q^{9}+2\beta q^{10}+\cdots\)
1078.2.a.q	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		1078.2.a.q	$2$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{5}-\beta q^{6}-q^{8}+\cdots\)
1078.2.a.r	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		1078.2.a.r	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}+5q^{9}+\cdots\)
1078.2.a.s	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{7}) \)	None	✓				154.2.e.f	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}+(1-\beta )q^{5}-\beta q^{6}+\cdots\)
1078.2.a.t	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{2}) \)	None	✓		✓	✓	154.2.e.e	$1$	$1$	\(2\)	\(-2\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-2-\beta )q^{5}+\cdots\)
1078.2.a.u	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{2}) \)	None	✓	✓			1078.2.a.u	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+3\beta q^{5}+\beta q^{6}+\cdots\)
1078.2.a.v	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{2}) \)	None	✓	✓			1078.2.a.v	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2\beta q^{3}+q^{4}+2\beta q^{6}+q^{8}+\cdots\)
1078.2.a.w	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{5}) \)	None	✓				154.2.a.d	$1$	$0$	\(2\)	\(2\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{5}+\cdots\)
1078.2.a.x	$1078$	$2$	1078.a	1.a	$1$	$2$	$2$	$8.608$	\(\Q(\sqrt{2}) \)	None	✓				154.2.e.e	$1$	$0$	\(2\)	\(2\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(2-\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1078))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1078)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(539))\)\(^{\oplus 2}\)