LMFDB - Newform orbit 9898.2.a.bi

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9898,2,Mod(1,9898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9898, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9898.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9898 = 2 \cdot 7^{2} \cdot 101 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9898.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:	$79.0359279207$
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} - 43 x^{18} + 78 x^{17} + 774 x^{16} - 1240 x^{15} - 7575 x^{14} + 10337 x^{13} + \cdots + 3160 $ x^20 - 2x^19 - 43x^18 + 78x^17 + 774x^16 - 1240x^15 - 7575x^14 + 10337x^13 + 43740x^12 - 48326x^11 - 150887x^10 + 125274x^9 + 299785x^8 - 166516x^7 - 316036x^6 + 97790x^5 + 163407x^4 - 22723x^3 - 38018x^2 + 1644*x + 3160
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1414)
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 + q^{2} - \beta_1 q^{3} + q^{4} - \beta_{7} q^{5} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) $ q + q^2 - b1 * q^3 + q^4 - b7 * q^5 - b1 * q^6 + q^8 + (b2 + 2) * q^9	$ q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_{7} q^{5} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 2) q^{9} - \beta_{7} q^{10} + (\beta_{11} + 1) q^{11} - \beta_1 q^{12} - \beta_{14} q^{13} + ( - \beta_{6} + 1) q^{15} + q^{16} + \beta_{5} q^{17} + (\beta_{2} + 2) q^{18} + \beta_{8} q^{19} - \beta_{7} q^{20} + (\beta_{11} + 1) q^{22} + ( - \beta_{15} + 2) q^{23} - \beta_1 q^{24} + ( - \beta_{19} - \beta_{14} + \beta_1 + 2) q^{25} - \beta_{14} q^{26} + ( - \beta_{19} - \beta_{17} - \beta_{13} + \cdots - 2) q^{27}+ \cdots + (\beta_{14} + \beta_{13} + \beta_{12} + \cdots + 2) q^{99}+O(q^{100}) $ q + q^2 - b1 * q^3 + q^4 - b7 * q^5 - b1 * q^6 + q^8 + (b2 + 2) * q^9 - b7 * q^10 + (b11 + 1) * q^11 - b1 * q^12 - b14 * q^13 + (-b6 + 1) * q^15 + q^16 + b5 * q^17 + (b2 + 2) * q^18 + b8 * q^19 - b7 * q^20 + (b11 + 1) * q^22 + (-b15 + 2) * q^23 - b1 * q^24 + (-b19 - b14 + b1 + 2) * q^25 - b14 * q^26 + (-b19 - b17 - b13 - b12 + b11 - b7 + b6 + b5 - b4 - b1 - 2) * q^27 + (-b18 + b11 + b7 + b6 + b5) * q^29 + (-b6 + 1) * q^30 - b17 * q^31 + q^32 + (-b10 - b5 + b4 - b1 + 1) * q^33 + b5 * q^34 + (b2 + 2) * q^36 + (b18 + b15 + b14 - b7 - b3 + 3) * q^37 + b8 * q^38 + (b19 - b13 - b11 - b6 - b5 + 1) * q^39 - b7 * q^40 + (b19 + b18 + b17 + b16 + b15 + b14 + b12 - b11 - b8 - b6 + b4 - b3 - b2 - b1 + 2) * q^41 + (-b17 + b11 - b10 + b6 - b1) * q^43 + (b11 + 1) * q^44 + (b19 + b17 - b12 - b10 - b7 - b5 - b3 - b2 - b1 - 1) * q^45 + (-b15 + 2) * q^46 + (b13 + b12 + b10 + b3 + b2) * q^47 - b1 * q^48 + (-b19 - b14 + b1 + 2) * q^50 + (b19 + b17 + b14 - b12 + b9 + b4) * q^51 - b14 * q^52 + (b16 + b15 - b11 + 3) * q^53 + (-b19 - b17 - b13 - b12 + b11 - b7 + b6 + b5 - b4 - b1 - 2) * q^54 + (-b18 - b16 - b15 + b10 - b9 - b7 - b6 - b5 + b3 - b1) * q^55 + (b18 + b17 + 2b13 + b12 - b11 + b10 + b7 - b6 - b5 + b3 + 1) q^57 + (-b18 + b11 + b7 + b6 + b5) * q^58 + (-b19 - b18 - b15 - b14 + b13 - b11 + b10 - b4 + b3 + b2 + 1) * q^59 + (-b6 + 1) * q^60 + (-b19 + b12 - b8 - b7 + b6 + b5 - b4 + b2) * q^61 - b17 * q^62 + q^64 + (-b19 - b18 - b17 - b14 - b11 + b10 - 2b9 - 2b7 + b5 - b4 + b3 + b2 + 1) * q^65 + (-b10 - b5 + b4 - b1 + 1) * q^66 + (-b18 - b16 - b15 - b13 - b12 + b11 + b8 + b6 + b5 - b4 + b3 - 2) * q^67 + b5 * q^68 + (2b17 - b16 - 2b12 + b11 + 2b9 + b8 - 2b1 - 1) * q^69 + (b18 - b16 + b13 + b12 - b5 - 2b1 + 2) q^71 + (b2 + 2) * q^72 + (b19 + b17 + b15 + b13 + b12 - b11 + b10 - b6 + b2 - b1 + 2) * q^73 + (b18 + b15 + b14 - b7 - b3 + 3) * q^74 + (b19 + b17 - b16 - b15 + b14 + 2b12 + b10 - b8 - b7 - b5 - b4 - 2b1 - 2) * q^75 + b8 * q^76 + (b19 - b13 - b11 - b6 - b5 + 1) * q^78 + (b13 - b12 - b11 - b7 - b5 - b4 + b3 - b1 + 1) * q^79 - b7 * q^80 + (b18 - b16 + 2b12 - 2b11 + b10 - 2b8 - b5 + b2 + b1 + 4) q^81 + (b19 + b18 + b17 + b16 + b15 + b14 + b12 - b11 - b8 - b6 + b4 - b3 - b2 - b1 + 2) * q^82 + (-b12 + b8 + b7 - b6 - 2b5 + b3 - b1) q^83 + (b19 + b17 + b16 + 2b12 - b11 - b8 + 2b7 + 2b4 - b3 - b2 - b1 + 2) q^85 + (-b17 + b11 - b10 + b6 - b1) * q^86 + (b18 + b14 + b13 + 2b12 - b11 - 2b8 + 2b7 - b5 + 2b4 + 3) * q^87 + (b11 + 1) * q^88 + (b19 - b18 - b13 + b11 - b10 + b4 - b3) * q^89 + (b19 + b17 - b12 - b10 - b7 - b5 - b3 - b2 - b1 - 1) * q^90 + (-b15 + 2) * q^92 + (b19 + b15 - b13 - b12 - b10 - b7 + b6 - b3 - 1) * q^93 + (b13 + b12 + b10 + b3 + b2) * q^94 + (-b19 + b18 - b16 - b12 + b8 - 2b7 - b4 - b2 + 2b1 - 1) * q^95 - b1 * q^96 + (b18 + b12 - 2b11 - 2b8 - b5 - b2 + 1) * q^97 + (b14 + b13 + b12 - b7 + b5 - b4 - b3 - 2b1 + 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{2} - 2 q^{3} + 20 q^{4} + q^{5} - 2 q^{6} + 20 q^{8} + 30 q^{9}+O(q^{10}) $ 20 * q + 20 * q^2 - 2 * q^3 + 20 * q^4 + q^5 - 2 * q^6 + 20 * q^8 + 30 * q^9	$ 20 q + 20 q^{2} - 2 q^{3} + 20 q^{4} + q^{5} - 2 q^{6} + 20 q^{8} + 30 q^{9} + q^{10} + 16 q^{11} - 2 q^{12} + 5 q^{13} + 10 q^{15} + 20 q^{16} + 3 q^{17} + 30 q^{18} + 2 q^{19} + q^{20} + 16 q^{22} + 38 q^{23} - 2 q^{24} + 43 q^{25} + 5 q^{26} - 20 q^{27} + 17 q^{29} + 10 q^{30} - q^{31} + 20 q^{32} + 14 q^{33} + 3 q^{34} + 30 q^{36} + 39 q^{37} + 2 q^{38} + 17 q^{39} + q^{40} - 2 q^{41} + 9 q^{43} + 16 q^{44} - 5 q^{45} + 38 q^{46} - 16 q^{47} - 2 q^{48} + 43 q^{50} + 5 q^{51} + 5 q^{52} + 56 q^{53} - 20 q^{54} + 3 q^{55} - 6 q^{57} + 17 q^{58} + 31 q^{59} + 10 q^{60} - 3 q^{61} - q^{62} + 20 q^{64} + 31 q^{65} + 14 q^{66} + 15 q^{67} + 3 q^{68} + 10 q^{69} + 24 q^{71} + 30 q^{72} + 13 q^{73} + 39 q^{74} - 55 q^{75} + 2 q^{76} + 17 q^{78} + 43 q^{79} + q^{80} + 52 q^{81} - 2 q^{82} + q^{83} + 3 q^{85} + 9 q^{86} + 9 q^{87} + 16 q^{88} - 5 q^{90} + 38 q^{92} + 3 q^{93} - 16 q^{94} + 10 q^{95} - 2 q^{96} + 14 q^{97} + 22 q^{99}+O(q^{100}) $ 20 * q + 20 * q^2 - 2 * q^3 + 20 * q^4 + q^5 - 2 * q^6 + 20 * q^8 + 30 * q^9 + q^10 + 16 * q^11 - 2 * q^12 + 5 * q^13 + 10 * q^15 + 20 * q^16 + 3 * q^17 + 30 * q^18 + 2 * q^19 + q^20 + 16 * q^22 + 38 * q^23 - 2 * q^24 + 43 * q^25 + 5 * q^26 - 20 * q^27 + 17 * q^29 + 10 * q^30 - q^31 + 20 * q^32 + 14 * q^33 + 3 * q^34 + 30 * q^36 + 39 * q^37 + 2 * q^38 + 17 * q^39 + q^40 - 2 * q^41 + 9 * q^43 + 16 * q^44 - 5 * q^45 + 38 * q^46 - 16 * q^47 - 2 * q^48 + 43 * q^50 + 5 * q^51 + 5 * q^52 + 56 * q^53 - 20 * q^54 + 3 * q^55 - 6 * q^57 + 17 * q^58 + 31 * q^59 + 10 * q^60 - 3 * q^61 - q^62 + 20 * q^64 + 31 * q^65 + 14 * q^66 + 15 * q^67 + 3 * q^68 + 10 * q^69 + 24 * q^71 + 30 * q^72 + 13 * q^73 + 39 * q^74 - 55 * q^75 + 2 * q^76 + 17 * q^78 + 43 * q^79 + q^80 + 52 * q^81 - 2 * q^82 + q^83 + 3 * q^85 + 9 * q^86 + 9 * q^87 + 16 * q^88 - 5 * q^90 + 38 * q^92 + 3 * q^93 - 16 * q^94 + 10 * q^95 - 2 * q^96 + 14 * q^97 + 22 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 2 x^{19} - 43 x^{18} + 78 x^{17} + 774 x^{16} - 1240 x^{15} - 7575 x^{14} + 10337 x^{13} + \cdots + 3160 $ x^20 - 2*x^19 - 43*x^18 + 78*x^17 + 774*x^16 - 1240*x^15 - 7575*x^14 + 10337*x^13 + 43740*x^12 - 48326*x^11 - 150887*x^10 + 125274*x^9 + 299785*x^8 - 166516*x^7 - 316036*x^6 + 97790*x^5 + 163407*x^4 - 22723*x^3 - 38018*x^2 + 1644*x + 3160 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 5 $ v^2 - 5
$\beta_{3}$	$=$	$ ( - 96\!\cdots\!79 \nu^{19} + \cdots + 88\!\cdots\!64 ) / 20\!\cdots\!16 $ (-967503253172564879v^19 + 3206661904104090414v^18 + 37638207996237088881v^17 - 126242786083741226804v^16 - 591056386073341037456v^15 + 2029243021816870996210v^14 + 4751283465372275588797v^13 - 17123401517985178476703v^12 - 20021269490342410643220v^11 + 80892112586031903860230v^10 + 36256492664845239694867v^9 - 209113288870627849311836v^8 + 15493072519547932546851v^7 + 260235674307496527401864v^6 - 141070135905325335664146v^5 - 98831141931197742224754v^4 + 129281250829928757843065v^3 - 17738146312339415144677v^2 - 27152142790636435141000*v + 8898544719048943539064) / 205883411862415803916
$\beta_{4}$	$=$	$ ( 22\!\cdots\!93 \nu^{19} + \cdots + 28\!\cdots\!48 ) / 41\!\cdots\!32 $ (2284337205501273793v^19 - 6249699411106878542v^18 - 93637536260310746179v^17 + 248988196922855027306v^16 + 1585199389920510446490v^15 - 4078677669792632422324v^14 - 14322229700702918808903v^13 + 35510425471431674317413v^12 + 74310487201766631495140v^11 - 177193745056010015243134v^10 - 220600019386445823803727v^9 + 508640256011543387753102v^8 + 348275419573403623543665v^7 - 799286105348495187180492v^6 - 245688243194266261434516v^5 + 622783842342109642924234v^4 + 65251604739436711035247v^3 - 224374010295483385211895v^2 - 5578677200452570385046*v + 28644169288918291753248) / 411766823724831607832
$\beta_{5}$	$=$	$ ( - 95\!\cdots\!71 \nu^{19} + \cdots - 27\!\cdots\!92 ) / 82\!\cdots\!64 $ (-9569408074536055271v^19 + 24649945171773632952v^18 + 396107601496966363061v^17 - 973892531275711320672v^16 - 6792695453771363119002v^15 + 15753965285126383562052v^14 + 62418103883777900610081v^13 - 134498152647163982587701v^12 - 331021848024658416411294v^11 + 650137905759004285695190v^10 + 1010262475384154246208661v^9 - 1767429843770061485193044v^8 - 1648995759554446180209991v^7 + 2511614789338314474236142v^6 + 1201809609390903876581888v^5 - 1594565671757926148434730v^4 - 312082778136137483040165v^3 + 390879569544140083771539v^2 + 19767626575477333918044*v - 27850777577496681374492) / 823533647449663215664
$\beta_{6}$	$=$	$ ( - 70\!\cdots\!53 \nu^{19} + \cdots - 44\!\cdots\!68 ) / 41\!\cdots\!32 $ (-7014692873478297653v^19 + 18207153078386711651v^18 + 292051905154766939146v^17 - 720801215654170257518v^16 - 5054640218514283792020v^15 + 11692720394271277335604v^14 + 47137916423067957109167v^13 - 100255246378520321737940v^12 - 256124290597711630152438v^11 + 488207780234346381370780v^10 + 815141403186918260650031v^9 - 1346803512278136647561641v^8 - 1441014449207144333331822v^7 + 1980376542031778051218532v^6 + 1255201550115397051832436v^5 - 1379688574031738026708282v^4 - 483544863739493251774509v^3 + 419492901198024146522232v^2 + 64938465473169035086524*v - 44697360709085211730968) / 411766823724831607832
$\beta_{7}$	$=$	$ ( - 14\!\cdots\!30 \nu^{19} + \cdots - 88\!\cdots\!44 ) / 41\!\cdots\!32 $ (-14275040358484190930v^19 + 35564773590446679513v^18 + 595619582336433498339v^17 - 1405505053116533831686v^16 - 10328080021812593522302v^15 + 22755690263034680545220v^14 + 96440710321246468959146v^13 - 194699008608719038752577v^12 - 524135018901578189540260v^11 + 945979890961818641035618v^10 + 1665710234336257735484130v^9 - 2603432809055666795214851v^8 - 2932639461590046530388409v^7 + 3818037069540497870231702v^6 + 2531050112702131713534948v^5 - 2651157746771566082877136v^4 - 952952945827088160590228v^3 + 807916605805329522276899v^2 + 123215583150827824254508*v - 88406631822517044975444) / 411766823724831607832
$\beta_{8}$	$=$	$ ( - 15\!\cdots\!04 \nu^{19} + \cdots - 10\!\cdots\!28 ) / 41\!\cdots\!32 $ (-15598566115602060704v^19 + 40100675851661506607v^18 + 648866972032840013977v^17 - 1588860537108020117572v^16 - 11210001972265994852870v^15 + 25809581622190907883400v^14 + 104207591705369920165872v^13 - 221783015790720698987897v^12 - 563243113391021769309474v^11 + 1083756203823788176878210v^10 + 1777800882462961290272348v^9 - 3005663271094123468595253v^8 - 3103108196575713780380951v^7 + 4453097988433135424957800v^6 + 2652655651432292468238784v^5 - 3126007682766600097140256v^4 - 1005503176616978442040206v^3 + 950612199055435304465763v^2 + 136220572442809428295318*v - 101389011487409166353728) / 411766823724831607832
$\beta_{9}$	$=$	$ ( - 16\!\cdots\!85 \nu^{19} + \cdots - 10\!\cdots\!24 ) / 41\!\cdots\!32 $ (-16677370577211688285v^19 + 46241967353496760846v^18 + 684247318919106117117v^17 - 1837717234647665757692v^16 - 11608215450964175851454v^15 + 29970040964890744491976v^14 + 105289314333143547337135v^13 - 258871383992174855369037v^12 - 549808238723446826490146v^11 + 1273468520208975608829794v^10 + 1648416722025751686099607v^9 - 3560167768375048317588914v^8 - 2642111048924859676071631v^7 + 5311227483651745139418158v^6 + 1915797204646636166150112v^5 - 3707198839878928377381098v^4 - 554803451843138843132659v^3 + 1085438279246541642542043v^2 + 55262068917007461318732*v - 106391233707116453968124) / 411766823724831607832
$\beta_{10}$	$=$	$ ( 36\!\cdots\!05 \nu^{19} + \cdots + 22\!\cdots\!92 ) / 82\!\cdots\!64 $ (36169893905645014705v^19 - 96377142452016339696v^18 - 1493586847924405703071v^17 + 3819113310872579890356v^16 + 25555144342263280636510v^15 - 62040797292486429076236v^14 - 234461103759540805419743v^13 + 533035762367024678055859v^12 + 1243878445415285171362222v^11 - 2602974461723750582706218v^10 - 3815994471115845136674363v^9 + 7204640498638581087737548v^8 + 6341947448788652507568757v^7 - 10613771393939837967261326v^6 - 4905882095300283035096144v^5 + 7332277533396015427895950v^4 + 1544917681848650860707475v^3 - 2165073540709606564816365v^2 - 165367804597871223455864*v + 222069905370513345911292) / 823533647449663215664
$\beta_{11}$	$=$	$ ( - 21\!\cdots\!77 \nu^{19} + \cdots - 10\!\cdots\!52 ) / 41\!\cdots\!32 $ (-21535958159451015477v^19 + 54087822028955236878v^18 + 896432301627379190681v^17 - 2134906823390588131032v^16 - 15495209422539506721662v^15 + 34500613172044707503744v^14 + 144070144723954051903507v^13 - 294316469731423680581677v^12 - 778224430505888743547314v^11 + 1422864525507176519921826v^10 + 2450271585878722239015719v^9 - 3880161600946466134294822v^8 - 4246192145408306261253295v^7 + 5584282033922844833303850v^6 + 3554345875927994567957072v^5 - 3712349348173138121318442v^4 - 1273054226484477089243763v^3 + 1040527424374025402482279v^2 + 156031082018758851094068*v - 102626527024881843828052) / 411766823724831607832
$\beta_{12}$	$=$	$ ( 25\!\cdots\!01 \nu^{19} + \cdots + 15\!\cdots\!20 ) / 41\!\cdots\!32 $ (25522351282354860301v^19 - 63178601718280668219v^18 - 1067323675605995599844v^17 + 2497526707637907741188v^16 + 18561196719587235628796v^15 - 40444768530822189247532v^14 - 173966469239894140248563v^13 + 346062837411689510867028v^12 + 950049188186007642023660v^11 - 1680901991375531776309396v^10 - 3038588582089573125561299v^9 + 4621712149890604895912341v^8 + 5396744628873085523992856v^7 - 6764330840294206868500202v^6 - 4716342162673262883768072v^5 + 4679735536457247317542350v^4 + 1798118431050338760290541v^3 - 1415680559898275822939960v^2 - 234598047542441457749410*v + 153747574958703904184820) / 411766823724831607832
$\beta_{13}$	$=$	$ ( - 61\!\cdots\!29 \nu^{19} + \cdots - 38\!\cdots\!56 ) / 82\!\cdots\!64 $ (-61664393577253693629v^19 + 156866291650753229794v^18 + 2567686724129744596713v^17 - 6208055068974971645548v^16 - 44414005982457902190962v^15 + 100686783357438426377284v^14 + 413464057631363747908259v^13 - 863364454393861191745013v^12 - 2238389812654166549854970v^11 + 4206316289685832769854542v^10 + 7076336256848384746997183v^9 - 11615689098371207918355754v^8 - 12360103553859761144564227v^7 + 17101058578782317223205550v^6 + 10522945016937225138366408v^5 - 11899371282152474347915414v^4 - 3888060877446157346137219v^3 + 3591039701967846269773531v^2 + 493874859524691603872924*v - 383209971567608496412556) / 823533647449663215664
$\beta_{14}$	$=$	$ ( 53\!\cdots\!03 \nu^{19} + \cdots + 28\!\cdots\!92 ) / 41\!\cdots\!32 $ (53194626553656950703v^19 - 135532726213421561083v^18 - 2212230382529471326700v^17 + 5358879169042303804118v^16 + 38197773608685737051440v^15 - 86795731153244046839680v^14 - 354677036074572054613013v^13 + 742664584865655915520354v^12 + 1912592989408242788640698v^11 - 3605457325397145416892540v^10 - 6007943841881849602422669v^9 + 9893110124989448304870481v^8 + 10375535925344342958711780v^7 - 14380554855505696072702604v^6 - 8637024511486738557306736v^5 + 9729334539479671127560978v^4 + 3079568686928735131061123v^3 - 2796801987796843014461162v^2 - 376216808750829779256888*v + 280827911430601061289792) / 411766823724831607832
$\beta_{15}$	$=$	$ ( 26\!\cdots\!35 \nu^{19} + \cdots + 16\!\cdots\!04 ) / 20\!\cdots\!16 $ (26650943069260124635v^19 - 68260365115207885059v^18 - 1108570898808718582360v^17 + 2702853281328041328094v^16 + 19147515713816010272110v^15 - 43863365594487087126966v^14 - 177887299292286853784101v^13 + 376364040496432144066808v^12 + 960192454997171971519520v^11 - 1834747749369066815120746v^10 - 3021885505798423876692129v^9 + 5067770981968503801591397v^8 + 5239719000322216233164230v^7 - 7452636421714885508198628v^6 - 4404742633514488311166600v^5 + 5158755128317550186576320v^4 + 1605670966256923306939131v^3 - 1541075643135095592073090v^2 - 203836207487735425137450*v + 162422662708870559304404) / 205883411862415803916
$\beta_{16}$	$=$	$ ( 62\!\cdots\!09 \nu^{19} + \cdots + 35\!\cdots\!60 ) / 41\!\cdots\!32 $ (62935427067167918509v^19 - 158092317093941784425v^18 - 2623804966054947371174v^17 + 6250198321263868147240v^16 + 45448524884245095835976v^15 - 101229950107530962215608v^14 - 423772778788615472015615v^13 + 866332013711001208832398v^12 + 2298324068958364780326656v^11 - 4208697367957870306416860v^10 - 7280247138878190734761227v^9 + 11570452621956829306570379v^8 + 12743687048208570340906658v^7 - 16907075327806329707508410v^6 - 10872632014287891311033868v^5 + 11607964201009981515858962v^4 + 4018803444506766803146369v^3 - 3435457872810729861496594v^2 - 511589199375724222446978*v + 357545261986736385724860) / 411766823724831607832
$\beta_{17}$	$=$	$ ( 77\!\cdots\!42 \nu^{19} + \cdots + 45\!\cdots\!80 ) / 41\!\cdots\!32 $ (77276218553989410142v^19 - 198141323389589916075v^18 - 3212043392456933398313v^17 + 7841912561375485501422v^16 + 55430084121472177605938v^15 - 127191287489237632214860v^14 - 514407799192517173913382v^13 + 1090630176168171280421671v^12 + 2772938958733090083095792v^11 - 5312634219135896337638522v^10 - 8712064367815250995619146v^9 + 14660184697568432437692237v^8 + 15070521387300273796857971v^7 - 21529871188704902552823130v^6 - 12619396732569385625013376v^5 + 14860815304449369682789964v^4 + 4566116042925290990646384v^3 - 4405038241145516756468341v^2 - 573172031075088255581860*v + 456908112228334478063580) / 411766823724831607832
$\beta_{18}$	$=$	$ ( - 85\!\cdots\!43 \nu^{19} + \cdots - 47\!\cdots\!12 ) / 41\!\cdots\!32 $ (-85247975037738489443v^19 + 217155425915748025307v^18 + 3546288157188168270896v^17 - 8588892736083309437858v^16 - 61258211242557700448436v^15 + 139177357831391053372888v^14 + 569155236371480105635181v^13 - 1191759589663760902362586v^12 - 3072159541927443323320998v^11 + 5792904257196500074002708v^10 + 9666203435234322266379017v^9 - 15930656593109880977499749v^8 - 16743874497677190134236920v^7 + 23259039489133120766764028v^6 + 14027901218124801984733004v^5 - 15891230769537235512700642v^4 - 5063048453789215951876471v^3 + 4652041380734368206495042v^2 + 634266153395244698909736*v - 478823638192010140572912) / 411766823724831607832
$\beta_{19}$	$=$	$ ( - 93\!\cdots\!57 \nu^{19} + \cdots - 52\!\cdots\!56 ) / 41\!\cdots\!32 $ (-93311024964931847257v^19 + 238191652796382933431v^18 + 3880079667452563258998v^17 - 9421549183014842664342v^16 - 66987594585877851136624v^15 + 152685967956977736771228v^14 + 621940872085096849375287v^13 - 1307642995679168824594496v^12 - 3353828354008132474052246v^11 + 6357733178383728147368364v^10 + 10537918833288537458771531v^9 - 17489939780338943491417937v^8 - 18214554671619509526414082v^7 + 25545387746039728248309800v^6 + 15199356748005014045578264v^5 - 17451812053484762608665794v^4 - 5444731406633213879243857v^3 + 5097116464894142898346812v^2 + 671166735918401389410136*v - 521361049039800440281856) / 411766823724831607832

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 5 $ b2 + 5
$\nu^{3}$	$=$	$ \beta_{19} + \beta_{17} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 7\beta _1 + 2 $ b19 + b17 + b13 + b12 - b11 + b7 - b6 - b5 + b4 + 7*b1 + 2
$\nu^{4}$	$=$	$ \beta_{18} - \beta_{16} + 2\beta_{12} - 2\beta_{11} + \beta_{10} - 2\beta_{8} - \beta_{5} + 10\beta_{2} + \beta _1 + 40 $ b18 - b16 + 2b12 - 2b11 + b10 - 2b8 - b5 + 10b2 + b1 + 40
$\nu^{5}$	$=$	$ 14 \beta_{19} + 4 \beta_{18} + 14 \beta_{17} + \beta_{15} + 3 \beta_{14} + 14 \beta_{13} + 14 \beta_{12} + \cdots + 32 $ 14b19 + 4b18 + 14b17 + b15 + 3b14 + 14b13 + 14b12 - 14b11 - 2b8 + 9b7 - 16b6 - 16b5 + 14b4 - b3 - b2 + 57*b1 + 32
$\nu^{6}$	$=$	$ \beta_{19} + 20 \beta_{18} + 3 \beta_{17} - 15 \beta_{16} + \beta_{14} + 6 \beta_{13} + 36 \beta_{12} + \cdots + 362 $ b19 + 20b18 + 3b17 - 15b16 + b14 + 6b13 + 36b12 - 35b11 + 19b10 - 34b8 + b7 - 3b6 - 19b5 + b3 + 98b2 + 16b1 + 362
$\nu^{7}$	$=$	$ 164 \beta_{19} + 75 \beta_{18} + 164 \beta_{17} - \beta_{16} + 17 \beta_{15} + 51 \beta_{14} + \cdots + 412 $ 164b19 + 75b18 + 164b17 - b16 + 17b15 + 51b14 + 170b13 + 175b12 - 169b11 + 5b10 - b9 - 48b8 + 71b7 - 199b6 - 199b5 + 161b4 - 23b3 - 14b2 + 503*b1 + 412
$\nu^{8}$	$=$	$ 23 \beta_{19} + 294 \beta_{18} + 72 \beta_{17} - 181 \beta_{16} + 22 \beta_{14} + 152 \beta_{13} + \cdots + 3499 $ 23b19 + 294b18 + 72b17 - 181b16 + 22b14 + 152b13 + 509b12 - 469b11 + 271b10 - 4b9 - 445b8 + 11b7 - 71b6 - 261b5 - 2b4 + 17b3 + 979b2 + 205b1 + 3499
$\nu^{9}$	$=$	$ 1820 \beta_{19} + 1072 \beta_{18} + 1820 \beta_{17} - 23 \beta_{16} + 226 \beta_{15} + 658 \beta_{14} + \cdots + 4964 $ 1820b19 + 1072b18 + 1820b17 - 23b16 + 226b15 + 658b14 + 1972b13 + 2116b12 - 1960b11 + 133b10 - 35b9 - 784b8 + 510b7 - 2287b6 - 2275b5 + 1732b4 - 378b3 - 131b2 + 4681*b1 + 4964
$\nu^{10}$	$=$	$ 392 \beta_{19} + 3851 \beta_{18} + 1204 \beta_{17} - 2046 \beta_{16} - 12 \beta_{15} + 362 \beta_{14} + \cdots + 35240 $ 392b19 + 3851b18 + 1204b17 - 2046b16 - 12b15 + 362b14 + 2596b13 + 6585b12 - 5721b11 + 3461b10 - 112b9 - 5367b8 + 35b7 - 1164b6 - 3205b5 - 43b4 + 173b3 + 9933b2 + 2488*b1 + 35240
$\nu^{11}$	$=$	$ 19791 \beta_{19} + 13860 \beta_{18} + 19798 \beta_{17} - 380 \beta_{16} + 2746 \beta_{15} + 7733 \beta_{14} + \cdots + 58227 $ 19791b19 + 13860b18 + 19798b17 - 380b16 + 2746b15 + 7733b14 + 22470b13 + 25146b12 - 22386b11 + 2421b10 - 739b9 - 10987b8 + 3117b7 - 25462b6 - 25125b5 + 18090b4 - 5325b3 - 845b2 + 45172*b1 + 58227
$\nu^{12}$	$=$	$ 5990 \beta_{19} + 47697 \beta_{18} + 17378 \beta_{17} - 22537 \beta_{16} - 352 \beta_{15} + \cdots + 364541 $ 5990b19 + 47697b18 + 17378b17 - 22537b16 - 352b15 + 5348b14 + 37644b13 + 81405b12 - 66926b11 + 41851b10 - 2072b9 - 62649b8 - 1080b7 - 16534b6 - 37549b5 - 540b4 + 1060b3 + 101939b2 + 29709*b1 + 364541
$\nu^{13}$	$=$	$ 213578 \beta_{19} + 170527 \beta_{18} + 213966 \beta_{17} - 5638 \beta_{16} + 31853 \beta_{15} + \cdots + 674271 $ 213578b19 + 170527b18 + 213966b17 - 5638b16 + 31853b15 + 87368b14 + 254168b13 + 295466b12 - 253854b11 + 37453b10 - 12384b9 - 142512b8 + 11621b7 - 279570b6 - 273525b5 + 186377b4 - 68897b3 - 731b2 + 447235*b1 + 674271
$\nu^{14}$	$=$	$ 86220 \beta_{19} + 572845 \beta_{18} + 232281 \beta_{17} - 245431 \beta_{16} - 6358 \beta_{15} + \cdots + 3840195 $ 86220b19 + 572845b18 + 232281b17 - 245431b16 - 6358b15 + 74366b14 + 501013b13 + 979451b12 - 767374b11 + 490905b10 - 32057b9 - 720434b8 - 30734b7 - 218859b6 - 431080b5 - 4357b4 - 2603b3 + 1055468b2 + 352274*b1 + 3840195
$\nu^{15}$	$=$	$ 2299959 \beta_{19} + 2039723 \beta_{18} + 2311415 \beta_{17} - 79862 \beta_{16} + 359435 \beta_{15} + \cdots + 7753209 $ 2299959b19 + 2039723b18 + 2311415b17 - 79862b16 + 359435b15 + 969792b14 + 2866427b13 + 3443350b12 - 2867564b11 + 529474b10 - 182181b9 - 1768735b8 - 77945b7 - 3051288b6 - 2962627b5 + 1909517b4 - 847038b3 + 97219b2 + 4512832*b1 + 7753209
$\nu^{16}$	$=$	$ 1189995 \beta_{19} + 6754747 \beta_{18} + 2966467 \beta_{17} - 2659394 \beta_{16} - 90863 \beta_{15} + \cdots + 40982314 $ 1189995b19 + 6754747b18 + 2966467b17 - 2659394b16 - 90863b15 + 991256b14 + 6339791b13 + 11579202b12 - 8713967b11 + 5654468b10 - 449924b9 - 8221540b8 - 556925b7 - 2784865b6 - 4909589b5 - 6152b4 - 217910b3 + 11008347b2 + 4157004*b1 + 40982314
$\nu^{17}$	$=$	$ 24777841 \beta_{19} + 23975472 \beta_{18} + 25028426 \beta_{17} - 1099132 \beta_{16} + 3985942 \beta_{15} + \cdots + 88770275 $ 24777841b19 + 23975472b18 + 25028426b17 - 1099132b16 + 3985942b15 + 10683465b14 + 32289323b13 + 39876925b12 - 32318489b11 + 7073667b10 - 2470656b9 - 21362157b8 - 2729714b7 - 33229607b6 - 32068832b5 + 19543601b4 - 10082253b3 + 2280350b2 + 46215262*b1 + 88770275
$\nu^{18}$	$=$	$ 15892346 \beta_{19} + 78726084 \beta_{18} + 36786988 \beta_{17} - 28763783 \beta_{16} - 1120939 \beta_{15} + \cdots + 441623570 $ 15892346b19 + 78726084b18 + 36786988b17 - 28763783b16 - 1120939b15 + 12794164b14 + 77696657b13 + 135250962b12 - 98509439b11 + 64394266b10 - 5949013b9 - 93420569b8 - 8548058b7 - 34580995b6 - 55790281b5 + 617386b4 - 4663205b3 + 115545147b2 + 48851039*b1 + 441623570
$\nu^{19}$	$=$	$ 267373082 \beta_{19} + 278607093 \beta_{18} + 271971867 \beta_{17} - 14771407 \beta_{16} + \cdots + 1013531650 $ 267373082b19 + 278607093b18 + 271971867b17 - 14771407b16 + 43709412b15 + 117389942b14 + 363589934b13 + 459520838b12 - 363728299b11 + 90957345b10 - 31744080b9 - 253447918b8 - 47129611b7 - 361797790b6 - 347656845b5 + 200356442b4 - 117481879b3 + 38563892b2 + 479017649*b1 + 1013531650

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

3.35871
3.05063
2.99500
2.52494
2.39224
2.24270
1.28125
0.775250
0.535014
0.424862
−0.523259
−0.548297
−0.586594
−0.894573
−1.94262
−2.05588
−2.29342
−2.58363
−2.96256
−3.18977

1.00000

−3.35871

1.00000

0.267686

−3.35871

1.00000

8.28096

0.267686

1.2

1.00000

−3.05063

1.00000

−4.21199

−3.05063

1.00000

6.30632

−4.21199

1.3

1.00000

−2.99500

1.00000

2.06577

−2.99500

1.00000

5.97000

2.06577

1.4

1.00000

−2.52494

1.00000

−0.967238

−2.52494

1.00000

3.37534

−0.967238

1.5

1.00000

−2.39224

1.00000

4.26305

−2.39224

1.00000

2.72280

4.26305

1.6

1.00000

−2.24270

1.00000

−4.16071

−2.24270

1.00000

2.02971

−4.16071

1.7

1.00000

−1.28125

1.00000

1.24467

−1.28125

1.00000

−1.35839

1.24467

1.8

1.00000

−0.775250

1.00000

−2.38408

−0.775250

1.00000

−2.39899

−2.38408

1.9

1.00000

−0.535014

1.00000

2.37602

−0.535014

1.00000

−2.71376

2.37602

1.10

1.00000

−0.424862

1.00000

3.40229

−0.424862

1.00000

−2.81949

3.40229

1.11

1.00000

0.523259

1.00000

2.58396

0.523259

1.00000

−2.72620

2.58396

1.12

1.00000

0.548297

1.00000

−1.26245

0.548297

1.00000

−2.69937

−1.26245

1.13

1.00000

0.586594

1.00000

−2.57677

0.586594

1.00000

−2.65591

−2.57677

1.14

1.00000

0.894573

1.00000

−3.87209

0.894573

1.00000

−2.19974

−3.87209

1.15

1.00000

1.94262

1.00000

0.645885

1.94262

1.00000

0.773770

0.645885

1.16

1.00000

2.05588

1.00000

−1.28904

2.05588

1.00000

1.22664

−1.28904

1.17

1.00000

2.29342

1.00000

3.57356

2.29342

1.00000

2.25977

3.57356

1.18

1.00000

2.58363

1.00000

1.60764

2.58363

1.00000

3.67513

1.60764

1.19

1.00000

2.96256

1.00000

2.36301

2.96256

1.00000

5.77679

2.36301

1.20

1.00000

3.18977

1.00000

−2.66918

3.18977

1.00000

7.17462

−2.66918

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$7$	$ -1 $
$101$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9898.2.a.bi		20
7.b	odd	2	1		9898.2.a.bj		20
7.d	odd	6	2		1414.2.e.g	✓	40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1414.2.e.g	✓	40	7.d	odd	6	2
9898.2.a.bi		20	1.a	even	1	1	trivial
9898.2.a.bj		20	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9898))$:

$ T_{3}^{20} + 2 T_{3}^{19} - 43 T_{3}^{18} - 78 T_{3}^{17} + 774 T_{3}^{16} + 1240 T_{3}^{15} + \cdots + 3160 $

$ T_{5}^{20} - T_{5}^{19} - 71 T_{5}^{18} + 84 T_{5}^{17} + 2074 T_{5}^{16} - 2807 T_{5}^{15} + \cdots - 941208 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 1)^{20} $ (T - 1)^20
$3$	$ T^{20} + 2 T^{19} + \cdots + 3160 $ T^20 + 2T^19 - 43T^18 - 78T^17 + 774T^16 + 1240T^15 - 7575T^14 - 10337T^13 + 43740T^12 + 48326T^11 - 150887T^10 - 125274T^9 + 299785T^8 + 166516T^7 - 316036T^6 - 97790T^5 + 163407T^4 + 22723T^3 - 38018T^2 - 1644*T + 3160
$5$	$ T^{20} - T^{19} + \cdots - 941208 $ T^20 - T^19 - 71T^18 + 84T^17 + 2074T^16 - 2807T^15 - 32298T^14 + 48624T^13 + 291077T^12 - 477013T^11 - 1546125T^10 + 2720716T^9 + 4716708T^8 - 8882137T^7 - 7658061T^6 + 15705269T^5 + 5549555T^4 - 13642919T^3 - 433352T^2 + 4435868T - 941208
$7$	$ T^{20} $ T^20
$11$	$ T^{20} - 16 T^{19} + \cdots - 7413760 $ T^20 - 16T^19 - 2T^18 + 1232T^17 - 4363T^16 - 33338T^15 + 192217T^14 + 345682T^13 - 3519764T^12 + 10298T^11 + 32566485T^10 - 26414580T^9 - 155549468T^8 + 187729612T^7 + 357374367T^6 - 479527130T^5 - 337464256T^4 + 377439976T^3 + 132365824T^2 - 32857088*T - 7413760
$13$	$ T^{20} + \cdots + 1343686456 $ T^20 - 5T^19 - 149T^18 + 615T^17 + 9687T^16 - 29598T^15 - 359427T^14 + 675745T^13 + 8268356T^12 - 6027809T^11 - 118588210T^10 - 36549577T^9 + 998601674T^8 + 1245475278T^7 - 4067000423T^6 - 9621157408T^5 + 1668550249T^4 + 23678289003T^3 + 26506221558T^2 + 11002593688*T + 1343686456
$17$	$ T^{20} - 3 T^{19} + \cdots - 57524184 $ T^20 - 3T^19 - 171T^18 + 553T^17 + 11333T^16 - 39085T^15 - 371331T^14 + 1367727T^13 + 6363180T^12 - 25542239T^11 - 55513917T^10 + 258753984T^9 + 210262808T^8 - 1387638796T^7 - 49907761T^6 + 3590525932T^5 - 1372023289T^4 - 3565422595T^3 + 1702930770T^2 + 880572924*T - 57524184
$19$	$ T^{20} + \cdots - 271706112 $ T^20 - 2T^19 - 196T^18 + 407T^17 + 15650T^16 - 34627T^15 - 654196T^14 + 1584931T^13 + 15228060T^12 - 41550143T^11 - 190966163T^10 + 610331321T^9 + 1086223716T^8 - 4501980944T^7 - 1195210576T^6 + 12775154896T^5 - 2234751040T^4 - 14981469120T^3 + 3142908928T^2 + 6532388864*T - 271706112
$23$	$ T^{20} + \cdots + 17863827451904 $ T^20 - 38T^19 + 400T^18 + 2451T^17 - 77108T^16 + 328790T^15 + 3739984T^14 - 37659441T^13 - 6710225T^12 + 1348216438T^11 - 3962103940T^10 - 19347207824T^9 + 112132838336T^8 + 53378510880T^7 - 1264850530816T^6 + 1160595805824T^5 + 6320879728384T^4 - 9060656165376T^3 - 14524751065088T^2 + 17664764723200*T + 17863827451904
$29$	$ T^{20} + \cdots + 1970600448 $ T^20 - 17T^19 - 176T^18 + 4481T^17 + 2946T^16 - 464379T^15 + 1401836T^14 + 22978097T^13 - 132412057T^12 - 467438915T^11 + 5052575057T^10 - 2608684141T^9 - 82481424856T^8 + 245111263688T^7 + 230056907224T^6 - 2446857323264T^5 + 5012906450240T^4 - 4314791396800T^3 + 1221418853888T^2 + 105219843584*T + 1970600448
$31$	$ T^{20} + T^{19} + \cdots + 61440 $ T^20 + T^19 - 256T^18 - 337T^17 + 25543T^16 + 41363T^15 - 1254034T^14 - 2427145T^13 + 31019181T^12 + 71251152T^11 - 338949348T^10 - 946044128T^9 + 857445984T^8 + 3841243936T^7 + 1693013440T^6 - 2071270016T^5 - 1182437120T^4 + 242839552T^3 + 106343424T^2 + 6475776T + 61440
$37$	$ T^{20} + \cdots + 36590983568600 $ T^20 - 39T^19 + 344T^18 + 5209T^17 - 102224T^16 + 49042T^15 + 9135783T^14 - 43124644T^13 - 352376976T^12 + 2949943618T^11 + 4559528011T^10 - 88798144712T^9 + 65802912836T^8 + 1292688418685T^7 - 2299239085593T^6 - 8606290781976T^5 + 18446514631364T^4 + 24806877452397T^3 - 47846066165750T^2 - 21477652793288*T + 36590983568600
$41$	$ T^{20} + \cdots + 1936065245184 $ T^20 + 2T^19 - 454T^18 - 220T^17 + 84963T^16 - 74584T^15 - 8380507T^14 + 17253173T^13 + 463034613T^12 - 1431327918T^11 - 14012985728T^10 + 57509739112T^9 + 204315430288T^8 - 1139381474144T^7 - 745528505088T^6 + 9684883321728T^5 - 8499518947584T^4 - 18330311275520T^3 + 31994647093248T^2 - 15392362696704*T + 1936065245184
$43$	$ T^{20} + \cdots + 152943034368 $ T^20 - 9T^19 - 307T^18 + 2544T^17 + 36331T^16 - 282273T^15 - 2066070T^14 + 16012837T^13 + 57399112T^12 - 501527190T^11 - 631249173T^10 + 8631448687T^9 - 2491402910T^8 - 74664620380T^7 + 105913977600T^6 + 235457987200T^5 - 644680136672T^4 + 229965453248T^3 + 576900469248T^2 - 587713384448*T + 152943034368
$47$	$ T^{20} + \cdots - 23753833803776 $ T^20 + 16T^19 - 293T^18 - 5866T^17 + 25471T^16 + 852442T^15 + 368895T^14 - 61797188T^13 - 190428539T^12 + 2260322880T^11 + 12103419880T^10 - 33012269952T^9 - 328839763248T^8 - 182574584960T^7 + 3705479035584T^6 + 9548027579648T^5 - 6026381878784T^4 - 61420681217536T^3 - 107490322900992T^2 - 81777543739392*T - 23753833803776
$53$	$ T^{20} + \cdots - 796895848448 $ T^20 - 56T^19 + 1013T^18 - 282T^17 - 216466T^16 + 2278449T^15 + 5846293T^14 - 235489569T^13 + 940813455T^12 + 7397754185T^11 - 65040599470T^10 - 3996701755T^9 + 1420659821110T^8 - 2815130154716T^7 - 12426379643744T^6 + 37238208528592T^5 + 42452594472832T^4 - 137391538167744T^3 - 80188936607232T^2 + 17397667180544*T - 796895848448
$59$	$ T^{20} + \cdots - 32595519651840 $ T^20 - 31T^19 - 191T^18 + 15302T^17 - 88401T^16 - 2356616T^15 + 28894534T^14 + 77916067T^13 - 2735412269T^12 + 8952417398T^11 + 77970537832T^10 - 589710193205T^9 + 490782721072T^8 + 5712921901349T^7 - 11479628742661T^6 - 23133544988306T^5 + 51931864650912T^4 + 64660896112568T^3 - 72275581331200T^2 - 110429872728064*T - 32595519651840
$61$	$ T^{20} + \cdots - 158972846819328 $ T^20 + 3T^19 - 675T^18 - 730T^17 + 185672T^16 - 145176T^15 - 26697527T^14 + 66761718T^13 + 2112293859T^12 - 8637920117T^11 - 87494255448T^10 + 507132192813T^9 + 1540630617534T^8 - 13921592785968T^7 - 564898867016T^6 + 165183170944512T^5 - 247913428279328T^4 - 587173165562432T^3 + 1698814074463744T^2 - 996682099325952*T - 158972846819328
$67$	$ T^{20} + \cdots + 679462165126584 $ T^20 - 15T^19 - 715T^18 + 10394T^17 + 207160T^16 - 2805893T^15 - 32795152T^14 + 381894656T^13 + 3256794745T^12 - 28371792311T^11 - 212426453813T^10 + 1120443256210T^9 + 8740344347776T^8 - 18885375595559T^7 - 198678904179323T^6 - 26402835182557T^5 + 1836735531255679T^4 + 2704813866996503T^3 - 2214440830269626T^2 - 3927987317386200*T + 679462165126584
$71$	$ T^{20} + \cdots + 48159926026240 $ T^20 - 24T^19 - 363T^18 + 11319T^17 + 48170T^16 - 2247231T^15 - 2465655T^14 + 243146453T^13 - 22871419T^12 - 15461189150T^11 + 8016831088T^10 + 578714454456T^9 - 367616176032T^8 - 12030197933984T^7 + 8204731866176T^6 + 120215581858816T^5 - 86654052595200T^4 - 400816500910592T^3 + 100608627578880T^2 + 294158037786624*T + 48159926026240
$73$	$ T^{20} + \cdots - 17\!\cdots\!44 $ T^20 - 13T^19 - 634T^18 + 8221T^17 + 166053T^16 - 2101315T^15 - 23871182T^14 + 281388125T^13 + 2129492973T^12 - 21363055818T^11 - 125871331040T^10 + 921891106512T^9 + 4924933176368T^8 - 20705507578304T^7 - 116133458129088T^6 + 171170645654656T^5 + 1306846006470656T^4 + 496943400324096T^3 - 3792547905699840T^2 - 4956752292970496*T - 1736658003103744
$79$	$ T^{20} + \cdots - 34\!\cdots\!12 $ T^20 - 43T^19 + 37T^18 + 24131T^17 - 315806T^16 - 3625969T^15 + 99798437T^14 - 159699402T^13 - 12072243811T^12 + 91587594164T^11 + 499547383452T^10 - 8833448273352T^9 + 16029377180512T^8 + 308907283068672T^7 - 1883488922383488T^6 - 389597785041024T^5 + 38537745794402560T^4 - 136992747698286080T^3 + 130176012102862848T^2 + 192324695351531520*T - 349345716577677312
$83$	$ T^{20} + \cdots - 64650141462088 $ T^20 - T^19 - 855T^18 + 1490T^17 + 287086T^16 - 624147T^15 - 48352322T^14 + 106865128T^13 + 4400234187T^12 - 7928461709T^11 - 223170032097T^10 + 234516804780T^9 + 6262050858112T^8 - 532699680813T^7 - 90967629667713T^6 - 81531172518929T^5 + 528888993808431T^4 + 947955449646589T^3 + 86006592550310T^2 - 244249069442872T - 64650141462088
$89$	$ T^{20} + \cdots + 161429430910976 $ T^20 - 749T^18 - 945T^17 + 206446T^16 + 468395T^15 - 26384771T^14 - 76255799T^13 + 1660794953T^12 + 5141414890T^11 - 51997222476T^10 - 145942207688T^9 + 891424069744T^8 + 2017726492096T^7 - 8729518395072T^6 - 14256496848512T^5 + 48333633926912T^4 + 48797596873216T^3 - 139618672996352T^2 - 63279898370048T + 161429430910976
$97$	$ T^{20} + \cdots + 58\!\cdots\!96 $ T^20 - 14T^19 - 722T^18 + 10035T^17 + 208080T^16 - 2867567T^15 - 31512258T^14 + 426909598T^13 + 2810530775T^12 - 36617063225T^11 - 155812321153T^10 + 1871673372989T^9 + 5456505728140T^8 - 56134953678005T^7 - 117889474366142T^6 + 913594258723838T^5 + 1433486958514814T^4 - 6499028261627193T^3 - 7533600221531978T^2 + 7848744676909748*T + 5887999287614696
show more
show less

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:	\( N \)	\(=\)	\( 9898 = 2 \cdot 7^{2} \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9898.a (trivial)

\(f(q)\)	\(=\)	\( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_{7} q^{5} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) q + q^2 - b1 * q^3 + q^4 - b7 * q^5 - b1 * q^6 + q^8 + (b2 + 2) * q^9	\( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_{7} q^{5} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 2) q^{9} - \beta_{7} q^{10} + (\beta_{11} + 1) q^{11} - \beta_1 q^{12} - \beta_{14} q^{13} + ( - \beta_{6} + 1) q^{15} + q^{16} + \beta_{5} q^{17} + (\beta_{2} + 2) q^{18} + \beta_{8} q^{19} - \beta_{7} q^{20} + (\beta_{11} + 1) q^{22} + ( - \beta_{15} + 2) q^{23} - \beta_1 q^{24} + ( - \beta_{19} - \beta_{14} + \beta_1 + 2) q^{25} - \beta_{14} q^{26} + ( - \beta_{19} - \beta_{17} - \beta_{13} + \cdots - 2) q^{27}+ \cdots + (\beta_{14} + \beta_{13} + \beta_{12} + \cdots + 2) q^{99}+O(q^{100}) \) q + q^2 - b1 * q^3 + q^4 - b7 * q^5 - b1 * q^6 + q^8 + (b2 + 2) * q^9 - b7 * q^10 + (b11 + 1) * q^11 - b1 * q^12 - b14 * q^13 + (-b6 + 1) * q^15 + q^16 + b5 * q^17 + (b2 + 2) * q^18 + b8 * q^19 - b7 * q^20 + (b11 + 1) * q^22 + (-b15 + 2) * q^23 - b1 * q^24 + (-b19 - b14 + b1 + 2) * q^25 - b14 * q^26 + (-b19 - b17 - b13 - b12 + b11 - b7 + b6 + b5 - b4 - b1 - 2) * q^27 + (-b18 + b11 + b7 + b6 + b5) * q^29 + (-b6 + 1) * q^30 - b17 * q^31 + q^32 + (-b10 - b5 + b4 - b1 + 1) * q^33 + b5 * q^34 + (b2 + 2) * q^36 + (b18 + b15 + b14 - b7 - b3 + 3) * q^37 + b8 * q^38 + (b19 - b13 - b11 - b6 - b5 + 1) * q^39 - b7 * q^40 + (b19 + b18 + b17 + b16 + b15 + b14 + b12 - b11 - b8 - b6 + b4 - b3 - b2 - b1 + 2) * q^41 + (-b17 + b11 - b10 + b6 - b1) * q^43 + (b11 + 1) * q^44 + (b19 + b17 - b12 - b10 - b7 - b5 - b3 - b2 - b1 - 1) * q^45 + (-b15 + 2) * q^46 + (b13 + b12 + b10 + b3 + b2) * q^47 - b1 * q^48 + (-b19 - b14 + b1 + 2) * q^50 + (b19 + b17 + b14 - b12 + b9 + b4) * q^51 - b14 * q^52 + (b16 + b15 - b11 + 3) * q^53 + (-b19 - b17 - b13 - b12 + b11 - b7 + b6 + b5 - b4 - b1 - 2) * q^54 + (-b18 - b16 - b15 + b10 - b9 - b7 - b6 - b5 + b3 - b1) * q^55 + (b18 + b17 + 2b13 + b12 - b11 + b10 + b7 - b6 - b5 + b3 + 1) q^57 + (-b18 + b11 + b7 + b6 + b5) * q^58 + (-b19 - b18 - b15 - b14 + b13 - b11 + b10 - b4 + b3 + b2 + 1) * q^59 + (-b6 + 1) * q^60 + (-b19 + b12 - b8 - b7 + b6 + b5 - b4 + b2) * q^61 - b17 * q^62 + q^64 + (-b19 - b18 - b17 - b14 - b11 + b10 - 2b9 - 2b7 + b5 - b4 + b3 + b2 + 1) * q^65 + (-b10 - b5 + b4 - b1 + 1) * q^66 + (-b18 - b16 - b15 - b13 - b12 + b11 + b8 + b6 + b5 - b4 + b3 - 2) * q^67 + b5 * q^68 + (2b17 - b16 - 2b12 + b11 + 2b9 + b8 - 2b1 - 1) * q^69 + (b18 - b16 + b13 + b12 - b5 - 2b1 + 2) q^71 + (b2 + 2) * q^72 + (b19 + b17 + b15 + b13 + b12 - b11 + b10 - b6 + b2 - b1 + 2) * q^73 + (b18 + b15 + b14 - b7 - b3 + 3) * q^74 + (b19 + b17 - b16 - b15 + b14 + 2b12 + b10 - b8 - b7 - b5 - b4 - 2b1 - 2) * q^75 + b8 * q^76 + (b19 - b13 - b11 - b6 - b5 + 1) * q^78 + (b13 - b12 - b11 - b7 - b5 - b4 + b3 - b1 + 1) * q^79 - b7 * q^80 + (b18 - b16 + 2b12 - 2b11 + b10 - 2b8 - b5 + b2 + b1 + 4) q^81 + (b19 + b18 + b17 + b16 + b15 + b14 + b12 - b11 - b8 - b6 + b4 - b3 - b2 - b1 + 2) * q^82 + (-b12 + b8 + b7 - b6 - 2b5 + b3 - b1) q^83 + (b19 + b17 + b16 + 2b12 - b11 - b8 + 2b7 + 2b4 - b3 - b2 - b1 + 2) q^85 + (-b17 + b11 - b10 + b6 - b1) * q^86 + (b18 + b14 + b13 + 2b12 - b11 - 2b8 + 2b7 - b5 + 2b4 + 3) * q^87 + (b11 + 1) * q^88 + (b19 - b18 - b13 + b11 - b10 + b4 - b3) * q^89 + (b19 + b17 - b12 - b10 - b7 - b5 - b3 - b2 - b1 - 1) * q^90 + (-b15 + 2) * q^92 + (b19 + b15 - b13 - b12 - b10 - b7 + b6 - b3 - 1) * q^93 + (b13 + b12 + b10 + b3 + b2) * q^94 + (-b19 + b18 - b16 - b12 + b8 - 2b7 - b4 - b2 + 2b1 - 1) * q^95 - b1 * q^96 + (b18 + b12 - 2b11 - 2b8 - b5 - b2 + 1) * q^97 + (b14 + b13 + b12 - b7 + b5 - b4 - b3 - 2b1 + 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{2} - 2 q^{3} + 20 q^{4} + q^{5} - 2 q^{6} + 20 q^{8} + 30 q^{9}+O(q^{10}) \) 20 * q + 20 * q^2 - 2 * q^3 + 20 * q^4 + q^5 - 2 * q^6 + 20 * q^8 + 30 * q^9	\( 20 q + 20 q^{2} - 2 q^{3} + 20 q^{4} + q^{5} - 2 q^{6} + 20 q^{8} + 30 q^{9} + q^{10} + 16 q^{11} - 2 q^{12} + 5 q^{13} + 10 q^{15} + 20 q^{16} + 3 q^{17} + 30 q^{18} + 2 q^{19} + q^{20} + 16 q^{22} + 38 q^{23} - 2 q^{24} + 43 q^{25} + 5 q^{26} - 20 q^{27} + 17 q^{29} + 10 q^{30} - q^{31} + 20 q^{32} + 14 q^{33} + 3 q^{34} + 30 q^{36} + 39 q^{37} + 2 q^{38} + 17 q^{39} + q^{40} - 2 q^{41} + 9 q^{43} + 16 q^{44} - 5 q^{45} + 38 q^{46} - 16 q^{47} - 2 q^{48} + 43 q^{50} + 5 q^{51} + 5 q^{52} + 56 q^{53} - 20 q^{54} + 3 q^{55} - 6 q^{57} + 17 q^{58} + 31 q^{59} + 10 q^{60} - 3 q^{61} - q^{62} + 20 q^{64} + 31 q^{65} + 14 q^{66} + 15 q^{67} + 3 q^{68} + 10 q^{69} + 24 q^{71} + 30 q^{72} + 13 q^{73} + 39 q^{74} - 55 q^{75} + 2 q^{76} + 17 q^{78} + 43 q^{79} + q^{80} + 52 q^{81} - 2 q^{82} + q^{83} + 3 q^{85} + 9 q^{86} + 9 q^{87} + 16 q^{88} - 5 q^{90} + 38 q^{92} + 3 q^{93} - 16 q^{94} + 10 q^{95} - 2 q^{96} + 14 q^{97} + 22 q^{99}+O(q^{100}) \) 20 * q + 20 * q^2 - 2 * q^3 + 20 * q^4 + q^5 - 2 * q^6 + 20 * q^8 + 30 * q^9 + q^10 + 16 * q^11 - 2 * q^12 + 5 * q^13 + 10 * q^15 + 20 * q^16 + 3 * q^17 + 30 * q^18 + 2 * q^19 + q^20 + 16 * q^22 + 38 * q^23 - 2 * q^24 + 43 * q^25 + 5 * q^26 - 20 * q^27 + 17 * q^29 + 10 * q^30 - q^31 + 20 * q^32 + 14 * q^33 + 3 * q^34 + 30 * q^36 + 39 * q^37 + 2 * q^38 + 17 * q^39 + q^40 - 2 * q^41 + 9 * q^43 + 16 * q^44 - 5 * q^45 + 38 * q^46 - 16 * q^47 - 2 * q^48 + 43 * q^50 + 5 * q^51 + 5 * q^52 + 56 * q^53 - 20 * q^54 + 3 * q^55 - 6 * q^57 + 17 * q^58 + 31 * q^59 + 10 * q^60 - 3 * q^61 - q^62 + 20 * q^64 + 31 * q^65 + 14 * q^66 + 15 * q^67 + 3 * q^68 + 10 * q^69 + 24 * q^71 + 30 * q^72 + 13 * q^73 + 39 * q^74 - 55 * q^75 + 2 * q^76 + 17 * q^78 + 43 * q^79 + q^80 + 52 * q^81 - 2 * q^82 + q^83 + 3 * q^85 + 9 * q^86 + 9 * q^87 + 16 * q^88 - 5 * q^90 + 38 * q^92 + 3 * q^93 - 16 * q^94 + 10 * q^95 - 2 * q^96 + 14 * q^97 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	9898.2.a.bi

Level	$9898$
Weight	$2$
Character orbit	9898.a
Self dual	yes
Analytic conductor	$79.036$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(79.0359279207\)
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:	\( x^{20} - 2 x^{19} - 43 x^{18} + 78 x^{17} + 774 x^{16} - 1240 x^{15} - 7575 x^{14} + 10337 x^{13} + \cdots + 3160 \) x^20 - 2x^19 - 43x^18 + 78x^17 + 774x^16 - 1240x^15 - 7575x^14 + 10337x^13 + 43740x^12 - 48326x^11 - 150887x^10 + 125274x^9 + 299785x^8 - 166516x^7 - 316036x^6 + 97790x^5 + 163407x^4 - 22723x^3 - 38018x^2 + 1644*x + 3160
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 1414)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 5 \) v^2 - 5
\(\beta_{3}\)	\(=\)	\( ( - 96\!\cdots\!79 \nu^{19} + \cdots + 88\!\cdots\!64 ) / 20\!\cdots\!16 \) (-967503253172564879v^19 + 3206661904104090414v^18 + 37638207996237088881v^17 - 126242786083741226804v^16 - 591056386073341037456v^15 + 2029243021816870996210v^14 + 4751283465372275588797v^13 - 17123401517985178476703v^12 - 20021269490342410643220v^11 + 80892112586031903860230v^10 + 36256492664845239694867v^9 - 209113288870627849311836v^8 + 15493072519547932546851v^7 + 260235674307496527401864v^6 - 141070135905325335664146v^5 - 98831141931197742224754v^4 + 129281250829928757843065v^3 - 17738146312339415144677v^2 - 27152142790636435141000*v + 8898544719048943539064) / 205883411862415803916
\(\beta_{4}\)	\(=\)	\( ( 22\!\cdots\!93 \nu^{19} + \cdots + 28\!\cdots\!48 ) / 41\!\cdots\!32 \) (2284337205501273793v^19 - 6249699411106878542v^18 - 93637536260310746179v^17 + 248988196922855027306v^16 + 1585199389920510446490v^15 - 4078677669792632422324v^14 - 14322229700702918808903v^13 + 35510425471431674317413v^12 + 74310487201766631495140v^11 - 177193745056010015243134v^10 - 220600019386445823803727v^9 + 508640256011543387753102v^8 + 348275419573403623543665v^7 - 799286105348495187180492v^6 - 245688243194266261434516v^5 + 622783842342109642924234v^4 + 65251604739436711035247v^3 - 224374010295483385211895v^2 - 5578677200452570385046*v + 28644169288918291753248) / 411766823724831607832
\(\beta_{5}\)	\(=\)	\( ( - 95\!\cdots\!71 \nu^{19} + \cdots - 27\!\cdots\!92 ) / 82\!\cdots\!64 \) (-9569408074536055271v^19 + 24649945171773632952v^18 + 396107601496966363061v^17 - 973892531275711320672v^16 - 6792695453771363119002v^15 + 15753965285126383562052v^14 + 62418103883777900610081v^13 - 134498152647163982587701v^12 - 331021848024658416411294v^11 + 650137905759004285695190v^10 + 1010262475384154246208661v^9 - 1767429843770061485193044v^8 - 1648995759554446180209991v^7 + 2511614789338314474236142v^6 + 1201809609390903876581888v^5 - 1594565671757926148434730v^4 - 312082778136137483040165v^3 + 390879569544140083771539v^2 + 19767626575477333918044*v - 27850777577496681374492) / 823533647449663215664
\(\beta_{6}\)	\(=\)	\( ( - 70\!\cdots\!53 \nu^{19} + \cdots - 44\!\cdots\!68 ) / 41\!\cdots\!32 \) (-7014692873478297653v^19 + 18207153078386711651v^18 + 292051905154766939146v^17 - 720801215654170257518v^16 - 5054640218514283792020v^15 + 11692720394271277335604v^14 + 47137916423067957109167v^13 - 100255246378520321737940v^12 - 256124290597711630152438v^11 + 488207780234346381370780v^10 + 815141403186918260650031v^9 - 1346803512278136647561641v^8 - 1441014449207144333331822v^7 + 1980376542031778051218532v^6 + 1255201550115397051832436v^5 - 1379688574031738026708282v^4 - 483544863739493251774509v^3 + 419492901198024146522232v^2 + 64938465473169035086524*v - 44697360709085211730968) / 411766823724831607832
\(\beta_{7}\)	\(=\)	\( ( - 14\!\cdots\!30 \nu^{19} + \cdots - 88\!\cdots\!44 ) / 41\!\cdots\!32 \) (-14275040358484190930v^19 + 35564773590446679513v^18 + 595619582336433498339v^17 - 1405505053116533831686v^16 - 10328080021812593522302v^15 + 22755690263034680545220v^14 + 96440710321246468959146v^13 - 194699008608719038752577v^12 - 524135018901578189540260v^11 + 945979890961818641035618v^10 + 1665710234336257735484130v^9 - 2603432809055666795214851v^8 - 2932639461590046530388409v^7 + 3818037069540497870231702v^6 + 2531050112702131713534948v^5 - 2651157746771566082877136v^4 - 952952945827088160590228v^3 + 807916605805329522276899v^2 + 123215583150827824254508*v - 88406631822517044975444) / 411766823724831607832
\(\beta_{8}\)	\(=\)	\( ( - 15\!\cdots\!04 \nu^{19} + \cdots - 10\!\cdots\!28 ) / 41\!\cdots\!32 \) (-15598566115602060704v^19 + 40100675851661506607v^18 + 648866972032840013977v^17 - 1588860537108020117572v^16 - 11210001972265994852870v^15 + 25809581622190907883400v^14 + 104207591705369920165872v^13 - 221783015790720698987897v^12 - 563243113391021769309474v^11 + 1083756203823788176878210v^10 + 1777800882462961290272348v^9 - 3005663271094123468595253v^8 - 3103108196575713780380951v^7 + 4453097988433135424957800v^6 + 2652655651432292468238784v^5 - 3126007682766600097140256v^4 - 1005503176616978442040206v^3 + 950612199055435304465763v^2 + 136220572442809428295318*v - 101389011487409166353728) / 411766823724831607832
\(\beta_{9}\)	\(=\)	\( ( - 16\!\cdots\!85 \nu^{19} + \cdots - 10\!\cdots\!24 ) / 41\!\cdots\!32 \) (-16677370577211688285v^19 + 46241967353496760846v^18 + 684247318919106117117v^17 - 1837717234647665757692v^16 - 11608215450964175851454v^15 + 29970040964890744491976v^14 + 105289314333143547337135v^13 - 258871383992174855369037v^12 - 549808238723446826490146v^11 + 1273468520208975608829794v^10 + 1648416722025751686099607v^9 - 3560167768375048317588914v^8 - 2642111048924859676071631v^7 + 5311227483651745139418158v^6 + 1915797204646636166150112v^5 - 3707198839878928377381098v^4 - 554803451843138843132659v^3 + 1085438279246541642542043v^2 + 55262068917007461318732*v - 106391233707116453968124) / 411766823724831607832
\(\beta_{10}\)	\(=\)	\( ( 36\!\cdots\!05 \nu^{19} + \cdots + 22\!\cdots\!92 ) / 82\!\cdots\!64 \) (36169893905645014705v^19 - 96377142452016339696v^18 - 1493586847924405703071v^17 + 3819113310872579890356v^16 + 25555144342263280636510v^15 - 62040797292486429076236v^14 - 234461103759540805419743v^13 + 533035762367024678055859v^12 + 1243878445415285171362222v^11 - 2602974461723750582706218v^10 - 3815994471115845136674363v^9 + 7204640498638581087737548v^8 + 6341947448788652507568757v^7 - 10613771393939837967261326v^6 - 4905882095300283035096144v^5 + 7332277533396015427895950v^4 + 1544917681848650860707475v^3 - 2165073540709606564816365v^2 - 165367804597871223455864*v + 222069905370513345911292) / 823533647449663215664
\(\beta_{11}\)	\(=\)	\( ( - 21\!\cdots\!77 \nu^{19} + \cdots - 10\!\cdots\!52 ) / 41\!\cdots\!32 \) (-21535958159451015477v^19 + 54087822028955236878v^18 + 896432301627379190681v^17 - 2134906823390588131032v^16 - 15495209422539506721662v^15 + 34500613172044707503744v^14 + 144070144723954051903507v^13 - 294316469731423680581677v^12 - 778224430505888743547314v^11 + 1422864525507176519921826v^10 + 2450271585878722239015719v^9 - 3880161600946466134294822v^8 - 4246192145408306261253295v^7 + 5584282033922844833303850v^6 + 3554345875927994567957072v^5 - 3712349348173138121318442v^4 - 1273054226484477089243763v^3 + 1040527424374025402482279v^2 + 156031082018758851094068*v - 102626527024881843828052) / 411766823724831607832
\(\beta_{12}\)	\(=\)	\( ( 25\!\cdots\!01 \nu^{19} + \cdots + 15\!\cdots\!20 ) / 41\!\cdots\!32 \) (25522351282354860301v^19 - 63178601718280668219v^18 - 1067323675605995599844v^17 + 2497526707637907741188v^16 + 18561196719587235628796v^15 - 40444768530822189247532v^14 - 173966469239894140248563v^13 + 346062837411689510867028v^12 + 950049188186007642023660v^11 - 1680901991375531776309396v^10 - 3038588582089573125561299v^9 + 4621712149890604895912341v^8 + 5396744628873085523992856v^7 - 6764330840294206868500202v^6 - 4716342162673262883768072v^5 + 4679735536457247317542350v^4 + 1798118431050338760290541v^3 - 1415680559898275822939960v^2 - 234598047542441457749410*v + 153747574958703904184820) / 411766823724831607832
\(\beta_{13}\)	\(=\)	\( ( - 61\!\cdots\!29 \nu^{19} + \cdots - 38\!\cdots\!56 ) / 82\!\cdots\!64 \) (-61664393577253693629v^19 + 156866291650753229794v^18 + 2567686724129744596713v^17 - 6208055068974971645548v^16 - 44414005982457902190962v^15 + 100686783357438426377284v^14 + 413464057631363747908259v^13 - 863364454393861191745013v^12 - 2238389812654166549854970v^11 + 4206316289685832769854542v^10 + 7076336256848384746997183v^9 - 11615689098371207918355754v^8 - 12360103553859761144564227v^7 + 17101058578782317223205550v^6 + 10522945016937225138366408v^5 - 11899371282152474347915414v^4 - 3888060877446157346137219v^3 + 3591039701967846269773531v^2 + 493874859524691603872924*v - 383209971567608496412556) / 823533647449663215664
\(\beta_{14}\)	\(=\)	\( ( 53\!\cdots\!03 \nu^{19} + \cdots + 28\!\cdots\!92 ) / 41\!\cdots\!32 \) (53194626553656950703v^19 - 135532726213421561083v^18 - 2212230382529471326700v^17 + 5358879169042303804118v^16 + 38197773608685737051440v^15 - 86795731153244046839680v^14 - 354677036074572054613013v^13 + 742664584865655915520354v^12 + 1912592989408242788640698v^11 - 3605457325397145416892540v^10 - 6007943841881849602422669v^9 + 9893110124989448304870481v^8 + 10375535925344342958711780v^7 - 14380554855505696072702604v^6 - 8637024511486738557306736v^5 + 9729334539479671127560978v^4 + 3079568686928735131061123v^3 - 2796801987796843014461162v^2 - 376216808750829779256888*v + 280827911430601061289792) / 411766823724831607832
\(\beta_{15}\)	\(=\)	\( ( 26\!\cdots\!35 \nu^{19} + \cdots + 16\!\cdots\!04 ) / 20\!\cdots\!16 \) (26650943069260124635v^19 - 68260365115207885059v^18 - 1108570898808718582360v^17 + 2702853281328041328094v^16 + 19147515713816010272110v^15 - 43863365594487087126966v^14 - 177887299292286853784101v^13 + 376364040496432144066808v^12 + 960192454997171971519520v^11 - 1834747749369066815120746v^10 - 3021885505798423876692129v^9 + 5067770981968503801591397v^8 + 5239719000322216233164230v^7 - 7452636421714885508198628v^6 - 4404742633514488311166600v^5 + 5158755128317550186576320v^4 + 1605670966256923306939131v^3 - 1541075643135095592073090v^2 - 203836207487735425137450*v + 162422662708870559304404) / 205883411862415803916
\(\beta_{16}\)	\(=\)	\( ( 62\!\cdots\!09 \nu^{19} + \cdots + 35\!\cdots\!60 ) / 41\!\cdots\!32 \) (62935427067167918509v^19 - 158092317093941784425v^18 - 2623804966054947371174v^17 + 6250198321263868147240v^16 + 45448524884245095835976v^15 - 101229950107530962215608v^14 - 423772778788615472015615v^13 + 866332013711001208832398v^12 + 2298324068958364780326656v^11 - 4208697367957870306416860v^10 - 7280247138878190734761227v^9 + 11570452621956829306570379v^8 + 12743687048208570340906658v^7 - 16907075327806329707508410v^6 - 10872632014287891311033868v^5 + 11607964201009981515858962v^4 + 4018803444506766803146369v^3 - 3435457872810729861496594v^2 - 511589199375724222446978*v + 357545261986736385724860) / 411766823724831607832
\(\beta_{17}\)	\(=\)	\( ( 77\!\cdots\!42 \nu^{19} + \cdots + 45\!\cdots\!80 ) / 41\!\cdots\!32 \) (77276218553989410142v^19 - 198141323389589916075v^18 - 3212043392456933398313v^17 + 7841912561375485501422v^16 + 55430084121472177605938v^15 - 127191287489237632214860v^14 - 514407799192517173913382v^13 + 1090630176168171280421671v^12 + 2772938958733090083095792v^11 - 5312634219135896337638522v^10 - 8712064367815250995619146v^9 + 14660184697568432437692237v^8 + 15070521387300273796857971v^7 - 21529871188704902552823130v^6 - 12619396732569385625013376v^5 + 14860815304449369682789964v^4 + 4566116042925290990646384v^3 - 4405038241145516756468341v^2 - 573172031075088255581860*v + 456908112228334478063580) / 411766823724831607832
\(\beta_{18}\)	\(=\)	\( ( - 85\!\cdots\!43 \nu^{19} + \cdots - 47\!\cdots\!12 ) / 41\!\cdots\!32 \) (-85247975037738489443v^19 + 217155425915748025307v^18 + 3546288157188168270896v^17 - 8588892736083309437858v^16 - 61258211242557700448436v^15 + 139177357831391053372888v^14 + 569155236371480105635181v^13 - 1191759589663760902362586v^12 - 3072159541927443323320998v^11 + 5792904257196500074002708v^10 + 9666203435234322266379017v^9 - 15930656593109880977499749v^8 - 16743874497677190134236920v^7 + 23259039489133120766764028v^6 + 14027901218124801984733004v^5 - 15891230769537235512700642v^4 - 5063048453789215951876471v^3 + 4652041380734368206495042v^2 + 634266153395244698909736*v - 478823638192010140572912) / 411766823724831607832
\(\beta_{19}\)	\(=\)	\( ( - 93\!\cdots\!57 \nu^{19} + \cdots - 52\!\cdots\!56 ) / 41\!\cdots\!32 \) (-93311024964931847257v^19 + 238191652796382933431v^18 + 3880079667452563258998v^17 - 9421549183014842664342v^16 - 66987594585877851136624v^15 + 152685967956977736771228v^14 + 621940872085096849375287v^13 - 1307642995679168824594496v^12 - 3353828354008132474052246v^11 + 6357733178383728147368364v^10 + 10537918833288537458771531v^9 - 17489939780338943491417937v^8 - 18214554671619509526414082v^7 + 25545387746039728248309800v^6 + 15199356748005014045578264v^5 - 17451812053484762608665794v^4 - 5444731406633213879243857v^3 + 5097116464894142898346812v^2 + 671166735918401389410136*v - 521361049039800440281856) / 411766823724831607832

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 5 \) b2 + 5
\(\nu^{3}\)	\(=\)	\( \beta_{19} + \beta_{17} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 7\beta _1 + 2 \) b19 + b17 + b13 + b12 - b11 + b7 - b6 - b5 + b4 + 7*b1 + 2
\(\nu^{4}\)	\(=\)	\( \beta_{18} - \beta_{16} + 2\beta_{12} - 2\beta_{11} + \beta_{10} - 2\beta_{8} - \beta_{5} + 10\beta_{2} + \beta _1 + 40 \) b18 - b16 + 2b12 - 2b11 + b10 - 2b8 - b5 + 10b2 + b1 + 40
\(\nu^{5}\)	\(=\)	\( 14 \beta_{19} + 4 \beta_{18} + 14 \beta_{17} + \beta_{15} + 3 \beta_{14} + 14 \beta_{13} + 14 \beta_{12} + \cdots + 32 \) 14b19 + 4b18 + 14b17 + b15 + 3b14 + 14b13 + 14b12 - 14b11 - 2b8 + 9b7 - 16b6 - 16b5 + 14b4 - b3 - b2 + 57*b1 + 32
\(\nu^{6}\)	\(=\)	\( \beta_{19} + 20 \beta_{18} + 3 \beta_{17} - 15 \beta_{16} + \beta_{14} + 6 \beta_{13} + 36 \beta_{12} + \cdots + 362 \) b19 + 20b18 + 3b17 - 15b16 + b14 + 6b13 + 36b12 - 35b11 + 19b10 - 34b8 + b7 - 3b6 - 19b5 + b3 + 98b2 + 16b1 + 362
\(\nu^{7}\)	\(=\)	\( 164 \beta_{19} + 75 \beta_{18} + 164 \beta_{17} - \beta_{16} + 17 \beta_{15} + 51 \beta_{14} + \cdots + 412 \) 164b19 + 75b18 + 164b17 - b16 + 17b15 + 51b14 + 170b13 + 175b12 - 169b11 + 5b10 - b9 - 48b8 + 71b7 - 199b6 - 199b5 + 161b4 - 23b3 - 14b2 + 503*b1 + 412
\(\nu^{8}\)	\(=\)	\( 23 \beta_{19} + 294 \beta_{18} + 72 \beta_{17} - 181 \beta_{16} + 22 \beta_{14} + 152 \beta_{13} + \cdots + 3499 \) 23b19 + 294b18 + 72b17 - 181b16 + 22b14 + 152b13 + 509b12 - 469b11 + 271b10 - 4b9 - 445b8 + 11b7 - 71b6 - 261b5 - 2b4 + 17b3 + 979b2 + 205b1 + 3499
\(\nu^{9}\)	\(=\)	\( 1820 \beta_{19} + 1072 \beta_{18} + 1820 \beta_{17} - 23 \beta_{16} + 226 \beta_{15} + 658 \beta_{14} + \cdots + 4964 \) 1820b19 + 1072b18 + 1820b17 - 23b16 + 226b15 + 658b14 + 1972b13 + 2116b12 - 1960b11 + 133b10 - 35b9 - 784b8 + 510b7 - 2287b6 - 2275b5 + 1732b4 - 378b3 - 131b2 + 4681*b1 + 4964
\(\nu^{10}\)	\(=\)	\( 392 \beta_{19} + 3851 \beta_{18} + 1204 \beta_{17} - 2046 \beta_{16} - 12 \beta_{15} + 362 \beta_{14} + \cdots + 35240 \) 392b19 + 3851b18 + 1204b17 - 2046b16 - 12b15 + 362b14 + 2596b13 + 6585b12 - 5721b11 + 3461b10 - 112b9 - 5367b8 + 35b7 - 1164b6 - 3205b5 - 43b4 + 173b3 + 9933b2 + 2488*b1 + 35240
\(\nu^{11}\)	\(=\)	\( 19791 \beta_{19} + 13860 \beta_{18} + 19798 \beta_{17} - 380 \beta_{16} + 2746 \beta_{15} + 7733 \beta_{14} + \cdots + 58227 \) 19791b19 + 13860b18 + 19798b17 - 380b16 + 2746b15 + 7733b14 + 22470b13 + 25146b12 - 22386b11 + 2421b10 - 739b9 - 10987b8 + 3117b7 - 25462b6 - 25125b5 + 18090b4 - 5325b3 - 845b2 + 45172*b1 + 58227
\(\nu^{12}\)	\(=\)	\( 5990 \beta_{19} + 47697 \beta_{18} + 17378 \beta_{17} - 22537 \beta_{16} - 352 \beta_{15} + \cdots + 364541 \) 5990b19 + 47697b18 + 17378b17 - 22537b16 - 352b15 + 5348b14 + 37644b13 + 81405b12 - 66926b11 + 41851b10 - 2072b9 - 62649b8 - 1080b7 - 16534b6 - 37549b5 - 540b4 + 1060b3 + 101939b2 + 29709*b1 + 364541
\(\nu^{13}\)	\(=\)	\( 213578 \beta_{19} + 170527 \beta_{18} + 213966 \beta_{17} - 5638 \beta_{16} + 31853 \beta_{15} + \cdots + 674271 \) 213578b19 + 170527b18 + 213966b17 - 5638b16 + 31853b15 + 87368b14 + 254168b13 + 295466b12 - 253854b11 + 37453b10 - 12384b9 - 142512b8 + 11621b7 - 279570b6 - 273525b5 + 186377b4 - 68897b3 - 731b2 + 447235*b1 + 674271
\(\nu^{14}\)	\(=\)	\( 86220 \beta_{19} + 572845 \beta_{18} + 232281 \beta_{17} - 245431 \beta_{16} - 6358 \beta_{15} + \cdots + 3840195 \) 86220b19 + 572845b18 + 232281b17 - 245431b16 - 6358b15 + 74366b14 + 501013b13 + 979451b12 - 767374b11 + 490905b10 - 32057b9 - 720434b8 - 30734b7 - 218859b6 - 431080b5 - 4357b4 - 2603b3 + 1055468b2 + 352274*b1 + 3840195
\(\nu^{15}\)	\(=\)	\( 2299959 \beta_{19} + 2039723 \beta_{18} + 2311415 \beta_{17} - 79862 \beta_{16} + 359435 \beta_{15} + \cdots + 7753209 \) 2299959b19 + 2039723b18 + 2311415b17 - 79862b16 + 359435b15 + 969792b14 + 2866427b13 + 3443350b12 - 2867564b11 + 529474b10 - 182181b9 - 1768735b8 - 77945b7 - 3051288b6 - 2962627b5 + 1909517b4 - 847038b3 + 97219b2 + 4512832*b1 + 7753209
\(\nu^{16}\)	\(=\)	\( 1189995 \beta_{19} + 6754747 \beta_{18} + 2966467 \beta_{17} - 2659394 \beta_{16} - 90863 \beta_{15} + \cdots + 40982314 \) 1189995b19 + 6754747b18 + 2966467b17 - 2659394b16 - 90863b15 + 991256b14 + 6339791b13 + 11579202b12 - 8713967b11 + 5654468b10 - 449924b9 - 8221540b8 - 556925b7 - 2784865b6 - 4909589b5 - 6152b4 - 217910b3 + 11008347b2 + 4157004*b1 + 40982314
\(\nu^{17}\)	\(=\)	\( 24777841 \beta_{19} + 23975472 \beta_{18} + 25028426 \beta_{17} - 1099132 \beta_{16} + 3985942 \beta_{15} + \cdots + 88770275 \) 24777841b19 + 23975472b18 + 25028426b17 - 1099132b16 + 3985942b15 + 10683465b14 + 32289323b13 + 39876925b12 - 32318489b11 + 7073667b10 - 2470656b9 - 21362157b8 - 2729714b7 - 33229607b6 - 32068832b5 + 19543601b4 - 10082253b3 + 2280350b2 + 46215262*b1 + 88770275
\(\nu^{18}\)	\(=\)	\( 15892346 \beta_{19} + 78726084 \beta_{18} + 36786988 \beta_{17} - 28763783 \beta_{16} - 1120939 \beta_{15} + \cdots + 441623570 \) 15892346b19 + 78726084b18 + 36786988b17 - 28763783b16 - 1120939b15 + 12794164b14 + 77696657b13 + 135250962b12 - 98509439b11 + 64394266b10 - 5949013b9 - 93420569b8 - 8548058b7 - 34580995b6 - 55790281b5 + 617386b4 - 4663205b3 + 115545147b2 + 48851039*b1 + 441623570
\(\nu^{19}\)	\(=\)	\( 267373082 \beta_{19} + 278607093 \beta_{18} + 271971867 \beta_{17} - 14771407 \beta_{16} + \cdots + 1013531650 \) 267373082b19 + 278607093b18 + 271971867b17 - 14771407b16 + 43709412b15 + 117389942b14 + 363589934b13 + 459520838b12 - 363728299b11 + 90957345b10 - 31744080b9 - 253447918b8 - 47129611b7 - 361797790b6 - 347656845b5 + 200356442b4 - 117481879b3 + 38563892b2 + 479017649*b1 + 1013531650

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

$p$	$F_p(T)$
$2$	\( (T - 1)^{20} \) (T - 1)^20
$3$	\( T^{20} + 2 T^{19} + \cdots + 3160 \) T^20 + 2T^19 - 43T^18 - 78T^17 + 774T^16 + 1240T^15 - 7575T^14 - 10337T^13 + 43740T^12 + 48326T^11 - 150887T^10 - 125274T^9 + 299785T^8 + 166516T^7 - 316036T^6 - 97790T^5 + 163407T^4 + 22723T^3 - 38018T^2 - 1644*T + 3160
$5$	\( T^{20} - T^{19} + \cdots - 941208 \) T^20 - T^19 - 71T^18 + 84T^17 + 2074T^16 - 2807T^15 - 32298T^14 + 48624T^13 + 291077T^12 - 477013T^11 - 1546125T^10 + 2720716T^9 + 4716708T^8 - 8882137T^7 - 7658061T^6 + 15705269T^5 + 5549555T^4 - 13642919T^3 - 433352T^2 + 4435868T - 941208
$7$	\( T^{20} \) T^20
$11$	\( T^{20} - 16 T^{19} + \cdots - 7413760 \) T^20 - 16T^19 - 2T^18 + 1232T^17 - 4363T^16 - 33338T^15 + 192217T^14 + 345682T^13 - 3519764T^12 + 10298T^11 + 32566485T^10 - 26414580T^9 - 155549468T^8 + 187729612T^7 + 357374367T^6 - 479527130T^5 - 337464256T^4 + 377439976T^3 + 132365824T^2 - 32857088*T - 7413760
$13$	\( T^{20} + \cdots + 1343686456 \) T^20 - 5T^19 - 149T^18 + 615T^17 + 9687T^16 - 29598T^15 - 359427T^14 + 675745T^13 + 8268356T^12 - 6027809T^11 - 118588210T^10 - 36549577T^9 + 998601674T^8 + 1245475278T^7 - 4067000423T^6 - 9621157408T^5 + 1668550249T^4 + 23678289003T^3 + 26506221558T^2 + 11002593688*T + 1343686456
$17$	\( T^{20} - 3 T^{19} + \cdots - 57524184 \) T^20 - 3T^19 - 171T^18 + 553T^17 + 11333T^16 - 39085T^15 - 371331T^14 + 1367727T^13 + 6363180T^12 - 25542239T^11 - 55513917T^10 + 258753984T^9 + 210262808T^8 - 1387638796T^7 - 49907761T^6 + 3590525932T^5 - 1372023289T^4 - 3565422595T^3 + 1702930770T^2 + 880572924*T - 57524184
$19$	\( T^{20} + \cdots - 271706112 \) T^20 - 2T^19 - 196T^18 + 407T^17 + 15650T^16 - 34627T^15 - 654196T^14 + 1584931T^13 + 15228060T^12 - 41550143T^11 - 190966163T^10 + 610331321T^9 + 1086223716T^8 - 4501980944T^7 - 1195210576T^6 + 12775154896T^5 - 2234751040T^4 - 14981469120T^3 + 3142908928T^2 + 6532388864*T - 271706112
$23$	\( T^{20} + \cdots + 17863827451904 \) T^20 - 38T^19 + 400T^18 + 2451T^17 - 77108T^16 + 328790T^15 + 3739984T^14 - 37659441T^13 - 6710225T^12 + 1348216438T^11 - 3962103940T^10 - 19347207824T^9 + 112132838336T^8 + 53378510880T^7 - 1264850530816T^6 + 1160595805824T^5 + 6320879728384T^4 - 9060656165376T^3 - 14524751065088T^2 + 17664764723200*T + 17863827451904
$29$	\( T^{20} + \cdots + 1970600448 \) T^20 - 17T^19 - 176T^18 + 4481T^17 + 2946T^16 - 464379T^15 + 1401836T^14 + 22978097T^13 - 132412057T^12 - 467438915T^11 + 5052575057T^10 - 2608684141T^9 - 82481424856T^8 + 245111263688T^7 + 230056907224T^6 - 2446857323264T^5 + 5012906450240T^4 - 4314791396800T^3 + 1221418853888T^2 + 105219843584*T + 1970600448
$31$	\( T^{20} + T^{19} + \cdots + 61440 \) T^20 + T^19 - 256T^18 - 337T^17 + 25543T^16 + 41363T^15 - 1254034T^14 - 2427145T^13 + 31019181T^12 + 71251152T^11 - 338949348T^10 - 946044128T^9 + 857445984T^8 + 3841243936T^7 + 1693013440T^6 - 2071270016T^5 - 1182437120T^4 + 242839552T^3 + 106343424T^2 + 6475776T + 61440
$37$	\( T^{20} + \cdots + 36590983568600 \) T^20 - 39T^19 + 344T^18 + 5209T^17 - 102224T^16 + 49042T^15 + 9135783T^14 - 43124644T^13 - 352376976T^12 + 2949943618T^11 + 4559528011T^10 - 88798144712T^9 + 65802912836T^8 + 1292688418685T^7 - 2299239085593T^6 - 8606290781976T^5 + 18446514631364T^4 + 24806877452397T^3 - 47846066165750T^2 - 21477652793288*T + 36590983568600
$41$	\( T^{20} + \cdots + 1936065245184 \) T^20 + 2T^19 - 454T^18 - 220T^17 + 84963T^16 - 74584T^15 - 8380507T^14 + 17253173T^13 + 463034613T^12 - 1431327918T^11 - 14012985728T^10 + 57509739112T^9 + 204315430288T^8 - 1139381474144T^7 - 745528505088T^6 + 9684883321728T^5 - 8499518947584T^4 - 18330311275520T^3 + 31994647093248T^2 - 15392362696704*T + 1936065245184
$43$	\( T^{20} + \cdots + 152943034368 \) T^20 - 9T^19 - 307T^18 + 2544T^17 + 36331T^16 - 282273T^15 - 2066070T^14 + 16012837T^13 + 57399112T^12 - 501527190T^11 - 631249173T^10 + 8631448687T^9 - 2491402910T^8 - 74664620380T^7 + 105913977600T^6 + 235457987200T^5 - 644680136672T^4 + 229965453248T^3 + 576900469248T^2 - 587713384448*T + 152943034368
$47$	\( T^{20} + \cdots - 23753833803776 \) T^20 + 16T^19 - 293T^18 - 5866T^17 + 25471T^16 + 852442T^15 + 368895T^14 - 61797188T^13 - 190428539T^12 + 2260322880T^11 + 12103419880T^10 - 33012269952T^9 - 328839763248T^8 - 182574584960T^7 + 3705479035584T^6 + 9548027579648T^5 - 6026381878784T^4 - 61420681217536T^3 - 107490322900992T^2 - 81777543739392*T - 23753833803776
$53$	\( T^{20} + \cdots - 796895848448 \) T^20 - 56T^19 + 1013T^18 - 282T^17 - 216466T^16 + 2278449T^15 + 5846293T^14 - 235489569T^13 + 940813455T^12 + 7397754185T^11 - 65040599470T^10 - 3996701755T^9 + 1420659821110T^8 - 2815130154716T^7 - 12426379643744T^6 + 37238208528592T^5 + 42452594472832T^4 - 137391538167744T^3 - 80188936607232T^2 + 17397667180544*T - 796895848448
$59$	\( T^{20} + \cdots - 32595519651840 \) T^20 - 31T^19 - 191T^18 + 15302T^17 - 88401T^16 - 2356616T^15 + 28894534T^14 + 77916067T^13 - 2735412269T^12 + 8952417398T^11 + 77970537832T^10 - 589710193205T^9 + 490782721072T^8 + 5712921901349T^7 - 11479628742661T^6 - 23133544988306T^5 + 51931864650912T^4 + 64660896112568T^3 - 72275581331200T^2 - 110429872728064*T - 32595519651840
$61$	\( T^{20} + \cdots - 158972846819328 \) T^20 + 3T^19 - 675T^18 - 730T^17 + 185672T^16 - 145176T^15 - 26697527T^14 + 66761718T^13 + 2112293859T^12 - 8637920117T^11 - 87494255448T^10 + 507132192813T^9 + 1540630617534T^8 - 13921592785968T^7 - 564898867016T^6 + 165183170944512T^5 - 247913428279328T^4 - 587173165562432T^3 + 1698814074463744T^2 - 996682099325952*T - 158972846819328
$67$	\( T^{20} + \cdots + 679462165126584 \) T^20 - 15T^19 - 715T^18 + 10394T^17 + 207160T^16 - 2805893T^15 - 32795152T^14 + 381894656T^13 + 3256794745T^12 - 28371792311T^11 - 212426453813T^10 + 1120443256210T^9 + 8740344347776T^8 - 18885375595559T^7 - 198678904179323T^6 - 26402835182557T^5 + 1836735531255679T^4 + 2704813866996503T^3 - 2214440830269626T^2 - 3927987317386200*T + 679462165126584
$71$	\( T^{20} + \cdots + 48159926026240 \) T^20 - 24T^19 - 363T^18 + 11319T^17 + 48170T^16 - 2247231T^15 - 2465655T^14 + 243146453T^13 - 22871419T^12 - 15461189150T^11 + 8016831088T^10 + 578714454456T^9 - 367616176032T^8 - 12030197933984T^7 + 8204731866176T^6 + 120215581858816T^5 - 86654052595200T^4 - 400816500910592T^3 + 100608627578880T^2 + 294158037786624*T + 48159926026240
$73$	\( T^{20} + \cdots - 17\!\cdots\!44 \) T^20 - 13T^19 - 634T^18 + 8221T^17 + 166053T^16 - 2101315T^15 - 23871182T^14 + 281388125T^13 + 2129492973T^12 - 21363055818T^11 - 125871331040T^10 + 921891106512T^9 + 4924933176368T^8 - 20705507578304T^7 - 116133458129088T^6 + 171170645654656T^5 + 1306846006470656T^4 + 496943400324096T^3 - 3792547905699840T^2 - 4956752292970496*T - 1736658003103744
$79$	\( T^{20} + \cdots - 34\!\cdots\!12 \) T^20 - 43T^19 + 37T^18 + 24131T^17 - 315806T^16 - 3625969T^15 + 99798437T^14 - 159699402T^13 - 12072243811T^12 + 91587594164T^11 + 499547383452T^10 - 8833448273352T^9 + 16029377180512T^8 + 308907283068672T^7 - 1883488922383488T^6 - 389597785041024T^5 + 38537745794402560T^4 - 136992747698286080T^3 + 130176012102862848T^2 + 192324695351531520*T - 349345716577677312
$83$	\( T^{20} + \cdots - 64650141462088 \) T^20 - T^19 - 855T^18 + 1490T^17 + 287086T^16 - 624147T^15 - 48352322T^14 + 106865128T^13 + 4400234187T^12 - 7928461709T^11 - 223170032097T^10 + 234516804780T^9 + 6262050858112T^8 - 532699680813T^7 - 90967629667713T^6 - 81531172518929T^5 + 528888993808431T^4 + 947955449646589T^3 + 86006592550310T^2 - 244249069442872T - 64650141462088
$89$	\( T^{20} + \cdots + 161429430910976 \) T^20 - 749T^18 - 945T^17 + 206446T^16 + 468395T^15 - 26384771T^14 - 76255799T^13 + 1660794953T^12 + 5141414890T^11 - 51997222476T^10 - 145942207688T^9 + 891424069744T^8 + 2017726492096T^7 - 8729518395072T^6 - 14256496848512T^5 + 48333633926912T^4 + 48797596873216T^3 - 139618672996352T^2 - 63279898370048T + 161429430910976
$97$	\( T^{20} + \cdots + 58\!\cdots\!96 \) T^20 - 14T^19 - 722T^18 + 10035T^17 + 208080T^16 - 2867567T^15 - 31512258T^14 + 426909598T^13 + 2810530775T^12 - 36617063225T^11 - 155812321153T^10 + 1871673372989T^9 + 5456505728140T^8 - 56134953678005T^7 - 117889474366142T^6 + 913594258723838T^5 + 1433486958514814T^4 - 6499028261627193T^3 - 7533600221531978T^2 + 7848744676909748*T + 5887999287614696
show more
show less

\( p \)	Sign
\(2\)	\( -1 \)
\(7\)	\( -1 \)
\(101\)	\( -1 \)