Space of modular forms of level 1, weight 38, and trivial character

• Label: 1.38.a
• Level: $1$
• Weight: $38$
• Character orbit: 1.a
• Rep. character: $\chi_{1}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $3$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 1 \)
Weight:	\( k \)	\(=\)	\( 38 \)
Character orbit:	\([\chi]\)	\(=\)	1.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(3\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(1))\).

	Total	New	Old
Modular forms	3	3	0
Cusp forms	2	2	0
Eisenstein series	1	1	0

Trace form

2 * q - 194400 * q^2 + 13991400 * q^3 + 37720269824 * q^4 + 5529584385900 * q^5 - 22506543847296 * q^6 - 3448443953486000 * q^7 - 34044043043635200 * q^8 - 898947378401769414 * q^9 - 9015609355013275200 * q^10 - 26734036354848538056 * q^11 + 4374774798370099200 * q^12 + 530581741653933178300 * q^13 + 3169419119584065186048 * q^14 + 649108943683113884400 * q^15 - 31152340106193176363008 * q^16 - 89439073881931767332700 * q^17 + 87081814924495841786400 * q^18 + 373276466572513713706120 * q^19 + 1752437909050980457420800 * q^20 - 228188865811378281195456 * q^21 - 6854650238581457086684800 * q^22 - 26241180149933881945462800 * q^23 + 1869795529093100955893760 * q^24 + 114502199615774855754278750 * q^25 + 161198290271390767044721344 * q^26 - 12567565757525316335881200 * q^27 - 616012501603352544390963200 * q^28 - 1270673827125882417951629220 * q^29 - 180869768268003018176083200 * q^30 + 261420819791418895481545024 * q^31 + 13400499670935825918040473600 * q^32 + 493606999076023001294056800 * q^33 + 15928395480391611359115494208 * q^34 - 91348258395203301140779687200 * q^35 - 16896751657649331450099821568 * q^36 - 68025363793820786588733238100 * q^37 + 343467860448446492149172419200 * q^38 - 11607709470242600846801346768 * q^39 + 751002277177488834411651072000 * q^40 - 1260483295466373133974684841836 * q^41 + 78468780705143776426791244800 * q^42 - 2570241023157918831169581605000 * q^43 + 1333494270055063548163396988928 * q^44 - 2476861984515775772921684271300 * q^45 + 12543759397110220887548257622784 * q^46 + 4252206875568934025158407583200 * q^47 - 627865474342240811187752140800 * q^48 - 3828013658149768338307153838286 * q^49 - 58010169823175993824907262180000 * q^50 - 1146602989088714802084517292976 * q^51 - 31355807203456647433862622464000 * q^52 + 159799736258590810071505678893900 * q^53 + 20246293327921369338050721342720 * q^54 + 198965756266726963295068969294800 * q^55 - 223825491222026073030759257210880 * q^56 - 24730693798747893221767420706400 * q^57 - 554449751247961885353825667003200 * q^58 - 237962459606090128758899369700840 * q^59 + 35138010316774983406295554252800 * q^60 + 101798841373038700200106255199644 * q^61 + 1873125324903360954810571368883200 * q^62 + 1547129676521036250657950915523600 * q^63 + 1874672702019432167416499183550464 * q^64 - 4674979790566994364848214315456600 * q^65 + 168556374577560093117347449548288 * q^66 - 10891923280981358108643809352546200 * q^67 - 3093300961636357481932215727257600 * q^68 - 903079825405531082422221799280448 * q^69 + 31333293246739547087546996653401600 * q^70 - 746133089793832492689962158868016 * q^71 + 15331394897876624646558687016550400 * q^72 + 19639851676496426023909382507487700 * q^73 - 67111659095025202293405215279979072 * q^74 + 4176423071329708963282786324335000 * q^75 - 66783420216030647132134009134018560 * q^76 - 45127994201424729507574115685000000 * q^77 - 2993244266642124681941208094368000 * q^78 + 272401638034095839035647945144674080 * q^79 - 250481015549362947010655598516633600 * q^80 + 403323837550038341003235827350619442 * q^81 + 293555739501144005521394805048475200 * q^82 - 470460275390909929308746217601073400 * q^83 - 15246219975764064504961836760399872 * q^84 - 456126475708535200141442602834276200 * q^85 - 1006428830786982641255996134716471936 * q^86 + 39923812909186495628003917423124400 * q^87 + 1397391845312343171300827514362265600 * q^88 - 1332868921711238117914343978794942860 * q^89 + 4050631020486902745715661556073886400 * q^90 + 1138398786311531726477856383010207584 * q^91 - 2437574081740573687511448195442483200 * q^92 - 134865729229255425577378938731155200 * q^93 + 703460887213042587108786164575999488 * q^94 - 9929993151899196579173099213128626000 * q^95 + 173258625718217609590386311471038464 * q^96 + 6002061888473229973039130090661237700 * q^97 - 9401601645347953932912572443881928800 * q^98 + 12025768918384639461946415841980309592 * q^99

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\) 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				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1.38.a.a	$1$	$38$	1.a	1.a	$1$	$2$	$2$	$8.671$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	1.38.a.a	$1$	$1$	\(-194400\)	\(13991400\)	\(55\!\cdots\!00\)	\(-34\!\cdots\!00\)	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-97200-\beta )q^{2}+(6995700+72\beta )q^{3}+\cdots\)