Space of modular forms of level 10, weight 16, and trivial character

• Label: 10.16.a
• Level: $10$
• Weight: $16$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $4$
• Sturm bound: $24$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(24\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	25	5	20
Cusp forms	21	5	16
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.

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(2\)

Trace form

5 * q + 128 * q^2 - 9632 * q^3 + 81920 * q^4 + 78125 * q^5 + 427520 * q^6 + 535804 * q^7 + 2097152 * q^8 + 11409985 * q^9 + 10000000 * q^10 - 21143940 * q^11 - 157810688 * q^12 + 744201478 * q^13 - 343946240 * q^14 - 405625000 * q^15 + 1342177280 * q^16 + 890757354 * q^17 + 654709376 * q^18 - 760960100 * q^19 + 1280000000 * q^20 + 7887208760 * q^21 + 13494391296 * q^22 + 1886355348 * q^23 + 7004487680 * q^24 + 30517578125 * q^25 - 30680218880 * q^26 - 33283167920 * q^27 + 8778612736 * q^28 - 134211118050 * q^29 + 41080000000 * q^30 - 288025204040 * q^31 + 34359738368 * q^32 - 152670425424 * q^33 - 101290755840 * q^34 + 205025312500 * q^35 + 186941194240 * q^36 + 1660154133694 * q^37 + 376608140800 * q^38 - 3672013210480 * q^39 + 163840000000 * q^40 - 349873701990 * q^41 + 4441635721216 * q^42 - 2804034484232 * q^43 - 346422312960 * q^44 + 4978887265625 * q^45 - 4509689748480 * q^46 + 10246116175284 * q^47 - 2585570312192 * q^48 - 7182896417835 * q^49 + 781250000000 * q^50 + 9093956417760 * q^51 + 12192997015552 * q^52 - 25057737033762 * q^53 - 7521119872000 * q^54 + 6415069687500 * q^55 - 5635215196160 * q^56 + 19903369887200 * q^57 - 10077353329920 * q^58 - 60966958907100 * q^59 - 6645760000000 * q^60 + 28196537181910 * q^61 + 24327579406336 * q^62 + 9069525437468 * q^63 + 21990232555520 * q^64 + 41972637343750 * q^65 - 130917115176960 * q^66 - 98148968872856 * q^67 + 14594168487936 * q^68 + 197582993684520 * q^69 - 44127760000000 * q^70 + 111220835423760 * q^71 + 10726758416384 * q^72 + 66038638669618 * q^73 + 179782580565760 * q^74 - 58789062500000 * q^75 - 12467570278400 * q^76 - 516867155383872 * q^77 + 63914271450112 * q^78 + 610332360628000 * q^79 + 20971520000000 * q^80 - 191631237239195 * q^81 - 248954943124224 * q^82 - 169647139364832 * q^83 + 129224028323840 * q^84 - 184603809843750 * q^85 + 18482801144320 * q^86 - 764001995922720 * q^87 + 221092106993664 * q^88 + 74973548668050 * q^89 + 34099570000000 * q^90 + 49650886540760 * q^91 + 30906046021632 * q^92 - 660512111777584 * q^93 + 331947449948160 * q^94 + 223816885937500 * q^95 + 114761526149120 * q^96 + 1420399684126474 * q^97 - 405707494390656 * q^98 + 1515440388237420 * q^99

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(10))\) 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	5
10.16.a.a	$10$	$16$	10.a	1.a	$1$	$1$	$1$	$14.269$	\(\Q\)	None	✓	✓	✓	✓	10.16.a.a	$1$	$1$	\(-128\)	\(-5568\)	\(78125\)	\(2564996\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{7}q^{2}-5568q^{3}+2^{14}q^{4}+5^{7}q^{5}+\cdots\)
10.16.a.b	$10$	$16$	10.a	1.a	$1$	$1$	$1$	$14.269$	\(\Q\)	None	✓	✓	✓	✓	10.16.a.b	$1$	$0$	\(-128\)	\(-918\)	\(-78125\)	\(-953554\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{7}q^{2}-918q^{3}+2^{14}q^{4}-5^{7}q^{5}+\cdots\)
10.16.a.c	$10$	$16$	10.a	1.a	$1$	$1$	$1$	$14.269$	\(\Q\)	None	✓	✓	✓	✓	10.16.a.c	$1$	$1$	\(128\)	\(-1302\)	\(-78125\)	\(-90706\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{7}q^{2}-1302q^{3}+2^{14}q^{4}-5^{7}q^{5}+\cdots\)
10.16.a.d	$10$	$16$	10.a	1.a	$1$	$2$	$2$	$14.269$	\(\Q(\sqrt{239569}) \)	None	✓	✓	✓	✓	10.16.a.d	$1$	$0$	\(256\)	\(-1844\)	\(156250\)	\(-984932\)		$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{7}q^{2}+(-922-\beta )q^{3}+2^{14}q^{4}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)