Space of modular forms of level 10, weight 12, and trivial character

• Label: 10.12.a
• Level: $10$
• Weight: $12$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $4$
• Sturm bound: $18$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(18\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	19	5	14
Cusp forms	15	5	10
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(10\)	\(3\)	\(7\)		\(8\)	\(3\)	\(5\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)

Trace form

5 * q + 32 * q^2 + 1012 * q^3 + 5120 * q^4 + 3125 * q^5 - 14080 * q^6 - 45224 * q^7 + 32768 * q^8 + 336385 * q^9 + 100000 * q^10 + 672660 * q^11 + 1036288 * q^12 - 2191418 * q^13 - 2176640 * q^14 + 537500 * q^15 + 5242880 * q^16 + 12748506 * q^17 - 1427296 * q^18 - 29636900 * q^19 + 3200000 * q^20 - 23916040 * q^21 + 20705664 * q^22 - 98981928 * q^23 - 14417920 * q^24 + 48828125 * q^25 + 46640320 * q^26 + 423112600 * q^27 - 46309376 * q^28 - 293845650 * q^29 + 167200000 * q^30 - 345535040 * q^31 + 33554432 * q^32 + 134433024 * q^33 - 431509440 * q^34 + 140800000 * q^35 + 344458240 * q^36 + 403327246 * q^37 - 247560320 * q^38 + 1094070920 * q^39 + 102400000 * q^40 - 788322390 * q^41 - 1984111616 * q^42 - 1784459588 * q^43 + 688803840 * q^44 - 770509375 * q^45 + 46024320 * q^46 - 1770523344 * q^47 + 1061158912 * q^48 + 8405476965 * q^49 + 312500000 * q^50 + 1793570760 * q^51 - 2244012032 * q^52 + 219799902 * q^53 + 4277004800 * q^54 - 4194337500 * q^55 - 2228879360 * q^56 - 13068201520 * q^57 + 4214028480 * q^58 + 6798539700 * q^59 + 550400000 * q^60 + 9056775910 * q^61 + 7243427584 * q^62 - 23253311128 * q^63 + 5368709120 * q^64 + 12004693750 * q^65 - 32607605760 * q^66 + 12541795636 * q^67 + 13054470144 * q^68 - 26524371480 * q^69 + 12455600000 * q^70 + 5366257560 * q^71 - 1461551104 * q^72 - 48488064638 * q^73 - 22356607040 * q^74 + 9882812500 * q^75 - 30348185600 * q^76 + 103920657552 * q^77 + 77720217088 * q^78 - 18957078800 * q^79 + 3276800000 * q^80 + 123956164405 * q^81 - 6817242816 * q^82 - 87174300588 * q^83 - 24490024960 * q^84 - 61838043750 * q^85 - 116121121280 * q^86 + 10614977880 * q^87 + 21202599936 * q^88 + 25259764050 * q^89 + 84243700000 * q^90 + 48640223360 * q^91 - 101357494272 * q^92 - 88457439856 * q^93 - 39614240640 * q^94 - 121922562500 * q^95 - 14763950080 * q^96 - 17065411814 * q^97 + 467353825056 * q^98 + 146930040420 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.12.a.a	$10$	$12$	10.a	1.a	$1$	$1$	$1$	$7.683$	\(\Q\)	None	✓	✓	✓	✓	10.12.a.a	$1$	$1$	\(-32\)	\(-12\)	\(3125\)	\(-14176\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}-12q^{3}+2^{10}q^{4}+5^{5}q^{5}+\cdots\)
10.12.a.b	$10$	$12$	10.a	1.a	$1$	$1$	$1$	$7.683$	\(\Q\)	None	✓	✓	✓	✓	10.12.a.b	$1$	$0$	\(-32\)	\(738\)	\(-3125\)	\(25574\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+738q^{3}+2^{10}q^{4}-5^{5}q^{5}+\cdots\)
10.12.a.c	$10$	$12$	10.a	1.a	$1$	$1$	$1$	$7.683$	\(\Q\)	None	✓	✓	✓	✓	10.12.a.c	$1$	$1$	\(32\)	\(-318\)	\(-3125\)	\(-70714\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}-318q^{3}+2^{10}q^{4}-5^{5}q^{5}+\cdots\)
10.12.a.d	$10$	$12$	10.a	1.a	$1$	$2$	$2$	$7.683$	\(\Q(\sqrt{1969}) \)	None	✓	✓	✓	✓	10.12.a.d	$1$	$0$	\(64\)	\(604\)	\(6250\)	\(14092\)		$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+(302-\beta )q^{3}+2^{10}q^{4}+5^{5}q^{5}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)