LMFDB - Space of modular forms of level 2, weight 78, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 2 $
Weight:	$ k $	$=$	$ 78 $
Character orbit:	$[\chi]$	$=$	2.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 2 $
Sturm bound:			$19$
Trace bound:			$2$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{78}(\Gamma_0(2))$.

	Total	New	Old
Modular forms	20	6	14
Cusp forms	18	6	12
Eisenstein series	2	0	2

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

$2$	Total			Cusp			Eisenstein
$2$	All	New	Old	All	New	Old	All	New	Old
$+$	$10$	$3$	$7$	$9$	$3$	$6$	$1$	$0$	$1$
$-$	$10$	$3$	$7$	$9$	$3$	$6$	$1$	$0$	$1$

Trace form

$ 6 q - 26\!\cdots\!80 q^{3} + 45\!\cdots\!16 q^{4} + 95\!\cdots\!60 q^{5} + 66\!\cdots\!24 q^{6} - 40\!\cdots\!60 q^{7} + 27\!\cdots\!58 q^{9} + 91\!\cdots\!60 q^{10} + 28\!\cdots\!52 q^{11} - 19\!\cdots\!80 q^{12}+ \cdots - 33\!\cdots\!64 q^{99}+O(q^{100}) $

6 * q - 2638706479849430280 * q^3 + 453347182355485940514816 * q^4 + 951948703777156427890806660 * q^5 + 664069599363895225939000295424 * q^6 - 404664187371826723533400217750160 * q^7 + 27635744030038056171543919721032526958 * q^9 + 91257727220733797922901686662159400960 * q^10 + 2813285456351650460858754578674470792552 * q^11 - 199375024617150342745880842780071249838080 * q^12 - 5387427944763819764030050276353666486056460 * q^13 + 16520452604894625405860749985152667870035968 * q^14 - 1135883271557496516132416112035892348266320560 * q^15 + 34253944624943037145398863266787883273185918976 * q^16 - 240351405602411934790934759670260169813104869140 * q^17 + 4839316985813546497882978970098276147197546332160 * q^18 - 45893971269809092729734590453917163953845035256360 * q^19 + 71927210434055167140260971699563051078232320245760 * q^20 + 1401012816099590795429365101383802989450997291970752 * q^21 - 11125248257094086183871982045171633101780341426749440 * q^22 + 48640348424149570925324498153599959856137326190461520 * q^23 + 50175680293259716553311739858033147080681785774833664 * q^24 - 398875278193186197856744600318615444390641586944858150 * q^25 + 3415403793959182394051815855026300616551980020178878464 * q^26 + 19349767246913973555813361436471974968223870568539877040 * q^27 - 30575561524198343424449603702271110059135213208611061760 * q^28 + 177509221507210771314192900454648567151613591454452548820 * q^29 + 1340514695285109270197692869004094029901201395971407216640 * q^30 - 5235559682219472906690176086938683491174471358818087437888 * q^31 + 40577551469543753340467332593881981549497732378992850805280 * q^33 - 28728510870588968139878894736118585268489508521741345882112 * q^34 - 123064352879350240342180780799622219421217658897628058701920 * q^35 + 2088097781385865762745322404541080951012278757890508653068288 * q^36 + 7156514143967091986133454180146950526241890550563178852648260 * q^37 + 2040897416681867651538648803119205521313170834634472538767360 * q^38 + 107456379433622082812502228370241930898258059873982872409735696 * q^39 + 6895238917280866375135335442099832497030641086794792760770560 * q^40 + 410465139749811625309074459022728334344890109972786982566849852 * q^41 + 805403647085810964014486128394288828918198462460989714800312320 * q^42 + 2320058003993125449433825043551023223127010133647134942508081960 * q^43 + 212565839133114693977856248066434581905269098221961512403075072 * q^44 + 10203244720734129287172794367624582396187024267861702888626955380 * q^45 + 2689960197998271719351769644493200790795055143703777841170612224 * q^46 - 38244595912715455466995651647529711267054630500403405513549841760 * q^47 - 15064350940373459149228584094629141708006792960481646191973498880 * q^48 - 389094542452427259916390029434866502780300547855292070198944383978 * q^49 - 831485774843342499680359664508624926672108521209001092974012006400 * q^50 - 930191979149192566325986756313185136870549452751754737819177754768 * q^51 - 407062546483647372579849484620264530532879283368214456976340418560 * q^52 - 695797257743603428807751534595932331958483278842484417070166762460 * q^53 + 1754690806597444293046334739806632909274959133479013419923611320320 * q^54 + 38765245534507901360456409072653869831426630634683147406804309076720 * q^55 + 1248250106611054411112330855294537584863889729514344296745059483648 * q^56 + 129362944731709088997142014082973350866062059161528333699670779875040 * q^57 + 50553365460753368965295588731110862830308083132310069555735581163520 * q^58 - 83738260671160073339274788272749998269062795139071833447932168007160 * q^59 - 85824913440887054944685196442844885241558241613465368296333414236160 * q^60 + 139900325275364299372802562013721613938014575020089549956553410097812 * q^61 - 2111006982576088507000264401233211102767594655185504995235583838126080 * q^62 - 6126200075624497706359484909456780492729528147181203469617380029598800 * q^63 + 2588154880046461420288033448353884544669165864563894958185946583924736 * q^64 - 12393134856502627186159140322763470190755840041406744302014774648849160 * q^65 + 15393885428335113682010960552095081769079893184032598792332776788983808 * q^66 - 12854778152408665924380911481075717372896438820035823065050865579327880 * q^67 - 18160438750839001419211575762557524388833159140012271964552214651863040 * q^68 + 359310602285066394235528572119300341447192465781666788376235147864668736 * q^69 + 103629847839080851842304847321434986509738469514573404043319760629268480 * q^70 - 239348329233499159281725256062739658510043600972130704323629438986077968 * q^71 + 365648453340602405398745089460256882904927686172564038282850307156213760 * q^72 + 590732221588297211993359202991552093313332614637765009735841839598898940 * q^73 - 1932561582707509952011992201876418120033979606336145283880984403039485952 * q^74 - 6894358829739875418006698611431046529908046354215975823843697141950004600 * q^75 - 3467650427045262567968315642170391976780739571258665204670493541489704960 * q^76 + 3027746567174723274672896523599217478828065756918059258896992435435562560 * q^77 + 5582523603240121029709723147076013923944333973942409491332660145569136640 * q^78 - 5776495766817453464284107181954195043305348051620600864517138288432413600 * q^79 + 5434666364161503150888916355684998022341974188838754175526045799006863360 * q^80 + 117563120151265824119033769345677085904590527052736653327730870160256552726 * q^81 + 19357494913548812795451232384100923643744067379118639730033395516288532480 * q^82 + 129264760599533898767518723678436374167477584530796383608431823494550778200 * q^83 + 105857535437112346177585855037195646219860979986395570285290974848349110272 * q^84 - 801241925385693765103426964254546576751329262200466456455724329000162401720 * q^85 + 53797180931623833539915540017975805817399081240325266040630423998645141504 * q^86 - 291328796825929144330520656517167227174892291079422928234202042149107063920 * q^87 - 840599991726480803409567601862008337142872621983882976102199012471173283840 * q^88 - 1193812277046235700760495300028009526099839740926801826997139607286011853860 * q^89 - 1609060304447922381867852111640472845776561247088529753708962342303925534720 * q^90 + 2492679381298712523448873417158446985376314632434513670973361519518001603872 * q^91 + 3675160817812884788672025368135229915457597531387580745772777698516239646720 * q^92 + 10687213444988864959111097540950099077730437025392413394337931784475766526720 * q^93 - 31395041641246475618760185950925797315969613155484425844934831292035366912 * q^94 + 23652710336299468471330383881491290651477153277385046023331615359669641952400 * q^95 + 3791167213953162498930463176849797580866121787939356866278082602690154594304 * q^96 + 34330669035023975921697546293905865217388554866066568629115439228943107591500 * q^97 + 26683820159041618494567864569402267280153427530555533481830490985535782256640 * q^98 - 338117328528106478398409250709483622397469797614871765540271738811872768465464 * q^99

Decomposition of $S_{78}^{\mathrm{new}}(\Gamma_0(2))$ 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
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	2	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
2.78.a.a	$2$	$78$	2.a	1.a	$1$	$3$	$3$	$75.096$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓	✓	2.78.a.a	$1$	$1$	$-824633720832$	$-25\!\cdots\!88$	$30\!\cdots\!10$	$-23\!\cdots\!16$	$+$	$+$	$2^{24}\cdot 3^{11}\cdot 5^{2}\cdot 7\cdot 11\cdot 13\cdot 19$	$\mathrm{SU}(2)$	$q-2^{38}q^{2}+(-842429601461527596+\cdots)q^{3}+\cdots$
2.78.a.b	$2$	$78$	2.a	1.a	$1$	$3$	$3$	$75.096$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓	✓	2.78.a.b	$1$	$0$	$824633720832$	$-11\!\cdots\!92$	$64\!\cdots\!50$	$-17\!\cdots\!44$	$-$	$-$	$2^{24}\cdot 3^{8}\cdot 5^{4}\cdot 7\cdot 11$	$\mathrm{SU}(2)$	$q+2^{38}q^{2}+(-37139225154949164+\cdots)q^{3}+\cdots$

Decomposition of $S_{78}^{\mathrm{old}}(\Gamma_0(2))$ into lower level spaces

$ S_{78}^{\mathrm{old}}(\Gamma_0(2)) \simeq $ $S_{78}^{\mathrm{new}}(\Gamma_0(1))$$^{\oplus 2}$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2 \)
Weight:	\( k \)	\(=\)	\( 78 \)
Character orbit:	\([\chi]\)	\(=\)	2.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(19\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

\(2\)	Total			Cusp			Eisenstein
\(2\)	All	New	Old	All	New	Old	All	New	Old
\(+\)	\(10\)	\(3\)	\(7\)	\(9\)	\(3\)	\(6\)	\(1\)	\(0\)	\(1\)
\(-\)	\(10\)	\(3\)	\(7\)	\(9\)	\(3\)	\(6\)	\(1\)	\(0\)	\(1\)

Introduction

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Modular forms

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Groups

Database

Properties

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{78}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Decomposition of \(S_{78}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2.78.a

Level	$2$
Weight	$78$
Character orbit	2.a
Rep. character	$\chi_{2}(1,\cdot)$
Character field	$\Q$
Dimension	$6$
Newform subspaces	$2$
Sturm bound	$19$
Trace bound	$2$