Space of modular forms of level 4, weight 36, and trivial character

• Label: 4.36.a
• Level: $4$
• Weight: $36$
• Character orbit: 4.a
• Rep. character: $\chi_{4}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $1$
• Sturm bound: $18$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(18\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{36}(\Gamma_0(4))\).

	Total	New	Old
Modular forms	19	3	16
Cusp forms	16	3	13
Eisenstein series	3	0	3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	Dim
\(-\)	\(3\)

Trace form

3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9 - 420463958869151460 * q^11 - 7370701315414736574 * q^13 + 35737802623493622360 * q^15 + 616802274448822103958 * q^17 - 3523177807465210008348 * q^19 - 35614415089970552016672 * q^21 - 97200569155131383968872 * q^23 - 1397444430311587979767275 * q^25 - 13813263585892101497770872 * q^27 - 72104888547394509330484398 * q^29 - 305306760334252700869662432 * q^31 - 1206118755141042924734246640 * q^33 - 4139993873456897291989313040 * q^35 - 10467805589252853184477978086 * q^37 - 21306312046147482048067771848 * q^39 - 28614809221544804827249675602 * q^41 - 20661331953141752160177135300 * q^43 + 109445534106618058172074201410 * q^45 + 431671100729347757519313060528 * q^47 + 1202271820749265164710734860219 * q^49 + 1812312130969213433267629787496 * q^51 + 2094590682902326557681205190442 * q^53 - 424018895462656971242660883000 * q^55 - 7056738639317962169249569687824 * q^57 - 21898322952557962385462881418676 * q^59 - 46553596324122391351852871731854 * q^61 - 60534256794589553738854966512984 * q^63 - 47301026213190020793582172528260 * q^65 + 75982094975394264303461043768084 * q^67 + 266168516453886472772566159456416 * q^69 + 591751031973420639599491172643144 * q^71 + 840249720714294658198657086588366 * q^73 + 844328642366224046122007927037900 * q^75 + 366867399909875520359123617491360 * q^77 - 1512021839065233136089826558058544 * q^79 - 4715323858342810015899488694567333 * q^81 - 8944825848970974146478286242709212 * q^83 - 11518175153366806194794571426747180 * q^85 - 9434346897547249340232797055364104 * q^87 + 1625382197094823895014829864559198 * q^89 + 28933989451866564710063788057094832 * q^91 + 43351982414907351295090224432607104 * q^93 + 96155354088435090124008638354927160 * q^95 + 83007021412015685382745763811847014 * q^97 + 91112511762343802272300511308928460 * q^99

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(4))\) 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
4.36.a.a	$4$	$36$	4.a	1.a	$1$	$3$	$3$	$31.038$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	4.36.a.a	$1$	$1$	\(0\)	\(50908884\)	\(280720890\)	\(-55\!\cdots\!16\)		$-$	$-$	$2^{24}\cdot 3^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(16969628+\beta _{1})q^{3}+(93573630+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_0(4))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_0(4)) \simeq \) \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)