LMFDB - Newform orbit 800.6.d.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(401,800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$800 = 2^{5} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	800.d (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$128.307055850$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776$ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2^{93}\cdot 3^{4}\cdot 5^{12}$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{3} + ( - \beta_{4} - 10) q^{7} + (\beta_{5} - \beta_{4} - 81) q^{9} + (\beta_{3} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{8} + 7 \beta_1) q^{13} + \beta_{7} q^{17} + ( - \beta_{15} + \beta_{3} + \cdots - 3 \beta_1) q^{19}+ \cdots + ( - 14 \beta_{19} - 14 \beta_{17} + \cdots + 101 \beta_1) q^{99}+O(q^{100})$ q - b1 * q^3 + (-b4 - 10) * q^7 + (b5 - b4 - 81) * q^9 + (b3 + b2 - b1) * q^11 + (-b8 + 7b1) q^13 + b7 * q^17 + (-b15 + b3 - 6b2 - 3b1) * q^19 + (b9 + b8 + b3 - b2 + 44b1) q^21 + (b18 + b13 - b12 - 3b5 - 2b4 - 235) * q^23 + (-b17 + b16 - b15 - b14 + b9 - b3 - 9b2 + 69b1) * q^27 + (-b19 - 2b17 - b16 + b15 + b14 - 2b8 + b3 - 2b2 - 53b1) * q^29 + (-b13 - b11 + b7 + 2b5 + 4b4 - 357) * q^31 + (b18 - 3b13 - b12 + b11 - 2b10 - b7 + 2b6 + b5 + 14b4 - 280) * q^33 + (-b19 - b16 - 6b14 + b9 + 4b3 - 3b2 + 21b1) * q^37 + (-2b18 + b13 + b12 - b11 + 4b10 + b7 - 4b6 - 7b5 - 25b4 + 2239) q^39 + (-3b18 - 4b13 + b12 + b10 - 2b7 - 5b6 - 14b5 - 18b4 + 575) * q^41 + (-3b19 + b17 + 7b16 - 5b9 + 11b8 - 3b3 - 49b2 + 117b1) q^43 + (4b18 + 4b13 - 5b12 + 2b10 + 6b7 + 5b6 + 24b5 + 3b4 + 2214) * q^47 + (-5b18 - 6b13 + 11b12 - 2b11 - 3b10 + b6 + 6b5 + 16b4 + 942) q^49 + (-4b19 - 8b17 + 6b16 + 2b15 + 15b14 - 6b9 + 3b8 + 2b3 - 65b2 - 58b1) * q^51 + (-9b19 - 6b17 + 3b16 + 11b15 - b14 - 9b9 + 6b8 - 16b3 + 4b2 + 250b1) q^53 + (-2b18 + 4b13 + 14b12 + 4b11 - 6b10 - b7 - 32b5 - 8b4 - 260) q^57 + (-6b16 - 2b15 + b14 + 6b9 + 21b8 - 3b3 + 72b2 + 231b1) q^59 + (-9b19 - 7b16 + 4b15 + 2b14 - 11b9 + 12b8 + 22b3 - 33b2 - 153b1) q^61 + (5b18 - 13b13 - b12 + 2b11 - 10b10 - 4b7 + 5b6 - 64b5 + 67b4 + 12033) * q^63 + (10b19 + b17 - b16 - 9b15 + 3b14 - 9b9 - 22b8 - 41b3 + 143b2 - 230b1) * q^67 + (-5b19 + 6b17 + 11b16 + 5b15 + 13b14 - 2b9 + 8b8 + 35b3 - 51b2 - 341b1) * q^69 + (6b18 - 12b13 - 7b12 - 2b11 - 4b10 - 26b7 + 2b6 + 41b5 + 7b4 + 10026) q^71 + (15b18 - 5b13 + 17b12 - b11 + 2b10 - 3b7 - 105b5 + 6b4 + 5238) q^73 + (3b19 + 22b17 - 3b16 + 17b15 - 3b14 + 5b9 + 2b8 - 32b3 - b2 - 780b1) q^77 + (6b18 - 11b13 - 4b12 + 3b11 + 8b10 - 15b7 + 10b6 - 16b5 + 16b4 - 14099) q^79 + (6b18 - 14b13 + 42b12 + 2b11 - 8b10 - 8b7 - 10b6 - 119b5 + 161b4 + 3269) q^81 + (-35b19 - 23b17 + 15b16 + 22b15 + 20b14 - 13b9 - 25b8 + 19b3 - 93b2 + 569b1) * q^83 + (9b18 + 11b13 - 20b12 - 2b11 + 26b10 - 28b7 + 7b6 + 41b5 + b4 - 16619) * q^87 + (13b18 + 22b13 - 67b12 + 2b11 - 21b10 - 8b7 + 11b6 - 83b5 - 43b4 - 191) q^89 + (-8b19 - 6b17 + 30b16 - b15 + 14b14 - 18b9 + 4b8 + 6b3 - 3b2 + 1348b1) q^91 + (-29b19 - 14b17 + 15b16 - 33b15 + 19b14 - 23b9 + 37b8 + 54b3 - 288b2 + 443b1) * q^93 + (-23b18 + b13 - 49b12 - 3b11 - 30b10 - b7 - 4b6 + 27b5 + 124b4 - 7346) q^97 + (-14b19 - 14b17 - 28b16 + 6b15 - 9b14 - 32b9 - 151b8 - 121b3 + 100b2 + 101b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q - 196 q^{7} - 1620 q^{9} - 4676 q^{23} - 7160 q^{31} - 5672 q^{33} + 44904 q^{39} + 11608 q^{41} + 44180 q^{47} + 18756 q^{49} - 5032 q^{57} + 240620 q^{63} + 200312 q^{71} + 105136 q^{73} - 282080 q^{79}+ \cdots - 147376 q^{97}+O(q^{100})$ 20 * q - 196 * q^7 - 1620 * q^9 - 4676 * q^23 - 7160 * q^31 - 5672 * q^33 + 44904 * q^39 + 11608 * q^41 + 44180 * q^47 + 18756 * q^49 - 5032 * q^57 + 240620 * q^63 + 200312 * q^71 + 105136 * q^73 - 282080 * q^79 + 65172 * q^81 - 332592 * q^87 - 3160 * q^89 - 147376 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776$ x^20 - 2*x^19 - 17*x^18 + 78*x^17 + 253*x^16 - 884*x^15 + 2396*x^14 + 19376*x^13 - 109104*x^12 - 96128*x^11 + 3580672*x^10 - 1538048*x^9 - 27930624*x^8 + 79364096*x^7 + 157024256*x^6 - 926941184*x^5 + 4244635648*x^4 + 20937965568*x^3 - 73014444032*x^2 - 137438953472*x + 1099511627776 :

$\beta_{1}$	$=$	$( - 79348087 \nu^{19} + 164467438 \nu^{18} + 2750967783 \nu^{17} + 2347892670 \nu^{16} + \cdots + 38\!\cdots\!76 ) / 71\!\cdots\!80$ (-79348087v^19 + 164467438v^18 + 2750967783v^17 + 2347892670v^16 - 17200470427v^15 + 69605863660v^14 - 24303499972v^13 - 719687080656v^12 + 16148446793808v^11 + 66989955357824v^10 - 209376218715392v^9 - 308378042677248v^8 + 2858199953264640v^7 + 4761986870738944v^6 + 7314399222824960v^5 + 210707338900799488v^4 + 19270179452092416v^3 - 2314359262031118336v^2 + 3379581800966782976*v + 38936417104430104576) / 713485464769658880
$\beta_{2}$	$=$	$( - 6673115 \nu^{19} - 30164170 \nu^{18} + 153706635 \nu^{17} + 266710470 \nu^{16} + \cdots + 15\!\cdots\!20 ) / 16\!\cdots\!28$ (-6673115v^19 - 30164170v^18 + 153706635v^17 + 266710470v^16 - 3862360175v^15 - 2106440260v^14 - 4737874100v^13 - 236780073360v^12 + 50299499280v^11 + 4828965765760v^10 - 12397823115520v^9 - 82271702046720v^8 + 93706914631680v^7 + 50248724971520v^6 - 1926810069893120v^5 + 7838825908797440v^4 + 6061205571502080v^3 - 180832148914176000v^2 - 275660321136312320*v + 1550266383906897920) / 16465049186992128
$\beta_{3}$	$=$	$( - 392347319 \nu^{19} + 3248676206 \nu^{18} + 37138350375 \nu^{17} + 56596678206 \nu^{16} + \cdots + 46\!\cdots\!32 ) / 71\!\cdots\!80$ (-392347319v^19 + 3248676206v^18 + 37138350375v^17 + 56596678206v^16 - 116030366171v^15 + 215726096876v^14 + 7354614456892v^13 + 28326961252656v^12 + 89620996541520v^11 + 695657667918976v^10 - 64356184347904v^9 - 4877709075462144v^8 + 35110862377537536v^7 + 176315402435624960v^6 + 252028719960162304v^5 + 172884124007137280v^4 + 5395871712027869184v^3 - 4882972880129753088v^2 + 18049481855350079488*v + 464189588174098399232) / 713485464769658880
$\beta_{4}$	$=$	$( - 13756221 \nu^{19} - 7338982 \nu^{18} + 200878797 \nu^{17} - 336652854 \nu^{16} + \cdots + 16\!\cdots\!56 ) / 13\!\cdots\!80$ (-13756221v^19 - 7338982v^18 + 200878797v^17 - 336652854v^16 - 2577873801v^15 + 10357630916v^14 - 40516598892v^13 - 198756032368v^12 + 1182900850544v^11 + 3523610475392v^10 - 19032642164480v^9 - 17076135725056v^8 + 214125830696960v^7 - 344410878705664v^6 - 66398979883008v^5 + 14093127287570432v^4 - 25707255580065792v^3 - 64272954002767872v^2 + 140142841543262208*v + 1695627043779117056) / 13989911073914880
$\beta_{5}$	$=$	$( 16634889 \nu^{19} + 110715086 \nu^{18} - 463943001 \nu^{17} - 1145810466 \nu^{16} + \cdots - 79\!\cdots\!76 ) / 13\!\cdots\!80$ (16634889v^19 + 110715086v^18 - 463943001v^17 - 1145810466v^16 + 6592520229v^15 + 24985916492v^14 - 39165383748v^13 + 407299798448v^12 - 444635751856v^11 - 16455359245696v^10 + 31229684808448v^9 + 254075443275776v^8 - 337690320547840v^7 - 1102678310453248v^6 + 1472660100612096v^5 - 18613792178962432v^4 - 49157013485125632v^3 + 692899524384915456v^2 - 81055567702917120*v - 7971915667421003776) / 13989911073914880
$\beta_{6}$	$=$	$( 916605 \nu^{19} - 557722 \nu^{18} - 72289293 \nu^{17} + 76715958 \nu^{16} + \cdots - 96\!\cdots\!52 ) / 269036751421440$ (916605v^19 - 557722v^18 - 72289293v^17 + 76715958v^16 + 527303625v^15 - 1517841604v^14 - 5348077716v^13 + 6403189616v^12 - 309416479600v^11 - 1362757426048v^10 + 4562500572928v^9 + 9275458783232v^8 - 85058051338240v^7 - 113845855780864v^6 - 443649509031936v^5 - 3038262008479744v^4 - 2652130812887040v^3 + 47702577185292288v^2 - 11760256111607808*v - 961224469802647552) / 269036751421440
$\beta_{7}$	$=$	$( 836343 \nu^{19} + 4825138 \nu^{18} + 26902617 \nu^{17} - 67997406 \nu^{16} + \cdots + 32\!\cdots\!04 ) / 241205363343360$ (836343v^19 + 4825138v^18 + 26902617v^17 - 67997406v^16 - 250315557v^15 + 785814196v^14 + 4487363652v^13 + 26267975248v^12 + 131794625968v^11 - 71168402048v^10 - 2500521196288v^9 - 1606553298944v^8 + 47753989304320v^7 + 65189977194496v^6 + 147694426521600v^5 + 212935867629568v^4 - 4662885082988544v^3 - 2307979059658752v^2 + 67795045155078144*v + 326224344046895104) / 241205363343360
$\beta_{8}$	$=$	$( - 2596081553 \nu^{19} + 2571847202 \nu^{18} + 86300109729 \nu^{17} + 14967625938 \nu^{16} + \cdots + 80\!\cdots\!24 ) / 71\!\cdots\!80$ (-2596081553v^19 + 2571847202v^18 + 86300109729v^17 + 14967625938v^16 - 666619683149v^15 + 1642767296948v^14 - 782367772700v^13 - 38659737685680v^12 + 498530511380784v^11 + 1629997537578880v^10 - 7025350735299328v^9 - 9046298791151616v^8 + 86064960731885568v^7 + 44913884879716352v^6 + 648708649148219392v^5 + 4944058766657060864v^4 - 6262791787983667200v^3 - 60595984488833482752v^2 + 195240255357360013312*v + 807519008638342529024) / 713485464769658880
$\beta_{9}$	$=$	$( - 93629767 \nu^{19} + 702882478 \nu^{18} + 981182583 \nu^{17} - 7979058114 \nu^{16} + \cdots + 40\!\cdots\!60 ) / 20\!\cdots\!60$ (-93629767v^19 + 702882478v^18 + 981182583v^17 - 7979058114v^16 + 8841499541v^15 + 16353862348v^14 + 200746076156v^13 + 2759221703088v^12 + 7122600385872v^11 - 28654941196672v^10 - 82030654354688v^9 + 1030240803366912v^8 + 717663035805696v^7 + 8838309116968960v^6 + 32711987570671616v^5 - 45524197232869376v^4 + 28883515361722368v^3 + 3115688246038757376v^2 + 7746605471233998848*v + 4026192365784924160) / 20581311483740160
$\beta_{10}$	$=$	$( 1017903 \nu^{19} - 5922366 \nu^{18} - 40945759 \nu^{17} - 18273230 \nu^{16} + \cdots - 10\!\cdots\!80 ) / 160803575562240$ (1017903v^19 - 5922366v^18 - 40945759v^17 - 18273230v^16 + 696329523v^15 - 1014965932v^14 - 12216570780v^13 + 10837650640v^12 - 233765807312v^11 - 1469673965184v^10 + 4388269305088v^9 + 10438841456640v^8 - 89504132157440v^7 - 232292172234752v^6 + 248550134120448v^5 - 3349064691220480v^4 - 5978665091334144v^3 + 59776185520881664v^2 - 93891988000079872*v - 1056826868398817280) / 160803575562240
$\beta_{11}$	$=$	$( - 139676643 \nu^{19} + 2240750630 \nu^{18} + 9107502675 \nu^{17} - 41092516746 \nu^{16} + \cdots + 52\!\cdots\!64 ) / 13\!\cdots\!80$ (-139676643v^19 + 2240750630v^18 + 9107502675v^17 - 41092516746v^16 - 13162704663v^15 + 616533384380v^14 - 662833163028v^13 + 8136429834608v^12 + 84170414663312v^11 - 76703819885440v^10 - 1045629245606144v^9 + 3811425548896256v^8 + 13439834111774720v^7 - 4451743495290880v^6 + 165804190316888064v^5 + 192389010578997248v^4 - 2272004378204307456v^3 + 1902422311336673280v^2 + 49540266551628791808*v + 52483153386656497664) / 13989911073914880
$\beta_{12}$	$=$	$( - 1494615 \nu^{19} - 139442 \nu^{18} + 12535687 \nu^{17} - 26796322 \nu^{16} + \cdots + 93\!\cdots\!08 ) / 145728240353280$ (-1494615v^19 - 139442v^18 + 12535687v^17 - 26796322v^16 - 279617595v^15 + 122543756v^14 - 5962000836v^13 - 23573933904v^12 + 90795397200v^11 + 199244947072v^10 - 2560414975232v^9 - 3962388420608v^8 - 2942057164800v^7 - 54493173448704v^6 - 2748377726976v^5 + 1091904015958016v^4 - 3594256216227840v^3 - 21742532933189632v^2 + 189051575468032*v + 93180380638609408) / 145728240353280
$\beta_{13}$	$=$	$( 55930047 \nu^{19} + 298875042 \nu^{18} - 471238127 \nu^{17} - 811773934 \nu^{16} + \cdots - 48\!\cdots\!04 ) / 46\!\cdots\!60$ (55930047v^19 + 298875042v^18 - 471238127v^17 - 811773934v^16 + 15047923587v^15 + 41419986964v^14 + 119568371748v^13 + 2209681571792v^12 + 3750568239152v^11 - 26612297711232v^10 + 56270384684288v^9 + 636654389581824v^8 + 526021163233280v^7 + 1598354895601664v^6 + 19328373443788800v^5 + 3209696181747712v^4 - 184920549774852096v^3 + 1157282097929388032v^2 + 4511770355974537216*v - 4887483048372731904) / 4663303691304960
$\beta_{14}$	$=$	$( - 3023485621 \nu^{19} - 3513130646 \nu^{18} + 23790813957 \nu^{17} - 48592005414 \nu^{16} + \cdots + 26\!\cdots\!04 ) / 23\!\cdots\!60$ (-3023485621v^19 - 3513130646v^18 + 23790813957v^17 - 48592005414v^16 - 454251774433v^15 - 1685917512956v^14 - 10616492811532v^13 - 82212079011696v^12 + 150528537739248v^11 + 630848422047104v^10 - 4082820043124480v^9 - 13324667699374080v^8 - 18908712196706304v^7 - 109324306413977600v^6 - 209664098101362688v^5 + 1978120501121253376v^4 - 10412771141222400v^3 - 63942789403647148032v^2 + 9488797279126028288*v + 26930691448508514304) / 237828488256552960
$\beta_{15}$	$=$	$( 13747776289 \nu^{19} - 4153513666 \nu^{18} - 348265483569 \nu^{17} + 147182976654 \nu^{16} + \cdots - 60\!\cdots\!48 ) / 10\!\cdots\!20$ (13747776289v^19 - 4153513666v^18 - 348265483569v^17 + 147182976654v^16 + 3752560744861v^15 - 3493958732404v^14 + 10808186492380v^13 + 197622701803440v^12 - 1619389489493040v^11 - 7763285463482240v^10 + 35050305071084288v^9 + 74859598874314752v^8 - 439256002670309376v^7 - 505390155029610496v^6 - 752678592968916992v^5 - 19196143597212663808v^4 - 4147643445751578624v^3 + 389752753841888034816v^2 - 410910124975767683072*v - 6067633191738249576448) / 1070228197154488320
$\beta_{16}$	$=$	$( 445476863 \nu^{19} + 755206978 \nu^{18} - 2379736431 \nu^{17} + 690187890 \nu^{16} + \cdots - 55\!\cdots\!20 ) / 31\!\cdots\!80$ (445476863v^19 + 755206978v^18 - 2379736431v^17 + 690187890v^16 + 41497627523v^15 + 424482480052v^14 + 2364490799780v^13 + 8491155454800v^12 - 2213726830032v^11 - 135462442775680v^10 + 443775654153472v^9 + 3113792848312320v^8 + 7601811276902400v^7 + 8147471708520448v^6 + 7364567046029312v^5 - 78951472682762240v^4 + 130657928091795456v^3 + 9577769516843139072v^2 + 24592412889225101312*v - 55918171747851960320) / 31477299916308480
$\beta_{17}$	$=$	$( - 279607999 \nu^{19} + 207980926 \nu^{18} + 4651875759 \nu^{17} - 5763844146 \nu^{16} + \cdots + 21\!\cdots\!60 ) / 18\!\cdots\!20$ (-279607999v^19 + 207980926v^18 + 4651875759v^17 - 5763844146v^16 - 69380146627v^15 - 15389892212v^14 - 180631027876v^13 - 5214217274448v^12 + 24807733624272v^11 + 58583395989632v^10 - 593749860183296v^9 - 880987800532992v^8 + 2653284081782784v^7 + 1255725749960704v^6 - 25940664486461440v^5 + 165537541947129856v^4 - 637830318060994560v^3 - 6394634195309690880v^2 + 7376531761657806848*v + 21477900337870274560) / 18294499096657920
$\beta_{18}$	$=$	$( - 244379043 \nu^{19} - 882778202 \nu^{18} + 3542156307 \nu^{17} - 8059440138 \nu^{16} + \cdots + 22\!\cdots\!32 ) / 13\!\cdots\!80$ (-244379043v^19 - 882778202v^18 + 3542156307v^17 - 8059440138v^16 - 57201457623v^15 + 58658586556v^14 - 822954486804v^13 - 9582542930576v^12 - 328114898288v^11 + 107071571539072v^10 - 393965912033536v^9 - 1797414647232512v^8 + 378330237276160v^7 - 19669188103307264v^6 - 52666172690399232v^5 + 217678425886818304v^4 + 326477872694820864v^3 - 5850063588600840192v^2 - 12678068546631106560*v + 22911574302609375232) / 13989911073914880
$\beta_{19}$	$=$	$( 57874591541 \nu^{19} - 28605234794 \nu^{18} - 940175038341 \nu^{17} + 1158073375782 \nu^{16} + \cdots - 96\!\cdots\!68 ) / 21\!\cdots\!40$ (57874591541v^19 - 28605234794v^18 - 940175038341v^17 + 1158073375782v^16 + 9774585318497v^15 + 8894804969212v^14 + 106021015360268v^13 + 809441452781424v^12 - 4038214998218736v^11 - 17586093478669696v^10 + 102290210682416896v^9 + 201502488566212608v^8 - 491180679633235968v^7 + 139445601868644352v^6 + 133034924197019648v^5 - 34196547037451780096v^4 + 8221824496189833216v^3 + 1058026525469259595776v^2 + 113411325758541922304*v - 9644851138866307923968) / 2140456394308976640

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( 28 \beta_{19} + 18 \beta_{18} - 2 \beta_{17} - 12 \beta_{16} - 3 \beta_{15} + 23 \beta_{14} + \cdots + 2538 ) / 25600$ (28b19 + 18b18 - 2b17 - 12b16 - 3b15 + 23b14 + 8b13 - 10b12 + 8b11 - 10b10 + 8b9 + 5b8 + 8b7 + 42b6 - 94b5 - 14b4 + 90b3 + 20b2 - 104*b1 + 2538) / 25600
$\nu^{2}$	$=$	$( - 28 \beta_{19} - 38 \beta_{18} - 78 \beta_{17} + 12 \beta_{16} + 83 \beta_{15} + 17 \beta_{14} + \cdots + 48802 ) / 25600$ (-28b19 - 38b18 - 78b17 + 12b16 + 83b15 + 17b14 - 88b13 - 26b12 - 8b11 - 10b10 + 72b9 + 195b8 + 112b7 + 18b6 + 10b5 + 738b4 - 250b3 - 236b2 + 2504*b1 + 48802) / 25600
$\nu^{3}$	$=$	$( - 18 \beta_{19} - 39 \beta_{18} - 108 \beta_{17} + 112 \beta_{16} - 32 \beta_{15} + 142 \beta_{14} + \cdots - 39819 ) / 6400$ (-18b19 - 39b18 - 108b17 + 112b16 - 32b15 + 142b14 - 84b13 + 137b12 + 16b11 + 5b10 - 28b9 - 210b8 - 34b7 - 41b6 + 155b5 - 971b4 + 220b3 + 58b2 + 2714*b1 - 39819) / 6400
$\nu^{4}$	$=$	$( 20 \beta_{19} + 34 \beta_{18} - 2 \beta_{17} + 20 \beta_{16} - 19 \beta_{15} + 23 \beta_{14} + \cdots - 39478 ) / 1024$ (20b19 + 34b18 - 2b17 + 20b16 - 19b15 + 23b14 + 72b13 + 54b12 - 24b11 + 22b10 + 72b9 - 91b8 + 56b7 - 14b6 - 182b5 + 10b4 - 86b3 - 28b2 + 3552*b1 - 39478) / 1024
$\nu^{5}$	$=$	$( 1308 \beta_{19} - 694 \beta_{18} - 1822 \beta_{17} + 268 \beta_{16} - 5733 \beta_{15} + \cdots - 3461694 ) / 25600$ (1308b19 - 694b18 - 1822b17 + 268b16 - 5733b15 - 1847b14 + 136b13 + 7574b12 - 1064b11 + 1030b10 + 2088b9 + 21355b8 - 4464b7 - 5686b6 + 24258b5 - 64294b4 + 390b3 + 10580b2 - 579344*b1 - 3461694) / 25600
$\nu^{6}$	$=$	$( 10606 \beta_{19} - 1491 \beta_{18} + 7856 \beta_{17} - 5024 \beta_{16} - 9566 \beta_{15} + \cdots - 9927991 ) / 6400$ (10606b19 - 1491b18 + 7856b17 - 5024b16 - 9566b15 - 284b14 - 3196b13 - 4171b12 - 296b11 + 2585b10 + 7356b9 - 7840b8 - 2066b7 - 4689b6 + 23499b5 - 11043b4 - 6400b3 + 21202b2 + 285842*b1 - 9927991) / 6400
$\nu^{7}$	$=$	$( - 99876 \beta_{19} - 28942 \beta_{18} - 28626 \beta_{17} + 38324 \beta_{16} - 106699 \beta_{15} + \cdots - 157676822 ) / 25600$ (-99876b19 - 28942b18 - 28626b17 + 38324b16 - 106699b15 - 35601b14 + 32648b13 - 112730b12 - 20952b11 + 19590b10 - 54776b9 + 46125b8 + 16248b7 + 30602b6 + 114306b5 + 919746b4 - 168950b3 - 147196b2 - 5629512*b1 - 157676822) / 25600
$\nu^{8}$	$=$	$( - 8172 \beta_{19} - 2390 \beta_{18} + 6946 \beta_{17} - 20 \beta_{16} - 677 \beta_{15} + \cdots + 13514578 ) / 1024$ (-8172b19 - 2390b18 + 6946b17 - 20b16 - 677b15 - 11511b14 + 5352b13 - 20554b12 + 3064b11 + 3846b10 + 2824b9 - 15637b8 - 28080b7 - 7982b6 + 13002b5 - 126526b4 - 57674b3 + 398036b2 + 454072*b1 + 13514578) / 1024
$\nu^{9}$	$=$	$( 91758 \beta_{19} + 145665 \beta_{18} + 136748 \beta_{17} - 210672 \beta_{16} + 11092 \beta_{15} + \cdots + 480581725 ) / 6400$ (91758b19 + 145665b18 + 136748b17 - 210672b16 + 11092b15 + 336298b14 + 525340b13 - 556831b12 - 16160b11 - 41875b10 - 194732b9 - 237790b8 - 71810b7 + 86535b6 + 350011b5 + 4892549b4 + 1566980b3 + 4021002b2 - 22020534*b1 + 480581725) / 6400
$\nu^{10}$	$=$	$( - 1462268 \beta_{19} + 1872850 \beta_{18} - 2420018 \beta_{17} - 7964428 \beta_{16} + \cdots - 5939140390 ) / 25600$ (-1462268b19 + 1872850b18 - 2420018b17 - 7964428b16 + 8935973b15 - 4606273b14 + 650760b13 - 11131322b12 + 2259880b11 - 5691610b10 - 440568b9 + 12127645b8 - 3361800b7 + 4649330b6 - 11983158b5 + 21796618b4 - 11322950b3 + 66720964b2 - 51171376*b1 - 5939140390) / 25600
$\nu^{11}$	$=$	$( - 11180916 \beta_{19} - 27219414 \beta_{18} - 9920766 \beta_{17} + 11878284 \beta_{16} + \cdots + 26439427586 ) / 25600$ (-11180916b19 - 27219414b18 - 9920766b17 + 11878284b16 + 32903691b15 + 28922009b14 - 27408184b13 + 9900214b12 + 2520216b11 - 6975770b10 - 29630616b9 - 14953125b8 + 1779216b7 - 27821766b6 + 12153778b5 + 248202506b4 - 1528250b3 - 299524716b2 + 373283008*b1 + 26439427586) / 25600
$\nu^{12}$	$=$	$( - 377730 \beta_{19} + 227613 \beta_{18} - 609080 \beta_{17} + 191424 \beta_{16} + 976198 \beta_{15} + \cdots - 1250502055 ) / 256$ (-377730b19 + 227613b18 - 609080b17 + 191424b16 + 976198b15 - 78688b14 + 526228b13 + 1576661b12 + 200168b11 - 325399b10 - 575860b9 - 3416428b8 + 293886b7 + 400887b6 - 7871133b5 + 3891253b4 + 2596520b3 - 6319342b2 + 25492274*b1 - 1250502055) / 256
$\nu^{13}$	$=$	$( - 206557204 \beta_{19} + 27478386 \beta_{18} + 91894446 \beta_{17} + 293929396 \beta_{16} + \cdots + 545833861386 ) / 25600$ (-206557204b19 + 27478386b18 + 91894446b17 + 293929396b16 + 520746229b15 - 235122129b14 - 29284984b13 + 586404134b12 - 108045784b11 - 119013370b10 + 38162696b9 + 771820525b8 + 30346616b7 - 335494566b6 - 1454940142b5 + 11526226b4 - 264856150b3 - 8638079484b2 - 16058946648*b1 + 545833861386) / 25600
$\nu^{14}$	$=$	$( 2382349028 \beta_{19} + 536563450 \beta_{18} + 872846578 \beta_{17} - 190007412 \beta_{16} + \cdots - 682288380990 ) / 25600$ (2382349028b19 + 536563450b18 + 872846578b17 - 190007412b16 - 976010733b15 - 583473167b14 - 1221698840b13 + 2968476518b12 + 216805880b11 - 207910410b10 - 299700472b9 - 99108445b8 - 856353200b7 - 159897870b6 + 4778661802b5 - 5770098142b4 + 7113996550b3 - 22288872204b2 + 14669896*b1 - 682288380990) / 25600
$\nu^{15}$	$=$	$( 160118542 \beta_{19} + 506109225 \beta_{18} + 880534852 \beta_{17} - 86269328 \beta_{16} + \cdots - 1625292214715 ) / 6400$ (160118542b19 + 506109225b18 + 880534852b17 - 86269328b16 - 717920792b15 - 1238466298b14 - 871984500b13 - 1003453191b12 - 437480400b11 + 620805525b10 + 653569732b9 + 1417850390b8 + 1615280350b7 + 869833975b6 - 4494279909b5 + 8270743509b4 - 2470103380b3 - 67271786502b2 + 99281887834*b1 - 1625292214715) / 6400
$\nu^{16}$	$=$	$( - 894641740 \beta_{19} - 747940190 \beta_{18} - 240157186 \beta_{17} + 1194117652 \beta_{16} + \cdots + 652260593290 ) / 1024$ (-894641740b19 - 747940190b18 - 240157186b17 + 1194117652b16 - 597604755b15 - 683230185b14 - 816918712b13 + 1212533046b12 - 125666328b11 + 574010134b10 - 754548408b9 + 1481800869b8 - 616024392b7 + 291804050b6 + 3017587562b5 - 10401614934b4 - 181680662b3 + 2240011108b2 + 27122287168*b1 + 652260593290) / 1024
$\nu^{17}$	$=$	$( 28884382716 \beta_{19} + 79191908618 \beta_{18} + 92698079266 \beta_{17} + 5046178316 \beta_{16} + \cdots + 109761795011138 ) / 25600$ (28884382716b19 + 79191908618b18 + 92698079266b17 + 5046178316b16 - 51476558341b15 + 14554110441b14 + 85337017608b13 - 140583824170b12 - 2715674792b11 + 110981128390b10 + 72861531816b9 - 157502044725b8 + 43867953808b7 + 41740713642b6 + 26701218466b5 - 800980856774b4 + 287813420550b3 + 1805311212756b2 + 7790063605392*b1 + 109761795011138) / 25600
$\nu^{18}$	$=$	$( - 72314538066 \beta_{19} + 46613619069 \beta_{18} - 46315267616 \beta_{17} - 50923610336 \beta_{16} + \cdots + 31192842076569 ) / 6400$ (-72314538066b19 + 46613619069b18 - 46315267616b17 - 50923610336b16 + 3388478026b15 - 357013476b14 + 79647000164b13 - 110661130011b12 - 15303372936b11 - 42593445015b10 + 4963779484b9 + 315142973640b8 + 81780472494b7 + 145785265551b6 + 699268734859b5 - 443605087363b4 + 162479678800b3 + 1273023881138b2 - 4383724445262*b1 + 31192842076569) / 6400
$\nu^{19}$	$=$	$( 800902948092 \beta_{19} - 238043720974 \beta_{18} - 362774695378 \beta_{17} - 1025926382668 \beta_{16} + \cdots + 510561238570026 ) / 25600$ (800902948092b19 - 238043720974b18 - 362774695378b17 - 1025926382668b16 + 202713121333b15 + 895426865647b14 - 835658829944b13 - 2851939019866b12 + 473227670056b11 + 125807492230b10 + 47456123912b9 + 1047275027245b8 + 762115259256b7 - 411223522006b6 + 7795692831138b5 - 23188791768094b4 - 5463208433590b3 + 42910274360580b2 + 113812977032344*b1 + 510561238570026) / 25600

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$ .

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

401.1

3.46430	+	1.99965i
−3.90102	−	0.884346i
0.236693	−	3.99299i
2.93366	+	2.71913i
3.18502	−	2.41984i
−3.80026	−	1.24819i
3.72553	−	1.45618i
−2.80358	+	2.85306i
0.593959	+	3.95566i
−2.63430	+	3.01006i
−2.63430	−	3.01006i
0.593959	−	3.95566i
−2.80358	−	2.85306i
3.72553	+	1.45618i
−3.80026	+	1.24819i
3.18502	+	2.41984i
2.93366	−	2.71913i
0.236693	+	3.99299i
−3.90102	+	0.884346i
3.46430	−	1.99965i

−

29.2080i

−168.173

−610.110

401.2

−

25.4343i

−56.4938

−403.904

401.3

−

25.0521i

103.624

−384.607

401.4

−

18.7876i

−107.536

−109.975

401.5

−

17.3148i

−9.19080

−56.8021

401.6

−

11.5927i

−231.529

108.609

401.7

−

10.8240i

163.706

125.841

401.8

−

10.7455i

198.733

127.535

401.9

−

6.93089i

47.1406

194.963

401.10

−

6.67450i

−38.2812

198.451

401.11

6.67450i

−38.2812

198.451

401.12

6.93089i

47.1406

194.963

401.13

10.7455i

198.733

127.535

401.14

10.8240i

163.706

125.841

401.15

11.5927i

−231.529

108.609

401.16

17.3148i

−9.19080

−56.8021

401.17

18.7876i

−107.536

−109.975

401.18

25.0521i

103.624

−384.607

401.19

25.4343i

−56.4938

−403.904

401.20

29.2080i

−168.173

−610.110

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	800.6.d.c		20
4.b	odd	2	1		200.6.d.b		20
5.b	even	2	1		160.6.d.a		20
5.c	odd	4	1		800.6.f.b		20
5.c	odd	4	1		800.6.f.c		20
8.b	even	2	1	inner	800.6.d.c		20
8.d	odd	2	1		200.6.d.b		20
20.d	odd	2	1		40.6.d.a	✓	20
20.e	even	4	1		200.6.f.b		20
20.e	even	4	1		200.6.f.c		20
40.e	odd	2	1		40.6.d.a	✓	20
40.f	even	2	1		160.6.d.a		20
40.i	odd	4	1		800.6.f.b		20
40.i	odd	4	1		800.6.f.c		20
40.k	even	4	1		200.6.f.b		20
40.k	even	4	1		200.6.f.c		20
60.h	even	2	1		360.6.k.b		20
120.m	even	2	1		360.6.k.b		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.6.d.a	✓	20	20.d	odd	2	1
40.6.d.a	✓	20	40.e	odd	2	1
160.6.d.a		20	5.b	even	2	1
160.6.d.a		20	40.f	even	2	1
200.6.d.b		20	4.b	odd	2	1
200.6.d.b		20	8.d	odd	2	1
200.6.f.b		20	20.e	even	4	1
200.6.f.b		20	40.k	even	4	1
200.6.f.c		20	20.e	even	4	1
200.6.f.c		20	40.k	even	4	1
360.6.k.b		20	60.h	even	2	1
360.6.k.b		20	120.m	even	2	1
800.6.d.c		20	1.a	even	1	1	trivial
800.6.d.c		20	8.b	even	2	1	inner
800.6.f.b		20	5.c	odd	4	1
800.6.f.b		20	40.i	odd	4	1
800.6.f.c		20	5.c	odd	4	1
800.6.f.c		20	40.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(800, [\chi])$ :

T_{3}^{20} + 3240 T_{3}^{18} + 4346772 T_{3}^{16} + 3151305344 T_{3}^{14} + 1355603009184 T_{3}^{12} + \cdots + 14\!\cdots\!84

T_{7}^{10} + 98 T_{7}^{9} - 83922 T_{7}^{8} - 6806560 T_{7}^{7} + 2129001128 T_{7}^{6} + \cdots + 13\!\cdots\!08

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20}$ T^20
$3$	$T^{20} + \cdots + 14\!\cdots\!84$ T^20 + 3240T^18 + 4346772T^16 + 3151305344T^14 + 1355603009184T^12 + 359218958085888T^10 + 59164106793818752T^8 + 5978764455444817920T^6 + 354916377079609533696T^4 + 11177338398394463004672*T^2 + 142602076715343595766784
$5$	$T^{20}$ T^20
$7$	$(T^{10} + \cdots + 13\!\cdots\!08)^{2}$ (T^10 + 98T^9 - 83922T^8 - 6806560T^7 + 2129001128T^6 + 149256489744T^5 - 16715051341136T^4 - 1088925123904640T^3 + 22278527346308496T^2 + 1721950139896063008*T + 13226564728477155808)^2
$11$	$T^{20} + \cdots + 70\!\cdots\!00$ T^20 + 1711016T^18 + 1187234134224T^16 + 438539966696586752T^14 + 94879747424584980517376T^12 + 12478678172254103334566031360T^10 + 1004907526767812027210486637469696T^8 + 48442269228536219186486307527193395200T^6 + 1306753581309652020110605216191306670080000T^4 + 16880002061984162758418768553020776095744000000*T^2 + 70167852897926169794298630715984656885760000000000
$13$	$T^{20} + \cdots + 16\!\cdots\!00$ T^20 + 3905528T^18 + 6136454126416T^16 + 4988469520919693824T^14 + 2258733925223246490825216T^12 + 577651295500612071992616407040T^10 + 82106651575366167887526352582877184T^8 + 6137788881525358860884565374533307596800T^6 + 215711768162188164741601449589076573429760000T^4 + 3217057431922814635360338733419212496494592000000*T^2 + 16511112125491170401455720758406419700190822400000000
$17$	$(T^{10} + \cdots + 29\!\cdots\!68)^{2}$ (T^10 - 8750908T^8 - 1859072000T^7 + 23998151145888T^6 + 11063772518400000T^5 - 21446330507601376128T^4 - 14317463194873602048000T^3 + 1988905722934233183245568T^2 + 2313907995449384505999360000T + 294580496690248704818606650368)^2
$19$	$T^{20} + \cdots + 23\!\cdots\!56$ T^20 + 25883288T^18 + 285046332695760T^16 + 1746271114538976571904T^14 + 6516200664459955340153573888T^12 + 15202268167893997261028763450249216T^10 + 21804659081583891282267655790665406685184T^8 + 17974632097588711413964184354778560588432932864T^6 + 7196396079747636051136014188955800980746248402042880T^4 + 834234046827879548853650177744267006656537537814315139072*T^2 + 23751782553356803907066925574188970785243151884587053540704256
$23$	$(T^{10} + \cdots - 88\!\cdots\!16)^{2}$ (T^10 + 2338T^9 - 28365770T^8 - 90426486752T^7 + 117354068596520T^6 + 637751007312995728T^5 + 365256046547394812400T^4 - 854744129145344502855552T^3 - 1245351955498942534596098160T^2 - 573732979732993040095696684512*T - 88471012791702948854644084275616)^2
$29$	$T^{20} + \cdots + 42\!\cdots\!00$ T^20 + 195130080T^18 + 15351673563972864T^16 + 640582862784030126800896T^14 + 15646238138232348012837933809664T^12 + 232336863610611750902129383428694474752T^10 + 2095588350906987898603680361094488537247514624T^8 + 11020729012782863909635144395483876077710638632140800T^6 + 30510345197779619183407906041426326120480519560636661760000T^4 + 34783649959736879549990698432082433127279711185333682962432000000*T^2 + 4257741901690041830921675255673030673204495643468955751179878400000000
$31$	$(T^{10} + \cdots - 42\!\cdots\!88)^{2}$ (T^10 + 3580T^9 - 153869896T^8 - 437494937344T^7 + 7341181528205952T^6 + 15797753164246990336T^5 - 108500883515612631520256T^4 - 149396246874032910324449280T^3 + 346432850950705074038378958848T^2 + 271599668713170267985848132943872*T - 42506672219643903529913221101682688)^2
$37$	$T^{20} + \cdots + 17\!\cdots\!36$ T^20 + 821506472T^18 + 283168092338137296T^16 + 53614409791166010231193088T^14 + 6122137772446843167011525370335744T^12 + 435476238395264394888571662139150206824448T^10 + 19258147448360316285191821612057874042851730464768T^8 + 513926248767002937244179018545905586444147911666188550144T^6 + 7784155353020183914146654116937200373273586250173081078993846272T^4 + 59973502819027341913874854008893787249829232173475324215775608825184256*T^2 + 178725925096385086823221609054630064267136066709897011138677253330512181723136
$41$	$(T^{10} + \cdots - 21\!\cdots\!00)^{2}$ (T^10 - 5804T^9 - 577824776T^8 + 1875250039808T^7 + 112673734070960256T^6 - 80442400781905077760T^5 - 8724196432503550885413888T^4 - 8072080537977110753976320000T^3 + 232931407962635431304496499200000T^2 + 199866639445197069256822776907776000*T - 2129594087346616896515164046253117440000)^2
$43$	$T^{20} + \cdots + 24\!\cdots\!76$ T^20 + 1732042120T^18 + 1264805616873797588T^16 + 508983510551730053880975488T^14 + 123765439745221626490204363571884704T^12 + 18741109459618538200143459210026335609594624T^10 + 1752931352996502522619923709865477273421215088243328T^8 + 96489295973780224138283590103208494998079285542187504732160T^6 + 2776189875669765470744881687914064251111731104099509007677874693376T^4 + 30596627988727262560117702312678421906091402339625222547883720417390790656*T^2 + 24226051988890617857696446673164212398296144306700432902045548976473856364594176
$47$	$(T^{10} + \cdots + 16\!\cdots\!92)^{2}$ (T^10 - 22090T^9 - 744939994T^8 + 19816630242400T^7 + 65043919811739304T^6 - 4220725961073437022160T^5 + 18381802218896663308929136T^4 + 177736154425140630725117301120T^3 - 1290246234067258465992384824144496T^2 + 1088469430275725328209235292887111520*T + 1633295114563215923585010454685317693792)^2
$53$	$T^{20} + \cdots + 82\!\cdots\!44$ T^20 + 4360240504T^18 + 7547899229497442640T^16 + 6586760769321828854564055552T^14 + 3055113607573734123194975509915267584T^12 + 748448778205602133781496299733062999342764032T^10 + 99289215064468658604196106516391089018876921962831872T^8 + 7104624154343797858245194153211702542985561006376336839278592T^6 + 256940821319584926627329786737352507382807947685669409504974584217600T^4 + 3804504188681598899657666365308957629805554311077278765485913150344893300736*T^2 + 8218543534197564144428436599117103779959729611227226932849035850171679617758265344
$59$	$T^{20} + \cdots + 40\!\cdots\!76$ T^20 + 4301536152T^18 + 7893526636419653840T^16 + 8057909856982563696503468544T^14 + 4997240412317258284245273221006938624T^12 + 1926809239516110941759681445082404257229869056T^10 + 452552672078695105663606535205093469091837361806450688T^8 + 60333291889295619630686363579727304955206280899012722800328704T^6 + 3885689727393451818198156298396423768861904695585144732307083831869440T^4 + 81120499824905058930649873005614923911871573274805097068089259040402178572288*T^2 + 403540028902294593914841834126903142619544297303781223921385131870339566130116427776
$61$	$T^{20} + \cdots + 29\!\cdots\!00$ T^20 + 9739073672T^18 + 39049057139936353872T^16 + 83604204136394781079545099776T^14 + 103864876633929765741157010751854555648T^12 + 76497978730975905181579976859465337533970722816T^10 + 32940149491321999386327126552634621600395737162649051136T^8 + 7867646420793382063636770322422120013417044016640913399778508800T^6 + 932336525459732023596754647103098594070787814585152355478000006471680000T^4 + 44543458855540294227023108297954067311982092847836388949181729965520265216000000*T^2 + 299537713907462762131805512876854462668349802239887200099961261832556558428160000000000
$67$	$T^{20} + \cdots + 50\!\cdots\!04$ T^20 + 15173983176T^18 + 91092031416369225300T^16 + 277467605920414066270535337088T^14 + 461485195273952838210263922149980158624T^12 + 426166422723961317457240497185137134478478162688T^10 + 214225754003586296081280925050694334801430561241165587072T^8 + 55621794683804803098661024739985475701241961075651216203962697728T^6 + 6666929836164768286085153987487191803105409505566468283608631589071709440T^4 + 318839816024018478222547961436447167288934485116688345883375545267211411597871104*T^2 + 5089672139612270986738224343903797791662025920318980266106715498794674280744413391995904
$71$	$(T^{10} + \cdots + 19\!\cdots\!32)^{2}$ (T^10 - 100156T^9 - 4593845064T^8 + 668893729933056T^7 - 3305303406670534016T^6 - 1029998039283869641104896T^5 + 17117410470145166387347434496T^4 + 243496951842460889215838375100416T^3 - 4045029571199781234215837031742468096T^2 - 11429043178618743826173449632452362878976*T + 199215962933176171034180132498264362490232832)^2
$73$	$(T^{10} + \cdots - 24\!\cdots\!16)^{2}$ (T^10 - 52568T^9 - 11351373036T^8 + 606266359383680T^7 + 32586480428083645088T^6 - 1815124743592703150563584T^5 - 25534871464500172795398923648T^4 + 1472196464690865551891671779665920T^3 + 11296022259716543464884773703751623936T^2 - 340283614301808839867207685493974187726848*T - 2484162303900341550147901680105283004478422016)^2
$79$	$(T^{10} + \cdots + 28\!\cdots\!00)^{2}$ (T^10 + 141040T^9 - 262124464T^8 - 878655619892224T^7 - 38639251881508836864T^6 + 368804259967459976519680T^5 + 46432072671986912096531685376T^4 + 246422077195686913044334639841280T^3 - 18993047487205440986435864422388531200T^2 - 116990025584986458985990635244300664832000*T + 2898380858667191038277431903243203835985920000)^2
$83$	$T^{20} + \cdots + 23\!\cdots\!36$ T^20 + 44012159480T^18 + 727681127398744111668T^16 + 5681169449754670235756897534848T^14 + 21878543541004798036916296912839587999904T^12 + 41275199782849078422880207380152868984441202162944T^10 + 36334764354335363723945978723444427299335734590758150059648T^8 + 13998863075049807812435566178223805110575070833853972557663735592960T^6 + 2036771244426971092848443599539582042165494837732410886300739271290843284736T^4 + 119627794582621301277879879485562069254756029387002744588319042504128263242759796736*T^2 + 2389527001683006690330779300263178280690821049985093737529872372090877116613739578206299136
$89$	$(T^{10} + \cdots + 21\!\cdots\!00)^{2}$ (T^10 + 1580T^9 - 28442400444T^8 + 380635615133504T^7 + 252005360584542180896T^6 - 7563365388046180744772480T^5 - 671540242766408546810621844864T^4 + 34610702600086976220464891122242560T^3 - 377933131026737681673796074215287980800T^2 - 1425549484010168477903499693504159761536000*T + 2148613791468302474557427046244603353019520000)^2
$97$	$(T^{10} + \cdots + 22\!\cdots\!48)^{2}$ (T^10 + 73688T^9 - 42200485708T^8 - 2877738584271488T^7 + 564180396810943351968T^6 + 37013898413631660974129408T^5 - 2458922010489390687930848405888T^4 - 177124349795910544583780048030803968T^3 + 1182400804278885328761347209268667876608T^2 + 193385696003934905939679749756550287788136448*T + 2230363856141118897530997543954042514224613762048)^2
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 401.20
Significant digits:
Format:

Introduction

L-functions

Modular forms

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Fields

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Groups

Database

Properties

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