LMFDB - Newform orbit 2240.2.k.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1791,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.k (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:	$17.8864900528$
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} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} + \cdots + 5956 $ x^16 - 4x^15 - 6x^14 + 68x^13 - 126x^12 - 148x^11 + 1006x^10 - 1516x^9 - 179x^8 + 3992x^7 - 5596x^6 - 488x^5 + 16080x^4 - 33776x^3 + 37344x^2 - 22336*x + 5956
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 1120)
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_{9} q^{3} + \beta_{4} q^{5} - \beta_{11} q^{7} + (\beta_{9} - \beta_{8} - \beta_1 + 1) q^{9}+O(q^{10}) $ q + b9 * q^3 + b4 * q^5 - b11 * q^7 + (b9 - b8 - b1 + 1) * q^9	$ q + \beta_{9} q^{3} + \beta_{4} q^{5} - \beta_{11} q^{7} + (\beta_{9} - \beta_{8} - \beta_1 + 1) q^{9} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{4}) q^{99}+O(q^{100}) $ q + b9 * q^3 + b4 * q^5 - b11 * q^7 + (b9 - b8 - b1 + 1) * q^9 + (-b14 + b11 - b7 + b3 - b2) * q^11 + (-b7 + b3) * q^13 + b7 * q^15 + (b14 - b13 - b11 + b7 + b6 - b3) * q^17 + (-b10 + b8 + b6) * q^19 + (-b12 + b11 - b9 - b6 - b4 + b3) * q^21 + (b13 - b6 + b3 + b2) * q^23 - q^25 + (-b15 + b12 + b9 + b6 + b4 + 2) * q^27 + (-b15 + b14 + b12 + b11 - b9 + b8 + b6 + b4 + b1 - 2) * q^29 + (-b14 + b13 - b11 + b10 + 2b9 - b8 - 2b1) * q^31 + (2b14 - 2b11 + b7 - 2b4 - b3 + 2b2) * q^33 - b13 * q^35 + (b14 + b13 + b11 + b10 + 2b9 + b8 + b1 - 2) q^37 + (-b15 - b12 - b7 - b6 - 3b4 + b3 + b2) q^39 + (-2b7 + b5 - 2b4 - b3 - b2) * q^41 + (-b15 - 2b13 - b12 + 2b7 + b6 - 2b5 + b4 - b3 - b2) q^43 + (b7 - b5 + b4 - b2) * q^45 + (-b15 - b14 + b12 - b11 + 2b10 - b9 - b6 + b4 + 2b1 - 2) * q^47 + (b14 - b13 + b12 - 2b9 - b8 - b5 + b4 - b2 + b1 - 1) q^49 + (-b15 - 2b14 - b12 + 2b11 - b7 - b6 + b4 + b3 - b2) * q^51 + (b15 + 2b14 + b13 - b12 + 2b11 - 2b10 + 2b8 + 2b6 - b4) q^53 + (b13 + b10 + b9 + b8) * q^55 + (b15 + b14 - b12 + b11 - b10 - 2b9 - b8 - b4 - b1 + 2) q^57 + (b14 - b13 + b11 - 2b10 - 2b8 + b6) * q^59 + (b15 + 3b14 + 2b13 + b12 - 3b11 + 2b7 - b6 + b5 + b4 - b3 + b2) * q^61 + (-b15 - b12 + b11 - b9 + 2b8 + b6 + b4 + b3 + b2 + 2b1 - 4) * q^63 + (b10 + b9 - b6) * q^65 + (b15 + b14 + b13 + b12 - b11 + 2b7 - b4) q^67 + (-b5 + 2b4 + b3 - b2) q^69 + (b15 + b14 + 2b13 + b12 - b11 - b6 + 4b5 - 5b4 - 2b3 + 2b2) q^71 + (b14 + b13 - b11 - 2b7 - b6 + 2b5 - 2b4) q^73 - b9 * q^75 + (b15 + 3b13 + b12 + b10 + b8 - b7 + b6 + 2b5 + b4 + b3 - b1) * q^77 + (-b14 + b11 - b7 - b3 - b2) * q^79 + (-b15 + b14 - 2b13 + b12 + b11 + 2b9 - b6 + b4 + 1) * q^81 + (b15 - 2b13 - b12 - 2b10 - 2b9 - 2b8 - b6 - b4 + 6) * q^83 + (b14 - b13 + b11 - b10 - b9) * q^85 + (-b15 - 2b13 + b12 - b9 - 2b8 - b6 + b4 - 2b1 - 2) q^87 + (-b15 + b14 - 2b13 - b12 - b11 + b6 - 3b5 + b4 - b3 - b2) * q^89 + (b15 + b14 + b13 + b12 + b11 - b10 - 2b9 + b8 - b7 + b6 + 3b4 + b3 - b2 - 4) * q^91 + (-b15 + b14 - 2b13 + b12 + b11 - b10 + 2b9 - b8 + b4 + b1 + 2) * q^93 + (b3 + b2) * q^95 + (b14 - b13 - b11 + b7 + b6 - 2b5 + 4b4 + b3 - 2b2) q^97 + (-b15 - b14 - b12 + b11 - 2b7 - b6 + b4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^7 + 16 * q^9	$ 16 q - 4 q^{7} + 16 q^{9} - 8 q^{19} - 4 q^{21} - 16 q^{25} + 48 q^{27} - 8 q^{29} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{53} + 8 q^{55} + 16 q^{57} - 8 q^{59} - 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} + 64 q^{83} - 16 q^{87} - 64 q^{91} + 48 q^{93}+O(q^{100}) $ 16 * q - 4 * q^7 + 16 * q^9 - 8 * q^19 - 4 * q^21 - 16 * q^25 + 48 * q^27 - 8 * q^29 - 16 * q^37 - 8 * q^47 - 4 * q^49 - 16 * q^53 + 8 * q^55 + 16 * q^57 - 8 * q^59 - 60 * q^63 + 8 * q^65 + 8 * q^77 + 40 * q^81 + 64 * q^83 - 16 * q^87 - 64 * q^91 + 48 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} + \cdots + 5956 $ x^16 - 4*x^15 - 6*x^14 + 68*x^13 - 126*x^12 - 148*x^11 + 1006*x^10 - 1516*x^9 - 179*x^8 + 3992*x^7 - 5596*x^6 - 488*x^5 + 16080*x^4 - 33776*x^3 + 37344*x^2 - 22336*x + 5956 :

$\beta_{1}$	$=$	$ ( - 12758377455 \nu^{15} - 17029973070 \nu^{14} + 255451555872 \nu^{13} + \cdots + 634026321716742 ) / 216025654874206 $ (-12758377455v^15 - 17029973070v^14 + 255451555872v^13 - 5782883602v^12 - 2428092530782v^11 + 2461866146935v^10 + 10812644798656v^9 - 20882203036101v^8 - 17931607412769v^7 + 69829732643649v^6 - 13633178133740v^5 - 109539021238893v^4 + 106762977907130v^3 + 7255442894980v^2 - 55212568945784*v + 634026321716742) / 216025654874206
$\beta_{2}$	$=$	$ ( 87\!\cdots\!74 \nu^{15} + \cdots - 34\!\cdots\!36 ) / 64\!\cdots\!44 $ (8765614024466674v^15 - 62502845947521146v^14 + 27930788945903613v^13 + 919541822694668216v^12 - 2758329551827048716v^11 - 505933606394946956v^10 + 16874220098891849632v^9 - 31541580432531125900v^8 + 2468356936003309764v^7 + 71827760037341267022v^6 - 102679857907601843147v^5 + 8812409656846222588v^4 + 240534886957138604770v^3 - 612154820371672153200v^2 + 735676972740082807394*v - 348289662532115458336) / 64532047726642313144
$\beta_{3}$	$=$	$ ( 13\!\cdots\!05 \nu^{15} + \cdots + 14\!\cdots\!08 ) / 64\!\cdots\!44 $ (13090040637728505v^15 + 211916728286895534v^14 - 725493091669104857v^13 - 1961008466169258786v^12 + 12493927886363852188v^11 - 10299818010203611792v^10 - 52349917243107468542v^9 + 149934458133588569104v^8 - 74189346620384657089v^7 - 263650896553140225122v^6 + 546458846557001737209v^5 - 205329180934372231738v^4 - 836795779768439795796v^3 + 2402362524933101244484v^2 - 2957390654429469965602*v + 1427468805432647561708) / 64532047726642313144
$\beta_{4}$	$=$	$ ( - 1210752049 \nu^{15} - 1110096336 \nu^{14} + 20403779147 \nu^{13} - 19050599618 \nu^{12} + \cdots - 17489786211320 ) / 4256018862188 $ (-1210752049v^15 - 1110096336v^14 + 20403779147v^13 - 19050599618v^12 - 146738265037v^11 + 351660968602v^10 + 157695428815v^9 - 1793479248628v^8 + 1916781072920v^7 + 1497184388698v^6 - 6855726785840v^5 + 5953850442270v^4 + 1659909458620v^3 - 20164201672372v^2 + 32148639488212*v - 17489786211320) / 4256018862188
$\beta_{5}$	$=$	$ ( - 3343558 \nu^{15} - 1238164 \nu^{14} + 52258753 \nu^{13} - 74255495 \nu^{12} + \cdots - 42814645170 ) / 3918178202 $ (-3343558v^15 - 1238164v^14 + 52258753v^13 - 74255495v^12 - 312877523v^11 + 939062890v^10 - 23564043v^9 - 3862442118v^8 + 4919305197v^7 + 1699089560v^6 - 14234786412v^5 + 16165231231v^4 - 5856993922v^3 - 37472206390v^2 + 73775229484*v - 42814645170) / 3918178202
$\beta_{6}$	$=$	$ ( - 12\!\cdots\!73 \nu^{15} + \cdots + 33\!\cdots\!40 ) / 12\!\cdots\!88 $ (-120223170196860373v^15 + 508409725558160580v^14 + 1001904479712438592v^13 - 9006400866078907244v^12 + 12390972778503174636v^11 + 31494971175260942572v^10 - 125744891278723942282v^9 + 118832891793713150884v^8 + 160017238475171890871v^7 - 495008163163051562664v^6 + 391831367090758893822v^5 + 538294739675650401264v^4 - 2039725726045424389930v^3 + 3212935115563632993032v^2 - 2117068501808309500776*v + 331964725632513472440) / 129064095453284626288
$\beta_{7}$	$=$	$ ( 12\!\cdots\!95 \nu^{15} + \cdots - 90\!\cdots\!04 ) / 12\!\cdots\!88 $ (126425665182626895v^15 - 402469074925196481v^14 - 1121956011275831728v^13 + 7733154061744690654v^12 - 9125291680193799000v^11 - 27685781977964858108v^10 + 103545419009722591998v^9 - 96472901443840712450v^8 - 114632157105709674301v^7 + 394060154218144271303v^6 - 329971656220426141850v^5 - 366229213299615695132v^4 + 1676643046921451745682v^3 - 2720002409034269825638v^2 + 2374127347238370132368*v - 908825391783385858004) / 129064095453284626288
$\beta_{8}$	$=$	$ ( 82\!\cdots\!25 \nu^{15} + \cdots - 39\!\cdots\!80 ) / 64\!\cdots\!44 $ (82713379168634325v^15 - 281075459079638160v^14 - 713483502120422931v^13 + 5336541030249633080v^12 - 6695141944048570864v^11 - 19147893177766628976v^10 + 73841771752378564118v^9 - 68228822294103924196v^8 - 91976769270153237685v^7 + 294773493670649913012v^6 - 220270109894291825885v^5 - 310073024820485450084v^4 + 1201588731759913835968v^3 - 1840543284179116451012v^2 + 1393734925792015031754*v - 394988529315488072880) / 64532047726642313144
$\beta_{9}$	$=$	$ ( 23\!\cdots\!35 \nu^{15} + \cdots - 10\!\cdots\!60 ) / 12\!\cdots\!88 $ (237140810440379735v^15 - 694839951210633181v^14 - 2223510643919990830v^13 + 13842456444243259968v^12 - 14492654803433050080v^11 - 53007947937536874348v^10 + 183385760798826199310v^9 - 153337931243226373826v^8 - 237649344704454034305v^7 + 719125568618894204991v^6 - 517634899815047395024v^5 - 821736504755162748938v^4 + 3021381137616653188334v^3 - 4556629515839990044858v^2 + 3426054626530645624092*v - 1020322781709869753760) / 129064095453284626288
$\beta_{10}$	$=$	$ ( 25\!\cdots\!15 \nu^{15} + \cdots - 13\!\cdots\!72 ) / 12\!\cdots\!88 $ (259293226844765315v^15 - 770370709610531096v^14 - 2305263887028898338v^13 + 15278820432296986152v^12 - 17498219145135478052v^11 - 55549252514026466004v^10 + 205855473162409076806v^9 - 189152648431021147452v^8 - 234485299461395158061v^7 + 807831922352873380748v^6 - 649630715069165603172v^5 - 728970739234563805676v^4 + 3399968126198016637530v^3 - 5424839780231186441632v^2 + 4533424520138778259396*v - 1310279850884661310672) / 129064095453284626288
$\beta_{11}$	$=$	$ ( - 40\!\cdots\!67 \nu^{15} + \cdots + 95\!\cdots\!72 ) / 12\!\cdots\!88 $ (-401514885070314967v^15 + 1118370145453479424v^14 + 3999070438841951874v^13 - 22929190127933666378v^12 + 20193867782511255404v^11 + 95284425048363017136v^10 - 293551487608489295942v^9 + 197696087447360859392v^8 + 449419601267458200241v^7 - 1107409032339395042580v^6 + 632064008668323899948v^5 + 1485218623630154093798v^4 - 4785700310140283855202v^3 + 6761685223629213815172v^2 - 4307250443939268057452*v + 958797510564913058372) / 129064095453284626288
$\beta_{12}$	$=$	$ ( - 20\!\cdots\!96 \nu^{15} + \cdots + 16\!\cdots\!32 ) / 64\!\cdots\!44 $ (-203138371139447396v^15 + 707031214890825801v^14 + 1605940222342174914v^13 - 12971112886976321418v^12 + 18772666473353955162v^11 + 40117807089961648770v^10 - 184898054661417274554v^9 + 212344977649881363120v^8 + 158314411051918264382v^7 - 748686664357395711185v^6 + 753019705595276023700v^5 + 546154222509972678626v^4 - 3060392513637979096840v^3 + 5296744972024004608390v^2 - 4644440188818386285000*v + 1678744867755471500432) / 64532047726642313144
$\beta_{13}$	$=$	$ ( - 62\!\cdots\!45 \nu^{15} + \cdots + 32\!\cdots\!36 ) / 12\!\cdots\!88 $ (-622859702467771945v^15 + 1867679735746360758v^14 + 5650730542224720896v^13 - 36854430938284936774v^12 + 41116165628762327132v^11 + 136242982429528198460v^10 - 492786652970957868114v^9 + 438963481118088233728v^8 + 586766801271558866667v^7 - 1921017645189911857042v^6 + 1497237271104639140046v^5 + 1916936457289342249746v^4 - 8097992310736121571546v^3 + 12701471705105446312128v^2 - 10075478303001224763280*v + 3239428024377916922436) / 129064095453284626288
$\beta_{14}$	$=$	$ ( - 62\!\cdots\!81 \nu^{15} + \cdots + 41\!\cdots\!20 ) / 12\!\cdots\!88 $ (-625676353274793981v^15 + 2046731257544845448v^14 + 5284503975155719586v^13 - 38894072028060292708v^12 + 50274671850866341900v^11 + 132799764376717123880v^10 - 538509129785875372338v^9 + 543808656389170520872v^8 + 561969906032701808571v^7 - 2136373410007872469824v^6 + 1862585432143242059608v^5 + 1900518047840479370596v^4 - 8817764898518887072334v^3 + 14421858692828249207256v^2 - 11871046424796416720100*v + 4139600485325221870320) / 129064095453284626288
$\beta_{15}$	$=$	$ ( 77\!\cdots\!49 \nu^{15} + \cdots - 31\!\cdots\!96 ) / 12\!\cdots\!88 $ (773830196453774649v^15 - 2257663355263442372v^14 - 7170009218278348232v^13 + 45269222028181278742v^12 - 47961230184040499388v^11 - 173776796765668563960v^10 + 601569813669377153970v^9 - 494725854442072439064v^8 - 777850522470759953827v^7 + 2319785730056291618072v^6 - 1659946859124546953910v^5 - 2510620628878174669634v^4 + 9819322295097294277378v^3 - 15062240195278282876764v^2 + 11281234388048318993104*v - 3185578392552200053596) / 129064095453284626288

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

$\nu$	$=$	$ ( \beta_{13} + \beta_{10} + \beta_{9} + \beta_{7} ) / 2 $ (b13 + b10 + b9 + b7) / 2
$\nu^{2}$	$=$	$ ( - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} + \cdots + 4 ) / 2 $ (-b14 + b11 - b10 - b9 + 2b8 + b7 + b6 + 2b4 + b3 - b2 + 4) / 2
$\nu^{3}$	$=$	$ ( - 3 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} - \beta_{12} + 2 \beta_{11} + 11 \beta_{10} - 5 \beta_{8} + \cdots - 22 ) / 4 $ (-3b15 - 8b14 + 9b13 - b12 + 2b11 + 11b10 - 5b8 + 4b7 - 5b6 - 5b4 + 3b3 - 3b2 + 3b1 - 22) / 4
$\nu^{4}$	$=$	$ ( \beta_{15} - 5 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 12 \beta_{10} - 11 \beta_{9} + \cdots + 22 ) / 2 $ (b15 - 5b14 - 7b13 - 3b12 + 13b11 - 12b10 - 11b9 + 13b8 - 5b7 + 4b5 - 5b4 + 8b3 - 2b2 - 3*b1 + 22) / 2
$\nu^{5}$	$=$	$ ( - 21 \beta_{15} - 46 \beta_{14} + 37 \beta_{13} + 13 \beta_{12} + 75 \beta_{10} + 24 \beta_{9} + \cdots - 182 ) / 4 $ (-21b15 - 46b14 + 37b13 + 13b12 + 75b10 + 24b9 - 79b8 - 8b7 - 39b6 + 22b5 - 75b4 + 7b3 - 13b2 + 31b1 - 182) / 4
$\nu^{6}$	$=$	$ ( 15 \beta_{15} - 82 \beta_{13} - 19 \beta_{12} + 98 \beta_{11} - 130 \beta_{10} - 43 \beta_{9} + \cdots + 214 ) / 2 $ (15b15 - 82b13 - 19b12 + 98b11 - 130b10 - 43b9 + 100b8 - 31b7 + 3b6 + 14b5 - 21b4 + 40b3 + 7b2 - 25b1 + 214) / 2
$\nu^{7}$	$=$	$ ( - 81 \beta_{15} - 144 \beta_{14} + 263 \beta_{13} + 151 \beta_{12} - 174 \beta_{11} + 581 \beta_{10} + \cdots - 1544 ) / 4 $ (-81b15 - 144b14 + 263b13 + 151b12 - 174b11 + 581b10 + 320b9 - 749b8 + 48b7 - 215b6 + 81b5 - 283b4 - 109b3 - 177b2 + 286*b1 - 1544) / 4
$\nu^{8}$	$=$	$ ( 179 \beta_{15} + 231 \beta_{14} - 757 \beta_{13} - 225 \beta_{12} + 653 \beta_{11} - 1232 \beta_{10} + \cdots + 2002 ) / 2 $ (179b15 + 231b14 - 757b13 - 225b12 + 653b11 - 1232b10 - 463b9 + 907b8 - 81b7 + 124b6 - 74b5 + 281b4 + 252b3 + 208b2 - 205*b1 + 2002) / 2
$\nu^{9}$	$=$	$ ( - 553 \beta_{15} + 46 \beta_{14} + 2417 \beta_{13} + 1181 \beta_{12} - 3100 \beta_{11} + 5499 \beta_{10} + \cdots - 14066 ) / 4 $ (-553b15 + 46b14 + 2417b13 + 1181b12 - 3100b11 + 5499b10 + 2580b9 - 6271b8 + 916b7 - 1191b6 + 316b5 - 1839b4 - 1881b3 - 1529b2 + 2449*b1 - 14066) / 4
$\nu^{10}$	$=$	$ ( 1161 \beta_{15} + 2128 \beta_{14} - 6430 \beta_{13} - 2141 \beta_{12} + 4718 \beta_{11} - 10312 \beta_{10} + \cdots + 18554 ) / 2 $ (1161b15 + 2128b14 - 6430b13 - 2141b12 + 4718b11 - 10312b10 - 4485b9 + 8492b8 - 1041b7 + 1531b6 - 1270b5 + 4613b4 + 2350b3 + 2981b2 - 2169*b1 + 18554) / 2
$\nu^{11}$	$=$	$ ( - 5851 \beta_{15} + 3292 \beta_{14} + 25033 \beta_{13} + 11801 \beta_{12} - 35758 \beta_{11} + \cdots - 121468 ) / 4 $ (-5851b15 + 3292b14 + 25033b13 + 11801b12 - 35758b11 + 54695b10 + 24584b9 - 53811b8 + 8872b7 - 7513b6 + 2785b5 - 21121b4 - 22107b3 - 15347b2 + 18288*b1 - 121468) / 4
$\nu^{12}$	$=$	$ ( 8415 \beta_{15} + 11375 \beta_{14} - 49763 \beta_{13} - 17893 \beta_{12} + 41993 \beta_{11} + \cdots + 174470 ) / 2 $ (8415b15 + 11375b14 - 49763b13 - 17893b12 + 41993b11 - 85620b10 - 38113b9 + 77845b8 - 12523b7 + 14390b6 - 12908b5 + 53341b4 + 25804b3 + 31616b2 - 23965*b1 + 174470) / 2
$\nu^{13}$	$=$	$ ( - 47307 \beta_{15} + 26662 \beta_{14} + 261031 \beta_{13} + 116311 \beta_{12} - 346640 \beta_{11} + \cdots - 1043830 ) / 4 $ (-47307b15 + 26662b14 + 261031b13 + 116311b12 - 346640b11 + 514361b10 + 228868b9 - 466245b8 + 95516b7 - 50433b6 + 39340b5 - 263453b4 - 226323b3 - 173323b2 + 141931*b1 - 1043830) / 4
$\nu^{14}$	$=$	$ ( 69001 \beta_{15} + 17414 \beta_{14} - 384702 \beta_{13} - 164929 \beta_{12} + 427592 \beta_{11} + \cdots + 1588582 ) / 2 $ (69001b15 + 17414b14 - 384702b13 - 164929b12 + 427592b11 - 738564b10 - 333975b9 + 706460b8 - 137783b7 + 115051b6 - 114152b5 + 544801b4 + 298730b3 + 313623b2 - 231165*b1 + 1588582) / 2
$\nu^{15}$	$=$	$ ( - 373861 \beta_{15} + 223888 \beta_{14} + 2459443 \beta_{13} + 1059647 \beta_{12} - 3146942 \beta_{11} + \cdots - 9147324 ) / 4 $ (-373861b15 + 223888b14 + 2459443b13 + 1059647b12 - 3146942b11 + 4620993b10 + 2034216b9 - 4088325b8 + 956432b7 - 388587b6 + 541235b5 - 3161903b4 - 2184813b3 - 1894373b2 + 1209820*b1 - 9147324) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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} $

1791.1

−0.00830917	+	1.38001i
−0.00830917	−	1.38001i
2.23037	+	0.887196i
2.23037	−	0.887196i
−1.65334	+	0.722492i
−1.65334	−	0.722492i
1.23885	+	0.371381i
1.23885	−	0.371381i
−3.01485	+	0.0876273i
−3.01485	−	0.0876273i
0.715495	−	0.510550i
0.715495	+	0.510550i
1.40610	−	1.29043i
1.40610	+	1.29043i
1.08568	−	1.64772i
1.08568	+	1.64772i

−2.76002

−

1.00000i

−2.62068

0.363381i

4.61769

1791.2

−2.76002

1.00000i

−2.62068

−

0.363381i

4.61769

1791.3

−1.77439

−

1.00000i

0.829909

−

2.51222i

0.148464

1791.4

−1.77439

1.00000i

0.829909

2.51222i

0.148464

1791.5

−1.44498

−

1.00000i

2.48080

−

0.919594i

−0.912023

1791.6

−1.44498

1.00000i

2.48080

0.919594i

−0.912023

1791.7

−0.742762

−

1.00000i

−2.53467

−

0.758567i

−2.44831

1791.8

−0.742762

1.00000i

−2.53467

0.758567i

−2.44831

1791.9

−0.175255

−

1.00000i

0.684520

2.55567i

−2.96929

1791.10

−0.175255

1.00000i

0.684520

−

2.55567i

−2.96929

1791.11

1.02110

−

1.00000i

1.49445

2.18326i

−1.95735

1791.12

1.02110

1.00000i

1.49445

−

2.18326i

−1.95735

1791.13

2.58086

−

1.00000i

−0.319247

−

2.62642i

3.66083

1791.14

2.58086

1.00000i

−0.319247

2.62642i

3.66083

1791.15

3.29545

−

1.00000i

−2.01507

1.71449i

7.85998

1791.16

3.29545

1.00000i

−2.01507

−

1.71449i

7.85998

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2240.2.k.f		16
4.b	odd	2	1		2240.2.k.g		16
7.b	odd	2	1		2240.2.k.g		16
8.b	even	2	1		1120.2.k.a	✓	16
8.d	odd	2	1		1120.2.k.b	yes	16
28.d	even	2	1	inner	2240.2.k.f		16
56.e	even	2	1		1120.2.k.a	✓	16
56.h	odd	2	1		1120.2.k.b	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1120.2.k.a	✓	16	8.b	even	2	1
1120.2.k.a	✓	16	56.e	even	2	1
1120.2.k.b	yes	16	8.d	odd	2	1
1120.2.k.b	yes	16	56.h	odd	2	1
2240.2.k.f		16	1.a	even	1	1	trivial
2240.2.k.f		16	28.d	even	2	1	inner
2240.2.k.g		16	4.b	odd	2	1
2240.2.k.g		16	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2240, [\chi])$:

$ T_{3}^{8} - 16T_{3}^{6} - 8T_{3}^{5} + 69T_{3}^{4} + 64T_{3}^{3} - 50T_{3}^{2} - 56T_{3} - 8 $

$ T_{19}^{8} + 4T_{19}^{7} - 60T_{19}^{6} - 256T_{19}^{5} + 384T_{19}^{4} + 1792T_{19}^{3} - 1152T_{19}^{2} - 3072T_{19} + 2048 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 16 T^{6} - 8 T^{5} + \cdots - 8)^{2} $ (T^8 - 16T^6 - 8T^5 + 69T^4 + 64T^3 - 50T^2 - 56T - 8)^2
$5$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$7$	$ T^{16} + 4 T^{15} + \cdots + 5764801 $ T^16 + 4T^15 + 10T^14 + 52T^13 + 128T^12 + 164T^11 + 598T^10 + 1076T^9 + 190T^8 + 7532T^7 + 29302T^6 + 56252T^5 + 307328T^4 + 873964T^3 + 1176490T^2 + 3294172*T + 5764801
$11$	$ T^{16} + 100 T^{14} + \cdots + 4096 $ T^16 + 100T^14 + 3542T^12 + 54956T^10 + 400289T^8 + 1334888T^6 + 1692944T^4 + 458240*T^2 + 4096
$13$	$ T^{16} + 100 T^{14} + \cdots + 20736 $ T^16 + 100T^14 + 3574T^12 + 53268T^10 + 289329T^8 + 539664T^6 + 438880T^4 + 158976*T^2 + 20736
$17$	$ T^{16} + \cdots + 131974144 $ T^16 + 124T^14 + 6134T^12 + 155700T^10 + 2177601T^8 + 16930648T^6 + 71941840T^4 + 155093760*T^2 + 131974144
$19$	$ (T^{8} + 4 T^{7} + \cdots + 2048)^{2} $ (T^8 + 4T^7 - 60T^6 - 256T^5 + 384T^4 + 1792T^3 - 1152T^2 - 3072*T + 2048)^2
$23$	$ T^{16} + 116 T^{14} + \cdots + 4734976 $ T^16 + 116T^14 + 4948T^12 + 98400T^10 + 995264T^8 + 5132288T^6 + 12598272T^4 + 13099008*T^2 + 4734976
$29$	$ (T^{8} + 4 T^{7} + \cdots + 272)^{2} $ (T^8 + 4T^7 - 138T^6 - 436T^5 + 4617T^4 + 9528T^3 - 30504T^2 - 4192*T + 272)^2
$31$	$ (T^{8} - 128 T^{6} + \cdots + 32768)^{2} $ (T^8 - 128T^6 - 32T^5 + 4320T^4 + 3584T^3 - 26368T^2 - 8192T + 32768)^2
$37$	$ (T^{8} + 8 T^{7} + \cdots + 122624)^{2} $ (T^8 + 8T^7 - 112T^6 - 640T^5 + 3840T^4 + 12672T^3 - 50944T^2 - 46080*T + 122624)^2
$41$	$ T^{16} + \cdots + 53557067776 $ T^16 + 344T^14 + 44048T^12 + 2714368T^10 + 85502976T^8 + 1357856768T^6 + 10589323264T^4 + 38860750848*T^2 + 53557067776
$43$	$ T^{16} + \cdots + 122466402304 $ T^16 + 340T^14 + 47156T^12 + 3445600T^10 + 142480448T^8 + 3304139776T^6 + 39227764736T^4 + 183466065920*T^2 + 122466402304
$47$	$ (T^{8} + 4 T^{7} + \cdots - 231568)^{2} $ (T^8 + 4T^7 - 204T^6 - 224T^5 + 13581T^4 - 14468T^3 - 287030T^2 + 766912*T - 231568)^2
$53$	$ (T^{8} + 8 T^{7} + \cdots - 665856)^{2} $ (T^8 + 8T^7 - 280T^6 - 2176T^5 + 20480T^4 + 129664T^3 - 464000T^2 - 1459200*T - 665856)^2
$59$	$ (T^{8} + 4 T^{7} + \cdots - 672768)^{2} $ (T^8 + 4T^7 - 268T^6 - 448T^5 + 22832T^4 - 21184T^3 - 637760T^2 + 1970688*T - 672768)^2
$61$	$ T^{16} + \cdots + 647532301582336 $ T^16 + 784T^14 + 256384T^12 + 45476608T^10 + 4750933504T^8 + 294979448832T^6 + 10287941124096T^4 + 168587098128384*T^2 + 647532301582336
$67$	$ T^{16} + 436 T^{14} + \cdots + 16384 $ T^16 + 436T^14 + 57492T^12 + 2491232T^10 + 21440192T^8 + 21862400T^6 + 7105536T^4 + 647168*T^2 + 16384
$71$	$ T^{16} + \cdots + 22419997720576 $ T^16 + 888T^14 + 307696T^12 + 53264640T^10 + 4963401472T^8 + 252540254208T^6 + 6607642955776T^4 + 70103100030976*T^2 + 22419997720576
$73$	$ T^{16} + \cdots + 113154195456 $ T^16 + 584T^14 + 124848T^12 + 11843584T^10 + 492927744T^8 + 9381095424T^6 + 79220592640T^4 + 245725986816*T^2 + 113154195456
$79$	$ T^{16} + \cdots + 1196606464 $ T^16 + 292T^14 + 30278T^12 + 1353180T^10 + 26891185T^8 + 249816632T^6 + 1075124688T^4 + 1999695104*T^2 + 1196606464
$83$	$ (T^{8} - 32 T^{7} + \cdots - 12271744)^{2} $ (T^8 - 32T^7 + 102T^6 + 5576T^5 - 50056T^4 - 141504T^3 + 2571168T^2 - 4208512*T - 12271744)^2
$89$	$ T^{16} + \cdots + 718886928384 $ T^16 + 704T^14 + 180256T^12 + 21196800T^10 + 1238941952T^8 + 36256497664T^6 + 510216503296T^4 + 2794048192512*T^2 + 718886928384
$97$	$ T^{16} + \cdots + 335984794092544 $ T^16 + 732T^14 + 210582T^12 + 31426644T^10 + 2676287009T^8 + 133405156696T^6 + 3798419954896T^4 + 56407587773184*T^2 + 335984794092544
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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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{9} q^{3} + \beta_{4} q^{5} - \beta_{11} q^{7} + (\beta_{9} - \beta_{8} - \beta_1 + 1) q^{9}+O(q^{10}) \) q + b9 * q^3 + b4 * q^5 - b11 * q^7 + (b9 - b8 - b1 + 1) * q^9	\( q + \beta_{9} q^{3} + \beta_{4} q^{5} - \beta_{11} q^{7} + (\beta_{9} - \beta_{8} - \beta_1 + 1) q^{9} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{4}) q^{99}+O(q^{100}) \) q + b9 * q^3 + b4 * q^5 - b11 * q^7 + (b9 - b8 - b1 + 1) * q^9 + (-b14 + b11 - b7 + b3 - b2) * q^11 + (-b7 + b3) * q^13 + b7 * q^15 + (b14 - b13 - b11 + b7 + b6 - b3) * q^17 + (-b10 + b8 + b6) * q^19 + (-b12 + b11 - b9 - b6 - b4 + b3) * q^21 + (b13 - b6 + b3 + b2) * q^23 - q^25 + (-b15 + b12 + b9 + b6 + b4 + 2) * q^27 + (-b15 + b14 + b12 + b11 - b9 + b8 + b6 + b4 + b1 - 2) * q^29 + (-b14 + b13 - b11 + b10 + 2b9 - b8 - 2b1) * q^31 + (2b14 - 2b11 + b7 - 2b4 - b3 + 2b2) * q^33 - b13 * q^35 + (b14 + b13 + b11 + b10 + 2b9 + b8 + b1 - 2) q^37 + (-b15 - b12 - b7 - b6 - 3b4 + b3 + b2) q^39 + (-2b7 + b5 - 2b4 - b3 - b2) * q^41 + (-b15 - 2b13 - b12 + 2b7 + b6 - 2b5 + b4 - b3 - b2) q^43 + (b7 - b5 + b4 - b2) * q^45 + (-b15 - b14 + b12 - b11 + 2b10 - b9 - b6 + b4 + 2b1 - 2) * q^47 + (b14 - b13 + b12 - 2b9 - b8 - b5 + b4 - b2 + b1 - 1) q^49 + (-b15 - 2b14 - b12 + 2b11 - b7 - b6 + b4 + b3 - b2) * q^51 + (b15 + 2b14 + b13 - b12 + 2b11 - 2b10 + 2b8 + 2b6 - b4) q^53 + (b13 + b10 + b9 + b8) * q^55 + (b15 + b14 - b12 + b11 - b10 - 2b9 - b8 - b4 - b1 + 2) q^57 + (b14 - b13 + b11 - 2b10 - 2b8 + b6) * q^59 + (b15 + 3b14 + 2b13 + b12 - 3b11 + 2b7 - b6 + b5 + b4 - b3 + b2) * q^61 + (-b15 - b12 + b11 - b9 + 2b8 + b6 + b4 + b3 + b2 + 2b1 - 4) * q^63 + (b10 + b9 - b6) * q^65 + (b15 + b14 + b13 + b12 - b11 + 2b7 - b4) q^67 + (-b5 + 2b4 + b3 - b2) q^69 + (b15 + b14 + 2b13 + b12 - b11 - b6 + 4b5 - 5b4 - 2b3 + 2b2) q^71 + (b14 + b13 - b11 - 2b7 - b6 + 2b5 - 2b4) q^73 - b9 * q^75 + (b15 + 3b13 + b12 + b10 + b8 - b7 + b6 + 2b5 + b4 + b3 - b1) * q^77 + (-b14 + b11 - b7 - b3 - b2) * q^79 + (-b15 + b14 - 2b13 + b12 + b11 + 2b9 - b6 + b4 + 1) * q^81 + (b15 - 2b13 - b12 - 2b10 - 2b9 - 2b8 - b6 - b4 + 6) * q^83 + (b14 - b13 + b11 - b10 - b9) * q^85 + (-b15 - 2b13 + b12 - b9 - 2b8 - b6 + b4 - 2b1 - 2) q^87 + (-b15 + b14 - 2b13 - b12 - b11 + b6 - 3b5 + b4 - b3 - b2) * q^89 + (b15 + b14 + b13 + b12 + b11 - b10 - 2b9 + b8 - b7 + b6 + 3b4 + b3 - b2 - 4) * q^91 + (-b15 + b14 - 2b13 + b12 + b11 - b10 + 2b9 - b8 + b4 + b1 + 2) * q^93 + (b3 + b2) * q^95 + (b14 - b13 - b11 + b7 + b6 - 2b5 + 4b4 + b3 - 2b2) q^97 + (-b15 - b14 - b12 + b11 - 2b7 - b6 + b4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^7 + 16 * q^9	\( 16 q - 4 q^{7} + 16 q^{9} - 8 q^{19} - 4 q^{21} - 16 q^{25} + 48 q^{27} - 8 q^{29} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{53} + 8 q^{55} + 16 q^{57} - 8 q^{59} - 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} + 64 q^{83} - 16 q^{87} - 64 q^{91} + 48 q^{93}+O(q^{100}) \) 16 * q - 4 * q^7 + 16 * q^9 - 8 * q^19 - 4 * q^21 - 16 * q^25 + 48 * q^27 - 8 * q^29 - 16 * q^37 - 8 * q^47 - 4 * q^49 - 16 * q^53 + 8 * q^55 + 16 * q^57 - 8 * q^59 - 60 * q^63 + 8 * q^65 + 8 * q^77 + 40 * q^81 + 64 * q^83 - 16 * q^87 - 64 * q^91 + 48 * q^93

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	2240.2.k.f

Level	$2240$
Weight	$2$
Character orbit	2240.k
Analytic conductor	$17.886$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(17.8864900528\)
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:	\( x^{16} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} + \cdots + 5956 \) x^16 - 4x^15 - 6x^14 + 68x^13 - 126x^12 - 148x^11 + 1006x^10 - 1516x^9 - 179x^8 + 3992x^7 - 5596x^6 - 488x^5 + 16080x^4 - 33776x^3 + 37344x^2 - 22336*x + 5956
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{11} \)
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 12758377455 \nu^{15} - 17029973070 \nu^{14} + 255451555872 \nu^{13} + \cdots + 634026321716742 ) / 216025654874206 \) (-12758377455v^15 - 17029973070v^14 + 255451555872v^13 - 5782883602v^12 - 2428092530782v^11 + 2461866146935v^10 + 10812644798656v^9 - 20882203036101v^8 - 17931607412769v^7 + 69829732643649v^6 - 13633178133740v^5 - 109539021238893v^4 + 106762977907130v^3 + 7255442894980v^2 - 55212568945784*v + 634026321716742) / 216025654874206
\(\beta_{2}\)	\(=\)	\( ( 87\!\cdots\!74 \nu^{15} + \cdots - 34\!\cdots\!36 ) / 64\!\cdots\!44 \) (8765614024466674v^15 - 62502845947521146v^14 + 27930788945903613v^13 + 919541822694668216v^12 - 2758329551827048716v^11 - 505933606394946956v^10 + 16874220098891849632v^9 - 31541580432531125900v^8 + 2468356936003309764v^7 + 71827760037341267022v^6 - 102679857907601843147v^5 + 8812409656846222588v^4 + 240534886957138604770v^3 - 612154820371672153200v^2 + 735676972740082807394*v - 348289662532115458336) / 64532047726642313144
\(\beta_{3}\)	\(=\)	\( ( 13\!\cdots\!05 \nu^{15} + \cdots + 14\!\cdots\!08 ) / 64\!\cdots\!44 \) (13090040637728505v^15 + 211916728286895534v^14 - 725493091669104857v^13 - 1961008466169258786v^12 + 12493927886363852188v^11 - 10299818010203611792v^10 - 52349917243107468542v^9 + 149934458133588569104v^8 - 74189346620384657089v^7 - 263650896553140225122v^6 + 546458846557001737209v^5 - 205329180934372231738v^4 - 836795779768439795796v^3 + 2402362524933101244484v^2 - 2957390654429469965602*v + 1427468805432647561708) / 64532047726642313144
\(\beta_{4}\)	\(=\)	\( ( - 1210752049 \nu^{15} - 1110096336 \nu^{14} + 20403779147 \nu^{13} - 19050599618 \nu^{12} + \cdots - 17489786211320 ) / 4256018862188 \) (-1210752049v^15 - 1110096336v^14 + 20403779147v^13 - 19050599618v^12 - 146738265037v^11 + 351660968602v^10 + 157695428815v^9 - 1793479248628v^8 + 1916781072920v^7 + 1497184388698v^6 - 6855726785840v^5 + 5953850442270v^4 + 1659909458620v^3 - 20164201672372v^2 + 32148639488212*v - 17489786211320) / 4256018862188
\(\beta_{5}\)	\(=\)	\( ( - 3343558 \nu^{15} - 1238164 \nu^{14} + 52258753 \nu^{13} - 74255495 \nu^{12} + \cdots - 42814645170 ) / 3918178202 \) (-3343558v^15 - 1238164v^14 + 52258753v^13 - 74255495v^12 - 312877523v^11 + 939062890v^10 - 23564043v^9 - 3862442118v^8 + 4919305197v^7 + 1699089560v^6 - 14234786412v^5 + 16165231231v^4 - 5856993922v^3 - 37472206390v^2 + 73775229484*v - 42814645170) / 3918178202
\(\beta_{6}\)	\(=\)	\( ( - 12\!\cdots\!73 \nu^{15} + \cdots + 33\!\cdots\!40 ) / 12\!\cdots\!88 \) (-120223170196860373v^15 + 508409725558160580v^14 + 1001904479712438592v^13 - 9006400866078907244v^12 + 12390972778503174636v^11 + 31494971175260942572v^10 - 125744891278723942282v^9 + 118832891793713150884v^8 + 160017238475171890871v^7 - 495008163163051562664v^6 + 391831367090758893822v^5 + 538294739675650401264v^4 - 2039725726045424389930v^3 + 3212935115563632993032v^2 - 2117068501808309500776*v + 331964725632513472440) / 129064095453284626288
\(\beta_{7}\)	\(=\)	\( ( 12\!\cdots\!95 \nu^{15} + \cdots - 90\!\cdots\!04 ) / 12\!\cdots\!88 \) (126425665182626895v^15 - 402469074925196481v^14 - 1121956011275831728v^13 + 7733154061744690654v^12 - 9125291680193799000v^11 - 27685781977964858108v^10 + 103545419009722591998v^9 - 96472901443840712450v^8 - 114632157105709674301v^7 + 394060154218144271303v^6 - 329971656220426141850v^5 - 366229213299615695132v^4 + 1676643046921451745682v^3 - 2720002409034269825638v^2 + 2374127347238370132368*v - 908825391783385858004) / 129064095453284626288
\(\beta_{8}\)	\(=\)	\( ( 82\!\cdots\!25 \nu^{15} + \cdots - 39\!\cdots\!80 ) / 64\!\cdots\!44 \) (82713379168634325v^15 - 281075459079638160v^14 - 713483502120422931v^13 + 5336541030249633080v^12 - 6695141944048570864v^11 - 19147893177766628976v^10 + 73841771752378564118v^9 - 68228822294103924196v^8 - 91976769270153237685v^7 + 294773493670649913012v^6 - 220270109894291825885v^5 - 310073024820485450084v^4 + 1201588731759913835968v^3 - 1840543284179116451012v^2 + 1393734925792015031754*v - 394988529315488072880) / 64532047726642313144
\(\beta_{9}\)	\(=\)	\( ( 23\!\cdots\!35 \nu^{15} + \cdots - 10\!\cdots\!60 ) / 12\!\cdots\!88 \) (237140810440379735v^15 - 694839951210633181v^14 - 2223510643919990830v^13 + 13842456444243259968v^12 - 14492654803433050080v^11 - 53007947937536874348v^10 + 183385760798826199310v^9 - 153337931243226373826v^8 - 237649344704454034305v^7 + 719125568618894204991v^6 - 517634899815047395024v^5 - 821736504755162748938v^4 + 3021381137616653188334v^3 - 4556629515839990044858v^2 + 3426054626530645624092*v - 1020322781709869753760) / 129064095453284626288
\(\beta_{10}\)	\(=\)	\( ( 25\!\cdots\!15 \nu^{15} + \cdots - 13\!\cdots\!72 ) / 12\!\cdots\!88 \) (259293226844765315v^15 - 770370709610531096v^14 - 2305263887028898338v^13 + 15278820432296986152v^12 - 17498219145135478052v^11 - 55549252514026466004v^10 + 205855473162409076806v^9 - 189152648431021147452v^8 - 234485299461395158061v^7 + 807831922352873380748v^6 - 649630715069165603172v^5 - 728970739234563805676v^4 + 3399968126198016637530v^3 - 5424839780231186441632v^2 + 4533424520138778259396*v - 1310279850884661310672) / 129064095453284626288
\(\beta_{11}\)	\(=\)	\( ( - 40\!\cdots\!67 \nu^{15} + \cdots + 95\!\cdots\!72 ) / 12\!\cdots\!88 \) (-401514885070314967v^15 + 1118370145453479424v^14 + 3999070438841951874v^13 - 22929190127933666378v^12 + 20193867782511255404v^11 + 95284425048363017136v^10 - 293551487608489295942v^9 + 197696087447360859392v^8 + 449419601267458200241v^7 - 1107409032339395042580v^6 + 632064008668323899948v^5 + 1485218623630154093798v^4 - 4785700310140283855202v^3 + 6761685223629213815172v^2 - 4307250443939268057452*v + 958797510564913058372) / 129064095453284626288
\(\beta_{12}\)	\(=\)	\( ( - 20\!\cdots\!96 \nu^{15} + \cdots + 16\!\cdots\!32 ) / 64\!\cdots\!44 \) (-203138371139447396v^15 + 707031214890825801v^14 + 1605940222342174914v^13 - 12971112886976321418v^12 + 18772666473353955162v^11 + 40117807089961648770v^10 - 184898054661417274554v^9 + 212344977649881363120v^8 + 158314411051918264382v^7 - 748686664357395711185v^6 + 753019705595276023700v^5 + 546154222509972678626v^4 - 3060392513637979096840v^3 + 5296744972024004608390v^2 - 4644440188818386285000*v + 1678744867755471500432) / 64532047726642313144
\(\beta_{13}\)	\(=\)	\( ( - 62\!\cdots\!45 \nu^{15} + \cdots + 32\!\cdots\!36 ) / 12\!\cdots\!88 \) (-622859702467771945v^15 + 1867679735746360758v^14 + 5650730542224720896v^13 - 36854430938284936774v^12 + 41116165628762327132v^11 + 136242982429528198460v^10 - 492786652970957868114v^9 + 438963481118088233728v^8 + 586766801271558866667v^7 - 1921017645189911857042v^6 + 1497237271104639140046v^5 + 1916936457289342249746v^4 - 8097992310736121571546v^3 + 12701471705105446312128v^2 - 10075478303001224763280*v + 3239428024377916922436) / 129064095453284626288
\(\beta_{14}\)	\(=\)	\( ( - 62\!\cdots\!81 \nu^{15} + \cdots + 41\!\cdots\!20 ) / 12\!\cdots\!88 \) (-625676353274793981v^15 + 2046731257544845448v^14 + 5284503975155719586v^13 - 38894072028060292708v^12 + 50274671850866341900v^11 + 132799764376717123880v^10 - 538509129785875372338v^9 + 543808656389170520872v^8 + 561969906032701808571v^7 - 2136373410007872469824v^6 + 1862585432143242059608v^5 + 1900518047840479370596v^4 - 8817764898518887072334v^3 + 14421858692828249207256v^2 - 11871046424796416720100*v + 4139600485325221870320) / 129064095453284626288
\(\beta_{15}\)	\(=\)	\( ( 77\!\cdots\!49 \nu^{15} + \cdots - 31\!\cdots\!96 ) / 12\!\cdots\!88 \) (773830196453774649v^15 - 2257663355263442372v^14 - 7170009218278348232v^13 + 45269222028181278742v^12 - 47961230184040499388v^11 - 173776796765668563960v^10 + 601569813669377153970v^9 - 494725854442072439064v^8 - 777850522470759953827v^7 + 2319785730056291618072v^6 - 1659946859124546953910v^5 - 2510620628878174669634v^4 + 9819322295097294277378v^3 - 15062240195278282876764v^2 + 11281234388048318993104*v - 3185578392552200053596) / 129064095453284626288

\(\nu\)	\(=\)	\( ( \beta_{13} + \beta_{10} + \beta_{9} + \beta_{7} ) / 2 \) (b13 + b10 + b9 + b7) / 2
\(\nu^{2}\)	\(=\)	\( ( - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} + \cdots + 4 ) / 2 \) (-b14 + b11 - b10 - b9 + 2b8 + b7 + b6 + 2b4 + b3 - b2 + 4) / 2
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} - \beta_{12} + 2 \beta_{11} + 11 \beta_{10} - 5 \beta_{8} + \cdots - 22 ) / 4 \) (-3b15 - 8b14 + 9b13 - b12 + 2b11 + 11b10 - 5b8 + 4b7 - 5b6 - 5b4 + 3b3 - 3b2 + 3b1 - 22) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{15} - 5 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 12 \beta_{10} - 11 \beta_{9} + \cdots + 22 ) / 2 \) (b15 - 5b14 - 7b13 - 3b12 + 13b11 - 12b10 - 11b9 + 13b8 - 5b7 + 4b5 - 5b4 + 8b3 - 2b2 - 3*b1 + 22) / 2
\(\nu^{5}\)	\(=\)	\( ( - 21 \beta_{15} - 46 \beta_{14} + 37 \beta_{13} + 13 \beta_{12} + 75 \beta_{10} + 24 \beta_{9} + \cdots - 182 ) / 4 \) (-21b15 - 46b14 + 37b13 + 13b12 + 75b10 + 24b9 - 79b8 - 8b7 - 39b6 + 22b5 - 75b4 + 7b3 - 13b2 + 31b1 - 182) / 4
\(\nu^{6}\)	\(=\)	\( ( 15 \beta_{15} - 82 \beta_{13} - 19 \beta_{12} + 98 \beta_{11} - 130 \beta_{10} - 43 \beta_{9} + \cdots + 214 ) / 2 \) (15b15 - 82b13 - 19b12 + 98b11 - 130b10 - 43b9 + 100b8 - 31b7 + 3b6 + 14b5 - 21b4 + 40b3 + 7b2 - 25b1 + 214) / 2
\(\nu^{7}\)	\(=\)	\( ( - 81 \beta_{15} - 144 \beta_{14} + 263 \beta_{13} + 151 \beta_{12} - 174 \beta_{11} + 581 \beta_{10} + \cdots - 1544 ) / 4 \) (-81b15 - 144b14 + 263b13 + 151b12 - 174b11 + 581b10 + 320b9 - 749b8 + 48b7 - 215b6 + 81b5 - 283b4 - 109b3 - 177b2 + 286*b1 - 1544) / 4
\(\nu^{8}\)	\(=\)	\( ( 179 \beta_{15} + 231 \beta_{14} - 757 \beta_{13} - 225 \beta_{12} + 653 \beta_{11} - 1232 \beta_{10} + \cdots + 2002 ) / 2 \) (179b15 + 231b14 - 757b13 - 225b12 + 653b11 - 1232b10 - 463b9 + 907b8 - 81b7 + 124b6 - 74b5 + 281b4 + 252b3 + 208b2 - 205*b1 + 2002) / 2
\(\nu^{9}\)	\(=\)	\( ( - 553 \beta_{15} + 46 \beta_{14} + 2417 \beta_{13} + 1181 \beta_{12} - 3100 \beta_{11} + 5499 \beta_{10} + \cdots - 14066 ) / 4 \) (-553b15 + 46b14 + 2417b13 + 1181b12 - 3100b11 + 5499b10 + 2580b9 - 6271b8 + 916b7 - 1191b6 + 316b5 - 1839b4 - 1881b3 - 1529b2 + 2449*b1 - 14066) / 4
\(\nu^{10}\)	\(=\)	\( ( 1161 \beta_{15} + 2128 \beta_{14} - 6430 \beta_{13} - 2141 \beta_{12} + 4718 \beta_{11} - 10312 \beta_{10} + \cdots + 18554 ) / 2 \) (1161b15 + 2128b14 - 6430b13 - 2141b12 + 4718b11 - 10312b10 - 4485b9 + 8492b8 - 1041b7 + 1531b6 - 1270b5 + 4613b4 + 2350b3 + 2981b2 - 2169*b1 + 18554) / 2
\(\nu^{11}\)	\(=\)	\( ( - 5851 \beta_{15} + 3292 \beta_{14} + 25033 \beta_{13} + 11801 \beta_{12} - 35758 \beta_{11} + \cdots - 121468 ) / 4 \) (-5851b15 + 3292b14 + 25033b13 + 11801b12 - 35758b11 + 54695b10 + 24584b9 - 53811b8 + 8872b7 - 7513b6 + 2785b5 - 21121b4 - 22107b3 - 15347b2 + 18288*b1 - 121468) / 4
\(\nu^{12}\)	\(=\)	\( ( 8415 \beta_{15} + 11375 \beta_{14} - 49763 \beta_{13} - 17893 \beta_{12} + 41993 \beta_{11} + \cdots + 174470 ) / 2 \) (8415b15 + 11375b14 - 49763b13 - 17893b12 + 41993b11 - 85620b10 - 38113b9 + 77845b8 - 12523b7 + 14390b6 - 12908b5 + 53341b4 + 25804b3 + 31616b2 - 23965*b1 + 174470) / 2
\(\nu^{13}\)	\(=\)	\( ( - 47307 \beta_{15} + 26662 \beta_{14} + 261031 \beta_{13} + 116311 \beta_{12} - 346640 \beta_{11} + \cdots - 1043830 ) / 4 \) (-47307b15 + 26662b14 + 261031b13 + 116311b12 - 346640b11 + 514361b10 + 228868b9 - 466245b8 + 95516b7 - 50433b6 + 39340b5 - 263453b4 - 226323b3 - 173323b2 + 141931*b1 - 1043830) / 4
\(\nu^{14}\)	\(=\)	\( ( 69001 \beta_{15} + 17414 \beta_{14} - 384702 \beta_{13} - 164929 \beta_{12} + 427592 \beta_{11} + \cdots + 1588582 ) / 2 \) (69001b15 + 17414b14 - 384702b13 - 164929b12 + 427592b11 - 738564b10 - 333975b9 + 706460b8 - 137783b7 + 115051b6 - 114152b5 + 544801b4 + 298730b3 + 313623b2 - 231165*b1 + 1588582) / 2
\(\nu^{15}\)	\(=\)	\( ( - 373861 \beta_{15} + 223888 \beta_{14} + 2459443 \beta_{13} + 1059647 \beta_{12} - 3146942 \beta_{11} + \cdots - 9147324 ) / 4 \) (-373861b15 + 223888b14 + 2459443b13 + 1059647b12 - 3146942b11 + 4620993b10 + 2034216b9 - 4088325b8 + 956432b7 - 388587b6 + 541235b5 - 3161903b4 - 2184813b3 - 1894373b2 + 1209820*b1 - 9147324) / 4

\(n\)	\(897\)	\(1471\)	\(1541\)	\(1921\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1791.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 16 T^{6} - 8 T^{5} + \cdots - 8)^{2} \) (T^8 - 16T^6 - 8T^5 + 69T^4 + 64T^3 - 50T^2 - 56T - 8)^2
$5$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$7$	\( T^{16} + 4 T^{15} + \cdots + 5764801 \) T^16 + 4T^15 + 10T^14 + 52T^13 + 128T^12 + 164T^11 + 598T^10 + 1076T^9 + 190T^8 + 7532T^7 + 29302T^6 + 56252T^5 + 307328T^4 + 873964T^3 + 1176490T^2 + 3294172*T + 5764801
$11$	\( T^{16} + 100 T^{14} + \cdots + 4096 \) T^16 + 100T^14 + 3542T^12 + 54956T^10 + 400289T^8 + 1334888T^6 + 1692944T^4 + 458240*T^2 + 4096
$13$	\( T^{16} + 100 T^{14} + \cdots + 20736 \) T^16 + 100T^14 + 3574T^12 + 53268T^10 + 289329T^8 + 539664T^6 + 438880T^4 + 158976*T^2 + 20736
$17$	\( T^{16} + \cdots + 131974144 \) T^16 + 124T^14 + 6134T^12 + 155700T^10 + 2177601T^8 + 16930648T^6 + 71941840T^4 + 155093760*T^2 + 131974144
$19$	\( (T^{8} + 4 T^{7} + \cdots + 2048)^{2} \) (T^8 + 4T^7 - 60T^6 - 256T^5 + 384T^4 + 1792T^3 - 1152T^2 - 3072*T + 2048)^2
$23$	\( T^{16} + 116 T^{14} + \cdots + 4734976 \) T^16 + 116T^14 + 4948T^12 + 98400T^10 + 995264T^8 + 5132288T^6 + 12598272T^4 + 13099008*T^2 + 4734976
$29$	\( (T^{8} + 4 T^{7} + \cdots + 272)^{2} \) (T^8 + 4T^7 - 138T^6 - 436T^5 + 4617T^4 + 9528T^3 - 30504T^2 - 4192*T + 272)^2
$31$	\( (T^{8} - 128 T^{6} + \cdots + 32768)^{2} \) (T^8 - 128T^6 - 32T^5 + 4320T^4 + 3584T^3 - 26368T^2 - 8192T + 32768)^2
$37$	\( (T^{8} + 8 T^{7} + \cdots + 122624)^{2} \) (T^8 + 8T^7 - 112T^6 - 640T^5 + 3840T^4 + 12672T^3 - 50944T^2 - 46080*T + 122624)^2
$41$	\( T^{16} + \cdots + 53557067776 \) T^16 + 344T^14 + 44048T^12 + 2714368T^10 + 85502976T^8 + 1357856768T^6 + 10589323264T^4 + 38860750848*T^2 + 53557067776
$43$	\( T^{16} + \cdots + 122466402304 \) T^16 + 340T^14 + 47156T^12 + 3445600T^10 + 142480448T^8 + 3304139776T^6 + 39227764736T^4 + 183466065920*T^2 + 122466402304
$47$	\( (T^{8} + 4 T^{7} + \cdots - 231568)^{2} \) (T^8 + 4T^7 - 204T^6 - 224T^5 + 13581T^4 - 14468T^3 - 287030T^2 + 766912*T - 231568)^2
$53$	\( (T^{8} + 8 T^{7} + \cdots - 665856)^{2} \) (T^8 + 8T^7 - 280T^6 - 2176T^5 + 20480T^4 + 129664T^3 - 464000T^2 - 1459200*T - 665856)^2
$59$	\( (T^{8} + 4 T^{7} + \cdots - 672768)^{2} \) (T^8 + 4T^7 - 268T^6 - 448T^5 + 22832T^4 - 21184T^3 - 637760T^2 + 1970688*T - 672768)^2
$61$	\( T^{16} + \cdots + 647532301582336 \) T^16 + 784T^14 + 256384T^12 + 45476608T^10 + 4750933504T^8 + 294979448832T^6 + 10287941124096T^4 + 168587098128384*T^2 + 647532301582336
$67$	\( T^{16} + 436 T^{14} + \cdots + 16384 \) T^16 + 436T^14 + 57492T^12 + 2491232T^10 + 21440192T^8 + 21862400T^6 + 7105536T^4 + 647168*T^2 + 16384
$71$	\( T^{16} + \cdots + 22419997720576 \) T^16 + 888T^14 + 307696T^12 + 53264640T^10 + 4963401472T^8 + 252540254208T^6 + 6607642955776T^4 + 70103100030976*T^2 + 22419997720576
$73$	\( T^{16} + \cdots + 113154195456 \) T^16 + 584T^14 + 124848T^12 + 11843584T^10 + 492927744T^8 + 9381095424T^6 + 79220592640T^4 + 245725986816*T^2 + 113154195456
$79$	\( T^{16} + \cdots + 1196606464 \) T^16 + 292T^14 + 30278T^12 + 1353180T^10 + 26891185T^8 + 249816632T^6 + 1075124688T^4 + 1999695104*T^2 + 1196606464
$83$	\( (T^{8} - 32 T^{7} + \cdots - 12271744)^{2} \) (T^8 - 32T^7 + 102T^6 + 5576T^5 - 50056T^4 - 141504T^3 + 2571168T^2 - 4208512*T - 12271744)^2
$89$	\( T^{16} + \cdots + 718886928384 \) T^16 + 704T^14 + 180256T^12 + 21196800T^10 + 1238941952T^8 + 36256497664T^6 + 510216503296T^4 + 2794048192512*T^2 + 718886928384
$97$	\( T^{16} + \cdots + 335984794092544 \) T^16 + 732T^14 + 210582T^12 + 31426644T^10 + 2676287009T^8 + 133405156696T^6 + 3798419954896T^4 + 56407587773184*T^2 + 335984794092544
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