LMFDB - Newform orbit 4719.2.a.bq

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4719,2,Mod(1,4719)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4719, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4719.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4719 = 3 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4719.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,-1,18,27,10,-1,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.6814047138$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 31 x^{16} + 29 x^{15} + 396 x^{14} - 348 x^{13} - 2689 x^{12} + 2242 x^{11} + \cdots + 55 $ x^18 - x^17 - 31x^16 + 29x^15 + 396x^14 - 348x^13 - 2689x^12 + 2242x^11 + 10427x^10 - 8392x^9 - 22969x^8 + 18293x^7 + 26541x^6 - 21589x^5 - 12506x^4 + 11016x^3 + 151x^2 - 630x + 55
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 $1,\beta_1,\ldots,\beta_{17}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{7} + 1) q^{5} - \beta_1 q^{6} - \beta_{16} q^{7} + ( - \beta_{3} - 2 \beta_1) q^{8} + q^{9} + (\beta_{17} + \beta_{16} + \cdots + \beta_{2}) q^{10}+ \cdots + ( - \beta_{17} - 2 \beta_{16} + \cdots - 4) q^{98}+O(q^{100}) $ q - b1 * q^2 + q^3 + (b2 + 2) * q^4 + (-b7 + 1) * q^5 - b1 * q^6 - b16 * q^7 + (-b3 - 2b1) q^8 + q^9 + (b17 + b16 - b15 + b14 + b6 + b5 + b4 + b2) * q^10 + (b2 + 2) * q^12 - q^13 + (b17 + b16 + b13 - b12 - b7 + b3 + 1) * q^14 + (-b7 + 1) * q^15 + (b12 + b11 + b10 + b9 + b7 + b6 + b2 + b1 + 1) * q^16 + (b13 - b5 - b1) * q^17 - b1 * q^18 + (b10 - b8 - b6 - b4 + b3 - b2) * q^19 + (-b15 - b11 - b10 - b8 - b7 + 2b6 + b5 - b3 + 2b2 - b1 + 2) * q^20 - b16 * q^21 + (-b16 + b15 - b14 - b11 - b7 - b6 + 2) * q^23 + (-b3 - 2b1) q^24 + (-b17 + b16 + b14 - b13 - b10 - b3 - b1 + 3) * q^25 + b1 * q^26 + q^27 + (-2b16 + b15 - b14 + b12 + b11 - b10 - b7 - 2b6 - b5 - b4 - b2 - b1 + 1) * q^28 + (b16 - b12 - b9 - b7 - b6 - b5 + b3 - b1 + 1) * q^29 + (b17 + b16 - b15 + b14 + b6 + b5 + b4 + b2) * q^30 + (-b17 + b15 - b14 + b11 + b10 - b8 - 2b6 - b4 + b3 - b2 + 2) q^31 + (-b17 - b14 - b8 - b7 + b5 - b3 - 2b1 - 2) q^32 + (-2b16 + b15 - b14 + b12 + b11 + b10 + b8 + b7 - 2b6 + 1) * q^34 + (-b16 + b15 + b14 - b11 - b10 + b8 + b7 - b5 - b3 - b1 + 2) * q^35 + (b2 + 2) * q^36 + (-2b15 + b14 + b7 + 2b6 - b5 + b4 + b2 + 1) * q^37 + (b17 + b16 - b15 + b14 - b12 - b11 - b10 - b9 + b8 - b7 + b6 + 1) * q^38 - q^39 + (b16 - b15 + b14 - b13 + b10 + b9 + b8 + 2b7 + 5b6 + 3b5 + b4 - b3 + 2b2) * q^40 + (b8 + b5 + b4 - b2 + b1 - 2) * q^41 + (b17 + b16 + b13 - b12 - b7 + b3 + 1) * q^42 + (-b16 - b14 + b9 + b5 - b4 + b2 + 1) * q^43 + (-b7 + 1) * q^45 + (b17 + b16 + b14 + b10 + 2b7 - b6 - b5 + 2b4 + b3 - 1) * q^46 + (b17 - b7 + b4 + b3 - b2) * q^47 + (b12 + b11 + b10 + b9 + b7 + b6 + b2 + b1 + 1) * q^48 + (b16 + b14 - b12 - b9 + b8 + b6 - b5 - b3 + 3) * q^49 + (2b16 - 3b15 + b14 - 2b13 - b11 - b10 + b7 + 4b6 + b5 + b4 - b3 + 3b2 - 2b1 + 2) * q^50 + (b13 - b5 - b1) * q^51 + (-b2 - 2) * q^52 + (-b16 + b12 - b10 - b9 + b8 - b7 + 1) * q^53 - b1 * q^54 + (b16 + b15 + b14 + b13 - 2b12 + b11 + 2b10 + b7 - 3b6 - b5 + 2b3 - 2b2 + b1 + 2) q^56 + (b10 - b8 - b6 - b4 + b3 - b2) * q^57 + (-2b16 + 2b15 + b12 - b10 + b8 - 5b6 - b5 - b4 - b2 - b1 + 5) q^58 + (b10 - 3b6 - b5 - b4 - b2 - b1 + 2) q^59 + (-b15 - b11 - b10 - b8 - b7 + 2b6 + b5 - b3 + 2b2 - b1 + 2) * q^60 + (-b15 - b13 - b12 - b11 - b10 - b9 + b8 - b7 + 2b6 + b4 + b2 + b1 + 1) q^61 + (-2b16 + b15 - b14 + b12 + b11 + b9 + 2b8 + 2b7 + 2b6 - b5 - b3 - 3b1 - 1) q^62 - b16 * q^63 + (b17 - b15 + b14 - b13 + b12 + b9 + 2b8 + 3b7 + 6b6 + 3b5 + 3b4 - b3 + 4b2 + 4b1) q^64 + (b7 - 1) * q^65 + (-b8 - 2b6 + b5 - b4 - b2 - b1 + 4) q^67 + (-b17 + 3b15 - 2b14 + b13 - b12 + b11 + 2b10 - b8 - b7 - 3b6 - 2b5 - 2b4 + b3 - 2b2 - 3b1 + 2) * q^68 + (-b16 + b15 - b14 - b11 - b7 - b6 + 2) * q^69 + (b16 - b15 - b14 - 2b12 - b11 - 2b9 - b8 - b7 - 3b6 - 3b5 - b4 + 2b3 - b2 - 3b1 + 2) * q^70 + (-b17 - 2b16 - b14 + b5 + b4 - b2 + 2b1 - 1) * q^71 + (-b3 - 2b1) q^72 + (-b16 + b15 - b14 + b12 - b11 + b10 - b8 - b7 + 2b5 - 1) q^73 + (b16 + b15 - b11 - b10 - 2b7 - 4b6 + 3b5 - b3 - 2b1 + 3) * q^74 + (-b17 + b16 + b14 - b13 - b10 - b3 - b1 + 3) * q^75 + (b16 - b15 + 2b14 - b12 - b11 - 3b10 - 2b9 + b8 - 2b7 + b1) * q^76 + b1 * q^78 + (-b17 + b15 + b14 - b13 + b12 + b11 - 2b10 - 2b6 - 2b5 - b3 + 2) q^79 + (-3b17 - b15 - b14 - 2b13 + 2b12 + b11 - b8 + b7 + 5b6 + b5 - b4 - 3b3 + b2 - 4b1 - 1) * q^80 + q^81 + (-b17 - b16 - b14 + b12 - b8 + 2b6 - b5 + b3 + 4b1 - 2) * q^82 + (-b17 - b16 - b14 + b12 + 2b11 + b10 + 2b9 + b7 + b6 - b5 - b4 - b2 - 1) * q^83 + (-2b16 + b15 - b14 + b12 + b11 - b10 - b7 - 2b6 - b5 - b4 - b2 - b1 + 1) * q^84 + (-2b17 - b16 - b15 - b14 - b12 + b11 + b7 + b6 + b5 + b3 + 3b1 - 3) * q^85 + (b16 - b15 + b14 - b12 + b11 - b8 + b7 + 3b6 - b3 - 3b1 - 1) * q^86 + (b16 - b12 - b9 - b7 - b6 - b5 + b3 - b1 + 1) * q^87 + (-b17 + b15 - 2b13 + b12 + b11 - b10 + 2b8 + b7 - 2b3 + b1 + 2) q^89 + (b17 + b16 - b15 + b14 + b6 + b5 + b4 + b2) * q^90 + b16 * q^91 + (-2b17 - 2b16 + 2b15 - 2b14 + b12 - 2b10 - b8 - 2b7 - 8b6 - 4b5 - b4 - b3 - b2 - 2b1 + 5) q^92 + (-b17 + b15 - b14 + b11 + b10 - b8 - 2b6 - b4 + b3 - b2 + 2) q^93 + (b17 + b15 + b13 - b12 - b11 - b10 - b8 - 3b7 - 4b6 + b4 + 2b3 + 3b1 + 7) * q^94 + (b17 + b16 + 2b15 - b10 - 2b8 - b7 - b6 - b4 + b3 - b2 + 2) * q^95 + (-b17 - b14 - b8 - b7 + b5 - b3 - 2b1 - 2) q^96 + (-b16 + b15 - b11 + 3b6 + b5 - b4 - b2 + 3) q^97 + (-b17 - 2b16 + b15 - b14 + 2b12 + b11 - 3b6 - b5 - 3b4 - b2 - 4b1 - 4) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 18 q^{3} + 27 q^{4} + 10 q^{5} - q^{6} - 7 q^{7} - 3 q^{8} + 18 q^{9} + 3 q^{10} + 27 q^{12} - 18 q^{13} + 10 q^{14} + 10 q^{15} + 41 q^{16} + 5 q^{17} - q^{18} - 10 q^{19} + 16 q^{20}+ \cdots - 93 q^{98}+O(q^{100}) $ 18 * q - q^2 + 18 * q^3 + 27 * q^4 + 10 * q^5 - q^6 - 7 * q^7 - 3 * q^8 + 18 * q^9 + 3 * q^10 + 27 * q^12 - 18 * q^13 + 10 * q^14 + 10 * q^15 + 41 * q^16 + 5 * q^17 - q^18 - 10 * q^19 + 16 * q^20 - 7 * q^21 + 9 * q^23 - 3 * q^24 + 52 * q^25 + q^26 + 18 * q^27 - 4 * q^28 + 3 * q^30 + 36 * q^31 - 37 * q^32 + 20 * q^34 + 23 * q^35 + 27 * q^36 + 37 * q^37 + 6 * q^38 - 18 * q^39 + 54 * q^40 - 12 * q^41 + 10 * q^42 + 2 * q^43 + 10 * q^45 - 6 * q^46 + 41 * q^48 + 61 * q^49 + 48 * q^50 + 5 * q^51 - 27 * q^52 + 18 * q^53 - q^54 + 36 * q^56 - 10 * q^57 + 40 * q^58 + 10 * q^59 + 16 * q^60 + 12 * q^61 + 25 * q^62 - 7 * q^63 + 67 * q^64 - 10 * q^65 + 52 * q^67 + 18 * q^68 + 9 * q^69 - 13 * q^70 + q^71 - 3 * q^72 - 29 * q^73 + 2 * q^74 + 52 * q^75 - 26 * q^76 + q^78 + 18 * q^79 + 48 * q^80 + 18 * q^81 - 6 * q^82 + 26 * q^83 - 4 * q^84 - 15 * q^85 + 9 * q^86 + 59 * q^89 + 3 * q^90 + 7 * q^91 + 4 * q^92 + 36 * q^93 + 53 * q^94 + 5 * q^95 - 37 * q^96 + 70 * q^97 - 93 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - x^{17} - 31 x^{16} + 29 x^{15} + 396 x^{14} - 348 x^{13} - 2689 x^{12} + 2242 x^{11} + \cdots + 55 $ x^18 - x^17 - 31*x^16 + 29*x^15 + 396*x^14 - 348*x^13 - 2689*x^12 + 2242*x^11 + 10427*x^10 - 8392*x^9 - 22969*x^8 + 18293*x^7 + 26541*x^6 - 21589*x^5 - 12506*x^4 + 11016*x^3 + 151*x^2 - 630*x + 55 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 4 $ v^2 - 4
$\beta_{3}$	$=$	$ \nu^{3} - 6\nu $ v^3 - 6*v
$\beta_{4}$	$=$	$ ( 1794316 \nu^{17} + 40117477 \nu^{16} - 121295666 \nu^{15} - 1140544478 \nu^{14} + \cdots - 2013758985 ) / 1855265315 $ (1794316v^17 + 40117477v^16 - 121295666v^15 - 1140544478v^14 + 2455273968v^13 + 12958646951v^12 - 23249070804v^11 - 74554608503v^10 + 117304780481v^9 + 225228527484v^8 - 323028890972v^7 - 326151717937v^6 + 456971721696v^5 + 146628206998v^4 - 256386569273v^3 + 55145841762v^2 - 6212666145*v - 2013758985) / 1855265315
$\beta_{5}$	$=$	$ ( 7494672 \nu^{17} + 9572099 \nu^{16} - 220222849 \nu^{15} - 225723188 \nu^{14} + \cdots - 1885514235 ) / 3710530630 $ (7494672v^17 + 9572099v^16 - 220222849v^15 - 225723188v^14 + 2624974410v^13 + 1929291318v^12 - 16482600428v^11 - 6757388867v^10 + 59589480036v^9 + 4624640560v^8 - 127362413246v^7 + 25136398721v^6 + 155074744519v^5 - 53687713782v^4 - 90044577870v^3 + 32995767138v^2 + 8608947330*v - 1885514235) / 3710530630
$\beta_{6}$	$=$	$ ( - 34282077 \nu^{17} + 26787405 \nu^{16} + 1053172288 \nu^{15} - 773957384 \nu^{14} + \cdots + 12988761180 ) / 3710530630 $ (-34282077v^17 + 26787405v^16 + 1053172288v^15 - 773957384v^14 - 13349979304v^13 + 9305188386v^12 + 90255213735v^11 - 60377816206v^10 - 350701828012v^9 + 228105710148v^8 + 782800386053v^7 - 499759621315v^6 - 935017004378v^5 + 585041015834v^4 + 482419368744v^3 - 287606782362v^2 - 38172360765*v + 12988761180) / 3710530630
$\beta_{7}$	$=$	$ ( 22768834 \nu^{17} - 83197933 \nu^{16} - 578141728 \nu^{15} + 2298516130 \nu^{14} + \cdots - 7596659310 ) / 1855265315 $ (22768834v^17 - 83197933v^16 - 578141728v^15 + 2298516130v^14 + 5631399805v^13 - 25578132025v^12 - 25599283324v^11 + 146909595751v^10 + 48395207000v^9 - 463072713130v^8 + 4925094585v^7 + 782447233153v^6 - 121502960412v^5 - 629431002480v^4 + 86830861630v^3 + 171050822747v^2 + 20363741090*v - 7596659310) / 1855265315
$\beta_{8}$	$=$	$ ( 58233332 \nu^{17} - 169924489 \nu^{16} - 1608424113 \nu^{15} + 4611815144 \nu^{14} + \cdots - 6979046035 ) / 3710530630 $ (58233332v^17 - 169924489v^16 - 1608424113v^15 + 4611815144v^14 + 18026238822v^13 - 49967370466v^12 - 105807297202v^11 + 275695509921v^10 + 350216230262v^9 - 819097544944v^8 - 650322872358v^7 + 1277086255749v^6 + 610962999323v^5 - 951252447354v^4 - 190982799652v^3 + 272518498362v^2 - 22986934480*v - 6979046035) / 3710530630
$\beta_{9}$	$=$	$ ( 62700640 \nu^{17} + 30228543 \nu^{16} - 2251325551 \nu^{15} - 529501840 \nu^{14} + \cdots - 21931031955 ) / 3710530630 $ (62700640v^17 + 30228543v^16 - 2251325551v^15 - 529501840v^14 + 32928162714v^13 + 1291601422v^12 - 252648637866v^11 + 28772998647v^10 + 1088720697380v^9 - 249754534558v^8 - 2608196748494v^7 + 812270473777v^6 + 3199451092551v^5 - 1166157887650v^4 - 1625164594964v^3 + 636369597688v^2 + 137795258290*v - 21931031955) / 3710530630
$\beta_{10}$	$=$	$ ( 63851057 \nu^{17} + 8776947 \nu^{16} - 1945929542 \nu^{15} - 574209510 \nu^{14} + \cdots + 8111535620 ) / 3710530630 $ (63851057v^17 + 8776947v^16 - 1945929542v^15 - 574209510v^14 + 24391477788v^13 + 10787243076v^12 - 162434416509v^11 - 91534278566v^10 + 619587083450v^9 + 393419836684v^8 - 1357342887117v^7 - 848508183097v^6 + 1600751696932v^5 + 811944616820v^4 - 812819729008v^3 - 235795201440v^2 + 56422799485*v + 8111535620) / 3710530630
$\beta_{11}$	$=$	$ ( - 32589642 \nu^{17} + 67028098 \nu^{16} + 937238049 \nu^{15} - 1806452155 \nu^{14} + \cdots + 11821918765 ) / 1855265315 $ (-32589642v^17 + 67028098v^16 + 937238049v^15 - 1806452155v^14 - 11112524948v^13 + 19516653465v^12 + 70669029464v^11 - 108433991633v^10 - 262569229895v^9 + 331813227551v^8 + 578176745795v^7 - 561408171288v^6 - 715877388944v^5 + 508723190210v^4 + 419507127993v^3 - 215275576641v^2 - 66796677570*v + 11821918765) / 1855265315
$\beta_{12}$	$=$	$ ( - 72628004 \nu^{17} - 33453225 \nu^{16} + 2425890163 \nu^{15} + 893540784 \nu^{14} + \cdots + 18353930655 ) / 3710530630 $ (-72628004v^17 - 33453225v^16 + 2425890163v^15 + 893540784v^14 - 33007410912v^13 - 9261075764v^12 + 234688348360v^11 + 46187887889v^10 - 929257907028v^9 - 109252041116v^8 + 2016535568798v^7 + 93919206905v^6 - 2190425086393v^5 + 14298410166v^4 + 942888975982v^3 - 50491820508v^2 - 66890354680*v + 18353930655) / 3710530630
$\beta_{13}$	$=$	$ ( 94494963 \nu^{17} - 185484818 \nu^{16} - 2805678553 \nu^{15} + 5218759922 \nu^{14} + \cdots - 56440849885 ) / 3710530630 $ (94494963v^17 - 185484818v^16 - 2805678553v^15 + 5218759922v^14 + 34466735510v^13 - 59840521738v^12 - 227518675579v^11 + 361739405471v^10 + 876522867636v^9 - 1249096533620v^8 - 1992342264249v^7 + 2497311728078v^6 + 2521668233933v^5 - 2752314000092v^4 - 1447133728430v^3 + 1344845725842v^2 + 143825671665*v - 56440849885) / 3710530630
$\beta_{14}$	$=$	$ ( 2631396 \nu^{17} + 2929809 \nu^{16} - 84836171 \nu^{15} - 83694562 \nu^{14} + 1120419802 \nu^{13} + \cdots - 479252305 ) / 86291410 $ (2631396v^17 + 2929809v^16 - 84836171v^15 - 83694562v^14 + 1120419802v^13 + 947599950v^12 - 7809176268v^11 - 5373611193v^10 + 30823730214v^9 + 15743358316v^8 - 68554722640v^7 - 21390846209v^6 + 79751081741v^5 + 7589369052v^4 - 38922886802v^3 + 5079885718v^2 + 2257187100*v - 479252305) / 86291410
$\beta_{15}$	$=$	$ ( - 120061537 \nu^{17} + 119261603 \nu^{16} + 3769620476 \nu^{15} - 3438745238 \nu^{14} + \cdots + 36700715240 ) / 3710530630 $ (-120061537v^17 + 119261603v^16 + 3769620476v^15 - 3438745238v^14 - 48862480042v^13 + 40891326124v^12 + 337460411449v^11 - 260068729732v^10 - 1335287384064v^9 + 957205548514v^8 + 3019237908037v^7 - 2042879830293v^6 - 3631953257486v^5 + 2343847853498v^4 + 1880111935362v^3 - 1139239346960v^2 - 149853313935*v + 36700715240) / 3710530630
$\beta_{16}$	$=$	$ ( - 121444634 \nu^{17} - 87841057 \nu^{16} + 3911217537 \nu^{15} + 2506477964 \nu^{14} + \cdots + 29096747885 ) / 3710530630 $ (-121444634v^17 - 87841057v^16 + 3911217537v^15 + 2506477964v^14 - 51693114648v^13 - 28061150712v^12 + 361386416174v^11 + 154416932401v^10 - 1435360749228v^9 - 420044583464v^8 + 3228621352814v^7 + 449269984297v^6 - 3832531928177v^5 + 105967198186v^4 + 1947087254988v^3 - 383154874860v^2 - 148316576330*v + 29096747885) / 3710530630
$\beta_{17}$	$=$	$ ( - 209426356 \nu^{17} + 219910667 \nu^{16} + 6192440073 \nu^{15} - 5835704426 \nu^{14} + \cdots + 33473638275 ) / 3710530630 $ (-209426356v^17 + 219910667v^16 + 6192440073v^15 - 5835704426v^14 - 74842115508v^13 + 62306127984v^12 + 476317842946v^11 - 345206808991v^10 - 1712837563428v^9 + 1072903204176v^8 + 3460963343462v^7 - 1897037936347v^6 - 3638468318213v^5 + 1830083869296v^4 + 1567565855338v^3 - 800059462592v^2 - 46822155590*v + 33473638275) / 3710530630

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 4 $ b2 + 4
$\nu^{3}$	$=$	$ \beta_{3} + 6\beta_1 $ b3 + 6*b1
$\nu^{4}$	$=$	$ \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + 7\beta_{2} + \beta _1 + 21 $ b12 + b11 + b10 + b9 + b7 + b6 + 7*b2 + b1 + 21
$\nu^{5}$	$=$	$ \beta_{17} + \beta_{14} + \beta_{8} + \beta_{7} - \beta_{5} + 9\beta_{3} + 38\beta _1 + 2 $ b17 + b14 + b8 + b7 - b5 + 9b3 + 38b1 + 2
$\nu^{6}$	$=$	$ \beta_{17} - \beta_{15} + \beta_{14} - \beta_{13} + 11 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} + \cdots + 122 $ b17 - b15 + b14 - b13 + 11b12 + 10b11 + 10b10 + 11b9 + 2b8 + 13b7 + 16b6 + 3b5 + 3b4 - b3 + 50b2 + 14*b1 + 122
$\nu^{7}$	$=$	$ 14 \beta_{17} + 13 \beta_{14} + 2 \beta_{11} + \beta_{10} + 15 \beta_{8} + 14 \beta_{7} - 3 \beta_{6} + \cdots + 26 $ 14b17 + 13b14 + 2b11 + b10 + 15b8 + 14b7 - 3b6 - 12b5 + b4 + 70b3 + b2 + 253*b1 + 26
$\nu^{8}$	$=$	$ 16 \beta_{17} + 4 \beta_{16} - 17 \beta_{15} + 17 \beta_{14} - 16 \beta_{13} + 95 \beta_{12} + \cdots + 757 $ 16b17 + 4b16 - 17b15 + 17b14 - 16b13 + 95b12 + 81b11 + 83b10 + 95b9 + 29b8 + 126b7 + 164b6 + 46b5 + 44b4 - 12b3 + 363b2 + 142*b1 + 757
$\nu^{9}$	$=$	$ 142 \beta_{17} - \beta_{16} + 3 \beta_{15} + 123 \beta_{14} + 2 \beta_{13} + \beta_{12} + 35 \beta_{11} + \cdots + 247 $ 142b17 - b16 + 3b15 + 123b14 + 2b13 + b12 + 35b11 + 21b10 + 4b9 + 156b8 + 147b7 - 46b6 - 105b5 + 18b4 + 529b3 + 16b2 + 1746*b1 + 247
$\nu^{10}$	$=$	$ 179 \beta_{17} + 81 \beta_{16} - 196 \beta_{15} + 204 \beta_{14} - 174 \beta_{13} + 755 \beta_{12} + \cdots + 4920 $ 179b17 + 81b16 - 196b15 + 204b14 - 174b13 + 755b12 + 616b11 + 653b10 + 756b9 + 300b8 + 1093b7 + 1445b6 + 493b5 + 457b4 - 102b3 + 2659b2 + 1275*b1 + 4920
$\nu^{11}$	$=$	$ 1270 \beta_{17} - 11 \beta_{16} + 67 \beta_{15} + 1041 \beta_{14} + 42 \beta_{13} + 19 \beta_{12} + \cdots + 2117 $ 1270b17 - 11b16 + 67b15 + 1041b14 + 42b13 + 19b12 + 415b11 + 277b10 + 84b9 + 1409b8 + 1373b7 - 502b6 - 824b5 + 220b4 + 3970b3 + 182b2 + 12349*b1 + 2117
$\nu^{12}$	$=$	$ 1730 \beta_{17} + 1069 \beta_{16} - 1917 \beta_{15} + 2098 \beta_{14} - 1612 \beta_{13} + 5772 \beta_{12} + \cdots + 33081 $ 1730b17 + 1069b16 - 1917b15 + 2098b14 - 1612b13 + 5772b12 + 4573b11 + 5034b10 + 5804b9 + 2721b8 + 8976b7 + 11893b6 + 4602b5 + 4144b4 - 738b3 + 19573b2 + 10813*b1 + 33081
$\nu^{13}$	$=$	$ 10668 \beta_{17} - 21 \beta_{16} + 951 \beta_{15} + 8392 \beta_{14} + 570 \beta_{13} + 227 \beta_{12} + \cdots + 17476 $ 10668b17 - 21b16 + 951b15 + 8392b14 + 570b13 + 227b12 + 4171b11 + 3004b10 + 1136b9 + 11867b8 + 12034b7 - 4859b6 - 6155b5 + 2280b4 + 29750b3 + 1820b2 + 88830*b1 + 17476
$\nu^{14}$	$=$	$ 15488 \beta_{17} + 11709 \beta_{16} - 17161 \beta_{15} + 19779 \beta_{14} - 13724 \beta_{13} + \cdots + 228164 $ 15488b17 + 11709b16 - 17161b15 + 19779b14 - 13724b13 + 43204b12 + 33628b11 + 38481b10 + 43785b9 + 23101b8 + 71492b7 + 94404b6 + 40152b5 + 35097b4 - 4729b3 + 144492b2 + 89005*b1 + 228164
$\nu^{15}$	$=$	$ 86523 \beta_{17} + 1130 \beta_{16} + 10994 \beta_{15} + 66128 \beta_{14} + 6378 \beta_{13} + \cdots + 142382 $ 86523b17 + 1130b16 + 10994b15 + 66128b14 + 6378b13 + 2179b12 + 38350b11 + 29355b10 + 12630b9 + 96098b8 + 101448b7 - 44497b6 - 44688b5 + 21623b4 + 222979b3 + 17063b2 + 646642*b1 + 142382
$\nu^{16}$	$=$	$ 132548 \beta_{17} + 115815 \beta_{16} - 145567 \beta_{15} + 176614 \beta_{14} - 111113 \beta_{13} + \cdots + 1604495 $ 132548b17 + 115815b16 - 145567b15 + 176614b14 - 111113b13 + 319323b12 + 246558b11 + 293180b10 + 327411b9 + 188873b8 + 559123b7 + 733278b6 + 337381b5 + 285987b4 - 26345b3 + 1068642b2 + 720237*b1 + 1604495
$\nu^{17}$	$=$	$ 687197 \beta_{17} + 24430 \beta_{16} + 113127 \beta_{15} + 515665 \beta_{14} + 64120 \beta_{13} + \cdots + 1154806 $ 687197b17 + 24430b16 + 113127b15 + 515665b14 + 64120b13 + 18271b12 + 334147b11 + 269470b10 + 125920b9 + 759945b8 + 833835b7 - 394567b6 - 317624b5 + 194381b4 + 1672500b3 + 153863b2 + 4748228*b1 + 1154806

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

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} $

1.1

2.78469
2.71642
2.33995
2.30337
1.61608
1.47269
1.00961
0.855437
0.207338
0.110220
−0.265598
−1.28332
−1.37009
−1.95234
−1.99876
−2.24423
−2.58601
−2.71545

−2.78469

1.00000

5.75448

−2.79004

−2.78469

−4.09953

−10.4550

1.00000

7.76939

1.2

−2.71642

1.00000

5.37894

0.225569

−2.71642

2.19622

−9.17863

1.00000

−0.612740

1.3

−2.33995

1.00000

3.47535

3.43996

−2.33995

1.93374

−3.45224

1.00000

−8.04932

1.4

−2.30337

1.00000

3.30550

2.22133

−2.30337

−4.39340

−3.00706

1.00000

−5.11655

1.5

−1.61608

1.00000

0.611726

−2.09951

−1.61608

4.41014

2.24357

1.00000

3.39298

1.6

−1.47269

1.00000

0.168814

1.13349

−1.47269

−4.08081

2.69677

1.00000

−1.66928

1.7

−1.00961

1.00000

−0.980693

3.79468

−1.00961

5.10168

3.00933

1.00000

−3.83114

1.8

−0.855437

1.00000

−1.26823

−2.89210

−0.855437

−3.88648

2.79576

1.00000

2.47401

1.9

−0.207338

1.00000

−1.95701

−0.942442

−0.207338

0.527618

0.820440

1.00000

0.195404

1.10

−0.110220

1.00000

−1.98785

2.75251

−0.110220

−2.87193

0.439542

1.00000

−0.303382

1.11

0.265598

1.00000

−1.92946

3.84309

0.265598

−2.81061

−1.04366

1.00000

1.02072

1.12

1.28332

1.00000

−0.353096

0.613240

1.28332

0.715509

−3.01977

1.00000

0.786982

1.13

1.37009

1.00000

−0.122856

−4.35041

1.37009

−1.02695

−2.90850

1.00000

−5.96045

1.14

1.95234

1.00000

1.81162

3.98212

1.95234

1.61365

−0.367789

1.00000

7.77443

1.15

1.99876

1.00000

1.99506

−3.43571

1.99876

−4.07684

−0.00987225

1.00000

−6.86718

1.16

2.24423

1.00000

3.03656

0.599319

2.24423

4.24044

2.32627

1.00000

1.34501

1.17

2.58601

1.00000

4.68746

−0.367017

2.58601

1.96460

6.94980

1.00000

−0.949111

1.18

2.71545

1.00000

5.37368

4.27193

2.71545

−2.45705

9.16108

1.00000

11.6002

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$11$	$ +1 $
$13$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4719.2.a.bq		18
11.b	odd	2	1		4719.2.a.br		18
11.c	even	5	2		429.2.n.d	✓	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
429.2.n.d	✓	36	11.c	even	5	2
4719.2.a.bq		18	1.a	even	1	1	trivial
4719.2.a.br		18	11.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 $S_{2}^{\mathrm{new}}(\Gamma_0(4719))$:

$ T_{2}^{18} + T_{2}^{17} - 31 T_{2}^{16} - 29 T_{2}^{15} + 396 T_{2}^{14} + 348 T_{2}^{13} - 2689 T_{2}^{12} + \cdots + 55 $

T2^18 + T2^17 - 31*T2^16 - 29*T2^15 + 396*T2^14 + 348*T2^13 - 2689*T2^12 - 2242*T2^11 + 10427*T2^10 + 8392*T2^9 - 22969*T2^8 - 18293*T2^7 + 26541*T2^6 + 21589*T2^5 - 12506*T2^4 - 11016*T2^3 + 151*T2^2 + 630*T2 + 55

$ T_{5}^{18} - 10 T_{5}^{17} - 21 T_{5}^{16} + 494 T_{5}^{15} - 590 T_{5}^{14} - 8923 T_{5}^{13} + \cdots - 42944 $

T5^18 - 10*T5^17 - 21*T5^16 + 494*T5^15 - 590*T5^14 - 8923*T5^13 + 22386*T5^12 + 70738*T5^11 - 262602*T5^10 - 196875*T5^9 + 1370062*T5^8 - 335670*T5^7 - 2976127*T5^6 + 2286223*T5^5 + 1599855*T5^4 - 1909436*T5^3 + 177072*T5^2 + 225056*T5 - 42944

$ T_{7}^{18} + 7 T_{7}^{17} - 69 T_{7}^{16} - 569 T_{7}^{15} + 1618 T_{7}^{14} + 18070 T_{7}^{13} + \cdots - 11501824 $

T7^18 + 7*T7^17 - 69*T7^16 - 569*T7^15 + 1618*T7^14 + 18070*T7^13 - 12146*T7^12 - 288683*T7^11 - 65522*T7^10 + 2536215*T7^9 + 1370817*T7^8 - 12956016*T7^7 - 6678188*T7^6 + 38445072*T7^5 + 10284736*T7^4 - 59460800*T7^3 + 5248128*T7^2 + 31615744*T7 - 11501824

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} + T^{17} + \cdots + 55 $ T^18 + T^17 - 31T^16 - 29T^15 + 396T^14 + 348T^13 - 2689T^12 - 2242T^11 + 10427T^10 + 8392T^9 - 22969T^8 - 18293T^7 + 26541T^6 + 21589T^5 - 12506T^4 - 11016T^3 + 151T^2 + 630T + 55
$3$	$ (T - 1)^{18} $ (T - 1)^18
$5$	$ T^{18} - 10 T^{17} + \cdots - 42944 $ T^18 - 10T^17 - 21T^16 + 494T^15 - 590T^14 - 8923T^13 + 22386T^12 + 70738T^11 - 262602T^10 - 196875T^9 + 1370062T^8 - 335670T^7 - 2976127T^6 + 2286223T^5 + 1599855T^4 - 1909436T^3 + 177072T^2 + 225056*T - 42944
$7$	$ T^{18} + 7 T^{17} + \cdots - 11501824 $ T^18 + 7T^17 - 69T^16 - 569T^15 + 1618T^14 + 18070T^13 - 12146T^12 - 288683T^11 - 65522T^10 + 2536215T^9 + 1370817T^8 - 12956016T^7 - 6678188T^6 + 38445072T^5 + 10284736T^4 - 59460800T^3 + 5248128T^2 + 31615744*T - 11501824
$11$	$ T^{18} $ T^18
$13$	$ (T + 1)^{18} $ (T + 1)^18
$17$	$ T^{18} + \cdots - 567398656 $ T^18 - 5T^17 - 188T^16 + 975T^15 + 13943T^14 - 76135T^13 - 514457T^12 + 3041932T^11 + 9703132T^10 - 65910842T^9 - 82878429T^8 + 756458628T^7 + 151000984T^6 - 4180186288T^5 + 1207493376T^4 + 8737926208T^3 - 1955930304T^2 - 4108622080*T - 567398656
$19$	$ T^{18} + \cdots - 117633024 $ T^18 + 10T^17 - 159T^16 - 1719T^15 + 9057T^14 + 113429T^13 - 208203T^12 - 3654521T^11 + 738533T^10 + 60704990T^9 + 44425977T^8 - 494741160T^7 - 664642568T^6 + 1575028192T^5 + 2941604224T^4 - 205303552T^3 - 1885797376T^2 - 872466432*T - 117633024
$23$	$ T^{18} + \cdots + 12466253824 $ T^18 - 9T^17 - 226T^16 + 2183T^15 + 18963T^14 - 208170T^13 - 694221T^12 + 9937288T^11 + 7908088T^10 - 251940896T^9 + 148890592T^8 + 3345936704T^7 - 4726624448T^6 - 21047083520T^5 + 40710692608T^4 + 44954033152T^3 - 103620557824T^2 - 1639460864*T + 12466253824
$29$	$ T^{18} + \cdots - 2567147264 $ T^18 - 289T^16 + 45T^15 + 32083T^14 - 14869T^13 - 1761253T^12 + 1629801T^11 + 50928159T^10 - 75237332T^9 - 741492231T^8 + 1554759612T^7 + 4419211464T^6 - 13342182016T^5 - 2602510368T^4 + 37103270080T^3 - 40523074112T^2 + 17121031680T - 2567147264
$31$	$ T^{18} + \cdots + 29764267264 $ T^18 - 36T^17 + 319T^16 + 3226T^15 - 68574T^14 + 151534T^13 + 3890650T^12 - 24994358T^11 - 57905224T^10 + 917742268T^9 - 1222930769T^8 - 12185580084T^7 + 39600858412T^6 + 31639114672T^5 - 271925016352T^4 + 262701910848T^3 + 155856694784T^2 - 213731399680*T + 29764267264
$37$	$ T^{18} + \cdots - 627994624 $ T^18 - 37T^17 + 313T^16 + 4594T^15 - 95984T^14 + 300191T^13 + 5515663T^12 - 52975828T^11 + 69533684T^10 + 1261315584T^9 - 7485073280T^8 + 14900551360T^7 + 544926976T^6 - 35400636160T^5 + 20518065152T^4 + 20570799104T^3 - 3588994048T^2 - 4190760960*T - 627994624
$41$	$ T^{18} + \cdots + 385454080 $ T^18 + 12T^17 - 228T^16 - 3419T^15 + 12998T^14 + 343330T^13 + 384409T^12 - 14244552T^11 - 54434876T^10 + 191396192T^9 + 1412596080T^8 + 1068670336T^7 - 8749555008T^6 - 21965899264T^5 - 11436977152T^4 + 16396993536T^3 + 21035544576T^2 + 6845112320*T + 385454080
$43$	$ T^{18} + \cdots + 5808956218736 $ T^18 - 2T^17 - 395T^16 + 1284T^15 + 63956T^14 - 280008T^13 - 5386114T^12 + 29592052T^11 + 245225022T^10 - 1677809168T^9 - 5511837056T^8 + 51054619752T^7 + 37681429009T^6 - 771544152946T^5 + 381442814909T^4 + 5014237373420T^3 - 4102018073540T^2 - 11846860912192*T + 5808956218736
$47$	$ T^{18} - 388 T^{16} + \cdots - 3117031 $ T^18 - 388T^16 - 676T^15 + 55894T^14 + 172581T^13 - 3594397T^12 - 14448835T^11 + 99331414T^10 + 448230848T^9 - 1052084948T^8 - 4775336179T^7 + 3031272645T^6 + 8343908788T^5 - 3637021904T^4 - 2901072939T^3 + 1289267392T^2 - 95302921T - 3117031
$53$	$ T^{18} + \cdots - 13263190902016 $ T^18 - 18T^17 - 495T^16 + 9626T^15 + 92073T^14 - 1991955T^13 - 8055327T^12 + 200931717T^11 + 358034125T^10 - 10273154953T^9 - 10208391717T^8 + 255895382968T^7 + 261098138708T^6 - 2806682192112T^5 - 3881020004848T^4 + 9661995940544T^3 + 15102799380800T^2 - 7719173792512*T - 13263190902016
$59$	$ T^{18} - 10 T^{17} + \cdots + 647429 $ T^18 - 10T^17 - 256T^16 + 2526T^15 + 20379T^14 - 213512T^13 - 511334T^12 + 7282998T^11 + 1269908T^10 - 108575776T^9 + 79013359T^8 + 718110166T^7 - 685325953T^6 - 2156093906T^5 + 1714764323T^4 + 2468108484T^3 - 916696083T^2 - 97120092*T + 647429
$61$	$ T^{18} + \cdots + 180385112801839 $ T^18 - 12T^17 - 497T^16 + 6087T^15 + 100770T^14 - 1279967T^13 - 10679966T^12 + 144209584T^11 + 627349081T^10 - 9398649365T^9 - 19708740427T^8 + 357340798104T^7 + 273264140004T^6 - 7616841176129T^5 + 17853017070T^4 + 82043961498417T^3 - 32594663645089T^2 - 340489724339574*T + 180385112801839
$67$	$ T^{18} + \cdots + 44563124224 $ T^18 - 52T^17 + 980T^16 - 5827T^15 - 54470T^14 + 948428T^13 - 2745211T^12 - 28857420T^11 + 205122224T^10 + 89339920T^9 - 4117614480T^8 + 7077632640T^7 + 30207969600T^6 - 88821255680T^5 - 48240659200T^4 + 264343298048T^3 - 11181654016T^2 - 226059489280*T + 44563124224
$71$	$ T^{18} + \cdots + 217467806374144 $ T^18 - T^17 - 791T^16 + 17T^15 + 251231T^14 + 165835T^13 - 41467315T^12 - 41591114T^11 + 3876885813T^10 + 4043236496T^9 - 209667631777T^8 - 172632118748T^7 + 6379461594008T^6 + 2379514058384T^5 - 99260020157216T^4 + 30826612249856T^3 + 599412806224064T^2 - 753333650501632T + 217467806374144
$73$	$ T^{18} + \cdots + 57371318726656 $ T^18 + 29T^17 - 322T^16 - 17629T^15 - 72359T^14 + 3355152T^13 + 38844467T^12 - 123251816T^11 - 4471056528T^10 - 21796036592T^9 + 87804789536T^8 + 1351701066176T^7 + 5609602742592T^6 + 7671771975936T^5 - 13482266350336T^4 - 55263292340224T^3 - 41349592734720T^2 + 44125476884480*T + 57371318726656
$79$	$ T^{18} + \cdots + 1752198906176 $ T^18 - 18T^17 - 563T^16 + 11309T^15 + 115645T^14 - 2792883T^13 - 10021066T^12 + 348864049T^11 + 185597013T^10 - 23437761985T^9 + 26263774412T^8 + 815599704739T^7 - 1710807352566T^6 - 12304717516087T^5 + 32909241932655T^4 + 40587450924484T^3 - 86455118796752T^2 - 84483534977632*T + 1752198906176
$83$	$ T^{18} + \cdots + 9652886263955 $ T^18 - 26T^17 - 459T^16 + 16186T^15 + 39417T^14 - 3737390T^13 + 9568709T^12 + 396544164T^11 - 2001333811T^10 - 19554725592T^9 + 132115250996T^8 + 392086057260T^7 - 3447336076428T^6 - 2330368940698T^5 + 34297852697139T^4 - 5628763181578T^3 - 86593266945904T^2 + 41289500800010*T + 9652886263955
$89$	$ T^{18} + \cdots - 14\!\cdots\!96 $ T^18 - 59T^17 + 796T^16 + 17881T^15 - 559079T^14 + 1349592T^13 + 102473265T^12 - 1007071920T^11 - 5534043900T^10 + 123867095056T^9 - 226960221328T^8 - 5575005151360T^7 + 29104201833920T^6 + 72192454689280T^5 - 768464032246784T^4 + 459234886311936T^3 + 6066517217542144T^2 - 6591405588033536*T - 14458029971664896
$97$	$ T^{18} + \cdots - 31840696422400 $ T^18 - 70T^17 + 1680T^16 - 6237T^15 - 442792T^14 + 8220354T^13 - 29956755T^12 - 713810400T^11 + 9994208252T^10 - 39612599088T^9 - 185662210048T^8 + 2693568277120T^7 - 12097472695680T^6 + 22856842219776T^5 + 4125675764736T^4 - 90274487352320T^3 + 126492759485440T^2 - 30052470476800*T - 31840696422400
show more
show less

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4719 = 3 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4719.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{7} + 1) q^{5} - \beta_1 q^{6} - \beta_{16} q^{7} + ( - \beta_{3} - 2 \beta_1) q^{8} + q^{9} + (\beta_{17} + \beta_{16} + \cdots + \beta_{2}) q^{10}+ \cdots + ( - \beta_{17} - 2 \beta_{16} + \cdots - 4) q^{98}+O(q^{100}) \) q - b1 * q^2 + q^3 + (b2 + 2) * q^4 + (-b7 + 1) * q^5 - b1 * q^6 - b16 * q^7 + (-b3 - 2b1) q^8 + q^9 + (b17 + b16 - b15 + b14 + b6 + b5 + b4 + b2) * q^10 + (b2 + 2) * q^12 - q^13 + (b17 + b16 + b13 - b12 - b7 + b3 + 1) * q^14 + (-b7 + 1) * q^15 + (b12 + b11 + b10 + b9 + b7 + b6 + b2 + b1 + 1) * q^16 + (b13 - b5 - b1) * q^17 - b1 * q^18 + (b10 - b8 - b6 - b4 + b3 - b2) * q^19 + (-b15 - b11 - b10 - b8 - b7 + 2b6 + b5 - b3 + 2b2 - b1 + 2) * q^20 - b16 * q^21 + (-b16 + b15 - b14 - b11 - b7 - b6 + 2) * q^23 + (-b3 - 2b1) q^24 + (-b17 + b16 + b14 - b13 - b10 - b3 - b1 + 3) * q^25 + b1 * q^26 + q^27 + (-2b16 + b15 - b14 + b12 + b11 - b10 - b7 - 2b6 - b5 - b4 - b2 - b1 + 1) * q^28 + (b16 - b12 - b9 - b7 - b6 - b5 + b3 - b1 + 1) * q^29 + (b17 + b16 - b15 + b14 + b6 + b5 + b4 + b2) * q^30 + (-b17 + b15 - b14 + b11 + b10 - b8 - 2b6 - b4 + b3 - b2 + 2) q^31 + (-b17 - b14 - b8 - b7 + b5 - b3 - 2b1 - 2) q^32 + (-2b16 + b15 - b14 + b12 + b11 + b10 + b8 + b7 - 2b6 + 1) * q^34 + (-b16 + b15 + b14 - b11 - b10 + b8 + b7 - b5 - b3 - b1 + 2) * q^35 + (b2 + 2) * q^36 + (-2b15 + b14 + b7 + 2b6 - b5 + b4 + b2 + 1) * q^37 + (b17 + b16 - b15 + b14 - b12 - b11 - b10 - b9 + b8 - b7 + b6 + 1) * q^38 - q^39 + (b16 - b15 + b14 - b13 + b10 + b9 + b8 + 2b7 + 5b6 + 3b5 + b4 - b3 + 2b2) * q^40 + (b8 + b5 + b4 - b2 + b1 - 2) * q^41 + (b17 + b16 + b13 - b12 - b7 + b3 + 1) * q^42 + (-b16 - b14 + b9 + b5 - b4 + b2 + 1) * q^43 + (-b7 + 1) * q^45 + (b17 + b16 + b14 + b10 + 2b7 - b6 - b5 + 2b4 + b3 - 1) * q^46 + (b17 - b7 + b4 + b3 - b2) * q^47 + (b12 + b11 + b10 + b9 + b7 + b6 + b2 + b1 + 1) * q^48 + (b16 + b14 - b12 - b9 + b8 + b6 - b5 - b3 + 3) * q^49 + (2b16 - 3b15 + b14 - 2b13 - b11 - b10 + b7 + 4b6 + b5 + b4 - b3 + 3b2 - 2b1 + 2) * q^50 + (b13 - b5 - b1) * q^51 + (-b2 - 2) * q^52 + (-b16 + b12 - b10 - b9 + b8 - b7 + 1) * q^53 - b1 * q^54 + (b16 + b15 + b14 + b13 - 2b12 + b11 + 2b10 + b7 - 3b6 - b5 + 2b3 - 2b2 + b1 + 2) q^56 + (b10 - b8 - b6 - b4 + b3 - b2) * q^57 + (-2b16 + 2b15 + b12 - b10 + b8 - 5b6 - b5 - b4 - b2 - b1 + 5) q^58 + (b10 - 3b6 - b5 - b4 - b2 - b1 + 2) q^59 + (-b15 - b11 - b10 - b8 - b7 + 2b6 + b5 - b3 + 2b2 - b1 + 2) * q^60 + (-b15 - b13 - b12 - b11 - b10 - b9 + b8 - b7 + 2b6 + b4 + b2 + b1 + 1) q^61 + (-2b16 + b15 - b14 + b12 + b11 + b9 + 2b8 + 2b7 + 2b6 - b5 - b3 - 3b1 - 1) q^62 - b16 * q^63 + (b17 - b15 + b14 - b13 + b12 + b9 + 2b8 + 3b7 + 6b6 + 3b5 + 3b4 - b3 + 4b2 + 4b1) q^64 + (b7 - 1) * q^65 + (-b8 - 2b6 + b5 - b4 - b2 - b1 + 4) q^67 + (-b17 + 3b15 - 2b14 + b13 - b12 + b11 + 2b10 - b8 - b7 - 3b6 - 2b5 - 2b4 + b3 - 2b2 - 3b1 + 2) * q^68 + (-b16 + b15 - b14 - b11 - b7 - b6 + 2) * q^69 + (b16 - b15 - b14 - 2b12 - b11 - 2b9 - b8 - b7 - 3b6 - 3b5 - b4 + 2b3 - b2 - 3b1 + 2) * q^70 + (-b17 - 2b16 - b14 + b5 + b4 - b2 + 2b1 - 1) * q^71 + (-b3 - 2b1) q^72 + (-b16 + b15 - b14 + b12 - b11 + b10 - b8 - b7 + 2b5 - 1) q^73 + (b16 + b15 - b11 - b10 - 2b7 - 4b6 + 3b5 - b3 - 2b1 + 3) * q^74 + (-b17 + b16 + b14 - b13 - b10 - b3 - b1 + 3) * q^75 + (b16 - b15 + 2b14 - b12 - b11 - 3b10 - 2b9 + b8 - 2b7 + b1) * q^76 + b1 * q^78 + (-b17 + b15 + b14 - b13 + b12 + b11 - 2b10 - 2b6 - 2b5 - b3 + 2) q^79 + (-3b17 - b15 - b14 - 2b13 + 2b12 + b11 - b8 + b7 + 5b6 + b5 - b4 - 3b3 + b2 - 4b1 - 1) * q^80 + q^81 + (-b17 - b16 - b14 + b12 - b8 + 2b6 - b5 + b3 + 4b1 - 2) * q^82 + (-b17 - b16 - b14 + b12 + 2b11 + b10 + 2b9 + b7 + b6 - b5 - b4 - b2 - 1) * q^83 + (-2b16 + b15 - b14 + b12 + b11 - b10 - b7 - 2b6 - b5 - b4 - b2 - b1 + 1) * q^84 + (-2b17 - b16 - b15 - b14 - b12 + b11 + b7 + b6 + b5 + b3 + 3b1 - 3) * q^85 + (b16 - b15 + b14 - b12 + b11 - b8 + b7 + 3b6 - b3 - 3b1 - 1) * q^86 + (b16 - b12 - b9 - b7 - b6 - b5 + b3 - b1 + 1) * q^87 + (-b17 + b15 - 2b13 + b12 + b11 - b10 + 2b8 + b7 - 2b3 + b1 + 2) q^89 + (b17 + b16 - b15 + b14 + b6 + b5 + b4 + b2) * q^90 + b16 * q^91 + (-2b17 - 2b16 + 2b15 - 2b14 + b12 - 2b10 - b8 - 2b7 - 8b6 - 4b5 - b4 - b3 - b2 - 2b1 + 5) q^92 + (-b17 + b15 - b14 + b11 + b10 - b8 - 2b6 - b4 + b3 - b2 + 2) q^93 + (b17 + b15 + b13 - b12 - b11 - b10 - b8 - 3b7 - 4b6 + b4 + 2b3 + 3b1 + 7) * q^94 + (b17 + b16 + 2b15 - b10 - 2b8 - b7 - b6 - b4 + b3 - b2 + 2) * q^95 + (-b17 - b14 - b8 - b7 + b5 - b3 - 2b1 - 2) q^96 + (-b16 + b15 - b11 + 3b6 + b5 - b4 - b2 + 3) q^97 + (-b17 - 2b16 + b15 - b14 + 2b12 + b11 - 3b6 - b5 - 3b4 - b2 - 4b1 - 4) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 18 q^{3} + 27 q^{4} + 10 q^{5} - q^{6} - 7 q^{7} - 3 q^{8} + 18 q^{9} + 3 q^{10} + 27 q^{12} - 18 q^{13} + 10 q^{14} + 10 q^{15} + 41 q^{16} + 5 q^{17} - q^{18} - 10 q^{19} + 16 q^{20}+ \cdots - 93 q^{98}+O(q^{100}) \) 18 * q - q^2 + 18 * q^3 + 27 * q^4 + 10 * q^5 - q^6 - 7 * q^7 - 3 * q^8 + 18 * q^9 + 3 * q^10 + 27 * q^12 - 18 * q^13 + 10 * q^14 + 10 * q^15 + 41 * q^16 + 5 * q^17 - q^18 - 10 * q^19 + 16 * q^20 - 7 * q^21 + 9 * q^23 - 3 * q^24 + 52 * q^25 + q^26 + 18 * q^27 - 4 * q^28 + 3 * q^30 + 36 * q^31 - 37 * q^32 + 20 * q^34 + 23 * q^35 + 27 * q^36 + 37 * q^37 + 6 * q^38 - 18 * q^39 + 54 * q^40 - 12 * q^41 + 10 * q^42 + 2 * q^43 + 10 * q^45 - 6 * q^46 + 41 * q^48 + 61 * q^49 + 48 * q^50 + 5 * q^51 - 27 * q^52 + 18 * q^53 - q^54 + 36 * q^56 - 10 * q^57 + 40 * q^58 + 10 * q^59 + 16 * q^60 + 12 * q^61 + 25 * q^62 - 7 * q^63 + 67 * q^64 - 10 * q^65 + 52 * q^67 + 18 * q^68 + 9 * q^69 - 13 * q^70 + q^71 - 3 * q^72 - 29 * q^73 + 2 * q^74 + 52 * q^75 - 26 * q^76 + q^78 + 18 * q^79 + 48 * q^80 + 18 * q^81 - 6 * q^82 + 26 * q^83 - 4 * q^84 - 15 * q^85 + 9 * q^86 + 59 * q^89 + 3 * q^90 + 7 * q^91 + 4 * q^92 + 36 * q^93 + 53 * q^94 + 5 * q^95 - 37 * q^96 + 70 * q^97 - 93 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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	4719.2.a.bq

Level	$4719$
Weight	$2$
Character orbit	4719.a
Self dual	yes
Analytic conductor	$37.681$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(37.6814047138\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} - x^{17} - 31 x^{16} + 29 x^{15} + 396 x^{14} - 348 x^{13} - 2689 x^{12} + 2242 x^{11} + \cdots + 55 \) x^18 - x^17 - 31x^16 + 29x^15 + 396x^14 - 348x^13 - 2689x^12 + 2242x^11 + 10427x^10 - 8392x^9 - 22969x^8 + 18293x^7 + 26541x^6 - 21589x^5 - 12506x^4 + 11016x^3 + 151x^2 - 630x + 55
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 4 \) v^2 - 4
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 6\nu \) v^3 - 6*v
\(\beta_{4}\)	\(=\)	\( ( 1794316 \nu^{17} + 40117477 \nu^{16} - 121295666 \nu^{15} - 1140544478 \nu^{14} + \cdots - 2013758985 ) / 1855265315 \) (1794316v^17 + 40117477v^16 - 121295666v^15 - 1140544478v^14 + 2455273968v^13 + 12958646951v^12 - 23249070804v^11 - 74554608503v^10 + 117304780481v^9 + 225228527484v^8 - 323028890972v^7 - 326151717937v^6 + 456971721696v^5 + 146628206998v^4 - 256386569273v^3 + 55145841762v^2 - 6212666145*v - 2013758985) / 1855265315
\(\beta_{5}\)	\(=\)	\( ( 7494672 \nu^{17} + 9572099 \nu^{16} - 220222849 \nu^{15} - 225723188 \nu^{14} + \cdots - 1885514235 ) / 3710530630 \) (7494672v^17 + 9572099v^16 - 220222849v^15 - 225723188v^14 + 2624974410v^13 + 1929291318v^12 - 16482600428v^11 - 6757388867v^10 + 59589480036v^9 + 4624640560v^8 - 127362413246v^7 + 25136398721v^6 + 155074744519v^5 - 53687713782v^4 - 90044577870v^3 + 32995767138v^2 + 8608947330*v - 1885514235) / 3710530630
\(\beta_{6}\)	\(=\)	\( ( - 34282077 \nu^{17} + 26787405 \nu^{16} + 1053172288 \nu^{15} - 773957384 \nu^{14} + \cdots + 12988761180 ) / 3710530630 \) (-34282077v^17 + 26787405v^16 + 1053172288v^15 - 773957384v^14 - 13349979304v^13 + 9305188386v^12 + 90255213735v^11 - 60377816206v^10 - 350701828012v^9 + 228105710148v^8 + 782800386053v^7 - 499759621315v^6 - 935017004378v^5 + 585041015834v^4 + 482419368744v^3 - 287606782362v^2 - 38172360765*v + 12988761180) / 3710530630
\(\beta_{7}\)	\(=\)	\( ( 22768834 \nu^{17} - 83197933 \nu^{16} - 578141728 \nu^{15} + 2298516130 \nu^{14} + \cdots - 7596659310 ) / 1855265315 \) (22768834v^17 - 83197933v^16 - 578141728v^15 + 2298516130v^14 + 5631399805v^13 - 25578132025v^12 - 25599283324v^11 + 146909595751v^10 + 48395207000v^9 - 463072713130v^8 + 4925094585v^7 + 782447233153v^6 - 121502960412v^5 - 629431002480v^4 + 86830861630v^3 + 171050822747v^2 + 20363741090*v - 7596659310) / 1855265315
\(\beta_{8}\)	\(=\)	\( ( 58233332 \nu^{17} - 169924489 \nu^{16} - 1608424113 \nu^{15} + 4611815144 \nu^{14} + \cdots - 6979046035 ) / 3710530630 \) (58233332v^17 - 169924489v^16 - 1608424113v^15 + 4611815144v^14 + 18026238822v^13 - 49967370466v^12 - 105807297202v^11 + 275695509921v^10 + 350216230262v^9 - 819097544944v^8 - 650322872358v^7 + 1277086255749v^6 + 610962999323v^5 - 951252447354v^4 - 190982799652v^3 + 272518498362v^2 - 22986934480*v - 6979046035) / 3710530630
\(\beta_{9}\)	\(=\)	\( ( 62700640 \nu^{17} + 30228543 \nu^{16} - 2251325551 \nu^{15} - 529501840 \nu^{14} + \cdots - 21931031955 ) / 3710530630 \) (62700640v^17 + 30228543v^16 - 2251325551v^15 - 529501840v^14 + 32928162714v^13 + 1291601422v^12 - 252648637866v^11 + 28772998647v^10 + 1088720697380v^9 - 249754534558v^8 - 2608196748494v^7 + 812270473777v^6 + 3199451092551v^5 - 1166157887650v^4 - 1625164594964v^3 + 636369597688v^2 + 137795258290*v - 21931031955) / 3710530630
\(\beta_{10}\)	\(=\)	\( ( 63851057 \nu^{17} + 8776947 \nu^{16} - 1945929542 \nu^{15} - 574209510 \nu^{14} + \cdots + 8111535620 ) / 3710530630 \) (63851057v^17 + 8776947v^16 - 1945929542v^15 - 574209510v^14 + 24391477788v^13 + 10787243076v^12 - 162434416509v^11 - 91534278566v^10 + 619587083450v^9 + 393419836684v^8 - 1357342887117v^7 - 848508183097v^6 + 1600751696932v^5 + 811944616820v^4 - 812819729008v^3 - 235795201440v^2 + 56422799485*v + 8111535620) / 3710530630
\(\beta_{11}\)	\(=\)	\( ( - 32589642 \nu^{17} + 67028098 \nu^{16} + 937238049 \nu^{15} - 1806452155 \nu^{14} + \cdots + 11821918765 ) / 1855265315 \) (-32589642v^17 + 67028098v^16 + 937238049v^15 - 1806452155v^14 - 11112524948v^13 + 19516653465v^12 + 70669029464v^11 - 108433991633v^10 - 262569229895v^9 + 331813227551v^8 + 578176745795v^7 - 561408171288v^6 - 715877388944v^5 + 508723190210v^4 + 419507127993v^3 - 215275576641v^2 - 66796677570*v + 11821918765) / 1855265315
\(\beta_{12}\)	\(=\)	\( ( - 72628004 \nu^{17} - 33453225 \nu^{16} + 2425890163 \nu^{15} + 893540784 \nu^{14} + \cdots + 18353930655 ) / 3710530630 \) (-72628004v^17 - 33453225v^16 + 2425890163v^15 + 893540784v^14 - 33007410912v^13 - 9261075764v^12 + 234688348360v^11 + 46187887889v^10 - 929257907028v^9 - 109252041116v^8 + 2016535568798v^7 + 93919206905v^6 - 2190425086393v^5 + 14298410166v^4 + 942888975982v^3 - 50491820508v^2 - 66890354680*v + 18353930655) / 3710530630
\(\beta_{13}\)	\(=\)	\( ( 94494963 \nu^{17} - 185484818 \nu^{16} - 2805678553 \nu^{15} + 5218759922 \nu^{14} + \cdots - 56440849885 ) / 3710530630 \) (94494963v^17 - 185484818v^16 - 2805678553v^15 + 5218759922v^14 + 34466735510v^13 - 59840521738v^12 - 227518675579v^11 + 361739405471v^10 + 876522867636v^9 - 1249096533620v^8 - 1992342264249v^7 + 2497311728078v^6 + 2521668233933v^5 - 2752314000092v^4 - 1447133728430v^3 + 1344845725842v^2 + 143825671665*v - 56440849885) / 3710530630
\(\beta_{14}\)	\(=\)	\( ( 2631396 \nu^{17} + 2929809 \nu^{16} - 84836171 \nu^{15} - 83694562 \nu^{14} + 1120419802 \nu^{13} + \cdots - 479252305 ) / 86291410 \) (2631396v^17 + 2929809v^16 - 84836171v^15 - 83694562v^14 + 1120419802v^13 + 947599950v^12 - 7809176268v^11 - 5373611193v^10 + 30823730214v^9 + 15743358316v^8 - 68554722640v^7 - 21390846209v^6 + 79751081741v^5 + 7589369052v^4 - 38922886802v^3 + 5079885718v^2 + 2257187100*v - 479252305) / 86291410
\(\beta_{15}\)	\(=\)	\( ( - 120061537 \nu^{17} + 119261603 \nu^{16} + 3769620476 \nu^{15} - 3438745238 \nu^{14} + \cdots + 36700715240 ) / 3710530630 \) (-120061537v^17 + 119261603v^16 + 3769620476v^15 - 3438745238v^14 - 48862480042v^13 + 40891326124v^12 + 337460411449v^11 - 260068729732v^10 - 1335287384064v^9 + 957205548514v^8 + 3019237908037v^7 - 2042879830293v^6 - 3631953257486v^5 + 2343847853498v^4 + 1880111935362v^3 - 1139239346960v^2 - 149853313935*v + 36700715240) / 3710530630
\(\beta_{16}\)	\(=\)	\( ( - 121444634 \nu^{17} - 87841057 \nu^{16} + 3911217537 \nu^{15} + 2506477964 \nu^{14} + \cdots + 29096747885 ) / 3710530630 \) (-121444634v^17 - 87841057v^16 + 3911217537v^15 + 2506477964v^14 - 51693114648v^13 - 28061150712v^12 + 361386416174v^11 + 154416932401v^10 - 1435360749228v^9 - 420044583464v^8 + 3228621352814v^7 + 449269984297v^6 - 3832531928177v^5 + 105967198186v^4 + 1947087254988v^3 - 383154874860v^2 - 148316576330*v + 29096747885) / 3710530630
\(\beta_{17}\)	\(=\)	\( ( - 209426356 \nu^{17} + 219910667 \nu^{16} + 6192440073 \nu^{15} - 5835704426 \nu^{14} + \cdots + 33473638275 ) / 3710530630 \) (-209426356v^17 + 219910667v^16 + 6192440073v^15 - 5835704426v^14 - 74842115508v^13 + 62306127984v^12 + 476317842946v^11 - 345206808991v^10 - 1712837563428v^9 + 1072903204176v^8 + 3460963343462v^7 - 1897037936347v^6 - 3638468318213v^5 + 1830083869296v^4 + 1567565855338v^3 - 800059462592v^2 - 46822155590*v + 33473638275) / 3710530630

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 4 \) b2 + 4
\(\nu^{3}\)	\(=\)	\( \beta_{3} + 6\beta_1 \) b3 + 6*b1
\(\nu^{4}\)	\(=\)	\( \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + 7\beta_{2} + \beta _1 + 21 \) b12 + b11 + b10 + b9 + b7 + b6 + 7*b2 + b1 + 21
\(\nu^{5}\)	\(=\)	\( \beta_{17} + \beta_{14} + \beta_{8} + \beta_{7} - \beta_{5} + 9\beta_{3} + 38\beta _1 + 2 \) b17 + b14 + b8 + b7 - b5 + 9b3 + 38b1 + 2
\(\nu^{6}\)	\(=\)	\( \beta_{17} - \beta_{15} + \beta_{14} - \beta_{13} + 11 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} + \cdots + 122 \) b17 - b15 + b14 - b13 + 11b12 + 10b11 + 10b10 + 11b9 + 2b8 + 13b7 + 16b6 + 3b5 + 3b4 - b3 + 50b2 + 14*b1 + 122
\(\nu^{7}\)	\(=\)	\( 14 \beta_{17} + 13 \beta_{14} + 2 \beta_{11} + \beta_{10} + 15 \beta_{8} + 14 \beta_{7} - 3 \beta_{6} + \cdots + 26 \) 14b17 + 13b14 + 2b11 + b10 + 15b8 + 14b7 - 3b6 - 12b5 + b4 + 70b3 + b2 + 253*b1 + 26
\(\nu^{8}\)	\(=\)	\( 16 \beta_{17} + 4 \beta_{16} - 17 \beta_{15} + 17 \beta_{14} - 16 \beta_{13} + 95 \beta_{12} + \cdots + 757 \) 16b17 + 4b16 - 17b15 + 17b14 - 16b13 + 95b12 + 81b11 + 83b10 + 95b9 + 29b8 + 126b7 + 164b6 + 46b5 + 44b4 - 12b3 + 363b2 + 142*b1 + 757
\(\nu^{9}\)	\(=\)	\( 142 \beta_{17} - \beta_{16} + 3 \beta_{15} + 123 \beta_{14} + 2 \beta_{13} + \beta_{12} + 35 \beta_{11} + \cdots + 247 \) 142b17 - b16 + 3b15 + 123b14 + 2b13 + b12 + 35b11 + 21b10 + 4b9 + 156b8 + 147b7 - 46b6 - 105b5 + 18b4 + 529b3 + 16b2 + 1746*b1 + 247
\(\nu^{10}\)	\(=\)	\( 179 \beta_{17} + 81 \beta_{16} - 196 \beta_{15} + 204 \beta_{14} - 174 \beta_{13} + 755 \beta_{12} + \cdots + 4920 \) 179b17 + 81b16 - 196b15 + 204b14 - 174b13 + 755b12 + 616b11 + 653b10 + 756b9 + 300b8 + 1093b7 + 1445b6 + 493b5 + 457b4 - 102b3 + 2659b2 + 1275*b1 + 4920
\(\nu^{11}\)	\(=\)	\( 1270 \beta_{17} - 11 \beta_{16} + 67 \beta_{15} + 1041 \beta_{14} + 42 \beta_{13} + 19 \beta_{12} + \cdots + 2117 \) 1270b17 - 11b16 + 67b15 + 1041b14 + 42b13 + 19b12 + 415b11 + 277b10 + 84b9 + 1409b8 + 1373b7 - 502b6 - 824b5 + 220b4 + 3970b3 + 182b2 + 12349*b1 + 2117
\(\nu^{12}\)	\(=\)	\( 1730 \beta_{17} + 1069 \beta_{16} - 1917 \beta_{15} + 2098 \beta_{14} - 1612 \beta_{13} + 5772 \beta_{12} + \cdots + 33081 \) 1730b17 + 1069b16 - 1917b15 + 2098b14 - 1612b13 + 5772b12 + 4573b11 + 5034b10 + 5804b9 + 2721b8 + 8976b7 + 11893b6 + 4602b5 + 4144b4 - 738b3 + 19573b2 + 10813*b1 + 33081
\(\nu^{13}\)	\(=\)	\( 10668 \beta_{17} - 21 \beta_{16} + 951 \beta_{15} + 8392 \beta_{14} + 570 \beta_{13} + 227 \beta_{12} + \cdots + 17476 \) 10668b17 - 21b16 + 951b15 + 8392b14 + 570b13 + 227b12 + 4171b11 + 3004b10 + 1136b9 + 11867b8 + 12034b7 - 4859b6 - 6155b5 + 2280b4 + 29750b3 + 1820b2 + 88830*b1 + 17476
\(\nu^{14}\)	\(=\)	\( 15488 \beta_{17} + 11709 \beta_{16} - 17161 \beta_{15} + 19779 \beta_{14} - 13724 \beta_{13} + \cdots + 228164 \) 15488b17 + 11709b16 - 17161b15 + 19779b14 - 13724b13 + 43204b12 + 33628b11 + 38481b10 + 43785b9 + 23101b8 + 71492b7 + 94404b6 + 40152b5 + 35097b4 - 4729b3 + 144492b2 + 89005*b1 + 228164
\(\nu^{15}\)	\(=\)	\( 86523 \beta_{17} + 1130 \beta_{16} + 10994 \beta_{15} + 66128 \beta_{14} + 6378 \beta_{13} + \cdots + 142382 \) 86523b17 + 1130b16 + 10994b15 + 66128b14 + 6378b13 + 2179b12 + 38350b11 + 29355b10 + 12630b9 + 96098b8 + 101448b7 - 44497b6 - 44688b5 + 21623b4 + 222979b3 + 17063b2 + 646642*b1 + 142382
\(\nu^{16}\)	\(=\)	\( 132548 \beta_{17} + 115815 \beta_{16} - 145567 \beta_{15} + 176614 \beta_{14} - 111113 \beta_{13} + \cdots + 1604495 \) 132548b17 + 115815b16 - 145567b15 + 176614b14 - 111113b13 + 319323b12 + 246558b11 + 293180b10 + 327411b9 + 188873b8 + 559123b7 + 733278b6 + 337381b5 + 285987b4 - 26345b3 + 1068642b2 + 720237*b1 + 1604495
\(\nu^{17}\)	\(=\)	\( 687197 \beta_{17} + 24430 \beta_{16} + 113127 \beta_{15} + 515665 \beta_{14} + 64120 \beta_{13} + \cdots + 1154806 \) 687197b17 + 24430b16 + 113127b15 + 515665b14 + 64120b13 + 18271b12 + 334147b11 + 269470b10 + 125920b9 + 759945b8 + 833835b7 - 394567b6 - 317624b5 + 194381b4 + 1672500b3 + 153863b2 + 4748228*b1 + 1154806

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.18
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{18} + T^{17} + \cdots + 55 \) T^18 + T^17 - 31T^16 - 29T^15 + 396T^14 + 348T^13 - 2689T^12 - 2242T^11 + 10427T^10 + 8392T^9 - 22969T^8 - 18293T^7 + 26541T^6 + 21589T^5 - 12506T^4 - 11016T^3 + 151T^2 + 630T + 55
$3$	\( (T - 1)^{18} \) (T - 1)^18
$5$	\( T^{18} - 10 T^{17} + \cdots - 42944 \) T^18 - 10T^17 - 21T^16 + 494T^15 - 590T^14 - 8923T^13 + 22386T^12 + 70738T^11 - 262602T^10 - 196875T^9 + 1370062T^8 - 335670T^7 - 2976127T^6 + 2286223T^5 + 1599855T^4 - 1909436T^3 + 177072T^2 + 225056*T - 42944
$7$	\( T^{18} + 7 T^{17} + \cdots - 11501824 \) T^18 + 7T^17 - 69T^16 - 569T^15 + 1618T^14 + 18070T^13 - 12146T^12 - 288683T^11 - 65522T^10 + 2536215T^9 + 1370817T^8 - 12956016T^7 - 6678188T^6 + 38445072T^5 + 10284736T^4 - 59460800T^3 + 5248128T^2 + 31615744*T - 11501824
$11$	\( T^{18} \) T^18
$13$	\( (T + 1)^{18} \) (T + 1)^18
$17$	\( T^{18} + \cdots - 567398656 \) T^18 - 5T^17 - 188T^16 + 975T^15 + 13943T^14 - 76135T^13 - 514457T^12 + 3041932T^11 + 9703132T^10 - 65910842T^9 - 82878429T^8 + 756458628T^7 + 151000984T^6 - 4180186288T^5 + 1207493376T^4 + 8737926208T^3 - 1955930304T^2 - 4108622080*T - 567398656
$19$	\( T^{18} + \cdots - 117633024 \) T^18 + 10T^17 - 159T^16 - 1719T^15 + 9057T^14 + 113429T^13 - 208203T^12 - 3654521T^11 + 738533T^10 + 60704990T^9 + 44425977T^8 - 494741160T^7 - 664642568T^6 + 1575028192T^5 + 2941604224T^4 - 205303552T^3 - 1885797376T^2 - 872466432*T - 117633024
$23$	\( T^{18} + \cdots + 12466253824 \) T^18 - 9T^17 - 226T^16 + 2183T^15 + 18963T^14 - 208170T^13 - 694221T^12 + 9937288T^11 + 7908088T^10 - 251940896T^9 + 148890592T^8 + 3345936704T^7 - 4726624448T^6 - 21047083520T^5 + 40710692608T^4 + 44954033152T^3 - 103620557824T^2 - 1639460864*T + 12466253824
$29$	\( T^{18} + \cdots - 2567147264 \) T^18 - 289T^16 + 45T^15 + 32083T^14 - 14869T^13 - 1761253T^12 + 1629801T^11 + 50928159T^10 - 75237332T^9 - 741492231T^8 + 1554759612T^7 + 4419211464T^6 - 13342182016T^5 - 2602510368T^4 + 37103270080T^3 - 40523074112T^2 + 17121031680T - 2567147264
$31$	\( T^{18} + \cdots + 29764267264 \) T^18 - 36T^17 + 319T^16 + 3226T^15 - 68574T^14 + 151534T^13 + 3890650T^12 - 24994358T^11 - 57905224T^10 + 917742268T^9 - 1222930769T^8 - 12185580084T^7 + 39600858412T^6 + 31639114672T^5 - 271925016352T^4 + 262701910848T^3 + 155856694784T^2 - 213731399680*T + 29764267264
$37$	\( T^{18} + \cdots - 627994624 \) T^18 - 37T^17 + 313T^16 + 4594T^15 - 95984T^14 + 300191T^13 + 5515663T^12 - 52975828T^11 + 69533684T^10 + 1261315584T^9 - 7485073280T^8 + 14900551360T^7 + 544926976T^6 - 35400636160T^5 + 20518065152T^4 + 20570799104T^3 - 3588994048T^2 - 4190760960*T - 627994624
$41$	\( T^{18} + \cdots + 385454080 \) T^18 + 12T^17 - 228T^16 - 3419T^15 + 12998T^14 + 343330T^13 + 384409T^12 - 14244552T^11 - 54434876T^10 + 191396192T^9 + 1412596080T^8 + 1068670336T^7 - 8749555008T^6 - 21965899264T^5 - 11436977152T^4 + 16396993536T^3 + 21035544576T^2 + 6845112320*T + 385454080
$43$	\( T^{18} + \cdots + 5808956218736 \) T^18 - 2T^17 - 395T^16 + 1284T^15 + 63956T^14 - 280008T^13 - 5386114T^12 + 29592052T^11 + 245225022T^10 - 1677809168T^9 - 5511837056T^8 + 51054619752T^7 + 37681429009T^6 - 771544152946T^5 + 381442814909T^4 + 5014237373420T^3 - 4102018073540T^2 - 11846860912192*T + 5808956218736
$47$	\( T^{18} - 388 T^{16} + \cdots - 3117031 \) T^18 - 388T^16 - 676T^15 + 55894T^14 + 172581T^13 - 3594397T^12 - 14448835T^11 + 99331414T^10 + 448230848T^9 - 1052084948T^8 - 4775336179T^7 + 3031272645T^6 + 8343908788T^5 - 3637021904T^4 - 2901072939T^3 + 1289267392T^2 - 95302921T - 3117031
$53$	\( T^{18} + \cdots - 13263190902016 \) T^18 - 18T^17 - 495T^16 + 9626T^15 + 92073T^14 - 1991955T^13 - 8055327T^12 + 200931717T^11 + 358034125T^10 - 10273154953T^9 - 10208391717T^8 + 255895382968T^7 + 261098138708T^6 - 2806682192112T^5 - 3881020004848T^4 + 9661995940544T^3 + 15102799380800T^2 - 7719173792512*T - 13263190902016
$59$	\( T^{18} - 10 T^{17} + \cdots + 647429 \) T^18 - 10T^17 - 256T^16 + 2526T^15 + 20379T^14 - 213512T^13 - 511334T^12 + 7282998T^11 + 1269908T^10 - 108575776T^9 + 79013359T^8 + 718110166T^7 - 685325953T^6 - 2156093906T^5 + 1714764323T^4 + 2468108484T^3 - 916696083T^2 - 97120092*T + 647429
$61$	\( T^{18} + \cdots + 180385112801839 \) T^18 - 12T^17 - 497T^16 + 6087T^15 + 100770T^14 - 1279967T^13 - 10679966T^12 + 144209584T^11 + 627349081T^10 - 9398649365T^9 - 19708740427T^8 + 357340798104T^7 + 273264140004T^6 - 7616841176129T^5 + 17853017070T^4 + 82043961498417T^3 - 32594663645089T^2 - 340489724339574*T + 180385112801839
$67$	\( T^{18} + \cdots + 44563124224 \) T^18 - 52T^17 + 980T^16 - 5827T^15 - 54470T^14 + 948428T^13 - 2745211T^12 - 28857420T^11 + 205122224T^10 + 89339920T^9 - 4117614480T^8 + 7077632640T^7 + 30207969600T^6 - 88821255680T^5 - 48240659200T^4 + 264343298048T^3 - 11181654016T^2 - 226059489280*T + 44563124224
$71$	\( T^{18} + \cdots + 217467806374144 \) T^18 - T^17 - 791T^16 + 17T^15 + 251231T^14 + 165835T^13 - 41467315T^12 - 41591114T^11 + 3876885813T^10 + 4043236496T^9 - 209667631777T^8 - 172632118748T^7 + 6379461594008T^6 + 2379514058384T^5 - 99260020157216T^4 + 30826612249856T^3 + 599412806224064T^2 - 753333650501632T + 217467806374144
$73$	\( T^{18} + \cdots + 57371318726656 \) T^18 + 29T^17 - 322T^16 - 17629T^15 - 72359T^14 + 3355152T^13 + 38844467T^12 - 123251816T^11 - 4471056528T^10 - 21796036592T^9 + 87804789536T^8 + 1351701066176T^7 + 5609602742592T^6 + 7671771975936T^5 - 13482266350336T^4 - 55263292340224T^3 - 41349592734720T^2 + 44125476884480*T + 57371318726656
$79$	\( T^{18} + \cdots + 1752198906176 \) T^18 - 18T^17 - 563T^16 + 11309T^15 + 115645T^14 - 2792883T^13 - 10021066T^12 + 348864049T^11 + 185597013T^10 - 23437761985T^9 + 26263774412T^8 + 815599704739T^7 - 1710807352566T^6 - 12304717516087T^5 + 32909241932655T^4 + 40587450924484T^3 - 86455118796752T^2 - 84483534977632*T + 1752198906176
$83$	\( T^{18} + \cdots + 9652886263955 \) T^18 - 26T^17 - 459T^16 + 16186T^15 + 39417T^14 - 3737390T^13 + 9568709T^12 + 396544164T^11 - 2001333811T^10 - 19554725592T^9 + 132115250996T^8 + 392086057260T^7 - 3447336076428T^6 - 2330368940698T^5 + 34297852697139T^4 - 5628763181578T^3 - 86593266945904T^2 + 41289500800010*T + 9652886263955
$89$	\( T^{18} + \cdots - 14\!\cdots\!96 \) T^18 - 59T^17 + 796T^16 + 17881T^15 - 559079T^14 + 1349592T^13 + 102473265T^12 - 1007071920T^11 - 5534043900T^10 + 123867095056T^9 - 226960221328T^8 - 5575005151360T^7 + 29104201833920T^6 + 72192454689280T^5 - 768464032246784T^4 + 459234886311936T^3 + 6066517217542144T^2 - 6591405588033536*T - 14458029971664896
$97$	\( T^{18} + \cdots - 31840696422400 \) T^18 - 70T^17 + 1680T^16 - 6237T^15 - 442792T^14 + 8220354T^13 - 29956755T^12 - 713810400T^11 + 9994208252T^10 - 39612599088T^9 - 185662210048T^8 + 2693568277120T^7 - 12097472695680T^6 + 22856842219776T^5 + 4125675764736T^4 - 90274487352320T^3 + 126492759485440T^2 - 30052470476800*T - 31840696422400
show more
show less

\( p \)	Sign
\(3\)	\( -1 \)
\(11\)	\( +1 \)
\(13\)	\( +1 \)