Space of modular forms of level 10, weight 28, and trivial character

• Label: 10.28.a
• Level: $10$
• Weight: $28$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $4$
• Sturm bound: $42$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 28 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(42\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{28}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	43	9	34
Cusp forms	39	9	30
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
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(3\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(4\)

Trace form

9 * q + 8192 * q^2 - 5585528 * q^3 + 603979776 * q^4 + 1220703125 * q^5 + 45769097216 * q^6 - 26881836804 * q^7 + 549755813888 * q^8 + 20176277678173 * q^9 + 10000000000000 * q^10 + 143740288335468 * q^11 - 374838438920192 * q^12 + 425308381788102 * q^13 + 2710940158885888 * q^14 + 10146083984375000 * q^15 + 40532396646334464 * q^16 + 6681364551758226 * q^17 + 201364589787521024 * q^18 + 532254106636691100 * q^19 + 81920000000000000 * q^20 + 109842772900221008 * q^21 + 3495399077188829184 * q^22 - 6982312948683377628 * q^23 + 3071512120471322624 * q^24 + 13411045074462890625 * q^25 - 2692381700589830144 * q^26 - 89283751298201073680 * q^27 - 1804009530149830656 * q^28 - 26057757232808972010 * q^29 + 105016720000000000000 * q^30 + 64079104901634815448 * q^31 + 36893488147419103232 * q^32 + 1084944057566163351744 * q^33 - 378036268430006403072 * q^34 + 400246228999023437500 * q^35 + 1354007074730747625472 * q^36 - 987703039883954851554 * q^37 + 5666409518834220236800 * q^38 - 12705465732007359561904 * q^39 + 671088640000000000000 * q^40 - 15149337479201963665302 * q^41 + 9863265465742102626304 * q^42 - 8827063906753261149168 * q^43 + 9646247461225708388352 * q^44 - 14616441454349365234375 * q^45 - 20179180495843385769984 * q^46 + 111594255111887773270836 * q^47 - 25154981819467471781888 * q^48 + 167299904677708060870257 * q^49 + 12207031250000000000000 * q^50 + 520719384147446634976848 * q^51 + 28541962351477813936128 * q^52 + 300330320174543209438302 * q^53 - 627821897253170118000640 * q^54 - 256575706211118164062500 * q^55 + 181928114434811449311232 * q^56 - 1424768382188793206159200 * q^57 + 210492666738415258091520 * q^58 + 443056034174127793436580 * q^59 + 680892170240000000000000 * q^60 + 168101771611321262304678 * q^61 + 1669578868622464251265024 * q^62 + 3149316348073265634939212 * q^63 + 2720083094132915643088896 * q^64 + 3844883233672487792968750 * q^65 - 3455825954049454216839168 * q^66 - 3523566238154064750244464 * q^67 + 448378785038363749515264 * q^68 + 12976001982227537903327856 * q^69 + 2049019307960000000000000 * q^70 - 31544603880291666410254992 * q^71 + 13513348870466537296756736 * q^72 - 10222732513042515682939398 * q^73 - 22970438132647801248038912 * q^74 - 8323085308074951171875000 * q^75 + 35718968455723200439910400 * q^76 - 19315729944095831187248208 * q^77 - 15765469973509183667044352 * q^78 - 143502833237962370375166240 * q^79 + 5497558138880000000000000 * q^80 + 127572881298658101229983409 * q^81 - 70431375105977767817035776 * q^82 + 151454670603188593724129592 * q^83 + 7371423707943817195814912 * q^84 + 160487328140725861816406250 * q^85 - 246153343382869441669627904 * q^86 - 21552465969231732641857680 * q^87 + 234572261296790640028286976 * q^88 - 84801790916321014336283910 * q^89 - 52847736494030000000000000 * q^90 - 1487675213492509595781913272 * q^91 - 468575090078631768298094592 * q^92 + 363824802843394270792921184 * q^93 + 382929062491295368645214208 * q^94 + 317538938993545776367187500 * q^95 + 206125689167061605874139136 * q^96 + 1466928433453302566766065346 * q^97 + 1845423955521133174021103616 * q^98 - 2397411903922276889582760804 * q^99

Decomposition of \(S_{28}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.28.a.a	$10$	$28$	10.a	1.a	$1$	$2$	$2$	$46.186$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.28.a.a	$1$	$1$	\(-16384\)	\(-3340644\)	\(2441406250\)	\(-58706842292\)		$+$	$-$	$-$	$2^{2}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{13}q^{2}+(-1670322-\beta )q^{3}+2^{26}q^{4}+\cdots\)
10.28.a.b	$10$	$28$	10.a	1.a	$1$	$2$	$2$	$46.186$	\(\Q(\sqrt{12929}) \)	None	✓	✓	✓	✓	10.28.a.b	$1$	$0$	\(-16384\)	\(-2245644\)	\(-2441406250\)	\(-120196732292\)		$+$	$+$	$+$	$2^{4}\cdot 3^{3}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{13}q^{2}+(-1122822-3\beta )q^{3}+\cdots\)
10.28.a.c	$10$	$28$	10.a	1.a	$1$	$2$	$2$	$46.186$	\(\Q(\sqrt{711649}) \)	None	✓	✓	✓	✓	10.28.a.c	$1$	$1$	\(16384\)	\(-4702956\)	\(-2441406250\)	\(-57185041508\)		$-$	$+$	$-$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{13}q^{2}+(-2351478-31\beta )q^{3}+\cdots\)
10.28.a.d	$10$	$28$	10.a	1.a	$1$	$3$	$3$	$46.186$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.28.a.d	$1$	$0$	\(24576\)	\(4703716\)	\(3662109375\)	\(209206779288\)		$-$	$-$	$+$	$2^{9}\cdot 3^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{13}q^{2}+(1567905-\beta _{1})q^{3}+2^{26}q^{4}+\cdots\)

Decomposition of \(S_{28}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{28}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{28}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)