LMFDB - Newform orbit 176.6.e.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,6,Mod(175,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("176.175");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 176 = 2^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	176.e (of order $2$, degree $1$, 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:	$28.2275522871$
Analytic rank:	$0$
Dimension:	$16$
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} + 5899 x^{14} + 31909903 x^{12} + 16672338574 x^{10} + 7262078908132 x^{8} + \cdots + 21\!\cdots\!96 $ x^16 + 5899x^14 + 31909903x^12 + 16672338574x^10 + 7262078908132x^8 + 510923309868544x^6 + 29217515417325568x^4 + 267327060014792704*x^2 + 2137083591258013696
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{50}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{3} + (\beta_{8} - 6) q^{5} + \beta_{9} q^{7} + (\beta_{8} + \beta_{3} - \beta_1 + 56) q^{9}+O(q^{10}) $ q + b6 * q^3 + (b8 - 6) * q^5 + b9 * q^7 + (b8 + b3 - b1 + 56) * q^9	$ q + \beta_{6} q^{3} + (\beta_{8} - 6) q^{5} + \beta_{9} q^{7} + (\beta_{8} + \beta_{3} - \beta_1 + 56) q^{9} + (\beta_{14} - \beta_{13} + \cdots - 4 \beta_{4}) q^{11}+ \cdots + (3 \beta_{15} - 77 \beta_{14} + \cdots + 20 \beta_{2}) q^{99}+O(q^{100}) $ q + b6 * q^3 + (b8 - 6) * q^5 + b9 * q^7 + (b8 + b3 - b1 + 56) * q^9 + (b14 - b13 + b6 + 2b5 - 4b4) * q^11 - b10 * q^13 + (-4b14 - 9b6 - 10b5 + b4) q^15 + (b10 + b7) * q^17 - b2 * q^19 + (-b12 + b10) * q^21 + (-5b14 + 33b6 - 41b5 - 5b4) * q^23 + (-30b8 - 35b1 + 776) * q^25 + (9b14 + 167b6 - 69b5 - 12b4) * q^27 + (-4b12 - b11 - 3b7) * q^29 + (19b14 - 123b6 + 158b5 - 25b4) * q^31 + (-2b12 + b11 - b10 + 20b8 + 2b7 - 10b3 + 5b1 - 553) q^33 + (-b15 - 11b14 + 22b13 - 7b9 + 2b2) * q^35 + (80b8 + 5b3 - 42b1 - 386) q^37 + (b15 - 6b14 + 12b13 + 34b9 + 3b2) * q^39 + (-3b12 - 7b11 - b10 + 6b7) q^41 + (b15 - 9b14 + 18b13 - 33b9 - b2) q^43 + (119b8 + 23b3 - 70b1 + 1094) q^45 + (15b14 - 498b6 + 384b5 - 200b4) * q^47 + (-b8 + 35b3 - 236b1 + 7791) * q^49 + (-21b14 + 42b13 - 30b9 + b2) q^51 + (109b8 - 71b3 - 132b1 + 120) q^53 + (-b15 - 60b14 + 18b13 - 76b9 - 851b6 + 533b5 - 171b4 - 3b2) q^55 + (-13b12 - 7b11 - 17b10 + 4b7) * q^57 + (130b14 + 393b6 + 31b5 - 50b4) * q^59 + (-6b12 - 19b11 + 18b10 + 13b7) * q^61 + (-b15 - 18b14 + 36b13 + 155b9 + 3b2) * q^63 + (-38b12 + 14b11 + 17b10 - 11b7) * q^65 + (-106b14 + 1181b6 - 1304b5 - 242b4) * q^67 + (-133b8 + 32b3 - 138b1 - 2826) q^69 + (-59b14 + 3077b6 - 1793b5 + 189b4) * q^71 + (-17b12 + 15b11 + b10 + 18b7) q^73 + (295b14 - 744b6 + 1630b5 + 110b4) * q^75 + (-30b12 + 11b11 - 17b10 - 543b8 - 37b7 - 99b3 - 148b1 + 9674) q^77 + (3b15 + 36b14 - 72b13 + 83b9 - 21b2) q^79 + (587b8 + 227b3 - 242b1 - 11819) q^81 + (-2b15 - 25b14 + 50b13 + 184b9 - 22b2) q^83 + (51b12 - 36b11 - 17b10 + 54b7) * q^85 + (-3b15 + 12b14 - 24b13 - 225b9 + 15b2) q^87 + (-726b8 + 100b3 - 359b1 + 15655) q^89 + (-600b14 - 4312b6 - 15b5 - 924b4) * q^91 + (554b8 - 205b3 + 478b1 + 7936) q^93 + (b15 + 72b14 - 144b13 - 254b9 - 3b2) * q^95 + (1125b8 - 87b3 + 463b1 + 3011) q^97 + (3b15 - 77b14 + 27b13 - 69b9 + 690b6 + 1066b5 - 802b4 + 20b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 100 q^{5} + 892 q^{9}+O(q^{10}) $ 16 * q - 100 * q^5 + 892 * q^9	$ 16 q - 100 q^{5} + 892 q^{9} + 12396 q^{25} - 8948 q^{33} - 6644 q^{37} + 16840 q^{45} + 123856 q^{49} + 672 q^{53} - 45108 q^{69} + 155968 q^{77} - 191512 q^{81} + 252348 q^{89} + 125852 q^{93} + 45180 q^{97}+O(q^{100}) $ 16 * q - 100 * q^5 + 892 * q^9 + 12396 * q^25 - 8948 * q^33 - 6644 * q^37 + 16840 * q^45 + 123856 * q^49 + 672 * q^53 - 45108 * q^69 + 155968 * q^77 - 191512 * q^81 + 252348 * q^89 + 125852 * q^93 + 45180 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 5899 x^{14} + 31909903 x^{12} + 16672338574 x^{10} + 7262078908132 x^{8} + \cdots + 21\!\cdots\!96 $ x^16 + 5899*x^14 + 31909903*x^12 + 16672338574*x^10 + 7262078908132*x^8 + 510923309868544*x^6 + 29217515417325568*x^4 + 267327060014792704*x^2 + 2137083591258013696 :

$\beta_{1}$	$=$	$ ( 11\!\cdots\!35 \nu^{14} + \cdots - 37\!\cdots\!84 ) / 41\!\cdots\!44 $ (110832354121059080457843735v^14 + 646356748826031823636878484349v^12 + 3495311982398828396891534837343705v^10 + 1624250950067336600153237576431812930v^8 + 755624510740503315273079329407348922236v^6 + 6603468463676240852089494323834051880960v^4 + 60430602749190798124051285969536712704000*v^2 - 3750907748780539485321512671075495388721577984) / 41068537224302103032743749959514404972396544
$\beta_{2}$	$=$	$ ( - 93\!\cdots\!19 \nu^{15} + \cdots - 31\!\cdots\!04 \nu ) / 36\!\cdots\!76 $ (-93059542045793761485601424300119v^15 - 544991786432899205772058585370590725v^13 - 2947702821365954777196292341884415415089v^11 - 1434167374540372546777175592585097386551674v^9 - 659502864877683544857065488859353848912586092v^7 - 41895989590460882962711853197767788635728064032v^5 - 11149518662509610983050923644798861557990584516608v^3 - 3122711435160074092317129166896986416392275656704v) / 3664376168808426407436714740918887358766432485376
$\beta_{3}$	$=$	$ ( 12\!\cdots\!71 \nu^{14} + \cdots - 57\!\cdots\!76 ) / 41\!\cdots\!44 $ (1219914155472199738961792571v^14 + 7119764082807469565357306063113v^12 + 38472345001916140060273424703540213v^10 + 17877872772259571045064010594915946298v^8 + 7478852597827763428391435961408350868364v^6 + 72683330765075407948208455125460260243456v^4 + 665150066531408588957547390828359378534400*v^2 - 574549165803361460439690648307520991695077376) / 41068537224302103032743749959514404972396544
$\beta_{4}$	$=$	$ ( 48\!\cdots\!97 \nu^{14} + \cdots + 23\!\cdots\!64 ) / 14\!\cdots\!04 $ (483869763203188439455963745229097v^14 + 2894608872945514555731993378757153539v^12 + 15672687945347165096806482177258040663335v^10 + 9322313943263215089291031684363854146582398v^8 + 4024826356269214327657428220055819192820297732v^6 + 459806618325271738506879895627394276041196471040v^4 + 1345348344588447238773063446712994464988627542016*v^2 + 23127602738455016040377969576859138828955180990464) / 14657504675233705629746858963675549435065729941504
$\beta_{5}$	$=$	$ ( 62\!\cdots\!37 \nu^{14} + \cdots + 92\!\cdots\!64 ) / 88\!\cdots\!48 $ (6226607898963750375737v^14 + 36673514127644629973448875v^12 + 198356422527061609288312894655v^10 + 102006756225472333513387677407974v^8 + 44379178245270147811792011059472340v^6 + 2819259306996168987337901015321628640v^4 + 178515263779246264113250158176857755648*v^2 + 921730466904652351631529558648258084864) / 88949673529126915265026995766446286848
$\beta_{6}$	$=$	$ ( 23\!\cdots\!33 \nu^{14} + \cdots + 32\!\cdots\!24 ) / 29\!\cdots\!08 $ (2301025959234559735631879695533733v^14 + 13566788768692202941302293624428873175v^12 + 73384081778218733819465890939453460478699v^10 + 38139122889115001818726715784844409183700934v^8 + 16582484330840258959011036494590713763736591732v^6 + 1117660099064239935017730460391468481770378962176v^4 + 62031599954998325838337084946451314846685868564480*v^2 + 325380529401920111985203542792788378755322022068224) / 29315009350467411259493717927351098870131459883008
$\beta_{7}$	$=$	$ ( 29\!\cdots\!47 \nu^{15} + \cdots + 24\!\cdots\!88 \nu ) / 14\!\cdots\!04 $ (2904929780796478681038476588612647v^15 + 17327778528997852330622433858831016541v^13 + 93820917020426595165022486262215080006873v^11 + 54514313673021997472484198376218307360787698v^9 + 24119650753238200589816791140885480954013819484v^7 + 2786528667353109682134569774629277276342601918144v^5 + 142838983882546977263015849048964184740725180409856v^3 + 2408194812968344661008845260407230651871304865087488v) / 14657504675233705629746858963675549435065729941504
$\beta_{8}$	$=$	$ ( 52\!\cdots\!45 \nu^{14} + \cdots - 38\!\cdots\!56 ) / 41\!\cdots\!44 $ (5237588385402992771557035645v^14 + 30565050740682663884554880894687v^12 + 165177448296151713147819089149044435v^10 + 76756990127272113099758024361928311510v^8 + 32572816381413557616335104006421422463060v^6 + 312059145571770769517841683986739662878720v^4 + 2855760175736664292173733091144776777728000*v^2 - 3839810826500673677439421750243017033786720256) / 41068537224302103032743749959514404972396544
$\beta_{9}$	$=$	$ ( - 24\!\cdots\!43 \nu^{15} + \cdots - 83\!\cdots\!92 \nu ) / 73\!\cdots\!52 $ (-2481850727676099885498550258663243v^15 - 14619429136953796828672963844682115425v^13 - 79072260512065833991884146187292674439197v^11 - 40711223839880232617942747517260416866457330v^9 - 17691193956891683904744873542993275634000120316v^7 - 1123861801568102754419249495357269719815057451936v^5 - 65321257346687435490380027032185337443295124362240v^3 - 83766874433618309360235282807582037810097742446592v) / 7328752337616852814873429481837774717532864970752
$\beta_{10}$	$=$	$ ( 52\!\cdots\!35 \nu^{15} + \cdots + 32\!\cdots\!84 \nu ) / 73\!\cdots\!52 $ (5217313934060260832575970885385335v^15 + 30888432480654811432661234200954028989v^13 + 167139631213064299263309162349150698974937v^11 + 90530335978529628488891490475726750024404578v^9 + 39680545248759395102571307411439090933840451772v^7 + 3436857065641346373752162970901979362298053605504v^5 + 193078711324301857514312741601762629637295394945024v^3 + 3278501706235894131544938169897502897169595845443584v) / 7328752337616852814873429481837774717532864970752
$\beta_{11}$	$=$	$ ( 37\!\cdots\!27 \nu^{15} + \cdots + 20\!\cdots\!52 \nu ) / 36\!\cdots\!76 $ (3752754696366987548755479927034427v^15 + 22158323665582119548525718095378469337v^13 + 119873068510382597357048895302699989890789v^11 + 63232901499437261635511057770942974217866602v^9 + 27606621981590260081487809255906069140725071084v^7 + 2074715323864268342452699097612039058002765458752v^5 + 122767989665248631371792113238604498723739430037504v^3 + 2092425541946824374998692257449423109922014678679552v) / 3664376168808426407436714740918887358766432485376
$\beta_{12}$	$=$	$ ( - 66\!\cdots\!69 \nu^{15} + \cdots - 39\!\cdots\!44 \nu ) / 48\!\cdots\!68 $ (-6656014104170412297776997002629169v^15 - 39349169255400885426844170872695961963v^13 - 212895072393215632263387577098798131124303v^11 - 113688044004657263891211603755914343220117470v^9 - 49724039107110825065655548586785919634222498820v^7 - 4004842995970693244923168793568255059383177836864v^5 - 230914649328838038031015815821253971799719953602560v^3 - 3928410173613087366074478598227922132005516363628544v) / 4885834891744568543248952987891849811688576647168
$\beta_{13}$	$=$	$ ( 17\!\cdots\!10 \nu^{15} + \cdots + 32\!\cdots\!88 ) / 29\!\cdots\!08 $ (17597006817243636292954787921484710v^15 + 23422090674908904609650442434287929v^14 + 103666259538305672782660358293224817650v^13 + 138100264079508362374170178162562518963v^12 + 560700790963466053137936703491287415679146v^11 + 747000663495437817139095091990211262194199v^10 + 288957992055921027009207355444851349794684740v^9 + 388358048612266974757417372276860065828934750v^8 + 125448120244439220670838905696294240486980466488v^7 + 168894030592236560593856022678327830972470085188v^6 + 7969295388699618018900859882793544669041237091648v^5 + 11446848317508552362993145090125946634756277539584v^4 + 435578977594545086024371185804486182073023848478720v^3 + 627216440071218380311643503480439666662121681657856v^2 + 593990262164063725166176976383316492114485544288256*v + 3295386927690577903735272797588739223127657411903488) / 29315009350467411259493717927351098870131459883008
$\beta_{14}$	$=$	$ ( 23\!\cdots\!29 \nu^{14} + \cdots + 32\!\cdots\!88 ) / 14\!\cdots\!04 $ (23422090674908904609650442434287929v^14 + 138100264079508362374170178162562518963v^12 + 747000663495437817139095091990211262194199v^10 + 388358048612266974757417372276860065828934750v^8 + 168894030592236560593856022678327830972470085188v^6 + 11446848317508552362993145090125946634756277539584v^4 + 627216440071218380311643503480439666662121681657856*v^2 + 3295386927690577903735272797588739223127657411903488) / 14657504675233705629746858963675549435065729941504
$\beta_{15}$	$=$	$ ( - 52\!\cdots\!09 \nu^{15} + \cdots - 17\!\cdots\!12 \nu ) / 24\!\cdots\!84 $ (-52766974808612984801506027263795809v^15 - 310875505312705734786660402168533824675v^13 - 1681435623281003747056143524426095255281367v^11 - 867013077283683121625311018240217089300541142v^9 - 376195185831979431027935064020291294864952409076v^7 - 23898409588442464806664438159917755386218086748896v^5 - 1266048840374698227144917131355333821551469584225280v^3 - 1781264451167424745305357599282761438018745298419712v) / 2442917445872284271624476493945924905844288323584

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

$\nu$	$=$	$ ( 2\beta_{14} - 4\beta_{13} - 3\beta_{12} - 3\beta_{11} - 6\beta_{9} - 3\beta_{7} + 2\beta_{2} ) / 192 $ (2b14 - 4b13 - 3b12 - 3b11 - 6b9 - 3b7 + 2*b2) / 192
$\nu^{2}$	$=$	$ ( -48\beta_{14} + 33\beta_{8} - 3528\beta_{6} + 4727\beta_{5} + 708\beta_{4} - 139\beta_{3} - 268\beta _1 - 23486 ) / 32 $ (-48b14 + 33b8 - 3528b6 + 4727b5 + 708b4 - 139b3 - 268*b1 - 23486) / 32
$\nu^{3}$	$=$	$ ( 417\beta_{15} - 33484\beta_{14} + 66968\beta_{13} + 93441\beta_{9} - 17779\beta_{2} ) / 192 $ (417b15 - 33484b14 + 66968b13 + 93441b9 - 17779*b2) / 192
$\nu^{4}$	$=$	$ ( 158520 \beta_{14} + 220947 \beta_{8} + 20459832 \beta_{6} - 24779465 \beta_{5} - 3775140 \beta_{4} + \cdots - 115483650 ) / 32 $ (158520b14 + 220947b8 + 20459832b6 - 24779465b5 - 3775140b4 - 826073b3 - 1364060*b1 - 115483650) / 32
$\nu^{5}$	$=$	$ ( - 2478219 \beta_{15} + 184941820 \beta_{14} - 369883640 \beta_{13} + 188252811 \beta_{12} + \cdots + 94053385 \beta_{2} ) / 384 $ (-2478219b15 + 184941820b14 - 369883640b13 + 188252811b12 + 189752217b11 + 59477256b10 - 504629715b9 + 99842595b7 + 94053385*b2) / 384
$\nu^{6}$	$=$	$ ( -1195666983\beta_{8} + 4472155845\beta_{3} + 7279149564\beta _1 + 615599156842 ) / 16 $ (-1195666983b8 + 4472155845b3 + 7279149564*b1 + 615599156842) / 16
$\nu^{7}$	$=$	$ ( - 13416467535 \beta_{15} + 994971021916 \beta_{14} - 1989942043832 \beta_{13} + \cdots + 504195664837 \beta_{2} ) / 384 $ (-13416467535b15 + 994971021916b14 - 1989942043832b13 - 1011672148551b12 - 1018523138109b11 - 321995220840b10 - 2709372212391b9 - 534313259007b7 + 504195664837*b2) / 384
$\nu^{8}$	$=$	$ ( - 4303532760744 \beta_{14} + 6421113321423 \beta_{8} - 593711426959704 \beta_{6} + \cdots - 33\!\cdots\!86 ) / 32 $ (-4303532760744b14 + 6421113321423b8 - 593711426959704b6 + 712785538583237b5 + 108539632996836b4 - 24020720663197b3 - 39048899506108*b1 - 3302139623840186) / 32
$\nu^{9}$	$=$	$ ( 72062161989591 \beta_{15} + \cdots - 27\!\cdots\!45 \beta_{2} ) / 192 $ (72062161989591b15 - 5341284195846556b14 + 10682568391693112b13 + 14542135292308431b9 - 2705841344218045*b2) / 192
$\nu^{10}$	$=$	$ ( 23\!\cdots\!44 \beta_{14} + \cdots - 17\!\cdots\!02 ) / 32 $ (23088461556135144b14 + 34465060305340863b8 + 3186725916533063640b6 - 3825627766564188149b5 - 582545535929128548b4 - 128932169596188461b3 - 209574153013799612*b1 - 17722371488226194202) / 32
$\nu^{11}$	$=$	$ ( - 38\!\cdots\!83 \beta_{15} + \cdots + 14\!\cdots\!25 \beta_{2} ) / 384 $ (-386796508788565383b15 + 28668233217875935132b14 - 57336466435751870264b13 + 29146521629187284751b12 + 29340632412621487797b11 + 9283116210925569192b10 - 78050724595706451135b9 + 15387748823277768327b7 + 14522663903177853325*b2) / 384
$\nu^{12}$	$=$	$ ( - 18\!\cdots\!03 \beta_{8} + \cdots + 95\!\cdots\!58 ) / 16 $ (-184981625416656325503b8 + 692008465129051631085b3 + 1124822182713506849340*b1 + 95119109263500387160858) / 16
$\nu^{13}$	$=$	$ ( - 20\!\cdots\!55 \beta_{15} + \cdots + 77\!\cdots\!73 \beta_{2} ) / 384 $ (-2076025395387154893255b15 + 153868362966394181734684b14 - 307736725932788363469368b13 - 156435330771069401076495b12 - 157477044591308553909045b11 - 49824609489291717438120b10 - 418913917551769875037311b9 - 82588974002667105967239b7 + 77945980802604731732173*b2) / 384
$\nu^{14}$	$=$	$ ( - 66\!\cdots\!80 \beta_{14} + \cdots - 51\!\cdots\!38 ) / 32 $ (-665080189108565288114280b14 + 992833696816455071533407b8 - 91799907208654775554543320b6 + 110204131510645331442076181b5 + 16781270153411190667318116b4 - 3714149435509188100086925b3 - 6037143510542773747848892*b1 - 510523083445536200908398938) / 32
$\nu^{15}$	$=$	$ ( 11\!\cdots\!75 \beta_{15} + \cdots - 41\!\cdots\!25 \beta_{2} ) / 192 $ (11142448306527564300260775b15 - 825842355428130903721522588b14 + 1651684710856261807443045176b13 + 2248394744347560097518556767b9 - 418351628050520793181372525*b2) / 192

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$.

$n$	$111$	$133$	$145$
$\chi(n)$	$-1$	$1$	$-1$

Embeddings

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

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

175.1

10.7118	−	18.5534i
−10.7118	+	18.5534i
−1.52874	+	2.64786i
1.52874	−	2.64786i
36.6306	+	63.4460i
−36.6306	−	63.4460i
−3.98377	−	6.90009i
3.98377	+	6.90009i
−3.98377	+	6.90009i
3.98377	−	6.90009i
36.6306	−	63.4460i
−36.6306	+	63.4460i
−1.52874	−	2.64786i
1.52874	+	2.64786i
10.7118	+	18.5534i
−10.7118	−	18.5534i

−

21.5024i

24.8335

−17.3632

−219.355

175.2

−

21.5024i

24.8335

17.3632

−219.355

175.3

−

14.2541i

−99.5140

−213.906

39.8219

175.4

−

14.2541i

−99.5140

213.906

39.8219

175.5

−

9.11187i

−19.0071

−78.3341

159.974

175.6

−

9.11187i

−19.0071

78.3341

159.974

175.7

−

0.663859i

68.6876

−214.472

242.559

175.8

−

0.663859i

68.6876

214.472

242.559

175.9

0.663859i

68.6876

−214.472

242.559

175.10

0.663859i

68.6876

214.472

242.559

175.11

9.11187i

−19.0071

−78.3341

159.974

175.12

9.11187i

−19.0071

78.3341

159.974

175.13

14.2541i

−99.5140

−213.906

39.8219

175.14

14.2541i

−99.5140

213.906

39.8219

175.15

21.5024i

24.8335

−17.3632

−219.355

175.16

21.5024i

24.8335

17.3632

−219.355

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
11.b	odd	2	1	inner
44.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	176.6.e.d	✓	16
4.b	odd	2	1	inner	176.6.e.d	✓	16
11.b	odd	2	1	inner	176.6.e.d	✓	16
44.c	even	2	1	inner	176.6.e.d	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
176.6.e.d	✓	16	1.a	even	1	1	trivial
176.6.e.d	✓	16	4.b	odd	2	1	inner
176.6.e.d	✓	16	11.b	odd	2	1	inner
176.6.e.d	✓	16	44.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 749T_{3}^{6} + 149527T_{3}^{4} + 7865271T_{3}^{2} + 3437316 $ T3^8 + 749*T3^6 + 149527*T3^4 + 7865271*T3^2 + 3437316 acting on $S_{6}^{\mathrm{new}}(176, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 749 T^{6} + \cdots + 3437316)^{2} $ (T^8 + 749T^6 + 149527T^4 + 7865271*T^2 + 3437316)^2
$5$	$ (T^{4} + 25 T^{3} + \cdots + 3226386)^{4} $ (T^4 + 25T^3 - 7487T^2 + 25275*T + 3226386)^4
$7$	$ (T^{8} + \cdots + 38\!\cdots\!64)^{2} $ (T^8 - 98192T^6 + 2697235456T^4 - 13719185965056*T^2 + 3893597583704064)^2
$11$	$ T^{16} + \cdots + 45\!\cdots\!01 $ T^16 - 153352T^14 + 26901500924T^12 + 1569620594315400T^10 - 79673084962285833370T^8 + 40711915817232496713155400T^6 + 18097984610299258405206150001724T^4 - 2675900736738268335115629673252890952*T^2 + 452592555681759518058893560348969204658401
$13$	$ (T^{8} + \cdots + 65\!\cdots\!56)^{2} $ (T^8 + 2497424T^6 + 1866034052608T^4 + 368069110335258624*T^2 + 65415298977464057856)^2
$17$	$ (T^{8} + \cdots + 32\!\cdots\!84)^{2} $ (T^8 + 6932144T^6 + 15331814339584T^4 + 12296048272013672448*T^2 + 3225259015955511911645184)^2
$19$	$ (T^{8} + \cdots + 22\!\cdots\!36)^{2} $ (T^8 - 11815856T^6 + 37403057712640T^4 - 32857691676194193408*T^2 + 229553478636099520167936)^2
$23$	$ (T^{8} + \cdots + 43\!\cdots\!84)^{2} $ (T^8 + 4610749T^6 + 6618776503543T^4 + 3061434327759005127*T^2 + 43767030235519992649284)^2
$29$	$ (T^{8} + \cdots + 17\!\cdots\!56)^{2} $ (T^8 + 106770912T^6 + 3334160301391872T^4 + 32314309456262760235008*T^2 + 17507527705440453094388269056)^2
$31$	$ (T^{8} + \cdots + 82\!\cdots\!84)^{2} $ (T^8 + 74844965T^6 + 1775678470272871T^4 + 13632788853215706146991*T^2 + 8245524816315664771930695684)^2
$37$	$ (T^{4} + \cdots - 373732020594182)^{4} $ (T^4 + 1661T^3 - 65400351T^2 - 318303368737*T - 373732020594182)^4
$41$	$ (T^{8} + \cdots + 46\!\cdots\!00)^{2} $ (T^8 + 597084464T^6 + 116259667079805952T^4 + 7538083166517818468352000*T^2 + 46359050090502759083523440640000)^2
$43$	$ (T^{8} + \cdots + 91\!\cdots\!64)^{2} $ (T^8 - 713746400T^6 + 144280519683893248T^4 - 5855662818777049289195520*T^2 + 9165930182735819517639643889664)^2
$47$	$ (T^{8} + \cdots + 15\!\cdots\!44)^{2} $ (T^8 + 766982544T^6 + 109760268112431328T^4 + 2729875289574004654755072*T^2 + 15402354549231846704242709158144)^2
$53$	$ (T^{4} + \cdots - 20\!\cdots\!56)^{4} $ (T^4 - 168T^3 - 758686728T^2 + 8980259357664*T - 20082173781081456)^4
$59$	$ (T^{8} + \cdots + 95\!\cdots\!84)^{2} $ (T^8 + 2415779149T^6 + 1046317786893426103T^4 + 102577414142967926989743447*T^2 + 95735975573840677560822392163684)^2
$61$	$ (T^{8} + \cdots + 20\!\cdots\!44)^{2} $ (T^8 + 4298651808T^6 + 4767497000779032576T^4 + 431200635628270806417014784*T^2 + 200903448014278531268219548729344)^2
$67$	$ (T^{8} + \cdots + 94\!\cdots\!96)^{2} $ (T^8 + 3425352693T^6 + 3707047451594526759T^4 + 1380741329565767824132832127*T^2 + 94576746814552676403932724974424996)^2
$71$	$ (T^{8} + \cdots + 25\!\cdots\!96)^{2} $ (T^8 + 6479535253T^6 + 13018847119731867175T^4 + 8287026065816202748667821407*T^2 + 25119205502860659483020072595317796)^2
$73$	$ (T^{8} + \cdots + 13\!\cdots\!44)^{2} $ (T^8 + 9911179568T^6 + 30510625147317959680T^4 + 27588804235634860560751804416*T^2 + 130746779812882316926119378098847744)^2
$79$	$ (T^{8} + \cdots + 27\!\cdots\!00)^{2} $ (T^8 - 11661013760T^6 + 43447920512964886528T^4 - 62385141853505734579952025600*T^2 + 27521940193681641979008293630115840000)^2
$83$	$ (T^{8} + \cdots + 54\!\cdots\!56)^{2} $ (T^8 - 11071443872T^6 + 31814735239773356032T^4 - 27240525931841392418161164288*T^2 + 542294792624212451337769748954873856)^2
$89$	$ (T^{4} + \cdots + 23\!\cdots\!74)^{4} $ (T^4 - 63087T^3 - 6158015403T^2 + 20851467662331*T + 231062952275493174)^4
$97$	$ (T^{4} + \cdots - 11\!\cdots\!06)^{4} $ (T^4 - 11295T^3 - 15416158499T^2 + 882134813743899*T - 11662574569410676706)^4
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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:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	176.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + (\beta_{8} - 6) q^{5} + \beta_{9} q^{7} + (\beta_{8} + \beta_{3} - \beta_1 + 56) q^{9}+O(q^{10}) \) q + b6 * q^3 + (b8 - 6) * q^5 + b9 * q^7 + (b8 + b3 - b1 + 56) * q^9	\( q + \beta_{6} q^{3} + (\beta_{8} - 6) q^{5} + \beta_{9} q^{7} + (\beta_{8} + \beta_{3} - \beta_1 + 56) q^{9} + (\beta_{14} - \beta_{13} + \cdots - 4 \beta_{4}) q^{11}+ \cdots + (3 \beta_{15} - 77 \beta_{14} + \cdots + 20 \beta_{2}) q^{99}+O(q^{100}) \) q + b6 * q^3 + (b8 - 6) * q^5 + b9 * q^7 + (b8 + b3 - b1 + 56) * q^9 + (b14 - b13 + b6 + 2b5 - 4b4) * q^11 - b10 * q^13 + (-4b14 - 9b6 - 10b5 + b4) q^15 + (b10 + b7) * q^17 - b2 * q^19 + (-b12 + b10) * q^21 + (-5b14 + 33b6 - 41b5 - 5b4) * q^23 + (-30b8 - 35b1 + 776) * q^25 + (9b14 + 167b6 - 69b5 - 12b4) * q^27 + (-4b12 - b11 - 3b7) * q^29 + (19b14 - 123b6 + 158b5 - 25b4) * q^31 + (-2b12 + b11 - b10 + 20b8 + 2b7 - 10b3 + 5b1 - 553) q^33 + (-b15 - 11b14 + 22b13 - 7b9 + 2b2) * q^35 + (80b8 + 5b3 - 42b1 - 386) q^37 + (b15 - 6b14 + 12b13 + 34b9 + 3b2) * q^39 + (-3b12 - 7b11 - b10 + 6b7) q^41 + (b15 - 9b14 + 18b13 - 33b9 - b2) q^43 + (119b8 + 23b3 - 70b1 + 1094) q^45 + (15b14 - 498b6 + 384b5 - 200b4) * q^47 + (-b8 + 35b3 - 236b1 + 7791) * q^49 + (-21b14 + 42b13 - 30b9 + b2) q^51 + (109b8 - 71b3 - 132b1 + 120) q^53 + (-b15 - 60b14 + 18b13 - 76b9 - 851b6 + 533b5 - 171b4 - 3b2) q^55 + (-13b12 - 7b11 - 17b10 + 4b7) * q^57 + (130b14 + 393b6 + 31b5 - 50b4) * q^59 + (-6b12 - 19b11 + 18b10 + 13b7) * q^61 + (-b15 - 18b14 + 36b13 + 155b9 + 3b2) * q^63 + (-38b12 + 14b11 + 17b10 - 11b7) * q^65 + (-106b14 + 1181b6 - 1304b5 - 242b4) * q^67 + (-133b8 + 32b3 - 138b1 - 2826) q^69 + (-59b14 + 3077b6 - 1793b5 + 189b4) * q^71 + (-17b12 + 15b11 + b10 + 18b7) q^73 + (295b14 - 744b6 + 1630b5 + 110b4) * q^75 + (-30b12 + 11b11 - 17b10 - 543b8 - 37b7 - 99b3 - 148b1 + 9674) q^77 + (3b15 + 36b14 - 72b13 + 83b9 - 21b2) q^79 + (587b8 + 227b3 - 242b1 - 11819) q^81 + (-2b15 - 25b14 + 50b13 + 184b9 - 22b2) q^83 + (51b12 - 36b11 - 17b10 + 54b7) * q^85 + (-3b15 + 12b14 - 24b13 - 225b9 + 15b2) q^87 + (-726b8 + 100b3 - 359b1 + 15655) q^89 + (-600b14 - 4312b6 - 15b5 - 924b4) * q^91 + (554b8 - 205b3 + 478b1 + 7936) q^93 + (b15 + 72b14 - 144b13 - 254b9 - 3b2) * q^95 + (1125b8 - 87b3 + 463b1 + 3011) q^97 + (3b15 - 77b14 + 27b13 - 69b9 + 690b6 + 1066b5 - 802b4 + 20b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 100 q^{5} + 892 q^{9}+O(q^{10}) \) 16 * q - 100 * q^5 + 892 * q^9	\( 16 q - 100 q^{5} + 892 q^{9} + 12396 q^{25} - 8948 q^{33} - 6644 q^{37} + 16840 q^{45} + 123856 q^{49} + 672 q^{53} - 45108 q^{69} + 155968 q^{77} - 191512 q^{81} + 252348 q^{89} + 125852 q^{93} + 45180 q^{97}+O(q^{100}) \) 16 * q - 100 * q^5 + 892 * q^9 + 12396 * q^25 - 8948 * q^33 - 6644 * q^37 + 16840 * q^45 + 123856 * q^49 + 672 * q^53 - 45108 * q^69 + 155968 * q^77 - 191512 * q^81 + 252348 * q^89 + 125852 * q^93 + 45180 * 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	176.6.e.d

Level	$176$
Weight	$6$
Character orbit	176.e
Analytic conductor	$28.228$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(28.2275522871\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 5899 x^{14} + 31909903 x^{12} + 16672338574 x^{10} + 7262078908132 x^{8} + \cdots + 21\!\cdots\!96 \) x^16 + 5899x^14 + 31909903x^12 + 16672338574x^10 + 7262078908132x^8 + 510923309868544x^6 + 29217515417325568x^4 + 267327060014792704*x^2 + 2137083591258013696
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{50}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 11\!\cdots\!35 \nu^{14} + \cdots - 37\!\cdots\!84 ) / 41\!\cdots\!44 \) (110832354121059080457843735v^14 + 646356748826031823636878484349v^12 + 3495311982398828396891534837343705v^10 + 1624250950067336600153237576431812930v^8 + 755624510740503315273079329407348922236v^6 + 6603468463676240852089494323834051880960v^4 + 60430602749190798124051285969536712704000*v^2 - 3750907748780539485321512671075495388721577984) / 41068537224302103032743749959514404972396544
\(\beta_{2}\)	\(=\)	\( ( - 93\!\cdots\!19 \nu^{15} + \cdots - 31\!\cdots\!04 \nu ) / 36\!\cdots\!76 \) (-93059542045793761485601424300119v^15 - 544991786432899205772058585370590725v^13 - 2947702821365954777196292341884415415089v^11 - 1434167374540372546777175592585097386551674v^9 - 659502864877683544857065488859353848912586092v^7 - 41895989590460882962711853197767788635728064032v^5 - 11149518662509610983050923644798861557990584516608v^3 - 3122711435160074092317129166896986416392275656704v) / 3664376168808426407436714740918887358766432485376
\(\beta_{3}\)	\(=\)	\( ( 12\!\cdots\!71 \nu^{14} + \cdots - 57\!\cdots\!76 ) / 41\!\cdots\!44 \) (1219914155472199738961792571v^14 + 7119764082807469565357306063113v^12 + 38472345001916140060273424703540213v^10 + 17877872772259571045064010594915946298v^8 + 7478852597827763428391435961408350868364v^6 + 72683330765075407948208455125460260243456v^4 + 665150066531408588957547390828359378534400*v^2 - 574549165803361460439690648307520991695077376) / 41068537224302103032743749959514404972396544
\(\beta_{4}\)	\(=\)	\( ( 48\!\cdots\!97 \nu^{14} + \cdots + 23\!\cdots\!64 ) / 14\!\cdots\!04 \) (483869763203188439455963745229097v^14 + 2894608872945514555731993378757153539v^12 + 15672687945347165096806482177258040663335v^10 + 9322313943263215089291031684363854146582398v^8 + 4024826356269214327657428220055819192820297732v^6 + 459806618325271738506879895627394276041196471040v^4 + 1345348344588447238773063446712994464988627542016*v^2 + 23127602738455016040377969576859138828955180990464) / 14657504675233705629746858963675549435065729941504
\(\beta_{5}\)	\(=\)	\( ( 62\!\cdots\!37 \nu^{14} + \cdots + 92\!\cdots\!64 ) / 88\!\cdots\!48 \) (6226607898963750375737v^14 + 36673514127644629973448875v^12 + 198356422527061609288312894655v^10 + 102006756225472333513387677407974v^8 + 44379178245270147811792011059472340v^6 + 2819259306996168987337901015321628640v^4 + 178515263779246264113250158176857755648*v^2 + 921730466904652351631529558648258084864) / 88949673529126915265026995766446286848
\(\beta_{6}\)	\(=\)	\( ( 23\!\cdots\!33 \nu^{14} + \cdots + 32\!\cdots\!24 ) / 29\!\cdots\!08 \) (2301025959234559735631879695533733v^14 + 13566788768692202941302293624428873175v^12 + 73384081778218733819465890939453460478699v^10 + 38139122889115001818726715784844409183700934v^8 + 16582484330840258959011036494590713763736591732v^6 + 1117660099064239935017730460391468481770378962176v^4 + 62031599954998325838337084946451314846685868564480*v^2 + 325380529401920111985203542792788378755322022068224) / 29315009350467411259493717927351098870131459883008
\(\beta_{7}\)	\(=\)	\( ( 29\!\cdots\!47 \nu^{15} + \cdots + 24\!\cdots\!88 \nu ) / 14\!\cdots\!04 \) (2904929780796478681038476588612647v^15 + 17327778528997852330622433858831016541v^13 + 93820917020426595165022486262215080006873v^11 + 54514313673021997472484198376218307360787698v^9 + 24119650753238200589816791140885480954013819484v^7 + 2786528667353109682134569774629277276342601918144v^5 + 142838983882546977263015849048964184740725180409856v^3 + 2408194812968344661008845260407230651871304865087488v) / 14657504675233705629746858963675549435065729941504
\(\beta_{8}\)	\(=\)	\( ( 52\!\cdots\!45 \nu^{14} + \cdots - 38\!\cdots\!56 ) / 41\!\cdots\!44 \) (5237588385402992771557035645v^14 + 30565050740682663884554880894687v^12 + 165177448296151713147819089149044435v^10 + 76756990127272113099758024361928311510v^8 + 32572816381413557616335104006421422463060v^6 + 312059145571770769517841683986739662878720v^4 + 2855760175736664292173733091144776777728000*v^2 - 3839810826500673677439421750243017033786720256) / 41068537224302103032743749959514404972396544
\(\beta_{9}\)	\(=\)	\( ( - 24\!\cdots\!43 \nu^{15} + \cdots - 83\!\cdots\!92 \nu ) / 73\!\cdots\!52 \) (-2481850727676099885498550258663243v^15 - 14619429136953796828672963844682115425v^13 - 79072260512065833991884146187292674439197v^11 - 40711223839880232617942747517260416866457330v^9 - 17691193956891683904744873542993275634000120316v^7 - 1123861801568102754419249495357269719815057451936v^5 - 65321257346687435490380027032185337443295124362240v^3 - 83766874433618309360235282807582037810097742446592v) / 7328752337616852814873429481837774717532864970752
\(\beta_{10}\)	\(=\)	\( ( 52\!\cdots\!35 \nu^{15} + \cdots + 32\!\cdots\!84 \nu ) / 73\!\cdots\!52 \) (5217313934060260832575970885385335v^15 + 30888432480654811432661234200954028989v^13 + 167139631213064299263309162349150698974937v^11 + 90530335978529628488891490475726750024404578v^9 + 39680545248759395102571307411439090933840451772v^7 + 3436857065641346373752162970901979362298053605504v^5 + 193078711324301857514312741601762629637295394945024v^3 + 3278501706235894131544938169897502897169595845443584v) / 7328752337616852814873429481837774717532864970752
\(\beta_{11}\)	\(=\)	\( ( 37\!\cdots\!27 \nu^{15} + \cdots + 20\!\cdots\!52 \nu ) / 36\!\cdots\!76 \) (3752754696366987548755479927034427v^15 + 22158323665582119548525718095378469337v^13 + 119873068510382597357048895302699989890789v^11 + 63232901499437261635511057770942974217866602v^9 + 27606621981590260081487809255906069140725071084v^7 + 2074715323864268342452699097612039058002765458752v^5 + 122767989665248631371792113238604498723739430037504v^3 + 2092425541946824374998692257449423109922014678679552v) / 3664376168808426407436714740918887358766432485376
\(\beta_{12}\)	\(=\)	\( ( - 66\!\cdots\!69 \nu^{15} + \cdots - 39\!\cdots\!44 \nu ) / 48\!\cdots\!68 \) (-6656014104170412297776997002629169v^15 - 39349169255400885426844170872695961963v^13 - 212895072393215632263387577098798131124303v^11 - 113688044004657263891211603755914343220117470v^9 - 49724039107110825065655548586785919634222498820v^7 - 4004842995970693244923168793568255059383177836864v^5 - 230914649328838038031015815821253971799719953602560v^3 - 3928410173613087366074478598227922132005516363628544v) / 4885834891744568543248952987891849811688576647168
\(\beta_{13}\)	\(=\)	\( ( 17\!\cdots\!10 \nu^{15} + \cdots + 32\!\cdots\!88 ) / 29\!\cdots\!08 \) (17597006817243636292954787921484710v^15 + 23422090674908904609650442434287929v^14 + 103666259538305672782660358293224817650v^13 + 138100264079508362374170178162562518963v^12 + 560700790963466053137936703491287415679146v^11 + 747000663495437817139095091990211262194199v^10 + 288957992055921027009207355444851349794684740v^9 + 388358048612266974757417372276860065828934750v^8 + 125448120244439220670838905696294240486980466488v^7 + 168894030592236560593856022678327830972470085188v^6 + 7969295388699618018900859882793544669041237091648v^5 + 11446848317508552362993145090125946634756277539584v^4 + 435578977594545086024371185804486182073023848478720v^3 + 627216440071218380311643503480439666662121681657856v^2 + 593990262164063725166176976383316492114485544288256*v + 3295386927690577903735272797588739223127657411903488) / 29315009350467411259493717927351098870131459883008
\(\beta_{14}\)	\(=\)	\( ( 23\!\cdots\!29 \nu^{14} + \cdots + 32\!\cdots\!88 ) / 14\!\cdots\!04 \) (23422090674908904609650442434287929v^14 + 138100264079508362374170178162562518963v^12 + 747000663495437817139095091990211262194199v^10 + 388358048612266974757417372276860065828934750v^8 + 168894030592236560593856022678327830972470085188v^6 + 11446848317508552362993145090125946634756277539584v^4 + 627216440071218380311643503480439666662121681657856*v^2 + 3295386927690577903735272797588739223127657411903488) / 14657504675233705629746858963675549435065729941504
\(\beta_{15}\)	\(=\)	\( ( - 52\!\cdots\!09 \nu^{15} + \cdots - 17\!\cdots\!12 \nu ) / 24\!\cdots\!84 \) (-52766974808612984801506027263795809v^15 - 310875505312705734786660402168533824675v^13 - 1681435623281003747056143524426095255281367v^11 - 867013077283683121625311018240217089300541142v^9 - 376195185831979431027935064020291294864952409076v^7 - 23898409588442464806664438159917755386218086748896v^5 - 1266048840374698227144917131355333821551469584225280v^3 - 1781264451167424745305357599282761438018745298419712v) / 2442917445872284271624476493945924905844288323584

\(\nu\)	\(=\)	\( ( 2\beta_{14} - 4\beta_{13} - 3\beta_{12} - 3\beta_{11} - 6\beta_{9} - 3\beta_{7} + 2\beta_{2} ) / 192 \) (2b14 - 4b13 - 3b12 - 3b11 - 6b9 - 3b7 + 2*b2) / 192
\(\nu^{2}\)	\(=\)	\( ( -48\beta_{14} + 33\beta_{8} - 3528\beta_{6} + 4727\beta_{5} + 708\beta_{4} - 139\beta_{3} - 268\beta _1 - 23486 ) / 32 \) (-48b14 + 33b8 - 3528b6 + 4727b5 + 708b4 - 139b3 - 268*b1 - 23486) / 32
\(\nu^{3}\)	\(=\)	\( ( 417\beta_{15} - 33484\beta_{14} + 66968\beta_{13} + 93441\beta_{9} - 17779\beta_{2} ) / 192 \) (417b15 - 33484b14 + 66968b13 + 93441b9 - 17779*b2) / 192
\(\nu^{4}\)	\(=\)	\( ( 158520 \beta_{14} + 220947 \beta_{8} + 20459832 \beta_{6} - 24779465 \beta_{5} - 3775140 \beta_{4} + \cdots - 115483650 ) / 32 \) (158520b14 + 220947b8 + 20459832b6 - 24779465b5 - 3775140b4 - 826073b3 - 1364060*b1 - 115483650) / 32
\(\nu^{5}\)	\(=\)	\( ( - 2478219 \beta_{15} + 184941820 \beta_{14} - 369883640 \beta_{13} + 188252811 \beta_{12} + \cdots + 94053385 \beta_{2} ) / 384 \) (-2478219b15 + 184941820b14 - 369883640b13 + 188252811b12 + 189752217b11 + 59477256b10 - 504629715b9 + 99842595b7 + 94053385*b2) / 384
\(\nu^{6}\)	\(=\)	\( ( -1195666983\beta_{8} + 4472155845\beta_{3} + 7279149564\beta _1 + 615599156842 ) / 16 \) (-1195666983b8 + 4472155845b3 + 7279149564*b1 + 615599156842) / 16
\(\nu^{7}\)	\(=\)	\( ( - 13416467535 \beta_{15} + 994971021916 \beta_{14} - 1989942043832 \beta_{13} + \cdots + 504195664837 \beta_{2} ) / 384 \) (-13416467535b15 + 994971021916b14 - 1989942043832b13 - 1011672148551b12 - 1018523138109b11 - 321995220840b10 - 2709372212391b9 - 534313259007b7 + 504195664837*b2) / 384
\(\nu^{8}\)	\(=\)	\( ( - 4303532760744 \beta_{14} + 6421113321423 \beta_{8} - 593711426959704 \beta_{6} + \cdots - 33\!\cdots\!86 ) / 32 \) (-4303532760744b14 + 6421113321423b8 - 593711426959704b6 + 712785538583237b5 + 108539632996836b4 - 24020720663197b3 - 39048899506108*b1 - 3302139623840186) / 32
\(\nu^{9}\)	\(=\)	\( ( 72062161989591 \beta_{15} + \cdots - 27\!\cdots\!45 \beta_{2} ) / 192 \) (72062161989591b15 - 5341284195846556b14 + 10682568391693112b13 + 14542135292308431b9 - 2705841344218045*b2) / 192
\(\nu^{10}\)	\(=\)	\( ( 23\!\cdots\!44 \beta_{14} + \cdots - 17\!\cdots\!02 ) / 32 \) (23088461556135144b14 + 34465060305340863b8 + 3186725916533063640b6 - 3825627766564188149b5 - 582545535929128548b4 - 128932169596188461b3 - 209574153013799612*b1 - 17722371488226194202) / 32
\(\nu^{11}\)	\(=\)	\( ( - 38\!\cdots\!83 \beta_{15} + \cdots + 14\!\cdots\!25 \beta_{2} ) / 384 \) (-386796508788565383b15 + 28668233217875935132b14 - 57336466435751870264b13 + 29146521629187284751b12 + 29340632412621487797b11 + 9283116210925569192b10 - 78050724595706451135b9 + 15387748823277768327b7 + 14522663903177853325*b2) / 384
\(\nu^{12}\)	\(=\)	\( ( - 18\!\cdots\!03 \beta_{8} + \cdots + 95\!\cdots\!58 ) / 16 \) (-184981625416656325503b8 + 692008465129051631085b3 + 1124822182713506849340*b1 + 95119109263500387160858) / 16
\(\nu^{13}\)	\(=\)	\( ( - 20\!\cdots\!55 \beta_{15} + \cdots + 77\!\cdots\!73 \beta_{2} ) / 384 \) (-2076025395387154893255b15 + 153868362966394181734684b14 - 307736725932788363469368b13 - 156435330771069401076495b12 - 157477044591308553909045b11 - 49824609489291717438120b10 - 418913917551769875037311b9 - 82588974002667105967239b7 + 77945980802604731732173*b2) / 384
\(\nu^{14}\)	\(=\)	\( ( - 66\!\cdots\!80 \beta_{14} + \cdots - 51\!\cdots\!38 ) / 32 \) (-665080189108565288114280b14 + 992833696816455071533407b8 - 91799907208654775554543320b6 + 110204131510645331442076181b5 + 16781270153411190667318116b4 - 3714149435509188100086925b3 - 6037143510542773747848892*b1 - 510523083445536200908398938) / 32
\(\nu^{15}\)	\(=\)	\( ( 11\!\cdots\!75 \beta_{15} + \cdots - 41\!\cdots\!25 \beta_{2} ) / 192 \) (11142448306527564300260775b15 - 825842355428130903721522588b14 + 1651684710856261807443045176b13 + 2248394744347560097518556767b9 - 418351628050520793181372525*b2) / 192

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 175.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 749 T^{6} + \cdots + 3437316)^{2} \) (T^8 + 749T^6 + 149527T^4 + 7865271*T^2 + 3437316)^2
$5$	\( (T^{4} + 25 T^{3} + \cdots + 3226386)^{4} \) (T^4 + 25T^3 - 7487T^2 + 25275*T + 3226386)^4
$7$	\( (T^{8} + \cdots + 38\!\cdots\!64)^{2} \) (T^8 - 98192T^6 + 2697235456T^4 - 13719185965056*T^2 + 3893597583704064)^2
$11$	\( T^{16} + \cdots + 45\!\cdots\!01 \) T^16 - 153352T^14 + 26901500924T^12 + 1569620594315400T^10 - 79673084962285833370T^8 + 40711915817232496713155400T^6 + 18097984610299258405206150001724T^4 - 2675900736738268335115629673252890952*T^2 + 452592555681759518058893560348969204658401
$13$	\( (T^{8} + \cdots + 65\!\cdots\!56)^{2} \) (T^8 + 2497424T^6 + 1866034052608T^4 + 368069110335258624*T^2 + 65415298977464057856)^2
$17$	\( (T^{8} + \cdots + 32\!\cdots\!84)^{2} \) (T^8 + 6932144T^6 + 15331814339584T^4 + 12296048272013672448*T^2 + 3225259015955511911645184)^2
$19$	\( (T^{8} + \cdots + 22\!\cdots\!36)^{2} \) (T^8 - 11815856T^6 + 37403057712640T^4 - 32857691676194193408*T^2 + 229553478636099520167936)^2
$23$	\( (T^{8} + \cdots + 43\!\cdots\!84)^{2} \) (T^8 + 4610749T^6 + 6618776503543T^4 + 3061434327759005127*T^2 + 43767030235519992649284)^2
$29$	\( (T^{8} + \cdots + 17\!\cdots\!56)^{2} \) (T^8 + 106770912T^6 + 3334160301391872T^4 + 32314309456262760235008*T^2 + 17507527705440453094388269056)^2
$31$	\( (T^{8} + \cdots + 82\!\cdots\!84)^{2} \) (T^8 + 74844965T^6 + 1775678470272871T^4 + 13632788853215706146991*T^2 + 8245524816315664771930695684)^2
$37$	\( (T^{4} + \cdots - 373732020594182)^{4} \) (T^4 + 1661T^3 - 65400351T^2 - 318303368737*T - 373732020594182)^4
$41$	\( (T^{8} + \cdots + 46\!\cdots\!00)^{2} \) (T^8 + 597084464T^6 + 116259667079805952T^4 + 7538083166517818468352000*T^2 + 46359050090502759083523440640000)^2
$43$	\( (T^{8} + \cdots + 91\!\cdots\!64)^{2} \) (T^8 - 713746400T^6 + 144280519683893248T^4 - 5855662818777049289195520*T^2 + 9165930182735819517639643889664)^2
$47$	\( (T^{8} + \cdots + 15\!\cdots\!44)^{2} \) (T^8 + 766982544T^6 + 109760268112431328T^4 + 2729875289574004654755072*T^2 + 15402354549231846704242709158144)^2
$53$	\( (T^{4} + \cdots - 20\!\cdots\!56)^{4} \) (T^4 - 168T^3 - 758686728T^2 + 8980259357664*T - 20082173781081456)^4
$59$	\( (T^{8} + \cdots + 95\!\cdots\!84)^{2} \) (T^8 + 2415779149T^6 + 1046317786893426103T^4 + 102577414142967926989743447*T^2 + 95735975573840677560822392163684)^2
$61$	\( (T^{8} + \cdots + 20\!\cdots\!44)^{2} \) (T^8 + 4298651808T^6 + 4767497000779032576T^4 + 431200635628270806417014784*T^2 + 200903448014278531268219548729344)^2
$67$	\( (T^{8} + \cdots + 94\!\cdots\!96)^{2} \) (T^8 + 3425352693T^6 + 3707047451594526759T^4 + 1380741329565767824132832127*T^2 + 94576746814552676403932724974424996)^2
$71$	\( (T^{8} + \cdots + 25\!\cdots\!96)^{2} \) (T^8 + 6479535253T^6 + 13018847119731867175T^4 + 8287026065816202748667821407*T^2 + 25119205502860659483020072595317796)^2
$73$	\( (T^{8} + \cdots + 13\!\cdots\!44)^{2} \) (T^8 + 9911179568T^6 + 30510625147317959680T^4 + 27588804235634860560751804416*T^2 + 130746779812882316926119378098847744)^2
$79$	\( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) (T^8 - 11661013760T^6 + 43447920512964886528T^4 - 62385141853505734579952025600*T^2 + 27521940193681641979008293630115840000)^2
$83$	\( (T^{8} + \cdots + 54\!\cdots\!56)^{2} \) (T^8 - 11071443872T^6 + 31814735239773356032T^4 - 27240525931841392418161164288*T^2 + 542294792624212451337769748954873856)^2
$89$	\( (T^{4} + \cdots + 23\!\cdots\!74)^{4} \) (T^4 - 63087T^3 - 6158015403T^2 + 20851467662331*T + 231062952275493174)^4
$97$	\( (T^{4} + \cdots - 11\!\cdots\!06)^{4} \) (T^4 - 11295T^3 - 15416158499T^2 + 882134813743899*T - 11662574569410676706)^4
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\(n\)	\(111\)	\(133\)	\(145\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)