Space of modular forms of level 538, weight 8, and trivial character

• Label: 538.8.a
• Level: $538$
• Weight: $8$
• Character orbit: 538.a
• Rep. character: $\chi_{538}(1,\cdot)$
• Character field: $\Q$
• Dimension: $155$
• Newform subspaces: $4$
• Sturm bound: $540$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$538 = 2 \cdot 269$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	538.a (trivial)
Character field:			$\Q$
Newform subspaces:			$4$
Sturm bound:			$540$
Trace bound:			$1$

Dimensions

The following table gives the dimensions of various subspaces of $M_{8}(\Gamma_0(538))$.

	Total	New	Old
Modular forms	475	155	320
Cusp forms	471	155	316
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
$+$	$+$	$+$	$37$
$+$	$-$	$-$	$40$
$-$	$+$	$-$	$35$
$-$	$-$	$+$	$43$
Plus space		$+$	$80$
Minus space		$-$	$75$

Trace form

155 * q + 8 * q^2 + 28 * q^3 + 9920 * q^4 - 4 * q^5 + 688 * q^6 - 684 * q^7 + 512 * q^8 + 111257 * q^9 + 2000 * q^10 - 7482 * q^11 + 1792 * q^12 - 3424 * q^13 - 8992 * q^14 + 32188 * q^15 + 634880 * q^16 + 2726 * q^17 - 11224 * q^18 + 130884 * q^19 - 256 * q^20 - 94012 * q^21 - 49216 * q^22 - 69004 * q^23 + 44032 * q^24 + 2229279 * q^25 + 93296 * q^26 + 92992 * q^27 - 43776 * q^28 + 96674 * q^29 - 208320 * q^30 - 254524 * q^31 + 32768 * q^32 - 509700 * q^33 + 291184 * q^34 + 870612 * q^35 + 7120448 * q^36 - 375508 * q^37 - 1101264 * q^38 + 639472 * q^39 + 128000 * q^40 + 54874 * q^41 + 507136 * q^42 + 898530 * q^43 - 478848 * q^44 + 1566840 * q^45 + 190848 * q^46 - 357144 * q^47 + 114688 * q^48 + 16961131 * q^49 + 434232 * q^50 - 2760332 * q^51 - 219136 * q^52 + 1977236 * q^53 + 1994752 * q^54 - 4041772 * q^55 - 575488 * q^56 + 2353408 * q^57 + 510240 * q^58 - 4385060 * q^59 + 2060032 * q^60 - 1452548 * q^61 + 2470240 * q^62 + 4342680 * q^63 + 40632320 * q^64 + 9905940 * q^65 - 4746048 * q^66 - 5011250 * q^67 + 174464 * q^68 + 534804 * q^69 + 3694976 * q^70 + 3270572 * q^71 - 718336 * q^72 + 911718 * q^73 + 1913008 * q^74 - 3652552 * q^75 + 8376576 * q^76 - 7420964 * q^77 + 10809024 * q^78 + 6263024 * q^79 - 16384 * q^80 + 80552011 * q^81 + 13081168 * q^82 - 9902292 * q^83 - 6016768 * q^84 - 26343524 * q^85 + 8311968 * q^86 + 16701900 * q^87 - 3149824 * q^88 + 17335290 * q^89 + 4549808 * q^90 + 22914308 * q^91 - 4416256 * q^92 + 15361704 * q^93 - 7844768 * q^94 + 30502200 * q^95 + 2818048 * q^96 - 3281414 * q^97 + 281032 * q^98 - 9954846 * q^99

Decomposition of $S_{8}^{\mathrm{new}}(\Gamma_0(538))$ 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
538.8.a.a	$538$	$8$	538.a	1.a	$1$	$35$	$35$	$168.063$		None	✓	✓	✓	✓	538.8.a.a	$1$	$1$	$280$	$-66$	$-1126$	$-3196$		$-$	$+$	$-$		$\mathrm{SU}(2)$
538.8.a.b	$538$	$8$	538.a	1.a	$1$	$37$	$37$	$168.063$		None	✓	✓	✓	✓	538.8.a.b	$1$	$0$	$-296$	$80$	$624$	$453$		$+$	$+$	$+$		$\mathrm{SU}(2)$
538.8.a.c	$538$	$8$	538.a	1.a	$1$	$40$	$40$	$168.063$		None	✓	✓	✓	✓	538.8.a.c	$1$	$1$	$-320$	$-109$	$-751$	$-233$		$+$	$-$	$-$		$\mathrm{SU}(2)$
538.8.a.d	$538$	$8$	538.a	1.a	$1$	$43$	$43$	$168.063$		None	✓	✓	✓	✓	538.8.a.d	$1$	$0$	$344$	$123$	$1249$	$2292$		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of $S_{8}^{\mathrm{old}}(\Gamma_0(538))$ into lower level spaces

$S_{8}^{\mathrm{old}}(\Gamma_0(538)) \simeq$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))$$^{\oplus 2}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(269))$$^{\oplus 2}$