LMFDB - Newform orbit 1452.2.b.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,2,Mod(725,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.725"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1452.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,6,0,0,0,0,0,-10,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,-32,0,6, 0,0,0,40,0,0,0,0,0,36,0,0,0,0,0,0,0,2,0,0,0,-28,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,44,0,16,0,0,0,0,0,-84,0,0,0,0,0,-58,0,0,0,0,0,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	no
Analytic conductor:	$11.5942783735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 23 x^{12} - 72 x^{11} + 146 x^{10} - 176 x^{9} + 223 x^{8} + \cdots + 6561 $ x^16 - 4x^15 + 8x^14 - 12x^13 + 23x^12 - 72x^11 + 146x^10 - 176x^9 + 223x^8 - 528x^7 + 1314x^6 - 1944x^5 + 1863x^4 - 2916x^3 + 5832x^2 - 8748*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{4} q^{3} + \beta_{15} q^{5} + (\beta_{14} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{9} + (\beta_{14} - \beta_{12} + \cdots + \beta_1) q^{13} + (\beta_{15} + \beta_{11} + 2 \beta_{8} + \cdots - 2) q^{15}+ \cdots + (\beta_{11} - 2 \beta_{6} + 2 \beta_{4} + 2) q^{97}+O(q^{100}) $ q + b4 * q^3 + b15 * q^5 + (b14 + b2 + b1) * q^7 + (-b11 + b10 + b4) * q^9 + (b14 - b12 - b5 + b2 + b1) * q^13 + (b15 + b11 + 2b8 - b3 - 2) q^15 + (b13 - b12 + b5) * q^17 + (-b12 + 2b9 - b5 + b1) q^19 + (-b13 - b9 + b2 + b1) * q^21 + (b11 + b10 + b8 + b6 + b4 + 2b3 + 1) q^23 + (-b11 + 2b6 - 2b4) * q^25 + (-b15 + b11 + b10 + b8 + 3b6 + b3 + 2) q^27 + (-b14 + b12 + b9 - b5 + b2) * q^29 + (-b8 + b3 + 3) * q^31 + (-2b12 + b9 + b7 + b5 + b2 + b1) q^35 + (-b11 + b8 + b6 - b4 - b3 + 3) * q^37 + (-b13 - b9 - b7 + b2 - 2b1) q^39 + (2b14 + b13 - b12 - b9 + b7 - b2 + b1) q^41 + (-b12 + b9 - b5 - 3b1) q^43 + (6b11 + 3b8 - b4 - b3 - 3) * q^45 + (-b15 - b10 + b6 + b4 - b3 - 1) * q^47 + (2b8 + b6 - b4 - 2b3 - 2) * q^49 + (b14 + b13 + b7 + 2b2 - b1) q^51 + (b15 - b11 - b10 - b8 - 3b6 - 3b4 - 2b3 - 1) q^53 + (-2b14 + 2b9 - b7 + b5 - 3b1) q^57 + (-b15 + 3b11 + 3b8 + 2b6 + 2b4 + 3b3) q^59 + (-2b12 + 2b9 - 2b5 + 3b1) * q^61 + (-b13 - 4b9 + b5 - 2b2 - 2b1) q^63 + (-b14 + 2b13 - 2b12 + 2b9 + b7 + b5 + 2b2 + b1) * q^65 + (3b8 + b6 - b4 - 3b3 + 2) * q^67 + (b15 + 6b11 + b10 - b8 + 3b6 - b4 + 2b3 + 1) q^69 + (-b15 + 2b11 - 2b10 + 2b8 + 2b6 + 2b4 - 2) q^71 + (-b14 + 3b12 - b9 + 3b5 - b2) * q^73 + (2b11 - 2b10 - 2b4 + b3 - 6) q^75 + (-b14 - b12 - 6b9 - b5 - b2 - b1) q^79 + (-b15 + 5b11 - 2b8 + 3b6 + 2b3 - 4) * q^81 + (-b13 + b9 + b7 - b5 + b2 + b1) * q^83 + (-3b14 + 2b12 + 2b9 + 2b5 - 3b2 + b1) q^85 + (-b14 + b13 - 4b9 - b7 + b5 - b2 + 2b1) * q^87 + (3b11 - 3b10 + 3b8 - 3b6 - 3b4 - 3) q^89 + (-8b11 - 1) q^91 + (b15 - 2b11 - b8 + 2b4 + 1) * q^93 + (-3b14 - b13 + 2b12 + b9 - 2b7 + b2 - 2b1) * q^95 + (b11 - 2b6 + 2b4 + 2) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{3} - 10 q^{9} - 16 q^{15} - 32 q^{25} + 6 q^{27} + 40 q^{31} + 36 q^{37} + 2 q^{45} - 28 q^{49} + 44 q^{67} + 16 q^{69} - 84 q^{75} - 58 q^{81} - 80 q^{91} + 12 q^{93} + 64 q^{97}+O(q^{100}) $ 16 * q + 6 * q^3 - 10 * q^9 - 16 * q^15 - 32 * q^25 + 6 * q^27 + 40 * q^31 + 36 * q^37 + 2 * q^45 - 28 * q^49 + 44 * q^67 + 16 * q^69 - 84 * q^75 - 58 * q^81 - 80 * q^91 + 12 * q^93 + 64 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 23 x^{12} - 72 x^{11} + 146 x^{10} - 176 x^{9} + 223 x^{8} + \cdots + 6561 $ x^16 - 4*x^15 + 8*x^14 - 12*x^13 + 23*x^12 - 72*x^11 + 146*x^10 - 176*x^9 + 223*x^8 - 528*x^7 + 1314*x^6 - 1944*x^5 + 1863*x^4 - 2916*x^3 + 5832*x^2 - 8748*x + 6561 :

$\beta_{1}$	$=$	$ ( 937 \nu^{15} - 8014 \nu^{14} + 29672 \nu^{13} - 8526 \nu^{12} + 44762 \nu^{11} - 204435 \nu^{10} + \cdots - 41380227 ) / 5677452 $ (937v^15 - 8014v^14 + 29672v^13 - 8526v^12 + 44762v^11 - 204435v^10 + 457526v^9 - 551984v^8 - 76448v^7 - 1436082v^6 + 5169285v^5 - 5389848v^4 + 3477168v^3 - 4273398v^2 + 24436080*v - 41380227) / 5677452
$\beta_{2}$	$=$	$ ( 153 \nu^{15} - 5006 \nu^{14} + 8285 \nu^{13} - 7402 \nu^{12} + 18621 \nu^{11} - 66493 \nu^{10} + \cdots - 5272128 ) / 630828 $ (153v^15 - 5006v^14 + 8285v^13 - 7402v^12 + 18621v^11 - 66493v^10 + 158097v^9 - 112495v^8 + 46471v^7 - 644111v^6 + 1315818v^5 - 1441737v^4 + 1016496v^3 - 2370303v^2 + 7884864*v - 5272128) / 630828
$\beta_{3}$	$=$	$ ( 692 \nu^{15} - 643 \nu^{14} - 9030 \nu^{13} + 1811 \nu^{12} + 16426 \nu^{11} - 11623 \nu^{10} + \cdots + 9073134 ) / 1892484 $ (692v^15 - 643v^14 - 9030v^13 + 1811v^12 + 16426v^11 - 11623v^10 - 72605v^9 + 185803v^8 + 208905v^7 - 332195v^6 - 882699v^5 + 1088856v^4 - 549477v^3 - 2405376v^2 - 578583*v + 9073134) / 1892484
$\beta_{4}$	$=$	$ ( 1000 \nu^{15} - 9107 \nu^{14} + 20985 \nu^{13} - 14687 \nu^{12} + 35729 \nu^{11} - 133517 \nu^{10} + \cdots - 14465547 ) / 1892484 $ (1000v^15 - 9107v^14 + 20985v^13 - 14687v^12 + 35729v^11 - 133517v^10 + 325478v^9 - 325840v^8 + 127548v^7 - 1095100v^6 + 2922270v^5 - 3215745v^4 + 2240973v^3 - 3801087v^2 + 13157721*v - 14465547) / 1892484
$\beta_{5}$	$=$	$ ( 3116 \nu^{15} - 18839 \nu^{14} + 68626 \nu^{13} - 67737 \nu^{12} + 70138 \nu^{11} + \cdots - 72639018 ) / 5677452 $ (3116v^15 - 18839v^14 + 68626v^13 - 67737v^12 + 70138v^11 - 338955v^10 + 975847v^9 - 1471201v^8 + 531101v^7 - 1744791v^6 + 9470385v^5 - 13359816v^4 + 11321127v^3 - 7923744v^2 + 32015493*v - 72639018) / 5677452
$\beta_{6}$	$=$	$ ( 353 \nu^{15} - 1777 \nu^{14} + 5697 \nu^{13} - 3637 \nu^{12} + 5695 \nu^{11} - 30562 \nu^{10} + \cdots - 4406076 ) / 630828 $ (353v^15 - 1777v^14 + 5697v^13 - 3637v^12 + 5695v^11 - 30562v^10 + 67468v^9 - 95240v^8 + 16320v^7 - 166754v^6 + 664635v^5 - 766413v^4 + 669465v^3 - 566109v^2 + 2005965*v - 4406076) / 630828
$\beta_{7}$	$=$	$ ( - 4919 \nu^{15} + 43211 \nu^{14} - 6295 \nu^{13} - 44319 \nu^{12} - 146995 \nu^{11} + \cdots - 14438574 ) / 5677452 $ (-4919v^15 + 43211v^14 - 6295v^13 - 44319v^12 - 146995v^11 + 470466v^10 - 692686v^9 - 821234v^8 + 155962v^7 + 4963584v^6 - 4565007v^5 + 267381v^4 + 4037607v^3 + 16023663v^2 - 38636271*v - 14438574) / 5677452
$\beta_{8}$	$=$	$ ( - 615 \nu^{15} + 1477 \nu^{14} - 4308 \nu^{13} + 4891 \nu^{12} - 10258 \nu^{11} + 33866 \nu^{10} + \cdots + 3440151 ) / 630828 $ (-615v^15 + 1477v^14 - 4308v^13 + 4891v^12 - 10258v^11 + 33866v^10 - 46189v^9 + 69683v^8 - 94149v^7 + 214319v^6 - 614072v^5 + 559422v^4 - 751329v^3 + 1441152v^2 - 1521423*v + 3440151) / 630828
$\beta_{9}$	$=$	$ ( 6901 \nu^{15} - 4852 \nu^{14} + 14816 \nu^{13} - 35004 \nu^{12} + 91394 \nu^{11} - 216117 \nu^{10} + \cdots - 24855255 ) / 5677452 $ (6901v^15 - 4852v^14 + 14816v^13 - 35004v^12 + 91394v^11 - 216117v^10 + 142016v^9 - 470258v^8 + 991390v^7 - 1068648v^6 + 2844567v^5 - 2468448v^4 + 6022350v^3 - 9696186v^2 + 5617674*v - 24855255) / 5677452
$\beta_{10}$	$=$	$ ( 2893 \nu^{15} - 9485 \nu^{14} + 19758 \nu^{13} - 17447 \nu^{12} + 46784 \nu^{11} - 216176 \nu^{10} + \cdots - 25811703 ) / 1892484 $ (2893v^15 - 9485v^14 + 19758v^13 - 17447v^12 + 46784v^11 - 216176v^10 + 381137v^9 - 296329v^8 + 202593v^7 - 1419913v^6 + 4132242v^5 - 3470256v^4 + 1874151v^3 - 5596614v^2 + 18912933*v - 25811703) / 1892484
$\beta_{11}$	$=$	$ ( - 21 \nu^{15} + 64 \nu^{14} - 66 \nu^{13} + 64 \nu^{12} - 316 \nu^{11} + 1061 \nu^{10} + \cdots + 53217 ) / 10692 $ (-21v^15 + 64v^14 - 66v^13 + 64v^12 - 316v^11 + 1061v^10 - 1426v^9 + 680v^8 - 1992v^7 + 8762v^6 - 14525v^5 + 11574v^4 - 9882v^3 + 35640v^2 - 67554*v + 53217) / 10692
$\beta_{12}$	$=$	$ ( 618 \nu^{15} - 1751 \nu^{14} + 2734 \nu^{13} - 2889 \nu^{12} + 9634 \nu^{11} - 32155 \nu^{10} + \cdots - 2193318 ) / 210276 $ (618v^15 - 1751v^14 + 2734v^13 - 2889v^12 + 9634v^11 - 32155v^10 + 43441v^9 - 36601v^8 + 70559v^7 - 243759v^6 + 490349v^5 - 414096v^4 + 444789v^3 - 1181304v^2 + 2256255*v - 2193318) / 210276
$\beta_{13}$	$=$	$ ( 107 \nu^{15} - 324 \nu^{14} + 374 \nu^{13} - 368 \nu^{12} + 1432 \nu^{11} - 5339 \nu^{10} + \cdots - 287955 ) / 32076 $ (107v^15 - 324v^14 + 374v^13 - 368v^12 + 1432v^11 - 5339v^10 + 7534v^9 - 3360v^8 + 8044v^7 - 39706v^6 + 78159v^5 - 62442v^4 + 26082v^3 - 185328v^2 + 363042*v - 287955) / 32076
$\beta_{14}$	$=$	$ ( - 19865 \nu^{15} + 57002 \nu^{14} - 78061 \nu^{13} + 114318 \nu^{12} - 356977 \nu^{11} + \cdots + 56599560 ) / 5677452 $ (-19865v^15 + 57002v^14 - 78061v^13 + 114318v^12 - 356977v^11 + 1003413v^10 - 1444819v^9 + 1069627v^8 - 2360825v^7 + 8518131v^6 - 14610996v^5 + 14405553v^4 - 13617882v^3 + 38362167v^2 - 72752742*v + 56599560) / 5677452
$\beta_{15}$	$=$	$ ( - 9409 \nu^{15} + 17150 \nu^{14} - 20316 \nu^{13} + 34454 \nu^{12} - 122678 \nu^{11} + \cdots + 17060787 ) / 1892484 $ (-9409v^15 + 17150v^14 - 20316v^13 + 34454v^12 - 122678v^11 + 373175v^10 - 387470v^9 + 226168v^8 - 1058160v^7 + 2894098v^6 - 4756521v^5 + 3418848v^4 - 4410828v^3 + 16407198v^2 - 20335212*v + 17060787) / 1892484

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

$\nu$	$=$	$ ( \beta_{12} + \beta_{11} + \beta_{8} ) / 2 $ (b12 + b11 + b8) / 2
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{2} - \beta _1 + 1 ) / 2 $ (b15 - b14 - b13 + b12 - b11 + b10 - b9 - b2 - b1 + 1) / 2
$\nu^{3}$	$=$	$ ( - \beta_{15} - 2 \beta_{13} + \beta_{12} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + \cdots + 1 ) / 2 $ (-b15 - 2b13 + b12 + b8 - b7 + b6 + b5 - 3b4 - 2*b3 + b2 - b1 + 1) / 2
$\nu^{4}$	$=$	$ ( - \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + \cdots - 6 ) / 2 $ (-b15 + 2b14 - b13 - b11 + b10 + 5b9 + 2b8 - b6 - 3b5 + 3b4 - 4b3 + 2b2 - 4b1 - 6) / 2
$\nu^{5}$	$=$	$ \beta_{15} - 3\beta_{11} - 4\beta_{8} - 6\beta_{6} - 4\beta_{3} + 5 $ b15 - 3b11 - 4b8 - 6b6 - 4b3 + 5
$\nu^{6}$	$=$	$ ( 3 \beta_{15} + 3 \beta_{13} + 7 \beta_{12} + 18 \beta_{11} + 3 \beta_{10} + 9 \beta_{9} + 3 \beta_{8} + \cdots + 18 ) / 2 $ (3b15 + 3b13 + 7b12 + 18b11 + 3b10 + 9b9 + 3b8 - 6b7 + b6 + 6b5 - 3b4 + 3b3 - 2b2 - 6*b1 + 18) / 2
$\nu^{7}$	$=$	$ ( 7 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + 20 \beta_{12} - \beta_{11} + 8 \beta_{10} - 14 \beta_{9} + \cdots - 9 ) / 2 $ (7b15 + 4b14 - 6b13 + 20b12 - b11 + 8b10 - 14b9 - 4b8 + b7 - 17b6 + 8b5 + 27b4 + 19b3 - 17b2 - 13*b1 - 9) / 2
$\nu^{8}$	$=$	$ ( - 3 \beta_{15} - 25 \beta_{14} - 23 \beta_{13} + 24 \beta_{12} + 42 \beta_{11} + 17 \beta_{10} + \cdots - 41 ) / 2 $ (-3b15 - 25b14 - 23b13 + 24b12 + 42b11 + 17b10 - 55b9 - 17b8 - 20b7 + 4b6 + 23b5 - 24b4 - b3 - 21b2 - 21b1 - 41) / 2
$\nu^{9}$	$=$	$ ( - 44 \beta_{15} + 16 \beta_{14} - 24 \beta_{13} - 38 \beta_{12} + 26 \beta_{11} + 24 \beta_{10} + \cdots - 46 ) / 2 $ (-44b15 + 16b14 - 24b13 - 38b12 + 26b11 + 24b10 + 34b9 + 58b8 - 20b7 + 24b6 + 11b5 + 60b4 - b3 - 58*b1 - 46) / 2
$\nu^{10}$	$=$	$ -24\beta_{15} - 2\beta_{11} - 38\beta_{10} + 12\beta_{8} - 96\beta_{6} + 60\beta_{4} - 86\beta_{3} - 157 $ -24b15 - 2b11 - 38b10 + 12b8 - 96b6 + 60b4 - 86*b3 - 157
$\nu^{11}$	$=$	$ ( 84 \beta_{15} - 156 \beta_{14} + 12 \beta_{13} - 49 \beta_{12} + 187 \beta_{11} + 12 \beta_{10} + \cdots + 126 ) / 2 $ (84b15 - 156b14 + 12b13 - 49b12 + 187b11 + 12b10 + 174b9 - 65b8 - 96b7 - 68b6 - 18b5 + 72b4 - 30b3 - 80b2 + 42*b1 + 126) / 2
$\nu^{12}$	$=$	$ ( - 49 \beta_{15} + 165 \beta_{14} - 19 \beta_{13} + 129 \beta_{12} + 13 \beta_{11} - 117 \beta_{10} + \cdots + 337 ) / 2 $ (-49b15 + 165b14 - 19b13 + 129b12 + 13b11 - 117b10 + 3b9 + 122b8 - 68b7 - 56b6 + 104b5 + 324b4 + 220b3 - 183b2 + 277*b1 + 337) / 2
$\nu^{13}$	$=$	$ ( - 73 \beta_{15} - 274 \beta_{14} - 202 \beta_{13} + 589 \beta_{12} + 1008 \beta_{11} + 56 \beta_{10} + \cdots - 991 ) / 2 $ (-73b15 - 274b14 - 202b13 + 589b12 + 1008b11 + 56b10 - 826b9 - b8 - 129b7 + 515b6 - 75b5 + 351b4 + 276b3 - 461b2 - 91b1 - 991) / 2
$\nu^{14}$	$=$	$ ( - 663 \beta_{15} - 276 \beta_{14} - 589 \beta_{13} - 548 \beta_{12} + 739 \beta_{11} + 589 \beta_{10} + \cdots + 1460 ) / 2 $ (-663b15 - 276b14 - 589b13 - 548b12 + 739b11 + 589b10 - 669b9 + 252b8 - 74b7 + 959b6 - 145b5 + 387b4 + 646b3 - 370b2 + 310*b1 + 1460) / 2
$\nu^{15}$	$=$	$ - 959 \beta_{15} + 487 \beta_{11} - 548 \beta_{10} + 1712 \beta_{8} + 132 \beta_{6} + 222 \beta_{4} + \cdots - 1469 $ -959b15 + 487b11 - 548b10 + 1712b8 + 132b6 + 222b4 - 1356*b3 - 1469

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$1333$
$\chi(n)$	$-1$	$1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

725.1

−0.982909	−	1.42615i
1.63346	+	0.576037i
1.63346	−	0.576037i
−0.982909	+	1.42615i
0.907906	+	1.47503i
−1.12228	+	1.31928i
−1.12228	−	1.31928i
0.907906	−	1.47503i
0.762305	−	1.55528i
−1.53089	+	0.810175i
−1.53089	−	0.810175i
0.762305	+	1.55528i
0.601787	+	1.62415i
1.73062	−	0.0704449i
1.73062	+	0.0704449i
0.601787	−	1.62415i

−1.05261

−

1.37550i

0.414160i

−

1.74829i

−0.784027

2.89574i

725.2

−1.05261

−

1.37550i

0.414160i

1.74829i

−0.784027

2.89574i

725.3

−1.05261

1.37550i

−

0.414160i

−

1.74829i

−0.784027

−

2.89574i

725.4

−1.05261

1.37550i

−

0.414160i

1.74829i

−0.784027

−

2.89574i

725.5

−0.132489

−

1.72698i

−

2.46740i

−

3.58170i

−2.96489

0.457609i

725.6

−0.132489

−

1.72698i

−

2.46740i

3.58170i

−2.96489

0.457609i

725.7

−0.132489

1.72698i

2.46740i

−

3.58170i

−2.96489

−

0.457609i

725.8

−0.132489

1.72698i

2.46740i

3.58170i

−2.96489

−

0.457609i

725.9

1.24359

−

1.20560i

−

3.05881i

−

3.65040i

0.0930440

−

2.99856i

725.10

1.24359

−

1.20560i

−

3.05881i

3.65040i

0.0930440

−

2.99856i

725.11

1.24359

1.20560i

3.05881i

−

3.65040i

0.0930440

2.99856i

725.12

1.24359

1.20560i

3.05881i

3.65040i

0.0930440

2.99856i

725.13

1.44151

−

0.960240i

3.51910i

−

2.40613i

1.15588

−

2.76838i

725.14

1.44151

−

0.960240i

3.51910i

2.40613i

1.15588

−

2.76838i

725.15

1.44151

0.960240i

−

3.51910i

−

2.40613i

1.15588

2.76838i

725.16

1.44151

0.960240i

−

3.51910i

2.40613i

1.15588

2.76838i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.b	odd	2	1	inner
33.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1452.2.b.e		16
3.b	odd	2	1	inner	1452.2.b.e		16
11.b	odd	2	1	inner	1452.2.b.e		16
11.c	even	5	1		132.2.p.a	✓	16
11.d	odd	10	1		132.2.p.a	✓	16
33.d	even	2	1	inner	1452.2.b.e		16
33.f	even	10	1		132.2.p.a	✓	16
33.h	odd	10	1		132.2.p.a	✓	16
44.g	even	10	1		528.2.bn.d		16
44.h	odd	10	1		528.2.bn.d		16
132.n	odd	10	1		528.2.bn.d		16
132.o	even	10	1		528.2.bn.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
132.2.p.a	✓	16	11.c	even	5	1
132.2.p.a	✓	16	11.d	odd	10	1
132.2.p.a	✓	16	33.f	even	10	1
132.2.p.a	✓	16	33.h	odd	10	1
528.2.bn.d		16	44.g	even	10	1
528.2.bn.d		16	44.h	odd	10	1
528.2.bn.d		16	132.n	odd	10	1
528.2.bn.d		16	132.o	even	10	1
1452.2.b.e		16	1.a	even	1	1	trivial
1452.2.b.e		16	3.b	odd	2	1	inner
1452.2.b.e		16	11.b	odd	2	1	inner
1452.2.b.e		16	33.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1452, [\chi])$:

$ T_{5}^{8} + 28T_{5}^{6} + 253T_{5}^{4} + 748T_{5}^{2} + 121 $

T5^8 + 28*T5^6 + 253*T5^4 + 748*T5^2 + 121

$ T_{7}^{8} + 35T_{7}^{6} + 420T_{7}^{4} + 1975T_{7}^{2} + 3025 $

T7^8 + 35*T7^6 + 420*T7^4 + 1975*T7^2 + 3025

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 3 T^{7} + 7 T^{6} + \cdots + 81)^{2} $ (T^8 - 3T^7 + 7T^6 - 13T^5 + 28T^4 - 39T^3 + 63T^2 - 81*T + 81)^2
$5$	$ (T^{8} + 28 T^{6} + \cdots + 121)^{2} $ (T^8 + 28T^6 + 253T^4 + 748*T^2 + 121)^2
$7$	$ (T^{8} + 35 T^{6} + \cdots + 3025)^{2} $ (T^8 + 35T^6 + 420T^4 + 1975*T^2 + 3025)^2
$11$	$ T^{16} $ T^16
$13$	$ (T^{8} + 55 T^{6} + \cdots + 3025)^{2} $ (T^8 + 55T^6 + 820T^4 + 3475*T^2 + 3025)^2
$17$	$ (T^{8} - 80 T^{6} + \cdots + 3025)^{2} $ (T^8 - 80T^6 + 1705T^4 - 4400*T^2 + 3025)^2
$19$	$ (T^{8} + 70 T^{6} + \cdots + 24025)^{2} $ (T^8 + 70T^6 + 1615T^4 + 13150*T^2 + 24025)^2
$23$	$ (T^{8} + 94 T^{6} + \cdots + 1936)^{2} $ (T^8 + 94T^6 + 2277T^4 + 4136*T^2 + 1936)^2
$29$	$ (T^{8} - 135 T^{6} + \cdots + 48400)^{2} $ (T^8 - 135T^6 + 5005T^4 - 34100*T^2 + 48400)^2
$31$	$ (T^{4} - 10 T^{3} + 29 T^{2} + \cdots - 1)^{4} $ (T^4 - 10T^3 + 29T^2 - 20*T - 1)^4
$37$	$ (T^{4} - 9 T^{3} + 10 T^{2} + \cdots - 1)^{4} $ (T^4 - 9T^3 + 10T^2 + 3*T - 1)^4
$41$	$ (T^{8} - 185 T^{6} + \cdots + 366025)^{2} $ (T^8 - 185T^6 + 8140T^4 - 99825*T^2 + 366025)^2
$43$	$ (T^{8} + 170 T^{6} + \cdots + 48400)^{2} $ (T^8 + 170T^6 + 5005T^4 + 36200*T^2 + 48400)^2
$47$	$ (T^{8} + 116 T^{6} + \cdots + 421201)^{2} $ (T^8 + 116T^6 + 4741T^4 + 78716*T^2 + 421201)^2
$53$	$ (T^{8} + 164 T^{6} + \cdots + 958441)^{2} $ (T^8 + 164T^6 + 9141T^4 + 191444*T^2 + 958441)^2
$59$	$ (T^{8} + 316 T^{6} + \cdots + 2758921)^{2} $ (T^8 + 316T^6 + 30261T^4 + 828476*T^2 + 2758921)^2
$61$	$ (T^{8} + 230 T^{6} + \cdots + 93025)^{2} $ (T^8 + 230T^6 + 15415T^4 + 309350*T^2 + 93025)^2
$67$	$ (T^{4} - 11 T^{3} + \cdots + 1024)^{4} $ (T^4 - 11T^3 - 35T^2 + 352*T + 1024)^4
$71$	$ (T^{8} + 272 T^{6} + \cdots + 121)^{2} $ (T^8 + 272T^6 + 2849T^4 + 1408*T^2 + 121)^2
$73$	$ (T^{8} + 405 T^{6} + \cdots + 36966400)^{2} $ (T^8 + 405T^6 + 53245T^4 + 2523200*T^2 + 36966400)^2
$79$	$ (T^{8} + 535 T^{6} + \cdots + 9579025)^{2} $ (T^8 + 535T^6 + 94620T^4 + 5597675*T^2 + 9579025)^2
$83$	$ (T^{8} - 185 T^{6} + \cdots + 3025)^{2} $ (T^8 - 185T^6 + 9020T^4 - 88825*T^2 + 3025)^2
$89$	$ (T^{8} + 558 T^{6} + \cdots + 203233536)^{2} $ (T^8 + 558T^6 + 106029T^4 + 8083152*T^2 + 203233536)^2
$97$	$ (T^{4} - 16 T^{3} + \cdots - 521)^{4} $ (T^4 - 16T^3 + 55T^2 + 122*T - 521)^4
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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

Level:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1452.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{3} + \beta_{15} q^{5} + (\beta_{14} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{9} + (\beta_{14} - \beta_{12} + \cdots + \beta_1) q^{13} + (\beta_{15} + \beta_{11} + 2 \beta_{8} + \cdots - 2) q^{15}+ \cdots + (\beta_{11} - 2 \beta_{6} + 2 \beta_{4} + 2) q^{97}+O(q^{100}) \) q + b4 * q^3 + b15 * q^5 + (b14 + b2 + b1) * q^7 + (-b11 + b10 + b4) * q^9 + (b14 - b12 - b5 + b2 + b1) * q^13 + (b15 + b11 + 2b8 - b3 - 2) q^15 + (b13 - b12 + b5) * q^17 + (-b12 + 2b9 - b5 + b1) q^19 + (-b13 - b9 + b2 + b1) * q^21 + (b11 + b10 + b8 + b6 + b4 + 2b3 + 1) q^23 + (-b11 + 2b6 - 2b4) * q^25 + (-b15 + b11 + b10 + b8 + 3b6 + b3 + 2) q^27 + (-b14 + b12 + b9 - b5 + b2) * q^29 + (-b8 + b3 + 3) * q^31 + (-2b12 + b9 + b7 + b5 + b2 + b1) q^35 + (-b11 + b8 + b6 - b4 - b3 + 3) * q^37 + (-b13 - b9 - b7 + b2 - 2b1) q^39 + (2b14 + b13 - b12 - b9 + b7 - b2 + b1) q^41 + (-b12 + b9 - b5 - 3b1) q^43 + (6b11 + 3b8 - b4 - b3 - 3) * q^45 + (-b15 - b10 + b6 + b4 - b3 - 1) * q^47 + (2b8 + b6 - b4 - 2b3 - 2) * q^49 + (b14 + b13 + b7 + 2b2 - b1) q^51 + (b15 - b11 - b10 - b8 - 3b6 - 3b4 - 2b3 - 1) q^53 + (-2b14 + 2b9 - b7 + b5 - 3b1) q^57 + (-b15 + 3b11 + 3b8 + 2b6 + 2b4 + 3b3) q^59 + (-2b12 + 2b9 - 2b5 + 3b1) * q^61 + (-b13 - 4b9 + b5 - 2b2 - 2b1) q^63 + (-b14 + 2b13 - 2b12 + 2b9 + b7 + b5 + 2b2 + b1) * q^65 + (3b8 + b6 - b4 - 3b3 + 2) * q^67 + (b15 + 6b11 + b10 - b8 + 3b6 - b4 + 2b3 + 1) q^69 + (-b15 + 2b11 - 2b10 + 2b8 + 2b6 + 2b4 - 2) q^71 + (-b14 + 3b12 - b9 + 3b5 - b2) * q^73 + (2b11 - 2b10 - 2b4 + b3 - 6) q^75 + (-b14 - b12 - 6b9 - b5 - b2 - b1) q^79 + (-b15 + 5b11 - 2b8 + 3b6 + 2b3 - 4) * q^81 + (-b13 + b9 + b7 - b5 + b2 + b1) * q^83 + (-3b14 + 2b12 + 2b9 + 2b5 - 3b2 + b1) q^85 + (-b14 + b13 - 4b9 - b7 + b5 - b2 + 2b1) * q^87 + (3b11 - 3b10 + 3b8 - 3b6 - 3b4 - 3) q^89 + (-8b11 - 1) q^91 + (b15 - 2b11 - b8 + 2b4 + 1) * q^93 + (-3b14 - b13 + 2b12 + b9 - 2b7 + b2 - 2b1) * q^95 + (b11 - 2b6 + 2b4 + 2) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{3} - 10 q^{9} - 16 q^{15} - 32 q^{25} + 6 q^{27} + 40 q^{31} + 36 q^{37} + 2 q^{45} - 28 q^{49} + 44 q^{67} + 16 q^{69} - 84 q^{75} - 58 q^{81} - 80 q^{91} + 12 q^{93} + 64 q^{97}+O(q^{100}) \) 16 * q + 6 * q^3 - 10 * q^9 - 16 * q^15 - 32 * q^25 + 6 * q^27 + 40 * q^31 + 36 * q^37 + 2 * q^45 - 28 * q^49 + 44 * q^67 + 16 * q^69 - 84 * q^75 - 58 * q^81 - 80 * q^91 + 12 * q^93 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1452.2.b.e

Level	$1452$
Weight	$2$
Character orbit	1452.b
Analytic conductor	$11.594$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(11.5942783735\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 23 x^{12} - 72 x^{11} + 146 x^{10} - 176 x^{9} + 223 x^{8} + \cdots + 6561 \) x^16 - 4x^15 + 8x^14 - 12x^13 + 23x^12 - 72x^11 + 146x^10 - 176x^9 + 223x^8 - 528x^7 + 1314x^6 - 1944x^5 + 1863x^4 - 2916x^3 + 5832x^2 - 8748*x + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 937 \nu^{15} - 8014 \nu^{14} + 29672 \nu^{13} - 8526 \nu^{12} + 44762 \nu^{11} - 204435 \nu^{10} + \cdots - 41380227 ) / 5677452 \) (937v^15 - 8014v^14 + 29672v^13 - 8526v^12 + 44762v^11 - 204435v^10 + 457526v^9 - 551984v^8 - 76448v^7 - 1436082v^6 + 5169285v^5 - 5389848v^4 + 3477168v^3 - 4273398v^2 + 24436080*v - 41380227) / 5677452
\(\beta_{2}\)	\(=\)	\( ( 153 \nu^{15} - 5006 \nu^{14} + 8285 \nu^{13} - 7402 \nu^{12} + 18621 \nu^{11} - 66493 \nu^{10} + \cdots - 5272128 ) / 630828 \) (153v^15 - 5006v^14 + 8285v^13 - 7402v^12 + 18621v^11 - 66493v^10 + 158097v^9 - 112495v^8 + 46471v^7 - 644111v^6 + 1315818v^5 - 1441737v^4 + 1016496v^3 - 2370303v^2 + 7884864*v - 5272128) / 630828
\(\beta_{3}\)	\(=\)	\( ( 692 \nu^{15} - 643 \nu^{14} - 9030 \nu^{13} + 1811 \nu^{12} + 16426 \nu^{11} - 11623 \nu^{10} + \cdots + 9073134 ) / 1892484 \) (692v^15 - 643v^14 - 9030v^13 + 1811v^12 + 16426v^11 - 11623v^10 - 72605v^9 + 185803v^8 + 208905v^7 - 332195v^6 - 882699v^5 + 1088856v^4 - 549477v^3 - 2405376v^2 - 578583*v + 9073134) / 1892484
\(\beta_{4}\)	\(=\)	\( ( 1000 \nu^{15} - 9107 \nu^{14} + 20985 \nu^{13} - 14687 \nu^{12} + 35729 \nu^{11} - 133517 \nu^{10} + \cdots - 14465547 ) / 1892484 \) (1000v^15 - 9107v^14 + 20985v^13 - 14687v^12 + 35729v^11 - 133517v^10 + 325478v^9 - 325840v^8 + 127548v^7 - 1095100v^6 + 2922270v^5 - 3215745v^4 + 2240973v^3 - 3801087v^2 + 13157721*v - 14465547) / 1892484
\(\beta_{5}\)	\(=\)	\( ( 3116 \nu^{15} - 18839 \nu^{14} + 68626 \nu^{13} - 67737 \nu^{12} + 70138 \nu^{11} + \cdots - 72639018 ) / 5677452 \) (3116v^15 - 18839v^14 + 68626v^13 - 67737v^12 + 70138v^11 - 338955v^10 + 975847v^9 - 1471201v^8 + 531101v^7 - 1744791v^6 + 9470385v^5 - 13359816v^4 + 11321127v^3 - 7923744v^2 + 32015493*v - 72639018) / 5677452
\(\beta_{6}\)	\(=\)	\( ( 353 \nu^{15} - 1777 \nu^{14} + 5697 \nu^{13} - 3637 \nu^{12} + 5695 \nu^{11} - 30562 \nu^{10} + \cdots - 4406076 ) / 630828 \) (353v^15 - 1777v^14 + 5697v^13 - 3637v^12 + 5695v^11 - 30562v^10 + 67468v^9 - 95240v^8 + 16320v^7 - 166754v^6 + 664635v^5 - 766413v^4 + 669465v^3 - 566109v^2 + 2005965*v - 4406076) / 630828
\(\beta_{7}\)	\(=\)	\( ( - 4919 \nu^{15} + 43211 \nu^{14} - 6295 \nu^{13} - 44319 \nu^{12} - 146995 \nu^{11} + \cdots - 14438574 ) / 5677452 \) (-4919v^15 + 43211v^14 - 6295v^13 - 44319v^12 - 146995v^11 + 470466v^10 - 692686v^9 - 821234v^8 + 155962v^7 + 4963584v^6 - 4565007v^5 + 267381v^4 + 4037607v^3 + 16023663v^2 - 38636271*v - 14438574) / 5677452
\(\beta_{8}\)	\(=\)	\( ( - 615 \nu^{15} + 1477 \nu^{14} - 4308 \nu^{13} + 4891 \nu^{12} - 10258 \nu^{11} + 33866 \nu^{10} + \cdots + 3440151 ) / 630828 \) (-615v^15 + 1477v^14 - 4308v^13 + 4891v^12 - 10258v^11 + 33866v^10 - 46189v^9 + 69683v^8 - 94149v^7 + 214319v^6 - 614072v^5 + 559422v^4 - 751329v^3 + 1441152v^2 - 1521423*v + 3440151) / 630828
\(\beta_{9}\)	\(=\)	\( ( 6901 \nu^{15} - 4852 \nu^{14} + 14816 \nu^{13} - 35004 \nu^{12} + 91394 \nu^{11} - 216117 \nu^{10} + \cdots - 24855255 ) / 5677452 \) (6901v^15 - 4852v^14 + 14816v^13 - 35004v^12 + 91394v^11 - 216117v^10 + 142016v^9 - 470258v^8 + 991390v^7 - 1068648v^6 + 2844567v^5 - 2468448v^4 + 6022350v^3 - 9696186v^2 + 5617674*v - 24855255) / 5677452
\(\beta_{10}\)	\(=\)	\( ( 2893 \nu^{15} - 9485 \nu^{14} + 19758 \nu^{13} - 17447 \nu^{12} + 46784 \nu^{11} - 216176 \nu^{10} + \cdots - 25811703 ) / 1892484 \) (2893v^15 - 9485v^14 + 19758v^13 - 17447v^12 + 46784v^11 - 216176v^10 + 381137v^9 - 296329v^8 + 202593v^7 - 1419913v^6 + 4132242v^5 - 3470256v^4 + 1874151v^3 - 5596614v^2 + 18912933*v - 25811703) / 1892484
\(\beta_{11}\)	\(=\)	\( ( - 21 \nu^{15} + 64 \nu^{14} - 66 \nu^{13} + 64 \nu^{12} - 316 \nu^{11} + 1061 \nu^{10} + \cdots + 53217 ) / 10692 \) (-21v^15 + 64v^14 - 66v^13 + 64v^12 - 316v^11 + 1061v^10 - 1426v^9 + 680v^8 - 1992v^7 + 8762v^6 - 14525v^5 + 11574v^4 - 9882v^3 + 35640v^2 - 67554*v + 53217) / 10692
\(\beta_{12}\)	\(=\)	\( ( 618 \nu^{15} - 1751 \nu^{14} + 2734 \nu^{13} - 2889 \nu^{12} + 9634 \nu^{11} - 32155 \nu^{10} + \cdots - 2193318 ) / 210276 \) (618v^15 - 1751v^14 + 2734v^13 - 2889v^12 + 9634v^11 - 32155v^10 + 43441v^9 - 36601v^8 + 70559v^7 - 243759v^6 + 490349v^5 - 414096v^4 + 444789v^3 - 1181304v^2 + 2256255*v - 2193318) / 210276
\(\beta_{13}\)	\(=\)	\( ( 107 \nu^{15} - 324 \nu^{14} + 374 \nu^{13} - 368 \nu^{12} + 1432 \nu^{11} - 5339 \nu^{10} + \cdots - 287955 ) / 32076 \) (107v^15 - 324v^14 + 374v^13 - 368v^12 + 1432v^11 - 5339v^10 + 7534v^9 - 3360v^8 + 8044v^7 - 39706v^6 + 78159v^5 - 62442v^4 + 26082v^3 - 185328v^2 + 363042*v - 287955) / 32076
\(\beta_{14}\)	\(=\)	\( ( - 19865 \nu^{15} + 57002 \nu^{14} - 78061 \nu^{13} + 114318 \nu^{12} - 356977 \nu^{11} + \cdots + 56599560 ) / 5677452 \) (-19865v^15 + 57002v^14 - 78061v^13 + 114318v^12 - 356977v^11 + 1003413v^10 - 1444819v^9 + 1069627v^8 - 2360825v^7 + 8518131v^6 - 14610996v^5 + 14405553v^4 - 13617882v^3 + 38362167v^2 - 72752742*v + 56599560) / 5677452
\(\beta_{15}\)	\(=\)	\( ( - 9409 \nu^{15} + 17150 \nu^{14} - 20316 \nu^{13} + 34454 \nu^{12} - 122678 \nu^{11} + \cdots + 17060787 ) / 1892484 \) (-9409v^15 + 17150v^14 - 20316v^13 + 34454v^12 - 122678v^11 + 373175v^10 - 387470v^9 + 226168v^8 - 1058160v^7 + 2894098v^6 - 4756521v^5 + 3418848v^4 - 4410828v^3 + 16407198v^2 - 20335212*v + 17060787) / 1892484

\(\nu\)	\(=\)	\( ( \beta_{12} + \beta_{11} + \beta_{8} ) / 2 \) (b12 + b11 + b8) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{2} - \beta _1 + 1 ) / 2 \) (b15 - b14 - b13 + b12 - b11 + b10 - b9 - b2 - b1 + 1) / 2
\(\nu^{3}\)	\(=\)	\( ( - \beta_{15} - 2 \beta_{13} + \beta_{12} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + \cdots + 1 ) / 2 \) (-b15 - 2b13 + b12 + b8 - b7 + b6 + b5 - 3b4 - 2*b3 + b2 - b1 + 1) / 2
\(\nu^{4}\)	\(=\)	\( ( - \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + \cdots - 6 ) / 2 \) (-b15 + 2b14 - b13 - b11 + b10 + 5b9 + 2b8 - b6 - 3b5 + 3b4 - 4b3 + 2b2 - 4b1 - 6) / 2
\(\nu^{5}\)	\(=\)	\( \beta_{15} - 3\beta_{11} - 4\beta_{8} - 6\beta_{6} - 4\beta_{3} + 5 \) b15 - 3b11 - 4b8 - 6b6 - 4b3 + 5
\(\nu^{6}\)	\(=\)	\( ( 3 \beta_{15} + 3 \beta_{13} + 7 \beta_{12} + 18 \beta_{11} + 3 \beta_{10} + 9 \beta_{9} + 3 \beta_{8} + \cdots + 18 ) / 2 \) (3b15 + 3b13 + 7b12 + 18b11 + 3b10 + 9b9 + 3b8 - 6b7 + b6 + 6b5 - 3b4 + 3b3 - 2b2 - 6*b1 + 18) / 2
\(\nu^{7}\)	\(=\)	\( ( 7 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + 20 \beta_{12} - \beta_{11} + 8 \beta_{10} - 14 \beta_{9} + \cdots - 9 ) / 2 \) (7b15 + 4b14 - 6b13 + 20b12 - b11 + 8b10 - 14b9 - 4b8 + b7 - 17b6 + 8b5 + 27b4 + 19b3 - 17b2 - 13*b1 - 9) / 2
\(\nu^{8}\)	\(=\)	\( ( - 3 \beta_{15} - 25 \beta_{14} - 23 \beta_{13} + 24 \beta_{12} + 42 \beta_{11} + 17 \beta_{10} + \cdots - 41 ) / 2 \) (-3b15 - 25b14 - 23b13 + 24b12 + 42b11 + 17b10 - 55b9 - 17b8 - 20b7 + 4b6 + 23b5 - 24b4 - b3 - 21b2 - 21b1 - 41) / 2
\(\nu^{9}\)	\(=\)	\( ( - 44 \beta_{15} + 16 \beta_{14} - 24 \beta_{13} - 38 \beta_{12} + 26 \beta_{11} + 24 \beta_{10} + \cdots - 46 ) / 2 \) (-44b15 + 16b14 - 24b13 - 38b12 + 26b11 + 24b10 + 34b9 + 58b8 - 20b7 + 24b6 + 11b5 + 60b4 - b3 - 58*b1 - 46) / 2
\(\nu^{10}\)	\(=\)	\( -24\beta_{15} - 2\beta_{11} - 38\beta_{10} + 12\beta_{8} - 96\beta_{6} + 60\beta_{4} - 86\beta_{3} - 157 \) -24b15 - 2b11 - 38b10 + 12b8 - 96b6 + 60b4 - 86*b3 - 157
\(\nu^{11}\)	\(=\)	\( ( 84 \beta_{15} - 156 \beta_{14} + 12 \beta_{13} - 49 \beta_{12} + 187 \beta_{11} + 12 \beta_{10} + \cdots + 126 ) / 2 \) (84b15 - 156b14 + 12b13 - 49b12 + 187b11 + 12b10 + 174b9 - 65b8 - 96b7 - 68b6 - 18b5 + 72b4 - 30b3 - 80b2 + 42*b1 + 126) / 2
\(\nu^{12}\)	\(=\)	\( ( - 49 \beta_{15} + 165 \beta_{14} - 19 \beta_{13} + 129 \beta_{12} + 13 \beta_{11} - 117 \beta_{10} + \cdots + 337 ) / 2 \) (-49b15 + 165b14 - 19b13 + 129b12 + 13b11 - 117b10 + 3b9 + 122b8 - 68b7 - 56b6 + 104b5 + 324b4 + 220b3 - 183b2 + 277*b1 + 337) / 2
\(\nu^{13}\)	\(=\)	\( ( - 73 \beta_{15} - 274 \beta_{14} - 202 \beta_{13} + 589 \beta_{12} + 1008 \beta_{11} + 56 \beta_{10} + \cdots - 991 ) / 2 \) (-73b15 - 274b14 - 202b13 + 589b12 + 1008b11 + 56b10 - 826b9 - b8 - 129b7 + 515b6 - 75b5 + 351b4 + 276b3 - 461b2 - 91b1 - 991) / 2
\(\nu^{14}\)	\(=\)	\( ( - 663 \beta_{15} - 276 \beta_{14} - 589 \beta_{13} - 548 \beta_{12} + 739 \beta_{11} + 589 \beta_{10} + \cdots + 1460 ) / 2 \) (-663b15 - 276b14 - 589b13 - 548b12 + 739b11 + 589b10 - 669b9 + 252b8 - 74b7 + 959b6 - 145b5 + 387b4 + 646b3 - 370b2 + 310*b1 + 1460) / 2
\(\nu^{15}\)	\(=\)	\( - 959 \beta_{15} + 487 \beta_{11} - 548 \beta_{10} + 1712 \beta_{8} + 132 \beta_{6} + 222 \beta_{4} + \cdots - 1469 \) -959b15 + 487b11 - 548b10 + 1712b8 + 132b6 + 222b4 - 1356*b3 - 1469

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 725.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 3 T^{7} + 7 T^{6} + \cdots + 81)^{2} \) (T^8 - 3T^7 + 7T^6 - 13T^5 + 28T^4 - 39T^3 + 63T^2 - 81*T + 81)^2
$5$	\( (T^{8} + 28 T^{6} + \cdots + 121)^{2} \) (T^8 + 28T^6 + 253T^4 + 748*T^2 + 121)^2
$7$	\( (T^{8} + 35 T^{6} + \cdots + 3025)^{2} \) (T^8 + 35T^6 + 420T^4 + 1975*T^2 + 3025)^2
$11$	\( T^{16} \) T^16
$13$	\( (T^{8} + 55 T^{6} + \cdots + 3025)^{2} \) (T^8 + 55T^6 + 820T^4 + 3475*T^2 + 3025)^2
$17$	\( (T^{8} - 80 T^{6} + \cdots + 3025)^{2} \) (T^8 - 80T^6 + 1705T^4 - 4400*T^2 + 3025)^2
$19$	\( (T^{8} + 70 T^{6} + \cdots + 24025)^{2} \) (T^8 + 70T^6 + 1615T^4 + 13150*T^2 + 24025)^2
$23$	\( (T^{8} + 94 T^{6} + \cdots + 1936)^{2} \) (T^8 + 94T^6 + 2277T^4 + 4136*T^2 + 1936)^2
$29$	\( (T^{8} - 135 T^{6} + \cdots + 48400)^{2} \) (T^8 - 135T^6 + 5005T^4 - 34100*T^2 + 48400)^2
$31$	\( (T^{4} - 10 T^{3} + 29 T^{2} + \cdots - 1)^{4} \) (T^4 - 10T^3 + 29T^2 - 20*T - 1)^4
$37$	\( (T^{4} - 9 T^{3} + 10 T^{2} + \cdots - 1)^{4} \) (T^4 - 9T^3 + 10T^2 + 3*T - 1)^4
$41$	\( (T^{8} - 185 T^{6} + \cdots + 366025)^{2} \) (T^8 - 185T^6 + 8140T^4 - 99825*T^2 + 366025)^2
$43$	\( (T^{8} + 170 T^{6} + \cdots + 48400)^{2} \) (T^8 + 170T^6 + 5005T^4 + 36200*T^2 + 48400)^2
$47$	\( (T^{8} + 116 T^{6} + \cdots + 421201)^{2} \) (T^8 + 116T^6 + 4741T^4 + 78716*T^2 + 421201)^2
$53$	\( (T^{8} + 164 T^{6} + \cdots + 958441)^{2} \) (T^8 + 164T^6 + 9141T^4 + 191444*T^2 + 958441)^2
$59$	\( (T^{8} + 316 T^{6} + \cdots + 2758921)^{2} \) (T^8 + 316T^6 + 30261T^4 + 828476*T^2 + 2758921)^2
$61$	\( (T^{8} + 230 T^{6} + \cdots + 93025)^{2} \) (T^8 + 230T^6 + 15415T^4 + 309350*T^2 + 93025)^2
$67$	\( (T^{4} - 11 T^{3} + \cdots + 1024)^{4} \) (T^4 - 11T^3 - 35T^2 + 352*T + 1024)^4
$71$	\( (T^{8} + 272 T^{6} + \cdots + 121)^{2} \) (T^8 + 272T^6 + 2849T^4 + 1408*T^2 + 121)^2
$73$	\( (T^{8} + 405 T^{6} + \cdots + 36966400)^{2} \) (T^8 + 405T^6 + 53245T^4 + 2523200*T^2 + 36966400)^2
$79$	\( (T^{8} + 535 T^{6} + \cdots + 9579025)^{2} \) (T^8 + 535T^6 + 94620T^4 + 5597675*T^2 + 9579025)^2
$83$	\( (T^{8} - 185 T^{6} + \cdots + 3025)^{2} \) (T^8 - 185T^6 + 9020T^4 - 88825*T^2 + 3025)^2
$89$	\( (T^{8} + 558 T^{6} + \cdots + 203233536)^{2} \) (T^8 + 558T^6 + 106029T^4 + 8083152*T^2 + 203233536)^2
$97$	\( (T^{4} - 16 T^{3} + \cdots - 521)^{4} \) (T^4 - 16T^3 + 55T^2 + 122*T - 521)^4
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\(n\)	\(485\)	\(727\)	\(1333\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)