Space of modular forms of level 10, weight 22, and trivial character

• Label: 10.22.a
• Level: $10$
• Weight: $22$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $4$
• Sturm bound: $33$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(33\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	33	7	26
Cusp forms	29	7	22
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
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(4\)

Trace form

7 * q - 1024 * q^2 - 17336 * q^3 + 7340032 * q^4 - 9765625 * q^5 + 36282368 * q^6 - 126415972 * q^7 - 1073741824 * q^8 + 41383462411 * q^9 - 10000000000 * q^10 + 135710538204 * q^11 - 18178113536 * q^12 + 1759096582034 * q^13 - 2251785457664 * q^14 + 1700234375000 * q^15 + 7696581394432 * q^16 - 1550055571602 * q^17 - 35099863954432 * q^18 - 51326075625940 * q^19 - 10240000000000 * q^20 + 357732759850064 * q^21 + 178816832827392 * q^22 - 504621232645116 * q^23 + 38044820307968 * q^24 + 667572021484375 * q^25 - 1219467350915072 * q^26 + 3273499683699760 * q^27 - 132556754255872 * q^28 - 3524415540119550 * q^29 - 2798960000000000 * q^30 - 4243482657818536 * q^31 - 1125899906842624 * q^32 + 20322682714697088 * q^33 + 27443506386794496 * q^34 + 13073243242187500 * q^35 + 43393705481076736 * q^36 - 13007505623316982 * q^37 + 39391290462515200 * q^38 + 20422945819665872 * q^39 - 10485760000000000 * q^40 + 120751027081467654 * q^41 - 356378085213175808 * q^42 - 243310821507893776 * q^43 + 142302813307797504 * q^44 - 91949683720703125 * q^45 + 49685501185118208 * q^46 + 881790342543934068 * q^47 - 19061133579124736 * q^48 - 859825119266747121 * q^49 - 97656250000000000 * q^50 - 333777283333304496 * q^51 + 1844546457602883584 * q^52 - 112497377888155926 * q^53 - 513621601477181440 * q^54 + 77936009648437500 * q^55 - 2361168188055486464 * q^56 - 15982097147052733600 * q^57 + 876669646172375040 * q^58 + 9351748568918022900 * q^59 + 1782824960000000000 * q^60 + 4644320675832264914 * q^61 + 128857993513787392 * q^62 + 5345077138568792204 * q^63 + 8070450532247928832 * q^64 - 25702340432480468750 * q^65 + 31854943282947588096 * q^66 - 22522900884194907472 * q^67 - 1625351071048138752 * q^68 + 69950075112175162992 * q^69 - 19042878920000000000 * q^70 - 42414559292477360016 * q^71 - 36804874945882488832 * q^72 + 63646881826441593494 * q^73 + 72923464532361127936 * q^74 - 1653289794921875000 * q^75 - 53819291075545661440 * q^76 - 184601145641795237424 * q^77 + 168867026027871256576 * q^78 + 129994339064062515680 * q^79 - 10737418240000000000 * q^80 + 250966569764430100207 * q^81 - 295910364070758279168 * q^82 + 5170514717007118584 * q^83 + 375109986392540708864 * q^84 - 313840038160957031250 * q^85 + 172059577347544956928 * q^86 - 1904616390843583644240 * q^87 + 187503039298815393792 * q^88 + 1185551134398546250230 * q^89 - 376093696130000000000 * q^90 - 506347891409805616856 * q^91 - 529133713642085154816 * q^92 + 1551870277213993198688 * q^93 + 166600909072914518016 * q^94 - 1106641585570507812500 * q^95 + 39892885499247853568 * q^96 - 1671611530660787780002 * q^97 - 1282164053311481422848 * q^98 + 5479397792014068082092 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.22.a.a	$10$	$22$	10.a	1.a	$1$	$1$	$1$	$27.948$	\(\Q\)	None	✓	✓	✓	✓	10.22.a.a	$1$	$1$	\(1024\)	\(-21924\)	\(9765625\)	\(-722753248\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}-21924q^{3}+2^{20}q^{4}+5^{10}q^{5}+\cdots\)
10.22.a.b	$10$	$22$	10.a	1.a	$1$	$2$	$2$	$27.948$	\(\Q(\sqrt{157921}) \)	None	✓	✓	✓	✓	10.22.a.b	$1$	$1$	\(-2048\)	\(-126692\)	\(-19531250\)	\(-292598684\)		$+$	$+$	$+$	$2^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+(-63346-\beta )q^{3}+2^{20}q^{4}+\cdots\)
10.22.a.c	$10$	$22$	10.a	1.a	$1$	$2$	$2$	$27.948$	\(\Q(\sqrt{474529}) \)	None	✓	✓	✓	✓	10.22.a.c	$1$	$0$	\(-2048\)	\(100308\)	\(19531250\)	\(1328895316\)		$+$	$-$	$-$	$2^{2}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+(50154-\beta )q^{3}+2^{20}q^{4}+\cdots\)
10.22.a.d	$10$	$22$	10.a	1.a	$1$	$2$	$2$	$27.948$	\(\Q(\sqrt{1179649}) \)	None	✓	✓	✓	✓	10.22.a.d	$1$	$0$	\(2048\)	\(30972\)	\(-19531250\)	\(-439959356\)		$-$	$+$	$-$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+(15486-\beta )q^{3}+2^{20}q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)