Space of modular forms of level 50, weight 12, and trivial character

• Label: 50.12.a
• Level: $50$
• Weight: $12$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $10$
• Sturm bound: $90$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(90\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(50))\).

	Total	New	Old
Modular forms	89	17	72
Cusp forms	77	17	60
Eisenstein series	12	0	12

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
\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(5\)
Plus space		\(+\)	\(10\)
Minus space		\(-\)	\(7\)

Trace form

17 * q - 32 * q^2 - 1012 * q^3 + 17408 * q^4 + 12608 * q^6 + 45224 * q^7 - 32768 * q^8 + 871259 * q^9 - 687626 * q^11 - 1036288 * q^12 + 2191418 * q^13 + 2201984 * q^14 + 17825792 * q^16 - 12748506 * q^17 + 1427296 * q^18 + 17372590 * q^19 + 47870684 * q^21 - 20705664 * q^22 + 98981928 * q^23 + 12910592 * q^24 - 11879232 * q^26 - 423112600 * q^27 + 46309376 * q^28 + 479170650 * q^29 + 18404884 * q^31 - 33554432 * q^32 - 134433024 * q^33 - 215054976 * q^34 + 892169216 * q^36 - 403327246 * q^37 + 247560320 * q^38 + 1203946368 * q^39 + 333408724 * q^41 + 1984111616 * q^42 + 1784459588 * q^43 - 704129024 * q^44 + 1409067648 * q^46 + 1770523344 * q^47 - 1061158912 * q^48 + 2758781601 * q^49 + 3574090274 * q^51 + 2244012032 * q^52 - 219799902 * q^53 + 4774475840 * q^54 + 2254831616 * q^56 + 13068201520 * q^57 - 4214028480 * q^58 + 5489660300 * q^59 + 13759505734 * q^61 - 7243427584 * q^62 + 23253311128 * q^63 + 18253611008 * q^64 + 6540563776 * q^66 - 12541795636 * q^67 - 13054470144 * q^68 - 22778315452 * q^69 + 6182405904 * q^71 + 1461551104 * q^72 + 48488064638 * q^73 + 30888037184 * q^74 + 17789532160 * q^76 - 103920657552 * q^77 - 77720217088 * q^78 + 80207971420 * q^79 - 15373187383 * q^81 + 6817242816 * q^82 + 87174300588 * q^83 + 49019580416 * q^84 + 97138064768 * q^86 - 10614977880 * q^87 - 21202599936 * q^88 - 196850296880 * q^89 - 232314413536 * q^91 + 101357494272 * q^92 + 88457439856 * q^93 - 109453380096 * q^94 + 13220446208 * q^96 + 17065411814 * q^97 - 467353825056 * q^98 - 368375403752 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(50))\) 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
50.12.a.a	$50$	$12$	50.a	1.a	$1$	$1$	$1$	$38.417$	\(\Q\)	None	✓	✓			50.12.a.a	$1$	$1$	\(-32\)	\(-207\)	\(0\)	\(19514\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}-207q^{3}+2^{10}q^{4}+6624q^{6}+\cdots\)
50.12.a.b	$50$	$12$	50.a	1.a	$1$	$1$	$1$	$38.417$	\(\Q\)	None	✓				10.12.a.c	$1$	$0$	\(-32\)	\(318\)	\(0\)	\(70714\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+318q^{3}+2^{10}q^{4}-10176q^{6}+\cdots\)
50.12.a.c	$50$	$12$	50.a	1.a	$1$	$1$	$1$	$38.417$	\(\Q\)	None	✓				10.12.a.b	$1$	$1$	\(32\)	\(-738\)	\(0\)	\(-25574\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}-738q^{3}+2^{10}q^{4}-23616q^{6}+\cdots\)
50.12.a.d	$50$	$12$	50.a	1.a	$1$	$1$	$1$	$38.417$	\(\Q\)	None	✓				10.12.a.a	$1$	$1$	\(32\)	\(12\)	\(0\)	\(14176\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+12q^{3}+2^{10}q^{4}+384q^{6}+\cdots\)
50.12.a.e	$50$	$12$	50.a	1.a	$1$	$1$	$1$	$38.417$	\(\Q\)	None	✓	✓			50.12.a.a	$1$	$1$	\(32\)	\(207\)	\(0\)	\(-19514\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+207q^{3}+2^{10}q^{4}+6624q^{6}+\cdots\)
50.12.a.f	$50$	$12$	50.a	1.a	$1$	$2$	$2$	$38.417$	\(\Q(\sqrt{1969}) \)	None	✓				10.12.a.d	$1$	$0$	\(-64\)	\(-604\)	\(0\)	\(-14092\)		$+$	$+$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+(-302-\beta )q^{3}+2^{10}q^{4}+\cdots\)
50.12.a.g	$50$	$12$	50.a	1.a	$1$	$2$	$2$	$38.417$	\(\Q(\sqrt{10129}) \)	None	✓	✓			50.12.a.g	$1$	$0$	\(-64\)	\(56\)	\(0\)	\(-54712\)		$+$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+(28-\beta )q^{3}+2^{10}q^{4}+(-896+\cdots)q^{6}+\cdots\)
50.12.a.h	$50$	$12$	50.a	1.a	$1$	$2$	$2$	$38.417$	\(\Q(\sqrt{10129}) \)	None	✓	✓			50.12.a.g	$1$	$0$	\(64\)	\(-56\)	\(0\)	\(54712\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+(-28-\beta )q^{3}+2^{10}q^{4}+\cdots\)
50.12.a.i	$50$	$12$	50.a	1.a	$1$	$3$	$3$	$38.417$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		10.12.b.a	$1$	$1$	\(-96\)	\(-266\)	\(0\)	\(-33218\)		$+$	$-$	$-$	$2^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+(-89+\beta _{1})q^{3}+2^{10}q^{4}+\cdots\)
50.12.a.j	$50$	$12$	50.a	1.a	$1$	$3$	$3$	$38.417$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		10.12.b.a	$1$	$0$	\(96\)	\(266\)	\(0\)	\(33218\)		$-$	$-$	$+$	$2^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+(89-\beta _{1})q^{3}+2^{10}q^{4}+(2848+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(50)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)