LMFDB - Newform orbit 3479.1.g.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3479,1,Mod(851,3479)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3479, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3479.851");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3479 = 7^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3479.g (of order $6$, degree $2$, 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:	$1.73624717895$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Coefficient field:	24.0.11935913115234869169771450911000887296.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} + 12 x^{22} + 91 x^{20} + 428 x^{18} + 1475 x^{16} + 3472 x^{14} + 5972 x^{12} + 6412 x^{10} + \cdots + 1 $ x^24 + 12x^22 + 91x^20 + 428x^18 + 1475x^16 + 3472x^14 + 5972x^12 + 6412x^10 + 4847x^8 + 1856x^6 + 490x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{28}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{28} - \cdots)$

$q$-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 $1,\beta_1,\ldots,\beta_{23}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{12} - \beta_{11} + \cdots + \beta_{4}) q^{2}+ \cdots + ( - \beta_{14} + \beta_{12} - 1) q^{9}+O(q^{10}) $ q + (b12 - b11 + b5 + b4) * q^2 - b6 * q^3 + (-b12 + b11) * q^4 + (b13 - b1) * q^5 + (b22 - b16 + b13) * q^6 + (b7 - b5) * q^8 + (-b14 + b12 - 1) * q^9	$ q + (\beta_{12} - \beta_{11} + \cdots + \beta_{4}) q^{2}+ \cdots + (\beta_{20} + \beta_{6}) q^{96}+O(q^{100}) $ q + (b12 - b11 + b5 + b4) * q^2 - b6 * q^3 + (-b12 + b11) * q^4 + (b13 - b1) * q^5 + (b22 - b16 + b13) * q^6 + (b7 - b5) * q^8 + (-b14 + b12 - 1) * q^9 + (b6 - b3) * q^10 + (-b23 + b20 - b19 + b6) * q^12 + (-b8 + b5) * q^15 + (b12 - b11 + b5 + b4) * q^16 + (b15 - b12 + b11 - b4) * q^18 + (b23 - b22 - b21 - b20 + b19 - b13 - b6 + b1) * q^19 + (-b22 - b21 + b16 - b13 - b10) * q^20 + (b18 + b16 - b10 - b1) * q^24 + (-b17 - b12 + b2) * q^25 + (-b23 - b20 + b18 + b16 - b10 + b6 - b3 - b1) * q^27 + b2 * q^29 + (-b17 + b14 - 2b12 + b11 + b7 - b5 + 1) q^30 - b12 * q^32 + (b7 - b5 - b2) * q^36 + b14 * q^37 + (-b18 - b16 - b6 + b3 + b1) * q^38 + (b22 + b21 + b13 - b1) * q^40 + (-b7 + b5 + 1) * q^43 + (b18 - b16 + b1) * q^45 + (b22 - b16 + b13) * q^48 + (b8 - b5 - b2) * q^50 + (b23 - b22 - b21 - b20 + b19 - 2b13 - b6 + 2b1) * q^54 + (b8 - b7) * q^57 + (-b17 + b15 + b8) * q^58 + (b17 - b15 + 2b12 - 2b11 + b4 - b2) * q^60 - q^71 + (b17 + b12 - b11 - b7 + b5) * q^72 + (b18 + b16 - b10 + b6 - b3 - b1) * q^73 - b15 * q^74 + (-b21 + b20 - b19 + b6) * q^75 + (-b23 + b21 + b20 + b18 + b16 + b13 + b6 - b3 - b1) * q^76 + (-b15 - b8) * q^79 + (b6 - b3) * q^80 + (b14 - b12 - b11 - b9) * q^81 + (b23 - b22 - b21 - b20 + b19 - b18 - b13 - b6 + b1) * q^83 + (-b12 + b11 + b7 - b5) * q^86 + (b10 - b3) * q^87 + (-b20 - b6) * q^89 + (-b21 + 2b20 - b19 - b10 + b3) q^90 + (-b14 + b9 - b4) * q^95 + (b20 + b6) * q^96
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{2} - 8 q^{4} + 16 q^{8} - 12 q^{9}+O(q^{10}) $ 24 * q - 4 * q^2 - 8 * q^4 + 16 * q^8 - 12 * q^9	$ 24 q - 4 q^{2} - 8 q^{4} + 16 q^{8} - 12 q^{9} - 8 q^{15} - 4 q^{16} - 4 q^{18} - 12 q^{25} + 20 q^{30} - 12 q^{32} + 16 q^{36} + 8 q^{43} + 8 q^{50} - 8 q^{57} + 12 q^{60} - 24 q^{71} - 8 q^{72} - 16 q^{81} + 8 q^{86} + 4 q^{95}+O(q^{100}) $ 24 * q - 4 * q^2 - 8 * q^4 + 16 * q^8 - 12 * q^9 - 8 * q^15 - 4 * q^16 - 4 * q^18 - 12 * q^25 + 20 * q^30 - 12 * q^32 + 16 * q^36 + 8 * q^43 + 8 * q^50 - 8 * q^57 + 12 * q^60 - 24 * q^71 - 8 * q^72 - 16 * q^81 + 8 * q^86 + 4 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{24} + 12 x^{22} + 91 x^{20} + 428 x^{18} + 1475 x^{16} + 3472 x^{14} + 5972 x^{12} + 6412 x^{10} + \cdots + 1 $ x^24 + 12*x^22 + 91*x^20 + 428*x^18 + 1475*x^16 + 3472*x^14 + 5972*x^12 + 6412*x^10 + 4847*x^8 + 1856*x^6 + 490*x^4 + 24*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 2022242251246 \nu^{22} - 23543224629428 \nu^{20} - 175703842562689 \nu^{18} + \cdots + 10\!\cdots\!54 ) / 537280551687823 $ (-2022242251246v^22 - 23543224629428v^20 - 175703842562689v^18 - 803670364449600v^16 - 2702306503482201v^14 - 6081260631639816v^12 - 9979354754987471v^10 - 9509239383753676v^8 - 6510026034453053v^6 - 1147985607993420v^4 - 56428876005167*v^2 + 1045550819836654) / 537280551687823
$\beta_{3}$	$=$	$ ( - 2022242251246 \nu^{23} - 23543224629428 \nu^{21} - 175703842562689 \nu^{19} + \cdots + 15\!\cdots\!77 \nu ) / 537280551687823 $ (-2022242251246v^23 - 23543224629428v^21 - 175703842562689v^19 - 803670364449600v^17 - 2702306503482201v^15 - 6081260631639816v^13 - 9979354754987471v^11 - 9509239383753676v^9 - 6510026034453053v^7 - 1147985607993420v^5 - 56428876005167v^3 + 1582831371524477v) / 537280551687823
$\beta_{4}$	$=$	$ ( - 7293209220672 \nu^{22} - 91741176925332 \nu^{20} - 713509324464216 \nu^{18} + \cdots - 297119999221338 ) / 537280551687823 $ (-7293209220672v^22 - 91741176925332v^20 - 713509324464216v^18 - 3494659013262481v^16 - 12480706751124384v^14 - 31136639527343574v^12 - 56772162185145568v^10 - 68324574302485914v^8 - 55912292177840232v^6 - 25866541978714946v^4 - 6057297284378760*v^2 - 297119999221338) / 537280551687823
$\beta_{5}$	$=$	$ ( 2530156860 \nu^{22} + 29492509865 \nu^{20} + 220187581988 \nu^{18} + 1007963119441 \nu^{16} + \cdots - 329969842484 ) / 184569066193 $ (2530156860v^22 + 29492509865v^20 + 220187581988v^18 + 1007963119441v^16 + 3390333751028v^14 + 7635715065993v^12 + 12524919645728v^10 + 11935821153445v^8 + 8050432197696v^6 + 1441000740830v^4 + 70832007396*v^2 - 329969842484) / 184569066193
$\beta_{6}$	$=$	$ ( - 9387528870706 \nu^{23} - 109395920846443 \nu^{21} - 816669893729757 \nu^{19} + \cdots + 25\!\cdots\!01 \nu ) / 537280551687823 $ (-9387528870706v^23 - 109395920846443v^21 - 816669893729757v^19 - 3737851005142351v^17 - 12571568052724709v^15 - 28308827188745439v^13 - 46439395843701679v^11 - 44254414761432071v^9 - 29944834161946109v^7 - 5342738764549550v^5 - 262620849534923v^3 + 2543373582995401v) / 537280551687823
$\beta_{7}$	$=$	$ ( - 3737203572 \nu^{22} - 44081883741 \nu^{20} - 330342886140 \nu^{18} - 1523964069237 \nu^{16} + \cdots + 78054196382 ) / 184569066193 $ (-3737203572v^22 - 44081883741v^20 - 330342886140v^18 - 1523964069237v^16 - 5141759003808v^14 - 11662089040826v^12 - 19063402326948v^10 - 18180202217577v^8 - 11425080870064v^6 - 2195890837350v^4 - 107939341176*v^2 + 78054196382) / 184569066193
$\beta_{8}$	$=$	$ ( 13858085948070 \nu^{22} + 161967807860670 \nu^{20} + \cdots - 825602005032194 ) / 537280551687823 $ (13858085948070v^22 + 161967807860670v^20 + 1210238668562073v^18 + 5550028377412506v^16 + 18680971743222033v^14 + 42146489868036498v^12 + 69069925207473183v^10 + 65832461509765230v^8 + 43495738799850364v^6 + 7948734819435480v^4 + 390718701101031*v^2 - 825602005032194) / 537280551687823
$\beta_{9}$	$=$	$ ( 17535332918916 \nu^{22} + 206817441757038 \nu^{20} + \cdots - 447138322115332 ) / 537280551687823 $ (17535332918916v^22 + 206817441757038v^20 + 1550090816292215v^18 + 7149295660464630v^16 + 24120703806999786v^14 + 54679565676396588v^12 + 89426611567954206v^10 + 85283030225490246v^8 + 54186475160364894v^6 + 10300851688802220v^4 + 506339866401462*v^2 - 447138322115332) / 537280551687823
$\beta_{10}$	$=$	$ ( 23245614818776 \nu^{23} + 271363728707113 \nu^{21} + \cdots - 33\!\cdots\!95 \nu ) / 537280551687823 $ (23245614818776v^23 + 271363728707113v^21 + 2026908562291830v^19 + 9287879382554857v^17 + 31252539795946742v^15 + 70455317056781937v^13 + 115509321051174862v^11 + 110086876271197301v^9 + 73440572961796473v^7 + 13291473583985030v^5 + 653339550635954v^3 - 3368975588027595v) / 537280551687823
$\beta_{11}$	$=$	$ ( - 28214523754680 \nu^{22} - 343669438624278 \nu^{20} + \cdots - 820896793996011 ) / 537280551687823 $ (-28214523754680v^22 - 343669438624278v^20 - 2627086531632304v^18 - 12520675067589968v^16 - 43657978592759180v^14 - 104832994944682333v^12 - 184022895828416092v^10 - 206322200064500443v^8 - 160975792969768568v^6 - 68070379208969542v^4 - 16749512899719460*v^2 - 820896793996011) / 537280551687823
$\beta_{12}$	$=$	$ ( 29010283538992 \nu^{22} + 346101160216658 \nu^{20} + \cdots + 639817928930641 ) / 537280551687823 $ (29010283538992v^22 + 346101160216658v^20 + 2616392577418844v^18 + 12240697512125887v^16 + 41986497855563600v^14 + 98021397943898023v^12 + 167168152663220408v^10 + 176034583297029233v^8 + 131103604929740548v^6 + 47333060213916099v^4 + 13067053326112660*v^2 + 639817928930641) / 537280551687823
$\beta_{13}$	$=$	$ ( - 29010283538992 \nu^{23} - 346101160216658 \nu^{21} + \cdots - 102537377242818 \nu ) / 537280551687823 $ (-29010283538992v^23 - 346101160216658v^21 - 2616392577418844v^19 - 12240697512125887v^17 - 41986497855563600v^15 - 98021397943898023v^13 - 167168152663220408v^11 - 176034583297029233v^9 - 131103604929740548v^7 - 47333060213916099v^5 - 13067053326112660v^3 - 102537377242818v) / 537280551687823
$\beta_{14}$	$=$	$ ( 34118786030475 \nu^{22} + 411766259117934 \nu^{20} + \cdots + 128929467508580 ) / 537280551687823 $ (34118786030475v^22 + 411766259117934v^20 + 3130322509601043v^18 + 14782972890522164v^16 + 51073189015688006v^14 + 120694140024316324v^12 + 207697875950183135v^10 + 223022185276509502v^8 + 164542220154660170v^6 + 59501062031579848v^4 + 12254705657061982*v^2 + 128929467508580) / 537280551687823
$\beta_{15}$	$=$	$ ( 40184373142361 \nu^{22} + 484319076767460 \nu^{20} + \cdots + 10\!\cdots\!78 ) / 537280551687823 $ (40184373142361v^22 + 484319076767460v^20 + 3680685815672110v^18 + 17375828798166978v^16 + 60067386452873078v^14 + 142175903478151354v^12 + 245870581830197156v^10 + 267388404652591054v^8 + 204025660229830790v^6 + 81445647484180328v^4 + 20806067937989740*v^2 + 1019266988395878) / 537280551687823
$\beta_{16}$	$=$	$ ( 41489901036328 \nu^{23} + 485538788494179 \nu^{21} + \cdots - 40\!\cdots\!98 \nu ) / 537280551687823 $ (41489901036328v^23 + 485538788494179v^21 + 3629502755012438v^19 + 16658319428796515v^17 + 56089461805274338v^15 + 126631224811732046v^13 + 207462926313634698v^11 + 197754620304242343v^9 + 130133791502045833v^7 + 23878464968067010v^5 + 1173742946329046v^3 - 4019453013478698v) / 537280551687823
$\beta_{17}$	$=$	$ ( - 58020567077984 \nu^{22} - 692202320433316 \nu^{20} + \cdots - 205074754485636 ) / 537280551687823 $ (-58020567077984v^22 - 692202320433316v^20 - 5232785154837688v^18 - 24481395024251774v^16 - 83972995711127200v^14 - 196042795887796046v^12 - 334336305326440816v^10 - 352069166594058466v^8 - 262207209859481096v^6 - 94666120427832198v^4 - 25596826100537497*v^2 - 205074754485636) / 537280551687823
$\beta_{18}$	$=$	$ ( 59025233955244 \nu^{23} + 692356230251217 \nu^{21} + \cdots - 44\!\cdots\!30 \nu ) / 537280551687823 $ (59025233955244v^23 + 692356230251217v^21 + 5179593571304653v^19 + 23807615089261145v^17 + 80210165612274124v^15 + 181310790488128634v^13 + 296889537881588904v^11 + 283037650529732589v^9 + 184320266662410727v^7 + 34179316656869230v^5 + 1680082812730508v^3 - 4466591335594030v) / 537280551687823
$\beta_{19}$	$=$	$ ( 85008608365730 \nu^{23} + \cdots + 18\!\cdots\!31 \nu ) / 537280551687823 $ (85008608365730v^23 + 1014760256020546v^21 + 7673473889693843v^19 + 35918422171928061v^17 + 123257187063208599v^15 + 287982933200054253v^13 + 491525103234673753v^11 + 518594510507334023v^9 + 386800788754768591v^7 + 140851195033754877v^5 + 38607450550644990v^3 + 1890443503252931v) / 537280551687823
$\beta_{20}$	$=$	$ ( 144327735309436 \nu^{23} + \cdots + 510664643962844 \nu ) / 537280551687823 $ (144327735309436v^23 + 1722185598782593v^21 + 13020113568010532v^19 + 60922986743523786v^17 + 208992524813131704v^15 + 488009513750036474v^13 + 832383385384866364v^11 + 876881134327809856v^9 + 652912728638383584v^7 + 235730831242475122v^5 + 62629340341633273v^3 + 510664643962844v) / 537280551687823
$\beta_{21}$	$=$	$ ( 175124579581091 \nu^{23} + \cdots + 40\!\cdots\!23 \nu ) / 537280551687823 $ (175124579581091v^23 + 2097108754703610v^21 + 15884129489952885v^19 + 74560964536548413v^17 + 256488343213280073v^15 + 601876590039442389v^13 + 1031814571371361841v^11 + 1100015124218968839v^9 + 826993554706268265v^7 + 311833739962105900v^5 + 83172787430088090v^3 + 4073305215354123v) / 537280551687823
$\beta_{22}$	$=$	$ ( - 203339103335771 \nu^{23} + \cdots - 48\!\cdots\!34 \nu ) / 537280551687823 $ (-203339103335771v^23 - 2440778193327888v^21 - 18511216021585189v^19 - 87081639604138381v^17 - 300146321806039253v^15 - 706709584984124722v^13 - 1215837467199777933v^11 - 1306337324283469282v^9 - 987969347676036833v^7 - 379904119171075442v^5 - 99922300329807550v^3 - 4894202009350134v) / 537280551687823
$\beta_{23}$	$=$	$ ( - 78054196382 \nu^{23} - 940387560156 \nu^{21} - 7147013754503 \nu^{19} + \cdots - 1981240054344 \nu ) / 184569066193 $ (-78054196382v^23 - 940387560156v^21 - 7147013754503v^19 - 33737538937636v^17 - 116653903732687v^15 - 276145928842112v^13 - 477801749834130v^11 - 519546909528332v^9 - 396508892081131v^7 - 156293669355056v^5 - 40442447064530v^3 - 1981240054344v) / 184569066193

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{17} + 2\beta_{12} - 2 $ b17 + 2*b12 - 2
$\nu^{3}$	$=$	$ -\beta_{19} - 3\beta_{13} + \beta_{3} $ -b19 - 3*b13 + b3
$\nu^{4}$	$=$	$ -4\beta_{17} - 7\beta_{12} + \beta_{11} - \beta_{4} + 4\beta_{2} $ -4b17 - 7b12 + b11 - b4 + 4*b2
$\nu^{5}$	$=$	$ -\beta_{20} + 5\beta_{19} + 10\beta_{13} - \beta_{6} - 10\beta_1 $ -b20 + 5b19 + 10b13 - b6 - 10*b1
$\nu^{6}$	$=$	$ \beta_{8} - 6\beta_{5} - 15\beta_{2} + 20 $ b8 - 6b5 - 15b2 + 20
$\nu^{7}$	$=$	$ \beta_{10} + 7\beta_{6} - 21\beta_{3} + 35\beta_1 $ b10 + 7b6 - 21b3 + 35*b1
$\nu^{8}$	$=$	$ 56\beta_{17} - 8\beta_{15} + 98\beta_{12} - 29\beta_{11} - 8\beta_{8} - \beta_{7} + 29\beta_{5} + 28\beta_{4} - 69 $ 56b17 - 8b15 + 98b12 - 29b11 - 8b8 - b7 + 29b5 + 28*b4 - 69
$\nu^{9}$	$=$	$ -\beta_{22} - 9\beta_{21} + 36\beta_{20} - 84\beta_{19} + \beta_{16} - 126\beta_{13} - 9\beta_{10} + 84\beta_{3} $ -b22 - 9b21 + 36b20 - 84b19 + b16 - 126b13 - 9b10 + 84b3
$\nu^{10}$	$=$	$ -210\beta_{17} + 45\beta_{15} + \beta_{14} - 372\beta_{12} + 130\beta_{11} - \beta_{9} - 120\beta_{4} + 210\beta_{2} $ -210b17 + 45b15 + b14 - 372b12 + 130b11 - b9 - 120b4 + 210b2
$\nu^{11}$	$=$	$ -\beta_{23} + 12\beta_{22} + 56\beta_{21} - 164\beta_{20} + 329\beta_{19} + 463\beta_{13} - 164\beta_{6} - 463\beta_1 $ -b23 + 12b22 + 56b21 - 164b20 + 329b19 + 463b13 - 164b6 - 463*b1
$\nu^{12}$	$=$	$ 12\beta_{9} + 220\beta_{8} + 67\beta_{7} - 561\beta_{5} - 792\beta_{2} + 858 $ 12b9 + 220b8 + 67b7 - 561b5 - 792*b2 + 858
$\nu^{13}$	$=$	$ 12\beta_{18} - 79\beta_{16} + 287\beta_{10} + 714\beta_{6} - 1286\beta_{3} + 1717\beta_1 $ 12b18 - 79b16 + 287b10 + 714b6 - 1286b3 + 1717b1
$\nu^{14}$	$=$	$ 3003 \beta_{17} - 1001 \beta_{15} - 91 \beta_{14} + 5434 \beta_{12} - 2366 \beta_{11} - 1001 \beta_{8} + \cdots - 3068 $ 3003b17 - 1001b15 - 91b14 + 5434b12 - 2366b11 - 1001b8 - 378b7 + 2366b5 + 1988*b4 - 3068
$\nu^{15}$	$=$	$ 91 \beta_{23} - 560 \beta_{22} - 1470 \beta_{21} + 2898 \beta_{20} - 4900 \beta_{19} - 91 \beta_{18} + \cdots + 91 \beta_1 $ 91b23 - 560b22 - 1470b21 + 2898b20 - 4900b19 - 91b18 + 469b16 - 6540b13 - 1379b10 - 91b6 + 4991b3 + 91b1
$\nu^{16}$	$=$	$ - 11440 \beta_{17} + 4368 \beta_{15} + 560 \beta_{14} - 20878 \beta_{12} + 9828 \beta_{11} + \cdots + 11440 \beta_{2} $ -11440b17 + 4368b15 + 560b14 - 20878b12 + 9828b11 - 560b9 - 7889b4 + 11440b2
$\nu^{17}$	$=$	$ - 560 \beta_{23} + 3059 \beta_{22} + 6867 \beta_{21} - 11697 \beta_{20} + 18769 \beta_{19} + \cdots - 24989 \beta_1 $ -560b23 + 3059b22 + 6867b21 - 11697b20 + 18769b19 + 24989b13 - 11697b6 - 24989b1
$\nu^{18}$	$=$	$ 3059\beta_{9} + 18564\beta_{8} + 9366\beta_{7} - 40392\beta_{5} - 43758\beta_{2} + 40052 $ 3059b9 + 18564b8 + 9366b7 - 40392b5 - 43758*b2 + 40052
$\nu^{19}$	$=$	$ 3059\beta_{18} - 12425\beta_{16} + 27930\beta_{10} + 49590\beta_{6} - 74784\beta_{3} + 93176\beta_1 $ 3059b18 - 12425b16 + 27930b10 + 49590b6 - 74784b3 + 93176b1
$\nu^{20}$	$=$	$ 167960 \beta_{17} - 77520 \beta_{15} - 15484 \beta_{14} + 310726 \beta_{12} - 164729 \beta_{11} + \cdots - 145997 $ 167960b17 - 77520b15 - 15484b14 + 310726b12 - 164729b11 - 77520b8 - 43414b7 + 164729b5 + 121315*b4 - 145997
$\nu^{21}$	$=$	$ 15484 \beta_{23} - 74382 \beta_{22} - 136418 \beta_{21} + 183351 \beta_{20} - 273791 \beta_{19} + \cdots + 15484 \beta_1 $ 15484b23 - 74382b22 - 136418b21 + 183351b20 - 273791b19 - 15484b18 + 58898b16 - 372855b13 - 120934b10 - 15484b6 + 289275b3 + 15484b1
$\nu^{22}$	$=$	$ - 646646 \beta_{17} + 319769 \beta_{15} + 74382 \beta_{14} - 1202852 \beta_{12} + \cdots + 646646 \beta_{2} $ -646646b17 + 319769b15 + 74382b14 - 1202852b12 + 667942b11 - 74382b9 - 472626b4 + 646646b2
$\nu^{23}$	$=$	$ - 74382 \beta_{23} + 344080 \beta_{22} + 589467 \beta_{21} - 718013 \beta_{20} + 1044890 \beta_{19} + \cdots - 1451254 \beta_1 $ -74382b23 + 344080b22 + 589467b21 - 718013b20 + 1044890b19 + 1451254b13 - 718013b6 - 1451254b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3479\mathbb{Z}\right)^\times$.

$n$	$640$	$1569$
$\chi(n)$	$-1 + \beta_{12}$	$-1$

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} $

851.1

0.993712	−	1.72116i
0.111964	−	0.193928i
−0.111964	+	0.193928i
−0.993712	+	1.72116i
0.330279	−	0.572060i
−0.943883	+	1.63485i
0.943883	−	1.63485i
−0.330279	+	0.572060i
−0.846724	+	1.46657i
0.532032	−	0.921507i
−0.532032	+	0.921507i
0.846724	−	1.46657i
0.993712	+	1.72116i
0.111964	+	0.193928i
−0.111964	−	0.193928i
−0.993712	−	1.72116i
0.330279	+	0.572060i
−0.943883	−	1.63485i
0.943883	+	1.63485i
−0.330279	−	0.572060i

−0.900969

−

1.56052i

−0.846724

1.46657i

−1.12349

1.94594i

0.993712

1.72116i

3.05149

2.24698

−0.933884

−

1.61753i

1.79061

−

3.10142i

851.2

−0.900969

−

1.56052i

−0.532032

0.921507i

−1.12349

1.94594i

0.111964

0.193928i

1.91738

2.24698

−0.0661163

−

0.114517i

0.201753

−

0.349446i

851.3

−0.900969

−

1.56052i

0.532032

−

0.921507i

−1.12349

1.94594i

−0.111964

−

0.193928i

−1.91738

2.24698

−0.0661163

−

0.114517i

−0.201753

0.349446i

851.4

−0.900969

−

1.56052i

0.846724

−

1.46657i

−1.12349

1.94594i

−0.993712

−

1.72116i

−3.05149

2.24698

−0.933884

−

1.61753i

−1.79061

3.10142i

851.5

−0.222521

−

0.385418i

−0.993712

1.72116i

0.400969

−

0.694498i

0.330279

0.572060i

0.884487

−0.801938

−1.47493

−

2.55465i

0.146988

−

0.254591i

851.6

−0.222521

−

0.385418i

−0.111964

0.193928i

0.400969

−

0.694498i

−0.943883

−

1.63485i

0.0996578

−0.801938

0.474928

0.822599i

−0.420068

0.727578i

851.7

−0.222521

−

0.385418i

0.111964

−

0.193928i

0.400969

−

0.694498i

0.943883

1.63485i

−0.0996578

−0.801938

0.474928

0.822599i

0.420068

−

0.727578i

851.8

−0.222521

−

0.385418i

0.993712

−

1.72116i

0.400969

−

0.694498i

−0.330279

−

0.572060i

−0.884487

−0.801938

−1.47493

−

2.55465i

−0.146988

0.254591i

851.9

0.623490

1.07992i

−0.943883

1.63485i

−0.277479

0.480608i

−0.846724

−

1.46657i

−2.35401

0.554958

−1.28183

−

2.22020i

1.05585

−

1.82878i

851.10

0.623490

1.07992i

−0.330279

0.572060i

−0.277479

0.480608i

0.532032

0.921507i

−0.823703

0.554958

0.281831

0.488146i

−0.663433

1.14910i

851.11

0.623490

1.07992i

0.330279

−

0.572060i

−0.277479

0.480608i

−0.532032

−

0.921507i

0.823703

0.554958

0.281831

0.488146i

0.663433

−

1.14910i

851.12

0.623490

1.07992i

0.943883

−

1.63485i

−0.277479

0.480608i

0.846724

1.46657i

2.35401

0.554958

−1.28183

−

2.22020i

−1.05585

1.82878i

1206.1

−0.900969

1.56052i

−0.846724

−

1.46657i

−1.12349

−

1.94594i

0.993712

−

1.72116i

3.05149

2.24698

−0.933884

1.61753i

1.79061

3.10142i

1206.2

−0.900969

1.56052i

−0.532032

−

0.921507i

−1.12349

−

1.94594i

0.111964

−

0.193928i

1.91738

2.24698

−0.0661163

0.114517i

0.201753

0.349446i

1206.3

−0.900969

1.56052i

0.532032

0.921507i

−1.12349

−

1.94594i

−0.111964

0.193928i

−1.91738

2.24698

−0.0661163

0.114517i

−0.201753

−

0.349446i

1206.4

−0.900969

1.56052i

0.846724

1.46657i

−1.12349

−

1.94594i

−0.993712

1.72116i

−3.05149

2.24698

−0.933884

1.61753i

−1.79061

−

3.10142i

1206.5

−0.222521

0.385418i

−0.993712

−

1.72116i

0.400969

0.694498i

0.330279

−

0.572060i

0.884487

−0.801938

−1.47493

2.55465i

0.146988

0.254591i

1206.6

−0.222521

0.385418i

−0.111964

−

0.193928i

0.400969

0.694498i

−0.943883

1.63485i

0.0996578

−0.801938

0.474928

−

0.822599i

−0.420068

−

0.727578i

1206.7

−0.222521

0.385418i

0.111964

0.193928i

0.400969

0.694498i

0.943883

−

1.63485i

−0.0996578

−0.801938

0.474928

−

0.822599i

0.420068

0.727578i

1206.8

−0.222521

0.385418i

0.993712

1.72116i

0.400969

0.694498i

−0.330279

0.572060i

−0.884487

−0.801938

−1.47493

2.55465i

−0.146988

−

0.254591i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.b	odd	2	1	CM by $\Q(\sqrt{-71}) $
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
497.b	even	2	1	inner
497.g	odd	6	1	inner
497.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3479.1.g.g		24
7.b	odd	2	1	inner	3479.1.g.g		24
7.c	even	3	1		3479.1.d.g	✓	12
7.c	even	3	1	inner	3479.1.g.g		24
7.d	odd	6	1		3479.1.d.g	✓	12
7.d	odd	6	1	inner	3479.1.g.g		24
71.b	odd	2	1	CM	3479.1.g.g		24
497.b	even	2	1	inner	3479.1.g.g		24
497.g	odd	6	1		3479.1.d.g	✓	12
497.g	odd	6	1	inner	3479.1.g.g		24
497.i	even	6	1		3479.1.d.g	✓	12
497.i	even	6	1	inner	3479.1.g.g		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3479.1.d.g	✓	12	7.c	even	3	1
3479.1.d.g	✓	12	7.d	odd	6	1
3479.1.d.g	✓	12	497.g	odd	6	1
3479.1.d.g	✓	12	497.i	even	6	1
3479.1.g.g		24	1.a	even	1	1	trivial
3479.1.g.g		24	7.b	odd	2	1	inner
3479.1.g.g		24	7.c	even	3	1	inner
3479.1.g.g		24	7.d	odd	6	1	inner
3479.1.g.g		24	71.b	odd	2	1	CM
3479.1.g.g		24	497.b	even	2	1	inner
3479.1.g.g		24	497.g	odd	6	1	inner
3479.1.g.g		24	497.i	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3479, [\chi])$:

$ T_{2}^{6} + T_{2}^{5} + 3T_{2}^{4} + 5T_{2}^{2} + 2T_{2} + 1 $

$ T_{3}^{24} + 12 T_{3}^{22} + 91 T_{3}^{20} + 428 T_{3}^{18} + 1475 T_{3}^{16} + 3472 T_{3}^{14} + \cdots + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} + T^{5} + 3 T^{4} + \cdots + 1)^{4} $ (T^6 + T^5 + 3T^4 + 5T^2 + 2*T + 1)^4
$3$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$5$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$7$	$ T^{24} $ T^24
$11$	$ T^{24} $ T^24
$13$	$ T^{24} $ T^24
$17$	$ T^{24} $ T^24
$19$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$23$	$ T^{24} $ T^24
$29$	$ (T^{6} - 7 T^{4} + 14 T^{2} - 7)^{4} $ (T^6 - 7T^4 + 14T^2 - 7)^4
$31$	$ T^{24} $ T^24
$37$	$ (T^{12} + 7 T^{10} + \cdots + 49)^{2} $ (T^12 + 7T^10 + 35T^8 + 84T^6 + 147T^4 + 98*T^2 + 49)^2
$41$	$ T^{24} $ T^24
$43$	$ (T^{3} - T^{2} - 2 T + 1)^{8} $ (T^3 - T^2 - 2*T + 1)^8
$47$	$ T^{24} $ T^24
$53$	$ T^{24} $ T^24
$59$	$ T^{24} $ T^24
$61$	$ T^{24} $ T^24
$67$	$ T^{24} $ T^24
$71$	$ (T + 1)^{24} $ (T + 1)^24
$73$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$79$	$ (T^{12} + 7 T^{10} + \cdots + 49)^{2} $ (T^12 + 7T^10 + 35T^8 + 84T^6 + 147T^4 + 98*T^2 + 49)^2
$83$	$ (T^{12} - 12 T^{10} + \cdots + 1)^{2} $ (T^12 - 12T^10 + 53T^8 - 104T^6 + 86T^4 - 24*T^2 + 1)^2
$89$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$97$	$ T^{24} $ T^24
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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:	\( N \)	\(=\)	\( 3479 = 7^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3479.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{12} - \beta_{11} + \cdots + \beta_{4}) q^{2}+ \cdots + ( - \beta_{14} + \beta_{12} - 1) q^{9}+O(q^{10}) \) q + (b12 - b11 + b5 + b4) * q^2 - b6 * q^3 + (-b12 + b11) * q^4 + (b13 - b1) * q^5 + (b22 - b16 + b13) * q^6 + (b7 - b5) * q^8 + (-b14 + b12 - 1) * q^9	\( q + (\beta_{12} - \beta_{11} + \cdots + \beta_{4}) q^{2}+ \cdots + (\beta_{20} + \beta_{6}) q^{96}+O(q^{100}) \) q + (b12 - b11 + b5 + b4) * q^2 - b6 * q^3 + (-b12 + b11) * q^4 + (b13 - b1) * q^5 + (b22 - b16 + b13) * q^6 + (b7 - b5) * q^8 + (-b14 + b12 - 1) * q^9 + (b6 - b3) * q^10 + (-b23 + b20 - b19 + b6) * q^12 + (-b8 + b5) * q^15 + (b12 - b11 + b5 + b4) * q^16 + (b15 - b12 + b11 - b4) * q^18 + (b23 - b22 - b21 - b20 + b19 - b13 - b6 + b1) * q^19 + (-b22 - b21 + b16 - b13 - b10) * q^20 + (b18 + b16 - b10 - b1) * q^24 + (-b17 - b12 + b2) * q^25 + (-b23 - b20 + b18 + b16 - b10 + b6 - b3 - b1) * q^27 + b2 * q^29 + (-b17 + b14 - 2b12 + b11 + b7 - b5 + 1) q^30 - b12 * q^32 + (b7 - b5 - b2) * q^36 + b14 * q^37 + (-b18 - b16 - b6 + b3 + b1) * q^38 + (b22 + b21 + b13 - b1) * q^40 + (-b7 + b5 + 1) * q^43 + (b18 - b16 + b1) * q^45 + (b22 - b16 + b13) * q^48 + (b8 - b5 - b2) * q^50 + (b23 - b22 - b21 - b20 + b19 - 2b13 - b6 + 2b1) * q^54 + (b8 - b7) * q^57 + (-b17 + b15 + b8) * q^58 + (b17 - b15 + 2b12 - 2b11 + b4 - b2) * q^60 - q^71 + (b17 + b12 - b11 - b7 + b5) * q^72 + (b18 + b16 - b10 + b6 - b3 - b1) * q^73 - b15 * q^74 + (-b21 + b20 - b19 + b6) * q^75 + (-b23 + b21 + b20 + b18 + b16 + b13 + b6 - b3 - b1) * q^76 + (-b15 - b8) * q^79 + (b6 - b3) * q^80 + (b14 - b12 - b11 - b9) * q^81 + (b23 - b22 - b21 - b20 + b19 - b18 - b13 - b6 + b1) * q^83 + (-b12 + b11 + b7 - b5) * q^86 + (b10 - b3) * q^87 + (-b20 - b6) * q^89 + (-b21 + 2b20 - b19 - b10 + b3) q^90 + (-b14 + b9 - b4) * q^95 + (b20 + b6) * q^96
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{2} - 8 q^{4} + 16 q^{8} - 12 q^{9}+O(q^{10}) \) 24 * q - 4 * q^2 - 8 * q^4 + 16 * q^8 - 12 * q^9	\( 24 q - 4 q^{2} - 8 q^{4} + 16 q^{8} - 12 q^{9} - 8 q^{15} - 4 q^{16} - 4 q^{18} - 12 q^{25} + 20 q^{30} - 12 q^{32} + 16 q^{36} + 8 q^{43} + 8 q^{50} - 8 q^{57} + 12 q^{60} - 24 q^{71} - 8 q^{72} - 16 q^{81} + 8 q^{86} + 4 q^{95}+O(q^{100}) \) 24 * q - 4 * q^2 - 8 * q^4 + 16 * q^8 - 12 * q^9 - 8 * q^15 - 4 * q^16 - 4 * q^18 - 12 * q^25 + 20 * q^30 - 12 * q^32 + 16 * q^36 + 8 * q^43 + 8 * q^50 - 8 * q^57 + 12 * q^60 - 24 * q^71 - 8 * q^72 - 16 * q^81 + 8 * q^86 + 4 * 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	3479.1.g.g

Level	$3479$
Weight	$1$
Character orbit	3479.g
Analytic conductor	$1.736$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{28}$
CM discriminant	-71
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.73624717895\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Coefficient field:	24.0.11935913115234869169771450911000887296.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} + 12 x^{22} + 91 x^{20} + 428 x^{18} + 1475 x^{16} + 3472 x^{14} + 5972 x^{12} + 6412 x^{10} + \cdots + 1 \) x^24 + 12x^22 + 91x^20 + 428x^18 + 1475x^16 + 3472x^14 + 5972x^12 + 6412x^10 + 4847x^8 + 1856x^6 + 490x^4 + 24*x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{28}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 2022242251246 \nu^{22} - 23543224629428 \nu^{20} - 175703842562689 \nu^{18} + \cdots + 10\!\cdots\!54 ) / 537280551687823 \) (-2022242251246v^22 - 23543224629428v^20 - 175703842562689v^18 - 803670364449600v^16 - 2702306503482201v^14 - 6081260631639816v^12 - 9979354754987471v^10 - 9509239383753676v^8 - 6510026034453053v^6 - 1147985607993420v^4 - 56428876005167*v^2 + 1045550819836654) / 537280551687823
\(\beta_{3}\)	\(=\)	\( ( - 2022242251246 \nu^{23} - 23543224629428 \nu^{21} - 175703842562689 \nu^{19} + \cdots + 15\!\cdots\!77 \nu ) / 537280551687823 \) (-2022242251246v^23 - 23543224629428v^21 - 175703842562689v^19 - 803670364449600v^17 - 2702306503482201v^15 - 6081260631639816v^13 - 9979354754987471v^11 - 9509239383753676v^9 - 6510026034453053v^7 - 1147985607993420v^5 - 56428876005167v^3 + 1582831371524477v) / 537280551687823
\(\beta_{4}\)	\(=\)	\( ( - 7293209220672 \nu^{22} - 91741176925332 \nu^{20} - 713509324464216 \nu^{18} + \cdots - 297119999221338 ) / 537280551687823 \) (-7293209220672v^22 - 91741176925332v^20 - 713509324464216v^18 - 3494659013262481v^16 - 12480706751124384v^14 - 31136639527343574v^12 - 56772162185145568v^10 - 68324574302485914v^8 - 55912292177840232v^6 - 25866541978714946v^4 - 6057297284378760*v^2 - 297119999221338) / 537280551687823
\(\beta_{5}\)	\(=\)	\( ( 2530156860 \nu^{22} + 29492509865 \nu^{20} + 220187581988 \nu^{18} + 1007963119441 \nu^{16} + \cdots - 329969842484 ) / 184569066193 \) (2530156860v^22 + 29492509865v^20 + 220187581988v^18 + 1007963119441v^16 + 3390333751028v^14 + 7635715065993v^12 + 12524919645728v^10 + 11935821153445v^8 + 8050432197696v^6 + 1441000740830v^4 + 70832007396*v^2 - 329969842484) / 184569066193
\(\beta_{6}\)	\(=\)	\( ( - 9387528870706 \nu^{23} - 109395920846443 \nu^{21} - 816669893729757 \nu^{19} + \cdots + 25\!\cdots\!01 \nu ) / 537280551687823 \) (-9387528870706v^23 - 109395920846443v^21 - 816669893729757v^19 - 3737851005142351v^17 - 12571568052724709v^15 - 28308827188745439v^13 - 46439395843701679v^11 - 44254414761432071v^9 - 29944834161946109v^7 - 5342738764549550v^5 - 262620849534923v^3 + 2543373582995401v) / 537280551687823
\(\beta_{7}\)	\(=\)	\( ( - 3737203572 \nu^{22} - 44081883741 \nu^{20} - 330342886140 \nu^{18} - 1523964069237 \nu^{16} + \cdots + 78054196382 ) / 184569066193 \) (-3737203572v^22 - 44081883741v^20 - 330342886140v^18 - 1523964069237v^16 - 5141759003808v^14 - 11662089040826v^12 - 19063402326948v^10 - 18180202217577v^8 - 11425080870064v^6 - 2195890837350v^4 - 107939341176*v^2 + 78054196382) / 184569066193
\(\beta_{8}\)	\(=\)	\( ( 13858085948070 \nu^{22} + 161967807860670 \nu^{20} + \cdots - 825602005032194 ) / 537280551687823 \) (13858085948070v^22 + 161967807860670v^20 + 1210238668562073v^18 + 5550028377412506v^16 + 18680971743222033v^14 + 42146489868036498v^12 + 69069925207473183v^10 + 65832461509765230v^8 + 43495738799850364v^6 + 7948734819435480v^4 + 390718701101031*v^2 - 825602005032194) / 537280551687823
\(\beta_{9}\)	\(=\)	\( ( 17535332918916 \nu^{22} + 206817441757038 \nu^{20} + \cdots - 447138322115332 ) / 537280551687823 \) (17535332918916v^22 + 206817441757038v^20 + 1550090816292215v^18 + 7149295660464630v^16 + 24120703806999786v^14 + 54679565676396588v^12 + 89426611567954206v^10 + 85283030225490246v^8 + 54186475160364894v^6 + 10300851688802220v^4 + 506339866401462*v^2 - 447138322115332) / 537280551687823
\(\beta_{10}\)	\(=\)	\( ( 23245614818776 \nu^{23} + 271363728707113 \nu^{21} + \cdots - 33\!\cdots\!95 \nu ) / 537280551687823 \) (23245614818776v^23 + 271363728707113v^21 + 2026908562291830v^19 + 9287879382554857v^17 + 31252539795946742v^15 + 70455317056781937v^13 + 115509321051174862v^11 + 110086876271197301v^9 + 73440572961796473v^7 + 13291473583985030v^5 + 653339550635954v^3 - 3368975588027595v) / 537280551687823
\(\beta_{11}\)	\(=\)	\( ( - 28214523754680 \nu^{22} - 343669438624278 \nu^{20} + \cdots - 820896793996011 ) / 537280551687823 \) (-28214523754680v^22 - 343669438624278v^20 - 2627086531632304v^18 - 12520675067589968v^16 - 43657978592759180v^14 - 104832994944682333v^12 - 184022895828416092v^10 - 206322200064500443v^8 - 160975792969768568v^6 - 68070379208969542v^4 - 16749512899719460*v^2 - 820896793996011) / 537280551687823
\(\beta_{12}\)	\(=\)	\( ( 29010283538992 \nu^{22} + 346101160216658 \nu^{20} + \cdots + 639817928930641 ) / 537280551687823 \) (29010283538992v^22 + 346101160216658v^20 + 2616392577418844v^18 + 12240697512125887v^16 + 41986497855563600v^14 + 98021397943898023v^12 + 167168152663220408v^10 + 176034583297029233v^8 + 131103604929740548v^6 + 47333060213916099v^4 + 13067053326112660*v^2 + 639817928930641) / 537280551687823
\(\beta_{13}\)	\(=\)	\( ( - 29010283538992 \nu^{23} - 346101160216658 \nu^{21} + \cdots - 102537377242818 \nu ) / 537280551687823 \) (-29010283538992v^23 - 346101160216658v^21 - 2616392577418844v^19 - 12240697512125887v^17 - 41986497855563600v^15 - 98021397943898023v^13 - 167168152663220408v^11 - 176034583297029233v^9 - 131103604929740548v^7 - 47333060213916099v^5 - 13067053326112660v^3 - 102537377242818v) / 537280551687823
\(\beta_{14}\)	\(=\)	\( ( 34118786030475 \nu^{22} + 411766259117934 \nu^{20} + \cdots + 128929467508580 ) / 537280551687823 \) (34118786030475v^22 + 411766259117934v^20 + 3130322509601043v^18 + 14782972890522164v^16 + 51073189015688006v^14 + 120694140024316324v^12 + 207697875950183135v^10 + 223022185276509502v^8 + 164542220154660170v^6 + 59501062031579848v^4 + 12254705657061982*v^2 + 128929467508580) / 537280551687823
\(\beta_{15}\)	\(=\)	\( ( 40184373142361 \nu^{22} + 484319076767460 \nu^{20} + \cdots + 10\!\cdots\!78 ) / 537280551687823 \) (40184373142361v^22 + 484319076767460v^20 + 3680685815672110v^18 + 17375828798166978v^16 + 60067386452873078v^14 + 142175903478151354v^12 + 245870581830197156v^10 + 267388404652591054v^8 + 204025660229830790v^6 + 81445647484180328v^4 + 20806067937989740*v^2 + 1019266988395878) / 537280551687823
\(\beta_{16}\)	\(=\)	\( ( 41489901036328 \nu^{23} + 485538788494179 \nu^{21} + \cdots - 40\!\cdots\!98 \nu ) / 537280551687823 \) (41489901036328v^23 + 485538788494179v^21 + 3629502755012438v^19 + 16658319428796515v^17 + 56089461805274338v^15 + 126631224811732046v^13 + 207462926313634698v^11 + 197754620304242343v^9 + 130133791502045833v^7 + 23878464968067010v^5 + 1173742946329046v^3 - 4019453013478698v) / 537280551687823
\(\beta_{17}\)	\(=\)	\( ( - 58020567077984 \nu^{22} - 692202320433316 \nu^{20} + \cdots - 205074754485636 ) / 537280551687823 \) (-58020567077984v^22 - 692202320433316v^20 - 5232785154837688v^18 - 24481395024251774v^16 - 83972995711127200v^14 - 196042795887796046v^12 - 334336305326440816v^10 - 352069166594058466v^8 - 262207209859481096v^6 - 94666120427832198v^4 - 25596826100537497*v^2 - 205074754485636) / 537280551687823
\(\beta_{18}\)	\(=\)	\( ( 59025233955244 \nu^{23} + 692356230251217 \nu^{21} + \cdots - 44\!\cdots\!30 \nu ) / 537280551687823 \) (59025233955244v^23 + 692356230251217v^21 + 5179593571304653v^19 + 23807615089261145v^17 + 80210165612274124v^15 + 181310790488128634v^13 + 296889537881588904v^11 + 283037650529732589v^9 + 184320266662410727v^7 + 34179316656869230v^5 + 1680082812730508v^3 - 4466591335594030v) / 537280551687823
\(\beta_{19}\)	\(=\)	\( ( 85008608365730 \nu^{23} + \cdots + 18\!\cdots\!31 \nu ) / 537280551687823 \) (85008608365730v^23 + 1014760256020546v^21 + 7673473889693843v^19 + 35918422171928061v^17 + 123257187063208599v^15 + 287982933200054253v^13 + 491525103234673753v^11 + 518594510507334023v^9 + 386800788754768591v^7 + 140851195033754877v^5 + 38607450550644990v^3 + 1890443503252931v) / 537280551687823
\(\beta_{20}\)	\(=\)	\( ( 144327735309436 \nu^{23} + \cdots + 510664643962844 \nu ) / 537280551687823 \) (144327735309436v^23 + 1722185598782593v^21 + 13020113568010532v^19 + 60922986743523786v^17 + 208992524813131704v^15 + 488009513750036474v^13 + 832383385384866364v^11 + 876881134327809856v^9 + 652912728638383584v^7 + 235730831242475122v^5 + 62629340341633273v^3 + 510664643962844v) / 537280551687823
\(\beta_{21}\)	\(=\)	\( ( 175124579581091 \nu^{23} + \cdots + 40\!\cdots\!23 \nu ) / 537280551687823 \) (175124579581091v^23 + 2097108754703610v^21 + 15884129489952885v^19 + 74560964536548413v^17 + 256488343213280073v^15 + 601876590039442389v^13 + 1031814571371361841v^11 + 1100015124218968839v^9 + 826993554706268265v^7 + 311833739962105900v^5 + 83172787430088090v^3 + 4073305215354123v) / 537280551687823
\(\beta_{22}\)	\(=\)	\( ( - 203339103335771 \nu^{23} + \cdots - 48\!\cdots\!34 \nu ) / 537280551687823 \) (-203339103335771v^23 - 2440778193327888v^21 - 18511216021585189v^19 - 87081639604138381v^17 - 300146321806039253v^15 - 706709584984124722v^13 - 1215837467199777933v^11 - 1306337324283469282v^9 - 987969347676036833v^7 - 379904119171075442v^5 - 99922300329807550v^3 - 4894202009350134v) / 537280551687823
\(\beta_{23}\)	\(=\)	\( ( - 78054196382 \nu^{23} - 940387560156 \nu^{21} - 7147013754503 \nu^{19} + \cdots - 1981240054344 \nu ) / 184569066193 \) (-78054196382v^23 - 940387560156v^21 - 7147013754503v^19 - 33737538937636v^17 - 116653903732687v^15 - 276145928842112v^13 - 477801749834130v^11 - 519546909528332v^9 - 396508892081131v^7 - 156293669355056v^5 - 40442447064530v^3 - 1981240054344v) / 184569066193

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{17} + 2\beta_{12} - 2 \) b17 + 2*b12 - 2
\(\nu^{3}\)	\(=\)	\( -\beta_{19} - 3\beta_{13} + \beta_{3} \) -b19 - 3*b13 + b3
\(\nu^{4}\)	\(=\)	\( -4\beta_{17} - 7\beta_{12} + \beta_{11} - \beta_{4} + 4\beta_{2} \) -4b17 - 7b12 + b11 - b4 + 4*b2
\(\nu^{5}\)	\(=\)	\( -\beta_{20} + 5\beta_{19} + 10\beta_{13} - \beta_{6} - 10\beta_1 \) -b20 + 5b19 + 10b13 - b6 - 10*b1
\(\nu^{6}\)	\(=\)	\( \beta_{8} - 6\beta_{5} - 15\beta_{2} + 20 \) b8 - 6b5 - 15b2 + 20
\(\nu^{7}\)	\(=\)	\( \beta_{10} + 7\beta_{6} - 21\beta_{3} + 35\beta_1 \) b10 + 7b6 - 21b3 + 35*b1
\(\nu^{8}\)	\(=\)	\( 56\beta_{17} - 8\beta_{15} + 98\beta_{12} - 29\beta_{11} - 8\beta_{8} - \beta_{7} + 29\beta_{5} + 28\beta_{4} - 69 \) 56b17 - 8b15 + 98b12 - 29b11 - 8b8 - b7 + 29b5 + 28*b4 - 69
\(\nu^{9}\)	\(=\)	\( -\beta_{22} - 9\beta_{21} + 36\beta_{20} - 84\beta_{19} + \beta_{16} - 126\beta_{13} - 9\beta_{10} + 84\beta_{3} \) -b22 - 9b21 + 36b20 - 84b19 + b16 - 126b13 - 9b10 + 84b3
\(\nu^{10}\)	\(=\)	\( -210\beta_{17} + 45\beta_{15} + \beta_{14} - 372\beta_{12} + 130\beta_{11} - \beta_{9} - 120\beta_{4} + 210\beta_{2} \) -210b17 + 45b15 + b14 - 372b12 + 130b11 - b9 - 120b4 + 210b2
\(\nu^{11}\)	\(=\)	\( -\beta_{23} + 12\beta_{22} + 56\beta_{21} - 164\beta_{20} + 329\beta_{19} + 463\beta_{13} - 164\beta_{6} - 463\beta_1 \) -b23 + 12b22 + 56b21 - 164b20 + 329b19 + 463b13 - 164b6 - 463*b1
\(\nu^{12}\)	\(=\)	\( 12\beta_{9} + 220\beta_{8} + 67\beta_{7} - 561\beta_{5} - 792\beta_{2} + 858 \) 12b9 + 220b8 + 67b7 - 561b5 - 792*b2 + 858
\(\nu^{13}\)	\(=\)	\( 12\beta_{18} - 79\beta_{16} + 287\beta_{10} + 714\beta_{6} - 1286\beta_{3} + 1717\beta_1 \) 12b18 - 79b16 + 287b10 + 714b6 - 1286b3 + 1717b1
\(\nu^{14}\)	\(=\)	\( 3003 \beta_{17} - 1001 \beta_{15} - 91 \beta_{14} + 5434 \beta_{12} - 2366 \beta_{11} - 1001 \beta_{8} + \cdots - 3068 \) 3003b17 - 1001b15 - 91b14 + 5434b12 - 2366b11 - 1001b8 - 378b7 + 2366b5 + 1988*b4 - 3068
\(\nu^{15}\)	\(=\)	\( 91 \beta_{23} - 560 \beta_{22} - 1470 \beta_{21} + 2898 \beta_{20} - 4900 \beta_{19} - 91 \beta_{18} + \cdots + 91 \beta_1 \) 91b23 - 560b22 - 1470b21 + 2898b20 - 4900b19 - 91b18 + 469b16 - 6540b13 - 1379b10 - 91b6 + 4991b3 + 91b1
\(\nu^{16}\)	\(=\)	\( - 11440 \beta_{17} + 4368 \beta_{15} + 560 \beta_{14} - 20878 \beta_{12} + 9828 \beta_{11} + \cdots + 11440 \beta_{2} \) -11440b17 + 4368b15 + 560b14 - 20878b12 + 9828b11 - 560b9 - 7889b4 + 11440b2
\(\nu^{17}\)	\(=\)	\( - 560 \beta_{23} + 3059 \beta_{22} + 6867 \beta_{21} - 11697 \beta_{20} + 18769 \beta_{19} + \cdots - 24989 \beta_1 \) -560b23 + 3059b22 + 6867b21 - 11697b20 + 18769b19 + 24989b13 - 11697b6 - 24989b1
\(\nu^{18}\)	\(=\)	\( 3059\beta_{9} + 18564\beta_{8} + 9366\beta_{7} - 40392\beta_{5} - 43758\beta_{2} + 40052 \) 3059b9 + 18564b8 + 9366b7 - 40392b5 - 43758*b2 + 40052
\(\nu^{19}\)	\(=\)	\( 3059\beta_{18} - 12425\beta_{16} + 27930\beta_{10} + 49590\beta_{6} - 74784\beta_{3} + 93176\beta_1 \) 3059b18 - 12425b16 + 27930b10 + 49590b6 - 74784b3 + 93176b1
\(\nu^{20}\)	\(=\)	\( 167960 \beta_{17} - 77520 \beta_{15} - 15484 \beta_{14} + 310726 \beta_{12} - 164729 \beta_{11} + \cdots - 145997 \) 167960b17 - 77520b15 - 15484b14 + 310726b12 - 164729b11 - 77520b8 - 43414b7 + 164729b5 + 121315*b4 - 145997
\(\nu^{21}\)	\(=\)	\( 15484 \beta_{23} - 74382 \beta_{22} - 136418 \beta_{21} + 183351 \beta_{20} - 273791 \beta_{19} + \cdots + 15484 \beta_1 \) 15484b23 - 74382b22 - 136418b21 + 183351b20 - 273791b19 - 15484b18 + 58898b16 - 372855b13 - 120934b10 - 15484b6 + 289275b3 + 15484b1
\(\nu^{22}\)	\(=\)	\( - 646646 \beta_{17} + 319769 \beta_{15} + 74382 \beta_{14} - 1202852 \beta_{12} + \cdots + 646646 \beta_{2} \) -646646b17 + 319769b15 + 74382b14 - 1202852b12 + 667942b11 - 74382b9 - 472626b4 + 646646b2
\(\nu^{23}\)	\(=\)	\( - 74382 \beta_{23} + 344080 \beta_{22} + 589467 \beta_{21} - 718013 \beta_{20} + 1044890 \beta_{19} + \cdots - 1451254 \beta_1 \) -74382b23 + 344080b22 + 589467b21 - 718013b20 + 1044890b19 + 1451254b13 - 718013b6 - 1451254b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 851.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{6} + T^{5} + 3 T^{4} + \cdots + 1)^{4} \) (T^6 + T^5 + 3T^4 + 5T^2 + 2*T + 1)^4
$3$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$5$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$7$	\( T^{24} \) T^24
$11$	\( T^{24} \) T^24
$13$	\( T^{24} \) T^24
$17$	\( T^{24} \) T^24
$19$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$23$	\( T^{24} \) T^24
$29$	\( (T^{6} - 7 T^{4} + 14 T^{2} - 7)^{4} \) (T^6 - 7T^4 + 14T^2 - 7)^4
$31$	\( T^{24} \) T^24
$37$	\( (T^{12} + 7 T^{10} + \cdots + 49)^{2} \) (T^12 + 7T^10 + 35T^8 + 84T^6 + 147T^4 + 98*T^2 + 49)^2
$41$	\( T^{24} \) T^24
$43$	\( (T^{3} - T^{2} - 2 T + 1)^{8} \) (T^3 - T^2 - 2*T + 1)^8
$47$	\( T^{24} \) T^24
$53$	\( T^{24} \) T^24
$59$	\( T^{24} \) T^24
$61$	\( T^{24} \) T^24
$67$	\( T^{24} \) T^24
$71$	\( (T + 1)^{24} \) (T + 1)^24
$73$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$79$	\( (T^{12} + 7 T^{10} + \cdots + 49)^{2} \) (T^12 + 7T^10 + 35T^8 + 84T^6 + 147T^4 + 98*T^2 + 49)^2
$83$	\( (T^{12} - 12 T^{10} + \cdots + 1)^{2} \) (T^12 - 12T^10 + 53T^8 - 104T^6 + 86T^4 - 24*T^2 + 1)^2
$89$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 + 91T^20 + 428T^18 + 1475T^16 + 3472T^14 + 5972T^12 + 6412T^10 + 4847T^8 + 1856T^6 + 490T^4 + 24*T^2 + 1
$97$	\( T^{24} \) T^24
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\(n\)	\(640\)	\(1569\)
\(\chi(n)\)	\(-1 + \beta_{12}\)	\(-1\)