LMFDB - Newform orbit 960.3.p.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(799,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.p (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:	$26.1581053786$
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} - 4 x^{15} + 8 x^{14} + 76 x^{13} - 505 x^{12} + 1474 x^{11} + 1032 x^{10} - 12054 x^{9} + \cdots + 100962304 $ x^16 - 4x^15 + 8x^14 + 76x^13 - 505x^12 + 1474x^11 + 1032x^10 - 12054x^9 + 121741x^8 - 65358x^7 - 21102x^6 + 2266852x^5 - 5268796x^4 + 8259424x^3 + 21411968x^2 - 65754112*x + 100962304
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{24} $
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_{5} q^{3} + \beta_{8} q^{5} + (\beta_{12} - 2) q^{7} - 3 q^{9}+O(q^{10}) $ q - b5 * q^3 + b8 * q^5 + (b12 - 2) * q^7 - 3 * q^9	$ q - \beta_{5} q^{3} + \beta_{8} q^{5} + (\beta_{12} - 2) q^{7} - 3 q^{9} + ( - \beta_{10} - \beta_1) q^{11} + (\beta_{9} - \beta_{8} + \beta_{7} - \beta_1) q^{13} + (\beta_{6} + 1) q^{15} + (\beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{17}+ \cdots + (3 \beta_{10} + 3 \beta_1) q^{99}+O(q^{100}) $ q - b5 * q^3 + b8 * q^5 + (b12 - 2) * q^7 - 3 * q^9 + (-b10 - b1) * q^11 + (b9 - b8 + b7 - b1) * q^13 + (b6 + 1) * q^15 + (b15 + b14 - b6 - b2) * q^17 + (-b9 + 2b8 - 2b7 + 2b1) q^19 + (b13 + b11 + b5) * q^21 + (b15 + b12 + b6 + 12) * q^23 + (2b15 + b14 + 2b12 + b4 - b3) * q^25 + 3b5 q^27 + (-2b13 + 4b11 + b8 + b7 - 8b5) q^29 + (b15 - 3b14 - b6 - 3b4 - 2b2) q^31 + (-4b4 - b2) q^33 + (-b13 + 2b9 - b8 - 5b7 - 2b5 - 3b1) * q^35 + (6b10 - b9 - b8 + b7 + 5b1) * q^37 + (2b15 + 3b14 - 2b6 - 3b4) * q^39 + (-4b12 + 2b3 + 8) * q^41 + (-4b11 - 6b8 - 6b7 + 8b5) * q^43 - 3b8 q^45 + (b15 - 3b12 + b6 + 2b3 - 22) * q^47 + (2b15 + 4b12 + 2b6 + 4b3 + 19) * q^49 + (3b10 - b9 + 2b8 - 2b7 - b1) q^51 + (6b10 - 4b9 + b8 - b7 - 4b1) q^53 + (b15 + 3b14 + b12 - b6 + 13b4 + 2b3 + 4) q^55 + (-3b15 - 3b14 + 3b6 + 6b4) * q^57 + (-b10 + 2b9 + 8b8 - 8b7 + 9b1) * q^59 + (2b13 + 8b11) * q^61 + (-3b12 + 6) q^63 + (-5b15 - 5b14 + 4b12 + b6 - 6b4 + 4b3 - b2 - 30) q^65 + (-6b8 - 6b7 - 12b5) q^67 + (b13 + b11 - 3b8 - 3b7 - 11b5) q^69 + (2b15 - 6b14 - 2b6 - 2b4 + 4b2) q^71 + (2b15 - 2b14 - 2b6 - 14b4) * q^73 + (3b13 - b9 - b8 - 5b7 + b5 - b1) * q^75 + (6b10 + 4b9 + 2b8 - 2b7 + 4b1) q^77 + (3b15 + 3b14 - 3b6 - 13b4 - 2b2) q^79 + 9 * q^81 + (2b13 + 12b11 - 6b8 - 6b7 + 4b5) q^83 + (-2b13 + 4b11 + 6b10 - b9 + b8 + 5b7 + 12b5 - 11b1) * q^85 + (b15 + b6 - 6b3 - 16) q^87 + (-4b15 + 4b12 - 4b6 - 2b3 + 20) * q^89 + (-6b10 - 4b9 + 8b8 - 8b7 - 14b1) q^91 + (6b10 + 3b9 + 6b8 - 6b7 + b1) * q^93 + (7b15 + 6b14 - 3b12 - b6 + 6b4 - 6b3 + 54) q^95 + (2b15 - 2b14 - 2b6 + 28b4 - 6b2) q^97 + (3b10 + 3b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{7} - 48 q^{9}+O(q^{10}) $ 16 * q - 24 * q^7 - 48 * q^9	$ 16 q - 24 q^{7} - 48 q^{9} + 12 q^{15} + 192 q^{23} + 8 q^{25} + 96 q^{41} - 384 q^{47} + 320 q^{49} + 72 q^{55} + 72 q^{63} - 432 q^{65} + 144 q^{81} - 264 q^{87} + 384 q^{89} + 816 q^{95}+O(q^{100}) $ 16 * q - 24 * q^7 - 48 * q^9 + 12 * q^15 + 192 * q^23 + 8 * q^25 + 96 * q^41 - 384 * q^47 + 320 * q^49 + 72 * q^55 + 72 * q^63 - 432 * q^65 + 144 * q^81 - 264 * q^87 + 384 * q^89 + 816 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 8 x^{14} + 76 x^{13} - 505 x^{12} + 1474 x^{11} + 1032 x^{10} - 12054 x^{9} + \cdots + 100962304 $ x^16 - 4*x^15 + 8*x^14 + 76*x^13 - 505*x^12 + 1474*x^11 + 1032*x^10 - 12054*x^9 + 121741*x^8 - 65358*x^7 - 21102*x^6 + 2266852*x^5 - 5268796*x^4 + 8259424*x^3 + 21411968*x^2 - 65754112*x + 100962304 :

$\beta_{1}$	$=$	$ ( - 43\!\cdots\!99 \nu^{15} + \cdots + 10\!\cdots\!40 ) / 32\!\cdots\!52 $ (-43548180222448799v^15 + 85071260460631324v^14 - 47830074944524952v^13 - 3688257682491664308v^12 + 14982174946153306151v^11 - 23335941524330413726v^10 - 139822953465898474136v^9 + 348960135343769431658v^8 - 4313838920442789556051v^7 - 7072527217125993651374v^6 + 896275713639751311602v^5 - 83734156215338734911388v^4 + 82341895135520046840708v^3 + 177727245318283674956128v^2 - 1106758802542082762046464*v + 1025051381396457651658240) / 328400558950166445917952
$\beta_{2}$	$=$	$ ( 11\!\cdots\!07 \nu^{15} + \cdots - 53\!\cdots\!96 ) / 41\!\cdots\!20 $ (11769846740674363730200007v^15 - 40017334874808281657926880v^14 + 78484011899072378454365512v^13 + 395023432302709349129128308v^12 - 154548604023569502748371903v^11 - 11092799229413810006182329230v^10 + 72489213127972777062435214512v^9 + 11717400975326291956077346374v^8 - 54331011818370983592935979149v^7 + 4649255656889451198188488203114v^6 - 9087125938257978081729063357642v^5 + 17998873549591588084113714949876v^4 + 53669634410502489312372791556556v^3 - 152956538243330381847078706638448v^2 + 638356107251695587437107080348672*v - 531175468772861999588167578256896) / 41083216979188440789962728485120
$\beta_{3}$	$=$	$ ( - 16\!\cdots\!99 \nu^{15} + \cdots + 22\!\cdots\!32 ) / 46\!\cdots\!20 $ (-160459291353318962699v^15 + 1162896733115644870640v^14 - 4920935532626420410184v^13 + 3495222559602118260124v^12 + 68163026798973447647971v^11 - 443228007402770320356730v^10 + 1231427692210403595925776v^9 - 1967892658727816712274478v^8 - 12285944707159078756174567v^7 + 48651563338872965638031502v^6 - 136950335895653591048734046v^5 + 81597524695678308179063228v^4 + 453866896609361918826554468v^3 - 2372213610103148470021995664v^2 + 4240790245172130619425471616*v + 2285281614721206024922281632) / 460698136036472153830209120
$\beta_{4}$	$=$	$ ( - 15\!\cdots\!23 \nu^{15} + \cdots - 56\!\cdots\!36 ) / 38\!\cdots\!60 $ (-151200404638176205523v^15 + 1154439074846286724820v^14 - 5156197413395128778008v^13 + 5418446039395078196508v^12 + 64446993321197663287707v^11 - 459775944421574296116910v^10 + 1390525339252314301326552v^9 - 2476490196221144664154206v^8 - 11281219772183407409045479v^7 + 53028668689002939539450994v^6 - 165640562842580929008102822v^5 + 175030788130266999128968676v^4 + 397908114291824749710277076v^3 - 2630601997078298386822168448v^2 + 5507343902863717092324221952*v - 5645754473100740005101390336) / 385732552594556609330492160
$\beta_{5}$	$=$	$ ( 95832546037 \nu^{15} - 678857865808 \nu^{14} + 2773747411768 \nu^{13} + \cdots + 20\!\cdots\!36 ) / 23\!\cdots\!80 $ (95832546037v^15 - 678857865808v^14 + 2773747411768v^13 - 1472045799332v^12 - 42433602180125v^11 + 260384919281606v^10 - 691344331530480v^9 + 1067381330297170v^8 + 7615674913103513v^7 - 28830072900817074v^6 + 78414911960732578v^5 - 70902759255972676v^4 - 283343645407142044v^3 + 1412271729348564848v^2 - 2501682767950898048*v + 2089869581524987136) / 235357349017370880
$\beta_{6}$	$=$	$ ( 18\!\cdots\!63 \nu^{15} + \cdots + 11\!\cdots\!76 ) / 41\!\cdots\!20 $ (18865389664737016707819763v^15 - 151527825413769150833820500v^14 + 645246061028076261822765928v^13 - 135277829598244000349648988v^12 - 12957529800559918474258965307v^11 + 77680791258726095811011296430v^10 - 200852163848495903328014380712v^9 + 131893945906489095716789745086v^8 + 2538949916500195962293033600519v^7 - 10483923722908243506740790331154v^6 + 26795935106336705266733118926582v^5 - 14459396389007165969165657124676v^4 - 122806420166246111195161950399156v^3 + 520714012721071353385109966891968v^2 - 974788895140544213052951223689152*v + 1187610520414394487275802244619776) / 41083216979188440789962728485120
$\beta_{7}$	$=$	$ ( - 21\!\cdots\!65 \nu^{15} + \cdots - 80\!\cdots\!16 ) / 41\!\cdots\!12 $ (-2168187784711879389130365v^15 + 16444171289121748671751548v^14 - 83595310125949315696870496v^13 + 130575816196018799207245316v^12 + 674834799026274538897943861v^11 - 6175485582052712949459221538v^10 + 20565772277918864981509174848v^9 - 45460971280537569610974184114v^8 - 120760792894903511497292904009v^7 + 526260756615511836785845919502v^6 - 2433850437290747212865090515378v^5 + 1940700535586225999858265133068v^4 + 817743053025454509303138792252v^3 - 26781644349393401248608573045280v^2 + 53190251348801152849381560520352*v - 80568015672398039872238070041216) / 4108321697918844078996272848512
$\beta_{8}$	$=$	$ ( 43\!\cdots\!03 \nu^{15} + \cdots + 65\!\cdots\!64 ) / 41\!\cdots\!20 $ (43276168777825898176096703v^15 - 290753416172837795642736872v^14 + 1119934022666410516193883192v^13 - 16068493832603534274160748v^12 - 20043285218098984938209282135v^11 + 110198506644142654393993503754v^10 - 259378586573951044580087543280v^9 + 345022947131351368873006465750v^8 + 3522370807652185699310614528507v^7 - 11311025708667144345346247174286v^6 + 29738036817885331411599053736742v^5 - 18940085446833388276963068527324v^4 - 122915357045494209320119290442196v^3 + 523414869641305492537869159450672v^2 - 849010084782215809937648034648512*v + 656161509212171076667300866116864) / 41083216979188440789962728485120
$\beta_{9}$	$=$	$ ( 46\!\cdots\!81 \nu^{15} + \cdots + 21\!\cdots\!28 ) / 28\!\cdots\!60 $ (461318172111006505392581v^15 - 4695291062222889131884104v^14 + 22057815410020156291241144v^13 - 38230943794095167546009636v^12 - 213986609460638152837918605v^11 + 1821365675385129516940424158v^10 - 6160461122405726654632999040v^9 + 11027724958421485664577524850v^8 + 26530951569171729644405522809v^7 - 235148486599515065668127392682v^6 + 605369184993413894930692851954v^5 - 930519737801566673790128778548v^4 - 933106280054276535866939664892v^3 + 9574342995644076729505095807504v^2 - 21543597971194602902814241383424*v + 21879982503594871438176152329728) / 289318429430904512605371327360
$\beta_{10}$	$=$	$ ( - 27\!\cdots\!16 \nu^{15} + \cdots - 51\!\cdots\!08 ) / 14\!\cdots\!80 $ (-270228772302814432708916v^15 + 1729647287006082174749959v^14 - 7247176352407538616247364v^13 + 2509641406124189533619016v^12 + 107962921798013344999884200v^11 - 649652350816015034121455623v^10 + 1637145216247867513780161430v^9 - 2823023647019440039657424320v^8 - 20388488421631249487215177054v^7 + 56247501542962715209991163667v^6 - 205002924477603006612129816274v^5 + 98406915586970184010399233758v^4 + 553168600528382673155438485212v^3 - 2996947285491692261650057413764v^2 + 4771394623464376006138438449664*v - 5146085032843587729272014212608) / 144659214715452256302685663680
$\beta_{11}$	$=$	$ ( - 31\!\cdots\!89 \nu^{15} + \cdots - 93\!\cdots\!32 ) / 16\!\cdots\!60 $ (-31906033696263786191389v^15 + 218763201927641860024476v^14 - 877755917335003650005536v^13 + 225596172067112758024684v^12 + 14627022323180903274419205v^11 - 86528320762410886194989282v^10 + 223307955681224237266494160v^9 - 339375043806675733087148970v^8 - 2628119822148917194247845961v^7 + 9573668994517841570046687838v^6 - 28073016013238015622341493666v^5 + 22400515293918140533379328292v^4 + 85033255611223830816246909548v^3 - 501089599655025865561842603696v^2 + 928086910773127578563190152576*v - 933410481914485093004446855632) / 16354783829294761460972423760
$\beta_{12}$	$=$	$ ( - 14\!\cdots\!87 \nu^{15} + \cdots - 56\!\cdots\!64 ) / 73\!\cdots\!92 $ (-1497719360903661901987v^15 + 10823317146426229674352v^14 - 49526064036218805074056v^13 + 61838208669191848977596v^12 + 524713548007080513031739v^11 - 4109527636676298472463306v^10 + 13346058582370271058620304v^9 - 29485010395674016350327070v^8 - 91140952916740552198046303v^7 + 431008040486957888879326590v^6 - 1596953244655382153818410190v^5 + 2027988255416834409532146844v^4 + 2068593628327353477956585092v^3 - 23058724299231577937406876944v^2 + 46505803290293851347748946048*v - 56618848994183288728435480064) / 737117017658355446128334592
$\beta_{13}$	$=$	$ ( - 74\!\cdots\!69 \nu^{15} + \cdots - 34\!\cdots\!44 ) / 26\!\cdots\!16 $ (-74420764408317934436069v^15 + 516543496373013012231696v^14 - 2258498187927868980025400v^13 + 1363091523928213709432804v^12 + 31891239520119270150003597v^11 - 211810216095130246344736678v^10 + 602762040825934520361922544v^9 - 1091927124728347421835477810v^8 - 5816029615069486901128414153v^7 + 22743334165512256792914717650v^6 - 83256487310164695168196445442v^5 + 65111205191383264121349194756v^4 + 122335231851309006993432868060v^3 - 1320542515731349174212323121264v^2 + 2681931570888411286691905760128*v - 3465886186497355216980884703744) / 26167654126871618337555878016
$\beta_{14}$	$=$	$ ( 12\!\cdots\!21 \nu^{15} + \cdots + 37\!\cdots\!72 ) / 41\!\cdots\!20 $ (122506639628173077308096821v^15 - 871076152205501352487841000v^14 + 3740853536976812172736693176v^13 - 2237713429920957926942638436v^12 - 56302811276471535670818099069v^11 + 363182570915611818744244688030v^10 - 1001625338829348598775494574144v^9 + 1521243712724374386926140499922v^8 + 10681692110516485747133809632233v^7 - 40852966075084334793273338969258v^6 + 124621902676085316395085891230994v^5 - 88275203937104132169959176185652v^4 - 377096087107670156903459851151292v^3 + 2164402105509652144673397987072656v^2 - 3683025414454878943053554883257344*v + 3706943277064444308723913819356672) / 41083216979188440789962728485120
$\beta_{15}$	$=$	$ ( - 12\!\cdots\!47 \nu^{15} + \cdots - 39\!\cdots\!76 ) / 41\!\cdots\!12 $ (-12499204634878389531036347v^15 + 92042854288932626542749370v^14 - 407293742809999309190705336v^13 + 380309513193190357685547276v^12 + 5307599733137085334096549067v^11 - 37041909536386839884990205556v^10 + 110100326811193213515166527532v^9 - 194762942649361258151436059854v^8 - 962993678068911614207422793203v^7 + 4171002560927544472192567892704v^6 - 13212569491033221355553330506354v^5 + 12583552444503426745186998603296v^4 + 32566641297825570350960438402172v^3 - 213750694101039201146787987861896v^2 + 410541596277811817723118288206752*v - 392584505833031091957235546674176) / 4108321697918844078996272848512

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

$\nu$	$=$	$ ( - \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{8} + 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + \cdots + 3 ) / 8 $ (-b12 + b11 + b10 + 2b8 + 2b6 - 3b5 + 2b4 - b3 + b2 - b1 + 3) / 8
$\nu^{2}$	$=$	$ ( - 2 \beta_{15} - \beta_{14} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 11 \beta_1 ) / 4 $ (-2b15 - b14 - b10 - b9 + b8 - b7 + 2b6 + 12b4 + b2 + 11b1) / 4
$\nu^{3}$	$=$	$ ( - 3 \beta_{15} - 3 \beta_{14} + 3 \beta_{13} - 15 \beta_{11} + 11 \beta_{10} - 3 \beta_{9} + 4 \beta_{8} + \cdots - 65 ) / 4 $ (-3b15 - 3b14 + 3b13 - 15b11 + 11b10 - 3b9 + 4b8 - 26b7 + 3b6 + b5 + 34b4 + 3b3 + 7b1 - 65) / 4
$\nu^{4}$	$=$	$ ( - 20 \beta_{15} + 25 \beta_{13} + 16 \beta_{12} - 82 \beta_{11} - 36 \beta_{8} - 36 \beta_{7} + \cdots + 151 ) / 4 $ (-20b15 + 25b13 + 16b12 - 82b11 - 36b8 - 36b7 - 20b6 - 206b5 + 15*b3 + 151) / 4
$\nu^{5}$	$=$	$ ( - 38 \beta_{15} - 72 \beta_{14} + 164 \beta_{13} - 71 \beta_{12} - 479 \beta_{11} - 185 \beta_{10} + \cdots - 195 ) / 8 $ (-38b15 - 72b14 + 164b13 - 71b12 - 479b11 - 185b10 + 164b9 - 664b8 + 294b7 + 248b6 - 1471b5 - 26b4 - 143b3 + 105b2 - 975*b1 - 195) / 8
$\nu^{6}$	$=$	$ ( - 370 \beta_{15} - 270 \beta_{14} - 497 \beta_{10} + 207 \beta_{9} - 789 \beta_{8} + 789 \beta_{7} + \cdots - 703 \beta_1 ) / 2 $ (-370b15 - 270b14 - 497b10 + 207b9 - 789b8 + 789b7 + 370b6 + 1596b4 + 108b2 - 703b1) / 2
$\nu^{7}$	$=$	$ ( - 8864 \beta_{15} - 5000 \beta_{14} - 1852 \beta_{13} + 1731 \beta_{12} + 5333 \beta_{11} + \cdots - 50881 ) / 8 $ (-8864b15 - 5000b14 - 1852b13 + 1731b12 + 5333b11 - 1451b10 + 1852b9 - 3882b8 + 6784b7 + 4598b6 + 19349b5 + 28806b4 + 6731b3 + 2133b2 - 12341*b1 - 50881) / 8
$\nu^{8}$	$=$	$ ( - 11380 \beta_{15} - 2369 \beta_{13} + 8580 \beta_{12} + 8022 \beta_{11} + 5124 \beta_{8} + \cdots - 147397 ) / 4 $ (-11380b15 - 2369b13 + 8580b12 + 8022b11 + 5124b8 + 5124b7 - 11380b6 + 8778b5 + 26755*b3 - 147397) / 4
$\nu^{9}$	$=$	$ ( 55551 \beta_{15} + 58383 \beta_{14} + 5333 \beta_{13} + 22332 \beta_{12} - 14707 \beta_{11} + \cdots - 605789 ) / 4 $ (55551b15 + 58383b14 + 5333b13 + 22332b12 - 14707b11 - 4855b10 + 5333b9 - 19562b8 + 9852b7 - 105879b6 - 48647b5 - 343252b4 + 80715b3 - 25164b2 - 31677*b1 - 605789) / 4
$\nu^{10}$	$=$	$ ( 468738 \beta_{15} + 331189 \beta_{14} - 202963 \beta_{10} + 107775 \beta_{9} - 364379 \beta_{8} + \cdots - 442297 \beta_1 ) / 4 $ (468738b15 + 331189b14 - 202963b10 + 107775b9 - 364379b8 + 364379b7 - 468738b6 - 1714558b4 - 180899b2 - 442297b1) / 4
$\nu^{11}$	$=$	$ ( 2984962 \beta_{15} + 1520522 \beta_{14} - 1058958 \beta_{13} - 653079 \beta_{12} + 3381477 \beta_{11} + \cdots + 13980171 ) / 8 $ (2984962b15 + 1520522b14 - 1058958b13 - 653079b12 + 3381477b11 - 1441439b10 + 1058958b9 - 1940038b8 + 4822916b7 - 1362240b6 + 7865261b5 - 8076886b4 - 2173601b3 - 811361b2 - 5623369*b1 + 13980171) / 8
$\nu^{12}$	$=$	$ ( 507444 \beta_{15} - 2098948 \beta_{13} - 461396 \beta_{12} + 6792836 \beta_{11} + 3353988 \beta_{8} + \cdots + 12695541 ) / 2 $ (507444b15 - 2098948b13 - 461396b12 + 6792836b11 + 3353988b8 + 3353988b7 + 507444b6 + 13046228b5 - 1667507*b3 + 12695541) / 2
$\nu^{13}$	$=$	$ ( 2594408 \beta_{15} + 1712274 \beta_{14} - 27904182 \beta_{13} + 538087 \beta_{12} + 90779413 \beta_{11} + \cdots - 32291635 ) / 8 $ (2594408b15 + 1712274b14 - 27904182b13 + 538087b12 + 90779413b11 + 43351999b10 - 27904182b9 + 134131412b8 - 47427414b7 - 1906314b6 + 180666621b5 - 17270998b4 + 2250361b3 + 344047b2 + 135723017*b1 - 32291635) / 8
$\nu^{14}$	$=$	$ ( 86751774 \beta_{15} + 61562339 \beta_{14} + 121521221 \beta_{10} - 81489257 \beta_{9} + \cdots + 407613775 \beta_1 ) / 4 $ (86751774b15 + 61562339b14 + 121521221b10 - 81489257b9 + 262907901b8 - 262907901b7 - 86751774b6 - 319048050b4 - 33380981b2 + 407613775b1) / 4
$\nu^{15}$	$=$	$ ( 658267335 \beta_{15} + 340805343 \beta_{14} + 203611805 \beta_{13} - 141955560 \beta_{12} + \cdots + 3196928881 ) / 4 $ (658267335b15 + 340805343b14 + 203611805b13 - 141955560b12 - 659765671b11 + 325385747b10 - 203611805b9 + 334379924b8 - 985151418b7 - 307254471b6 - 1266954939b5 - 1839844892b4 - 482760903b3 - 175506432b2 + 963360305*b1 + 3196928881) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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} $

799.1

−3.18491	+	0.121343i
1.83186	+	4.10454i
1.70711	−	1.18947i
−0.167934	−	3.35879i
−3.35879	+	0.167934i
1.18947	+	1.70711i
4.10454	−	1.83186i
−0.121343	−	3.18491i
−3.18491	−	0.121343i
1.83186	−	4.10454i
1.70711	+	1.18947i
−0.167934	+	3.35879i
−3.35879	−	0.167934i
1.18947	−	1.70711i
4.10454	+	1.83186i
−0.121343	+	3.18491i

−

1.73205i

−4.96716

0.572085i

11.5235

−3.00000

799.2

−

1.73205i

−4.76614

1.51127i

−2.10450

−3.00000

799.3

−

1.73205i

−1.86987

4.63720i

−10.8645

−3.00000

799.4

−

1.73205i

−0.338848

−

4.98850i

−4.55449

−3.00000

799.5

−

1.73205i

0.338848

−

4.98850i

−4.55449

−3.00000

799.6

−

1.73205i

1.86987

4.63720i

−10.8645

−3.00000

799.7

−

1.73205i

4.76614

1.51127i

−2.10450

−3.00000

799.8

−

1.73205i

4.96716

0.572085i

11.5235

−3.00000

799.9

1.73205i

−4.96716

−

0.572085i

11.5235

−3.00000

799.10

1.73205i

−4.76614

−

1.51127i

−2.10450

−3.00000

799.11

1.73205i

−1.86987

−

4.63720i

−10.8645

−3.00000

799.12

1.73205i

−0.338848

4.98850i

−4.55449

−3.00000

799.13

1.73205i

0.338848

4.98850i

−4.55449

−3.00000

799.14

1.73205i

1.86987

−

4.63720i

−10.8645

−3.00000

799.15

1.73205i

4.76614

−

1.51127i

−2.10450

−3.00000

799.16

1.73205i

4.96716

−

0.572085i

11.5235

−3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
20.d	odd	2	1	inner
40.e	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.3.p.c	✓	16
4.b	odd	2	1		960.3.p.d	yes	16
5.b	even	2	1		960.3.p.d	yes	16
8.b	even	2	1	inner	960.3.p.c	✓	16
8.d	odd	2	1		960.3.p.d	yes	16
20.d	odd	2	1	inner	960.3.p.c	✓	16
40.e	odd	2	1	inner	960.3.p.c	✓	16
40.f	even	2	1		960.3.p.d	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
960.3.p.c	✓	16	1.a	even	1	1	trivial
960.3.p.c	✓	16	8.b	even	2	1	inner
960.3.p.c	✓	16	20.d	odd	2	1	inner
960.3.p.c	✓	16	40.e	odd	2	1	inner
960.3.p.d	yes	16	4.b	odd	2	1
960.3.p.d	yes	16	5.b	even	2	1
960.3.p.d	yes	16	8.d	odd	2	1
960.3.p.d	yes	16	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{4} + 6T_{7}^{3} - 120T_{7}^{2} - 840T_{7} - 1200 $ T7^4 + 6*T7^3 - 120*T7^2 - 840*T7 - 1200 acting on $S_{3}^{\mathrm{new}}(960, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ T^{16} + \cdots + 152587890625 $ T^16 - 4T^14 - 1388T^12 + 2948T^10 + 1033750T^8 + 1842500T^6 - 542187500T^4 - 976562500*T^2 + 152587890625
$7$	$ (T^{4} + 6 T^{3} + \cdots - 1200)^{4} $ (T^4 + 6T^3 - 120T^2 - 840*T - 1200)^4
$11$	$ (T^{8} - 268 T^{6} + \cdots + 6150400)^{2} $ (T^8 - 268T^6 + 20736T^4 - 618688*T^2 + 6150400)^2
$13$	$ (T^{8} - 924 T^{6} + \cdots + 1440000)^{2} $ (T^8 - 924T^6 + 230592T^4 - 9266112*T^2 + 1440000)^2
$17$	$ (T^{8} + 828 T^{6} + \cdots + 3686400)^{2} $ (T^8 + 828T^6 + 177168T^4 + 2543616*T^2 + 3686400)^2
$19$	$ (T^{8} - 1656 T^{6} + \cdots + 2985984)^{2} $ (T^8 - 1656T^6 + 516240T^4 - 10077696*T^2 + 2985984)^2
$23$	$ (T^{4} - 48 T^{3} + \cdots - 65280)^{4} $ (T^4 - 48T^3 + 516T^2 + 4032*T - 65280)^4
$29$	$ (T^{8} + \cdots + 1169166438400)^{2} $ (T^8 + 6196T^6 + 11772432T^4 + 6756249088*T^2 + 1169166438400)^2
$31$	$ (T^{8} + \cdots + 3815302758400)^{2} $ (T^8 + 7048T^6 + 16378128T^4 + 13942521856*T^2 + 3815302758400)^2
$37$	$ (T^{8} + \cdots + 20649826347264)^{2} $ (T^8 - 9036T^6 + 29540160T^4 - 41288584896*T^2 + 20649826347264)^2
$41$	$ (T^{4} - 24 T^{3} + \cdots + 1471824)^{4} $ (T^4 - 24T^3 - 3672T^2 - 6048*T + 1471824)^4
$43$	$ (T^{8} + \cdots + 18043872804864)^{2} $ (T^8 + 9264T^6 + 29614080T^4 + 39030534144*T^2 + 18043872804864)^2
$47$	$ (T^{4} + 96 T^{3} + \cdots - 4740672)^{4} $ (T^4 + 96T^3 - 108T^2 - 216000*T - 4740672)^4
$53$	$ (T^{8} + \cdots + 40936451385600)^{2} $ (T^8 - 13212T^6 + 51796416T^4 - 78614741952*T^2 + 40936451385600)^2
$59$	$ (T^{8} + \cdots + 114174866749696)^{2} $ (T^8 - 17020T^6 + 96116352T^4 - 199569357760*T^2 + 114174866749696)^2
$61$	$ (T^{8} + \cdots + 15341322240000)^{2} $ (T^8 + 14880T^6 + 60569856T^4 + 57541017600*T^2 + 15341322240000)^2
$67$	$ (T^{8} + \cdots + 10330310246400)^{2} $ (T^8 + 9648T^6 + 26873856T^4 + 28865507328*T^2 + 10330310246400)^2
$71$	$ (T^{8} + \cdots + 10703890022400)^{2} $ (T^8 + 22944T^6 + 138150144T^4 + 93544022016*T^2 + 10703890022400)^2
$73$	$ (T^{8} + \cdots + 5668208640000)^{2} $ (T^8 + 7200T^6 + 16898304T^4 + 16360243200*T^2 + 5668208640000)^2
$79$	$ (T^{8} + 7240 T^{6} + \cdots + 27180138496)^{2} $ (T^8 + 7240T^6 + 3878928T^4 + 637788160*T^2 + 27180138496)^2
$83$	$ (T^{8} + \cdots + 22\!\cdots\!00)^{2} $ (T^8 + 38640T^6 + 504293376T^4 + 2404717424640*T^2 + 2222526563942400)^2
$89$	$ (T^{4} - 96 T^{3} + \cdots - 24665328)^{4} $ (T^4 - 96T^3 - 7368T^2 + 994176*T - 24665328)^4
$97$	$ (T^{8} + \cdots + 4624220160000)^{2} $ (T^8 + 41904T^6 + 418126080T^4 + 414557798400*T^2 + 4624220160000)^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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{3} + \beta_{8} q^{5} + (\beta_{12} - 2) q^{7} - 3 q^{9}+O(q^{10}) \) q - b5 * q^3 + b8 * q^5 + (b12 - 2) * q^7 - 3 * q^9	\( q - \beta_{5} q^{3} + \beta_{8} q^{5} + (\beta_{12} - 2) q^{7} - 3 q^{9} + ( - \beta_{10} - \beta_1) q^{11} + (\beta_{9} - \beta_{8} + \beta_{7} - \beta_1) q^{13} + (\beta_{6} + 1) q^{15} + (\beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{17}+ \cdots + (3 \beta_{10} + 3 \beta_1) q^{99}+O(q^{100}) \) q - b5 * q^3 + b8 * q^5 + (b12 - 2) * q^7 - 3 * q^9 + (-b10 - b1) * q^11 + (b9 - b8 + b7 - b1) * q^13 + (b6 + 1) * q^15 + (b15 + b14 - b6 - b2) * q^17 + (-b9 + 2b8 - 2b7 + 2b1) q^19 + (b13 + b11 + b5) * q^21 + (b15 + b12 + b6 + 12) * q^23 + (2b15 + b14 + 2b12 + b4 - b3) * q^25 + 3b5 q^27 + (-2b13 + 4b11 + b8 + b7 - 8b5) q^29 + (b15 - 3b14 - b6 - 3b4 - 2b2) q^31 + (-4b4 - b2) q^33 + (-b13 + 2b9 - b8 - 5b7 - 2b5 - 3b1) * q^35 + (6b10 - b9 - b8 + b7 + 5b1) * q^37 + (2b15 + 3b14 - 2b6 - 3b4) * q^39 + (-4b12 + 2b3 + 8) * q^41 + (-4b11 - 6b8 - 6b7 + 8b5) * q^43 - 3b8 q^45 + (b15 - 3b12 + b6 + 2b3 - 22) * q^47 + (2b15 + 4b12 + 2b6 + 4b3 + 19) * q^49 + (3b10 - b9 + 2b8 - 2b7 - b1) q^51 + (6b10 - 4b9 + b8 - b7 - 4b1) q^53 + (b15 + 3b14 + b12 - b6 + 13b4 + 2b3 + 4) q^55 + (-3b15 - 3b14 + 3b6 + 6b4) * q^57 + (-b10 + 2b9 + 8b8 - 8b7 + 9b1) * q^59 + (2b13 + 8b11) * q^61 + (-3b12 + 6) q^63 + (-5b15 - 5b14 + 4b12 + b6 - 6b4 + 4b3 - b2 - 30) q^65 + (-6b8 - 6b7 - 12b5) q^67 + (b13 + b11 - 3b8 - 3b7 - 11b5) q^69 + (2b15 - 6b14 - 2b6 - 2b4 + 4b2) q^71 + (2b15 - 2b14 - 2b6 - 14b4) * q^73 + (3b13 - b9 - b8 - 5b7 + b5 - b1) * q^75 + (6b10 + 4b9 + 2b8 - 2b7 + 4b1) q^77 + (3b15 + 3b14 - 3b6 - 13b4 - 2b2) q^79 + 9 * q^81 + (2b13 + 12b11 - 6b8 - 6b7 + 4b5) q^83 + (-2b13 + 4b11 + 6b10 - b9 + b8 + 5b7 + 12b5 - 11b1) * q^85 + (b15 + b6 - 6b3 - 16) q^87 + (-4b15 + 4b12 - 4b6 - 2b3 + 20) * q^89 + (-6b10 - 4b9 + 8b8 - 8b7 - 14b1) q^91 + (6b10 + 3b9 + 6b8 - 6b7 + b1) * q^93 + (7b15 + 6b14 - 3b12 - b6 + 6b4 - 6b3 + 54) q^95 + (2b15 - 2b14 - 2b6 + 28b4 - 6b2) q^97 + (3b10 + 3b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{7} - 48 q^{9}+O(q^{10}) \) 16 * q - 24 * q^7 - 48 * q^9	\( 16 q - 24 q^{7} - 48 q^{9} + 12 q^{15} + 192 q^{23} + 8 q^{25} + 96 q^{41} - 384 q^{47} + 320 q^{49} + 72 q^{55} + 72 q^{63} - 432 q^{65} + 144 q^{81} - 264 q^{87} + 384 q^{89} + 816 q^{95}+O(q^{100}) \) 16 * q - 24 * q^7 - 48 * q^9 + 12 * q^15 + 192 * q^23 + 8 * q^25 + 96 * q^41 - 384 * q^47 + 320 * q^49 + 72 * q^55 + 72 * q^63 - 432 * q^65 + 144 * q^81 - 264 * q^87 + 384 * q^89 + 816 * q^95

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	960.3.p.c

Level	$960$
Weight	$3$
Character orbit	960.p
Analytic conductor	$26.158$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(26.1581053786\)
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:	\( x^{16} - 4 x^{15} + 8 x^{14} + 76 x^{13} - 505 x^{12} + 1474 x^{11} + 1032 x^{10} - 12054 x^{9} + \cdots + 100962304 \) x^16 - 4x^15 + 8x^14 + 76x^13 - 505x^12 + 1474x^11 + 1032x^10 - 12054x^9 + 121741x^8 - 65358x^7 - 21102x^6 + 2266852x^5 - 5268796x^4 + 8259424x^3 + 21411968x^2 - 65754112*x + 100962304
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{24} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 43\!\cdots\!99 \nu^{15} + \cdots + 10\!\cdots\!40 ) / 32\!\cdots\!52 \) (-43548180222448799v^15 + 85071260460631324v^14 - 47830074944524952v^13 - 3688257682491664308v^12 + 14982174946153306151v^11 - 23335941524330413726v^10 - 139822953465898474136v^9 + 348960135343769431658v^8 - 4313838920442789556051v^7 - 7072527217125993651374v^6 + 896275713639751311602v^5 - 83734156215338734911388v^4 + 82341895135520046840708v^3 + 177727245318283674956128v^2 - 1106758802542082762046464*v + 1025051381396457651658240) / 328400558950166445917952
\(\beta_{2}\)	\(=\)	\( ( 11\!\cdots\!07 \nu^{15} + \cdots - 53\!\cdots\!96 ) / 41\!\cdots\!20 \) (11769846740674363730200007v^15 - 40017334874808281657926880v^14 + 78484011899072378454365512v^13 + 395023432302709349129128308v^12 - 154548604023569502748371903v^11 - 11092799229413810006182329230v^10 + 72489213127972777062435214512v^9 + 11717400975326291956077346374v^8 - 54331011818370983592935979149v^7 + 4649255656889451198188488203114v^6 - 9087125938257978081729063357642v^5 + 17998873549591588084113714949876v^4 + 53669634410502489312372791556556v^3 - 152956538243330381847078706638448v^2 + 638356107251695587437107080348672*v - 531175468772861999588167578256896) / 41083216979188440789962728485120
\(\beta_{3}\)	\(=\)	\( ( - 16\!\cdots\!99 \nu^{15} + \cdots + 22\!\cdots\!32 ) / 46\!\cdots\!20 \) (-160459291353318962699v^15 + 1162896733115644870640v^14 - 4920935532626420410184v^13 + 3495222559602118260124v^12 + 68163026798973447647971v^11 - 443228007402770320356730v^10 + 1231427692210403595925776v^9 - 1967892658727816712274478v^8 - 12285944707159078756174567v^7 + 48651563338872965638031502v^6 - 136950335895653591048734046v^5 + 81597524695678308179063228v^4 + 453866896609361918826554468v^3 - 2372213610103148470021995664v^2 + 4240790245172130619425471616*v + 2285281614721206024922281632) / 460698136036472153830209120
\(\beta_{4}\)	\(=\)	\( ( - 15\!\cdots\!23 \nu^{15} + \cdots - 56\!\cdots\!36 ) / 38\!\cdots\!60 \) (-151200404638176205523v^15 + 1154439074846286724820v^14 - 5156197413395128778008v^13 + 5418446039395078196508v^12 + 64446993321197663287707v^11 - 459775944421574296116910v^10 + 1390525339252314301326552v^9 - 2476490196221144664154206v^8 - 11281219772183407409045479v^7 + 53028668689002939539450994v^6 - 165640562842580929008102822v^5 + 175030788130266999128968676v^4 + 397908114291824749710277076v^3 - 2630601997078298386822168448v^2 + 5507343902863717092324221952*v - 5645754473100740005101390336) / 385732552594556609330492160
\(\beta_{5}\)	\(=\)	\( ( 95832546037 \nu^{15} - 678857865808 \nu^{14} + 2773747411768 \nu^{13} + \cdots + 20\!\cdots\!36 ) / 23\!\cdots\!80 \) (95832546037v^15 - 678857865808v^14 + 2773747411768v^13 - 1472045799332v^12 - 42433602180125v^11 + 260384919281606v^10 - 691344331530480v^9 + 1067381330297170v^8 + 7615674913103513v^7 - 28830072900817074v^6 + 78414911960732578v^5 - 70902759255972676v^4 - 283343645407142044v^3 + 1412271729348564848v^2 - 2501682767950898048*v + 2089869581524987136) / 235357349017370880
\(\beta_{6}\)	\(=\)	\( ( 18\!\cdots\!63 \nu^{15} + \cdots + 11\!\cdots\!76 ) / 41\!\cdots\!20 \) (18865389664737016707819763v^15 - 151527825413769150833820500v^14 + 645246061028076261822765928v^13 - 135277829598244000349648988v^12 - 12957529800559918474258965307v^11 + 77680791258726095811011296430v^10 - 200852163848495903328014380712v^9 + 131893945906489095716789745086v^8 + 2538949916500195962293033600519v^7 - 10483923722908243506740790331154v^6 + 26795935106336705266733118926582v^5 - 14459396389007165969165657124676v^4 - 122806420166246111195161950399156v^3 + 520714012721071353385109966891968v^2 - 974788895140544213052951223689152*v + 1187610520414394487275802244619776) / 41083216979188440789962728485120
\(\beta_{7}\)	\(=\)	\( ( - 21\!\cdots\!65 \nu^{15} + \cdots - 80\!\cdots\!16 ) / 41\!\cdots\!12 \) (-2168187784711879389130365v^15 + 16444171289121748671751548v^14 - 83595310125949315696870496v^13 + 130575816196018799207245316v^12 + 674834799026274538897943861v^11 - 6175485582052712949459221538v^10 + 20565772277918864981509174848v^9 - 45460971280537569610974184114v^8 - 120760792894903511497292904009v^7 + 526260756615511836785845919502v^6 - 2433850437290747212865090515378v^5 + 1940700535586225999858265133068v^4 + 817743053025454509303138792252v^3 - 26781644349393401248608573045280v^2 + 53190251348801152849381560520352*v - 80568015672398039872238070041216) / 4108321697918844078996272848512
\(\beta_{8}\)	\(=\)	\( ( 43\!\cdots\!03 \nu^{15} + \cdots + 65\!\cdots\!64 ) / 41\!\cdots\!20 \) (43276168777825898176096703v^15 - 290753416172837795642736872v^14 + 1119934022666410516193883192v^13 - 16068493832603534274160748v^12 - 20043285218098984938209282135v^11 + 110198506644142654393993503754v^10 - 259378586573951044580087543280v^9 + 345022947131351368873006465750v^8 + 3522370807652185699310614528507v^7 - 11311025708667144345346247174286v^6 + 29738036817885331411599053736742v^5 - 18940085446833388276963068527324v^4 - 122915357045494209320119290442196v^3 + 523414869641305492537869159450672v^2 - 849010084782215809937648034648512*v + 656161509212171076667300866116864) / 41083216979188440789962728485120
\(\beta_{9}\)	\(=\)	\( ( 46\!\cdots\!81 \nu^{15} + \cdots + 21\!\cdots\!28 ) / 28\!\cdots\!60 \) (461318172111006505392581v^15 - 4695291062222889131884104v^14 + 22057815410020156291241144v^13 - 38230943794095167546009636v^12 - 213986609460638152837918605v^11 + 1821365675385129516940424158v^10 - 6160461122405726654632999040v^9 + 11027724958421485664577524850v^8 + 26530951569171729644405522809v^7 - 235148486599515065668127392682v^6 + 605369184993413894930692851954v^5 - 930519737801566673790128778548v^4 - 933106280054276535866939664892v^3 + 9574342995644076729505095807504v^2 - 21543597971194602902814241383424*v + 21879982503594871438176152329728) / 289318429430904512605371327360
\(\beta_{10}\)	\(=\)	\( ( - 27\!\cdots\!16 \nu^{15} + \cdots - 51\!\cdots\!08 ) / 14\!\cdots\!80 \) (-270228772302814432708916v^15 + 1729647287006082174749959v^14 - 7247176352407538616247364v^13 + 2509641406124189533619016v^12 + 107962921798013344999884200v^11 - 649652350816015034121455623v^10 + 1637145216247867513780161430v^9 - 2823023647019440039657424320v^8 - 20388488421631249487215177054v^7 + 56247501542962715209991163667v^6 - 205002924477603006612129816274v^5 + 98406915586970184010399233758v^4 + 553168600528382673155438485212v^3 - 2996947285491692261650057413764v^2 + 4771394623464376006138438449664*v - 5146085032843587729272014212608) / 144659214715452256302685663680
\(\beta_{11}\)	\(=\)	\( ( - 31\!\cdots\!89 \nu^{15} + \cdots - 93\!\cdots\!32 ) / 16\!\cdots\!60 \) (-31906033696263786191389v^15 + 218763201927641860024476v^14 - 877755917335003650005536v^13 + 225596172067112758024684v^12 + 14627022323180903274419205v^11 - 86528320762410886194989282v^10 + 223307955681224237266494160v^9 - 339375043806675733087148970v^8 - 2628119822148917194247845961v^7 + 9573668994517841570046687838v^6 - 28073016013238015622341493666v^5 + 22400515293918140533379328292v^4 + 85033255611223830816246909548v^3 - 501089599655025865561842603696v^2 + 928086910773127578563190152576*v - 933410481914485093004446855632) / 16354783829294761460972423760
\(\beta_{12}\)	\(=\)	\( ( - 14\!\cdots\!87 \nu^{15} + \cdots - 56\!\cdots\!64 ) / 73\!\cdots\!92 \) (-1497719360903661901987v^15 + 10823317146426229674352v^14 - 49526064036218805074056v^13 + 61838208669191848977596v^12 + 524713548007080513031739v^11 - 4109527636676298472463306v^10 + 13346058582370271058620304v^9 - 29485010395674016350327070v^8 - 91140952916740552198046303v^7 + 431008040486957888879326590v^6 - 1596953244655382153818410190v^5 + 2027988255416834409532146844v^4 + 2068593628327353477956585092v^3 - 23058724299231577937406876944v^2 + 46505803290293851347748946048*v - 56618848994183288728435480064) / 737117017658355446128334592
\(\beta_{13}\)	\(=\)	\( ( - 74\!\cdots\!69 \nu^{15} + \cdots - 34\!\cdots\!44 ) / 26\!\cdots\!16 \) (-74420764408317934436069v^15 + 516543496373013012231696v^14 - 2258498187927868980025400v^13 + 1363091523928213709432804v^12 + 31891239520119270150003597v^11 - 211810216095130246344736678v^10 + 602762040825934520361922544v^9 - 1091927124728347421835477810v^8 - 5816029615069486901128414153v^7 + 22743334165512256792914717650v^6 - 83256487310164695168196445442v^5 + 65111205191383264121349194756v^4 + 122335231851309006993432868060v^3 - 1320542515731349174212323121264v^2 + 2681931570888411286691905760128*v - 3465886186497355216980884703744) / 26167654126871618337555878016
\(\beta_{14}\)	\(=\)	\( ( 12\!\cdots\!21 \nu^{15} + \cdots + 37\!\cdots\!72 ) / 41\!\cdots\!20 \) (122506639628173077308096821v^15 - 871076152205501352487841000v^14 + 3740853536976812172736693176v^13 - 2237713429920957926942638436v^12 - 56302811276471535670818099069v^11 + 363182570915611818744244688030v^10 - 1001625338829348598775494574144v^9 + 1521243712724374386926140499922v^8 + 10681692110516485747133809632233v^7 - 40852966075084334793273338969258v^6 + 124621902676085316395085891230994v^5 - 88275203937104132169959176185652v^4 - 377096087107670156903459851151292v^3 + 2164402105509652144673397987072656v^2 - 3683025414454878943053554883257344*v + 3706943277064444308723913819356672) / 41083216979188440789962728485120
\(\beta_{15}\)	\(=\)	\( ( - 12\!\cdots\!47 \nu^{15} + \cdots - 39\!\cdots\!76 ) / 41\!\cdots\!12 \) (-12499204634878389531036347v^15 + 92042854288932626542749370v^14 - 407293742809999309190705336v^13 + 380309513193190357685547276v^12 + 5307599733137085334096549067v^11 - 37041909536386839884990205556v^10 + 110100326811193213515166527532v^9 - 194762942649361258151436059854v^8 - 962993678068911614207422793203v^7 + 4171002560927544472192567892704v^6 - 13212569491033221355553330506354v^5 + 12583552444503426745186998603296v^4 + 32566641297825570350960438402172v^3 - 213750694101039201146787987861896v^2 + 410541596277811817723118288206752*v - 392584505833031091957235546674176) / 4108321697918844078996272848512

\(\nu\)	\(=\)	\( ( - \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{8} + 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + \cdots + 3 ) / 8 \) (-b12 + b11 + b10 + 2b8 + 2b6 - 3b5 + 2b4 - b3 + b2 - b1 + 3) / 8
\(\nu^{2}\)	\(=\)	\( ( - 2 \beta_{15} - \beta_{14} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 11 \beta_1 ) / 4 \) (-2b15 - b14 - b10 - b9 + b8 - b7 + 2b6 + 12b4 + b2 + 11b1) / 4
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{15} - 3 \beta_{14} + 3 \beta_{13} - 15 \beta_{11} + 11 \beta_{10} - 3 \beta_{9} + 4 \beta_{8} + \cdots - 65 ) / 4 \) (-3b15 - 3b14 + 3b13 - 15b11 + 11b10 - 3b9 + 4b8 - 26b7 + 3b6 + b5 + 34b4 + 3b3 + 7b1 - 65) / 4
\(\nu^{4}\)	\(=\)	\( ( - 20 \beta_{15} + 25 \beta_{13} + 16 \beta_{12} - 82 \beta_{11} - 36 \beta_{8} - 36 \beta_{7} + \cdots + 151 ) / 4 \) (-20b15 + 25b13 + 16b12 - 82b11 - 36b8 - 36b7 - 20b6 - 206b5 + 15*b3 + 151) / 4
\(\nu^{5}\)	\(=\)	\( ( - 38 \beta_{15} - 72 \beta_{14} + 164 \beta_{13} - 71 \beta_{12} - 479 \beta_{11} - 185 \beta_{10} + \cdots - 195 ) / 8 \) (-38b15 - 72b14 + 164b13 - 71b12 - 479b11 - 185b10 + 164b9 - 664b8 + 294b7 + 248b6 - 1471b5 - 26b4 - 143b3 + 105b2 - 975*b1 - 195) / 8
\(\nu^{6}\)	\(=\)	\( ( - 370 \beta_{15} - 270 \beta_{14} - 497 \beta_{10} + 207 \beta_{9} - 789 \beta_{8} + 789 \beta_{7} + \cdots - 703 \beta_1 ) / 2 \) (-370b15 - 270b14 - 497b10 + 207b9 - 789b8 + 789b7 + 370b6 + 1596b4 + 108b2 - 703b1) / 2
\(\nu^{7}\)	\(=\)	\( ( - 8864 \beta_{15} - 5000 \beta_{14} - 1852 \beta_{13} + 1731 \beta_{12} + 5333 \beta_{11} + \cdots - 50881 ) / 8 \) (-8864b15 - 5000b14 - 1852b13 + 1731b12 + 5333b11 - 1451b10 + 1852b9 - 3882b8 + 6784b7 + 4598b6 + 19349b5 + 28806b4 + 6731b3 + 2133b2 - 12341*b1 - 50881) / 8
\(\nu^{8}\)	\(=\)	\( ( - 11380 \beta_{15} - 2369 \beta_{13} + 8580 \beta_{12} + 8022 \beta_{11} + 5124 \beta_{8} + \cdots - 147397 ) / 4 \) (-11380b15 - 2369b13 + 8580b12 + 8022b11 + 5124b8 + 5124b7 - 11380b6 + 8778b5 + 26755*b3 - 147397) / 4
\(\nu^{9}\)	\(=\)	\( ( 55551 \beta_{15} + 58383 \beta_{14} + 5333 \beta_{13} + 22332 \beta_{12} - 14707 \beta_{11} + \cdots - 605789 ) / 4 \) (55551b15 + 58383b14 + 5333b13 + 22332b12 - 14707b11 - 4855b10 + 5333b9 - 19562b8 + 9852b7 - 105879b6 - 48647b5 - 343252b4 + 80715b3 - 25164b2 - 31677*b1 - 605789) / 4
\(\nu^{10}\)	\(=\)	\( ( 468738 \beta_{15} + 331189 \beta_{14} - 202963 \beta_{10} + 107775 \beta_{9} - 364379 \beta_{8} + \cdots - 442297 \beta_1 ) / 4 \) (468738b15 + 331189b14 - 202963b10 + 107775b9 - 364379b8 + 364379b7 - 468738b6 - 1714558b4 - 180899b2 - 442297b1) / 4
\(\nu^{11}\)	\(=\)	\( ( 2984962 \beta_{15} + 1520522 \beta_{14} - 1058958 \beta_{13} - 653079 \beta_{12} + 3381477 \beta_{11} + \cdots + 13980171 ) / 8 \) (2984962b15 + 1520522b14 - 1058958b13 - 653079b12 + 3381477b11 - 1441439b10 + 1058958b9 - 1940038b8 + 4822916b7 - 1362240b6 + 7865261b5 - 8076886b4 - 2173601b3 - 811361b2 - 5623369*b1 + 13980171) / 8
\(\nu^{12}\)	\(=\)	\( ( 507444 \beta_{15} - 2098948 \beta_{13} - 461396 \beta_{12} + 6792836 \beta_{11} + 3353988 \beta_{8} + \cdots + 12695541 ) / 2 \) (507444b15 - 2098948b13 - 461396b12 + 6792836b11 + 3353988b8 + 3353988b7 + 507444b6 + 13046228b5 - 1667507*b3 + 12695541) / 2
\(\nu^{13}\)	\(=\)	\( ( 2594408 \beta_{15} + 1712274 \beta_{14} - 27904182 \beta_{13} + 538087 \beta_{12} + 90779413 \beta_{11} + \cdots - 32291635 ) / 8 \) (2594408b15 + 1712274b14 - 27904182b13 + 538087b12 + 90779413b11 + 43351999b10 - 27904182b9 + 134131412b8 - 47427414b7 - 1906314b6 + 180666621b5 - 17270998b4 + 2250361b3 + 344047b2 + 135723017*b1 - 32291635) / 8
\(\nu^{14}\)	\(=\)	\( ( 86751774 \beta_{15} + 61562339 \beta_{14} + 121521221 \beta_{10} - 81489257 \beta_{9} + \cdots + 407613775 \beta_1 ) / 4 \) (86751774b15 + 61562339b14 + 121521221b10 - 81489257b9 + 262907901b8 - 262907901b7 - 86751774b6 - 319048050b4 - 33380981b2 + 407613775b1) / 4
\(\nu^{15}\)	\(=\)	\( ( 658267335 \beta_{15} + 340805343 \beta_{14} + 203611805 \beta_{13} - 141955560 \beta_{12} + \cdots + 3196928881 ) / 4 \) (658267335b15 + 340805343b14 + 203611805b13 - 141955560b12 - 659765671b11 + 325385747b10 - 203611805b9 + 334379924b8 - 985151418b7 - 307254471b6 - 1266954939b5 - 1839844892b4 - 482760903b3 - 175506432b2 + 963360305*b1 + 3196928881) / 4

\(n\)	\(511\)	\(577\)	\(641\)	\(901\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 799.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( T^{16} + \cdots + 152587890625 \) T^16 - 4T^14 - 1388T^12 + 2948T^10 + 1033750T^8 + 1842500T^6 - 542187500T^4 - 976562500*T^2 + 152587890625
$7$	\( (T^{4} + 6 T^{3} + \cdots - 1200)^{4} \) (T^4 + 6T^3 - 120T^2 - 840*T - 1200)^4
$11$	\( (T^{8} - 268 T^{6} + \cdots + 6150400)^{2} \) (T^8 - 268T^6 + 20736T^4 - 618688*T^2 + 6150400)^2
$13$	\( (T^{8} - 924 T^{6} + \cdots + 1440000)^{2} \) (T^8 - 924T^6 + 230592T^4 - 9266112*T^2 + 1440000)^2
$17$	\( (T^{8} + 828 T^{6} + \cdots + 3686400)^{2} \) (T^8 + 828T^6 + 177168T^4 + 2543616*T^2 + 3686400)^2
$19$	\( (T^{8} - 1656 T^{6} + \cdots + 2985984)^{2} \) (T^8 - 1656T^6 + 516240T^4 - 10077696*T^2 + 2985984)^2
$23$	\( (T^{4} - 48 T^{3} + \cdots - 65280)^{4} \) (T^4 - 48T^3 + 516T^2 + 4032*T - 65280)^4
$29$	\( (T^{8} + \cdots + 1169166438400)^{2} \) (T^8 + 6196T^6 + 11772432T^4 + 6756249088*T^2 + 1169166438400)^2
$31$	\( (T^{8} + \cdots + 3815302758400)^{2} \) (T^8 + 7048T^6 + 16378128T^4 + 13942521856*T^2 + 3815302758400)^2
$37$	\( (T^{8} + \cdots + 20649826347264)^{2} \) (T^8 - 9036T^6 + 29540160T^4 - 41288584896*T^2 + 20649826347264)^2
$41$	\( (T^{4} - 24 T^{3} + \cdots + 1471824)^{4} \) (T^4 - 24T^3 - 3672T^2 - 6048*T + 1471824)^4
$43$	\( (T^{8} + \cdots + 18043872804864)^{2} \) (T^8 + 9264T^6 + 29614080T^4 + 39030534144*T^2 + 18043872804864)^2
$47$	\( (T^{4} + 96 T^{3} + \cdots - 4740672)^{4} \) (T^4 + 96T^3 - 108T^2 - 216000*T - 4740672)^4
$53$	\( (T^{8} + \cdots + 40936451385600)^{2} \) (T^8 - 13212T^6 + 51796416T^4 - 78614741952*T^2 + 40936451385600)^2
$59$	\( (T^{8} + \cdots + 114174866749696)^{2} \) (T^8 - 17020T^6 + 96116352T^4 - 199569357760*T^2 + 114174866749696)^2
$61$	\( (T^{8} + \cdots + 15341322240000)^{2} \) (T^8 + 14880T^6 + 60569856T^4 + 57541017600*T^2 + 15341322240000)^2
$67$	\( (T^{8} + \cdots + 10330310246400)^{2} \) (T^8 + 9648T^6 + 26873856T^4 + 28865507328*T^2 + 10330310246400)^2
$71$	\( (T^{8} + \cdots + 10703890022400)^{2} \) (T^8 + 22944T^6 + 138150144T^4 + 93544022016*T^2 + 10703890022400)^2
$73$	\( (T^{8} + \cdots + 5668208640000)^{2} \) (T^8 + 7200T^6 + 16898304T^4 + 16360243200*T^2 + 5668208640000)^2
$79$	\( (T^{8} + 7240 T^{6} + \cdots + 27180138496)^{2} \) (T^8 + 7240T^6 + 3878928T^4 + 637788160*T^2 + 27180138496)^2
$83$	\( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) (T^8 + 38640T^6 + 504293376T^4 + 2404717424640*T^2 + 2222526563942400)^2
$89$	\( (T^{4} - 96 T^{3} + \cdots - 24665328)^{4} \) (T^4 - 96T^3 - 7368T^2 + 994176*T - 24665328)^4
$97$	\( (T^{8} + \cdots + 4624220160000)^{2} \) (T^8 + 41904T^6 + 418126080T^4 + 414557798400*T^2 + 4624220160000)^2
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