Space of modular forms of level 539, weight 8, and trivial character

• Label: 539.8.a
• Level: $539$
• Weight: $8$
• Character orbit: 539.a
• Rep. character: $\chi_{539}(1,\cdot)$
• Character field: $\Q$
• Dimension: $238$
• Newform subspaces: $14$
• Sturm bound: $448$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(448\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(539))\).

	Total	New	Old
Modular forms	400	238	162
Cusp forms	384	238	146
Eisenstein series	16	0	16

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

\(7\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(104\)	\(61\)	\(43\)		\(100\)	\(61\)	\(39\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(96\)	\(53\)	\(43\)		\(92\)	\(53\)	\(39\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(96\)	\(59\)	\(37\)		\(92\)	\(59\)	\(33\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(104\)	\(65\)	\(39\)		\(100\)	\(65\)	\(35\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(208\)	\(126\)	\(82\)		\(200\)	\(126\)	\(74\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(192\)	\(112\)	\(80\)		\(184\)	\(112\)	\(72\)		\(8\)	\(0\)	\(8\)

Trace form

238 * q - 8 * q^2 + 67 * q^3 + 15092 * q^4 + 567 * q^5 - 994 * q^6 + 2064 * q^8 + 171015 * q^9 - 7198 * q^10 - 2662 * q^11 + 10908 * q^12 - 9426 * q^13 + 4815 * q^15 + 915680 * q^16 - 8612 * q^17 - 210994 * q^18 + 111280 * q^19 + 230340 * q^20 - 10648 * q^22 + 6947 * q^23 + 188632 * q^24 + 3454899 * q^25 + 339480 * q^26 + 424141 * q^27 - 29146 * q^29 + 119598 * q^30 - 375841 * q^31 + 195144 * q^32 + 182347 * q^33 - 471088 * q^34 + 10031656 * q^36 - 821939 * q^37 - 1360580 * q^38 + 511488 * q^39 + 904716 * q^40 - 1348206 * q^41 + 1248802 * q^43 + 90508 * q^44 + 3992276 * q^45 + 1921042 * q^46 - 1453312 * q^47 - 2670796 * q^48 + 93386 * q^50 + 863378 * q^51 - 6090932 * q^52 + 5623892 * q^53 - 2183802 * q^54 - 9317 * q^55 + 3187304 * q^57 + 14176808 * q^58 - 1144215 * q^59 + 10610056 * q^60 - 3641110 * q^61 + 1534038 * q^62 + 50486716 * q^64 + 5319004 * q^65 + 3000074 * q^66 - 6643711 * q^67 + 3353508 * q^68 + 9044225 * q^69 - 5495291 * q^71 - 1752748 * q^72 - 3957906 * q^73 + 6748770 * q^74 - 2329784 * q^75 + 5757640 * q^76 + 21278040 * q^78 - 16248626 * q^79 + 27800444 * q^80 + 114286246 * q^81 + 7688420 * q^82 + 7428434 * q^83 - 1480534 * q^85 + 7062864 * q^86 + 17023740 * q^87 - 5286732 * q^88 + 7801745 * q^89 - 51639728 * q^90 + 34185676 * q^92 - 54772739 * q^93 + 12550340 * q^94 + 3807608 * q^95 + 58381716 * q^96 + 20985475 * q^97 - 2474329 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(539))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	11
539.8.a.a	$539$	$8$	539.a	1.a	$1$	$2$	$2$	$168.376$	\(\Q(\sqrt{15}) \)	None	✓				11.8.a.a	$1$	$0$	\(-8\)	\(6\)	\(470\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{2}+(3+6\beta )q^{3}+(-52+\cdots)q^{4}+\cdots\)
539.8.a.b	$539$	$8$	539.a	1.a	$1$	$4$	$4$	$168.376$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				11.8.a.b	$1$	$1$	\(0\)	\(35\)	\(-537\)	\(0\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(9+\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(152+\cdots)q^{4}+\cdots\)
539.8.a.c	$539$	$8$	539.a	1.a	$1$	$7$	$7$	$168.376$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				77.8.a.a	$1$	$1$	\(0\)	\(46\)	\(580\)	\(0\)		$-$	$+$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(7-\beta _{1}+\beta _{2})q^{3}+(33+\beta _{1}+\cdots)q^{4}+\cdots\)
539.8.a.d	$539$	$8$	539.a	1.a	$1$	$9$	$9$	$168.376$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				77.8.a.b	$1$	$0$	\(-8\)	\(129\)	\(-13\)	\(0\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(14+\beta _{5})q^{3}+(61+\cdots)q^{4}+\cdots\)
539.8.a.e	$539$	$8$	539.a	1.a	$1$	$9$	$9$	$168.376$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				77.8.a.c	$1$	$1$	\(0\)	\(-116\)	\(244\)	\(0\)		$-$	$+$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-13-\beta _{2})q^{3}+(68+\beta _{1}+\cdots)q^{4}+\cdots\)
539.8.a.f	$539$	$8$	539.a	1.a	$1$	$11$	$11$	$168.376$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				77.8.a.d	$1$	$0$	\(24\)	\(-33\)	\(-177\)	\(0\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(-3+\beta _{3})q^{3}+(85+\beta _{1}+\cdots)q^{4}+\cdots\)
539.8.a.g	$539$	$8$	539.a	1.a	$1$	$16$	$16$	$168.376$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓			539.8.a.g	$2$	$1$	\(-16\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{20}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{2}+\beta _{1}q^{3}+(59+\beta _{3}+\cdots)q^{4}+\cdots\)
539.8.a.h	$539$	$8$	539.a	1.a	$1$	$20$	$20$	$168.376$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	✓			539.8.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{21}\cdot 5^{3}\cdot 7^{8}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-\beta _{1}q^{3}+(79-\beta _{4})q^{4}-\beta _{12}q^{5}+\cdots\)
539.8.a.i	$539$	$8$	539.a	1.a	$1$	$23$	$23$	$168.376$		None	✓				77.8.e.b	$1$	$1$	\(0\)	\(-81\)	\(-750\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
539.8.a.j	$539$	$8$	539.a	1.a	$1$	$23$	$23$	$168.376$		None	✓		✓		77.8.e.a	$1$	$1$	\(0\)	\(-27\)	\(-250\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
539.8.a.k	$539$	$8$	539.a	1.a	$1$	$23$	$23$	$168.376$		None	✓				77.8.e.a	$1$	$0$	\(0\)	\(27\)	\(250\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
539.8.a.l	$539$	$8$	539.a	1.a	$1$	$23$	$23$	$168.376$		None	✓		✓		77.8.e.b	$1$	$0$	\(0\)	\(81\)	\(750\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
539.8.a.m	$539$	$8$	539.a	1.a	$1$	$30$	$30$	$168.376$		None	✓	✓	✓		539.8.a.m	$2$	$1$	\(-16\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
539.8.a.n	$539$	$8$	539.a	1.a	$1$	$38$	$38$	$168.376$		None	✓	✓	✓		539.8.a.n	$2$	$0$	\(16\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(539))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(539)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)