LMFDB - Newform orbit 3150.3.c.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.21117268020957937382075662336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 60 x^{14} - 12 x^{13} + 1608 x^{12} + 168 x^{11} - 24918 x^{10} + 3672 x^{9} + 247945 x^{8} + \cdots + 17199904 $ x^16 - 60x^14 - 12x^13 + 1608x^12 + 168x^11 - 24918x^10 + 3672x^9 + 247945x^8 - 118296x^7 - 1573338x^6 + 1049124x^5 + 6205602x^4 - 4320384x^3 - 12992160x^2 + 10002240x + 17199904
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{4} $
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_{2} q^{2} + 2 q^{4} - \beta_{10} q^{7} + 2 \beta_{2} q^{8}+O(q^{10}) $ q + b2 * q^2 + 2 * q^4 - b10 * q^7 + 2b2 q^8	$ q + \beta_{2} q^{2} + 2 q^{4} - \beta_{10} q^{7} + 2 \beta_{2} q^{8} + (\beta_{14} + \beta_{12} - \beta_{8}) q^{11} + (\beta_{13} + 2 \beta_{6}) q^{13} - \beta_{12} q^{14} + 4 q^{16} + ( - \beta_{5} + 3 \beta_{2} - 2 \beta_1) q^{17} + (5 \beta_{3} + 5) q^{19} + (\beta_{15} + 2 \beta_{10} + \beta_{6}) q^{22} + ( - \beta_{7} + 2 \beta_{5} + \cdots - 3 \beta_1) q^{23}+ \cdots - 7 \beta_{2} q^{98}+O(q^{100}) $ q + b2 * q^2 + 2 * q^4 - b10 * q^7 + 2b2 q^8 + (b14 + b12 - b8) * q^11 + (b13 + 2b6) q^13 - b12 * q^14 + 4 * q^16 + (-b5 + 3b2 - 2b1) * q^17 + (5b3 + 5) q^19 + (b15 + 2b10 + b6) q^22 + (-b7 + 2b5 - 11b2 - 3b1) q^23 + (b12 + 2b11 - 3b8) * q^26 - 2b10 q^28 + (-3b14 + 3b12 + 2b11 + b8) q^29 + (b9 - 2b4 + 7b3 - 11) * q^31 + 4b2 q^32 + (b9 + 5b3 + 7) q^34 + (-b15 + 16b6) q^37 + (5b2 - 5b1) * q^38 + (-4b14 - b12 + 13b8) * q^41 + (2b15 - 2b13 + 4b10 + b6) q^43 + (2b14 + 2b12 - 2b8) q^44 + (-2b9 + b4 + 4b3 - 25) * q^46 + (-2b7 + 3b5 - 19b2 - 8b1) * q^47 - 7 * q^49 + (2b13 + 4b6) * q^52 + (-2b5 - 7b2 - 2b1) q^53 - 2b12 q^56 + (-3b15 + 2b13 + 4b10 - b6) q^58 + (6b14 - 7b12 - 5b11 - 14b8) * q^59 + (b9 + 2b4 + 3b3 + 35) * q^61 + (4b7 - 2b5 - 14b2 - 6b1) * q^62 + 8 * q^64 + (b15 - 6b13 + 2b10 - 10b6) q^67 + (-2b5 + 6b2 - 4b1) q^68 + (3b14 + 17b12 - 4b11 + 6b8) * q^71 + (-2b15 - 2b13 - 7b10 - 3b6) * q^73 + (-2b14 - 15b8) * q^74 + (10b3 + 10) q^76 + (b7 - 3b5 + 5b2 + b1) * q^77 + (-b4 + 46b3 + 6) q^79 + (-4b15 - 2b10 - 22b6) q^82 + (6b7 + 2b5 + 4b2 - 10b1) * q^83 + (4b14 + 2b12 - 4b11 - b8) q^86 + (2b15 + 4b10 + 2b6) q^88 + (-2b14 - 10b12 + 3b11 + 27b8) * q^89 + (b9 + 2b4 + 2b3) * q^91 + (-2b7 + 4b5 - 22b2 - 6b1) * q^92 + (-3b9 + 2b4 + 13b3 - 43) q^94 + (2b15 + 6b13 + 12b10 + 16b6) * q^97 - 7b2 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 64 q^{16} + 80 q^{19} - 176 q^{31} + 112 q^{34} - 400 q^{46} - 112 q^{49} + 560 q^{61} + 128 q^{64} + 160 q^{76} + 96 q^{79} - 688 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 + 80 * q^19 - 176 * q^31 + 112 * q^34 - 400 * q^46 - 112 * q^49 + 560 * q^61 + 128 * q^64 + 160 * q^76 + 96 * q^79 - 688 * q^94

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 60 x^{14} - 12 x^{13} + 1608 x^{12} + 168 x^{11} - 24918 x^{10} + 3672 x^{9} + 247945 x^{8} + \cdots + 17199904 $ x^16 - 60*x^14 - 12*x^13 + 1608*x^12 + 168*x^11 - 24918*x^10 + 3672*x^9 + 247945*x^8 - 118296*x^7 - 1573338*x^6 + 1049124*x^5 + 6205602*x^4 - 4320384*x^3 - 12992160*x^2 + 10002240*x + 17199904 :

$\beta_{1}$	$=$	$ ( 209617739265 \nu^{15} + 3460726499946 \nu^{14} - 776213140398 \nu^{13} + \cdots - 16\!\cdots\!52 ) / 78\!\cdots\!20 $ (209617739265v^15 + 3460726499946v^14 - 776213140398v^13 - 194646657956414v^12 - 317880440928946v^11 + 4434704743276218v^10 + 9187033650282572v^9 - 56545919398901278v^8 - 112776079145575365v^7 + 472105720357459584v^6 + 846957682055160642v^5 - 2697073513983399616v^4 - 5220613323620674274v^3 + 7329247020739137552v^2 + 21749614413942573592*v - 16592269397792169152) / 7867624423016203920
$\beta_{2}$	$=$	$ ( - 21202330917883 \nu^{15} + 36639193224930 \nu^{14} + \cdots - 80\!\cdots\!00 ) / 65\!\cdots\!52 $ (-21202330917883v^15 + 36639193224930v^14 + 1130065232079710v^13 - 1846973066981430v^12 - 26677700362546486v^11 + 50291068741773330v^10 + 338625348479757320v^9 - 815218376206613910v^8 - 2345738444860854029v^7 + 8133535809943978560v^6 + 5264434145934418498v^5 - 39663472455474864960v^4 + 15232405850000035198v^3 + 87124253016047554800v^2 - 109428041664104727128*v - 80038715235732676800) / 65066393577741736752
$\beta_{3}$	$=$	$ ( - 191379784 \nu^{15} + 649530061 \nu^{14} + 10226480494 \nu^{13} - 35042249472 \nu^{12} + \cdots - 653883012899808 ) / 404949379812120 $ (-191379784v^15 + 649530061v^14 + 10226480494v^13 - 35042249472v^12 - 249230543606v^11 + 913863983084v^10 + 3306732290630v^9 - 13754039864982v^8 - 24035269775606v^7 + 127669580861033v^6 + 59019518529272v^5 - 621248763200598v^4 + 82225274038616v^3 + 1418819372410690v^2 - 629798083551104*v - 653883012899808) / 404949379812120
$\beta_{4}$	$=$	$ ( 32\!\cdots\!40 \nu^{15} + \cdots + 27\!\cdots\!36 ) / 57\!\cdots\!60 $ (32987548810881831943540v^15 + 6629957735174230040387v^14 - 2000312774933794259393266v^13 - 802742276128093065307728v^12 + 54276195202822532633768018v^11 + 17024218877716830345120076v^10 - 857737663332618279655507106v^9 - 64282303981255610445902346v^8 + 8818977448464585224398008230v^7 - 2055614506055641006919865377v^6 - 59625226259080726368273702416v^5 + 23783162976228582817751290758v^4 + 268559820911721968103472375912v^3 - 106176954170552005652312803186v^2 - 933899829656525683396551271936*v + 275489204059934470200216781536) / 57404880798862848533765731560
$\beta_{5}$	$=$	$ ( 29\!\cdots\!18 \nu^{15} + \cdots + 21\!\cdots\!80 ) / 21\!\cdots\!20 $ (29315520245815894270763118v^15 - 29973449115914935308295209v^14 - 1588065816722533543419718122v^13 + 1426198355830027436731806197v^12 + 37533662116165918405753982914v^11 - 43698666102498725166401283069v^10 - 485103901716341940260790232934v^9 + 813265889458541852318385540235v^8 + 3684231740693743915208716279212v^7 - 8671621975047580366129204933044v^6 - 13228754027096283938654888369148v^5 + 41356717016097064151957128161958v^4 + 8654380590426560419251648956696v^3 - 95885619822796402710658180606224v^2 + 44624641622794089355705273391144*v + 215044828803680929787917123203680) / 21785152263168451018564095127020
$\beta_{6}$	$=$	$ ( 27\!\cdots\!12 \nu^{15} + \cdots + 31\!\cdots\!04 ) / 17\!\cdots\!84 $ (2736391757227112v^15 - 3908087917670223v^14 - 156031472704321882v^13 + 196933316376732416v^12 + 4002355457696803218v^11 - 5575425284783155716v^10 - 57913304133316223490v^9 + 98225049198879882826v^8 + 510638411997268976578v^7 - 1100312290495019079891v^6 - 2510254821499062805256v^5 + 6687966672025499036554v^4 + 6388830417012380231352v^3 - 21855892568425721943414v^2 - 2376297405026161565568*v + 31364755670033315254304) / 1711938588536655687384
$\beta_{7}$	$=$	$ ( 35\!\cdots\!91 \nu^{15} + \cdots + 11\!\cdots\!72 ) / 18\!\cdots\!80 $ (3570800758810541019322391v^15 - 14397851552667607898589789v^14 - 199360055102040542657956006v^13 + 749174319785662577122543293v^12 + 5006117026546612893830313644v^11 - 18878299110759418714079361021v^10 - 67045539289135966066145579230v^9 + 280159444947294682182589665243v^8 + 492398203341710919797820709999v^7 - 2580189088536337452492376742712v^6 - 1304741660131863819944496972188v^5 + 12450487076313605983032478148142v^4 - 1132927889851518963669014777654v^3 - 28540821492251170983441624880200v^2 + 10685431263437175504341962322596*v + 11997767153678285360493810221472) / 1894361066362474001614269141480
$\beta_{8}$	$=$	$ ( - 46749969802459 \nu^{15} + 61501364552194 \nu^{14} + \cdots - 50\!\cdots\!68 ) / 19\!\cdots\!24 $ (-46749969802459v^15 + 61501364552194v^14 + 2705892674154538v^13 - 3052972203533526v^12 - 70291990688719082v^11 + 86687765015234882v^10 + 1030177594534026188v^9 - 1557356368283291382v^8 - 9216767715111291017v^7 + 17871000625066738736v^6 + 46626175366848148394v^5 - 110518645901712875904v^4 - 122193635554482200362v^3 + 355600217684265306448v^2 + 58518069242555283832*v - 502830303087937426368) / 19246244780391627024
$\beta_{9}$	$=$	$ ( 15\!\cdots\!40 \nu^{15} + \cdots + 60\!\cdots\!08 ) / 43\!\cdots\!40 $ (152209467405939379128882340v^15 - 297647359225331902032443209v^14 - 8374666594295643147769187818v^13 + 14056925400637039602895139376v^12 + 202423701270372773800928189834v^11 - 359141736804499820417099828612v^10 - 2621917333874613407335772353178v^9 + 5726535954117977361393315068142v^8 + 19030369180464952505860862453390v^7 - 57382623926774270651870551776701v^6 - 55227604785946511888712076692368v^5 + 280325143574270647322083249290894v^4 - 28554000151003000820963397896504v^3 - 630595559902846084641810896223658v^2 + 499663200502383283072504172652032*v + 603457247672926026853434241319208) / 43570304526336902037128190254040
$\beta_{10}$	$=$	$ ( 9565672389788 \nu^{15} - 13906827630132 \nu^{14} - 551431787898358 \nu^{13} + \cdots + 11\!\cdots\!91 ) / 22\!\cdots\!45 $ (9565672389788v^15 - 13906827630132v^14 - 551431787898358v^13 + 707209970048949v^12 + 14283391711034972v^11 - 20085699021103833v^10 - 208456235703013420v^9 + 356886630516299724v^8 + 1850115569470362592v^7 - 4041779185886963301v^6 - 9168260028379261334v^5 + 24943077017523727101v^4 + 22936297422629350528v^3 - 81012958651470321360v^2 - 7040026108041366752*v + 115926347647593279291) / 2244930299838472545
$\beta_{11}$	$=$	$ ( 11\!\cdots\!56 \nu^{15} + \cdots + 17\!\cdots\!00 ) / 24\!\cdots\!80 $ (11349876629504220215744356v^15 - 31968992093986984225221538v^14 - 668824300218198113686301709v^13 + 1602197809598169643284679934v^12 + 17716468346575146832764671133v^11 - 40538643328780583382193423078v^10 - 259150626363491918083266543793v^9 + 633610856233064196372395561030v^8 + 2252543977449102724548546088199v^7 - 6483257610677675903516964780548v^6 - 10545011051492785894248273864276v^5 + 37420988592877317897210729448776v^4 + 25673680743473555022084665838102v^3 - 120504816999361557936381321913088v^2 - 634849241726716815665613053792*v + 176818470306648064957371684395200) / 2420572473685383446507121680780
$\beta_{12}$	$=$	$ ( 372038894450805 \nu^{15} - 456602532799598 \nu^{14} + \cdots + 38\!\cdots\!96 ) / 58\!\cdots\!20 $ (372038894450805v^15 - 456602532799598v^14 - 21263089354146866v^13 + 22400238248110762v^12 + 545530274353175738v^11 - 645025736034324014v^10 - 7905874706362463496v^9 + 11712163647222087594v^8 + 70155395902346522355v^7 - 134679159338158912432v^6 - 352124916291588013726v^5 + 824533255174307490848v^4 + 937721069914876183582v^3 - 2669774821677024026896v^2 - 480507247350245476056*v + 3806612152604935500096) / 58483111405184011920
$\beta_{13}$	$=$	$ ( 675270703284860 \nu^{15} - 899666301224643 \nu^{14} + \cdots + 73\!\cdots\!56 ) / 42\!\cdots\!40 $ (675270703284860v^15 - 899666301224643v^14 - 39450049336121566v^13 + 44182910376938392v^12 + 1030907950522013998v^11 - 1253759640290762484v^10 - 15191393559171166046v^9 + 22688895069043466114v^8 + 136675465434350799850v^7 - 261932679218183660727v^6 - 694779469189509794336v^5 + 1635654342916473933098v^4 + 1807951980987536352152v^3 - 5257120440850155524046v^2 - 852243865197032125696*v + 7399081872775699494856) / 42429381168574497240
$\beta_{14}$	$=$	$ ( - 78\!\cdots\!71 \nu^{15} + \cdots - 97\!\cdots\!12 ) / 39\!\cdots\!40 $ (-78467263898622377905643971v^15 + 124070441726087764066231129v^14 + 4557524364558005125851776446v^13 - 6228974168020326718476270633v^12 - 118517509659669986333812309904v^11 + 173433328998944865071475749321v^10 + 1731830428768568542946953415810v^9 - 3039175253247094107437738381583v^8 - 15374810451203206066862532314639v^7 + 34110239416431049812338278969112v^6 + 76113215559173368909700502565328v^5 - 209764980188404917327380000195382v^4 - 190482135659013732544761313400626v^3 + 678587821102509772256231337087640v^2 + 55401365893196277054555968473204*v - 972087203598549701223265482983712) / 3960936775121536548829835477640
$\beta_{15}$	$=$	$ ( - 510631260 \nu^{15} + 841672826 \nu^{14} + 29664431922 \nu^{13} - 41084970559 \nu^{12} + \cdots - 57\!\cdots\!37 ) / 19317200603415 $ (-510631260v^15 + 841672826v^14 + 29664431922v^13 - 41084970559v^12 - 771658273476v^11 + 1106815550093v^10 + 11253181143972v^9 - 18939214754638v^8 - 99753677667720v^7 + 209799774300979v^6 + 497628080653842v^5 - 1264430436805631v^4 - 1315427709940224v^3 + 4038397490080432v^2 + 565253419665312*v - 5790717417620837) / 19317200603415

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

$\nu$	$=$	$ ( 3\beta_{12} + 3\beta_{8} - 6\beta_{6} - 2\beta_{4} + 3\beta_{2} - 3\beta_1 ) / 12 $ (3b12 + 3b8 - 6b6 - 2b4 + 3b2 - 3b1) / 12
$\nu^{2}$	$=$	$ ( - \beta_{15} - 3 \beta_{12} - 3 \beta_{11} + 3 \beta_{8} + 2 \beta_{7} + 9 \beta_{6} - 3 \beta_{5} + \cdots + 45 ) / 6 $ (-b15 - 3b12 - 3b11 + 3b8 + 2b7 + 9b6 - 3b5 + 3b3 - 4b2 + 45) / 6
$\nu^{3}$	$=$	$ ( 9 \beta_{15} - 6 \beta_{14} + 75 \beta_{12} + 9 \beta_{11} - 9 \beta_{10} + 9 \beta_{9} + 51 \beta_{8} + \cdots + 27 ) / 12 $ (9b15 - 6b14 + 75b12 + 9b11 - 9b10 + 9b9 + 51b8 - 138b6 - 9b5 - 17b4 + 78b2 - 60b1 + 27) / 12
$\nu^{4}$	$=$	$ ( - 6 \beta_{15} + 4 \beta_{14} + 3 \beta_{13} - 30 \beta_{12} - 16 \beta_{11} + 3 \beta_{10} + 34 \beta_{8} + \cdots + 96 ) / 2 $ (-6b15 + 4b14 + 3b13 - 30b12 - 16b11 + 3b10 + 34b8 + 12b7 + 135b6 - 12b5 - 3b4 + 45b3 - 30b2 - 18b1 + 96) / 2
$\nu^{5}$	$=$	$ ( 225 \beta_{15} - 190 \beta_{14} - 45 \beta_{13} + 1053 \beta_{12} + 195 \beta_{11} - 690 \beta_{10} + \cdots + 2070 ) / 12 $ (225b15 - 190b14 - 45b13 + 1053b12 + 195b11 - 690b10 + 210b9 - 433b8 + 60b7 - 1671b6 - 255b5 - 147b4 + 45b3 + 1662b2 - 378*b1 + 2070) / 12
$\nu^{6}$	$=$	$ ( - 532 \beta_{15} + 912 \beta_{14} + 675 \beta_{13} - 2436 \beta_{12} - 1176 \beta_{11} + 2025 \beta_{10} + \cdots + 2295 ) / 12 $ (-532b15 + 912b14 + 675b13 - 2436b12 - 1176b11 + 2025b10 + 135b9 + 6012b8 + 992b7 + 13149b6 - 336b5 - 720b4 + 4335b3 + 356b2 - 3060*b1 + 2295) / 12
$\nu^{7}$	$=$	$ ( 3381 \beta_{15} - 4144 \beta_{14} - 2205 \beta_{13} + 11718 \beta_{12} + 1869 \beta_{11} - 17598 \beta_{10} + \cdots + 53298 ) / 12 $ (3381b15 - 4144b14 - 2205b13 + 11718b12 + 1869b11 - 17598b10 + 2478b9 - 24286b8 + 3402b7 - 14043b6 - 5019b5 - 915b4 + 7245b3 + 21213b2 + 1953*b1 + 53298) / 12
$\nu^{8}$	$=$	$ ( - 1038 \beta_{15} + 3072 \beta_{14} + 2184 \beta_{13} - 2880 \beta_{12} - 2040 \beta_{11} + 10182 \beta_{10} + \cdots - 2209 ) / 2 $ (-1038b15 + 3072b14 + 2184b13 - 2880b12 - 2040b11 + 10182b10 + 1575b9 + 21072b8 + 1296b7 + 22680b6 + 312b5 - 2821b4 + 5880b3 + 17208b2 - 9912*b1 - 2209) / 2
$\nu^{9}$	$=$	$ ( 38799 \beta_{15} - 60666 \beta_{14} - 46413 \beta_{13} + 104565 \beta_{12} + 3627 \beta_{11} + \cdots + 940896 ) / 12 $ (38799b15 - 60666b14 - 46413b13 + 104565b12 + 3627b11 - 267264b10 + 9216b9 - 458049b8 + 103608b7 - 31911b6 - 86283b5 + 2611b4 + 318573b3 + 21126b2 + 76470*b1 + 940896) / 12
$\nu^{10}$	$=$	$ ( - 37562 \beta_{15} + 239064 \beta_{14} + 149175 \beta_{13} + 17004 \beta_{12} - 104676 \beta_{11} + \cdots - 455175 ) / 12 $ (-37562b15 + 239064b14 + 149175b13 + 17004b12 - 104676b11 + 929475b10 + 303885b9 + 1526628b8 - 42560b7 + 1042533b6 + 50844b5 - 326070b4 - 141321b3 + 3282640b2 - 1001520*b1 - 455175) / 12
$\nu^{11}$	$=$	$ ( 312081 \beta_{15} - 369732 \beta_{14} - 485595 \beta_{13} + 625908 \beta_{12} - 223641 \beta_{11} + \cdots + 14968008 ) / 12 $ (312081b15 - 369732b14 - 485595b13 + 625908b12 - 223641b11 - 1873872b10 - 304128b9 - 4586040b8 + 2302674b7 + 2037327b6 - 1424709b5 + 217237b4 + 8112555b3 - 4983135b2 + 1249113*b1 + 14968008) / 12
$\nu^{12}$	$=$	$ ( 129195 \beta_{15} + 202616 \beta_{14} + 7920 \beta_{13} + 586740 \beta_{12} - 78628 \beta_{11} + \cdots - 1324656 ) / 2 $ (129195b15 + 202616b14 + 7920b13 + 586740b12 - 78628b11 + 831555b10 + 1069200b9 + 118796b8 - 401760b7 + 700920b6 + 122220b5 - 959112b4 - 1591920b3 + 11372760b2 - 2761560*b1 - 1324656) / 2
$\nu^{13}$	$=$	$ ( - 867789 \beta_{15} + 9238658 \beta_{14} + 1242423 \beta_{13} - 5496813 \beta_{12} - 7502547 \beta_{11} + \cdots + 246611664 ) / 12 $ (-867789b15 + 9238658b14 + 1242423b13 - 5496813b12 - 7502547b11 + 30605952b10 - 7283328b9 + 12800993b8 + 40447680b7 + 76737837b6 - 23860317b5 + 4336815b4 + 145784457b3 - 109440750b2 + 19914138*b1 + 246611664) / 12
$\nu^{14}$	$=$	$ ( 40359800 \beta_{15} - 34755576 \beta_{14} - 44279235 \beta_{13} + 121279116 \beta_{12} + \cdots - 117713835 ) / 12 $ (40359800b15 - 34755576b14 - 44279235b13 + 121279116b12 + 14199276b11 - 134508465b10 + 99429057b9 - 451234044b8 - 34181968b7 - 146644929b6 + 10678668b5 - 93368184b4 - 134262219b3 + 1053486656b2 - 275950080*b1 - 117713835) / 12
$\nu^{15}$	$=$	$ ( - 123359751 \beta_{15} + 393056300 \beta_{14} + 163332585 \beta_{13} - 421841880 \beta_{12} + \cdots + 4020941412 ) / 12 $ (-123359751b15 + 393056300b14 + 163332585b13 - 421841880b12 - 220681731b11 + 1421527884b10 - 51660924b9 + 1604945108b8 + 565029246b7 + 2311253187b6 - 387302919b5 + 37146789b4 + 1951063335b3 - 1023039825b2 + 279281295*b1 + 4020941412) / 12

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

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} $

449.1

1.49491	−	1.08186i
−2.20202	−	0.788968i
−4.07285	−	1.08186i
3.36574	−	0.788968i
3.36574	+	0.788968i
−4.07285	+	1.08186i
−2.20202	+	0.788968i
1.49491	+	1.08186i
2.20202	+	1.78897i
−1.49491	+	0.0818610i
−3.36574	+	1.78897i
4.07285	+	0.0818610i
4.07285	−	0.0818610i
−3.36574	−	1.78897i
−1.49491	−	0.0818610i
2.20202	−	1.78897i

−1.41421

2.00000

−

2.64575i

−2.82843

449.2

−1.41421

2.00000

−

2.64575i

−2.82843

449.3

−1.41421

2.00000

−

2.64575i

−2.82843

449.4

−1.41421

2.00000

−

2.64575i

−2.82843

449.5

−1.41421

2.00000

2.64575i

−2.82843

449.6

−1.41421

2.00000

2.64575i

−2.82843

449.7

−1.41421

2.00000

2.64575i

−2.82843

449.8

−1.41421

2.00000

2.64575i

−2.82843

449.9

1.41421

2.00000

−

2.64575i

2.82843

449.10

1.41421

2.00000

−

2.64575i

2.82843

449.11

1.41421

2.00000

−

2.64575i

2.82843

449.12

1.41421

2.00000

−

2.64575i

2.82843

449.13

1.41421

2.00000

2.64575i

2.82843

449.14

1.41421

2.00000

2.64575i

2.82843

449.15

1.41421

2.00000

2.64575i

2.82843

449.16

1.41421

2.00000

2.64575i

2.82843

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3150.3.c.e		16
3.b	odd	2	1	inner	3150.3.c.e		16
5.b	even	2	1	inner	3150.3.c.e		16
5.c	odd	4	1		3150.3.e.g	✓	8
5.c	odd	4	1		3150.3.e.h	yes	8
15.d	odd	2	1	inner	3150.3.c.e		16
15.e	even	4	1		3150.3.e.g	✓	8
15.e	even	4	1		3150.3.e.h	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3150.3.c.e		16	1.a	even	1	1	trivial
3150.3.c.e		16	3.b	odd	2	1	inner
3150.3.c.e		16	5.b	even	2	1	inner
3150.3.c.e		16	15.d	odd	2	1	inner
3150.3.e.g	✓	8	5.c	odd	4	1
3150.3.e.g	✓	8	15.e	even	4	1
3150.3.e.h	yes	8	5.c	odd	4	1
3150.3.e.h	yes	8	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{8} + 616T_{11}^{6} + 126058T_{11}^{4} + 9670248T_{11}^{2} + 243266409 $ T11^8 + 616*T11^6 + 126058*T11^4 + 9670248*T11^2 + 243266409 acting on $S_{3}^{\mathrm{new}}(3150, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 2)^{8} $ (T^2 - 2)^8
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ (T^{2} + 7)^{8} $ (T^2 + 7)^8
$11$	$ (T^{8} + 616 T^{6} + \cdots + 243266409)^{2} $ (T^8 + 616T^6 + 126058T^4 + 9670248*T^2 + 243266409)^2
$13$	$ (T^{8} + 1008 T^{6} + \cdots + 1064586384)^{2} $ (T^8 + 1008T^6 + 319272T^4 + 34611136*T^2 + 1064586384)^2
$17$	$ (T^{8} - 944 T^{6} + \cdots + 921001104)^{2} $ (T^8 - 944T^6 + 283608T^4 - 29097792*T^2 + 921001104)^2
$19$	$ (T^{2} - 10 T - 150)^{8} $ (T^2 - 10*T - 150)^8
$23$	$ (T^{8} - 3272 T^{6} + \cdots + 31381059609)^{2} $ (T^8 - 3272T^6 + 3016602T^4 - 650483784*T^2 + 31381059609)^2
$29$	$ (T^{8} + 5000 T^{6} + \cdots + 2513919321)^{2} $ (T^8 + 5000T^6 + 6344154T^4 + 1186400520*T^2 + 2513919321)^2
$31$	$ (T^{4} + 44 T^{3} + \cdots - 1228288)^{4} $ (T^4 + 44T^3 - 1944T^2 - 114176*T - 1228288)^4
$37$	$ (T^{4} + 1070 T^{2} + 529)^{4} $ (T^4 + 1070*T^2 + 529)^4
$41$	$ (T^{8} + \cdots + 15140067768576)^{2} $ (T^8 + 9952T^6 + 32542624T^4 + 39207439872*T^2 + 15140067768576)^2
$43$	$ (T^{8} + 5908 T^{6} + \cdots + 10405428049)^{2} $ (T^8 + 5908T^6 + 8077686T^4 + 1864295860*T^2 + 10405428049)^2
$47$	$ (T^{8} + \cdots + 22616758581264)^{2} $ (T^8 - 10528T^6 + 35665912T^4 - 48181117824*T^2 + 22616758581264)^2
$53$	$ (T^{8} - 2776 T^{6} + \cdots + 4882655376)^{2} $ (T^8 - 2776T^6 + 1066360T^4 - 128979936*T^2 + 4882655376)^2
$59$	$ (T^{8} + \cdots + 189065030008464)^{2} $ (T^8 + 23920T^6 + 170381656T^4 + 328980442176*T^2 + 189065030008464)^2
$61$	$ (T^{4} - 140 T^{3} + \cdots - 2702448)^{4} $ (T^4 - 140T^3 + 3752T^2 + 116816*T - 2702448)^4
$67$	$ (T^{8} + \cdots + 30\!\cdots\!69)^{2} $ (T^8 + 32876T^6 + 375818038T^4 + 1796417503308*T^2 + 3058636056573969)^2
$71$	$ (T^{8} + \cdots + 1971140184729)^{2} $ (T^8 + 28952T^6 + 241388442T^4 + 512485939224*T^2 + 1971140184729)^2
$73$	$ (T^{8} + \cdots + 1522983064464)^{2} $ (T^8 + 12816T^6 + 43260648T^4 + 38494123072*T^2 + 1522983064464)^2
$79$	$ (T^{4} - 24 T^{3} + \cdots + 210122833)^{4} $ (T^4 - 24T^3 - 29966T^2 + 361320*T + 210122833)^4
$83$	$ (T^{8} + \cdots + 213850612871424)^{2} $ (T^8 - 31360T^6 + 305107744T^4 - 936134535168*T^2 + 213850612871424)^2
$89$	$ (T^{8} + \cdots + 3069041489424)^{2} $ (T^8 + 17920T^6 + 37249144T^4 + 20861180928*T^2 + 3069041489424)^2
$97$	$ (T^{8} + \cdots + 18\!\cdots\!56)^{2} $ (T^8 + 54160T^6 + 838798944T^4 + 3280489335040*T^2 + 1871218573517056)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} + 2 q^{4} - \beta_{10} q^{7} + 2 \beta_{2} q^{8}+O(q^{10}) \) q + b2 * q^2 + 2 * q^4 - b10 * q^7 + 2b2 q^8	\( q + \beta_{2} q^{2} + 2 q^{4} - \beta_{10} q^{7} + 2 \beta_{2} q^{8} + (\beta_{14} + \beta_{12} - \beta_{8}) q^{11} + (\beta_{13} + 2 \beta_{6}) q^{13} - \beta_{12} q^{14} + 4 q^{16} + ( - \beta_{5} + 3 \beta_{2} - 2 \beta_1) q^{17} + (5 \beta_{3} + 5) q^{19} + (\beta_{15} + 2 \beta_{10} + \beta_{6}) q^{22} + ( - \beta_{7} + 2 \beta_{5} + \cdots - 3 \beta_1) q^{23}+ \cdots - 7 \beta_{2} q^{98}+O(q^{100}) \) q + b2 * q^2 + 2 * q^4 - b10 * q^7 + 2b2 q^8 + (b14 + b12 - b8) * q^11 + (b13 + 2b6) q^13 - b12 * q^14 + 4 * q^16 + (-b5 + 3b2 - 2b1) * q^17 + (5b3 + 5) q^19 + (b15 + 2b10 + b6) q^22 + (-b7 + 2b5 - 11b2 - 3b1) q^23 + (b12 + 2b11 - 3b8) * q^26 - 2b10 q^28 + (-3b14 + 3b12 + 2b11 + b8) q^29 + (b9 - 2b4 + 7b3 - 11) * q^31 + 4b2 q^32 + (b9 + 5b3 + 7) q^34 + (-b15 + 16b6) q^37 + (5b2 - 5b1) * q^38 + (-4b14 - b12 + 13b8) * q^41 + (2b15 - 2b13 + 4b10 + b6) q^43 + (2b14 + 2b12 - 2b8) q^44 + (-2b9 + b4 + 4b3 - 25) * q^46 + (-2b7 + 3b5 - 19b2 - 8b1) * q^47 - 7 * q^49 + (2b13 + 4b6) * q^52 + (-2b5 - 7b2 - 2b1) q^53 - 2b12 q^56 + (-3b15 + 2b13 + 4b10 - b6) q^58 + (6b14 - 7b12 - 5b11 - 14b8) * q^59 + (b9 + 2b4 + 3b3 + 35) * q^61 + (4b7 - 2b5 - 14b2 - 6b1) * q^62 + 8 * q^64 + (b15 - 6b13 + 2b10 - 10b6) q^67 + (-2b5 + 6b2 - 4b1) q^68 + (3b14 + 17b12 - 4b11 + 6b8) * q^71 + (-2b15 - 2b13 - 7b10 - 3b6) * q^73 + (-2b14 - 15b8) * q^74 + (10b3 + 10) q^76 + (b7 - 3b5 + 5b2 + b1) * q^77 + (-b4 + 46b3 + 6) q^79 + (-4b15 - 2b10 - 22b6) q^82 + (6b7 + 2b5 + 4b2 - 10b1) * q^83 + (4b14 + 2b12 - 4b11 - b8) q^86 + (2b15 + 4b10 + 2b6) q^88 + (-2b14 - 10b12 + 3b11 + 27b8) * q^89 + (b9 + 2b4 + 2b3) * q^91 + (-2b7 + 4b5 - 22b2 - 6b1) * q^92 + (-3b9 + 2b4 + 13b3 - 43) q^94 + (2b15 + 6b13 + 12b10 + 16b6) * q^97 - 7b2 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} + 80 q^{19} - 176 q^{31} + 112 q^{34} - 400 q^{46} - 112 q^{49} + 560 q^{61} + 128 q^{64} + 160 q^{76} + 96 q^{79} - 688 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 + 80 * q^19 - 176 * q^31 + 112 * q^34 - 400 * q^46 - 112 * q^49 + 560 * q^61 + 128 * q^64 + 160 * q^76 + 96 * q^79 - 688 * q^94

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	3150.3.c.e

Level	$3150$
Weight	$3$
Character orbit	3150.c
Analytic conductor	$85.831$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(85.8312832735\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.21117268020957937382075662336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 60 x^{14} - 12 x^{13} + 1608 x^{12} + 168 x^{11} - 24918 x^{10} + 3672 x^{9} + 247945 x^{8} + \cdots + 17199904 \) x^16 - 60x^14 - 12x^13 + 1608x^12 + 168x^11 - 24918x^10 + 3672x^9 + 247945x^8 - 118296x^7 - 1573338x^6 + 1049124x^5 + 6205602x^4 - 4320384x^3 - 12992160x^2 + 10002240x + 17199904
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{16}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 209617739265 \nu^{15} + 3460726499946 \nu^{14} - 776213140398 \nu^{13} + \cdots - 16\!\cdots\!52 ) / 78\!\cdots\!20 \) (209617739265v^15 + 3460726499946v^14 - 776213140398v^13 - 194646657956414v^12 - 317880440928946v^11 + 4434704743276218v^10 + 9187033650282572v^9 - 56545919398901278v^8 - 112776079145575365v^7 + 472105720357459584v^6 + 846957682055160642v^5 - 2697073513983399616v^4 - 5220613323620674274v^3 + 7329247020739137552v^2 + 21749614413942573592*v - 16592269397792169152) / 7867624423016203920
\(\beta_{2}\)	\(=\)	\( ( - 21202330917883 \nu^{15} + 36639193224930 \nu^{14} + \cdots - 80\!\cdots\!00 ) / 65\!\cdots\!52 \) (-21202330917883v^15 + 36639193224930v^14 + 1130065232079710v^13 - 1846973066981430v^12 - 26677700362546486v^11 + 50291068741773330v^10 + 338625348479757320v^9 - 815218376206613910v^8 - 2345738444860854029v^7 + 8133535809943978560v^6 + 5264434145934418498v^5 - 39663472455474864960v^4 + 15232405850000035198v^3 + 87124253016047554800v^2 - 109428041664104727128*v - 80038715235732676800) / 65066393577741736752
\(\beta_{3}\)	\(=\)	\( ( - 191379784 \nu^{15} + 649530061 \nu^{14} + 10226480494 \nu^{13} - 35042249472 \nu^{12} + \cdots - 653883012899808 ) / 404949379812120 \) (-191379784v^15 + 649530061v^14 + 10226480494v^13 - 35042249472v^12 - 249230543606v^11 + 913863983084v^10 + 3306732290630v^9 - 13754039864982v^8 - 24035269775606v^7 + 127669580861033v^6 + 59019518529272v^5 - 621248763200598v^4 + 82225274038616v^3 + 1418819372410690v^2 - 629798083551104*v - 653883012899808) / 404949379812120
\(\beta_{4}\)	\(=\)	\( ( 32\!\cdots\!40 \nu^{15} + \cdots + 27\!\cdots\!36 ) / 57\!\cdots\!60 \) (32987548810881831943540v^15 + 6629957735174230040387v^14 - 2000312774933794259393266v^13 - 802742276128093065307728v^12 + 54276195202822532633768018v^11 + 17024218877716830345120076v^10 - 857737663332618279655507106v^9 - 64282303981255610445902346v^8 + 8818977448464585224398008230v^7 - 2055614506055641006919865377v^6 - 59625226259080726368273702416v^5 + 23783162976228582817751290758v^4 + 268559820911721968103472375912v^3 - 106176954170552005652312803186v^2 - 933899829656525683396551271936*v + 275489204059934470200216781536) / 57404880798862848533765731560
\(\beta_{5}\)	\(=\)	\( ( 29\!\cdots\!18 \nu^{15} + \cdots + 21\!\cdots\!80 ) / 21\!\cdots\!20 \) (29315520245815894270763118v^15 - 29973449115914935308295209v^14 - 1588065816722533543419718122v^13 + 1426198355830027436731806197v^12 + 37533662116165918405753982914v^11 - 43698666102498725166401283069v^10 - 485103901716341940260790232934v^9 + 813265889458541852318385540235v^8 + 3684231740693743915208716279212v^7 - 8671621975047580366129204933044v^6 - 13228754027096283938654888369148v^5 + 41356717016097064151957128161958v^4 + 8654380590426560419251648956696v^3 - 95885619822796402710658180606224v^2 + 44624641622794089355705273391144*v + 215044828803680929787917123203680) / 21785152263168451018564095127020
\(\beta_{6}\)	\(=\)	\( ( 27\!\cdots\!12 \nu^{15} + \cdots + 31\!\cdots\!04 ) / 17\!\cdots\!84 \) (2736391757227112v^15 - 3908087917670223v^14 - 156031472704321882v^13 + 196933316376732416v^12 + 4002355457696803218v^11 - 5575425284783155716v^10 - 57913304133316223490v^9 + 98225049198879882826v^8 + 510638411997268976578v^7 - 1100312290495019079891v^6 - 2510254821499062805256v^5 + 6687966672025499036554v^4 + 6388830417012380231352v^3 - 21855892568425721943414v^2 - 2376297405026161565568*v + 31364755670033315254304) / 1711938588536655687384
\(\beta_{7}\)	\(=\)	\( ( 35\!\cdots\!91 \nu^{15} + \cdots + 11\!\cdots\!72 ) / 18\!\cdots\!80 \) (3570800758810541019322391v^15 - 14397851552667607898589789v^14 - 199360055102040542657956006v^13 + 749174319785662577122543293v^12 + 5006117026546612893830313644v^11 - 18878299110759418714079361021v^10 - 67045539289135966066145579230v^9 + 280159444947294682182589665243v^8 + 492398203341710919797820709999v^7 - 2580189088536337452492376742712v^6 - 1304741660131863819944496972188v^5 + 12450487076313605983032478148142v^4 - 1132927889851518963669014777654v^3 - 28540821492251170983441624880200v^2 + 10685431263437175504341962322596*v + 11997767153678285360493810221472) / 1894361066362474001614269141480
\(\beta_{8}\)	\(=\)	\( ( - 46749969802459 \nu^{15} + 61501364552194 \nu^{14} + \cdots - 50\!\cdots\!68 ) / 19\!\cdots\!24 \) (-46749969802459v^15 + 61501364552194v^14 + 2705892674154538v^13 - 3052972203533526v^12 - 70291990688719082v^11 + 86687765015234882v^10 + 1030177594534026188v^9 - 1557356368283291382v^8 - 9216767715111291017v^7 + 17871000625066738736v^6 + 46626175366848148394v^5 - 110518645901712875904v^4 - 122193635554482200362v^3 + 355600217684265306448v^2 + 58518069242555283832*v - 502830303087937426368) / 19246244780391627024
\(\beta_{9}\)	\(=\)	\( ( 15\!\cdots\!40 \nu^{15} + \cdots + 60\!\cdots\!08 ) / 43\!\cdots\!40 \) (152209467405939379128882340v^15 - 297647359225331902032443209v^14 - 8374666594295643147769187818v^13 + 14056925400637039602895139376v^12 + 202423701270372773800928189834v^11 - 359141736804499820417099828612v^10 - 2621917333874613407335772353178v^9 + 5726535954117977361393315068142v^8 + 19030369180464952505860862453390v^7 - 57382623926774270651870551776701v^6 - 55227604785946511888712076692368v^5 + 280325143574270647322083249290894v^4 - 28554000151003000820963397896504v^3 - 630595559902846084641810896223658v^2 + 499663200502383283072504172652032*v + 603457247672926026853434241319208) / 43570304526336902037128190254040
\(\beta_{10}\)	\(=\)	\( ( 9565672389788 \nu^{15} - 13906827630132 \nu^{14} - 551431787898358 \nu^{13} + \cdots + 11\!\cdots\!91 ) / 22\!\cdots\!45 \) (9565672389788v^15 - 13906827630132v^14 - 551431787898358v^13 + 707209970048949v^12 + 14283391711034972v^11 - 20085699021103833v^10 - 208456235703013420v^9 + 356886630516299724v^8 + 1850115569470362592v^7 - 4041779185886963301v^6 - 9168260028379261334v^5 + 24943077017523727101v^4 + 22936297422629350528v^3 - 81012958651470321360v^2 - 7040026108041366752*v + 115926347647593279291) / 2244930299838472545
\(\beta_{11}\)	\(=\)	\( ( 11\!\cdots\!56 \nu^{15} + \cdots + 17\!\cdots\!00 ) / 24\!\cdots\!80 \) (11349876629504220215744356v^15 - 31968992093986984225221538v^14 - 668824300218198113686301709v^13 + 1602197809598169643284679934v^12 + 17716468346575146832764671133v^11 - 40538643328780583382193423078v^10 - 259150626363491918083266543793v^9 + 633610856233064196372395561030v^8 + 2252543977449102724548546088199v^7 - 6483257610677675903516964780548v^6 - 10545011051492785894248273864276v^5 + 37420988592877317897210729448776v^4 + 25673680743473555022084665838102v^3 - 120504816999361557936381321913088v^2 - 634849241726716815665613053792*v + 176818470306648064957371684395200) / 2420572473685383446507121680780
\(\beta_{12}\)	\(=\)	\( ( 372038894450805 \nu^{15} - 456602532799598 \nu^{14} + \cdots + 38\!\cdots\!96 ) / 58\!\cdots\!20 \) (372038894450805v^15 - 456602532799598v^14 - 21263089354146866v^13 + 22400238248110762v^12 + 545530274353175738v^11 - 645025736034324014v^10 - 7905874706362463496v^9 + 11712163647222087594v^8 + 70155395902346522355v^7 - 134679159338158912432v^6 - 352124916291588013726v^5 + 824533255174307490848v^4 + 937721069914876183582v^3 - 2669774821677024026896v^2 - 480507247350245476056*v + 3806612152604935500096) / 58483111405184011920
\(\beta_{13}\)	\(=\)	\( ( 675270703284860 \nu^{15} - 899666301224643 \nu^{14} + \cdots + 73\!\cdots\!56 ) / 42\!\cdots\!40 \) (675270703284860v^15 - 899666301224643v^14 - 39450049336121566v^13 + 44182910376938392v^12 + 1030907950522013998v^11 - 1253759640290762484v^10 - 15191393559171166046v^9 + 22688895069043466114v^8 + 136675465434350799850v^7 - 261932679218183660727v^6 - 694779469189509794336v^5 + 1635654342916473933098v^4 + 1807951980987536352152v^3 - 5257120440850155524046v^2 - 852243865197032125696*v + 7399081872775699494856) / 42429381168574497240
\(\beta_{14}\)	\(=\)	\( ( - 78\!\cdots\!71 \nu^{15} + \cdots - 97\!\cdots\!12 ) / 39\!\cdots\!40 \) (-78467263898622377905643971v^15 + 124070441726087764066231129v^14 + 4557524364558005125851776446v^13 - 6228974168020326718476270633v^12 - 118517509659669986333812309904v^11 + 173433328998944865071475749321v^10 + 1731830428768568542946953415810v^9 - 3039175253247094107437738381583v^8 - 15374810451203206066862532314639v^7 + 34110239416431049812338278969112v^6 + 76113215559173368909700502565328v^5 - 209764980188404917327380000195382v^4 - 190482135659013732544761313400626v^3 + 678587821102509772256231337087640v^2 + 55401365893196277054555968473204*v - 972087203598549701223265482983712) / 3960936775121536548829835477640
\(\beta_{15}\)	\(=\)	\( ( - 510631260 \nu^{15} + 841672826 \nu^{14} + 29664431922 \nu^{13} - 41084970559 \nu^{12} + \cdots - 57\!\cdots\!37 ) / 19317200603415 \) (-510631260v^15 + 841672826v^14 + 29664431922v^13 - 41084970559v^12 - 771658273476v^11 + 1106815550093v^10 + 11253181143972v^9 - 18939214754638v^8 - 99753677667720v^7 + 209799774300979v^6 + 497628080653842v^5 - 1264430436805631v^4 - 1315427709940224v^3 + 4038397490080432v^2 + 565253419665312*v - 5790717417620837) / 19317200603415

\(\nu\)	\(=\)	\( ( 3\beta_{12} + 3\beta_{8} - 6\beta_{6} - 2\beta_{4} + 3\beta_{2} - 3\beta_1 ) / 12 \) (3b12 + 3b8 - 6b6 - 2b4 + 3b2 - 3b1) / 12
\(\nu^{2}\)	\(=\)	\( ( - \beta_{15} - 3 \beta_{12} - 3 \beta_{11} + 3 \beta_{8} + 2 \beta_{7} + 9 \beta_{6} - 3 \beta_{5} + \cdots + 45 ) / 6 \) (-b15 - 3b12 - 3b11 + 3b8 + 2b7 + 9b6 - 3b5 + 3b3 - 4b2 + 45) / 6
\(\nu^{3}\)	\(=\)	\( ( 9 \beta_{15} - 6 \beta_{14} + 75 \beta_{12} + 9 \beta_{11} - 9 \beta_{10} + 9 \beta_{9} + 51 \beta_{8} + \cdots + 27 ) / 12 \) (9b15 - 6b14 + 75b12 + 9b11 - 9b10 + 9b9 + 51b8 - 138b6 - 9b5 - 17b4 + 78b2 - 60b1 + 27) / 12
\(\nu^{4}\)	\(=\)	\( ( - 6 \beta_{15} + 4 \beta_{14} + 3 \beta_{13} - 30 \beta_{12} - 16 \beta_{11} + 3 \beta_{10} + 34 \beta_{8} + \cdots + 96 ) / 2 \) (-6b15 + 4b14 + 3b13 - 30b12 - 16b11 + 3b10 + 34b8 + 12b7 + 135b6 - 12b5 - 3b4 + 45b3 - 30b2 - 18b1 + 96) / 2
\(\nu^{5}\)	\(=\)	\( ( 225 \beta_{15} - 190 \beta_{14} - 45 \beta_{13} + 1053 \beta_{12} + 195 \beta_{11} - 690 \beta_{10} + \cdots + 2070 ) / 12 \) (225b15 - 190b14 - 45b13 + 1053b12 + 195b11 - 690b10 + 210b9 - 433b8 + 60b7 - 1671b6 - 255b5 - 147b4 + 45b3 + 1662b2 - 378*b1 + 2070) / 12
\(\nu^{6}\)	\(=\)	\( ( - 532 \beta_{15} + 912 \beta_{14} + 675 \beta_{13} - 2436 \beta_{12} - 1176 \beta_{11} + 2025 \beta_{10} + \cdots + 2295 ) / 12 \) (-532b15 + 912b14 + 675b13 - 2436b12 - 1176b11 + 2025b10 + 135b9 + 6012b8 + 992b7 + 13149b6 - 336b5 - 720b4 + 4335b3 + 356b2 - 3060*b1 + 2295) / 12
\(\nu^{7}\)	\(=\)	\( ( 3381 \beta_{15} - 4144 \beta_{14} - 2205 \beta_{13} + 11718 \beta_{12} + 1869 \beta_{11} - 17598 \beta_{10} + \cdots + 53298 ) / 12 \) (3381b15 - 4144b14 - 2205b13 + 11718b12 + 1869b11 - 17598b10 + 2478b9 - 24286b8 + 3402b7 - 14043b6 - 5019b5 - 915b4 + 7245b3 + 21213b2 + 1953*b1 + 53298) / 12
\(\nu^{8}\)	\(=\)	\( ( - 1038 \beta_{15} + 3072 \beta_{14} + 2184 \beta_{13} - 2880 \beta_{12} - 2040 \beta_{11} + 10182 \beta_{10} + \cdots - 2209 ) / 2 \) (-1038b15 + 3072b14 + 2184b13 - 2880b12 - 2040b11 + 10182b10 + 1575b9 + 21072b8 + 1296b7 + 22680b6 + 312b5 - 2821b4 + 5880b3 + 17208b2 - 9912*b1 - 2209) / 2
\(\nu^{9}\)	\(=\)	\( ( 38799 \beta_{15} - 60666 \beta_{14} - 46413 \beta_{13} + 104565 \beta_{12} + 3627 \beta_{11} + \cdots + 940896 ) / 12 \) (38799b15 - 60666b14 - 46413b13 + 104565b12 + 3627b11 - 267264b10 + 9216b9 - 458049b8 + 103608b7 - 31911b6 - 86283b5 + 2611b4 + 318573b3 + 21126b2 + 76470*b1 + 940896) / 12
\(\nu^{10}\)	\(=\)	\( ( - 37562 \beta_{15} + 239064 \beta_{14} + 149175 \beta_{13} + 17004 \beta_{12} - 104676 \beta_{11} + \cdots - 455175 ) / 12 \) (-37562b15 + 239064b14 + 149175b13 + 17004b12 - 104676b11 + 929475b10 + 303885b9 + 1526628b8 - 42560b7 + 1042533b6 + 50844b5 - 326070b4 - 141321b3 + 3282640b2 - 1001520*b1 - 455175) / 12
\(\nu^{11}\)	\(=\)	\( ( 312081 \beta_{15} - 369732 \beta_{14} - 485595 \beta_{13} + 625908 \beta_{12} - 223641 \beta_{11} + \cdots + 14968008 ) / 12 \) (312081b15 - 369732b14 - 485595b13 + 625908b12 - 223641b11 - 1873872b10 - 304128b9 - 4586040b8 + 2302674b7 + 2037327b6 - 1424709b5 + 217237b4 + 8112555b3 - 4983135b2 + 1249113*b1 + 14968008) / 12
\(\nu^{12}\)	\(=\)	\( ( 129195 \beta_{15} + 202616 \beta_{14} + 7920 \beta_{13} + 586740 \beta_{12} - 78628 \beta_{11} + \cdots - 1324656 ) / 2 \) (129195b15 + 202616b14 + 7920b13 + 586740b12 - 78628b11 + 831555b10 + 1069200b9 + 118796b8 - 401760b7 + 700920b6 + 122220b5 - 959112b4 - 1591920b3 + 11372760b2 - 2761560*b1 - 1324656) / 2
\(\nu^{13}\)	\(=\)	\( ( - 867789 \beta_{15} + 9238658 \beta_{14} + 1242423 \beta_{13} - 5496813 \beta_{12} - 7502547 \beta_{11} + \cdots + 246611664 ) / 12 \) (-867789b15 + 9238658b14 + 1242423b13 - 5496813b12 - 7502547b11 + 30605952b10 - 7283328b9 + 12800993b8 + 40447680b7 + 76737837b6 - 23860317b5 + 4336815b4 + 145784457b3 - 109440750b2 + 19914138*b1 + 246611664) / 12
\(\nu^{14}\)	\(=\)	\( ( 40359800 \beta_{15} - 34755576 \beta_{14} - 44279235 \beta_{13} + 121279116 \beta_{12} + \cdots - 117713835 ) / 12 \) (40359800b15 - 34755576b14 - 44279235b13 + 121279116b12 + 14199276b11 - 134508465b10 + 99429057b9 - 451234044b8 - 34181968b7 - 146644929b6 + 10678668b5 - 93368184b4 - 134262219b3 + 1053486656b2 - 275950080*b1 - 117713835) / 12
\(\nu^{15}\)	\(=\)	\( ( - 123359751 \beta_{15} + 393056300 \beta_{14} + 163332585 \beta_{13} - 421841880 \beta_{12} + \cdots + 4020941412 ) / 12 \) (-123359751b15 + 393056300b14 + 163332585b13 - 421841880b12 - 220681731b11 + 1421527884b10 - 51660924b9 + 1604945108b8 + 565029246b7 + 2311253187b6 - 387302919b5 + 37146789b4 + 1951063335b3 - 1023039825b2 + 279281295*b1 + 4020941412) / 12

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 449.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{8} \) (T^2 - 2)^8
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( (T^{2} + 7)^{8} \) (T^2 + 7)^8
$11$	\( (T^{8} + 616 T^{6} + \cdots + 243266409)^{2} \) (T^8 + 616T^6 + 126058T^4 + 9670248*T^2 + 243266409)^2
$13$	\( (T^{8} + 1008 T^{6} + \cdots + 1064586384)^{2} \) (T^8 + 1008T^6 + 319272T^4 + 34611136*T^2 + 1064586384)^2
$17$	\( (T^{8} - 944 T^{6} + \cdots + 921001104)^{2} \) (T^8 - 944T^6 + 283608T^4 - 29097792*T^2 + 921001104)^2
$19$	\( (T^{2} - 10 T - 150)^{8} \) (T^2 - 10*T - 150)^8
$23$	\( (T^{8} - 3272 T^{6} + \cdots + 31381059609)^{2} \) (T^8 - 3272T^6 + 3016602T^4 - 650483784*T^2 + 31381059609)^2
$29$	\( (T^{8} + 5000 T^{6} + \cdots + 2513919321)^{2} \) (T^8 + 5000T^6 + 6344154T^4 + 1186400520*T^2 + 2513919321)^2
$31$	\( (T^{4} + 44 T^{3} + \cdots - 1228288)^{4} \) (T^4 + 44T^3 - 1944T^2 - 114176*T - 1228288)^4
$37$	\( (T^{4} + 1070 T^{2} + 529)^{4} \) (T^4 + 1070*T^2 + 529)^4
$41$	\( (T^{8} + \cdots + 15140067768576)^{2} \) (T^8 + 9952T^6 + 32542624T^4 + 39207439872*T^2 + 15140067768576)^2
$43$	\( (T^{8} + 5908 T^{6} + \cdots + 10405428049)^{2} \) (T^8 + 5908T^6 + 8077686T^4 + 1864295860*T^2 + 10405428049)^2
$47$	\( (T^{8} + \cdots + 22616758581264)^{2} \) (T^8 - 10528T^6 + 35665912T^4 - 48181117824*T^2 + 22616758581264)^2
$53$	\( (T^{8} - 2776 T^{6} + \cdots + 4882655376)^{2} \) (T^8 - 2776T^6 + 1066360T^4 - 128979936*T^2 + 4882655376)^2
$59$	\( (T^{8} + \cdots + 189065030008464)^{2} \) (T^8 + 23920T^6 + 170381656T^4 + 328980442176*T^2 + 189065030008464)^2
$61$	\( (T^{4} - 140 T^{3} + \cdots - 2702448)^{4} \) (T^4 - 140T^3 + 3752T^2 + 116816*T - 2702448)^4
$67$	\( (T^{8} + \cdots + 30\!\cdots\!69)^{2} \) (T^8 + 32876T^6 + 375818038T^4 + 1796417503308*T^2 + 3058636056573969)^2
$71$	\( (T^{8} + \cdots + 1971140184729)^{2} \) (T^8 + 28952T^6 + 241388442T^4 + 512485939224*T^2 + 1971140184729)^2
$73$	\( (T^{8} + \cdots + 1522983064464)^{2} \) (T^8 + 12816T^6 + 43260648T^4 + 38494123072*T^2 + 1522983064464)^2
$79$	\( (T^{4} - 24 T^{3} + \cdots + 210122833)^{4} \) (T^4 - 24T^3 - 29966T^2 + 361320*T + 210122833)^4
$83$	\( (T^{8} + \cdots + 213850612871424)^{2} \) (T^8 - 31360T^6 + 305107744T^4 - 936134535168*T^2 + 213850612871424)^2
$89$	\( (T^{8} + \cdots + 3069041489424)^{2} \) (T^8 + 17920T^6 + 37249144T^4 + 20861180928*T^2 + 3069041489424)^2
$97$	\( (T^{8} + \cdots + 18\!\cdots\!56)^{2} \) (T^8 + 54160T^6 + 838798944T^4 + 3280489335040*T^2 + 1871218573517056)^2
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\(n\)	\(127\)	\(451\)	\(2801\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)