Space of modular forms of level 50, weight 26, and trivial character

• Label: 50.26.a
• Level: $50$
• Weight: $26$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $12$
• Sturm bound: $195$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(195\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{26}(\Gamma_0(50))\).

	Total	New	Old
Modular forms	193	40	153
Cusp forms	181	40	141
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
\(+\)	\(+\)	\(+\)	\(9\)
\(+\)	\(-\)	\(-\)	\(11\)
\(-\)	\(+\)	\(-\)	\(10\)
\(-\)	\(-\)	\(+\)	\(10\)
Plus space		\(+\)	\(19\)
Minus space		\(-\)	\(21\)

Trace form

40 * q - 650200 * q^3 + 671088640 * q^4 + 1224335360 * q^6 + 70437967600 * q^7 + 10458624046970 * q^9 - 18707778902670 * q^11 - 10908545843200 * q^12 + 198216091044100 * q^13 + 42768283320320 * q^14 + 11258999068426240 * q^16 + 1605739651605900 * q^17 + 1098834470502400 * q^18 - 20023409075258350 * q^19 - 56706249527472220 * q^21 - 61603721370009600 * q^22 + 86916833205634800 * q^23 + 20540938791157760 * q^24 - 263273455332720640 * q^26 - 513936624127560400 * q^27 + 1181752997026201600 * q^28 - 4730388255669483900 * q^29 - 11278704068507006620 * q^31 + 28499228373440563200 * q^33 + 12049323277195960320 * q^34 + 175466594698809835520 * q^36 - 69169745772744397100 * q^37 - 37597143276394905600 * q^38 + 156260544056513859240 * q^39 + 15440800552334868630 * q^41 + 352194775496150220800 * q^42 - 1014624232049991470600 * q^43 - 313864447530337566720 * q^44 + 1273583309218679439360 * q^46 - 1320197927635672538400 * q^47 - 183015029857268531200 * q^48 + 7018512984421226560080 * q^49 - 9478887970144164009670 * q^51 + 3325514174122531225600 * q^52 + 6412508670052823786100 * q^53 - 6439047059727332761600 * q^54 + 717532727214205829120 * q^56 - 19014710394882097834400 * q^57 - 1256208523445482291200 * q^58 + 22827125128747487956200 * q^59 - 31589309165314865219320 * q^61 + 27217180041222065356800 * q^62 + 6624249377519303338000 * q^63 + 188894659314785808547840 * q^64 + 210144496222115325829120 * q^66 + 67303472848802721797800 * q^67 + 26939840974756931174400 * q^68 - 251748267511348504123060 * q^69 + 183247231517833619973480 * q^71 + 18435383259864393318400 * q^72 - 433839349884348717596900 * q^73 + 329320494924528282091520 * q^74 - 335937059111969593753600 * q^76 + 882051511906070620034400 * q^77 + 673757427289000696217600 * q^78 + 576591261105879412825100 * q^79 - 2196403505919457259226760 * q^81 - 885281386180090252492800 * q^82 + 53838218051264128113000 * q^83 - 951372996872299368939520 * q^84 + 831826389124349091676160 * q^86 + 5084218837797884597353200 * q^87 - 1033538939828466981273600 * q^88 + 8947295140305550016609850 * q^89 - 1730930407114521641051120 * q^91 + 1458222484726907456716800 * q^92 - 2994759256810704600279200 * q^93 - 13123017057111509812346880 * q^94 + 344619766942032629596160 * q^96 + 26725602399357926295527500 * q^97 - 19273576994176012281446400 * q^98 - 59055310562592409299402860 * q^99

Decomposition of \(S_{26}^{\mathrm{new}}(\Gamma_0(50))\) 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
50.26.a.a	$50$	$26$	50.a	1.a	$1$	$1$	$1$	$197.998$	\(\Q\)	None	✓				10.26.a.a	$1$	$1$	\(-4096\)	\(-162864\)	\(0\)	\(17600893492\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}-162864q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.b	$50$	$26$	50.a	1.a	$1$	$1$	$1$	$197.998$	\(\Q\)	None	✓				2.26.a.a	$1$	$0$	\(4096\)	\(-97956\)	\(0\)	\(40882637368\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}-97956q^{3}+2^{24}q^{4}-401227776q^{6}+\cdots\)
50.26.a.c	$50$	$26$	50.a	1.a	$1$	$2$	$2$	$197.998$	\(\Q(\sqrt{106705}) \)	None	✓				2.26.a.b	$1$	$1$	\(-8192\)	\(-379848\)	\(0\)	\(376536944\)		$+$	$+$	$+$	$2^{7}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}+(-189924-\beta )q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.d	$50$	$26$	50.a	1.a	$1$	$2$	$2$	$197.998$	\(\Q(\sqrt{95351}) \)	None	✓				10.26.a.d	$1$	$1$	\(-8192\)	\(97092\)	\(0\)	\(16137160124\)		$+$	$+$	$+$	$2^{4}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}+(48546+\beta )q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.e	$50$	$26$	50.a	1.a	$1$	$2$	$2$	$197.998$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓				10.26.a.c	$1$	$0$	\(8192\)	\(-545212\)	\(0\)	\(-38567856964\)		$-$	$+$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}+(-272606+\beta )q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.f	$50$	$26$	50.a	1.a	$1$	$2$	$2$	$197.998$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓				10.26.a.b	$1$	$0$	\(8192\)	\(438588\)	\(0\)	\(34008596636\)		$-$	$+$	$-$	$2^{2}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}+(219294-\beta )q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.g	$50$	$26$	50.a	1.a	$1$	$4$	$4$	$197.998$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		50.26.a.g	$1$	$1$	\(-16384\)	\(-106356\)	\(0\)	\(11094836368\)		$+$	$+$	$+$	$2^{10}\cdot 3^{5}\cdot 5^{7}$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}+(-26589+\beta _{1})q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.h	$50$	$26$	50.a	1.a	$1$	$4$	$4$	$197.998$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			50.26.a.g	$1$	$1$	\(16384\)	\(106356\)	\(0\)	\(-11094836368\)		$-$	$-$	$+$	$2^{10}\cdot 3^{5}\cdot 5^{7}$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}+(26589-\beta _{1})q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.i	$50$	$26$	50.a	1.a	$1$	$5$	$5$	$197.998$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			50.26.a.i	$1$	$0$	\(-20480\)	\(-723995\)	\(0\)	\(-49218886190\)		$+$	$-$	$-$	$2^{19}\cdot 3^{5}\cdot 5^{12}$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}+(-144799-\beta _{1})q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.j	$50$	$26$	50.a	1.a	$1$	$5$	$5$	$197.998$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		50.26.a.i	$1$	$0$	\(20480\)	\(723995\)	\(0\)	\(49218886190\)		$-$	$+$	$-$	$2^{19}\cdot 3^{5}\cdot 5^{12}$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}+(144799+\beta _{1})q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.k	$50$	$26$	50.a	1.a	$1$	$6$	$6$	$197.998$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		10.26.b.a	$1$	$0$	\(-24576\)	\(801416\)	\(0\)	\(34007705352\)		$+$	$-$	$-$	$2^{20}\cdot 3^{4}\cdot 5^{18}$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}+(133569+\beta _{1})q^{3}+2^{24}q^{4}+\cdots\)
50.26.a.l	$50$	$26$	50.a	1.a	$1$	$6$	$6$	$197.998$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		10.26.b.a	$1$	$1$	\(24576\)	\(-801416\)	\(0\)	\(-34007705352\)		$-$	$-$	$+$	$2^{20}\cdot 3^{4}\cdot 5^{18}$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}+(-133569-\beta _{1})q^{3}+2^{24}q^{4}+\cdots\)

Decomposition of \(S_{26}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{26}^{\mathrm{old}}(\Gamma_0(50)) \simeq \) \(S_{26}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)