Space of modular forms of level 1014, weight 4, and trivial character

• Label: 1014.4.a
• Level: $1014$
• Weight: $4$
• Character orbit: 1014.a
• Rep. character: $\chi_{1014}(1,\cdot)$
• Character field: $\Q$
• Dimension: $77$
• Newform subspaces: $31$
• Sturm bound: $728$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1014.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 31 \)
Sturm bound:			\(728\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1014))\).

	Total	New	Old
Modular forms	574	77	497
Cusp forms	518	77	441
Eisenstein series	56	0	56

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

\(2\)	\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(77\)	\(11\)	\(66\)		\(70\)	\(11\)	\(59\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(67\)	\(8\)	\(59\)		\(60\)	\(8\)	\(52\)		\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(+\)	\(-\)		\(70\)	\(9\)	\(61\)		\(63\)	\(9\)	\(54\)		\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(-\)	\(+\)		\(73\)	\(10\)	\(63\)		\(66\)	\(10\)	\(56\)		\(7\)	\(0\)	\(7\)
\(-\)	\(+\)	\(+\)	\(-\)		\(70\)	\(8\)	\(62\)		\(63\)	\(8\)	\(55\)		\(7\)	\(0\)	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)		\(74\)	\(11\)	\(63\)		\(67\)	\(11\)	\(56\)		\(7\)	\(0\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)		\(70\)	\(13\)	\(57\)		\(63\)	\(13\)	\(50\)		\(7\)	\(0\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)		\(73\)	\(7\)	\(66\)		\(66\)	\(7\)	\(59\)		\(7\)	\(0\)	\(7\)
Plus space			\(+\)		\(294\)	\(45\)	\(249\)		\(266\)	\(45\)	\(221\)		\(28\)	\(0\)	\(28\)
Minus space			\(-\)		\(280\)	\(32\)	\(248\)		\(252\)	\(32\)	\(220\)		\(28\)	\(0\)	\(28\)

Trace form

77 * q + 2 * q^2 + 3 * q^3 + 308 * q^4 + 26 * q^5 + 6 * q^6 + 12 * q^7 + 8 * q^8 + 693 * q^9 - 12 * q^10 - 124 * q^11 + 12 * q^12 - 32 * q^14 - 66 * q^15 + 1232 * q^16 - 2 * q^17 + 18 * q^18 + 136 * q^19 + 104 * q^20 + 36 * q^21 - 128 * q^23 + 24 * q^24 + 1747 * q^25 + 27 * q^27 + 48 * q^28 + 258 * q^29 - 84 * q^30 + 404 * q^31 + 32 * q^32 + 348 * q^33 + 260 * q^34 - 168 * q^35 + 2772 * q^36 - 410 * q^37 + 664 * q^38 - 48 * q^40 + 350 * q^41 - 264 * q^42 + 724 * q^43 - 496 * q^44 + 234 * q^45 + 352 * q^46 + 128 * q^47 + 48 * q^48 + 3905 * q^49 + 142 * q^50 - 618 * q^51 + 194 * q^53 + 54 * q^54 + 72 * q^55 - 128 * q^56 + 360 * q^57 + 556 * q^58 - 1268 * q^59 - 264 * q^60 - 478 * q^61 - 144 * q^62 + 108 * q^63 + 4928 * q^64 + 168 * q^66 + 224 * q^67 - 8 * q^68 + 384 * q^69 - 2544 * q^70 - 1712 * q^71 + 72 * q^72 - 1454 * q^73 + 2636 * q^74 + 693 * q^75 + 544 * q^76 + 1176 * q^77 + 3908 * q^79 + 416 * q^80 + 6237 * q^81 - 1228 * q^82 - 12 * q^83 + 144 * q^84 + 20 * q^85 + 680 * q^86 - 1914 * q^87 + 374 * q^89 - 108 * q^90 - 512 * q^92 + 348 * q^93 + 1056 * q^94 - 584 * q^95 + 96 * q^96 - 502 * q^97 - 1230 * q^98 - 1116 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1014))\) 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	3	13
1014.4.a.a	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.e.a	$1$	$0$	\(-2\)	\(-3\)	\(-7\)	\(-16\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}-7q^{5}+6q^{6}+\cdots\)
1014.4.a.b	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.a.e	$1$	$0$	\(-2\)	\(-3\)	\(-6\)	\(-20\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}-6q^{5}+6q^{6}+\cdots\)
1014.4.a.c	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.b.a	$1$	$1$	\(-2\)	\(-3\)	\(8\)	\(-14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+8q^{5}+6q^{6}+\cdots\)
1014.4.a.d	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.a.d	$1$	$0$	\(-2\)	\(-3\)	\(20\)	\(32\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+20q^{5}+6q^{6}+\cdots\)
1014.4.a.e	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.a.f	$1$	$1$	\(-2\)	\(3\)	\(-4\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-4q^{5}-6q^{6}+\cdots\)
1014.4.a.f	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.b.a	$1$	$0$	\(2\)	\(-3\)	\(-8\)	\(14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-8q^{5}-6q^{6}+\cdots\)
1014.4.a.g	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				6.4.a.a	$1$	$1$	\(2\)	\(-3\)	\(-6\)	\(16\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{5}-6q^{6}+\cdots\)
1014.4.a.h	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.e.a	$1$	$1$	\(2\)	\(-3\)	\(7\)	\(16\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+7q^{5}-6q^{6}+\cdots\)
1014.4.a.i	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.a.a	$1$	$1$	\(2\)	\(-3\)	\(16\)	\(-28\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+2^{4}q^{5}-6q^{6}+\cdots\)
1014.4.a.j	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.a.c	$1$	$0$	\(2\)	\(3\)	\(-10\)	\(8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}-10q^{5}+6q^{6}+\cdots\)
1014.4.a.k	$1014$	$4$	1014.a	1.a	$1$	$1$	$1$	$59.828$	\(\Q\)	None	✓				78.4.a.b	$1$	$0$	\(2\)	\(3\)	\(16\)	\(8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+2^{4}q^{5}+6q^{6}+\cdots\)
1014.4.a.l	$1014$	$4$	1014.a	1.a	$1$	$2$	$2$	$59.828$	\(\Q(\sqrt{61}) \)	None	✓				78.4.e.c	$1$	$0$	\(-4\)	\(-6\)	\(2\)	\(18\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+(1+2\beta )q^{5}+\cdots\)
1014.4.a.m	$1014$	$4$	1014.a	1.a	$1$	$2$	$2$	$59.828$	\(\Q(\sqrt{673}) \)	None	✓				78.4.e.b	$1$	$1$	\(-4\)	\(6\)	\(-13\)	\(-9\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(-6-\beta )q^{5}+\cdots\)
1014.4.a.n	$1014$	$4$	1014.a	1.a	$1$	$2$	$2$	$59.828$	\(\Q(\sqrt{3}) \)	None	✓				78.4.i.a	$1$	$0$	\(-4\)	\(6\)	\(12\)	\(6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(6+\beta )q^{5}-6q^{6}+\cdots\)
1014.4.a.o	$1014$	$4$	1014.a	1.a	$1$	$2$	$2$	$59.828$	\(\Q(\sqrt{17}) \)	None	✓				78.4.b.b	$1$	$0$	\(-4\)	\(6\)	\(18\)	\(0\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(9+\beta )q^{5}-6q^{6}+\cdots\)
1014.4.a.p	$1014$	$4$	1014.a	1.a	$1$	$2$	$2$	$59.828$	\(\Q(\sqrt{61}) \)	None	✓				78.4.e.c	$1$	$1$	\(4\)	\(-6\)	\(-2\)	\(-18\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(-1-2\beta )q^{5}+\cdots\)
1014.4.a.q	$1014$	$4$	1014.a	1.a	$1$	$2$	$2$	$59.828$	\(\Q(\sqrt{17}) \)	None	✓				78.4.b.b	$1$	$1$	\(4\)	\(6\)	\(-18\)	\(0\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(-9-\beta )q^{5}+\cdots\)
1014.4.a.r	$1014$	$4$	1014.a	1.a	$1$	$2$	$2$	$59.828$	\(\Q(\sqrt{3}) \)	None	✓				78.4.i.a	$1$	$1$	\(4\)	\(6\)	\(-12\)	\(-6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(-6+\beta )q^{5}+\cdots\)
1014.4.a.s	$1014$	$4$	1014.a	1.a	$1$	$2$	$2$	$59.828$	\(\Q(\sqrt{673}) \)	None	✓				78.4.e.b	$1$	$0$	\(4\)	\(6\)	\(13\)	\(9\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(7-\beta )q^{5}+6q^{6}+\cdots\)
1014.4.a.t	$1014$	$4$	1014.a	1.a	$1$	$3$	$3$	$59.828$	\(\Q(\zeta_{14})^+\)	None	✓	✓			1014.4.a.t	$1$	$1$	\(-6\)	\(-9\)	\(8\)	\(28\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+(3-3\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1014.4.a.u	$1014$	$4$	1014.a	1.a	$1$	$3$	$3$	$59.828$	3.3.1106005.1	None	✓				78.4.e.d	$1$	$1$	\(-6\)	\(9\)	\(-12\)	\(-17\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(-4+\beta _{1})q^{5}+\cdots\)
1014.4.a.v	$1014$	$4$	1014.a	1.a	$1$	$3$	$3$	$59.828$	\(\Q(\zeta_{14})^+\)	None	✓	✓			1014.4.a.v	$1$	$1$	\(-6\)	\(9\)	\(2\)	\(44\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(5\beta _{1}+3\beta _{2})q^{5}+\cdots\)
1014.4.a.w	$1014$	$4$	1014.a	1.a	$1$	$3$	$3$	$59.828$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		1014.4.a.t	$1$	$1$	\(6\)	\(-9\)	\(-8\)	\(-28\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(-5\beta _{1}+3\beta _{2})q^{5}+\cdots\)
1014.4.a.x	$1014$	$4$	1014.a	1.a	$1$	$3$	$3$	$59.828$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		1014.4.a.v	$1$	$1$	\(6\)	\(9\)	\(-2\)	\(-44\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(-5\beta _{1}-3\beta _{2})q^{5}+\cdots\)
1014.4.a.y	$1014$	$4$	1014.a	1.a	$1$	$3$	$3$	$59.828$	3.3.1106005.1	None	✓				78.4.e.d	$1$	$0$	\(6\)	\(9\)	\(12\)	\(17\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(4-\beta _{1})q^{5}+\cdots\)
1014.4.a.z	$1014$	$4$	1014.a	1.a	$1$	$4$	$4$	$59.828$	4.4.30907152.1	None	✓		✓		78.4.i.b	$1$	$1$	\(-8\)	\(-12\)	\(-8\)	\(-22\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+(-2-\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots\)
1014.4.a.ba	$1014$	$4$	1014.a	1.a	$1$	$4$	$4$	$59.828$	4.4.30907152.1	None	✓				78.4.i.b	$1$	$0$	\(8\)	\(-12\)	\(8\)	\(22\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(2-\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
1014.4.a.bb	$1014$	$4$	1014.a	1.a	$1$	$6$	$6$	$59.828$	6.6.\(\cdots\).1	None	✓	✓	✓		1014.4.a.bb	$1$	$0$	\(-12\)	\(-18\)	\(-7\)	\(-13\)		$+$	$+$	$+$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+(-1-\beta _{4})q^{5}+\cdots\)
1014.4.a.bc	$1014$	$4$	1014.a	1.a	$1$	$6$	$6$	$59.828$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		1014.4.a.bc	$1$	$0$	\(-12\)	\(18\)	\(3\)	\(1\)		$+$	$-$	$-$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(1-\beta _{1})q^{5}+\cdots\)
1014.4.a.bd	$1014$	$4$	1014.a	1.a	$1$	$6$	$6$	$59.828$	6.6.\(\cdots\).1	None	✓	✓	✓		1014.4.a.bb	$1$	$0$	\(12\)	\(-18\)	\(7\)	\(13\)		$-$	$+$	$-$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(1-\beta _{4})q^{5}+\cdots\)
1014.4.a.be	$1014$	$4$	1014.a	1.a	$1$	$6$	$6$	$59.828$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		1014.4.a.bc	$1$	$0$	\(12\)	\(18\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(-1+\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1014))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1014)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 2}\)