Space of modular forms of level 578, weight 6, and trivial character

• Label: 578.6.a
• Level: $578$
• Weight: $6$
• Character orbit: 578.a
• Rep. character: $\chi_{578}(1,\cdot)$
• Character field: $\Q$
• Dimension: $112$
• Newform subspaces: $18$
• Sturm bound: $459$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 578 = 2 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	578.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(459\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(578))\).

	Total	New	Old
Modular forms	400	112	288
Cusp forms	364	112	252
Eisenstein series	36	0	36

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

\(2\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(97\)	\(27\)	\(70\)		\(88\)	\(27\)	\(61\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)		\(103\)	\(29\)	\(74\)		\(94\)	\(29\)	\(65\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)		\(101\)	\(31\)	\(70\)		\(92\)	\(31\)	\(61\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)		\(99\)	\(25\)	\(74\)		\(90\)	\(25\)	\(65\)		\(9\)	\(0\)	\(9\)
Plus space		\(+\)		\(196\)	\(52\)	\(144\)		\(178\)	\(52\)	\(126\)		\(18\)	\(0\)	\(18\)
Minus space		\(-\)		\(204\)	\(60\)	\(144\)		\(186\)	\(60\)	\(126\)		\(18\)	\(0\)	\(18\)

Trace form

112 * q + 22 * q^3 + 1792 * q^4 - 66 * q^5 - 8 * q^6 + 236 * q^7 + 8492 * q^9 - 296 * q^10 + 262 * q^11 + 352 * q^12 - 48 * q^13 - 176 * q^14 + 820 * q^15 + 28672 * q^16 - 2272 * q^18 + 2180 * q^19 - 1056 * q^20 + 4244 * q^21 + 2392 * q^22 + 5496 * q^23 - 128 * q^24 + 73224 * q^25 - 2144 * q^26 + 13156 * q^27 + 3776 * q^28 - 2938 * q^29 + 8160 * q^30 - 13388 * q^31 + 18108 * q^33 + 11420 * q^35 + 135872 * q^36 + 13986 * q^37 - 19376 * q^38 - 39428 * q^39 - 4736 * q^40 + 33972 * q^41 + 37840 * q^42 - 21092 * q^43 + 4192 * q^44 - 56922 * q^45 + 33120 * q^46 - 45876 * q^47 + 5632 * q^48 + 265908 * q^49 - 9728 * q^50 - 768 * q^52 - 82564 * q^53 + 2800 * q^54 + 8064 * q^55 - 2816 * q^56 - 15304 * q^57 + 37752 * q^58 + 159092 * q^59 + 13120 * q^60 + 65402 * q^61 + 21904 * q^62 + 73660 * q^63 + 458752 * q^64 - 113836 * q^65 - 18528 * q^66 - 4668 * q^67 + 118544 * q^69 - 93968 * q^70 + 9192 * q^71 - 36352 * q^72 - 85800 * q^73 + 110248 * q^74 + 432850 * q^75 + 34880 * q^76 - 186648 * q^77 + 76880 * q^78 - 33800 * q^79 - 16896 * q^80 + 426488 * q^81 - 82160 * q^82 - 147128 * q^83 + 67904 * q^84 + 50976 * q^86 - 216356 * q^87 + 38272 * q^88 - 178112 * q^89 - 426248 * q^90 + 267768 * q^91 + 87936 * q^92 + 489276 * q^93 - 89712 * q^94 + 172440 * q^95 - 2048 * q^96 + 228356 * q^97 - 243680 * q^98 - 458714 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(578))\) 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	17
578.6.a.a	$578$	$6$	578.a	1.a	$1$	$1$	$1$	$92.702$	\(\Q\)	None	✓				34.6.a.a	$1$	$0$	\(4\)	\(14\)	\(74\)	\(78\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+14q^{3}+2^{4}q^{4}+74q^{5}+56q^{6}+\cdots\)
578.6.a.b	$578$	$6$	578.a	1.a	$1$	$2$	$2$	$92.702$	\(\Q(\sqrt{43}) \)	None	✓				34.6.a.c	$1$	$1$	\(-8\)	\(6\)	\(-32\)	\(138\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+(3+\beta )q^{3}+2^{4}q^{4}+(-2^{4}+\cdots)q^{5}+\cdots\)
578.6.a.c	$578$	$6$	578.a	1.a	$1$	$2$	$2$	$92.702$	\(\Q(\sqrt{69}) \)	None	✓				34.6.a.b	$1$	$1$	\(-8\)	\(6\)	\(36\)	\(2\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-4q^{2}+(3+3\beta )q^{3}+2^{4}q^{4}+(18+4\beta )q^{5}+\cdots\)
578.6.a.d	$578$	$6$	578.a	1.a	$1$	$3$	$3$	$92.702$	3.3.1505580.1	None	✓				34.6.a.d	$1$	$0$	\(12\)	\(-4\)	\(-144\)	\(18\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-1-\beta _{2})q^{3}+2^{4}q^{4}+(-7^{2}+\cdots)q^{5}+\cdots\)
578.6.a.e	$578$	$6$	578.a	1.a	$1$	$4$	$4$	$92.702$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			578.6.a.e	$1$	$1$	\(-16\)	\(-6\)	\(-2\)	\(-274\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-1-\beta _{1})q^{3}+2^{4}q^{4}+(-1+\cdots)q^{5}+\cdots\)
578.6.a.f	$578$	$6$	578.a	1.a	$1$	$4$	$4$	$92.702$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				34.6.b.b	$2$	$1$	\(-16\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-4q^{2}+\beta _{2}q^{3}+2^{4}q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
578.6.a.g	$578$	$6$	578.a	1.a	$1$	$4$	$4$	$92.702$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			578.6.a.e	$1$	$0$	\(-16\)	\(6\)	\(2\)	\(274\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(1+\beta _{1})q^{3}+2^{4}q^{4}+(1-2\beta _{1}+\cdots)q^{5}+\cdots\)
578.6.a.h	$578$	$6$	578.a	1.a	$1$	$4$	$4$	$92.702$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			578.6.a.h	$1$	$1$	\(16\)	\(-22\)	\(-110\)	\(-42\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-5-\beta _{1})q^{3}+2^{4}q^{4}+(-28+\cdots)q^{5}+\cdots\)
578.6.a.i	$578$	$6$	578.a	1.a	$1$	$4$	$4$	$92.702$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				34.6.b.a	$2$	$0$	\(16\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
578.6.a.j	$578$	$6$	578.a	1.a	$1$	$4$	$4$	$92.702$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			578.6.a.h	$1$	$0$	\(16\)	\(22\)	\(110\)	\(42\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+(5+\beta _{1})q^{3}+2^{4}q^{4}+(28+\beta _{2}+\cdots)q^{5}+\cdots\)
578.6.a.k	$578$	$6$	578.a	1.a	$1$	$6$	$6$	$92.702$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				34.6.c.a	$2$	$1$	\(-24\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-2\beta _{1}+\beta _{2})q^{3}+2^{4}q^{4}+\cdots\)
578.6.a.l	$578$	$6$	578.a	1.a	$1$	$9$	$9$	$92.702$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		578.6.a.l	$1$	$1$	\(-36\)	\(-27\)	\(0\)	\(-237\)		$+$	$+$	$+$	$2^{3}\cdot 17^{3}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-3+\beta _{5})q^{3}+2^{4}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
578.6.a.m	$578$	$6$	578.a	1.a	$1$	$9$	$9$	$92.702$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			578.6.a.l	$1$	$0$	\(-36\)	\(27\)	\(0\)	\(237\)		$+$	$-$	$-$	$2^{3}\cdot 17^{3}$	$\mathrm{SU}(2)$	\(q-4q^{2}+(3-\beta _{5})q^{3}+2^{4}q^{4}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
578.6.a.n	$578$	$6$	578.a	1.a	$1$	$9$	$9$	$92.702$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			578.6.a.n	$1$	$1$	\(36\)	\(-27\)	\(-150\)	\(-351\)		$-$	$-$	$+$	$2^{3}\cdot 17^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-3-\beta _{2})q^{3}+2^{4}q^{4}+(-17+\cdots)q^{5}+\cdots\)
578.6.a.o	$578$	$6$	578.a	1.a	$1$	$9$	$9$	$92.702$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			578.6.a.n	$1$	$0$	\(36\)	\(27\)	\(150\)	\(351\)		$-$	$+$	$-$	$2^{3}\cdot 17^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+(3+\beta _{2})q^{3}+2^{4}q^{4}+(17+2\beta _{1}+\cdots)q^{5}+\cdots\)
578.6.a.p	$578$	$6$	578.a	1.a	$1$	$10$	$10$	$92.702$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		34.6.c.b	$2$	$0$	\(40\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{7}\cdot 17^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+\beta _{5}q^{5}+4\beta _{1}q^{6}+\cdots\)
578.6.a.q	$578$	$6$	578.a	1.a	$1$	$12$	$12$	$92.702$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		34.6.d.a	$2$	$1$	\(48\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{5}\cdot 17^{4}$	$\mathrm{SU}(2)$	\(q+4q^{2}+\beta _{5}q^{3}+2^{4}q^{4}+(-\beta _{1}+\beta _{6}+\cdots)q^{5}+\cdots\)
578.6.a.r	$578$	$6$	578.a	1.a	$1$	$16$	$16$	$92.702$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		34.6.d.b	$2$	$0$	\(-64\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{6}\cdot 17^{10}$	$\mathrm{SU}(2)$	\(q-4q^{2}+\beta _{2}q^{3}+2^{4}q^{4}+\beta _{5}q^{5}-4\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(578))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(578)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 2}\)