Space of modular forms of level 8978, weight 2, and trivial character

• Label: 8978.2.a
• Level: $8978$
• Weight: $2$
• Character orbit: 8978.a
• Rep. character: $\chi_{8978}(1,\cdot)$
• Character field: $\Q$
• Dimension: $368$
• Newform subspaces: $24$
• Sturm bound: $2278$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 8978 = 2 \cdot 67^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8978.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(2278\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8978))\).

	Total	New	Old
Modular forms	1207	368	839
Cusp forms	1072	368	704
Eisenstein series	135	0	135

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

\(2\)	\(67\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(88\)
\(+\)	\(-\)	\(-\)	\(96\)
\(-\)	\(+\)	\(-\)	\(104\)
\(-\)	\(-\)	\(+\)	\(80\)
Plus space		\(+\)	\(168\)
Minus space		\(-\)	\(200\)

Trace form

\( 368 q - 4 q^{3} + 368 q^{4} - 2 q^{6} + 366 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8978))\) 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	67
8978.2.a.a	$8978$	$2$	8978.a	1.a	$1$	$1$	$1$	$71.690$	\(\Q\)	None	✓				134.2.c.a	$1$	$0$	\(-1\)	\(1\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-2q^{5}-q^{6}+4q^{7}+\cdots\)
8978.2.a.b	$8978$	$2$	8978.a	1.a	$1$	$1$	$1$	$71.690$	\(\Q\)	None	✓				134.2.c.a	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}-4q^{7}+\cdots\)
8978.2.a.c	$8978$	$2$	8978.a	1.a	$1$	$2$	$2$	$71.690$	\(\Q(\sqrt{21}) \)	None	✓				134.2.c.b	$1$	$0$	\(-2\)	\(-1\)	\(2\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+(-1+\cdots)q^{7}+\cdots\)
8978.2.a.d	$8978$	$2$	8978.a	1.a	$1$	$2$	$2$	$71.690$	\(\Q(\sqrt{5}) \)	None	✓				134.2.c.c	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+(1-2\beta )q^{5}+\cdots\)
8978.2.a.e	$8978$	$2$	8978.a	1.a	$1$	$2$	$2$	$71.690$	\(\Q(\sqrt{5}) \)	None	✓				134.2.c.c	$1$	$1$	\(2\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+2\beta )q^{5}+\cdots\)
8978.2.a.f	$8978$	$2$	8978.a	1.a	$1$	$2$	$2$	$71.690$	\(\Q(\sqrt{21}) \)	None	✓				134.2.c.b	$1$	$0$	\(2\)	\(1\)	\(-2\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
8978.2.a.g	$8978$	$2$	8978.a	1.a	$1$	$3$	$3$	$71.690$	\(\Q(\zeta_{18})^+\)	None	✓				134.2.a.b	$1$	$0$	\(-3\)	\(-3\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta _{1})q^{3}+q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
8978.2.a.h	$8978$	$2$	8978.a	1.a	$1$	$3$	$3$	$71.690$	3.3.257.1	None	✓	✓			8978.2.a.h	$1$	$1$	\(-3\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-\beta _{1}+\beta _{2})q^{5}+\cdots\)
8978.2.a.i	$8978$	$2$	8978.a	1.a	$1$	$3$	$3$	$71.690$	3.3.473.1	None	✓				134.2.a.a	$1$	$1$	\(3\)	\(-1\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{2}q^{3}+q^{4}+(-1-\beta _{1})q^{5}+\cdots\)
8978.2.a.j	$8978$	$2$	8978.a	1.a	$1$	$3$	$3$	$71.690$	3.3.257.1	None	✓	✓			8978.2.a.h	$1$	$0$	\(3\)	\(1\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
8978.2.a.k	$8978$	$2$	8978.a	1.a	$1$	$15$	$15$	$71.690$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓				134.2.e.a	$1$	$1$	\(-15\)	\(5\)	\(4\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{8}q^{5}-\beta _{1}q^{6}+\cdots\)
8978.2.a.l	$8978$	$2$	8978.a	1.a	$1$	$15$	$15$	$71.690$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓				134.2.e.a	$1$	$1$	\(15\)	\(-5\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{8}q^{5}-\beta _{1}q^{6}+\cdots\)
8978.2.a.m	$8978$	$2$	8978.a	1.a	$1$	$16$	$16$	$71.690$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓			8978.2.a.m	$1$	$0$	\(-16\)	\(8\)	\(4\)	\(13\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+(-\beta _{8}+\beta _{14}+\cdots)q^{5}+\cdots\)
8978.2.a.n	$8978$	$2$	8978.a	1.a	$1$	$16$	$16$	$71.690$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓			8978.2.a.m	$1$	$1$	\(16\)	\(-8\)	\(-4\)	\(-13\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(\beta _{8}-\beta _{14}+\cdots)q^{5}+\cdots\)
8978.2.a.o	$8978$	$2$	8978.a	1.a	$1$	$20$	$20$	$71.690$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓				134.2.g.a	$1$	$1$	\(-20\)	\(-3\)	\(11\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta _{1}+\beta _{6}-\beta _{7}-\beta _{9}+\cdots)q^{3}+\cdots\)
8978.2.a.p	$8978$	$2$	8978.a	1.a	$1$	$20$	$20$	$71.690$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓				134.2.e.b	$1$	$0$	\(-20\)	\(-2\)	\(-4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{16}q^{5}+\beta _{1}q^{6}+\cdots\)
8978.2.a.q	$8978$	$2$	8978.a	1.a	$1$	$20$	$20$	$71.690$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓				134.2.e.b	$1$	$0$	\(20\)	\(2\)	\(4\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{16}q^{5}+\beta _{1}q^{6}+\cdots\)
8978.2.a.r	$8978$	$2$	8978.a	1.a	$1$	$20$	$20$	$71.690$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓				134.2.g.a	$1$	$1$	\(20\)	\(3\)	\(-11\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1}-\beta _{6}+\beta _{7}+\beta _{9}+\beta _{13}+\cdots)q^{3}+\cdots\)
8978.2.a.s	$8978$	$2$	8978.a	1.a	$1$	$24$	$24$	$71.690$		None	✓	✓			8978.2.a.s	$1$	$0$	\(-24\)	\(8\)	\(7\)	\(17\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
8978.2.a.t	$8978$	$2$	8978.a	1.a	$1$	$24$	$24$	$71.690$		None	✓	✓	✓		8978.2.a.s	$1$	$1$	\(24\)	\(-8\)	\(-7\)	\(-17\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
8978.2.a.u	$8978$	$2$	8978.a	1.a	$1$	$30$	$30$	$71.690$		None	✓		✓		134.2.g.b	$1$	$0$	\(-30\)	\(0\)	\(-11\)	\(-3\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
8978.2.a.v	$8978$	$2$	8978.a	1.a	$1$	$30$	$30$	$71.690$		None	✓				134.2.g.b	$1$	$0$	\(30\)	\(0\)	\(11\)	\(3\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
8978.2.a.w	$8978$	$2$	8978.a	1.a	$1$	$48$	$48$	$71.690$		None	✓	✓	✓		8978.2.a.w	$1$	$1$	\(-48\)	\(-16\)	\(-16\)	\(-34\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
8978.2.a.x	$8978$	$2$	8978.a	1.a	$1$	$48$	$48$	$71.690$		None	✓	✓	✓		8978.2.a.w	$1$	$0$	\(48\)	\(16\)	\(16\)	\(34\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8978))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8978)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4489))\)\(^{\oplus 2}\)