LMFDB - Newform orbit 666.2.e.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [666,2,Mod(223,666)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 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("666.223"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$666 = 2 \cdot 3^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	666.e (of order $3$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [20,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$5.31803677462$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} + 28 x^{18} + 304 x^{16} + 1654 x^{14} + 4950 x^{12} + 8429 x^{10} + 8179 x^{8} + 4427 x^{6} + \cdots + 9$ x^20 + 28x^18 + 304x^16 + 1654x^14 + 4950x^12 + 8429x^10 + 8179x^8 + 4427x^6 + 1273x^4 + 177*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$3^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{11} q^{2} + \beta_{7} q^{3} + (\beta_{11} - 1) q^{4} - \beta_{12} q^{5} + \beta_{14} q^{6} + (\beta_{17} + \beta_{15} + \beta_{9} + \cdots - 1) q^{7} + q^{8} + ( - \beta_{15} + \beta_{11} + \cdots - \beta_{2}) q^{9}+ \cdots + ( - 2 \beta_{19} - 3 \beta_{18} + \cdots + 5) q^{99}+O(q^{100})$ q - b11 * q^2 + b7 * q^3 + (b11 - 1) * q^4 - b12 * q^5 + b14 * q^6 + (b17 + b15 + b9 + b7 + b2 + b1 - 1) * q^7 + q^8 + (-b15 + b11 - b8 - b5 - b2) * q^9 + b6 * q^10 + (-b18 + b16 - b11 - b10 - b9 + b8 - b4 - b3 - b2 + 1) * q^11 + (-b14 - b7) * q^12 + (-b19 - b18 + b17 - b15 - b13 + b11 + b10 - b8 - b7 + 2b6 - b5 - b3 - b1) q^13 + (-b19 - b18 - b15 + b14 - b13 + b10 - b9 - b8 + 2b6 - b5 - b3 - b2 - b1 + 1) q^14 + (b19 - b18 + b17 - b13 - b11 + b9 + b8 - b5 - 2b3 - b2 + b1 - 1) q^15 - b11 * q^16 + (-2b17 - b15 - b14 + b13 - b12 - 2b9 - b8 + b6 - b5 - b4 + b3 - b2 - b1 + 2) * q^17 + (-b11 + b5 + b2 + 1) * q^18 + (-b10 + 2b8 - b7 - b6 + 2) q^19 + (b12 - b6) * q^20 + (2b19 + b18 + b17 + b13 + b12 + 2b11 + 2b9 - b8 - 2b6 - b5 - b3 + b1 - 3) * q^21 + (-b19 + b11 - b9 - b8 + b6 + b4 + b3 - b1) * q^22 + (-b19 + b15 + b14 - b13 - b11 - 2b10 - 3b9 + b8 + 2b7 + b6 + b5 - b4 + 1) q^23 + b7 * q^24 + (b17 + b16 + b15 + b14 - b13 + b12 - 4b11 - b10 + b8 + 2b7 - b4 - 2b3 + b2 + 1) q^25 + (b19 + b18 - b17 - b14 + b13 - b6 + b5 + b3 + 1) * q^26 + (-b16 - b13 - 2b11 + b10 - b5 + b4) q^27 + (b19 + b18 - b17 - b14 + b13 - b10 + b8 - b7 - 2b6 + b5 + b3) q^28 + (-b18 - b17 - b16 - b11 - b9 + b8 - b5 - b2 - b1) * q^29 + (-b19 - b18 - b17 - b15 + b13 + b11 - 2b9 - b8 + b6 + 2b3 - 2b1 + 1) q^30 + (-2b17 - 2b14 - b12) * q^31 + (b11 - 1) * q^32 + (b16 - b15 + b14 - 2b11 - b10 - 2b9 + b8 + 2b6 - b5 - 2b4 - 3b3 - b2 - b1 + 3) q^33 + (b18 + b16 + b14 + b12 - b11 - b10 + b9 + b8 + b7 - 2b6 + b5 - b3 + b1) q^34 + (-b18 + 2b17 + b16 + b14 - b13 + b12 - b10 + b8 + b7 + b6 - 2b5 - 3b3 - 2b2 - 1) * q^35 + (b15 + b8 - 1) * q^36 - q^37 + (b17 - b14 - 2b11 + b10 + b9 - 2b8 + b6 + b3 + b2) * q^38 + (-b18 + b15 - b14 - b13 - b12 + 2b11 - b9 + 2b8 - b7 + b1 - 3) * q^39 - b12 * q^40 + (b19 + b17 - b15 + b13 + b11 + b10 + 2b9 - b8 - b7 - b6 - b5 + b4 - 1) q^41 + (-b19 - b18 - b17 - b15 - b12 + b10 - b9 + b6 + 2) * q^42 + (-b18 - b17 - 2b16 - b15 + b13 - b12 - b11 + 2b10 + b9 + b7 + b6 - b5 + b4 + b3 + b2 - b1 - 1) * q^43 + (b19 + b18 - b16 + b10 + 2b9 - b6 + b2 + b1 - 1) q^44 + (-b19 - b18 + b16 - 2b15 + 2b14 + b12 - 2b11 - b10 - 3b9 + 2b8 - b5 - 2b4 - 4b3 - 2b2 - 2b1 + 4) q^45 + (b19 + b16 + b14 + b9 - b7 - b6 - b5 - b3) * q^46 + (b18 + b17 + b16 - b14 + b12 - b11 + 2b8 - 2b7 - b6 + b5 - 3b3 + b1) q^47 + b14 * q^48 + (-b18 + 2b17 + b16 - 3b15 - b13 + 2b12 + b11 - b10 + b6 - 2b5 - 3b4 - 5b3) * q^49 + (b19 - b15 + b14 + b13 + 4b11 + 2b10 + 2b9 - b8 - b7 - b6 - b5 + b4 - b2 - 4) q^50 + (-b19 - b18 - b17 - b16 + b15 - b14 - 3b12 - b11 + b10 - 2b9 - 3b8 + b7 + 4b6 - b5 + 2b4 + 4b3 - b2 - b1 + 1) * q^51 + (b15 + b14 - b11 - b10 + b8 + b7 - b6 + b1 - 1) * q^52 + (-2b19 - b18 + b17 - b16 + b14 - b13 + 2b10 + b8 + 2b7 + 2b6 + b2 - 2) * q^53 + (b19 - b17 - b15 + b13 - b12 + b11 + b9 - b6 - b4 + b3 - 2) * q^54 + (-b19 - b18 - b15 + 3b14 + b10 + b9 - 4b8 + 3b6 - 2b5 - b4 - b3 + b2 - b1 + 2) * q^55 + (b17 + b15 + b9 + b7 + b2 + b1 - 1) * q^56 + (-b19 + b18 + b17 + 3b15 - b13 + b12 - 2b11 - 3b10 - 2b9 + 4b8 + 3b7 - b6 + 3b5 + b2 + b1 - 2) q^57 + (b18 + b16 - b15 + b13 + b11 - b10 - b6 - b4) * q^58 + (3b19 + 2b18 - 2b17 + b16 - 2b15 - b14 + 4b13 + 4b11 + b10 + 4b9 - b8 - b7 - 5b6 - b5 + b4 + b3 - b2 + b1 - 4) * q^59 + (2b18 + b15 + b9 - b6 + b5 + b2 + b1) q^60 + (b18 + b15 + 2b14 - b13 - b12 - b11 - b10 + 2b7 + b6 + b2) * q^61 + (2b17 + 2b14 + 2b12 + 2b9 + 2b8 + 2b7 - b6 - 2b3) q^62 + (2b19 + b18 - b17 - 2b14 + b10 - 2b8 - 3b7 - 2b6 + b4 + b3 - b2 - b1 + 2) q^63 + q^64 + (2b18 + b17 + b16 - b13 + 2b12 - 3b11 - b9 - 2b7 - b6 + b5 - 2b3 + b2 + 1) q^65 + (b19 + b18 - b17 + b16 - b14 + b13 + b12 + b11 + b9 - b7 - 3b6 + b5 + b4 + 2b3 + b1 - 2) * q^66 + (b19 - 2b14 + b12 + 2b9 + b8 - b7 - b6 - b4 + b2 + b1 - 1) * q^67 + (-b18 + 2b17 - b16 + b15 - b13 + b11 + b10 + b9 - b7 + b6 + b4 + b2 - 2) q^68 + (-b19 + b17 - b16 - b15 - b14 - 3b11 + b10 - 3b8 + 2b6 - b4 + 2b3 + b2 + 3) * q^69 + (-b17 - b16 - 2b15 + b13 + 2b11 + 2b10 - 3b8 - b7 + b4 + 3b3 + b2 - b1) q^70 + (-b19 + 2b17 - b16 + b15 - b13 + b12 - b10 + b9 + 2b5 + b4 + b2 + b1 + 2) * q^71 + (-b15 + b11 - b8 - b5 - b2) * q^72 + (2b19 + b18 - b17 - b15 + b14 + b13 + 2b9 - b8 - b7 - 2b6 - 2b5 - b4 - b3) * q^73 + b11 * q^74 + (-b19 + b16 - b15 + 3b14 - b13 + 4b11 - b10 - 3b9 - 3b8 + b7 + 2b6 - b5 - 2b4 - b3 - 2b2 - 2) q^75 + (-b17 + b14 + 2b11 - b9 + b7 - b3 - b2 - 2) q^76 + (-b19 + b18 + b17 + b16 + 2b14 + 3b12 - 3b10 - 3b9 + b8 + 2b7 + b5 - 2b4 - 2b3 + 1) q^77 + (-b19 - b18 - b17 - b15 - b12 - b9 - 2b8 + b7 + 2b6 - b5 + 2b3 - b2 - 2b1 + 4) * q^78 + (-2b18 - b17 + b16 - 2b15 + b14 + b13 - b12 - 3b11 + 2b7 + b6 - b4 - b1 + 2) * q^79 + b6 * q^80 + (-b19 + b18 + 2b15 + 2b14 - b11 - b10 + b8 + 2b7 - 3b6 + 2b5 + b4 + b3 + 2b1) * q^81 + (-b19 - b17 - b16 - b14 - b12 + b10 - b9 - b8 + 2b6 + b5 + 2b3 + b2) * q^82 + (-2b17 - b16 - 3b15 - b14 + b13 + b12 + 4b11 + 2b10 - 3b8 - 2b7 - b6 + b4 + 3b3 - b2 - 2b1 + 1) * q^83 + (-b19 + b15 - b13 - 2b11 - b10 - b9 + b8 + b6 + b5 + b3 - b1 + 1) q^84 + (2b19 - 2b17 - b16 + b15 - 4b14 + b13 - 3b12 + b11 + b10 + 3b9 - b7 - 2b6 + 2b4 + 2b3 + b2 + b1 - 3) * q^85 + (-b19 + b18 + b16 + b14 + b12 + b11 - b10 - b9 + b8 + b5 - 2b4) q^86 + (-2b19 + b18 + b17 + b16 + b15 - b9 + b8 - b7 + b6 + 3b5 + b4 + 2) * q^87 + (-b18 + b16 - b11 - b10 - b9 + b8 - b4 - b3 - b2 + 1) * q^88 + (-b18 + 2b17 + b16 + b15 + b14 - 2b13 + 2b10 + b9 - b8 + b7 + 2b6 - b5 + b4 - 2b3 + b1 + 1) q^89 + (b19 + 2b18 - b17 + b16 + b15 - 2b14 + b13 - 2b12 + b11 - b10 - 2b7 - b6 + 2b5 + b4 + 4b3 + b2 + b1 - 2) * q^90 + (2b19 + 2b18 - b17 + 2b16 + 2b15 - b14 + 2b13 + b12 - 2b10 + b9 + 5b8 - 3b7 - 7b6 + 6b5 + 2b4 + 3b3 + 2b2 + 5) q^91 + (-b16 - b15 - 2b14 + b13 + b11 + 2b10 + 2b9 - b8 - b7 + b4 + b3 - 1) q^92 + (-b19 - b18 - b17 - 2b15 + b13 + 3b11 + 2b10 + b9 - b8 + 2b6 - 3b5 - 2b3 - 3b2 - b1 - 7) q^93 + (-b18 - b17 - b16 + b15 - b14 - b13 - 2b12 + b11 - b9 + b7 + b6 + b4 + 4b3 - 2) * q^94 + (-2b19 + b17 + 2b15 + b14 - 2b13 - 2b12 - 4b11 - b10 - 2b9 + 2b8 + 2b6 + 2b5 - 2b4 - 4b3 + b2 + 4) q^95 + (-b14 - b7) * q^96 + (b18 - b17 - 3b16 - b15 - 2b11 + 3b10 + 3b9 - 3b8 + 2b7 + b6 - b5 + 2b4 + 3b3 + 2b2 - 2b1 - 1) * q^97 + (-b17 + 2b16 + b15 + b13 + b10 - b9 - b6 + 3b5 + b4 + 2b3 + b2 - b1 + 4) q^98 + (-2b19 - 3b18 - 3b17 - b16 - b14 - 3b12 - b10 - 8b9 - 2b8 + b7 + 5b6 + b5 + 6b3 - 2b2 - 3b1 + 5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q - 10 q^{2} + 2 q^{3} - 10 q^{4} - q^{5} + 2 q^{6} - q^{7} + 20 q^{8} + 2 q^{9} + 2 q^{10} - 3 q^{11} - 4 q^{12} - 12 q^{13} - q^{14} - 13 q^{15} - 10 q^{16} + 24 q^{17} + 2 q^{18} + 48 q^{19} - q^{20}+ \cdots + 49 q^{99}+O(q^{100})$ 20 * q - 10 * q^2 + 2 * q^3 - 10 * q^4 - q^5 + 2 * q^6 - q^7 + 20 * q^8 + 2 * q^9 + 2 * q^10 - 3 * q^11 - 4 * q^12 - 12 * q^13 - q^14 - 13 * q^15 - 10 * q^16 + 24 * q^17 + 2 * q^18 + 48 * q^19 - q^20 - 18 * q^21 - 3 * q^22 - 3 * q^23 + 2 * q^24 - 21 * q^25 + 24 * q^26 - q^27 + 2 * q^28 - 4 * q^29 + 2 * q^30 - 11 * q^31 - 10 * q^32 + 19 * q^33 - 12 * q^34 - 14 * q^35 - 4 * q^36 - 20 * q^37 - 24 * q^38 - 18 * q^39 - q^40 + 3 * q^41 + 30 * q^42 - 6 * q^43 + 6 * q^44 + 18 * q^45 + 6 * q^46 - 8 * q^47 + 2 * q^48 - 35 * q^49 - 21 * q^50 + 17 * q^51 - 12 * q^52 - 20 * q^53 - 31 * q^54 + 4 * q^55 - q^56 - 41 * q^57 - 4 * q^58 - 10 * q^59 + 11 * q^60 - 2 * q^61 + 22 * q^62 + 29 * q^63 + 20 * q^64 - 26 * q^65 - 17 * q^66 - 7 * q^67 - 12 * q^68 + 9 * q^69 + 7 * q^70 + 30 * q^71 + 2 * q^72 + 6 * q^73 + 10 * q^74 - 48 * q^75 - 24 * q^76 - 9 * q^77 + 48 * q^78 - 16 * q^79 + 2 * q^80 + 10 * q^81 - 6 * q^82 + 23 * q^83 - 12 * q^84 - 15 * q^85 - 6 * q^86 + 25 * q^87 - 3 * q^88 + 42 * q^89 - 15 * q^90 + 118 * q^91 - 3 * q^92 - 117 * q^93 - 8 * q^94 + 6 * q^95 - 4 * q^96 - 24 * q^97 + 70 * q^98 + 49 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} + 28 x^{18} + 304 x^{16} + 1654 x^{14} + 4950 x^{12} + 8429 x^{10} + 8179 x^{8} + 4427 x^{6} + \cdots + 9$ x^20 + 28*x^18 + 304*x^16 + 1654*x^14 + 4950*x^12 + 8429*x^10 + 8179*x^8 + 4427*x^6 + 1273*x^4 + 177*x^2 + 9 :

$\beta_{1}$	$=$	$( 1243960 \nu^{19} - 3832932 \nu^{18} + 34732066 \nu^{17} - 106226166 \nu^{16} + 375273154 \nu^{15} + \cdots - 119249415 ) / 1496898$ (1243960v^19 - 3832932v^18 + 34732066v^17 - 106226166v^16 + 375273154v^15 - 1134797115v^14 + 2024152318v^13 - 6014145252v^12 + 5960540679v^11 - 17243527959v^10 + 9832277465v^9 - 27335203143v^8 + 8929304521v^7 - 23448404514v^6 + 4171790087v^5 - 10186923813v^4 + 840386974v^3 - 1937526024v^2 + 45630822*v - 119249415) / 1496898
$\beta_{2}$	$=$	$( - 1199963 \nu^{19} - 5507991 \nu^{18} - 33315647 \nu^{17} - 152699049 \nu^{16} - 357040457 \nu^{15} + \cdots - 189596025 ) / 1496898$ (-1199963v^19 - 5507991v^18 - 33315647v^17 - 152699049v^16 - 357040457v^15 - 1632198150v^14 - 1903612616v^13 - 8659429002v^12 - 5522834892v^11 - 24877989306v^10 - 8971196626v^9 - 39592230579v^8 - 8109105437v^7 - 34225375662v^6 - 3943532764v^5 - 15093932157v^4 - 939528392v^3 - 2953635138v^2 - 77102907*v - 189596025) / 1496898
$\beta_{3}$	$=$	$( - 6397 \nu^{19} - 210889 \nu^{18} - 179116 \nu^{17} - 5844133 \nu^{16} - 1944688 \nu^{15} + \cdots - 6540174 ) / 38382$ (-6397v^19 - 210889v^18 - 179116v^17 - 5844133v^16 - 1944688v^15 - 62427025v^14 - 10580638v^13 - 330838624v^12 - 31665150v^11 - 948727305v^10 - 53920313v^9 - 1504963352v^8 - 52321063v^7 - 1293086080v^6 - 28319519v^5 - 563316611v^4 - 8143381v^3 - 107319691v^2 - 1132269*v - 6540174) / 38382
$\beta_{4}$	$=$	$( - 3327062 \nu^{19} + 1024341 \nu^{18} - 92566007 \nu^{17} + 28558644 \nu^{16} - 994870559 \nu^{15} + \cdots + 49564404 ) / 1496898$ (-3327062v^19 + 1024341v^18 - 92566007v^17 + 28558644v^16 - 994870559v^15 + 307901508v^14 - 5323523240v^13 + 1655398536v^12 - 15496844808v^11 + 4852403703v^10 - 25164465523v^9 + 7956805656v^8 - 22428964913v^7 + 7182600036v^6 - 10352571010v^5 + 3359143119v^4 - 2169073943v^3 + 706165872v^2 - 149076978*v + 49564404) / 1496898
$\beta_{5}$	$=$	$( - 29882 \nu^{19} + 10555605 \nu^{18} - 672746 \nu^{17} + 292799061 \nu^{16} - 4524830 \nu^{15} + \cdots + 352293300 ) / 1496898$ (-29882v^19 + 10555605v^18 - 672746v^17 + 292799061v^16 - 4524830v^15 + 3132308268v^14 - 489869v^13 + 16638072246v^12 + 112963317v^11 + 47878400142v^10 + 500196302v^9 + 76348367802v^8 + 946870018v^7 + 66115546065v^6 + 864928592v^5 + 29140511652v^4 + 354701920v^3 + 5653569960v^2 + 47144739*v + 352293300) / 1496898
$\beta_{6}$	$=$	$( - 150130 \nu^{18} - 4160902 \nu^{16} - 44455243 \nu^{14} - 235665379 \nu^{12} - 676107276 \nu^{10} + \cdots - 4661364 ) / 19191$ (-150130v^18 - 4160902v^16 - 44455243v^14 - 235665379v^12 - 676107276v^10 - 1073188301v^8 - 922797088v^6 - 402174605v^4 - 76532512*v^2 - 4661364) / 19191
$\beta_{7}$	$=$	$( - 4829060 \nu^{19} - 905205 \nu^{18} - 133914299 \nu^{17} - 25078143 \nu^{16} - 1432013381 \nu^{15} + \cdots - 19834947 ) / 1496898$ (-4829060v^19 - 905205v^18 - 133914299v^17 - 25078143v^16 - 1432013381v^15 - 267726972v^14 - 7602302582v^13 - 1416954738v^12 - 21862337520v^11 - 4050770643v^10 - 34844871592v^9 - 6379044657v^8 - 30192180482v^7 - 5385422298v^6 - 13368331000v^5 - 2247005733v^4 - 2637749507v^3 - 386599008v^2 - 171848328*v - 19834947) / 1496898
$\beta_{8}$	$=$	$( - 210889 \nu^{18} - 5844133 \nu^{16} - 62427025 \nu^{14} - 330838624 \nu^{12} - 948727305 \nu^{10} + \cdots - 6540174 ) / 19191$ (-210889v^18 - 5844133v^16 - 62427025v^14 - 330838624v^12 - 948727305v^10 - 1504963352v^8 - 1293086080v^6 - 563316611v^4 - 107319691*v^2 - 6540174) / 19191
$\beta_{9}$	$=$	$( 6504119 \nu^{19} + 3282018 \nu^{18} + 180387182 \nu^{17} + 91022523 \nu^{16} + 1929414416 \nu^{15} + \cdots + 101051352 ) / 1496898$ (6504119v^19 + 3282018v^18 + 180387182v^17 + 91022523v^16 + 1929414416v^15 + 973416597v^14 + 10247586332v^13 + 5167169187v^12 + 29496798867v^11 + 14849308170v^10 + 47101899028v^9 + 23611561806v^8 + 40969151630v^7 + 20319364275v^6 + 18275339344v^5 + 8833920798v^4 + 3654607070v^3 + 1667088285v^2 + 242943387*v + 101051352) / 1496898
$\beta_{10}$	$=$	$( 4829060 \nu^{19} - 13731597 \nu^{18} + 133914299 \nu^{17} - 380632527 \nu^{16} + 1432013381 \nu^{15} + \cdots - 438696891 ) / 1496898$ (4829060v^19 - 13731597v^18 + 133914299v^17 - 380632527v^16 + 1432013381v^15 - 4067577216v^14 + 7602302582v^13 - 21569722218v^12 + 21862337520v^11 - 61908290541v^10 + 34844871592v^9 - 98321595129v^8 + 30192180482v^7 - 84607018218v^6 + 13368331000v^5 - 36928881513v^4 + 2637749507v^3 - 7067067300v^2 + 171848328*v - 438696891) / 1496898
$\beta_{11}$	$=$	$( - 210889 \nu^{19} - 5844133 \nu^{17} - 62427025 \nu^{15} - 330838624 \nu^{13} - 948727305 \nu^{11} + \cdots + 19191 ) / 38382$ (-210889v^19 - 5844133v^17 - 62427025v^15 - 330838624v^13 - 948727305v^11 - 1504963352v^9 - 1293086080v^7 - 563316611v^5 - 107319691v^3 - 6540174v + 19191) / 38382
$\beta_{12}$	$=$	$( - 217286 \nu^{19} - 150130 \nu^{18} - 6023249 \nu^{17} - 4160902 \nu^{16} - 64371713 \nu^{15} + \cdots - 4661364 ) / 38382$ (-217286v^19 - 150130v^18 - 6023249v^17 - 4160902v^16 - 64371713v^15 - 44455243v^14 - 341419262v^13 - 235665379v^12 - 980392455v^11 - 676107276v^10 - 1558883665v^9 - 1073188301v^8 - 1345407143v^7 - 922797088v^6 - 591636130v^5 - 402174605v^4 - 115463072v^3 - 76532512v^2 - 7614870*v - 4661364) / 38382
$\beta_{13}$	$=$	$( 2912306 \nu^{19} + 2083082 \nu^{18} + 80785565 \nu^{17} + 57739298 \nu^{16} + 864263096 \nu^{15} + \cdots + 70219644 ) / 498966$ (2912306v^19 + 2083082v^18 + 80785565v^17 + 57739298v^16 + 864263096v^15 + 616996721v^14 + 4591097252v^13 + 3271834841v^12 + 13213479600v^11 + 9392229423v^10 + 21077594050v^9 + 14926666780v^8 + 18267771872v^7 + 12871193918v^6 + 8069908294v^5 + 5649398728v^4 + 1576681253v^3 + 1095933431v^2 + 102066387*v + 70219644) / 498966
$\beta_{14}$	$=$	$( - 10133216 \nu^{19} - 3131178 \nu^{18} - 280985834 \nu^{17} - 86754669 \nu^{16} + \cdots - 108379620 ) / 1496898$ (-10133216v^19 - 3131178v^18 - 280985834v^17 - 86754669v^16 - 3004387340v^15 - 926508525v^14 - 15946276298v^13 - 4909214553v^12 - 45836301495v^11 - 14079451779v^10 - 72975573994v^9 - 22359713430v^8 - 63052226675v^7 - 19291433685v^6 - 27700137580v^5 - 8507017092v^4 - 5352828185v^3 - 1673145861v^2 - 337688808*v - 108379620) / 1496898
$\beta_{15}$	$=$	$( - 15966142 \nu^{19} + 4181076 \nu^{18} - 442556590 \nu^{17} + 115790004 \nu^{16} + \cdots + 121903200 ) / 1496898$ (-15966142v^19 + 4181076v^18 - 442556590v^17 + 115790004v^16 - 4729116097v^15 + 1235628351v^14 - 25076822635v^13 + 6538079151v^12 - 71974726926v^11 + 18703786077v^10 - 114322917374v^9 + 29561522481v^8 - 98401733332v^7 + 25260829848v^6 - 42937037432v^5 + 10912948173v^4 - 8170237948v^3 + 2046682233v^2 - 494603418*v + 121903200) / 1496898
$\beta_{16}$	$=$	$( 15973354 \nu^{19} - 9432450 \nu^{18} + 442854961 \nu^{17} - 261349731 \nu^{16} + 4733906224 \nu^{15} + \cdots - 294527052 ) / 1496898$ (15973354v^19 - 9432450v^18 + 442854961v^17 - 261349731v^16 + 4733906224v^15 - 2791049238v^14 + 25115280529v^13 - 14785612389v^12 + 72141083322v^11 - 42372935064v^10 + 114716207534v^9 - 67146517827v^8 + 98894404621v^7 - 57597725196v^6 + 43239414263v^5 - 25038194427v^4 + 8249278795v^3 - 4772853990v^2 + 498803988*v - 294527052) / 1496898
$\beta_{17}$	$=$	$( 16682828 \nu^{19} + 3138390 \nu^{18} + 462434138 \nu^{17} + 87053040 \nu^{16} + 4941640280 \nu^{15} + \cdots + 116322435 ) / 1496898$ (16682828v^19 + 3138390v^18 + 462434138v^17 + 87053040v^16 + 4941640280v^15 + 931298652v^14 + 26203698884v^13 + 4947672447v^12 + 75202205043v^11 + 14245808175v^10 + 119412117286v^9 + 22753003590v^8 + 102705612647v^7 + 19784104974v^6 + 44762477065v^5 + 8809393923v^4 + 8521438571v^3 + 1752935157v^2 + 519415188*v + 116322435) / 1496898
$\beta_{18}$	$=$	$( 15544741 \nu^{19} - 15852549 \nu^{18} + 430578757 \nu^{17} - 439572225 \nu^{16} + 4596145657 \nu^{15} + \cdots - 512436312 ) / 1496898$ (15544741v^19 - 15852549v^18 + 430578757v^17 - 439572225v^16 + 4596145657v^15 - 4699892100v^14 + 24329293477v^13 - 24943588068v^12 + 69633338028v^11 - 71686007721v^10 + 110096557112v^9 - 114085780044v^8 + 94040883292v^7 - 98482920969v^6 + 40514626928v^5 - 43169941401v^4 + 7553153692v^3 - 8289607791v^2 + 445461618*v - 512436312) / 1496898
$\beta_{19}$	$=$	$( - 17452682 \nu^{19} - 15038871 \nu^{18} - 483828878 \nu^{17} - 416918598 \nu^{16} + \cdots - 481276773 ) / 1496898$ (-17452682v^19 - 15038871v^18 - 483828878v^17 - 416918598v^16 - 5171314343v^15 - 4456219101v^14 - 27432027896v^13 - 23638671465v^12 - 78785895747v^11 - 67886654877v^10 - 125290100065v^9 - 107925819492v^8 - 108097161497v^7 - 93021443691v^6 - 47422479676v^5 - 40680274125v^4 - 9151697126v^3 - 7786198509v^2 - 572920647*v - 481276773) / 1496898

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

$\nu$	$=$	$( 2\beta_{12} - 2\beta_{11} + \beta_{8} - \beta_{6} - 2\beta_{3} + 1 ) / 3$ (2b12 - 2b11 + b8 - b6 - 2*b3 + 1) / 3
$\nu^{2}$	$=$	$( - \beta_{18} + 2 \beta_{16} - \beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 8 ) / 3$ (-b18 + 2b16 - b15 + 2b14 + b13 + b12 - b10 - b9 + 2b8 + b7 - 2b6 - 2b5 - b4 - 2b3 - 2b2 - 2b1 - 8) / 3
$\nu^{3}$	$=$	$( \beta_{19} + 4 \beta_{18} - \beta_{17} - \beta_{15} + \beta_{14} + 4 \beta_{13} - 13 \beta_{12} + \cdots - 14 ) / 3$ (b19 + 4b18 - b17 - b15 + b14 + 4b13 - 13b12 + 24b11 + b10 + 3b9 - 7b8 - b7 + 2b6 + b5 + 3b4 + 13b3 + 2b1 - 14) / 3
$\nu^{4}$	$=$	$( - 3 \beta_{19} + 5 \beta_{18} + \beta_{17} - 16 \beta_{16} + 8 \beta_{15} - 16 \beta_{14} - 11 \beta_{13} + \cdots + 52 ) / 3$ (-3b19 + 5b18 + b17 - 16b16 + 8b15 - 16b14 - 11b13 - 10b12 + 6b10 + 5b9 - 19b8 - 11b7 + 28b6 + 13b5 + 8b4 + 15b3 + 15b2 + 16*b1 + 52) / 3
$\nu^{5}$	$=$	$( - 8 \beta_{19} - 46 \beta_{18} + 9 \beta_{17} - 2 \beta_{16} + 12 \beta_{15} - 13 \beta_{14} + \cdots + 136 ) / 3$ (-8b19 - 46b18 + 9b17 - 2b16 + 12b15 - 13b14 - 42b13 + 97b12 - 236b11 - 12b10 - 29b9 + 61b8 + 19b7 - 3b6 - 12b5 - 32b4 - 105b3 + 4b2 - 20*b1 + 136) / 3
$\nu^{6}$	$=$	$( 27 \beta_{19} - 31 \beta_{18} - 24 \beta_{17} + 128 \beta_{16} - 73 \beta_{15} + 134 \beta_{14} + \cdots - 410 ) / 3$ (27b19 - 31b18 - 24b17 + 128b16 - 73b15 + 134b14 + 112b13 + 88b12 - 37b10 - 40b9 + 167b8 + 109b7 - 272b6 - 92b5 - 73b4 - 107b3 - 116b2 - 143b1 - 410) / 3
$\nu^{7}$	$=$	$( 50 \beta_{19} + 455 \beta_{18} - 79 \beta_{17} + 48 \beta_{16} - 110 \beta_{15} + 158 \beta_{14} + \cdots - 1215 ) / 3$ (50b19 + 455b18 - 79b17 + 48b16 - 110b15 + 158b14 + 374b13 - 791b12 + 2164b11 + 87b10 + 216b9 - 520b8 - 212b7 - 12b6 + 131b5 + 276b4 + 866b3 - 73b2 + 181*b1 - 1215) / 3
$\nu^{8}$	$=$	$( - 174 \beta_{19} + 244 \beta_{18} + 318 \beta_{17} - 1061 \beta_{16} + 697 \beta_{15} - 1205 \beta_{14} + \cdots + 3419 ) / 3$ (-174b19 + 244b18 + 318b17 - 1061b16 + 697b15 - 1205b14 - 1096b13 - 778b12 + 208b10 + 379b9 - 1421b8 - 1084b7 + 2420b6 + 716b5 + 697b4 + 797b3 + 932b2 + 1340b1 + 3419) / 3
$\nu^{9}$	$=$	$( - 264 \beta_{19} - 4362 \beta_{18} + 724 \beta_{17} - 714 \beta_{16} + 900 \beta_{15} - 1785 \beta_{14} + \cdots + 10640 ) / 3$ (-264b19 - 4362b18 + 724b17 - 714b16 + 900b15 - 1785b14 - 3234b13 + 6728b12 - 19396b11 - 446b10 - 1449b9 + 4358b8 + 2073b7 + 271b6 - 1392b5 - 2256b4 - 7226b3 + 959b2 - 1656*b1 + 10640) / 3
$\nu^{10}$	$=$	$( 822 \beta_{19} - 2170 \beta_{18} - 3527 \beta_{17} + 8978 \beta_{16} - 6706 \beta_{15} + 11210 \beta_{14} + \cdots - 29180 ) / 3$ (822b19 - 2170b18 - 3527b17 + 8978b16 - 6706b15 + 11210b14 + 10522b13 + 6995b12 - 876b10 - 3652b9 + 11966b8 + 10780b7 - 21008b6 - 5882b5 - 6706b4 - 6171b3 - 7653b2 - 12692b1 - 29180) / 3
$\nu^{11}$	$=$	$( 905 \beta_{19} + 41415 \beta_{18} - 6777 \beta_{17} + 8824 \beta_{16} - 6883 \beta_{15} + 19086 \beta_{14} + \cdots - 92939 ) / 3$ (905b19 + 41415b18 - 6777b17 + 8824b16 - 6883b15 + 19086b14 + 27868b13 - 58387b12 + 172812b11 + 695b10 + 8782b9 - 36277b8 - 19425b7 - 3486b6 + 14452b5 + 18139b4 + 60842b3 - 11055b2 + 15357*b1 - 92939) / 3
$\nu^{12}$	$=$	$( - 807 \beta_{19} + 20202 \beta_{18} + 36368 \beta_{17} - 76842 \beta_{16} + 64341 \beta_{15} + \cdots + 252117 ) / 3$ (-807b19 + 20202b18 + 36368b17 - 76842b16 + 64341b15 - 105528b14 - 99972b13 - 63604b12 - 220b10 + 34848b9 - 100638b8 - 106443b7 + 181371b6 + 49719b5 + 64341b4 + 48982b3 + 63628b2 + 120174b1 + 252117) / 3
$\nu^{13}$	$=$	$( 3593 \beta_{19} - 391307 \beta_{18} + 63848 \beta_{17} - 99385 \beta_{16} + 50007 \beta_{15} + \cdots + 813860 ) / 3$ (3593b19 - 391307b18 + 63848b17 - 99385b16 + 50007b15 - 196721b14 - 240819b13 + 512173b12 - 1539886b11 + 22877b10 - 44098b9 + 301643b8 + 179336b7 + 39549b6 - 146895b5 - 145027b4 - 516017b3 + 119166b2 - 143302*b1 + 813860) / 3
$\nu^{14}$	$=$	$( - 48300 \beta_{19} - 190659 \beta_{18} - 362035 \beta_{17} + 662643 \beta_{16} - 614226 \beta_{15} + \cdots - 2195742 ) / 3$ (-48300b19 - 190659b18 - 362035b17 + 662643b16 - 614226b15 + 995697b14 + 943851b13 + 581816b12 + 68384b10 - 329472b9 + 848454b8 + 1041204b7 - 1567518b6 - 426960b5 - 614226b4 - 394550b3 - 533363b2 - 1134510b1 - 2195742) / 3
$\nu^{15}$	$=$	$( - 127295 \beta_{19} + 3684204 \beta_{18} - 601747 \beta_{17} + 1060535 \beta_{16} - 344822 \beta_{15} + \cdots - 7155188 ) / 3$ (-127295b19 + 3684204b18 - 601747b17 + 1060535b16 - 344822b15 + 1977885b14 + 2090270b13 - 4523705b12 + 13752758b11 - 451968b10 + 112385b9 - 2512581b8 - 1647303b7 - 423500b6 + 1466639b5 + 1157030b4 + 4403866b3 - 1234379b2 + 1339344*b1 - 7155188) / 3
$\nu^{16}$	$=$	$298764 \beta_{19} + 601045 \beta_{18} + 1178229 \beta_{17} - 1916161 \beta_{16} + 1944809 \beta_{15} + \cdots + 6413313$ 298764b19 + 601045b18 + 1178229b17 - 1916161b16 + 1944809b15 - 3127670b14 - 2957644b13 - 1779415b12 - 387181b10 + 1032064b9 - 2393825b8 - 3364195b7 + 4531525b6 + 1234255b5 + 1944809b4 + 1068861b3 + 1499880b2 + 3558689b1 + 6413313
$\nu^{17}$	$=$	$( 1921470 \beta_{19} - 34585536 \beta_{18} + 5661183 \beta_{17} - 10930767 \beta_{16} + 2215845 \beta_{15} + \cdots + 63174532 ) / 3$ (1921470b19 - 34585536b18 + 5661183b17 - 10930767b16 + 2215845b15 - 19539954b14 - 18230580b13 + 40160651b12 - 123186566b11 + 6119046b10 + 1309374b9 + 20996860b8 + 15109941b7 + 4377572b6 - 14433486b5 - 9221283b4 - 37799024b3 + 12456684b2 - 12512016*b1 + 63174532) / 3
$\nu^{18}$	$=$	$( - 11888013 \beta_{19} - 17024050 \beta_{18} - 34095180 \beta_{17} + 50134172 \beta_{16} + \cdots - 169459091 ) / 3$ (-11888013b19 - 17024050b18 - 34095180b17 + 50134172b16 - 55176664b15 + 88215449b14 + 83150749b13 + 49055569b12 + 14958632b10 - 28983400b9 + 61068137b8 + 97055035b7 - 118415555b6 - 32338616b5 - 55176664b4 - 26217521b3 - 38176871b2 - 100174799b1 - 169459091) / 3
$\nu^{19}$	$=$	$( - 23675603 \beta_{19} + 323858923 \beta_{18} - 53135503 \beta_{17} + 109997394 \beta_{16} + \cdots - 560117636 ) / 3$ (-23675603b19 + 323858923b18 - 53135503b17 + 109997394b16 - 12596761b15 + 190552267b14 + 159763468b13 - 358064347b12 + 1106741562b11 - 72114212b10 - 30788799b9 - 176175634b8 - 138588439b7 - 44161561b6 + 140419852b5 + 73441677b4 + 326195188b3 - 123418302b2 + 116744249*b1 - 560117636) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/666\mathbb{Z}\right)^\times$ .

$n$	$371$	$631$
$\chi(n)$	$-1 + \beta_{11}$	$1$

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}

223.1

		0.531449i
		3.03729i
	−	0.519589i
	−	1.94427i
	−	0.832956i
		2.82646i
		1.35695i
	−	0.340848i
		1.01469i
	−	1.66508i
	−	0.531449i
	−	3.03729i
		0.519589i
		1.94427i
		0.832956i
	−	2.82646i
	−	1.35695i
		0.340848i
	−	1.01469i
		1.66508i

−0.500000

0.866025i

−1.70319

0.314889i

−0.500000

−

0.866025i

1.66931

2.89132i

0.578891

−

1.63245i

−0.0979808

0.169708i

1.00000

2.80169

−

1.07263i

−3.33861

223.2

−0.500000

0.866025i

−1.51552

0.838574i

−0.500000

−

0.866025i

−1.84524

−

3.19605i

0.0315329

−

1.73176i

−0.269564

0.466898i

1.00000

1.59359

−

2.54175i

3.69049

223.3

−0.500000

0.866025i

−1.24741

−

1.20166i

−0.500000

−

0.866025i

−0.716775

−

1.24149i

1.66437

−

0.479458i

−0.348992

0.604472i

1.00000

0.112048

2.99791i

1.43355

223.4

−0.500000

0.866025i

−0.524756

1.65065i

−0.500000

−

0.866025i

1.73836

3.01093i

−1.16712

−

1.27977i

−2.36295

4.09274i

1.00000

−2.44926

−

1.73237i

−3.47672

223.5

−0.500000

0.866025i

−0.120566

−

1.72785i

−0.500000

−

0.866025i

0.181659

0.314643i

1.55664

0.759512i

2.51911

−

4.36323i

1.00000

−2.97093

0.416639i

−0.363318

223.6

−0.500000

0.866025i

0.688178

−

1.58947i

−0.500000

−

0.866025i

−1.64139

−

2.84297i

1.03243

1.39071i

1.56646

−

2.71318i

1.00000

−2.05282

−

2.18768i

3.28277

223.7

−0.500000

0.866025i

0.695998

−

1.58606i

−0.500000

−

0.866025i

−0.0369337

−

0.0639710i

1.02557

1.39578i

−2.33901

4.05129i

1.00000

−2.03117

−

2.20779i

0.0738674

223.8

−0.500000

0.866025i

1.41345

1.00108i

−0.500000

−

0.866025i

−1.74561

−

3.02349i

−1.57368

0.723546i

−1.43053

2.47776i

1.00000

0.995687

2.82995i

3.49123

223.9

−0.500000

0.866025i

1.58644

0.695127i

−0.500000

−

0.866025i

0.474737

0.822268i

−1.39522

1.02634i

2.02158

−

3.50148i

1.00000

2.03360

2.20556i

−0.949474

223.10

−0.500000

0.866025i

1.72736

−

0.127329i

−0.500000

−

0.866025i

1.42189

2.46278i

−0.753412

1.55961i

0.241881

−

0.418950i

1.00000

2.96757

−

0.439888i

−2.84378

445.1

−0.500000

−

0.866025i

−1.70319

−

0.314889i

−0.500000

0.866025i

1.66931

−

2.89132i

0.578891

1.63245i

−0.0979808

−

0.169708i

1.00000

2.80169

1.07263i

−3.33861

445.2

−0.500000

−

0.866025i

−1.51552

−

0.838574i

−0.500000

0.866025i

−1.84524

3.19605i

0.0315329

1.73176i

−0.269564

−

0.466898i

1.00000

1.59359

2.54175i

3.69049

445.3

−0.500000

−

0.866025i

−1.24741

1.20166i

−0.500000

0.866025i

−0.716775

1.24149i

1.66437

0.479458i

−0.348992

−

0.604472i

1.00000

0.112048

−

2.99791i

1.43355

445.4

−0.500000

−

0.866025i

−0.524756

−

1.65065i

−0.500000

0.866025i

1.73836

−

3.01093i

−1.16712

1.27977i

−2.36295

−

4.09274i

1.00000

−2.44926

1.73237i

−3.47672

445.5

−0.500000

−

0.866025i

−0.120566

1.72785i

−0.500000

0.866025i

0.181659

−

0.314643i

1.55664

−

0.759512i

2.51911

4.36323i

1.00000

−2.97093

−

0.416639i

−0.363318

445.6

−0.500000

−

0.866025i

0.688178

1.58947i

−0.500000

0.866025i

−1.64139

2.84297i

1.03243

−

1.39071i

1.56646

2.71318i

1.00000

−2.05282

2.18768i

3.28277

445.7

−0.500000

−

0.866025i

0.695998

1.58606i

−0.500000

0.866025i

−0.0369337

0.0639710i

1.02557

−

1.39578i

−2.33901

−

4.05129i

1.00000

−2.03117

2.20779i

0.0738674

445.8

−0.500000

−

0.866025i

1.41345

−

1.00108i

−0.500000

0.866025i

−1.74561

3.02349i

−1.57368

−

0.723546i

−1.43053

−

2.47776i

1.00000

0.995687

−

2.82995i

3.49123

445.9

−0.500000

−

0.866025i

1.58644

−

0.695127i

−0.500000

0.866025i

0.474737

−

0.822268i

−1.39522

−

1.02634i

2.02158

3.50148i

1.00000

2.03360

−

2.20556i

−0.949474

445.10

−0.500000

−

0.866025i

1.72736

0.127329i

−0.500000

0.866025i

1.42189

−

2.46278i

−0.753412

−

1.55961i

0.241881

0.418950i

1.00000

2.96757

0.439888i

−2.84378

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	666.2.e.e	✓	20
3.b	odd	2	1		1998.2.e.e		20
9.c	even	3	1	inner	666.2.e.e	✓	20
9.c	even	3	1		5994.2.a.bb		10
9.d	odd	6	1		1998.2.e.e		20
9.d	odd	6	1		5994.2.a.ba		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
666.2.e.e	✓	20	1.a	even	1	1	trivial
666.2.e.e	✓	20	9.c	even	3	1	inner
1998.2.e.e		20	3.b	odd	2	1
1998.2.e.e		20	9.d	odd	6	1
5994.2.a.ba		10	9.d	odd	6	1
5994.2.a.bb		10	9.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{20} + T_{5}^{19} + 36 T_{5}^{18} + 17 T_{5}^{17} + 827 T_{5}^{16} + 258 T_{5}^{15} + 11404 T_{5}^{14} + \cdots + 2601$ T5^20 + T5^19 + 36*T5^18 + 17*T5^17 + 827*T5^16 + 258*T5^15 + 11404*T5^14 + 445*T5^13 + 113568*T5^12 - 5213*T5^11 + 708409*T5^10 - 173610*T5^9 + 2989151*T5^8 - 506548*T5^7 + 4430193*T5^6 - 2718728*T5^5 + 4532782*T5^4 - 1185150*T5^3 + 390372*T5^2 + 26928*T5 + 2601 acting on $S_{2}^{\mathrm{new}}(666, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} + T + 1)^{10}$ (T^2 + T + 1)^10
$3$	$T^{20} - 2 T^{19} + \cdots + 59049$ T^20 - 2T^19 + T^18 + T^17 - 4T^16 + 13T^14 - 21T^13 + 42T^12 + 18T^11 - 261T^10 + 54T^9 + 378T^8 - 567T^7 + 1053T^6 - 2916T^4 + 2187T^3 + 6561T^2 - 39366*T + 59049
$5$	$T^{20} + T^{19} + \cdots + 2601$ T^20 + T^19 + 36T^18 + 17T^17 + 827T^16 + 258T^15 + 11404T^14 + 445T^13 + 113568T^12 - 5213T^11 + 708409T^10 - 173610T^9 + 2989151T^8 - 506548T^7 + 4430193T^6 - 2718728T^5 + 4532782T^4 - 1185150T^3 + 390372T^2 + 26928T + 2601
$7$	$T^{20} + T^{19} + \cdots + 20736$ T^20 + T^19 + 53T^18 + 48T^17 + 1913T^16 + 1685T^15 + 37195T^14 + 29565T^13 + 522295T^12 + 438939T^11 + 3834748T^10 + 3525019T^9 + 19860839T^8 + 16944702T^7 + 16844828T^6 + 6527648T^5 + 4338256T^4 + 1412544T^3 + 791424T^2 + 131328T + 20736
$11$	$T^{20} + 3 T^{19} + \cdots + 25391521$ T^20 + 3T^19 + 76T^18 + 207T^17 + 3635T^16 + 9215T^15 + 103567T^14 + 218469T^13 + 2047561T^12 + 3605502T^11 + 25498661T^10 + 26606690T^9 + 197343614T^8 + 140511144T^7 + 1018622939T^6 + 155652821T^5 + 2350716583T^4 + 1210508976T^3 + 2443030677T^2 - 240818849*T + 25391521
$13$	$T^{20} + 12 T^{19} + \cdots + 28561$ T^20 + 12T^19 + 146T^18 + 970T^17 + 7425T^16 + 39872T^15 + 238613T^14 + 1003686T^13 + 4492819T^12 + 14659260T^11 + 53219055T^10 + 134679336T^9 + 348727525T^8 + 468692037T^7 + 636288419T^6 - 32264878T^5 + 423570228T^4 + 104183560T^3 + 38702183T^2 - 1008423*T + 28561
$17$	$(T^{10} - 12 T^{9} + \cdots + 769872)^{2}$ (T^10 - 12T^9 - 50T^8 + 996T^7 - 516T^6 - 25749T^5 + 42613T^4 + 252366T^3 - 426800T^2 - 872688*T + 769872)^2
$19$	$(T^{10} - 24 T^{9} + \cdots + 7248)^{2}$ (T^10 - 24T^9 + 166T^8 + 318T^7 - 8877T^6 + 31926T^5 + 46267T^4 - 568350T^3 + 1358224T^2 - 1096752*T + 7248)^2
$23$	$T^{20} + \cdots + 1675837969$ T^20 + 3T^19 + 128T^18 + 527T^17 + 11301T^16 + 48205T^15 + 574115T^14 + 2700519T^13 + 22016395T^12 + 95613048T^11 + 541156419T^10 + 2138767884T^9 + 9310233208T^8 + 28581491154T^7 + 77772588425T^6 + 135383414467T^5 + 183306954171T^4 + 109544846138T^3 + 47395560563T^2 + 10701627729*T + 1675837969
$29$	$T^{20} + \cdots + 64892977081$ T^20 + 4T^19 + 125T^18 + 342T^17 + 9458T^16 + 23757T^15 + 428014T^14 + 949809T^13 + 13790057T^12 + 30132702T^11 + 284048458T^10 + 563228931T^9 + 4072779395T^8 + 6694878099T^7 + 28854722281T^6 + 11742687324T^5 + 68771278232T^4 - 34280197410T^3 + 172988893187T^2 - 94442168858*T + 64892977081
$31$	$T^{20} + \cdots + 5236596066321$ T^20 + 11T^19 + 284T^18 + 2331T^17 + 43937T^16 + 320830T^15 + 4031290T^14 + 20717613T^13 + 206277772T^12 + 840935439T^11 + 7406450593T^10 + 21690640370T^9 + 155668484975T^8 + 353137410714T^7 + 2215752998165T^6 + 2948845322230T^5 + 10889870314492T^4 - 1005486263538T^3 + 27130564632282T^2 + 10999181061936*T + 5236596066321
$37$	$(T + 1)^{20}$ (T + 1)^20
$41$	$T^{20} + \cdots + 11609416009$ T^20 - 3T^19 + 148T^18 - 1341T^17 + 20141T^16 - 137834T^15 + 1070905T^14 - 5958048T^13 + 35506438T^12 - 160434444T^11 + 645986291T^10 - 1974168440T^9 + 5286130283T^8 - 11398077831T^7 + 22979360291T^6 - 38706474059T^5 + 60279996514T^4 - 69641335308T^3 + 65499420051T^2 - 32587433668*T + 11609416009
$43$	$T^{20} + \cdots + 115915735296$ T^20 + 6T^19 + 281T^18 + 1668T^17 + 52207T^16 + 292602T^15 + 5376617T^14 + 25155786T^13 + 375467656T^12 + 1565992032T^11 + 15551814839T^10 + 45404228499T^9 + 377930189083T^8 + 1019705158842T^7 + 4769013044504T^6 + 3630812548872T^5 + 4203726780976T^4 + 190649094240T^3 + 1594944379392T^2 + 366859492992*T + 115915735296
$47$	$T^{20} + \cdots + 102012262412544$ T^20 + 8T^19 + 287T^18 + 588T^17 + 41942T^16 + 35209T^15 + 3818059T^14 - 3999480T^13 + 226493914T^12 - 346010583T^11 + 8961964834T^10 - 20867349169T^9 + 212747408621T^8 - 305461326060T^7 + 2310737780816T^6 - 2601872949248T^5 + 17361706364272T^4 - 9193932673344T^3 + 48945265510848T^2 + 7609262779008*T + 102012262412544
$53$	$(T^{10} + 10 T^{9} + \cdots + 1011376)^{2}$ (T^10 + 10T^9 - 233T^8 - 1780T^7 + 21508T^6 + 95744T^5 - 884923T^4 - 1183782T^3 + 11567060T^2 - 7756392*T + 1011376)^2
$59$	$T^{20} + \cdots + 197916277065984$ T^20 + 10T^19 + 422T^18 + 3006T^17 + 103040T^16 + 678146T^15 + 15070015T^14 + 82689024T^13 + 1503157006T^12 + 7629136506T^11 + 97611154198T^10 + 391259080552T^9 + 4049329774145T^8 + 14011556760660T^7 + 91798510655360T^6 + 110651580543584T^5 + 243100760735248T^4 + 155901372328320T^3 + 400779419232768T^2 + 241533772667136*T + 197916277065984
$61$	$T^{20} + \cdots + 1367774691361$ T^20 + 2T^19 + 213T^18 + 894T^17 + 33199T^16 + 128035T^15 + 2441780T^14 + 7388217T^13 + 122132309T^12 + 291084067T^11 + 3048968636T^10 + 2157428217T^9 + 40709087338T^8 + 27330276144T^7 + 280293195376T^6 - 93409776944T^5 + 817903139420T^4 - 322204899081T^3 + 1631240448994T^2 - 948576979077*T + 1367774691361
$67$	$T^{20} + \cdots + 1626438551761$ T^20 + 7T^19 + 242T^18 + 1881T^17 + 39365T^16 + 287367T^15 + 3712297T^14 + 24050541T^13 + 239538275T^12 + 1318184712T^11 + 9055948633T^10 + 34008798000T^9 + 171606921026T^8 + 475527857106T^7 + 2091276396883T^6 + 3567227397327T^5 + 9770092713197T^4 - 2695541578320T^3 + 10668459787085T^2 + 3526148632885*T + 1626438551761
$71$	$(T^{10} - 15 T^{9} + \cdots + 11950032)^{2}$ (T^10 - 15T^9 - 215T^8 + 3930T^7 + 8811T^6 - 321042T^5 + 535423T^4 + 7707282T^3 - 25937264T^2 + 2764416*T + 11950032)^2
$73$	$(T^{10} - 3 T^{9} + \cdots - 15936)^{2}$ (T^10 - 3T^9 - 242T^8 + 348T^7 + 13701T^6 - 9270T^5 - 194480T^4 + 62448T^3 + 727552T^2 + 183552*T - 15936)^2
$79$	$T^{20} + \cdots + 14\!\cdots\!61$ T^20 + 16T^19 + 462T^18 + 4300T^17 + 89341T^16 + 657057T^15 + 11484852T^14 + 61829937T^13 + 970599861T^12 + 3830617035T^11 + 60174355977T^10 + 150721130619T^9 + 2530486242252T^8 + 2881933042194T^7 + 77923147879350T^6 - 1391184062685T^5 + 1446355689854058T^4 - 1930881312266418T^3 + 18178152826909509T^2 - 15712309769954967*T + 14053540815712161
$83$	$T^{20} + \cdots + 48\!\cdots\!69$ T^20 - 23T^19 + 629T^18 - 6744T^17 + 117479T^16 - 896887T^15 + 14254171T^14 - 73386585T^13 + 997289455T^12 - 2736210975T^11 + 47912891737T^10 - 59334692039T^9 + 1592341715645T^8 + 777746361552T^7 + 36869974581680T^6 + 45960399904409T^5 + 552307186354957T^4 + 1249422689294352T^3 + 4649378049792564T^2 + 4847793500212434*T + 4830001610193369
$89$	$(T^{10} - 21 T^{9} + \cdots - 3312)^{2}$ (T^10 - 21T^9 - 101T^8 + 3024T^7 + 1203T^6 - 50496T^5 - 46679T^4 + 81606T^3 + 27724T^2 - 26520*T - 3312)^2
$97$	$T^{20} + \cdots + 32\!\cdots\!16$ T^20 + 24T^19 + 919T^18 + 14376T^17 + 368756T^16 + 4831471T^15 + 97216321T^14 + 1056267762T^13 + 17543565058T^12 + 163579879179T^11 + 2333688891494T^10 + 18381070032913T^9 + 223949635061825T^8 + 1481609310552816T^7 + 15733256218080128T^6 + 84260424373715968T^5 + 744431435964682240T^4 + 2993354225957535744T^3 + 22910511547820015616T^2 + 63940941411862446080*T + 327684985205467119616
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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 $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 223.10
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	666.2.e.e

Level	$666$
Weight	$2$
Character orbit	666.e
Analytic conductor	$5.318$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$