LMFDB - Newform orbit 2254.4.a.bc

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.a (trivial)

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:	yes
Analytic conductor:	$132.990305153$
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} - 398 x^{18} + 65501 x^{16} - 5759932 x^{14} + 291300980 x^{12} - 8455560298 x^{10} + \cdots + 1023755428864 $ x^20 - 398x^18 + 65501x^16 - 5759932x^14 + 291300980x^12 - 8455560298x^10 + 131887696352x^8 - 958301668480x^6 + 2985915503888x^4 - 3158776583552*x^2 + 1023755428864
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{2}\cdot 7^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + (\beta_{12} + \beta_1) q^{5} + 2 \beta_1 q^{6} + 8 q^{8} + (\beta_{2} + 13) q^{9} + (2 \beta_{12} + 2 \beta_1) q^{10} + (\beta_{3} + 11) q^{11} + 4 \beta_1 q^{12}+ \cdots + (2 \beta_{10} + 7 \beta_{8} + \cdots + 372) q^{99}+O(q^{100}) $ q + 2 * q^2 + b1 * q^3 + 4 * q^4 + (b12 + b1) * q^5 + 2b1 q^6 + 8 * q^8 + (b2 + 13) * q^9 + (2b12 + 2b1) * q^10 + (b3 + 11) * q^11 + 4b1 q^12 + b13 * q^13 + (b4 + b2 + 24) * q^15 + 16 * q^16 + (-b18 + b12 + 3b1) q^17 + (2b2 + 26) q^18 + (b16 - b14 + b13 + b1) * q^19 + (4b12 + 4b1) * q^20 + (2b3 + 22) q^22 + 23 * q^23 + 8b1 q^24 + (b6 + b5 + b3 + 40) * q^25 + 2b13 q^26 + (-b17 + b16 + b14 - b13 + 3b12 + 16b1) * q^27 + (-b8 + b7 - b6 + 2b2 + 40) q^29 + (2b4 + 2b2 + 48) * q^30 + (b17 - 2b16 - b15 + 2b12 + b11 + 4b1) q^31 + 32 * q^32 + (b18 - b16 + 2b15 - b13 + 4b12 - 2b11 + 18b1) * q^33 + (-2b18 + 2b12 + 6b1) q^34 + (4b2 + 52) q^36 + (b10 + b9 + 2b8 + b7 - 2b3 + 27) * q^37 + (2b16 - 2b14 + 2b13 + 2b1) * q^38 + (-3b10 - 2b9 - b8 - 5b7 - b6 - 3b5 - 3b4 + 2b3 - 5b2 - 5) q^39 + (8b12 + 8b1) * q^40 + (-b19 + 2b18 + 2b17 - b16 - b15 + 2b14 + 8b12 + 5b11 + 6b1) * q^41 + (2b10 + b9 + 2b8 + 2b7 + b6 - b4 - 3b3 + b2 + 40) * q^43 + (4b3 + 44) q^44 + (b19 + b18 - 3b17 - b15 - b14 - 4b13 + 9b12 - 4b11 + 37b1) q^45 + 46 * q^46 + (b19 + b18 + 2b17 - b15 + b14 - b11 + 10b1) * q^47 + 16b1 q^48 + (2b6 + 2b5 + 2b3 + 80) q^50 + (3b10 + 2b9 - 3b8 - b7 - b6 - 5b3 + 5b2 + 86) q^51 + 4b13 q^52 + (b10 - b9 + 2b8 + 4b7 - 2b6 - 2b2 + 28) * q^53 + (-2b17 + 2b16 + 2b14 - 2b13 + 6b12 + 32b1) * q^54 + (b18 - 2b17 - 4b16 + b15 - 3b13 + 23b12 - 4b11 + 26b1) * q^55 + (-3b10 - 2b9 + b8 - 4b7 + b6 - 4b5 - 3b4 - b3 - 4b2 + 27) * q^57 + (-2b8 + 2b7 - 2b6 + 4b2 + 80) * q^58 + (-2b19 + b18 + b17 + 2b15 - b14 - 3b12 + 6b11 + 25b1) q^59 + (4b4 + 4b2 + 96) * q^60 + (b19 - 2b18 + b17 + 2b16 + b14 - 3b13 - 2b12 - 2b11 + 21b1) * q^61 + (2b17 - 4b16 - 2b15 + 4b12 + 2b11 + 8b1) * q^62 + 64 * q^64 + (-4b10 + 2b9 - b8 - 8b7 - 4b4 + 2b3 - 13b2 + 59) * q^65 + (2b18 - 2b16 + 4b15 - 2b13 + 8b12 - 4b11 + 36b1) q^66 + (-b10 - b9 + 3b7 + 4b5 - b4 - 2b3 - 5b2 + 101) * q^67 + (-4b18 + 4b12 + 12b1) q^68 + 23b1 q^69 + (2b10 + 2b8 + 10b7 + 6b6 + 5b5 + 2b4 - 5b3 + 3b2 + 177) * q^71 + (8b2 + 104) q^72 + (-2b19 + 3b18 + 3b17 + b16 - 3b15 + 4b14 - 20b12 + 15b11 + 29b1) * q^73 + (2b10 + 2b9 + 4b8 + 2b7 - 4b3 + 54) q^74 + (3b18 + 2b17 - 5b16 + 2b15 + b14 - 7b13 + 12b12 - 6b11 + 40b1) * q^75 + (4b16 - 4b14 + 4b13 + 4b1) * q^76 + (-6b10 - 4b9 - 2b8 - 10b7 - 2b6 - 6b5 - 6b4 + 4b3 - 10b2 - 10) q^78 + (-8b10 - b7 - 8b6 - 3b4 + 14b3 - 13b2 + 67) q^79 + (16b12 + 16b1) * q^80 + (10b10 - 7b8 + 6b7 - 3b6 + 2b5 + 11b4 - 10b3 + 16b2 + 252) * q^81 + (-2b19 + 4b18 + 4b17 - 2b16 - 2b15 + 4b14 + 16b12 + 10b11 + 12b1) q^82 + (2b19 - 4b18 - 4b17 + 4b16 + b15 + b13 + 9b12 - 2b11 + 28b1) q^83 + (4b10 - 2b9 - 11b8 + 8b6 + b4 - 2b3 - 6b2 + 233) * q^85 + (4b10 + 2b9 + 4b8 + 4b7 + 2b6 - 2b4 - 6b3 + 2b2 + 80) * q^86 + (-b19 - 5b18 - 6b17 + 3b16 + 4b15 + 7b14 - b13 - 19b12 + 79b1) q^87 + (8b3 + 88) q^88 + (2b19 - 3b18 - 10b17 - 2b15 - 11b14 + 5b12 - 20b11 - 6b1) * q^89 + (2b19 + 2b18 - 6b17 - 2b15 - 2b14 - 8b13 + 18b12 - 8b11 + 74b1) q^90 + 92 * q^92 + (-9b10 + 2b9 + 5b8 - 15b7 - b6 - b5 - 4b4 + 2b3 - 18b2 + 141) q^93 + (2b19 + 2b18 + 4b17 - 2b15 + 2b14 - 2b11 + 20b1) q^94 + (4b10 + 2b9 + 7b8 - 5b7 + 3b6 + 3b5 - 5b4 - 15b3 - 4b2 + 75) q^95 + 32b1 q^96 + (6b17 + 10b16 + 3b15 - 9b14 + 5b13 - 4b12 + 8b11 - 20b1) * q^97 + (2b10 + 7b8 + 30b7 + 6b6 + 10b5 + 10b4 - b3 + 15b2 + 372) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 40 q^{2} + 80 q^{4} + 160 q^{8} + 256 q^{9} + 216 q^{11} + 468 q^{15} + 320 q^{16} + 512 q^{18} + 432 q^{22} + 460 q^{23} + 788 q^{25} + 800 q^{29} + 936 q^{30} + 640 q^{32} + 1024 q^{36} + 552 q^{37}+ \cdots + 7396 q^{99}+O(q^{100}) $ 20 * q + 40 * q^2 + 80 * q^4 + 160 * q^8 + 256 * q^9 + 216 * q^11 + 468 * q^15 + 320 * q^16 + 512 * q^18 + 432 * q^22 + 460 * q^23 + 788 * q^25 + 800 * q^29 + 936 * q^30 + 640 * q^32 + 1024 * q^36 + 552 * q^37 - 60 * q^39 + 812 * q^43 + 864 * q^44 + 920 * q^46 + 1576 * q^50 + 1692 * q^51 + 608 * q^53 + 584 * q^57 + 1600 * q^58 + 1872 * q^60 + 1280 * q^64 + 1228 * q^65 + 2076 * q^67 + 3524 * q^71 + 2048 * q^72 + 1104 * q^74 - 120 * q^78 + 1452 * q^79 + 4908 * q^81 + 4568 * q^85 + 1624 * q^86 + 1728 * q^88 + 1840 * q^92 + 2912 * q^93 + 1576 * q^95 + 7396 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 398 x^{18} + 65501 x^{16} - 5759932 x^{14} + 291300980 x^{12} - 8455560298 x^{10} + \cdots + 1023755428864 $ x^20 - 398*x^18 + 65501*x^16 - 5759932*x^14 + 291300980*x^12 - 8455560298*x^10 + 131887696352*x^8 - 958301668480*x^6 + 2985915503888*x^4 - 3158776583552*x^2 + 1023755428864 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 40 $ v^2 - 40
$\beta_{3}$	$=$	$ ( 15\!\cdots\!31 \nu^{18} + \cdots - 32\!\cdots\!36 ) / 15\!\cdots\!76 $ (1537482213375960032831v^18 - 617523398820884856461226v^16 + 102585153514374425472726683v^14 - 9100696502018889530395368604v^12 + 463343562641324513556103243668v^10 - 13469512697865438826412019884038v^8 + 207840455174712872261034523410128v^6 - 1449465000097922351051751093292848v^4 + 4204170783043650494627585446269680*v^2 - 3210498936803041619705204715018336) / 15167779790642126608042416823776
$\beta_{4}$	$=$	$ ( - 62\!\cdots\!93 \nu^{18} + \cdots + 43\!\cdots\!28 ) / 50\!\cdots\!92 $ (-624234491463937099393v^18 + 243156374368768524494550v^16 - 39042751383974944776361333v^14 + 3335534959710462850939967540v^12 - 162719547822306914099665564668v^10 + 4486103480781023169053580456458v^8 - 63831009503464739634670400964304v^6 + 373926378889209343850942483024208v^4 - 779433490087580448062266123967056*v^2 + 435318635874590837703408002694528) / 5055926596880708869347472274592
$\beta_{5}$	$=$	$ ( 27\!\cdots\!23 \nu^{18} + \cdots - 60\!\cdots\!00 ) / 11\!\cdots\!88 $ (27545345746738795123v^18 - 11309226962995570118086v^16 + 1929203788912140304306759v^14 - 176845193499089037153294360v^12 + 9390654147000991393058694164v^10 - 289137445117201246368617284790v^8 + 4858113566780531583346305231240v^6 - 38597078357287653094105936397056v^4 + 116285542262852875802747730860128*v^2 - 60364576962255363594989237711200) / 112077190571247240946126232688
$\beta_{6}$	$=$	$ ( 47\!\cdots\!41 \nu^{18} + \cdots - 74\!\cdots\!36 ) / 15\!\cdots\!76 $ (4729442795690863252941v^18 - 1869060187427867442371158v^16 + 305230559464691279739978417v^14 - 26625101207984510910946254364v^12 + 1336199803050585506042431487036v^10 - 38553427862431500534057812197890v^8 + 599938049743544743266242588744288v^6 - 4351788387200967026965854026394928v^4 + 12484673399419159323392287398767952*v^2 - 7487356528754892161512769041092736) / 15167779790642126608042416823776
$\beta_{7}$	$=$	$ ( - 78\!\cdots\!73 \nu^{18} + \cdots + 11\!\cdots\!72 ) / 22\!\cdots\!64 $ (-7812998156576520117173v^18 + 3115609158968471858607170v^16 - 513458553300969199313822585v^14 + 45163809004512674528047692216v^12 - 2279966430898975090306613606596v^10 + 65790943444434027172817140968850v^8 - 1010472306474435168504094375889688v^6 + 7022834177223809685741337909336016v^4 - 19189596237840626396433947353501904*v^2 + 11462663464145734220601465769440272) / 22751669685963189912063625235664
$\beta_{8}$	$=$	$ ( 29\!\cdots\!47 \nu^{18} + \cdots - 11\!\cdots\!88 ) / 75\!\cdots\!88 $ (2914311677918031545747v^18 - 1149501421550930357034850v^16 + 186534139915217499115813199v^14 - 16036947521921832033692531516v^12 + 781004609218164176269935810980v^10 - 21181495213361775672201520901614v^8 + 287371170105249482304297838487968v^6 - 1459869086871078135164880335345152v^4 + 2224097395863977196904174265823296*v^2 - 1103718896127567641567743530624688) / 7583889895321063304021208411888
$\beta_{9}$	$=$	$ ( - 48\!\cdots\!67 \nu^{18} + \cdots + 40\!\cdots\!92 ) / 65\!\cdots\!04 $ (-4848409643722412828467v^18 + 1916249594041221183889546v^16 - 312468892591451419826143831v^14 + 27129805893894443848906067700v^12 - 1346684698779857150096604000812v^10 + 37933325337408345803152863386078v^8 - 559548242862769069173480021893952v^6 + 3579458506896589974231528436314976v^4 - 8552006831434195533496824677190064*v^2 + 4028651659949801997109647564661792) / 6500477053132339974875321495904
$\beta_{10}$	$=$	$ ( 86\!\cdots\!22 \nu^{18} + \cdots - 72\!\cdots\!84 ) / 11\!\cdots\!32 $ (8606972845563750194722v^18 - 3397575050819844274075039v^16 + 552982126677027031055857882v^14 - 47869599344934296198415443595v^12 + 2364297998292229523319621981866v^10 - 65988124778860086389753442643400v^8 + 955109866241605167855300764396814v^6 - 5846773445188989900726785156184604v^4 + 13512516521215339453498342072554976*v^2 - 7271565261244582526694746820217384) / 11375834842981594956031812617832
$\beta_{11}$	$=$	$ ( 15\!\cdots\!95 \nu^{19} + \cdots - 23\!\cdots\!12 \nu ) / 11\!\cdots\!92 $ (1514060796158695v^19 - 598880713785092938v^17 + 97559744276608529371v^15 - 8433438236999579481324v^13 + 414034620720498520219340v^11 - 11377376114972455176624134v^9 + 158538485825105666390700720v^7 - 879441400756528777024652032v^5 + 1961535875530242021580623088v^3 - 2391581223798041403668806912v) / 117598010450800593011496192
$\beta_{12}$	$=$	$ ( - 12\!\cdots\!19 \nu^{19} + \cdots + 11\!\cdots\!72 \nu ) / 18\!\cdots\!12 $ (-12510271934868059940855819v^19 + 4956530892493947025436950514v^17 - 810648623261602974784861941519v^15 + 70647466782337135626814946115020v^13 - 3523721583389464745888475446377180v^11 + 99901325109547126856279176433831214v^9 - 1487841110845323157426007151909706000v^7 + 9682017108905377898544331857570795776v^5 - 23842411300636999065699832700779143984v^3 + 11351545443924191972684846468165353472v) / 182700963504881295702740258114656512
$\beta_{13}$	$=$	$ ( 29\!\cdots\!31 \nu^{19} + \cdots - 64\!\cdots\!52 \nu ) / 41\!\cdots\!52 $ (29155576088520172756250731v^19 - 11571588066098111305698136150v^17 + 1898840463502065456573683103391v^15 - 166503484485667626864413023893504v^13 + 8400997128341418491807502944234276v^11 - 243611284002145702215390016866653678v^9 + 3807845657637206665472906021879515704v^7 - 27882592697686299119014367611620404464v^5 + 85732961289888401555928113734267715344v^3 - 64256081479506150594329584588359386752v) / 411077167885982915331165580757977152
$\beta_{14}$	$=$	$ ( 12\!\cdots\!95 \nu^{19} + \cdots - 31\!\cdots\!60 \nu ) / 82\!\cdots\!04 $ (127425294573475705706155295v^19 - 50867990931567088260531239546v^17 + 8408760711221867587745154880307v^15 - 744371552028841694978021437957932v^13 + 38034077747775411852973119981992812v^11 - 1122327630805438294609910760795311926v^9 + 17993996114363799529728463562215040880v^7 - 136562468647940801286548627320919345408v^5 + 421060343260373856100098912313305940592v^3 - 316952769965000031927210646906236421760v) / 822154335771965830662331161515954304
$\beta_{15}$	$=$	$ ( - 14\!\cdots\!75 \nu^{19} + \cdots + 18\!\cdots\!80 \nu ) / 82\!\cdots\!04 $ (-149277393928940532574390175v^19 + 59351883914836072505852961338v^17 - 9755716231758623539653384164915v^15 + 856568614136195821555682125862124v^13 - 43233399479998302964136482424936812v^11 + 1251058580093387232750120246939197878v^9 - 19377140067620358645492170192161838064v^7 + 137050485288107551416102667134463271168v^5 - 371245641071980966127817928174594999280v^3 + 181288634756628252626394862407082446080v) / 822154335771965830662331161515954304
$\beta_{16}$	$=$	$ ( - 37\!\cdots\!13 \nu^{19} + \cdots + 56\!\cdots\!84 \nu ) / 16\!\cdots\!08 $ (-373269813889668782044524313v^19 + 149076285380769954890597795494v^17 - 24636771004323032785124125921477v^15 + 2177478150360579563717321729995716v^13 - 110812437282414329712488865052806644v^11 + 3241592271789694325576136989184362810v^9 - 51019885133974135189951917329180986032v^7 + 371595404150688726802792697758069365760v^5 - 1064755915655261733905340210793414296592v^3 + 561285184303823708375624600489232553984v) / 1644308671543931661324662323031908608
$\beta_{17}$	$=$	$ ( - 35\!\cdots\!35 \nu^{19} + \cdots + 37\!\cdots\!36 \nu ) / 10\!\cdots\!88 $ (-35801179458639730003770235v^19 + 14215811867460299579945345305v^17 - 2331382766496927484694988812215v^15 + 203889411710541833946433405922463v^13 - 10218047065734045507047592119871374v^11 + 291794870259073310646088331173362278v^9 - 4402186595538693002680550462114439318v^7 + 29463456224124845247903117107320236172v^5 - 75684780505175658397754686646746379560v^3 + 37874831517867290403358678892680962736v) / 102769291971495728832791395189494288
$\beta_{18}$	$=$	$ ( 99\!\cdots\!53 \nu^{19} + \cdots - 12\!\cdots\!56 \nu ) / 16\!\cdots\!08 $ (999847795917138456400952953v^19 - 396744834336802839710976924694v^17 + 65014106643475932325774123175653v^15 - 5680547664749538445600127149768500v^13 + 284387714464362705194643734440420052v^11 - 8111905078058638457270927616521154554v^9 + 122247419722793913953883669102940135824v^7 - 818416621810434039479324549931440348224v^5 + 2132105213822712375215189435637664622992v^3 - 1255983590891304868591724708072953363456v) / 1644308671543931661324662323031908608
$\beta_{19}$	$=$	$ ( 16\!\cdots\!47 \nu^{19} + \cdots - 20\!\cdots\!56 \nu ) / 16\!\cdots\!08 $ (1639473814007037923431836247v^19 - 649404825675824016324312719866v^17 + 106176069314592978834752007568747v^15 - 9249086036576110788221557112435580v^13 + 461085706511798078681944580065041740v^11 - 13067635125968633580915423193477151654v^9 + 194775037787612659209698849527997642832v^7 - 1277033632685175280872702093451770339328v^5 + 3289185533560281810236993849835378041584v^3 - 2027080109516318752671618862189290067456v) / 1644308671543931661324662323031908608

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 40 $ b2 + 40
$\nu^{3}$	$=$	$ -\beta_{17} + \beta_{16} + \beta_{14} - \beta_{13} + 3\beta_{12} + 70\beta_1 $ -b17 + b16 + b14 - b13 + 3b12 + 70b1
$\nu^{4}$	$=$	$ 10\beta_{10} - 7\beta_{8} + 6\beta_{7} - 3\beta_{6} + 2\beta_{5} + 11\beta_{4} - 10\beta_{3} + 97\beta_{2} + 2763 $ 10b10 - 7b8 + 6b7 - 3b6 + 2b5 + 11b4 - 10b3 + 97b2 + 2763
$\nu^{5}$	$=$	$ 16 \beta_{19} - 38 \beta_{18} - 151 \beta_{17} + 106 \beta_{16} + 2 \beta_{15} + 111 \beta_{14} + \cdots + 5470 \beta_1 $ 16b19 - 38b18 - 151b17 + 106b16 + 2b15 + 111b14 - 125b13 + 410b12 - 100b11 + 5470b1
$\nu^{6}$	$=$	$ 1540 \beta_{10} + 120 \beta_{9} - 982 \beta_{8} + 972 \beta_{7} - 327 \beta_{6} + 323 \beta_{5} + \cdots + 214011 $ 1540b10 + 120b9 - 982b8 + 972b7 - 327b6 + 323b5 + 1616b4 - 1405b3 + 9015*b2 + 214011
$\nu^{7}$	$=$	$ 2746 \beta_{19} - 5861 \beta_{18} - 16497 \beta_{17} + 9996 \beta_{16} + 320 \beta_{15} + \cdots + 455947 \beta_1 $ 2746b19 - 5861b18 - 16497b17 + 9996b16 + 320b15 + 11246b14 - 14254b13 + 45545b12 - 16090b11 + 455947b1
$\nu^{8}$	$=$	$ 180420 \beta_{10} + 23712 \beta_{9} - 104493 \beta_{8} + 124773 \beta_{7} - 23661 \beta_{6} + \cdots + 17708482 $ 180420b10 + 23712b9 - 104493b8 + 124773b7 - 23661b6 + 41964b5 + 184473b4 - 157632b3 + 836853*b2 + 17708482
$\nu^{9}$	$=$	$ 346692 \beta_{19} - 671808 \beta_{18} - 1633971 \beta_{17} + 915081 \beta_{16} + 49491 \beta_{15} + \cdots + 39667441 \beta_1 $ 346692b19 - 671808b18 - 1633971b17 + 915081b16 + 49491b15 + 1100040b14 - 1545915b13 + 4730787b12 - 2064924b11 + 39667441b1
$\nu^{10}$	$=$	$ 19151226 \beta_{10} + 3247656 \beta_{9} - 10131027 \beta_{8} + 14579223 \beta_{7} - 1122039 \beta_{6} + \cdots + 1531670131 $ 19151226b10 + 3247656b9 - 10131027b8 + 14579223b7 - 1122039b6 + 4911213b5 + 19342914b4 - 16148367b3 + 77864452*b2 + 1531670131
$\nu^{11}$	$=$	$ 38956200 \beta_{19} - 68929893 \beta_{18} - 155810101 \beta_{17} + 83110309 \beta_{16} + \cdots + 3549472489 \beta_1 $ 38956200b19 - 68929893b18 - 155810101b17 + 83110309b16 + 7539531b15 + 105688480b14 - 161814670b13 + 478004724b12 - 244189056b11 + 3549472489b1
$\nu^{12}$	$=$	$ 1941902632 \beta_{10} + 383606124 \beta_{9} - 943983031 \beta_{8} + 1619438781 \beta_{7} + \cdots + 136417048980 $ 1941902632b10 + 383606124b9 - 943983031b8 + 1619438781b7 + 12031107b6 + 541820894b5 + 1953802277b4 - 1577942632b3 + 7265211769*b2 + 136417048980
$\nu^{13}$	$=$	$ 4133127370 \beta_{19} - 6706153730 \beta_{18} - 14625603799 \beta_{17} + 7546784833 \beta_{16} + \cdots + 323480394991 \beta_1 $ 4133127370b19 - 6706153730b18 - 14625603799b17 + 7546784833b16 + 1069502849b15 + 10062011262b14 - 16540861169b13 + 47634986315b12 - 27608851084b11 + 323480394991b1
$\nu^{14}$	$=$	$ 192305727682 \beta_{10} + 42041586420 \beta_{9} - 86343948877 \beta_{8} + 174342321267 \beta_{7} + \cdots + 12385457509095 $ 192305727682b10 + 42041586420b9 - 86343948877b8 + 174342321267b7 + 12386912163b6 + 57683689331b5 + 193680829526b4 - 150136152685b3 + 679772077596*b2 + 12385457509095
$\nu^{15}$	$=$	$ 424772952880 \beta_{19} - 634274176955 \beta_{18} - 1364317536627 \beta_{17} + \cdots + 29838120025069 \beta_1 $ 424772952880b19 - 634274176955b18 - 1364317536627b17 + 687063104829b16 + 140551574495b15 + 953895397490b14 - 1664727705538b13 + 4708571979686b12 - 3028671861064b11 + 29838120025069b1
$\nu^{16}$	$=$	$ 18800707939788 \beta_{10} + 4418006409012 \beta_{9} - 7829243835999 \beta_{8} + 18375635932833 \beta_{7} + \cdots + 11\!\cdots\!92 $ 18800707939788b10 + 4418006409012b9 - 7829243835999b8 + 18375635932833b7 + 2200952010213b6 + 6003093593910b5 + 19004030847135b4 - 14067150745548b3 + 63768779192613*b2 + 1138892974309792
$\nu^{17}$	$=$	$ 42838610021250 \beta_{19} - 59066789623134 \beta_{18} - 127025221912521 \beta_{17} + \cdots + 27\!\cdots\!53 \beta_1 $ 42838610021250b19 - 59066789623134b18 - 127025221912521b17 + 62768951613273b16 + 17337405756717b15 + 90292435187106b14 - 165829083579015b13 + 462950651052819b12 - 324911435770008b11 + 2774849953289353b1
$\nu^{18}$	$=$	$ 18\!\cdots\!02 \beta_{10} + 452772962706756 \beta_{9} - 707265194294595 \beta_{8} + \cdots + 10\!\cdots\!77 $ 1824946970670702b10 + 452772962706756b9 - 707265194294595b8 + 1907397978029913b7 + 300959823224493b6 + 615246346457673b5 + 1853704897634802b4 - 1306603363922595b3 + 5996426710400680*b2 + 105636383438027677
$\nu^{19}$	$=$	$ 42\!\cdots\!36 \beta_{19} + \cdots + 25\!\cdots\!43 \beta_1 $ 4269911270485836b19 - 5454564857988489b18 - 11828826964129093b17 + 5755344522499543b16 + 2036196914393361b15 + 8546267942210554b14 - 16404561662781226b13 + 45347810821294254b12 - 34260128105602128b11 + 259536523193690143b1

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

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} $

1.1

−9.82712
−9.33099
−7.77654
−7.54446
−6.73523
−6.19234
−2.52045
−2.30645
−0.978857
−0.792466
0.792466
0.978857
2.30645
2.52045
6.19234
6.73523
7.54446
7.77654
9.33099
9.82712

2.00000

−9.82712

4.00000

−15.1246

−19.6542

8.00000

69.5723

−30.2493

1.2

2.00000

−9.33099

4.00000

−5.33673

−18.6620

8.00000

60.0673

−10.6735

1.3

2.00000

−7.77654

4.00000

11.0355

−15.5531

8.00000

33.4745

22.0710

1.4

2.00000

−7.54446

4.00000

−16.7547

−15.0889

8.00000

29.9188

−33.5094

1.5

2.00000

−6.73523

4.00000

−11.4128

−13.4705

8.00000

18.3633

−22.8256

1.6

2.00000

−6.19234

4.00000

15.4237

−12.3847

8.00000

11.3451

30.8474

1.7

2.00000

−2.52045

4.00000

1.00052

−5.04090

8.00000

−20.6473

2.00104

1.8

2.00000

−2.30645

4.00000

−15.2594

−4.61291

8.00000

−21.6803

−30.5187

1.9

2.00000

−0.978857

4.00000

19.5514

−1.95771

8.00000

−26.0418

39.1028

1.10

2.00000

−0.792466

4.00000

−0.114160

−1.58493

8.00000

−26.3720

−0.228320

1.11

2.00000

0.792466

4.00000

0.114160

1.58493

8.00000

−26.3720

0.228320

1.12

2.00000

0.978857

4.00000

−19.5514

1.95771

8.00000

−26.0418

−39.1028

1.13

2.00000

2.30645

4.00000

15.2594

4.61291

8.00000

−21.6803

30.5187

1.14

2.00000

2.52045

4.00000

−1.00052

5.04090

8.00000

−20.6473

−2.00104

1.15

2.00000

6.19234

4.00000

−15.4237

12.3847

8.00000

11.3451

−30.8474

1.16

2.00000

6.73523

4.00000

11.4128

13.4705

8.00000

18.3633

22.8256

1.17

2.00000

7.54446

4.00000

16.7547

15.0889

8.00000

29.9188

33.5094

1.18

2.00000

7.77654

4.00000

−11.0355

15.5531

8.00000

33.4745

−22.0710

1.19

2.00000

9.33099

4.00000

5.33673

18.6620

8.00000

60.0673

10.6735

1.20

2.00000

9.82712

4.00000

15.1246

19.6542

8.00000

69.5723

30.2493

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$7$	$ +1 $
$23$	$ -1 $

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2254.4.a.bc	✓	20
7.b	odd	2	1	inner	2254.4.a.bc	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2254.4.a.bc	✓	20	1.a	even	1	1	trivial
2254.4.a.bc	✓	20	7.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{20} - 398 T_{3}^{18} + 65501 T_{3}^{16} - 5759932 T_{3}^{14} + 291300980 T_{3}^{12} + \cdots + 1023755428864 $ T3^20 - 398*T3^18 + 65501*T3^16 - 5759932*T3^14 + 291300980*T3^12 - 8455560298*T3^10 + 131887696352*T3^8 - 958301668480*T3^6 + 2985915503888*T3^4 - 3158776583552*T3^2 + 1023755428864 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2254))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 2)^{20} $ (T - 2)^20
$3$	$ T^{20} + \cdots + 1023755428864 $ T^20 - 398T^18 + 65501T^16 - 5759932T^14 + 291300980T^12 - 8455560298T^10 + 131887696352T^8 - 958301668480T^6 + 2985915503888T^4 - 3158776583552*T^2 + 1023755428864
$5$	$ T^{20} + \cdots + 80\!\cdots\!44 $ T^20 - 1644T^18 + 1141032T^16 - 434782034T^14 + 98808318676T^12 - 13568917447008T^10 + 1080310052257872T^8 - 44111373997993120T^6 + 657932327737079552T^4 - 623487438034335744*T^2 + 8013922068742144
$7$	$ T^{20} $ T^20
$11$	$ (T^{10} + \cdots + 22\!\cdots\!28)^{2} $ (T^10 - 108T^9 - 2794T^8 + 642318T^7 - 9872364T^6 - 1015638952T^5 + 28522536384T^4 + 457592196000T^3 - 17874066854208T^2 - 14964432996224*T + 2220935560329728)^2
$13$	$ T^{20} + \cdots + 20\!\cdots\!84 $ T^20 - 28988T^18 + 348845377T^16 - 2261857074688T^14 + 8572518523344776T^12 - 19284029650238647584T^10 + 25075813581398145339664T^8 - 17654144251350646169521536T^6 + 5854001595534101547732661248T^4 - 654690171953458453586868273152*T^2 + 20337422407201494203782669533184
$17$	$ T^{20} + \cdots + 69\!\cdots\!96 $ T^20 - 67426T^18 + 1906224432T^16 - 29365380338602T^14 + 267928755880218108T^12 - 1474456305441944072104T^10 + 4794733787970428937888064T^8 - 8796390224264977950788427776T^6 + 8632816159818821123915953033216T^4 - 4069779318018384425684640624443392*T^2 + 694429581320740780613207521462583296
$19$	$ T^{20} + \cdots + 13\!\cdots\!76 $ T^20 - 69416T^18 + 2062548472T^16 - 34414486693312T^14 + 354635703431072528T^12 - 2330984269738459931392T^10 + 9700880728996036476985856T^8 - 24324291705087036425113350656T^6 + 32535932915975597052425886896128T^4 - 17078631311923842190371045051334656*T^2 + 1385300620930716693532395669396914176
$23$	$ (T - 23)^{20} $ (T - 23)^20
$29$	$ (T^{10} + \cdots + 72\!\cdots\!96)^{2} $ (T^10 - 400T^9 - 41635T^8 + 35669028T^7 - 2634098136T^6 - 696868785214T^5 + 94739847605132T^4 + 1828736871395672T^3 - 613905959979735088T^2 + 8102363100852644352*T + 728531648254327274496)^2
$31$	$ T^{20} + \cdots + 10\!\cdots\!96 $ T^20 - 316340T^18 + 40846302857T^16 - 2797774611019874T^14 + 109978281721989290640T^12 - 2487890397922316852922802T^10 + 30298780795070548045273849304T^8 - 163357444106941992999255642926312T^6 + 196434723731963977487677065796857616T^4 - 83239655197154822925062376895889166144*T^2 + 10771516025190766119582922345325101394496
$37$	$ (T^{10} + \cdots + 47\!\cdots\!84)^{2} $ (T^10 - 276T^9 - 153360T^8 + 42334268T^7 + 6106723536T^6 - 1544902639408T^5 - 107255095257440T^4 + 17269877659055872T^3 + 630229676053047168T^2 - 48178853885595678720*T + 475987266652446889984)^2
$41$	$ T^{20} + \cdots + 36\!\cdots\!00 $ T^20 - 786108T^18 + 245573542889T^16 - 39597794183753584T^14 + 3592944521202310227056T^12 - 187987055652678096209935552T^10 + 5550958360434772556581177784544T^8 - 85453253855365213298226532835227904T^6 + 564935777020105073384597869376244319744T^4 - 852911293719787203979634037312224275200000*T^2 + 366484859377377603732547981709379043220640000
$43$	$ (T^{10} + \cdots + 20\!\cdots\!44)^{2} $ (T^10 - 406T^9 - 223588T^8 + 120178196T^7 - 1427189968T^6 - 5715826935440T^5 + 661368093210880T^4 + 15900633422843648T^3 - 4443068301742864384T^2 + 104028348845822951424*T + 2053801899893136031744)^2
$47$	$ T^{20} + \cdots + 21\!\cdots\!96 $ T^20 - 654496T^18 + 121197788437T^16 - 9070249311645278T^14 + 343235688840636872220T^12 - 7282337276063638707172658T^10 + 89561170157552753728960995336T^8 - 625229159859654388496164660058824T^6 + 2263671097743809347295095247373585424T^4 - 3274987745878579605197653126322198527680*T^2 + 21562328556656713064496383277225803692096
$53$	$ (T^{10} + \cdots - 14\!\cdots\!64)^{2} $ (T^10 - 304T^9 - 586980T^8 + 168843800T^7 + 109692234464T^6 - 28653209621728T^5 - 6475857619124608T^4 + 1369401720082254848T^3 + 27457116059203808256T^2 - 1249160614054054109184*T - 14486817670637418020864)^2
$59$	$ T^{20} + \cdots + 85\!\cdots\!00 $ T^20 - 1604954T^18 + 948179186948T^16 - 264462107135893586T^14 + 39848851424807779549512T^12 - 3433427829077653891101073864T^10 + 169391073349759057792537970866128T^8 - 4505280927466616834895199375755838912T^6 + 54248228848350022313580547231781029627456T^4 - 171100612986947531281517966977147743312064000*T^2 + 85780875899791173143849078976326019132828160000
$61$	$ T^{20} + \cdots + 28\!\cdots\!36 $ T^20 - 1469608T^18 + 798641334420T^16 - 206628822100014442T^14 + 27415101373329886693252T^12 - 1924962286712586379362342784T^10 + 70239950887067357690231072872784T^8 - 1252897637567069250738668986134312480T^6 + 10678666733347182048205619435465380081408T^4 - 37885830747333386130310243792276497292702208*T^2 + 28948171160377095776469022435423729772385550336
$67$	$ (T^{10} + \cdots + 23\!\cdots\!08)^{2} $ (T^10 - 1038T^9 - 910366T^8 + 986761378T^7 + 270900603572T^6 - 322294197457624T^5 - 21220723708504096T^4 + 42522537683685810560T^3 - 2152898454168230035968T^2 - 1928105598935041290649600*T + 239531886872509212476407808)^2
$71$	$ (T^{10} + \cdots - 18\!\cdots\!00)^{2} $ (T^10 - 1762T^9 - 299859T^8 + 1852695268T^7 - 574228412256T^6 - 450032847446952T^5 + 261739233226322896T^4 - 17878013642427552768T^3 - 13246107147399600157824T^2 + 3004032822134220365175680*T - 188114056400132938228966400)^2
$73$	$ T^{20} + \cdots + 83\!\cdots\!24 $ T^20 - 4219084T^18 + 7422619528729T^16 - 7139708084516858992T^14 + 4133795599471077930466960T^12 - 1491781484437071864933593379520T^10 + 335049231430898925546317062569269088T^8 - 45188979941012595842392385134017790271744T^6 + 3357422019322323628770904399610063798037408768T^4 - 112527241063119392973551265446501810683654250735616*T^2 + 837874731940401509987589391730724460872186118106093824
$79$	$ (T^{10} + \cdots + 85\!\cdots\!00)^{2} $ (T^10 - 726T^9 - 4010154T^8 + 3069780066T^7 + 5490480373848T^6 - 4468674343937800T^5 - 2871487600268813408T^4 + 2656797793025832651968T^3 + 307547680330268680128512T^2 - 533826636737444873771274240*T + 85184230871865688759357952000)^2
$83$	$ T^{20} + \cdots + 20\!\cdots\!04 $ T^20 - 3584840T^18 + 4668540173336T^16 - 2932083107682448160T^14 + 958823685200889269733904T^12 - 162532695440134924444773356032T^10 + 13516317861219563607652406417301504T^8 - 481300405734994946294080701933843728384T^6 + 6201540299350358848586092420609399018684416T^4 - 7356296231747106974536796866397914091957518336*T^2 + 2014612339063829654420391748030762824452961992704
$89$	$ T^{20} + \cdots + 81\!\cdots\!00 $ T^20 - 7520518T^18 + 22843240761924T^16 - 36234543910137414594T^14 + 32208656266324278235639228T^12 - 15961192916527755313094134330664T^10 + 4183093367274243528223721372054735552T^8 - 553197718539340992184810696366161535662592T^6 + 35162088311637075372991951882338829154411167744T^4 - 962534650857495222281233343567306495926242407219200*T^2 + 8165724803838970322334264062009238557231703654400000000
$97$	$ T^{20} + \cdots + 41\!\cdots\!84 $ T^20 - 9646868T^18 + 38709897606800T^16 - 84797226274850517434T^14 + 111809339039541917954040764T^12 - 92141182175677716271580154443176T^10 + 47587014502411936741555337776672635200T^8 - 14936026786171606170141602150745270423362688T^6 + 2628057384051796596269312466811022693926654312448T^4 - 213798570467015840854478980218073350206909532472934400*T^2 + 4177696151196683060314041053067772654230994584358012125184
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\( q + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + (\beta_{12} + \beta_1) q^{5} + 2 \beta_1 q^{6} + 8 q^{8} + (\beta_{2} + 13) q^{9} + (2 \beta_{12} + 2 \beta_1) q^{10} + (\beta_{3} + 11) q^{11} + 4 \beta_1 q^{12}+ \cdots + (2 \beta_{10} + 7 \beta_{8} + \cdots + 372) q^{99}+O(q^{100}) \) q + 2 * q^2 + b1 * q^3 + 4 * q^4 + (b12 + b1) * q^5 + 2b1 q^6 + 8 * q^8 + (b2 + 13) * q^9 + (2b12 + 2b1) * q^10 + (b3 + 11) * q^11 + 4b1 q^12 + b13 * q^13 + (b4 + b2 + 24) * q^15 + 16 * q^16 + (-b18 + b12 + 3b1) q^17 + (2b2 + 26) q^18 + (b16 - b14 + b13 + b1) * q^19 + (4b12 + 4b1) * q^20 + (2b3 + 22) q^22 + 23 * q^23 + 8b1 q^24 + (b6 + b5 + b3 + 40) * q^25 + 2b13 q^26 + (-b17 + b16 + b14 - b13 + 3b12 + 16b1) * q^27 + (-b8 + b7 - b6 + 2b2 + 40) q^29 + (2b4 + 2b2 + 48) * q^30 + (b17 - 2b16 - b15 + 2b12 + b11 + 4b1) q^31 + 32 * q^32 + (b18 - b16 + 2b15 - b13 + 4b12 - 2b11 + 18b1) * q^33 + (-2b18 + 2b12 + 6b1) q^34 + (4b2 + 52) q^36 + (b10 + b9 + 2b8 + b7 - 2b3 + 27) * q^37 + (2b16 - 2b14 + 2b13 + 2b1) * q^38 + (-3b10 - 2b9 - b8 - 5b7 - b6 - 3b5 - 3b4 + 2b3 - 5b2 - 5) q^39 + (8b12 + 8b1) * q^40 + (-b19 + 2b18 + 2b17 - b16 - b15 + 2b14 + 8b12 + 5b11 + 6b1) * q^41 + (2b10 + b9 + 2b8 + 2b7 + b6 - b4 - 3b3 + b2 + 40) * q^43 + (4b3 + 44) q^44 + (b19 + b18 - 3b17 - b15 - b14 - 4b13 + 9b12 - 4b11 + 37b1) q^45 + 46 * q^46 + (b19 + b18 + 2b17 - b15 + b14 - b11 + 10b1) * q^47 + 16b1 q^48 + (2b6 + 2b5 + 2b3 + 80) q^50 + (3b10 + 2b9 - 3b8 - b7 - b6 - 5b3 + 5b2 + 86) q^51 + 4b13 q^52 + (b10 - b9 + 2b8 + 4b7 - 2b6 - 2b2 + 28) * q^53 + (-2b17 + 2b16 + 2b14 - 2b13 + 6b12 + 32b1) * q^54 + (b18 - 2b17 - 4b16 + b15 - 3b13 + 23b12 - 4b11 + 26b1) * q^55 + (-3b10 - 2b9 + b8 - 4b7 + b6 - 4b5 - 3b4 - b3 - 4b2 + 27) * q^57 + (-2b8 + 2b7 - 2b6 + 4b2 + 80) * q^58 + (-2b19 + b18 + b17 + 2b15 - b14 - 3b12 + 6b11 + 25b1) q^59 + (4b4 + 4b2 + 96) * q^60 + (b19 - 2b18 + b17 + 2b16 + b14 - 3b13 - 2b12 - 2b11 + 21b1) * q^61 + (2b17 - 4b16 - 2b15 + 4b12 + 2b11 + 8b1) * q^62 + 64 * q^64 + (-4b10 + 2b9 - b8 - 8b7 - 4b4 + 2b3 - 13b2 + 59) * q^65 + (2b18 - 2b16 + 4b15 - 2b13 + 8b12 - 4b11 + 36b1) q^66 + (-b10 - b9 + 3b7 + 4b5 - b4 - 2b3 - 5b2 + 101) * q^67 + (-4b18 + 4b12 + 12b1) q^68 + 23b1 q^69 + (2b10 + 2b8 + 10b7 + 6b6 + 5b5 + 2b4 - 5b3 + 3b2 + 177) * q^71 + (8b2 + 104) q^72 + (-2b19 + 3b18 + 3b17 + b16 - 3b15 + 4b14 - 20b12 + 15b11 + 29b1) * q^73 + (2b10 + 2b9 + 4b8 + 2b7 - 4b3 + 54) q^74 + (3b18 + 2b17 - 5b16 + 2b15 + b14 - 7b13 + 12b12 - 6b11 + 40b1) * q^75 + (4b16 - 4b14 + 4b13 + 4b1) * q^76 + (-6b10 - 4b9 - 2b8 - 10b7 - 2b6 - 6b5 - 6b4 + 4b3 - 10b2 - 10) q^78 + (-8b10 - b7 - 8b6 - 3b4 + 14b3 - 13b2 + 67) q^79 + (16b12 + 16b1) * q^80 + (10b10 - 7b8 + 6b7 - 3b6 + 2b5 + 11b4 - 10b3 + 16b2 + 252) * q^81 + (-2b19 + 4b18 + 4b17 - 2b16 - 2b15 + 4b14 + 16b12 + 10b11 + 12b1) q^82 + (2b19 - 4b18 - 4b17 + 4b16 + b15 + b13 + 9b12 - 2b11 + 28b1) q^83 + (4b10 - 2b9 - 11b8 + 8b6 + b4 - 2b3 - 6b2 + 233) * q^85 + (4b10 + 2b9 + 4b8 + 4b7 + 2b6 - 2b4 - 6b3 + 2b2 + 80) * q^86 + (-b19 - 5b18 - 6b17 + 3b16 + 4b15 + 7b14 - b13 - 19b12 + 79b1) q^87 + (8b3 + 88) q^88 + (2b19 - 3b18 - 10b17 - 2b15 - 11b14 + 5b12 - 20b11 - 6b1) * q^89 + (2b19 + 2b18 - 6b17 - 2b15 - 2b14 - 8b13 + 18b12 - 8b11 + 74b1) q^90 + 92 * q^92 + (-9b10 + 2b9 + 5b8 - 15b7 - b6 - b5 - 4b4 + 2b3 - 18b2 + 141) q^93 + (2b19 + 2b18 + 4b17 - 2b15 + 2b14 - 2b11 + 20b1) q^94 + (4b10 + 2b9 + 7b8 - 5b7 + 3b6 + 3b5 - 5b4 - 15b3 - 4b2 + 75) q^95 + 32b1 q^96 + (6b17 + 10b16 + 3b15 - 9b14 + 5b13 - 4b12 + 8b11 - 20b1) * q^97 + (2b10 + 7b8 + 30b7 + 6b6 + 10b5 + 10b4 - b3 + 15b2 + 372) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 40 q^{2} + 80 q^{4} + 160 q^{8} + 256 q^{9} + 216 q^{11} + 468 q^{15} + 320 q^{16} + 512 q^{18} + 432 q^{22} + 460 q^{23} + 788 q^{25} + 800 q^{29} + 936 q^{30} + 640 q^{32} + 1024 q^{36} + 552 q^{37}+ \cdots + 7396 q^{99}+O(q^{100}) \) 20 * q + 40 * q^2 + 80 * q^4 + 160 * q^8 + 256 * q^9 + 216 * q^11 + 468 * q^15 + 320 * q^16 + 512 * q^18 + 432 * q^22 + 460 * q^23 + 788 * q^25 + 800 * q^29 + 936 * q^30 + 640 * q^32 + 1024 * q^36 + 552 * q^37 - 60 * q^39 + 812 * q^43 + 864 * q^44 + 920 * q^46 + 1576 * q^50 + 1692 * q^51 + 608 * q^53 + 584 * q^57 + 1600 * q^58 + 1872 * q^60 + 1280 * q^64 + 1228 * q^65 + 2076 * q^67 + 3524 * q^71 + 2048 * q^72 + 1104 * q^74 - 120 * q^78 + 1452 * q^79 + 4908 * q^81 + 4568 * q^85 + 1624 * q^86 + 1728 * q^88 + 1840 * q^92 + 2912 * q^93 + 1576 * q^95 + 7396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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	2254.4.a.bc

Level	$2254$
Weight	$4$
Character orbit	2254.a
Self dual	yes
Analytic conductor	$132.990$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$

Self dual:	yes
Analytic conductor:	\(132.990305153\)
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:	\( x^{20} - 398 x^{18} + 65501 x^{16} - 5759932 x^{14} + 291300980 x^{12} - 8455560298 x^{10} + \cdots + 1023755428864 \) x^20 - 398x^18 + 65501x^16 - 5759932x^14 + 291300980x^12 - 8455560298x^10 + 131887696352x^8 - 958301668480x^6 + 2985915503888x^4 - 3158776583552*x^2 + 1023755428864
Coefficient ring:	\(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index:	\( 2^{9}\cdot 5^{2}\cdot 7^{4} \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 40 \) v^2 - 40
\(\beta_{3}\)	\(=\)	\( ( 15\!\cdots\!31 \nu^{18} + \cdots - 32\!\cdots\!36 ) / 15\!\cdots\!76 \) (1537482213375960032831v^18 - 617523398820884856461226v^16 + 102585153514374425472726683v^14 - 9100696502018889530395368604v^12 + 463343562641324513556103243668v^10 - 13469512697865438826412019884038v^8 + 207840455174712872261034523410128v^6 - 1449465000097922351051751093292848v^4 + 4204170783043650494627585446269680*v^2 - 3210498936803041619705204715018336) / 15167779790642126608042416823776
\(\beta_{4}\)	\(=\)	\( ( - 62\!\cdots\!93 \nu^{18} + \cdots + 43\!\cdots\!28 ) / 50\!\cdots\!92 \) (-624234491463937099393v^18 + 243156374368768524494550v^16 - 39042751383974944776361333v^14 + 3335534959710462850939967540v^12 - 162719547822306914099665564668v^10 + 4486103480781023169053580456458v^8 - 63831009503464739634670400964304v^6 + 373926378889209343850942483024208v^4 - 779433490087580448062266123967056*v^2 + 435318635874590837703408002694528) / 5055926596880708869347472274592
\(\beta_{5}\)	\(=\)	\( ( 27\!\cdots\!23 \nu^{18} + \cdots - 60\!\cdots\!00 ) / 11\!\cdots\!88 \) (27545345746738795123v^18 - 11309226962995570118086v^16 + 1929203788912140304306759v^14 - 176845193499089037153294360v^12 + 9390654147000991393058694164v^10 - 289137445117201246368617284790v^8 + 4858113566780531583346305231240v^6 - 38597078357287653094105936397056v^4 + 116285542262852875802747730860128*v^2 - 60364576962255363594989237711200) / 112077190571247240946126232688
\(\beta_{6}\)	\(=\)	\( ( 47\!\cdots\!41 \nu^{18} + \cdots - 74\!\cdots\!36 ) / 15\!\cdots\!76 \) (4729442795690863252941v^18 - 1869060187427867442371158v^16 + 305230559464691279739978417v^14 - 26625101207984510910946254364v^12 + 1336199803050585506042431487036v^10 - 38553427862431500534057812197890v^8 + 599938049743544743266242588744288v^6 - 4351788387200967026965854026394928v^4 + 12484673399419159323392287398767952*v^2 - 7487356528754892161512769041092736) / 15167779790642126608042416823776
\(\beta_{7}\)	\(=\)	\( ( - 78\!\cdots\!73 \nu^{18} + \cdots + 11\!\cdots\!72 ) / 22\!\cdots\!64 \) (-7812998156576520117173v^18 + 3115609158968471858607170v^16 - 513458553300969199313822585v^14 + 45163809004512674528047692216v^12 - 2279966430898975090306613606596v^10 + 65790943444434027172817140968850v^8 - 1010472306474435168504094375889688v^6 + 7022834177223809685741337909336016v^4 - 19189596237840626396433947353501904*v^2 + 11462663464145734220601465769440272) / 22751669685963189912063625235664
\(\beta_{8}\)	\(=\)	\( ( 29\!\cdots\!47 \nu^{18} + \cdots - 11\!\cdots\!88 ) / 75\!\cdots\!88 \) (2914311677918031545747v^18 - 1149501421550930357034850v^16 + 186534139915217499115813199v^14 - 16036947521921832033692531516v^12 + 781004609218164176269935810980v^10 - 21181495213361775672201520901614v^8 + 287371170105249482304297838487968v^6 - 1459869086871078135164880335345152v^4 + 2224097395863977196904174265823296*v^2 - 1103718896127567641567743530624688) / 7583889895321063304021208411888
\(\beta_{9}\)	\(=\)	\( ( - 48\!\cdots\!67 \nu^{18} + \cdots + 40\!\cdots\!92 ) / 65\!\cdots\!04 \) (-4848409643722412828467v^18 + 1916249594041221183889546v^16 - 312468892591451419826143831v^14 + 27129805893894443848906067700v^12 - 1346684698779857150096604000812v^10 + 37933325337408345803152863386078v^8 - 559548242862769069173480021893952v^6 + 3579458506896589974231528436314976v^4 - 8552006831434195533496824677190064*v^2 + 4028651659949801997109647564661792) / 6500477053132339974875321495904
\(\beta_{10}\)	\(=\)	\( ( 86\!\cdots\!22 \nu^{18} + \cdots - 72\!\cdots\!84 ) / 11\!\cdots\!32 \) (8606972845563750194722v^18 - 3397575050819844274075039v^16 + 552982126677027031055857882v^14 - 47869599344934296198415443595v^12 + 2364297998292229523319621981866v^10 - 65988124778860086389753442643400v^8 + 955109866241605167855300764396814v^6 - 5846773445188989900726785156184604v^4 + 13512516521215339453498342072554976*v^2 - 7271565261244582526694746820217384) / 11375834842981594956031812617832
\(\beta_{11}\)	\(=\)	\( ( 15\!\cdots\!95 \nu^{19} + \cdots - 23\!\cdots\!12 \nu ) / 11\!\cdots\!92 \) (1514060796158695v^19 - 598880713785092938v^17 + 97559744276608529371v^15 - 8433438236999579481324v^13 + 414034620720498520219340v^11 - 11377376114972455176624134v^9 + 158538485825105666390700720v^7 - 879441400756528777024652032v^5 + 1961535875530242021580623088v^3 - 2391581223798041403668806912v) / 117598010450800593011496192
\(\beta_{12}\)	\(=\)	\( ( - 12\!\cdots\!19 \nu^{19} + \cdots + 11\!\cdots\!72 \nu ) / 18\!\cdots\!12 \) (-12510271934868059940855819v^19 + 4956530892493947025436950514v^17 - 810648623261602974784861941519v^15 + 70647466782337135626814946115020v^13 - 3523721583389464745888475446377180v^11 + 99901325109547126856279176433831214v^9 - 1487841110845323157426007151909706000v^7 + 9682017108905377898544331857570795776v^5 - 23842411300636999065699832700779143984v^3 + 11351545443924191972684846468165353472v) / 182700963504881295702740258114656512
\(\beta_{13}\)	\(=\)	\( ( 29\!\cdots\!31 \nu^{19} + \cdots - 64\!\cdots\!52 \nu ) / 41\!\cdots\!52 \) (29155576088520172756250731v^19 - 11571588066098111305698136150v^17 + 1898840463502065456573683103391v^15 - 166503484485667626864413023893504v^13 + 8400997128341418491807502944234276v^11 - 243611284002145702215390016866653678v^9 + 3807845657637206665472906021879515704v^7 - 27882592697686299119014367611620404464v^5 + 85732961289888401555928113734267715344v^3 - 64256081479506150594329584588359386752v) / 411077167885982915331165580757977152
\(\beta_{14}\)	\(=\)	\( ( 12\!\cdots\!95 \nu^{19} + \cdots - 31\!\cdots\!60 \nu ) / 82\!\cdots\!04 \) (127425294573475705706155295v^19 - 50867990931567088260531239546v^17 + 8408760711221867587745154880307v^15 - 744371552028841694978021437957932v^13 + 38034077747775411852973119981992812v^11 - 1122327630805438294609910760795311926v^9 + 17993996114363799529728463562215040880v^7 - 136562468647940801286548627320919345408v^5 + 421060343260373856100098912313305940592v^3 - 316952769965000031927210646906236421760v) / 822154335771965830662331161515954304
\(\beta_{15}\)	\(=\)	\( ( - 14\!\cdots\!75 \nu^{19} + \cdots + 18\!\cdots\!80 \nu ) / 82\!\cdots\!04 \) (-149277393928940532574390175v^19 + 59351883914836072505852961338v^17 - 9755716231758623539653384164915v^15 + 856568614136195821555682125862124v^13 - 43233399479998302964136482424936812v^11 + 1251058580093387232750120246939197878v^9 - 19377140067620358645492170192161838064v^7 + 137050485288107551416102667134463271168v^5 - 371245641071980966127817928174594999280v^3 + 181288634756628252626394862407082446080v) / 822154335771965830662331161515954304
\(\beta_{16}\)	\(=\)	\( ( - 37\!\cdots\!13 \nu^{19} + \cdots + 56\!\cdots\!84 \nu ) / 16\!\cdots\!08 \) (-373269813889668782044524313v^19 + 149076285380769954890597795494v^17 - 24636771004323032785124125921477v^15 + 2177478150360579563717321729995716v^13 - 110812437282414329712488865052806644v^11 + 3241592271789694325576136989184362810v^9 - 51019885133974135189951917329180986032v^7 + 371595404150688726802792697758069365760v^5 - 1064755915655261733905340210793414296592v^3 + 561285184303823708375624600489232553984v) / 1644308671543931661324662323031908608
\(\beta_{17}\)	\(=\)	\( ( - 35\!\cdots\!35 \nu^{19} + \cdots + 37\!\cdots\!36 \nu ) / 10\!\cdots\!88 \) (-35801179458639730003770235v^19 + 14215811867460299579945345305v^17 - 2331382766496927484694988812215v^15 + 203889411710541833946433405922463v^13 - 10218047065734045507047592119871374v^11 + 291794870259073310646088331173362278v^9 - 4402186595538693002680550462114439318v^7 + 29463456224124845247903117107320236172v^5 - 75684780505175658397754686646746379560v^3 + 37874831517867290403358678892680962736v) / 102769291971495728832791395189494288
\(\beta_{18}\)	\(=\)	\( ( 99\!\cdots\!53 \nu^{19} + \cdots - 12\!\cdots\!56 \nu ) / 16\!\cdots\!08 \) (999847795917138456400952953v^19 - 396744834336802839710976924694v^17 + 65014106643475932325774123175653v^15 - 5680547664749538445600127149768500v^13 + 284387714464362705194643734440420052v^11 - 8111905078058638457270927616521154554v^9 + 122247419722793913953883669102940135824v^7 - 818416621810434039479324549931440348224v^5 + 2132105213822712375215189435637664622992v^3 - 1255983590891304868591724708072953363456v) / 1644308671543931661324662323031908608
\(\beta_{19}\)	\(=\)	\( ( 16\!\cdots\!47 \nu^{19} + \cdots - 20\!\cdots\!56 \nu ) / 16\!\cdots\!08 \) (1639473814007037923431836247v^19 - 649404825675824016324312719866v^17 + 106176069314592978834752007568747v^15 - 9249086036576110788221557112435580v^13 + 461085706511798078681944580065041740v^11 - 13067635125968633580915423193477151654v^9 + 194775037787612659209698849527997642832v^7 - 1277033632685175280872702093451770339328v^5 + 3289185533560281810236993849835378041584v^3 - 2027080109516318752671618862189290067456v) / 1644308671543931661324662323031908608

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 40 \) b2 + 40
\(\nu^{3}\)	\(=\)	\( -\beta_{17} + \beta_{16} + \beta_{14} - \beta_{13} + 3\beta_{12} + 70\beta_1 \) -b17 + b16 + b14 - b13 + 3b12 + 70b1
\(\nu^{4}\)	\(=\)	\( 10\beta_{10} - 7\beta_{8} + 6\beta_{7} - 3\beta_{6} + 2\beta_{5} + 11\beta_{4} - 10\beta_{3} + 97\beta_{2} + 2763 \) 10b10 - 7b8 + 6b7 - 3b6 + 2b5 + 11b4 - 10b3 + 97b2 + 2763
\(\nu^{5}\)	\(=\)	\( 16 \beta_{19} - 38 \beta_{18} - 151 \beta_{17} + 106 \beta_{16} + 2 \beta_{15} + 111 \beta_{14} + \cdots + 5470 \beta_1 \) 16b19 - 38b18 - 151b17 + 106b16 + 2b15 + 111b14 - 125b13 + 410b12 - 100b11 + 5470b1
\(\nu^{6}\)	\(=\)	\( 1540 \beta_{10} + 120 \beta_{9} - 982 \beta_{8} + 972 \beta_{7} - 327 \beta_{6} + 323 \beta_{5} + \cdots + 214011 \) 1540b10 + 120b9 - 982b8 + 972b7 - 327b6 + 323b5 + 1616b4 - 1405b3 + 9015*b2 + 214011
\(\nu^{7}\)	\(=\)	\( 2746 \beta_{19} - 5861 \beta_{18} - 16497 \beta_{17} + 9996 \beta_{16} + 320 \beta_{15} + \cdots + 455947 \beta_1 \) 2746b19 - 5861b18 - 16497b17 + 9996b16 + 320b15 + 11246b14 - 14254b13 + 45545b12 - 16090b11 + 455947b1
\(\nu^{8}\)	\(=\)	\( 180420 \beta_{10} + 23712 \beta_{9} - 104493 \beta_{8} + 124773 \beta_{7} - 23661 \beta_{6} + \cdots + 17708482 \) 180420b10 + 23712b9 - 104493b8 + 124773b7 - 23661b6 + 41964b5 + 184473b4 - 157632b3 + 836853*b2 + 17708482
\(\nu^{9}\)	\(=\)	\( 346692 \beta_{19} - 671808 \beta_{18} - 1633971 \beta_{17} + 915081 \beta_{16} + 49491 \beta_{15} + \cdots + 39667441 \beta_1 \) 346692b19 - 671808b18 - 1633971b17 + 915081b16 + 49491b15 + 1100040b14 - 1545915b13 + 4730787b12 - 2064924b11 + 39667441b1
\(\nu^{10}\)	\(=\)	\( 19151226 \beta_{10} + 3247656 \beta_{9} - 10131027 \beta_{8} + 14579223 \beta_{7} - 1122039 \beta_{6} + \cdots + 1531670131 \) 19151226b10 + 3247656b9 - 10131027b8 + 14579223b7 - 1122039b6 + 4911213b5 + 19342914b4 - 16148367b3 + 77864452*b2 + 1531670131
\(\nu^{11}\)	\(=\)	\( 38956200 \beta_{19} - 68929893 \beta_{18} - 155810101 \beta_{17} + 83110309 \beta_{16} + \cdots + 3549472489 \beta_1 \) 38956200b19 - 68929893b18 - 155810101b17 + 83110309b16 + 7539531b15 + 105688480b14 - 161814670b13 + 478004724b12 - 244189056b11 + 3549472489b1
\(\nu^{12}\)	\(=\)	\( 1941902632 \beta_{10} + 383606124 \beta_{9} - 943983031 \beta_{8} + 1619438781 \beta_{7} + \cdots + 136417048980 \) 1941902632b10 + 383606124b9 - 943983031b8 + 1619438781b7 + 12031107b6 + 541820894b5 + 1953802277b4 - 1577942632b3 + 7265211769*b2 + 136417048980
\(\nu^{13}\)	\(=\)	\( 4133127370 \beta_{19} - 6706153730 \beta_{18} - 14625603799 \beta_{17} + 7546784833 \beta_{16} + \cdots + 323480394991 \beta_1 \) 4133127370b19 - 6706153730b18 - 14625603799b17 + 7546784833b16 + 1069502849b15 + 10062011262b14 - 16540861169b13 + 47634986315b12 - 27608851084b11 + 323480394991b1
\(\nu^{14}\)	\(=\)	\( 192305727682 \beta_{10} + 42041586420 \beta_{9} - 86343948877 \beta_{8} + 174342321267 \beta_{7} + \cdots + 12385457509095 \) 192305727682b10 + 42041586420b9 - 86343948877b8 + 174342321267b7 + 12386912163b6 + 57683689331b5 + 193680829526b4 - 150136152685b3 + 679772077596*b2 + 12385457509095
\(\nu^{15}\)	\(=\)	\( 424772952880 \beta_{19} - 634274176955 \beta_{18} - 1364317536627 \beta_{17} + \cdots + 29838120025069 \beta_1 \) 424772952880b19 - 634274176955b18 - 1364317536627b17 + 687063104829b16 + 140551574495b15 + 953895397490b14 - 1664727705538b13 + 4708571979686b12 - 3028671861064b11 + 29838120025069b1
\(\nu^{16}\)	\(=\)	\( 18800707939788 \beta_{10} + 4418006409012 \beta_{9} - 7829243835999 \beta_{8} + 18375635932833 \beta_{7} + \cdots + 11\!\cdots\!92 \) 18800707939788b10 + 4418006409012b9 - 7829243835999b8 + 18375635932833b7 + 2200952010213b6 + 6003093593910b5 + 19004030847135b4 - 14067150745548b3 + 63768779192613*b2 + 1138892974309792
\(\nu^{17}\)	\(=\)	\( 42838610021250 \beta_{19} - 59066789623134 \beta_{18} - 127025221912521 \beta_{17} + \cdots + 27\!\cdots\!53 \beta_1 \) 42838610021250b19 - 59066789623134b18 - 127025221912521b17 + 62768951613273b16 + 17337405756717b15 + 90292435187106b14 - 165829083579015b13 + 462950651052819b12 - 324911435770008b11 + 2774849953289353b1
\(\nu^{18}\)	\(=\)	\( 18\!\cdots\!02 \beta_{10} + 452772962706756 \beta_{9} - 707265194294595 \beta_{8} + \cdots + 10\!\cdots\!77 \) 1824946970670702b10 + 452772962706756b9 - 707265194294595b8 + 1907397978029913b7 + 300959823224493b6 + 615246346457673b5 + 1853704897634802b4 - 1306603363922595b3 + 5996426710400680*b2 + 105636383438027677
\(\nu^{19}\)	\(=\)	\( 42\!\cdots\!36 \beta_{19} + \cdots + 25\!\cdots\!43 \beta_1 \) 4269911270485836b19 - 5454564857988489b18 - 11828826964129093b17 + 5755344522499543b16 + 2036196914393361b15 + 8546267942210554b14 - 16404561662781226b13 + 45347810821294254b12 - 34260128105602128b11 + 259536523193690143b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T - 2)^{20} \) (T - 2)^20
$3$	\( T^{20} + \cdots + 1023755428864 \) T^20 - 398T^18 + 65501T^16 - 5759932T^14 + 291300980T^12 - 8455560298T^10 + 131887696352T^8 - 958301668480T^6 + 2985915503888T^4 - 3158776583552*T^2 + 1023755428864
$5$	\( T^{20} + \cdots + 80\!\cdots\!44 \) T^20 - 1644T^18 + 1141032T^16 - 434782034T^14 + 98808318676T^12 - 13568917447008T^10 + 1080310052257872T^8 - 44111373997993120T^6 + 657932327737079552T^4 - 623487438034335744*T^2 + 8013922068742144
$7$	\( T^{20} \) T^20
$11$	\( (T^{10} + \cdots + 22\!\cdots\!28)^{2} \) (T^10 - 108T^9 - 2794T^8 + 642318T^7 - 9872364T^6 - 1015638952T^5 + 28522536384T^4 + 457592196000T^3 - 17874066854208T^2 - 14964432996224*T + 2220935560329728)^2
$13$	\( T^{20} + \cdots + 20\!\cdots\!84 \) T^20 - 28988T^18 + 348845377T^16 - 2261857074688T^14 + 8572518523344776T^12 - 19284029650238647584T^10 + 25075813581398145339664T^8 - 17654144251350646169521536T^6 + 5854001595534101547732661248T^4 - 654690171953458453586868273152*T^2 + 20337422407201494203782669533184
$17$	\( T^{20} + \cdots + 69\!\cdots\!96 \) T^20 - 67426T^18 + 1906224432T^16 - 29365380338602T^14 + 267928755880218108T^12 - 1474456305441944072104T^10 + 4794733787970428937888064T^8 - 8796390224264977950788427776T^6 + 8632816159818821123915953033216T^4 - 4069779318018384425684640624443392*T^2 + 694429581320740780613207521462583296
$19$	\( T^{20} + \cdots + 13\!\cdots\!76 \) T^20 - 69416T^18 + 2062548472T^16 - 34414486693312T^14 + 354635703431072528T^12 - 2330984269738459931392T^10 + 9700880728996036476985856T^8 - 24324291705087036425113350656T^6 + 32535932915975597052425886896128T^4 - 17078631311923842190371045051334656*T^2 + 1385300620930716693532395669396914176
$23$	\( (T - 23)^{20} \) (T - 23)^20
$29$	\( (T^{10} + \cdots + 72\!\cdots\!96)^{2} \) (T^10 - 400T^9 - 41635T^8 + 35669028T^7 - 2634098136T^6 - 696868785214T^5 + 94739847605132T^4 + 1828736871395672T^3 - 613905959979735088T^2 + 8102363100852644352*T + 728531648254327274496)^2
$31$	\( T^{20} + \cdots + 10\!\cdots\!96 \) T^20 - 316340T^18 + 40846302857T^16 - 2797774611019874T^14 + 109978281721989290640T^12 - 2487890397922316852922802T^10 + 30298780795070548045273849304T^8 - 163357444106941992999255642926312T^6 + 196434723731963977487677065796857616T^4 - 83239655197154822925062376895889166144*T^2 + 10771516025190766119582922345325101394496
$37$	\( (T^{10} + \cdots + 47\!\cdots\!84)^{2} \) (T^10 - 276T^9 - 153360T^8 + 42334268T^7 + 6106723536T^6 - 1544902639408T^5 - 107255095257440T^4 + 17269877659055872T^3 + 630229676053047168T^2 - 48178853885595678720*T + 475987266652446889984)^2
$41$	\( T^{20} + \cdots + 36\!\cdots\!00 \) T^20 - 786108T^18 + 245573542889T^16 - 39597794183753584T^14 + 3592944521202310227056T^12 - 187987055652678096209935552T^10 + 5550958360434772556581177784544T^8 - 85453253855365213298226532835227904T^6 + 564935777020105073384597869376244319744T^4 - 852911293719787203979634037312224275200000*T^2 + 366484859377377603732547981709379043220640000
$43$	\( (T^{10} + \cdots + 20\!\cdots\!44)^{2} \) (T^10 - 406T^9 - 223588T^8 + 120178196T^7 - 1427189968T^6 - 5715826935440T^5 + 661368093210880T^4 + 15900633422843648T^3 - 4443068301742864384T^2 + 104028348845822951424*T + 2053801899893136031744)^2
$47$	\( T^{20} + \cdots + 21\!\cdots\!96 \) T^20 - 654496T^18 + 121197788437T^16 - 9070249311645278T^14 + 343235688840636872220T^12 - 7282337276063638707172658T^10 + 89561170157552753728960995336T^8 - 625229159859654388496164660058824T^6 + 2263671097743809347295095247373585424T^4 - 3274987745878579605197653126322198527680*T^2 + 21562328556656713064496383277225803692096
$53$	\( (T^{10} + \cdots - 14\!\cdots\!64)^{2} \) (T^10 - 304T^9 - 586980T^8 + 168843800T^7 + 109692234464T^6 - 28653209621728T^5 - 6475857619124608T^4 + 1369401720082254848T^3 + 27457116059203808256T^2 - 1249160614054054109184*T - 14486817670637418020864)^2
$59$	\( T^{20} + \cdots + 85\!\cdots\!00 \) T^20 - 1604954T^18 + 948179186948T^16 - 264462107135893586T^14 + 39848851424807779549512T^12 - 3433427829077653891101073864T^10 + 169391073349759057792537970866128T^8 - 4505280927466616834895199375755838912T^6 + 54248228848350022313580547231781029627456T^4 - 171100612986947531281517966977147743312064000*T^2 + 85780875899791173143849078976326019132828160000
$61$	\( T^{20} + \cdots + 28\!\cdots\!36 \) T^20 - 1469608T^18 + 798641334420T^16 - 206628822100014442T^14 + 27415101373329886693252T^12 - 1924962286712586379362342784T^10 + 70239950887067357690231072872784T^8 - 1252897637567069250738668986134312480T^6 + 10678666733347182048205619435465380081408T^4 - 37885830747333386130310243792276497292702208*T^2 + 28948171160377095776469022435423729772385550336
$67$	\( (T^{10} + \cdots + 23\!\cdots\!08)^{2} \) (T^10 - 1038T^9 - 910366T^8 + 986761378T^7 + 270900603572T^6 - 322294197457624T^5 - 21220723708504096T^4 + 42522537683685810560T^3 - 2152898454168230035968T^2 - 1928105598935041290649600*T + 239531886872509212476407808)^2
$71$	\( (T^{10} + \cdots - 18\!\cdots\!00)^{2} \) (T^10 - 1762T^9 - 299859T^8 + 1852695268T^7 - 574228412256T^6 - 450032847446952T^5 + 261739233226322896T^4 - 17878013642427552768T^3 - 13246107147399600157824T^2 + 3004032822134220365175680*T - 188114056400132938228966400)^2
$73$	\( T^{20} + \cdots + 83\!\cdots\!24 \) T^20 - 4219084T^18 + 7422619528729T^16 - 7139708084516858992T^14 + 4133795599471077930466960T^12 - 1491781484437071864933593379520T^10 + 335049231430898925546317062569269088T^8 - 45188979941012595842392385134017790271744T^6 + 3357422019322323628770904399610063798037408768T^4 - 112527241063119392973551265446501810683654250735616*T^2 + 837874731940401509987589391730724460872186118106093824
$79$	\( (T^{10} + \cdots + 85\!\cdots\!00)^{2} \) (T^10 - 726T^9 - 4010154T^8 + 3069780066T^7 + 5490480373848T^6 - 4468674343937800T^5 - 2871487600268813408T^4 + 2656797793025832651968T^3 + 307547680330268680128512T^2 - 533826636737444873771274240*T + 85184230871865688759357952000)^2
$83$	\( T^{20} + \cdots + 20\!\cdots\!04 \) T^20 - 3584840T^18 + 4668540173336T^16 - 2932083107682448160T^14 + 958823685200889269733904T^12 - 162532695440134924444773356032T^10 + 13516317861219563607652406417301504T^8 - 481300405734994946294080701933843728384T^6 + 6201540299350358848586092420609399018684416T^4 - 7356296231747106974536796866397914091957518336*T^2 + 2014612339063829654420391748030762824452961992704
$89$	\( T^{20} + \cdots + 81\!\cdots\!00 \) T^20 - 7520518T^18 + 22843240761924T^16 - 36234543910137414594T^14 + 32208656266324278235639228T^12 - 15961192916527755313094134330664T^10 + 4183093367274243528223721372054735552T^8 - 553197718539340992184810696366161535662592T^6 + 35162088311637075372991951882338829154411167744T^4 - 962534650857495222281233343567306495926242407219200*T^2 + 8165724803838970322334264062009238557231703654400000000
$97$	\( T^{20} + \cdots + 41\!\cdots\!84 \) T^20 - 9646868T^18 + 38709897606800T^16 - 84797226274850517434T^14 + 111809339039541917954040764T^12 - 92141182175677716271580154443176T^10 + 47587014502411936741555337776672635200T^8 - 14936026786171606170141602150745270423362688T^6 + 2628057384051796596269312466811022693926654312448T^4 - 213798570467015840854478980218073350206909532472934400*T^2 + 4177696151196683060314041053067772654230994584358012125184
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\( p \)	Sign
\(2\)	\( -1 \)
\(7\)	\( +1 \)
\(23\)	\( -1 \)