Space of modular forms of level 538, weight 6, and trivial character

• Label: 538.6.a
• Level: $538$
• Weight: $6$
• Character orbit: 538.a
• Rep. character: $\chi_{538}(1,\cdot)$
• Character field: $\Q$
• Dimension: $113$
• Newform subspaces: $4$
• Sturm bound: $405$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(405\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(538))\).

	Total	New	Old
Modular forms	339	113	226
Cusp forms	335	113	222
Eisenstein series	4	0	4

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

\(2\)	\(269\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(30\)
\(+\)	\(-\)	\(-\)	\(27\)
\(-\)	\(+\)	\(-\)	\(32\)
\(-\)	\(-\)	\(+\)	\(24\)
Plus space		\(+\)	\(54\)
Minus space		\(-\)	\(59\)

Trace form

113 * q - 4 * q^2 + 4 * q^3 + 1808 * q^4 - 44 * q^5 - 8 * q^6 - 76 * q^7 - 64 * q^8 + 9059 * q^9 - 200 * q^10 + 598 * q^11 + 64 * q^12 + 48 * q^13 + 880 * q^14 + 28 * q^15 + 28928 * q^16 - 350 * q^17 - 3892 * q^18 - 3796 * q^19 - 704 * q^20 + 4604 * q^21 + 928 * q^22 - 3356 * q^23 - 128 * q^24 + 70933 * q^25 + 4776 * q^26 - 464 * q^27 - 1216 * q^28 - 3882 * q^29 - 3264 * q^30 + 4212 * q^31 - 1024 * q^32 + 27156 * q^33 + 1544 * q^34 + 7588 * q^35 + 144944 * q^36 - 23820 * q^37 + 11864 * q^38 - 15344 * q^39 - 3200 * q^40 + 11246 * q^41 + 37888 * q^42 - 1246 * q^43 + 9568 * q^44 - 17784 * q^45 - 21696 * q^46 - 8296 * q^47 + 1024 * q^48 + 259697 * q^49 - 20348 * q^50 + 49588 * q^51 + 768 * q^52 + 23772 * q^53 + 29344 * q^54 + 73300 * q^55 + 14080 * q^56 - 26240 * q^57 + 65520 * q^58 + 24292 * q^59 + 448 * q^60 - 37436 * q^61 + 58160 * q^62 - 120840 * q^63 + 462848 * q^64 - 6916 * q^65 - 11040 * q^66 + 56686 * q^67 - 5600 * q^68 + 192060 * q^69 - 65216 * q^70 + 8412 * q^71 - 62272 * q^72 + 36530 * q^73 - 111480 * q^74 + 240416 * q^75 - 60736 * q^76 + 81380 * q^77 - 46464 * q^78 - 128688 * q^79 - 11264 * q^80 + 560401 * q^81 + 5720 * q^82 + 26660 * q^83 + 73664 * q^84 + 464132 * q^85 - 44912 * q^86 + 44796 * q^87 + 14848 * q^88 - 140866 * q^89 - 114232 * q^90 - 181612 * q^91 - 53696 * q^92 + 21576 * q^93 - 12304 * q^94 - 299896 * q^95 - 2048 * q^96 + 41086 * q^97 - 282596 * q^98 + 230226 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(538))\) 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	269
538.6.a.a	$538$	$6$	538.a	1.a	$1$	$24$	$24$	$86.286$		None	✓	✓	✓	✓	538.6.a.a	$1$	$1$	\(96\)	\(-31\)	\(-261\)	\(-356\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
538.6.a.b	$538$	$6$	538.a	1.a	$1$	$27$	$27$	$86.286$		None	✓	✓	✓	✓	538.6.a.b	$1$	$0$	\(-108\)	\(33\)	\(139\)	\(-25\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
538.6.a.c	$538$	$6$	538.a	1.a	$1$	$30$	$30$	$86.286$		None	✓	✓	✓	✓	538.6.a.c	$1$	$1$	\(-120\)	\(-30\)	\(-136\)	\(-123\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
538.6.a.d	$538$	$6$	538.a	1.a	$1$	$32$	$32$	$86.286$		None	✓	✓	✓	✓	538.6.a.d	$1$	$0$	\(128\)	\(32\)	\(214\)	\(428\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(538))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(538)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(269))\)\(^{\oplus 2}\)