LMFDB - Newform orbit 1248.2.bv.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(673,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

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

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

chi := DirichletCharacter("1248.673");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.bv (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 28 x^{14} - 66 x^{13} + 152 x^{12} - 256 x^{11} + 402 x^{10} - 92 x^{9} - 1652 x^{8} + \cdots + 916 $ x^16 - 6x^15 + 28x^14 - 66x^13 + 152x^12 - 256x^11 + 402x^10 - 92x^9 - 1652x^8 + 2326x^7 + 1814x^6 - 3560x^5 - 2307x^4 + 4090x^3 + 1610x^2 - 3204*x + 916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{9} q^{3} + (\beta_{13} - \beta_{11}) q^{5} - \beta_{6} q^{7} + ( - \beta_{9} - 1) q^{9}+O(q^{10}) $ q + b9 * q^3 + (b13 - b11) * q^5 - b6 * q^7 + (-b9 - 1) * q^9	$ q + \beta_{9} q^{3} + (\beta_{13} - \beta_{11}) q^{5} - \beta_{6} q^{7} + ( - \beta_{9} - 1) q^{9} + ( - \beta_{13} + \beta_{5} - \beta_{2}) q^{11} + (\beta_{14} - \beta_{9} - 1) q^{13} - \beta_{13} q^{15} + (\beta_{9} + \beta_{2} + 1) q^{17} + (\beta_{15} - \beta_{14} - \beta_{13} + \cdots + 1) q^{19}+ \cdots + (\beta_{13} - \beta_{11} + \cdots + \beta_1) q^{99}+O(q^{100}) $ q + b9 * q^3 + (b13 - b11) * q^5 - b6 * q^7 + (-b9 - 1) * q^9 + (-b13 + b5 - b2) * q^11 + (b14 - b9 - 1) * q^13 - b13 * q^15 + (b9 + b2 + 1) * q^17 + (b15 - b14 - b13 - b12 - 2b8 + 2b7 + b4 + b1 + 1) * q^19 + (b6 - b3) * q^21 + (b14 + b8 - b7 - b6 + b5 + b4 + b3 - b2 - 2) * q^23 + (b12 - b11 - b3 - 1) * q^25 + q^27 + (-b14 + 2b9 - b8 - b7 - 2b6 - b4 + b1 + 2) * q^29 + (b12 + b11 - 2b9 + 2b7 + 2b6 + 2b4 + b3 - 2b1 - 3) q^31 + (b11 + b7 + b6 - b5 + b2 - b1) * q^33 + (-2b14 - b13 + b12 + b11 + b9 + b7 - 2b5 + b4 + b3 - b2 + b1) * q^35 + (-b14 + b13 + 2b12 + b8 + b7 + b6 - 3b5 + b4 + b3 + b2 - 2) * q^37 + (b8 - b7 - b6 + b3 - b2) * q^39 + (-b14 + b13 + b10 + b8 + b5 + 2b3 - b2 - 1) q^41 + (-b15 + b14 - 2b13 + b11 - 2b10 + 3b9 - b7 - b3 + 3) q^43 + b11 * q^45 + (2b14 - 2b13 + 2b11 + 2b9 + b8 - 2b7 + b5 - b2 - b1) q^47 + (2b15 - b14 + b10 - 2b9 - b8 + 2b7 + 2b6 + 2b4 - 1) q^49 + (-b6 - b2 + b1 - 1) * q^51 + (-2b13 - 2b12 - b8 - b7 - b6 + 3b5 + b3 - 2b2 + b1) * q^53 + (2b14 - b13 + 2b11 - 3b9 + 2b8 - 3b7 + 2b5 + b4 + b3 - 2b2 - b1 - 3) q^55 + (-b15 + 2b14 - b13 - b12 - b10 + b8 - 3b7 - 2b6 + b5 - 2b4 - b2 + 1) * q^57 + (-b13 - b12 - b9 + 2b7 + b6 - b5 + b4 + b3) q^59 + (-b13 + b12 + b11 + b9 + 2b7 + 2b6 + b5 + b4 + b3 - 2b1) q^61 + b3 * q^63 + (-b15 + b14 + b11 + b9 + 2b8 - b5 - b4 - b3 - b1) q^65 + (-b14 + b13 + b10 + b9 + b8 - 2b6 - b5 + 2b3 - 3b2 + 2b1 + 1) * q^67 + (-b14 - b13 - b12 - b6 - b4 - b2 + b1 + 1) * q^69 + (2b15 - b14 - b13 - b12 - 2b9 - 2b8 + 2b7 - b6 + b4 + b2 - b1 + 3) * q^71 + (-b15 + 2b14 + 2b13 + b12 - b11 - b10 - 2b9 + b8 - 2b7 + b6 + b5 + 2b4 - b2 - b1 - 4) q^73 + (-b13 + 2b11 - b9 + b7 + 2b6 + b4 - b1 - 1) * q^75 + (-b12 + b11 - 3b8 + 2b7 + b6 - b5 - 2b3 + 3b2 - b1 + 3) * q^77 + (-2b13 - b12 - b11 - 2b8 + 2b5 - b3 + 1) q^79 + b9 * q^81 + (-4b14 + b12 + b11 + 2b9 - 2b8 + 4b7 + b6 - 2b5 + 2b4 + b3 + b2 + b1 + 1) * q^83 + (-2b11 - 2b9 + b7 - 2b6 - b5 + b1 + 2) q^85 + (b14 + b13 + b12 - 2b9 + b7 + 3b6 + b5 + b4 - 2b3 + b2 - 2b1 - 3) * q^87 + (-2b13 - 2b12 - 2b10 + b9 - b7 - 3b6 - b5 - b4 - b2 + 2b1 + 3) q^89 + (b15 - 2b14 - 2b13 + b12 + 2b11 + b10 + b9 + b8 + 2b7 - b5 + b4 + 2b3 - b2 + b1 - 4) q^91 + (-b13 - 2b12 + b9 - b7 - 2b6 - b4 + b1 + 3) * q^93 + (-2b15 + 5b14 - 2b13 - 2b12 - 4b10 - b9 - 4b7 - 2b6 + 3b5 - 2b4 - b3 - b2 - b1 + 1) q^95 + (b15 - b14 - 2b11 + b9 - 2b8 - 3b6 + b5 - b3 - b2 + 3b1 + 1) * q^97 + (b13 - b11 - b7 - b6 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{3} - 6 q^{7} - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^3 - 6 * q^7 - 8 * q^9	$ 16 q - 8 q^{3} - 6 q^{7} - 8 q^{9} - 6 q^{15} + 6 q^{17} - 8 q^{23} - 20 q^{25} + 16 q^{27} - 10 q^{29} - 6 q^{37} + 6 q^{39} + 6 q^{41} + 18 q^{43} + 6 q^{45} + 14 q^{49} - 12 q^{51} - 12 q^{53} + 20 q^{55} + 12 q^{59} + 16 q^{61} + 6 q^{63} - 6 q^{65} + 30 q^{67} - 8 q^{69} + 24 q^{71} + 10 q^{75} + 8 q^{77} - 20 q^{79} - 8 q^{81} + 30 q^{85} - 10 q^{87} - 46 q^{91} + 6 q^{93} + 4 q^{95} - 18 q^{97}+O(q^{100}) $ 16 * q - 8 * q^3 - 6 * q^7 - 8 * q^9 - 6 * q^15 + 6 * q^17 - 8 * q^23 - 20 * q^25 + 16 * q^27 - 10 * q^29 - 6 * q^37 + 6 * q^39 + 6 * q^41 + 18 * q^43 + 6 * q^45 + 14 * q^49 - 12 * q^51 - 12 * q^53 + 20 * q^55 + 12 * q^59 + 16 * q^61 + 6 * q^63 - 6 * q^65 + 30 * q^67 - 8 * q^69 + 24 * q^71 + 10 * q^75 + 8 * q^77 - 20 * q^79 - 8 * q^81 + 30 * q^85 - 10 * q^87 - 46 * q^91 + 6 * q^93 + 4 * q^95 - 18 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 6 x^{15} + 28 x^{14} - 66 x^{13} + 152 x^{12} - 256 x^{11} + 402 x^{10} - 92 x^{9} - 1652 x^{8} + \cdots + 916 $ x^16 - 6*x^15 + 28*x^14 - 66*x^13 + 152*x^12 - 256*x^11 + 402*x^10 - 92*x^9 - 1652*x^8 + 2326*x^7 + 1814*x^6 - 3560*x^5 - 2307*x^4 + 4090*x^3 + 1610*x^2 - 3204*x + 916 :

$\beta_{1}$	$=$	$ ( 22\!\cdots\!81 \nu^{15} + \cdots - 14\!\cdots\!28 ) / 61\!\cdots\!73 $ (22025466988810458291781v^15 - 185713764819774321198002v^14 + 799942467133718734564931v^13 - 2253005315257952881151510v^12 + 3481560843460023541937425v^11 - 6666644634592414145602033v^10 + 2396738044331130539498292v^9 + 8651765194286259603728880v^8 - 97737718020984482344623339v^7 + 170574111596721546993965153v^6 + 24660247491338905801172857v^5 - 229424657141467119406393650v^4 - 72897990854567801961990799v^3 + 224726262376709725041222283v^2 + 187215413539774625067528258*v - 149379828403731999838219128) / 61643197265233151426841073
$\beta_{2}$	$=$	$ ( 15\!\cdots\!75 \nu^{15} + \cdots - 53\!\cdots\!32 ) / 12\!\cdots\!46 $ (158431676444308138210875v^15 - 771342034499023974976072v^14 + 3575738889937302957221935v^13 - 6152880799341903156854300v^12 + 15467308976199302255790913v^11 - 14006410228127809448680332v^10 + 24059175832060078941755907v^9 + 77998793718427035464881928v^8 - 303452369834938135481718019v^7 + 280961804837598270149852062v^6 + 309235292233667933048102949v^5 - 306293715532909132864614110v^4 - 348471599268278495752134460v^3 + 298925888691998890697699726v^2 + 857122773488001284323074424*v - 530107668790934374795119132) / 123286394530466302853682146
$\beta_{3}$	$=$	$ ( - 27\!\cdots\!97 \nu^{15} + \cdots + 13\!\cdots\!84 ) / 12\!\cdots\!46 $ (-275907785496358370695397v^15 + 1441996231784354464944626v^14 - 6343935341045048722116045v^13 + 12039730166183615036045150v^12 - 26647597524701419053372351v^11 + 39046290534104769189498784v^10 - 49949964955774243851996151v^9 - 54562354425709284074641842v^8 + 491089919160344627161503785v^7 - 234764583166611698519591766v^6 - 1033581623147196377488220411v^5 + 328681891465027144972067468v^4 + 1269010591462621791971745862v^3 + 5464283194453790159683756v^2 - 637585285366456237662788908*v + 134408189972762856179253384) / 123286394530466302853682146
$\beta_{4}$	$=$	$ ( 27\!\cdots\!97 \nu^{15} + \cdots - 13\!\cdots\!84 ) / 12\!\cdots\!46 $ (275907785496358370695397v^15 - 1441996231784354464944626v^14 + 6343935341045048722116045v^13 - 12039730166183615036045150v^12 + 26647597524701419053372351v^11 - 39046290534104769189498784v^10 + 49949964955774243851996151v^9 + 54562354425709284074641842v^8 - 491089919160344627161503785v^7 + 234764583166611698519591766v^6 + 1033581623147196377488220411v^5 - 328681891465027144972067468v^4 - 1269010591462621791971745862v^3 - 5464283194453790159683756v^2 + 884158074427388843370153200*v - 134408189972762856179253384) / 123286394530466302853682146
$\beta_{5}$	$=$	$ ( 43\!\cdots\!54 \nu^{15} + \cdots - 90\!\cdots\!76 ) / 12\!\cdots\!46 $ (431944536547067909218354v^15 - 2346152632946727160590803v^14 + 11071641641227351312218301v^13 - 23567559375987226279293828v^12 + 58521825384246455416293775v^11 - 86426730317296962812259838v^10 + 150961149326854446167214695v^9 + 24447448340461290917359364v^8 - 657471679588651205308721277v^7 + 761884123524190499486983758v^6 + 715754340916210259754291809v^5 - 994247903556661476531651692v^4 - 939501837205534170045679731v^3 + 1288944953366776165444878925v^2 + 1139240240176020328161874670*v - 904067971300908434925642076) / 123286394530466302853682146
$\beta_{6}$	$=$	$ ( - 66\!\cdots\!95 \nu^{15} + \cdots + 46\!\cdots\!96 ) / 12\!\cdots\!46 $ (-662007638649936654806195v^15 + 3171913765072528018860570v^14 - 14727196583675265761947301v^13 + 26045679817851981116891820v^12 - 69673003954139844889043551v^11 + 86095443977338359669829416v^10 - 161645178432414751096639381v^9 - 135742299897787054941727880v^8 + 941994224533795483823818053v^7 - 425031707887208015731287358v^6 - 1615034310944710766542652459v^5 + 390822706206632125266258618v^4 + 1698533496546299207451026750v^3 - 273354544478007624769974792v^2 - 1149623700247367918444867860*v + 464976251780042329301375496) / 123286394530466302853682146
$\beta_{7}$	$=$	$ ( 14\!\cdots\!39 \nu^{15} + \cdots - 17\!\cdots\!74 ) / 12\!\cdots\!46 $ (1455135901748594541504139v^15 - 7526230630674308870090510v^14 + 34945133106618830384261607v^13 - 69270361475756550571436820v^12 + 173781603078464217547501325v^11 - 247509800997590738857118568v^10 + 428021789327673015670358503v^9 + 152824012086985130412239752v^8 - 2163333791950872702474637627v^7 + 1639876585186101684218227210v^6 + 3366735170917851885827342509v^5 - 2103995747464682994563666950v^4 - 4033281216512824853168518448v^3 + 2249701974543790645263483680v^2 + 3119661663781697014886361548*v - 1747713238004962778012603974) / 123286394530466302853682146
$\beta_{8}$	$=$	$ ( 15\!\cdots\!73 \nu^{15} + \cdots - 12\!\cdots\!12 ) / 12\!\cdots\!46 $ (1578224720414536942673873v^15 - 7889393214113164921341231v^14 + 36012848648621255616465294v^13 - 66719590507413214879578332v^12 + 166160411447907054804356778v^11 - 223060944128051202740705964v^10 + 367709004566933522350241204v^9 + 290767814623212028376300088v^8 - 2462011976762887710906012430v^7 + 1275225875994523509878270566v^6 + 4333588287773850369053348448v^5 - 1447946811327876437067725106v^4 - 5223694767630021889619033301v^3 + 961244113361353301900935309v^2 + 3985836787145985347835564498*v - 1287845329985228460656905412) / 123286394530466302853682146
$\beta_{9}$	$=$	$ ( - 46951055886214 \nu^{15} + 257032746536189 \nu^{14} + \cdots + 75\!\cdots\!76 ) / 34\!\cdots\!42 $ (-46951055886214v^15 + 257032746536189v^14 - 1176314795701089v^13 + 2465685193393304v^12 - 5769715610579071v^11 + 8860177294202080v^10 - 13838422313939069v^9 - 3485025483031720v^8 + 76849575188612505v^7 - 68878801246918828v^6 - 124508041677473661v^5 + 101359752535599028v^4 + 167504208396733877v^3 - 100414409469165505v^2 - 135720127040329650*v + 75200015167695276) / 3497569387271042
$\beta_{10}$	$=$	$ ( - 15\!\cdots\!12 \nu^{15} + \cdots + 29\!\cdots\!44 ) / 61\!\cdots\!09 $ (-1539578758935525738312v^15 + 8441957746091932878520v^14 - 38740786127224027742716v^13 + 81333425248276730200694v^12 - 190661225462608568738424v^11 + 289710297569014776913130v^10 - 457010603900253484025808v^9 - 125536899509183652710922v^8 + 2526250998237719136070048v^7 - 2356045582033203159539570v^6 - 4011296247143239946556684v^5 + 3723928606511433394404070v^4 + 5270136222198310906668340v^3 - 3955104084962523293048646v^2 - 4461016114890993851414024*v + 2956844482511742322681544) / 61828683315178687489309
$\beta_{11}$	$=$	$ ( 31\!\cdots\!25 \nu^{15} + \cdots - 49\!\cdots\!50 ) / 12\!\cdots\!46 $ (3137862819106083689160325v^15 - 17262882616152875742381323v^14 + 79089288245895737534838348v^13 - 166779868064650769547323560v^12 + 389871353462393256762958658v^11 - 600736247635700802292513000v^10 + 943334700449688038821403766v^9 + 214886820433435090338422336v^8 - 5121438063869242659093592080v^7 + 4754041370819597794114964774v^6 + 8325551518246613389414288252v^5 - 7117762696983641211330709914v^4 - 11305583445794313174209647663v^3 + 7108297204894458689484729625v^2 + 9327569939812682107766786946*v - 4992050113344034376128069050) / 123286394530466302853682146
$\beta_{12}$	$=$	$ ( 36\!\cdots\!12 \nu^{15} + \cdots - 59\!\cdots\!72 ) / 12\!\cdots\!46 $ (3630359091771684646349212v^15 - 19730476345868321941143357v^14 + 90797641844521220042080043v^13 - 189903538203075092850457548v^12 + 451899560148424181132137179v^11 - 689654391144845841083424496v^10 + 1106983252034229096193785451v^9 + 231959617693905570817426104v^8 - 5777926135096327048824197723v^7 + 5186909235959146688347423848v^6 + 9014902449852256600580000513v^5 - 7429967498970432320328556392v^4 - 12027770073639510395476504265v^3 + 7760983297320547738838421497v^2 + 9468298931218622644042325230*v - 5968538367712588107440943972) / 123286394530466302853682146
$\beta_{13}$	$=$	$ ( - 36\!\cdots\!29 \nu^{15} + \cdots + 59\!\cdots\!48 ) / 12\!\cdots\!46 $ (-3635362423362086268676829v^15 + 19717867039416526571025497v^14 - 90369244263251086439455402v^13 + 187754643974363038728551400v^12 - 443617891860852261369255560v^11 + 675557146998215839165588150v^10 - 1067556222861213916898880160v^9 - 276648001823191190532154704v^8 + 5864597854314759435514596222v^7 - 5044015135041195977870776180v^6 - 9480312005264190424237930522v^5 + 7334379103597741483828554402v^4 + 12641818831176614369048166077v^3 - 7319575488847416887411187277v^2 - 9996672451584157414767776066*v + 5990078023184889863938269648) / 123286394530466302853682146
$\beta_{14}$	$=$	$ ( 21\!\cdots\!44 \nu^{15} + \cdots - 35\!\cdots\!95 ) / 61\!\cdots\!73 $ (2178818650005461712930544v^15 - 11868279567445867401674576v^14 + 54625820274211500803417695v^13 - 114509346148879528552647699v^12 + 272205439913995563681676986v^11 - 415952293978370851874808566v^10 + 669520087298167926693694079v^9 + 134995869435111142342455085v^8 - 3457132618087682392917641901v^7 + 3151623624193745277129597486v^6 + 5489389138886426015564165112v^5 - 4579629137840375709523610651v^4 - 7237042258795166110702221553v^3 + 4831652390036039992812406800v^2 + 5851494438949027510008829082*v - 3528985670518049538479928495) / 61643197265233151426841073
$\beta_{15}$	$=$	$ ( - 22\!\cdots\!12 \nu^{15} + \cdots + 31\!\cdots\!92 ) / 61\!\cdots\!09 $ (-2274550404808901490212v^15 + 12033319727491866708414v^14 - 55218049088125161967702v^13 + 111227917547308479075596v^12 - 268147177743543095581222v^11 + 393838335053286249299488v^10 - 640611730585334674278502v^9 - 240617864551049127684452v^8 + 3577440187969011090727902v^7 - 2780758853277706854224464v^6 - 5987923215820554548932902v^5 + 3830094885701049707215764v^4 + 7896073221885649768028010v^3 - 3892274981877633201680966v^2 - 6199034970862445514643424*v + 3118332418160600492165492) / 61828683315178687489309

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

$\nu$	$=$	$ ( \beta_{4} + \beta_{3} ) / 2 $ (b4 + b3) / 2
$\nu^{2}$	$=$	$ ( \beta_{14} + \beta_{13} - \beta_{9} + \beta_{8} - 4 \beta_{7} - 4 \beta_{6} + \beta_{5} + 3 \beta_{3} + \cdots - 5 ) / 2 $ (b14 + b13 - b9 + b8 - 4b7 - 4b6 + b5 + 3b3 - 3b2 + 2*b1 - 5) / 2
$\nu^{3}$	$=$	$ ( 3 \beta_{15} + 4 \beta_{14} + 12 \beta_{13} + 8 \beta_{12} + 4 \beta_{11} - \beta_{10} + 4 \beta_{9} + \cdots - 34 ) / 4 $ (3b15 + 4b14 + 12b13 + 8b12 + 4b11 - b10 + 4b9 + 12b8 - 14b7 - 2b6 + 8b5 - 4b4 - 2b3 - 6b2 - 2b1 - 34) / 4
$\nu^{4}$	$=$	$ ( - \beta_{15} - 3 \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} + \cdots - 21 ) / 2 $ (-b15 - 3b14 + b13 - 3b12 + 3b11 - 2b10 + 8b9 + 4b8 + 13b7 + 23b6 + 15b5 - 5b4 - 22b3 - b2 - 6b1 - 21) / 2
$\nu^{5}$	$=$	$ ( - 14 \beta_{15} - 36 \beta_{14} - 76 \beta_{13} - 104 \beta_{12} - 64 \beta_{11} + 3 \beta_{10} + \cdots + 182 ) / 4 $ (-14b15 - 36b14 - 76b13 - 104b12 - 64b11 + 3b10 - 196b9 - 68b8 + 170b7 + 106b6 - 8b5 - 66b4 - 92b3 + 54b2 - 42*b1 + 182) / 4
$\nu^{6}$	$=$	$ ( - 21 \beta_{15} - 44 \beta_{14} - 23 \beta_{13} - 131 \beta_{11} + 7 \beta_{10} - 280 \beta_{9} + \cdots + 560 ) / 2 $ (-21b15 - 44b14 - 23b13 - 131b11 + 7b10 - 280b9 - 137b8 + 251b7 + 153b6 - 257b5 + 16b4 - 24b3 + 243b2 - 96b1 + 560) / 2
$\nu^{7}$	$=$	$ ( - 43 \beta_{15} + 184 \beta_{14} + 106 \beta_{13} + 634 \beta_{12} + 24 \beta_{11} + 64 \beta_{10} + \cdots + 1792 ) / 4 $ (-43b15 + 184b14 + 106b13 + 634b12 + 24b11 + 64b10 + 1102b9 - 328b8 - 48b7 - 294b6 - 1574b5 + 832b4 + 944b3 + 784b2 + 256*b1 + 1792) / 4
$\nu^{8}$	$=$	$ ( 282 \beta_{15} + 1157 \beta_{14} - 141 \beta_{13} + 494 \beta_{12} + 1304 \beta_{11} + 189 \beta_{10} + \cdots - 1854 ) / 2 $ (282b15 + 1157b14 - 141b13 + 494b12 + 1304b11 + 189b10 + 4444b9 + 1009b8 - 3905b7 - 3426b6 + 558b5 + 379b4 + 2004b3 - 1706b2 + 1677*b1 - 1854) / 2
$\nu^{9}$	$=$	$ ( 2295 \beta_{15} + 5080 \beta_{14} + 216 \beta_{13} - 388 \beta_{12} + 8272 \beta_{11} + 9 \beta_{10} + \cdots - 29594 ) / 4 $ (2295b15 + 5080b14 + 216b13 - 388b12 + 8272b11 + 9b10 + 19264b9 + 10664b8 - 25742b7 - 17398b6 + 22116b5 - 3544b4 + 1474b3 - 18270b2 + 5794*b1 - 29594) / 4
$\nu^{10}$	$=$	$ ( - 93 \beta_{15} - 6223 \beta_{14} + 3156 \beta_{13} - 6432 \beta_{12} + 423 \beta_{11} - 2572 \beta_{10} + \cdots - 43949 ) / 2 $ (-93b15 - 6223b14 + 3156b13 - 6432b12 + 423b11 - 2572b10 - 18721b9 + 5980b8 + 2065b7 + 8331b6 + 32908b5 - 5340b4 - 16095b3 - 15886b2 - 6336*b1 - 43949) / 2
$\nu^{11}$	$=$	$ ( - 23072 \beta_{15} - 97536 \beta_{14} + 30696 \beta_{13} - 43616 \beta_{12} - 93924 \beta_{11} + \cdots - 27018 ) / 4 $ (-23072b15 - 97536b14 + 30696b13 - 43616b12 - 93924b11 - 16077b10 - 395028b9 - 70172b8 + 292538b7 + 235862b6 - 27072b5 - 19258b4 - 114168b3 + 83166b2 - 103166*b1 - 27018) / 4
$\nu^{12}$	$=$	$ ( - 36441 \beta_{15} - 98272 \beta_{14} + 37770 \beta_{13} + 23779 \beta_{12} - 166417 \beta_{11} + \cdots + 485421 ) / 2 $ (-36441b15 - 98272b14 + 37770b13 + 23779b12 - 166417b11 - 1935b10 - 445536b9 - 179243b8 + 509906b7 + 373327b6 - 443041b5 + 14496b4 - 58664b3 + 366723b2 - 136929*b1 + 485421) / 2
$\nu^{13}$	$=$	$ ( - 17947 \beta_{15} + 421968 \beta_{14} - 285198 \beta_{13} + 543910 \beta_{12} - 60520 \beta_{11} + \cdots + 4119420 ) / 4 $ (-17947b15 + 421968b14 - 285198b13 + 543910b12 - 60520b11 + 176250b10 + 1718354b9 - 540172b8 + 398948b7 + 56354b6 - 2795650b5 + 337176b4 + 767044b3 + 1691756b2 + 242848*b1 + 4119420) / 4
$\nu^{14}$	$=$	$ ( 386080 \beta_{15} + 1922393 \beta_{14} - 1218787 \beta_{13} + 425004 \beta_{12} + 1710132 \beta_{11} + \cdots + 2272183 ) / 2 $ (386080b15 + 1922393b14 - 1218787b13 + 425004b12 + 1710132b11 + 381137b10 + 7926247b9 + 987885b8 - 5309788b7 - 4555154b6 + 231045b5 + 314448b4 + 2156313b3 - 1327903b2 + 2102356*b1 + 2272183) / 2
$\nu^{15}$	$=$	$ ( 2965963 \beta_{15} + 8645508 \beta_{14} - 4370400 \beta_{13} - 2173804 \beta_{12} + 12451988 \beta_{11} + \cdots - 30991414 ) / 4 $ (2965963b15 + 8645508b14 - 4370400b13 - 2173804b12 + 12451988b11 + 592327b10 + 35275512b9 + 13212132b8 - 42821026b7 - 33380498b6 + 33472204b5 - 694504b4 + 7192322b3 - 28570270b2 + 11652638*b1 - 30991414) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$1 + \beta_{9}$	$1$	$1$

Embeddings

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

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

673.1

1.10545	−	0.463134i
0.570735	+	0.0355035i
−0.791982	+	0.726156i
−1.18206	−	0.128024i
−1.00880	−	2.24651i
1.16409	+	2.58125i
1.32930	+	0.617465i
1.81326	−	2.85475i
1.81326	+	2.85475i
1.32930	−	0.617465i
1.16409	−	2.58125i
−1.00880	+	2.24651i
−1.18206	+	0.128024i
−0.791982	−	0.726156i
0.570735	−	0.0355035i
1.10545	+	0.463134i

−0.500000

−

0.866025i

−

3.89452i

−3.97395

−

2.29436i

−0.500000

0.866025i

673.2

−0.500000

−

0.866025i

−

3.51799i

0.163186

0.0942157i

−0.500000

0.866025i

673.3

−0.500000

−

0.866025i

−

2.12383i

3.18859

1.84094i

−0.500000

0.866025i

673.4

−0.500000

−

0.866025i

−

1.01353i

−0.385170

−

0.222378i

−0.500000

0.866025i

673.5

−0.500000

−

0.866025i

−

0.0205602i

1.31495

0.759185i

−0.500000

0.866025i

673.6

−0.500000

−

0.866025i

2.20096i

2.50556

1.44658i

−0.500000

0.866025i

673.7

−0.500000

−

0.866025i

2.30686i

−3.76164

−

2.17178i

−0.500000

0.866025i

673.8

−0.500000

−

0.866025i

2.59851i

−2.05152

−

1.18445i

−0.500000

0.866025i

1057.1

−0.500000

0.866025i

−

2.59851i

−2.05152

1.18445i

−0.500000

−

0.866025i

1057.2

−0.500000

0.866025i

−

2.30686i

−3.76164

2.17178i

−0.500000

−

0.866025i

1057.3

−0.500000

0.866025i

−

2.20096i

2.50556

−

1.44658i

−0.500000

−

0.866025i

1057.4

−0.500000

0.866025i

0.0205602i

1.31495

−

0.759185i

−0.500000

−

0.866025i

1057.5

−0.500000

0.866025i

1.01353i

−0.385170

0.222378i

−0.500000

−

0.866025i

1057.6

−0.500000

0.866025i

2.12383i

3.18859

−

1.84094i

−0.500000

−

0.866025i

1057.7

−0.500000

0.866025i

3.51799i

0.163186

−

0.0942157i

−0.500000

−

0.866025i

1057.8

−0.500000

0.866025i

3.89452i

−3.97395

2.29436i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1248.2.bv.c	✓	16
4.b	odd	2	1		1248.2.bv.d	yes	16
13.e	even	6	1	inner	1248.2.bv.c	✓	16
52.i	odd	6	1		1248.2.bv.d	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1248.2.bv.c	✓	16	1.a	even	1	1	trivial
1248.2.bv.c	✓	16	13.e	even	6	1	inner
1248.2.bv.d	yes	16	4.b	odd	2	1
1248.2.bv.d	yes	16	52.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1248, [\chi])$:

$ T_{5}^{16} + 50 T_{5}^{14} + 999 T_{5}^{12} + 10300 T_{5}^{10} + 58943 T_{5}^{8} + 184978 T_{5}^{6} + \cdots + 64 $

$ T_{7}^{16} + 6 T_{7}^{15} - 17 T_{7}^{14} - 174 T_{7}^{13} + 309 T_{7}^{12} + 3156 T_{7}^{11} + \cdots + 4096 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + T + 1)^{8} $ (T^2 + T + 1)^8
$5$	$ T^{16} + 50 T^{14} + \cdots + 64 $ T^16 + 50T^14 + 999T^12 + 10300T^10 + 58943T^8 + 184978T^6 + 285369T^4 + 151520*T^2 + 64
$7$	$ T^{16} + 6 T^{15} + \cdots + 4096 $ T^16 + 6T^15 - 17T^14 - 174T^13 + 309T^12 + 3156T^11 - 1892T^10 - 30744T^9 + 23196T^8 + 168576T^7 - 52400T^6 - 467328T^5 + 398096T^4 + 271872T^3 + 3840T^2 - 24576*T + 4096
$11$	$ T^{16} - 52 T^{14} + \cdots + 65536 $ T^16 - 52T^14 + 1932T^12 + 432T^11 - 34544T^10 - 15456T^9 + 448592T^8 + 470208T^7 - 1863488T^6 - 2200320T^5 + 6278400T^4 + 8306688T^3 + 2453504T^2 - 786432*T + 65536
$13$	$ T^{16} + \cdots + 815730721 $ T^16 - 8T^14 + 24T^13 - 222T^12 + 312T^11 + 368T^10 - 6696T^9 + 54467T^8 - 87048T^7 + 62192T^6 + 685464T^5 - 6340542T^4 + 8911032T^3 - 38614472*T^2 + 815730721
$17$	$ T^{16} - 6 T^{15} + \cdots + 8042896 $ T^16 - 6T^15 + 77T^14 - 450T^13 + 3898T^12 - 19998T^11 + 113501T^10 - 466158T^9 + 2056186T^8 - 6993954T^7 + 22208537T^6 - 49223934T^5 + 88221709T^4 - 90260808T^3 + 67860860T^2 - 28382688*T + 8042896
$19$	$ T^{16} - 116 T^{14} + \cdots + 44302336 $ T^16 - 116T^14 + 9932T^12 - 12528T^11 - 380144T^10 + 732384T^9 + 10764112T^8 - 40807872T^7 + 25638080T^6 + 83420928T^5 - 53989120T^4 - 188467200T^3 + 320036864T^2 - 191692800*T + 44302336
$23$	$ T^{16} + \cdots + 60902342656 $ T^16 + 8T^15 + 144T^14 + 432T^13 + 7916T^12 + 9936T^11 + 325568T^10 - 96736T^9 + 8785680T^8 - 10448576T^7 + 182391680T^6 - 285055488T^5 + 2210309120T^4 - 2525995008T^3 + 15912763392T^2 - 16615473152*T + 60902342656
$29$	$ T^{16} + \cdots + 8333498944 $ T^16 + 10T^15 + 213T^14 + 1222T^13 + 22026T^12 + 112446T^11 + 1349253T^10 + 3509474T^9 + 37077786T^8 + 66756930T^7 + 718925853T^6 + 285420810T^5 + 3985037825T^4 - 3993473572T^3 + 20183543544T^2 - 12666757728*T + 8333498944
$31$	$ T^{16} + \cdots + 3650093056 $ T^16 + 294T^14 + 32393T^12 + 1694752T^10 + 45376896T^8 + 620978176T^6 + 4085649408T^4 + 10396499968*T^2 + 3650093056
$37$	$ T^{16} + \cdots + 28407079936 $ T^16 + 6T^15 - 211T^14 - 1338T^13 + 33702T^12 + 181830T^11 - 2449151T^10 - 11187678T^9 + 142509590T^8 + 236342322T^7 - 3413220899T^6 - 3268475718T^5 + 62352711273T^4 - 50297177688T^3 - 29388006688T^2 + 33545649408*T + 28407079936
$41$	$ T^{16} - 6 T^{15} + \cdots + 541696 $ T^16 - 6T^15 - 187T^14 + 1194T^13 + 26526T^12 - 164214T^11 - 1615519T^10 + 10997526T^9 + 72898158T^8 - 550005594T^7 - 145859755T^6 + 7039882254T^5 - 625033879T^4 - 71304077160T^3 + 134088652896T^2 + 466947840*T + 541696
$43$	$ T^{16} + \cdots + 23377186816 $ T^16 - 18T^15 + 387T^14 - 4522T^13 + 68405T^12 - 695076T^11 + 7191348T^10 - 47656152T^9 + 299864364T^8 - 1220606560T^7 + 5935858032T^6 - 19059703296T^5 + 74826730512T^4 - 107366120960T^3 + 115715187456T^2 - 61197541376*T + 23377186816
$47$	$ T^{16} + \cdots + 4498653184 $ T^16 + 376T^14 + 49912T^12 + 2984768T^10 + 84060432T^8 + 1106772352T^6 + 6082339072T^4 + 9640312832*T^2 + 4498653184
$53$	$ (T^{8} + 6 T^{7} + \cdots + 2513872)^{2} $ (T^8 + 6T^7 - 225T^6 - 980T^5 + 16159T^4 + 48822T^3 - 415343T^2 - 753736*T + 2513872)^2
$59$	$ T^{16} + \cdots + 17179869184 $ T^16 - 12T^15 - 168T^14 + 2592T^13 + 31808T^12 - 491520T^11 + 950272T^10 + 12312576T^9 - 39469056T^8 - 366477312T^7 + 3033530368T^6 - 9317646336T^5 + 12318670848T^4 - 14227079168*T^2 + 17179869184
$61$	$ T^{16} + \cdots + 1561222761121 $ T^16 - 16T^15 + 392T^14 - 2544T^13 + 48786T^12 - 192080T^11 + 4469056T^10 - 7848712T^9 + 237523507T^8 - 232668720T^7 + 9212868672T^6 - 2274551080T^5 + 204224387634T^4 - 17457725472T^3 + 2958059526904T^2 + 2067124613864*T + 1561222761121
$67$	$ T^{16} + \cdots + 176076339429376 $ T^16 - 30T^15 + 55T^14 + 7350T^13 - 36675T^12 - 1320300T^11 + 10913764T^10 + 115753848T^9 - 1050643956T^8 - 8019989280T^7 + 67307177392T^6 + 371886788928T^5 - 1662307094896T^4 - 9403054148736T^3 + 30624458595840T^2 + 148541004214272*T + 176076339429376
$71$	$ T^{16} + \cdots + 165864725807104 $ T^16 - 24T^15 - 96T^14 + 6912T^13 + 5228T^12 - 1424592T^11 + 5501056T^10 + 132139872T^9 - 591321072T^8 - 8862859200T^7 + 44015147392T^6 + 344276691456T^5 - 1162472758272T^4 - 8679934648320T^3 + 21333006057472T^2 + 127425176666112*T + 165864725807104
$73$	$ T^{16} + \cdots + 493543502473921 $ T^16 + 824T^14 + 267100T^12 + 44123656T^10 + 4042608774T^8 + 209677618568T^6 + 6026158395868T^4 + 87829815920824*T^2 + 493543502473921
$79$	$ (T^{8} + 10 T^{7} + \cdots + 4096)^{2} $ (T^8 + 10T^7 - 143T^6 - 1464T^5 + 160T^4 + 13760T^3 + 1344T^2 - 24576*T + 4096)^2
$83$	$ T^{16} + \cdots + 1479278657536 $ T^16 + 632T^14 + 144472T^12 + 16101760T^10 + 951002256T^8 + 29333368448T^6 + 416177971456T^4 + 1767953047552*T^2 + 1479278657536
$89$	$ T^{16} + \cdots + 268435456 $ T^16 - 308T^14 + 88144T^12 + 408960T^11 - 1351424T^10 - 9603072T^9 + 24374272T^8 + 209534976T^7 + 281657344T^6 - 445317120T^5 - 775356416T^4 + 1000341504T^3 + 1589641216T^2 - 1207959552*T + 268435456
$97$	$ T^{16} + \cdots + 177209344 $ T^16 + 18T^15 - 225T^14 - 5994T^13 + 55889T^12 + 1228560T^11 - 4163320T^10 - 124659936T^9 + 390001968T^8 + 7259720448T^7 + 20161006208T^6 - 17724817920T^5 - 376907520T^4 + 3812327424T^3 + 101433344T^2 - 736100352*T + 177209344
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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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.bv (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{9} q^{3} + (\beta_{13} - \beta_{11}) q^{5} - \beta_{6} q^{7} + ( - \beta_{9} - 1) q^{9}+O(q^{10}) \) q + b9 * q^3 + (b13 - b11) * q^5 - b6 * q^7 + (-b9 - 1) * q^9	\( q + \beta_{9} q^{3} + (\beta_{13} - \beta_{11}) q^{5} - \beta_{6} q^{7} + ( - \beta_{9} - 1) q^{9} + ( - \beta_{13} + \beta_{5} - \beta_{2}) q^{11} + (\beta_{14} - \beta_{9} - 1) q^{13} - \beta_{13} q^{15} + (\beta_{9} + \beta_{2} + 1) q^{17} + (\beta_{15} - \beta_{14} - \beta_{13} + \cdots + 1) q^{19}+ \cdots + (\beta_{13} - \beta_{11} + \cdots + \beta_1) q^{99}+O(q^{100}) \) q + b9 * q^3 + (b13 - b11) * q^5 - b6 * q^7 + (-b9 - 1) * q^9 + (-b13 + b5 - b2) * q^11 + (b14 - b9 - 1) * q^13 - b13 * q^15 + (b9 + b2 + 1) * q^17 + (b15 - b14 - b13 - b12 - 2b8 + 2b7 + b4 + b1 + 1) * q^19 + (b6 - b3) * q^21 + (b14 + b8 - b7 - b6 + b5 + b4 + b3 - b2 - 2) * q^23 + (b12 - b11 - b3 - 1) * q^25 + q^27 + (-b14 + 2b9 - b8 - b7 - 2b6 - b4 + b1 + 2) * q^29 + (b12 + b11 - 2b9 + 2b7 + 2b6 + 2b4 + b3 - 2b1 - 3) q^31 + (b11 + b7 + b6 - b5 + b2 - b1) * q^33 + (-2b14 - b13 + b12 + b11 + b9 + b7 - 2b5 + b4 + b3 - b2 + b1) * q^35 + (-b14 + b13 + 2b12 + b8 + b7 + b6 - 3b5 + b4 + b3 + b2 - 2) * q^37 + (b8 - b7 - b6 + b3 - b2) * q^39 + (-b14 + b13 + b10 + b8 + b5 + 2b3 - b2 - 1) q^41 + (-b15 + b14 - 2b13 + b11 - 2b10 + 3b9 - b7 - b3 + 3) q^43 + b11 * q^45 + (2b14 - 2b13 + 2b11 + 2b9 + b8 - 2b7 + b5 - b2 - b1) q^47 + (2b15 - b14 + b10 - 2b9 - b8 + 2b7 + 2b6 + 2b4 - 1) q^49 + (-b6 - b2 + b1 - 1) * q^51 + (-2b13 - 2b12 - b8 - b7 - b6 + 3b5 + b3 - 2b2 + b1) * q^53 + (2b14 - b13 + 2b11 - 3b9 + 2b8 - 3b7 + 2b5 + b4 + b3 - 2b2 - b1 - 3) q^55 + (-b15 + 2b14 - b13 - b12 - b10 + b8 - 3b7 - 2b6 + b5 - 2b4 - b2 + 1) * q^57 + (-b13 - b12 - b9 + 2b7 + b6 - b5 + b4 + b3) q^59 + (-b13 + b12 + b11 + b9 + 2b7 + 2b6 + b5 + b4 + b3 - 2b1) q^61 + b3 * q^63 + (-b15 + b14 + b11 + b9 + 2b8 - b5 - b4 - b3 - b1) q^65 + (-b14 + b13 + b10 + b9 + b8 - 2b6 - b5 + 2b3 - 3b2 + 2b1 + 1) * q^67 + (-b14 - b13 - b12 - b6 - b4 - b2 + b1 + 1) * q^69 + (2b15 - b14 - b13 - b12 - 2b9 - 2b8 + 2b7 - b6 + b4 + b2 - b1 + 3) * q^71 + (-b15 + 2b14 + 2b13 + b12 - b11 - b10 - 2b9 + b8 - 2b7 + b6 + b5 + 2b4 - b2 - b1 - 4) q^73 + (-b13 + 2b11 - b9 + b7 + 2b6 + b4 - b1 - 1) * q^75 + (-b12 + b11 - 3b8 + 2b7 + b6 - b5 - 2b3 + 3b2 - b1 + 3) * q^77 + (-2b13 - b12 - b11 - 2b8 + 2b5 - b3 + 1) q^79 + b9 * q^81 + (-4b14 + b12 + b11 + 2b9 - 2b8 + 4b7 + b6 - 2b5 + 2b4 + b3 + b2 + b1 + 1) * q^83 + (-2b11 - 2b9 + b7 - 2b6 - b5 + b1 + 2) q^85 + (b14 + b13 + b12 - 2b9 + b7 + 3b6 + b5 + b4 - 2b3 + b2 - 2b1 - 3) * q^87 + (-2b13 - 2b12 - 2b10 + b9 - b7 - 3b6 - b5 - b4 - b2 + 2b1 + 3) q^89 + (b15 - 2b14 - 2b13 + b12 + 2b11 + b10 + b9 + b8 + 2b7 - b5 + b4 + 2b3 - b2 + b1 - 4) q^91 + (-b13 - 2b12 + b9 - b7 - 2b6 - b4 + b1 + 3) * q^93 + (-2b15 + 5b14 - 2b13 - 2b12 - 4b10 - b9 - 4b7 - 2b6 + 3b5 - 2b4 - b3 - b2 - b1 + 1) q^95 + (b15 - b14 - 2b11 + b9 - 2b8 - 3b6 + b5 - b3 - b2 + 3b1 + 1) * q^97 + (b13 - b11 - b7 - b6 + b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} - 6 q^{7} - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^3 - 6 * q^7 - 8 * q^9	\( 16 q - 8 q^{3} - 6 q^{7} - 8 q^{9} - 6 q^{15} + 6 q^{17} - 8 q^{23} - 20 q^{25} + 16 q^{27} - 10 q^{29} - 6 q^{37} + 6 q^{39} + 6 q^{41} + 18 q^{43} + 6 q^{45} + 14 q^{49} - 12 q^{51} - 12 q^{53} + 20 q^{55} + 12 q^{59} + 16 q^{61} + 6 q^{63} - 6 q^{65} + 30 q^{67} - 8 q^{69} + 24 q^{71} + 10 q^{75} + 8 q^{77} - 20 q^{79} - 8 q^{81} + 30 q^{85} - 10 q^{87} - 46 q^{91} + 6 q^{93} + 4 q^{95} - 18 q^{97}+O(q^{100}) \) 16 * q - 8 * q^3 - 6 * q^7 - 8 * q^9 - 6 * q^15 + 6 * q^17 - 8 * q^23 - 20 * q^25 + 16 * q^27 - 10 * q^29 - 6 * q^37 + 6 * q^39 + 6 * q^41 + 18 * q^43 + 6 * q^45 + 14 * q^49 - 12 * q^51 - 12 * q^53 + 20 * q^55 + 12 * q^59 + 16 * q^61 + 6 * q^63 - 6 * q^65 + 30 * q^67 - 8 * q^69 + 24 * q^71 + 10 * q^75 + 8 * q^77 - 20 * q^79 - 8 * q^81 + 30 * q^85 - 10 * q^87 - 46 * q^91 + 6 * q^93 + 4 * q^95 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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	1248.2.bv.c

Level	$1248$
Weight	$2$
Character orbit	1248.bv
Analytic conductor	$9.965$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(9.96533017226\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 6 x^{15} + 28 x^{14} - 66 x^{13} + 152 x^{12} - 256 x^{11} + 402 x^{10} - 92 x^{9} - 1652 x^{8} + \cdots + 916 \) x^16 - 6x^15 + 28x^14 - 66x^13 + 152x^12 - 256x^11 + 402x^10 - 92x^9 - 1652x^8 + 2326x^7 + 1814x^6 - 3560x^5 - 2307x^4 + 4090x^3 + 1610x^2 - 3204*x + 916
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 22\!\cdots\!81 \nu^{15} + \cdots - 14\!\cdots\!28 ) / 61\!\cdots\!73 \) (22025466988810458291781v^15 - 185713764819774321198002v^14 + 799942467133718734564931v^13 - 2253005315257952881151510v^12 + 3481560843460023541937425v^11 - 6666644634592414145602033v^10 + 2396738044331130539498292v^9 + 8651765194286259603728880v^8 - 97737718020984482344623339v^7 + 170574111596721546993965153v^6 + 24660247491338905801172857v^5 - 229424657141467119406393650v^4 - 72897990854567801961990799v^3 + 224726262376709725041222283v^2 + 187215413539774625067528258*v - 149379828403731999838219128) / 61643197265233151426841073
\(\beta_{2}\)	\(=\)	\( ( 15\!\cdots\!75 \nu^{15} + \cdots - 53\!\cdots\!32 ) / 12\!\cdots\!46 \) (158431676444308138210875v^15 - 771342034499023974976072v^14 + 3575738889937302957221935v^13 - 6152880799341903156854300v^12 + 15467308976199302255790913v^11 - 14006410228127809448680332v^10 + 24059175832060078941755907v^9 + 77998793718427035464881928v^8 - 303452369834938135481718019v^7 + 280961804837598270149852062v^6 + 309235292233667933048102949v^5 - 306293715532909132864614110v^4 - 348471599268278495752134460v^3 + 298925888691998890697699726v^2 + 857122773488001284323074424*v - 530107668790934374795119132) / 123286394530466302853682146
\(\beta_{3}\)	\(=\)	\( ( - 27\!\cdots\!97 \nu^{15} + \cdots + 13\!\cdots\!84 ) / 12\!\cdots\!46 \) (-275907785496358370695397v^15 + 1441996231784354464944626v^14 - 6343935341045048722116045v^13 + 12039730166183615036045150v^12 - 26647597524701419053372351v^11 + 39046290534104769189498784v^10 - 49949964955774243851996151v^9 - 54562354425709284074641842v^8 + 491089919160344627161503785v^7 - 234764583166611698519591766v^6 - 1033581623147196377488220411v^5 + 328681891465027144972067468v^4 + 1269010591462621791971745862v^3 + 5464283194453790159683756v^2 - 637585285366456237662788908*v + 134408189972762856179253384) / 123286394530466302853682146
\(\beta_{4}\)	\(=\)	\( ( 27\!\cdots\!97 \nu^{15} + \cdots - 13\!\cdots\!84 ) / 12\!\cdots\!46 \) (275907785496358370695397v^15 - 1441996231784354464944626v^14 + 6343935341045048722116045v^13 - 12039730166183615036045150v^12 + 26647597524701419053372351v^11 - 39046290534104769189498784v^10 + 49949964955774243851996151v^9 + 54562354425709284074641842v^8 - 491089919160344627161503785v^7 + 234764583166611698519591766v^6 + 1033581623147196377488220411v^5 - 328681891465027144972067468v^4 - 1269010591462621791971745862v^3 - 5464283194453790159683756v^2 + 884158074427388843370153200*v - 134408189972762856179253384) / 123286394530466302853682146
\(\beta_{5}\)	\(=\)	\( ( 43\!\cdots\!54 \nu^{15} + \cdots - 90\!\cdots\!76 ) / 12\!\cdots\!46 \) (431944536547067909218354v^15 - 2346152632946727160590803v^14 + 11071641641227351312218301v^13 - 23567559375987226279293828v^12 + 58521825384246455416293775v^11 - 86426730317296962812259838v^10 + 150961149326854446167214695v^9 + 24447448340461290917359364v^8 - 657471679588651205308721277v^7 + 761884123524190499486983758v^6 + 715754340916210259754291809v^5 - 994247903556661476531651692v^4 - 939501837205534170045679731v^3 + 1288944953366776165444878925v^2 + 1139240240176020328161874670*v - 904067971300908434925642076) / 123286394530466302853682146
\(\beta_{6}\)	\(=\)	\( ( - 66\!\cdots\!95 \nu^{15} + \cdots + 46\!\cdots\!96 ) / 12\!\cdots\!46 \) (-662007638649936654806195v^15 + 3171913765072528018860570v^14 - 14727196583675265761947301v^13 + 26045679817851981116891820v^12 - 69673003954139844889043551v^11 + 86095443977338359669829416v^10 - 161645178432414751096639381v^9 - 135742299897787054941727880v^8 + 941994224533795483823818053v^7 - 425031707887208015731287358v^6 - 1615034310944710766542652459v^5 + 390822706206632125266258618v^4 + 1698533496546299207451026750v^3 - 273354544478007624769974792v^2 - 1149623700247367918444867860*v + 464976251780042329301375496) / 123286394530466302853682146
\(\beta_{7}\)	\(=\)	\( ( 14\!\cdots\!39 \nu^{15} + \cdots - 17\!\cdots\!74 ) / 12\!\cdots\!46 \) (1455135901748594541504139v^15 - 7526230630674308870090510v^14 + 34945133106618830384261607v^13 - 69270361475756550571436820v^12 + 173781603078464217547501325v^11 - 247509800997590738857118568v^10 + 428021789327673015670358503v^9 + 152824012086985130412239752v^8 - 2163333791950872702474637627v^7 + 1639876585186101684218227210v^6 + 3366735170917851885827342509v^5 - 2103995747464682994563666950v^4 - 4033281216512824853168518448v^3 + 2249701974543790645263483680v^2 + 3119661663781697014886361548*v - 1747713238004962778012603974) / 123286394530466302853682146
\(\beta_{8}\)	\(=\)	\( ( 15\!\cdots\!73 \nu^{15} + \cdots - 12\!\cdots\!12 ) / 12\!\cdots\!46 \) (1578224720414536942673873v^15 - 7889393214113164921341231v^14 + 36012848648621255616465294v^13 - 66719590507413214879578332v^12 + 166160411447907054804356778v^11 - 223060944128051202740705964v^10 + 367709004566933522350241204v^9 + 290767814623212028376300088v^8 - 2462011976762887710906012430v^7 + 1275225875994523509878270566v^6 + 4333588287773850369053348448v^5 - 1447946811327876437067725106v^4 - 5223694767630021889619033301v^3 + 961244113361353301900935309v^2 + 3985836787145985347835564498*v - 1287845329985228460656905412) / 123286394530466302853682146
\(\beta_{9}\)	\(=\)	\( ( - 46951055886214 \nu^{15} + 257032746536189 \nu^{14} + \cdots + 75\!\cdots\!76 ) / 34\!\cdots\!42 \) (-46951055886214v^15 + 257032746536189v^14 - 1176314795701089v^13 + 2465685193393304v^12 - 5769715610579071v^11 + 8860177294202080v^10 - 13838422313939069v^9 - 3485025483031720v^8 + 76849575188612505v^7 - 68878801246918828v^6 - 124508041677473661v^5 + 101359752535599028v^4 + 167504208396733877v^3 - 100414409469165505v^2 - 135720127040329650*v + 75200015167695276) / 3497569387271042
\(\beta_{10}\)	\(=\)	\( ( - 15\!\cdots\!12 \nu^{15} + \cdots + 29\!\cdots\!44 ) / 61\!\cdots\!09 \) (-1539578758935525738312v^15 + 8441957746091932878520v^14 - 38740786127224027742716v^13 + 81333425248276730200694v^12 - 190661225462608568738424v^11 + 289710297569014776913130v^10 - 457010603900253484025808v^9 - 125536899509183652710922v^8 + 2526250998237719136070048v^7 - 2356045582033203159539570v^6 - 4011296247143239946556684v^5 + 3723928606511433394404070v^4 + 5270136222198310906668340v^3 - 3955104084962523293048646v^2 - 4461016114890993851414024*v + 2956844482511742322681544) / 61828683315178687489309
\(\beta_{11}\)	\(=\)	\( ( 31\!\cdots\!25 \nu^{15} + \cdots - 49\!\cdots\!50 ) / 12\!\cdots\!46 \) (3137862819106083689160325v^15 - 17262882616152875742381323v^14 + 79089288245895737534838348v^13 - 166779868064650769547323560v^12 + 389871353462393256762958658v^11 - 600736247635700802292513000v^10 + 943334700449688038821403766v^9 + 214886820433435090338422336v^8 - 5121438063869242659093592080v^7 + 4754041370819597794114964774v^6 + 8325551518246613389414288252v^5 - 7117762696983641211330709914v^4 - 11305583445794313174209647663v^3 + 7108297204894458689484729625v^2 + 9327569939812682107766786946*v - 4992050113344034376128069050) / 123286394530466302853682146
\(\beta_{12}\)	\(=\)	\( ( 36\!\cdots\!12 \nu^{15} + \cdots - 59\!\cdots\!72 ) / 12\!\cdots\!46 \) (3630359091771684646349212v^15 - 19730476345868321941143357v^14 + 90797641844521220042080043v^13 - 189903538203075092850457548v^12 + 451899560148424181132137179v^11 - 689654391144845841083424496v^10 + 1106983252034229096193785451v^9 + 231959617693905570817426104v^8 - 5777926135096327048824197723v^7 + 5186909235959146688347423848v^6 + 9014902449852256600580000513v^5 - 7429967498970432320328556392v^4 - 12027770073639510395476504265v^3 + 7760983297320547738838421497v^2 + 9468298931218622644042325230*v - 5968538367712588107440943972) / 123286394530466302853682146
\(\beta_{13}\)	\(=\)	\( ( - 36\!\cdots\!29 \nu^{15} + \cdots + 59\!\cdots\!48 ) / 12\!\cdots\!46 \) (-3635362423362086268676829v^15 + 19717867039416526571025497v^14 - 90369244263251086439455402v^13 + 187754643974363038728551400v^12 - 443617891860852261369255560v^11 + 675557146998215839165588150v^10 - 1067556222861213916898880160v^9 - 276648001823191190532154704v^8 + 5864597854314759435514596222v^7 - 5044015135041195977870776180v^6 - 9480312005264190424237930522v^5 + 7334379103597741483828554402v^4 + 12641818831176614369048166077v^3 - 7319575488847416887411187277v^2 - 9996672451584157414767776066*v + 5990078023184889863938269648) / 123286394530466302853682146
\(\beta_{14}\)	\(=\)	\( ( 21\!\cdots\!44 \nu^{15} + \cdots - 35\!\cdots\!95 ) / 61\!\cdots\!73 \) (2178818650005461712930544v^15 - 11868279567445867401674576v^14 + 54625820274211500803417695v^13 - 114509346148879528552647699v^12 + 272205439913995563681676986v^11 - 415952293978370851874808566v^10 + 669520087298167926693694079v^9 + 134995869435111142342455085v^8 - 3457132618087682392917641901v^7 + 3151623624193745277129597486v^6 + 5489389138886426015564165112v^5 - 4579629137840375709523610651v^4 - 7237042258795166110702221553v^3 + 4831652390036039992812406800v^2 + 5851494438949027510008829082*v - 3528985670518049538479928495) / 61643197265233151426841073
\(\beta_{15}\)	\(=\)	\( ( - 22\!\cdots\!12 \nu^{15} + \cdots + 31\!\cdots\!92 ) / 61\!\cdots\!09 \) (-2274550404808901490212v^15 + 12033319727491866708414v^14 - 55218049088125161967702v^13 + 111227917547308479075596v^12 - 268147177743543095581222v^11 + 393838335053286249299488v^10 - 640611730585334674278502v^9 - 240617864551049127684452v^8 + 3577440187969011090727902v^7 - 2780758853277706854224464v^6 - 5987923215820554548932902v^5 + 3830094885701049707215764v^4 + 7896073221885649768028010v^3 - 3892274981877633201680966v^2 - 6199034970862445514643424*v + 3118332418160600492165492) / 61828683315178687489309

\(\nu\)	\(=\)	\( ( \beta_{4} + \beta_{3} ) / 2 \) (b4 + b3) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{14} + \beta_{13} - \beta_{9} + \beta_{8} - 4 \beta_{7} - 4 \beta_{6} + \beta_{5} + 3 \beta_{3} + \cdots - 5 ) / 2 \) (b14 + b13 - b9 + b8 - 4b7 - 4b6 + b5 + 3b3 - 3b2 + 2*b1 - 5) / 2
\(\nu^{3}\)	\(=\)	\( ( 3 \beta_{15} + 4 \beta_{14} + 12 \beta_{13} + 8 \beta_{12} + 4 \beta_{11} - \beta_{10} + 4 \beta_{9} + \cdots - 34 ) / 4 \) (3b15 + 4b14 + 12b13 + 8b12 + 4b11 - b10 + 4b9 + 12b8 - 14b7 - 2b6 + 8b5 - 4b4 - 2b3 - 6b2 - 2b1 - 34) / 4
\(\nu^{4}\)	\(=\)	\( ( - \beta_{15} - 3 \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} + \cdots - 21 ) / 2 \) (-b15 - 3b14 + b13 - 3b12 + 3b11 - 2b10 + 8b9 + 4b8 + 13b7 + 23b6 + 15b5 - 5b4 - 22b3 - b2 - 6b1 - 21) / 2
\(\nu^{5}\)	\(=\)	\( ( - 14 \beta_{15} - 36 \beta_{14} - 76 \beta_{13} - 104 \beta_{12} - 64 \beta_{11} + 3 \beta_{10} + \cdots + 182 ) / 4 \) (-14b15 - 36b14 - 76b13 - 104b12 - 64b11 + 3b10 - 196b9 - 68b8 + 170b7 + 106b6 - 8b5 - 66b4 - 92b3 + 54b2 - 42*b1 + 182) / 4
\(\nu^{6}\)	\(=\)	\( ( - 21 \beta_{15} - 44 \beta_{14} - 23 \beta_{13} - 131 \beta_{11} + 7 \beta_{10} - 280 \beta_{9} + \cdots + 560 ) / 2 \) (-21b15 - 44b14 - 23b13 - 131b11 + 7b10 - 280b9 - 137b8 + 251b7 + 153b6 - 257b5 + 16b4 - 24b3 + 243b2 - 96b1 + 560) / 2
\(\nu^{7}\)	\(=\)	\( ( - 43 \beta_{15} + 184 \beta_{14} + 106 \beta_{13} + 634 \beta_{12} + 24 \beta_{11} + 64 \beta_{10} + \cdots + 1792 ) / 4 \) (-43b15 + 184b14 + 106b13 + 634b12 + 24b11 + 64b10 + 1102b9 - 328b8 - 48b7 - 294b6 - 1574b5 + 832b4 + 944b3 + 784b2 + 256*b1 + 1792) / 4
\(\nu^{8}\)	\(=\)	\( ( 282 \beta_{15} + 1157 \beta_{14} - 141 \beta_{13} + 494 \beta_{12} + 1304 \beta_{11} + 189 \beta_{10} + \cdots - 1854 ) / 2 \) (282b15 + 1157b14 - 141b13 + 494b12 + 1304b11 + 189b10 + 4444b9 + 1009b8 - 3905b7 - 3426b6 + 558b5 + 379b4 + 2004b3 - 1706b2 + 1677*b1 - 1854) / 2
\(\nu^{9}\)	\(=\)	\( ( 2295 \beta_{15} + 5080 \beta_{14} + 216 \beta_{13} - 388 \beta_{12} + 8272 \beta_{11} + 9 \beta_{10} + \cdots - 29594 ) / 4 \) (2295b15 + 5080b14 + 216b13 - 388b12 + 8272b11 + 9b10 + 19264b9 + 10664b8 - 25742b7 - 17398b6 + 22116b5 - 3544b4 + 1474b3 - 18270b2 + 5794*b1 - 29594) / 4
\(\nu^{10}\)	\(=\)	\( ( - 93 \beta_{15} - 6223 \beta_{14} + 3156 \beta_{13} - 6432 \beta_{12} + 423 \beta_{11} - 2572 \beta_{10} + \cdots - 43949 ) / 2 \) (-93b15 - 6223b14 + 3156b13 - 6432b12 + 423b11 - 2572b10 - 18721b9 + 5980b8 + 2065b7 + 8331b6 + 32908b5 - 5340b4 - 16095b3 - 15886b2 - 6336*b1 - 43949) / 2
\(\nu^{11}\)	\(=\)	\( ( - 23072 \beta_{15} - 97536 \beta_{14} + 30696 \beta_{13} - 43616 \beta_{12} - 93924 \beta_{11} + \cdots - 27018 ) / 4 \) (-23072b15 - 97536b14 + 30696b13 - 43616b12 - 93924b11 - 16077b10 - 395028b9 - 70172b8 + 292538b7 + 235862b6 - 27072b5 - 19258b4 - 114168b3 + 83166b2 - 103166*b1 - 27018) / 4
\(\nu^{12}\)	\(=\)	\( ( - 36441 \beta_{15} - 98272 \beta_{14} + 37770 \beta_{13} + 23779 \beta_{12} - 166417 \beta_{11} + \cdots + 485421 ) / 2 \) (-36441b15 - 98272b14 + 37770b13 + 23779b12 - 166417b11 - 1935b10 - 445536b9 - 179243b8 + 509906b7 + 373327b6 - 443041b5 + 14496b4 - 58664b3 + 366723b2 - 136929*b1 + 485421) / 2
\(\nu^{13}\)	\(=\)	\( ( - 17947 \beta_{15} + 421968 \beta_{14} - 285198 \beta_{13} + 543910 \beta_{12} - 60520 \beta_{11} + \cdots + 4119420 ) / 4 \) (-17947b15 + 421968b14 - 285198b13 + 543910b12 - 60520b11 + 176250b10 + 1718354b9 - 540172b8 + 398948b7 + 56354b6 - 2795650b5 + 337176b4 + 767044b3 + 1691756b2 + 242848*b1 + 4119420) / 4
\(\nu^{14}\)	\(=\)	\( ( 386080 \beta_{15} + 1922393 \beta_{14} - 1218787 \beta_{13} + 425004 \beta_{12} + 1710132 \beta_{11} + \cdots + 2272183 ) / 2 \) (386080b15 + 1922393b14 - 1218787b13 + 425004b12 + 1710132b11 + 381137b10 + 7926247b9 + 987885b8 - 5309788b7 - 4555154b6 + 231045b5 + 314448b4 + 2156313b3 - 1327903b2 + 2102356*b1 + 2272183) / 2
\(\nu^{15}\)	\(=\)	\( ( 2965963 \beta_{15} + 8645508 \beta_{14} - 4370400 \beta_{13} - 2173804 \beta_{12} + 12451988 \beta_{11} + \cdots - 30991414 ) / 4 \) (2965963b15 + 8645508b14 - 4370400b13 - 2173804b12 + 12451988b11 + 592327b10 + 35275512b9 + 13212132b8 - 42821026b7 - 33380498b6 + 33472204b5 - 694504b4 + 7192322b3 - 28570270b2 + 11652638*b1 - 30991414) / 4

\(n\)	\(703\)	\(769\)	\(833\)	\(1093\)
\(\chi(n)\)	\(1\)	\(1 + \beta_{9}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 673.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + T + 1)^{8} \) (T^2 + T + 1)^8
$5$	\( T^{16} + 50 T^{14} + \cdots + 64 \) T^16 + 50T^14 + 999T^12 + 10300T^10 + 58943T^8 + 184978T^6 + 285369T^4 + 151520*T^2 + 64
$7$	\( T^{16} + 6 T^{15} + \cdots + 4096 \) T^16 + 6T^15 - 17T^14 - 174T^13 + 309T^12 + 3156T^11 - 1892T^10 - 30744T^9 + 23196T^8 + 168576T^7 - 52400T^6 - 467328T^5 + 398096T^4 + 271872T^3 + 3840T^2 - 24576*T + 4096
$11$	\( T^{16} - 52 T^{14} + \cdots + 65536 \) T^16 - 52T^14 + 1932T^12 + 432T^11 - 34544T^10 - 15456T^9 + 448592T^8 + 470208T^7 - 1863488T^6 - 2200320T^5 + 6278400T^4 + 8306688T^3 + 2453504T^2 - 786432*T + 65536
$13$	\( T^{16} + \cdots + 815730721 \) T^16 - 8T^14 + 24T^13 - 222T^12 + 312T^11 + 368T^10 - 6696T^9 + 54467T^8 - 87048T^7 + 62192T^6 + 685464T^5 - 6340542T^4 + 8911032T^3 - 38614472*T^2 + 815730721
$17$	\( T^{16} - 6 T^{15} + \cdots + 8042896 \) T^16 - 6T^15 + 77T^14 - 450T^13 + 3898T^12 - 19998T^11 + 113501T^10 - 466158T^9 + 2056186T^8 - 6993954T^7 + 22208537T^6 - 49223934T^5 + 88221709T^4 - 90260808T^3 + 67860860T^2 - 28382688*T + 8042896
$19$	\( T^{16} - 116 T^{14} + \cdots + 44302336 \) T^16 - 116T^14 + 9932T^12 - 12528T^11 - 380144T^10 + 732384T^9 + 10764112T^8 - 40807872T^7 + 25638080T^6 + 83420928T^5 - 53989120T^4 - 188467200T^3 + 320036864T^2 - 191692800*T + 44302336
$23$	\( T^{16} + \cdots + 60902342656 \) T^16 + 8T^15 + 144T^14 + 432T^13 + 7916T^12 + 9936T^11 + 325568T^10 - 96736T^9 + 8785680T^8 - 10448576T^7 + 182391680T^6 - 285055488T^5 + 2210309120T^4 - 2525995008T^3 + 15912763392T^2 - 16615473152*T + 60902342656
$29$	\( T^{16} + \cdots + 8333498944 \) T^16 + 10T^15 + 213T^14 + 1222T^13 + 22026T^12 + 112446T^11 + 1349253T^10 + 3509474T^9 + 37077786T^8 + 66756930T^7 + 718925853T^6 + 285420810T^5 + 3985037825T^4 - 3993473572T^3 + 20183543544T^2 - 12666757728*T + 8333498944
$31$	\( T^{16} + \cdots + 3650093056 \) T^16 + 294T^14 + 32393T^12 + 1694752T^10 + 45376896T^8 + 620978176T^6 + 4085649408T^4 + 10396499968*T^2 + 3650093056
$37$	\( T^{16} + \cdots + 28407079936 \) T^16 + 6T^15 - 211T^14 - 1338T^13 + 33702T^12 + 181830T^11 - 2449151T^10 - 11187678T^9 + 142509590T^8 + 236342322T^7 - 3413220899T^6 - 3268475718T^5 + 62352711273T^4 - 50297177688T^3 - 29388006688T^2 + 33545649408*T + 28407079936
$41$	\( T^{16} - 6 T^{15} + \cdots + 541696 \) T^16 - 6T^15 - 187T^14 + 1194T^13 + 26526T^12 - 164214T^11 - 1615519T^10 + 10997526T^9 + 72898158T^8 - 550005594T^7 - 145859755T^6 + 7039882254T^5 - 625033879T^4 - 71304077160T^3 + 134088652896T^2 + 466947840*T + 541696
$43$	\( T^{16} + \cdots + 23377186816 \) T^16 - 18T^15 + 387T^14 - 4522T^13 + 68405T^12 - 695076T^11 + 7191348T^10 - 47656152T^9 + 299864364T^8 - 1220606560T^7 + 5935858032T^6 - 19059703296T^5 + 74826730512T^4 - 107366120960T^3 + 115715187456T^2 - 61197541376*T + 23377186816
$47$	\( T^{16} + \cdots + 4498653184 \) T^16 + 376T^14 + 49912T^12 + 2984768T^10 + 84060432T^8 + 1106772352T^6 + 6082339072T^4 + 9640312832*T^2 + 4498653184
$53$	\( (T^{8} + 6 T^{7} + \cdots + 2513872)^{2} \) (T^8 + 6T^7 - 225T^6 - 980T^5 + 16159T^4 + 48822T^3 - 415343T^2 - 753736*T + 2513872)^2
$59$	\( T^{16} + \cdots + 17179869184 \) T^16 - 12T^15 - 168T^14 + 2592T^13 + 31808T^12 - 491520T^11 + 950272T^10 + 12312576T^9 - 39469056T^8 - 366477312T^7 + 3033530368T^6 - 9317646336T^5 + 12318670848T^4 - 14227079168*T^2 + 17179869184
$61$	\( T^{16} + \cdots + 1561222761121 \) T^16 - 16T^15 + 392T^14 - 2544T^13 + 48786T^12 - 192080T^11 + 4469056T^10 - 7848712T^9 + 237523507T^8 - 232668720T^7 + 9212868672T^6 - 2274551080T^5 + 204224387634T^4 - 17457725472T^3 + 2958059526904T^2 + 2067124613864*T + 1561222761121
$67$	\( T^{16} + \cdots + 176076339429376 \) T^16 - 30T^15 + 55T^14 + 7350T^13 - 36675T^12 - 1320300T^11 + 10913764T^10 + 115753848T^9 - 1050643956T^8 - 8019989280T^7 + 67307177392T^6 + 371886788928T^5 - 1662307094896T^4 - 9403054148736T^3 + 30624458595840T^2 + 148541004214272*T + 176076339429376
$71$	\( T^{16} + \cdots + 165864725807104 \) T^16 - 24T^15 - 96T^14 + 6912T^13 + 5228T^12 - 1424592T^11 + 5501056T^10 + 132139872T^9 - 591321072T^8 - 8862859200T^7 + 44015147392T^6 + 344276691456T^5 - 1162472758272T^4 - 8679934648320T^3 + 21333006057472T^2 + 127425176666112*T + 165864725807104
$73$	\( T^{16} + \cdots + 493543502473921 \) T^16 + 824T^14 + 267100T^12 + 44123656T^10 + 4042608774T^8 + 209677618568T^6 + 6026158395868T^4 + 87829815920824*T^2 + 493543502473921
$79$	\( (T^{8} + 10 T^{7} + \cdots + 4096)^{2} \) (T^8 + 10T^7 - 143T^6 - 1464T^5 + 160T^4 + 13760T^3 + 1344T^2 - 24576*T + 4096)^2
$83$	\( T^{16} + \cdots + 1479278657536 \) T^16 + 632T^14 + 144472T^12 + 16101760T^10 + 951002256T^8 + 29333368448T^6 + 416177971456T^4 + 1767953047552*T^2 + 1479278657536
$89$	\( T^{16} + \cdots + 268435456 \) T^16 - 308T^14 + 88144T^12 + 408960T^11 - 1351424T^10 - 9603072T^9 + 24374272T^8 + 209534976T^7 + 281657344T^6 - 445317120T^5 - 775356416T^4 + 1000341504T^3 + 1589641216T^2 - 1207959552*T + 268435456
$97$	\( T^{16} + \cdots + 177209344 \) T^16 + 18T^15 - 225T^14 - 5994T^13 + 55889T^12 + 1228560T^11 - 4163320T^10 - 124659936T^9 + 390001968T^8 + 7259720448T^7 + 20161006208T^6 - 17724817920T^5 - 376907520T^4 + 3812327424T^3 + 101433344T^2 - 736100352*T + 177209344
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