LMFDB - Newform orbit 2016.2.cs.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(703,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2016.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.cs (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.2353561680715186176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 $ x^16 - 4x^15 + 2x^14 + 41x^12 - 92x^11 + 66x^10 - 104x^9 + 291x^8 - 388x^7 + 366x^6 - 344x^5 + 286x^4 - 184x^3 + 84x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{10} q^{5} + \beta_{15} q^{7}+O(q^{10}) $ q + b10 * q^5 + b15 * q^7	$ q + \beta_{10} q^{5} + \beta_{15} q^{7} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - 2 \beta_{12} - 2 \beta_{10} + \cdots + 3) q^{97}+O(q^{100}) $ q + b10 * q^5 + b15 * q^7 + (-b15 + b13 - b7 + b3 + b2) * q^11 + (b14 - b8 + 2b6 - 1) q^13 - b8 * q^17 + (-b15 - b7 - b5 + b2) * q^19 + (-b13 + b2) * q^23 + (-b14 - b9 + b8 - 2b6 + 2) q^25 + (-b4 - 1) * q^29 + (-b13 - 2b11 - b5 - b3 + b2 - b1) q^31 + (-3b13 - 3b7 - b5 + b2 - 2b1) q^35 + (b14 - 2b12 - b10 - b9 - b6 + b4) q^37 + (-b14 + b8 - 2b6 + 1) q^41 + (-2b11 + 2b7) * q^43 + (-b15 + b11 + b7 - b5 - b3 + b2 + b1) * q^47 + (-2b12 - b9 - 2) q^49 + (2b14 - b12 - 2b10 - b8 - b6 + 1) * q^53 + (-3b15 + 3b13 - b5 - b2 + 6b1) q^55 + (2b15 + 2b13 + 3b7 - 2b2 + 3b1) q^59 + (b14 + b10 - b9 + b6 - b4 - 2) * q^61 + (-b14 - 4b12 - 2b10 + 2b8 - b6) q^65 + (-b15 + 2b13 + b7 + b5 - b3 + 2b2) * q^67 + (b15 + b13 - b11 + b7 + 3b5 - 3b2) * q^71 + (b14 - b9 + b8 - b6 + 2b4 - 1) q^73 + (3b14 + b12 - 2b9 + b8 + 2b6 + 2b4 - 5) * q^77 + (-b13 + 2b7 + b2 + 4b1) * q^79 + (-b15 + b13 - b11 + b7 + b5 - 2b3 + b2 + 4b1) * q^83 + (b14 - b12 + b10 + b8 - 1) * q^85 + (-2b14 + 2b10 - 3b6 + 6) q^89 + (b15 - b11 + b7 - 3b3 - 3b2 + 3b1) q^91 + (-4b15 + 2b13 - 2b7 - 2b5 + b3 + 2b2 + b1) q^95 + (-2b12 - 2b10 + 2b9 - 2b8 - 6b6 - b4 + 3) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{25} - 16 q^{29} - 8 q^{37} - 32 q^{49} + 8 q^{53} - 24 q^{61} - 8 q^{65} - 24 q^{73} - 64 q^{77} - 16 q^{85} + 72 q^{89}+O(q^{100}) $ 16 * q + 16 * q^25 - 16 * q^29 - 8 * q^37 - 32 * q^49 + 8 * q^53 - 24 * q^61 - 8 * q^65 - 24 * q^73 - 64 * q^77 - 16 * q^85 + 72 * q^89

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 $ x^16 - 4*x^15 + 2*x^14 + 41*x^12 - 92*x^11 + 66*x^10 - 104*x^9 + 291*x^8 - 388*x^7 + 366*x^6 - 344*x^5 + 286*x^4 - 184*x^3 + 84*x^2 - 24*x + 4 :

$\beta_{1}$	$=$	$ ( - 508009675 \nu^{15} + 17835297016 \nu^{14} - 56517662852 \nu^{13} + 582270848 \nu^{12} + \cdots + 242647661018 ) / 20109322702 $ (-508009675v^15 + 17835297016v^14 - 56517662852v^13 + 582270848v^12 - 7088890831v^11 + 702274874462v^10 - 1170090037720v^9 + 377376368526v^8 - 1364027322083v^7 + 4137630712600v^6 - 4084528421066v^5 + 3083496853086v^4 - 3236291968028v^3 + 2419000098258v^2 - 1055996592624*v + 242647661018) / 20109322702
$\beta_{2}$	$=$	$ ( 2306431404 \nu^{15} - 37444380512 \nu^{14} + 95714273392 \nu^{13} + 18886793845 \nu^{12} + \cdots - 394015237814 ) / 20109322702 $ (2306431404v^15 - 37444380512v^14 + 95714273392v^13 + 18886793845v^12 + 90029143982v^11 - 1367786602618v^10 + 1863018589666v^9 - 575541348219v^8 + 2921911891676v^7 - 7182761898450v^6 + 6570657118770v^5 - 5342350769881v^4 + 5286105515374v^3 - 3757872246264v^2 + 1675584716836*v - 394015237814) / 20109322702
$\beta_{3}$	$=$	$ ( 2358518199 \nu^{15} + 7744007704 \nu^{14} - 40220309508 \nu^{13} - 37698238811 \nu^{12} + \cdots + 6869804700 ) / 20109322702 $ (2358518199v^15 + 7744007704v^14 - 40220309508v^13 - 37698238811v^12 + 72348596101v^11 + 470758796196v^10 - 465557954766v^9 - 370921530677v^8 - 836020034165v^7 + 2010052783442v^6 - 927678153432v^5 + 1113778618237v^4 - 1251623532730v^3 + 664629352550v^2 - 193341101940*v + 6869804700) / 20109322702
$\beta_{4}$	$=$	$ ( 32131571 \nu^{15} - 338443886 \nu^{14} + 801340954 \nu^{13} - 7179161 \nu^{12} + \cdots - 4131081209 ) / 137735087 $ (32131571v^15 - 338443886v^14 + 801340954v^13 - 7179161v^12 + 1133962371v^11 - 11663672814v^10 + 17217925190v^9 - 7634184353v^8 + 25474337063v^7 - 64850621098v^6 + 63560694880v^5 - 49589483157v^4 + 50404045920v^3 - 36993277864v^2 + 15863090132*v - 4131081209) / 137735087
$\beta_{5}$	$=$	$ ( 10020520327 \nu^{15} - 45461900870 \nu^{14} + 44547235448 \nu^{13} - 12981941615 \nu^{12} + \cdots - 298001022024 ) / 20109322702 $ (10020520327v^15 - 45461900870v^14 + 44547235448v^13 - 12981941615v^12 + 385912971961v^11 - 1152439387714v^10 + 1281760102330v^9 - 1264840720461v^8 + 3117754162047v^7 - 5736727389604v^6 + 5819705362052v^5 - 4432128787003v^4 + 4137228095470v^3 - 2995942660194v^2 + 1162324265412*v - 298001022024) / 20109322702
$\beta_{6}$	$=$	$ ( 11175608166 \nu^{15} - 37005033535 \nu^{14} - 2381595084 \nu^{13} - 3936897701 \nu^{12} + \cdots - 67263229296 ) / 20109322702 $ (11175608166v^15 - 37005033535v^14 - 2381595084v^13 - 3936897701v^12 + 454687961278v^11 - 715591444243v^10 + 275560834120v^9 - 1011875313139v^8 + 2563606006170v^7 - 2641228538245v^6 + 2425489864376v^5 - 2311962665959v^4 + 1743124162188v^3 - 985628049642v^2 + 363876573956*v - 67263229296) / 20109322702
$\beta_{7}$	$=$	$ ( - 23323681491 \nu^{15} + 83201471534 \nu^{14} - 1607368764 \nu^{13} - 31036989629 \nu^{12} + \cdots - 57823456800 ) / 20109322702 $ (-23323681491v^15 + 83201471534v^14 - 1607368764v^13 - 31036989629v^12 - 972730040921v^11 + 1728562328958v^10 - 418212216102v^9 + 1650986994057v^8 - 5926250882147v^7 + 5756424025160v^6 - 3904479730020v^5 + 4320814620631v^4 - 3222266417878v^3 + 1215012220410v^2 - 194003175156*v - 57823456800) / 20109322702
$\beta_{8}$	$=$	$ ( 26335970486 \nu^{15} - 70391872069 \nu^{14} - 65246694428 \nu^{13} - 2526173993 \nu^{12} + \cdots + 188442633362 ) / 20109322702 $ (26335970486v^15 - 70391872069v^14 - 65246694428v^13 - 2526173993v^12 + 1078964554130v^11 - 1005388669869v^10 - 601570882444v^9 - 1845135723059v^8 + 4716559684762v^7 - 2071430507171v^6 + 1009660171492v^5 - 1797402444403v^4 + 670277831416v^3 + 568901432862v^2 - 555412302964*v + 188442633362) / 20109322702
$\beta_{9}$	$=$	$ ( 20583835651 \nu^{15} - 78924515133 \nu^{14} + 24486137978 \nu^{13} + 16944509126 \nu^{12} + \cdots - 137133576807 ) / 10054661351 $ (20583835651v^15 - 78924515133v^14 + 24486137978v^13 + 16944509126v^12 + 846855475659v^11 - 1758926814389v^10 + 913477887258v^9 - 1721634118436v^8 + 5659512709199v^7 - 6784243698523v^6 + 5445073635924v^5 - 5329410473936v^4 + 4424303976488v^3 - 2300351404182v^2 + 807236148784*v - 137133576807) / 10054661351
$\beta_{10}$	$=$	$ ( - 46688945564 \nu^{15} + 219887457941 \nu^{14} - 194424781504 \nu^{13} - 50488884423 \nu^{12} + \cdots + 828605565416 ) / 20109322702 $ (-46688945564v^15 + 219887457941v^14 - 194424781504v^13 - 50488884423v^12 - 1890422689576v^11 + 5678839565545v^10 - 4832040026616v^9 + 4520765331063v^8 - 15902360689892v^7 + 25178802231927v^6 - 21652899765320v^5 + 18807008211767v^4 - 17224972534876v^3 + 10719181869982v^2 - 4138992241108*v + 828605565416) / 20109322702
$\beta_{11}$	$=$	$ ( - 48745618686 \nu^{15} + 142220971434 \nu^{14} + 68706464252 \nu^{13} + 38916237257 \nu^{12} + \cdots + 132752818922 ) / 20109322702 $ (-48745618686v^15 + 142220971434v^14 + 68706464252v^13 + 38916237257v^12 - 1976323335252v^11 + 2339384761444v^10 - 183551886138v^9 + 4297225454217v^8 - 9571930476062v^7 + 7523047520356v^6 - 7276603158718v^5 + 7211236725035v^4 - 4647700997594v^3 + 2259841492452v^2 - 613753670388*v + 132752818922) / 20109322702
$\beta_{12}$	$=$	$ ( - 51918966616 \nu^{15} + 156637786453 \nu^{14} + 91292310268 \nu^{13} - 53280838773 \nu^{12} + \cdots - 435752795150 ) / 20109322702 $ (-51918966616v^15 + 156637786453v^14 + 91292310268v^13 - 53280838773v^12 - 2183806181780v^11 + 2664420234629v^10 + 895817893144v^9 + 3375808124601v^8 - 11013464257600v^7 + 6196852490403v^6 - 2797053807220v^5 + 5190337377625v^4 - 2414367784536v^3 - 578312257270v^2 + 892787498524*v - 435752795150) / 20109322702
$\beta_{13}$	$=$	$ ( - 52362709317 \nu^{15} + 193624330268 \nu^{14} - 47845583320 \nu^{13} - 16301340676 \nu^{12} + \cdots + 388645444070 ) / 20109322702 $ (-52362709317v^15 + 193624330268v^14 - 47845583320v^13 - 16301340676v^12 - 2131250452977v^11 + 4188622968602v^10 - 2252824563400v^9 + 4571066636438v^8 - 13594781396441v^7 + 16438115098564v^6 - 13965682547094v^5 + 12831594190630v^4 - 10686788900436v^3 + 6015340435046v^2 - 2040124934368*v + 388645444070) / 20109322702
$\beta_{14}$	$=$	$ ( - 62373751860 \nu^{15} + 229249674509 \nu^{14} - 55369811856 \nu^{13} - 7335930805 \nu^{12} + \cdots + 492025369408 ) / 20109322702 $ (-62373751860v^15 + 229249674509v^14 - 55369811856v^13 - 7335930805v^12 - 2539366718576v^11 + 4921777999277v^10 - 2727452903728v^9 + 5672835053989v^8 - 16079561020912v^7 + 19425764769699v^6 - 17215318352204v^5 + 15709615329357v^4 - 12784129629776v^3 + 7555243431778v^2 - 2624472816964*v + 492025369408) / 20109322702
$\beta_{15}$	$=$	$ ( - 65140863369 \nu^{15} + 213460867466 \nu^{14} + 34856893632 \nu^{13} - 15203334586 \nu^{12} + \cdots + 26610324654 ) / 20109322702 $ (-65140863369v^15 + 213460867466v^14 + 34856893632v^13 - 15203334586v^12 - 2674354185421v^11 + 4073707574976v^10 - 905327150412v^9 + 5240912244508v^8 - 14837104039053v^7 + 13747516506566v^6 - 11020742479438v^5 + 11296537134384v^4 - 8172670591536v^3 + 3778071443838v^2 - 923061465008*v + 26610324654) / 20109322702

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

$\nu$	$=$	$ ( 2\beta_{14} - \beta_{13} - \beta_{11} - \beta_{9} - \beta_{7} + 4\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 3\beta_1 ) / 8 $ (2b14 - b13 - b11 - b9 - b7 + 4b6 - b4 - b3 + b2 - 3*b1) / 8
$\nu^{2}$	$=$	$ ( -\beta_{13} - \beta_{11} + \beta_{9} + 3\beta_{7} - 6\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 6 ) / 4 $ (-b13 - b11 + b9 + 3b7 - 6b6 - b4 - b3 + b2 + b1 + 6) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} + 9 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 8 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + \cdots + 20 ) / 8 $ (b15 + 9b14 - 2b13 - 2b12 - 8b11 - 2b10 - 5b9 - b8 - 6b7 + b5 + 4b4 - 3b3 + 2b2 - b1 + 20) / 8
$\nu^{4}$	$=$	$ ( - 2 \beta_{15} + 6 \beta_{14} - 4 \beta_{13} - 8 \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 10 \beta_1 ) / 4 $ (-2b15 + 6b14 - 4b13 - 8b11 - 2b10 - b9 - 4b8 + 4b7 - 14b6 - 2b5 - 5b4 - 6b3 - 10b1) / 4
$\nu^{5}$	$=$	$ ( - 14 \beta_{15} + 18 \beta_{14} - 15 \beta_{13} - 10 \beta_{12} - 15 \beta_{11} + 7 \beta_{9} + \cdots + 104 ) / 8 $ (-14b15 + 18b14 - 15b13 - 10b12 - 15b11 + 7b9 - 18b8 + 27b7 - 104b6 + 6b5 + 11b4 + 9b3 + 15b2 + 29b1 + 104) / 8
$\nu^{6}$	$=$	$ ( 7 \beta_{15} + 44 \beta_{14} - 2 \beta_{13} - 20 \beta_{12} - 42 \beta_{11} - 20 \beta_{10} - 32 \beta_{9} + \cdots - 12 ) / 4 $ (7b15 + 44b14 - 2b13 - 20b12 - 42b11 - 20b10 - 32b9 - 20b8 - 38b7 - 3b5 + 12b4 + b3 - 8b2 - 47*b1 - 12) / 4
$\nu^{7}$	$=$	$ ( - 63 \beta_{15} + 11 \beta_{14} - 111 \beta_{13} - 41 \beta_{11} + 6 \beta_{10} + 68 \beta_{9} + \cdots - 60 \beta_1 ) / 8 $ (-63b15 + 11b14 - 111b13 - 41b11 + 6b10 + 68b9 - 147b8 + 155b7 - 472b6 - 63b5 - 79b4 + 22b3 - 29b2 - 60b1) / 8
$\nu^{8}$	$=$	$ ( 91 \beta_{14} - 24 \beta_{13} - 110 \beta_{12} - 24 \beta_{11} - 51 \beta_{9} - 91 \beta_{8} + \cdots + 166 ) / 4 $ (91b14 - 24b13 - 110b12 - 24b11 - 51b9 - 91b8 - 104b7 - 166b6 + 32b5 + 142b4 + 184b3 + 24b2 + 24*b1 + 166) / 4
$\nu^{9}$	$=$	$ ( 241 \beta_{15} + 177 \beta_{14} - 164 \beta_{13} - 222 \beta_{12} - 326 \beta_{11} - 222 \beta_{10} + \cdots - 1816 ) / 8 $ (241b15 + 177b14 - 164b13 - 222b12 - 326b11 - 222b10 - 329b9 - 481b8 - 480b7 - 395b5 - 152b4 + 231b3 - 472b2 - 1183b1 - 1816) / 8
$\nu^{10}$	$=$	$ ( - 213 \beta_{15} - 498 \beta_{14} - 487 \beta_{13} + 593 \beta_{11} + 420 \beta_{10} + 512 \beta_{9} + \cdots + 342 \beta_1 ) / 4 $ (-213b15 - 498b14 - 487b13 + 593b11 + 420b10 + 512b9 - 526b8 + 449b7 - 1624b6 - 213b5 - 14b4 + 806b3 - 23b2 + 342b1) / 4
$\nu^{11}$	$=$	$ ( 1836 \beta_{15} + 296 \beta_{14} + 649 \beta_{13} - 1982 \beta_{12} + 649 \beta_{11} - 2221 \beta_{9} + \cdots - 5232 ) / 8 $ (1836b15 + 296b14 + 649b13 - 1982b12 + 649b11 - 2221b9 - 296b8 - 5249b7 + 5232b6 - 144b5 + 2517b4 + 3807b3 - 649b2 - 2485b1 - 5232) / 8
$\nu^{12}$	$=$	$ ( 788 \beta_{15} - 3359 \beta_{14} - 1416 \beta_{13} + 962 \beta_{12} + 2176 \beta_{11} + 962 \beta_{10} + \cdots - 10066 ) / 4 $ (788b15 - 3359b14 - 1416b13 + 962b12 + 2176b11 + 962b10 + 1078b9 - 1203b8 + 264b7 - 2700b5 - 2281b4 + 1284b3 - 2072b2 - 3092b1 - 10066) / 4
$\nu^{13}$	$=$	$ ( 533 \beta_{15} - 13907 \beta_{14} - 1923 \beta_{13} + 19683 \beta_{11} + 11418 \beta_{10} + \cdots + 13172 \beta_1 ) / 8 $ (533b15 - 13907b14 - 1923b13 + 19683b11 + 11418b10 + 5972b9 + 1963b8 - 5121b7 + 4632b6 + 533b5 + 7935b4 + 19150b3 + 4055b2 + 13172b1) / 8
$\nu^{14}$	$=$	$ ( 12180 \beta_{15} - 7760 \beta_{14} + 6223 \beta_{13} - 1340 \beta_{12} + 6223 \beta_{11} - 9634 \beta_{9} + \cdots - 49232 ) / 4 $ (12180b15 - 7760b14 + 6223b13 - 1340b12 + 6223b11 - 9634b9 + 7760b8 - 23497b7 + 49232b6 - 5398b5 + 1874b4 + 5653b3 - 6223b2 - 18403b1 - 49232) / 4
$\nu^{15}$	$=$	$ ( - 6773 \beta_{15} - 105043 \beta_{14} - 22984 \beta_{13} + 51026 \beta_{12} + 90770 \beta_{11} + \cdots - 75680 ) / 8 $ (-6773b15 - 105043b14 - 22984b13 + 51026b12 + 90770b11 + 51026b10 + 61371b9 + 17699b8 + 51324b7 - 32673b5 - 43672b4 + 9689b3 - 2916b2 + 38719b1 - 75680) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$\beta_{6}$	$1$	$1$

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} $

703.1

0.224274	+	0.447866i
−1.00047	+	1.15497i
2.07391	−	0.620024i
0.849168	+	0.0870829i
−0.349168	+	0.778942i
−1.57391	+	1.48605i
0.275726	+	0.418160i
1.50047	−	0.288947i
0.224274	−	0.447866i
−1.00047	−	1.15497i
2.07391	+	0.620024i
0.849168	−	0.0870829i
−0.349168	−	0.778942i
−1.57391	−	1.48605i
0.275726	−	0.418160i
1.50047	+	0.288947i

−3.08101

1.77882i

−0.912798

2.48330i

703.2

−3.08101

1.77882i

0.912798

−

2.48330i

703.3

−1.00367

0.579471i

−0.286555

2.63019i

703.4

−1.00367

0.579471i

0.286555

−

2.63019i

703.5

1.00367

−

0.579471i

−1.44550

−

2.21597i

703.6

1.00367

−

0.579471i

1.44550

2.21597i

703.7

3.08101

−

1.77882i

−2.64485

0.0690906i

703.8

3.08101

−

1.77882i

2.64485

−

0.0690906i

1279.1

−3.08101

−

1.77882i

−0.912798

−

2.48330i

1279.2

−3.08101

−

1.77882i

0.912798

2.48330i

1279.3

−1.00367

−

0.579471i

−0.286555

−

2.63019i

1279.4

−1.00367

−

0.579471i

0.286555

2.63019i

1279.5

1.00367

0.579471i

−1.44550

2.21597i

1279.6

1.00367

0.579471i

1.44550

−

2.21597i

1279.7

3.08101

1.77882i

−2.64485

−

0.0690906i

1279.8

3.08101

1.77882i

2.64485

0.0690906i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
28.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2016.2.cs.b		16
3.b	odd	2	1		224.2.p.a	✓	16
4.b	odd	2	1	inner	2016.2.cs.b		16
7.d	odd	6	1	inner	2016.2.cs.b		16
12.b	even	2	1		224.2.p.a	✓	16
21.c	even	2	1		1568.2.p.b		16
21.g	even	6	1		224.2.p.a	✓	16
21.g	even	6	1		1568.2.f.b		16
21.h	odd	6	1		1568.2.f.b		16
21.h	odd	6	1		1568.2.p.b		16
24.f	even	2	1		448.2.p.e		16
24.h	odd	2	1		448.2.p.e		16
28.f	even	6	1	inner	2016.2.cs.b		16
84.h	odd	2	1		1568.2.p.b		16
84.j	odd	6	1		224.2.p.a	✓	16
84.j	odd	6	1		1568.2.f.b		16
84.n	even	6	1		1568.2.f.b		16
84.n	even	6	1		1568.2.p.b		16
168.s	odd	6	1		3136.2.f.j		16
168.v	even	6	1		3136.2.f.j		16
168.ba	even	6	1		448.2.p.e		16
168.ba	even	6	1		3136.2.f.j		16
168.be	odd	6	1		448.2.p.e		16
168.be	odd	6	1		3136.2.f.j		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.2.p.a	✓	16	3.b	odd	2	1
224.2.p.a	✓	16	12.b	even	2	1
224.2.p.a	✓	16	21.g	even	6	1
224.2.p.a	✓	16	84.j	odd	6	1
448.2.p.e		16	24.f	even	2	1
448.2.p.e		16	24.h	odd	2	1
448.2.p.e		16	168.ba	even	6	1
448.2.p.e		16	168.be	odd	6	1
1568.2.f.b		16	21.g	even	6	1
1568.2.f.b		16	21.h	odd	6	1
1568.2.f.b		16	84.j	odd	6	1
1568.2.f.b		16	84.n	even	6	1
1568.2.p.b		16	21.c	even	2	1
1568.2.p.b		16	21.h	odd	6	1
1568.2.p.b		16	84.h	odd	2	1
1568.2.p.b		16	84.n	even	6	1
2016.2.cs.b		16	1.a	even	1	1	trivial
2016.2.cs.b		16	4.b	odd	2	1	inner
2016.2.cs.b		16	7.d	odd	6	1	inner
2016.2.cs.b		16	28.f	even	6	1	inner
3136.2.f.j		16	168.s	odd	6	1
3136.2.f.j		16	168.v	even	6	1
3136.2.f.j		16	168.ba	even	6	1
3136.2.f.j		16	168.be	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2016, [\chi])$:

$ T_{5}^{8} - 14T_{5}^{6} + 179T_{5}^{4} - 238T_{5}^{2} + 289 $

$ T_{11}^{16} - 64 T_{11}^{14} + 2770 T_{11}^{12} - 65248 T_{11}^{10} + 1111795 T_{11}^{8} + \cdots + 352275361 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 14 T^{6} + \cdots + 289)^{2} $ (T^8 - 14T^6 + 179T^4 - 238*T^2 + 289)^2
$7$	$ T^{16} + 16 T^{14} + \cdots + 5764801 $ T^16 + 16T^14 + 60T^12 - 784T^10 - 10426T^8 - 38416T^6 + 144060T^4 + 1882384*T^2 + 5764801
$11$	$ T^{16} + \cdots + 352275361 $ T^16 - 64T^14 + 2770T^12 - 65248T^10 + 1111795T^8 - 10602976T^6 + 71309170T^4 - 184086352*T^2 + 352275361
$13$	$ (T^{8} + 56 T^{6} + \cdots + 12544)^{2} $ (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$17$	$ (T^{8} - 22 T^{6} + \cdots + 49)^{2} $ (T^8 - 22T^6 + 491T^4 - 1056T^3 + 922T^2 - 336*T + 49)^2
$19$	$ T^{16} + 32 T^{14} + \cdots + 1 $ T^16 + 32T^14 + 962T^12 + 1952T^10 + 3331T^8 + 928T^6 + 194T^4 + 16*T^2 + 1
$23$	$ T^{16} - 48 T^{14} + \cdots + 2825761 $ T^16 - 48T^14 + 1618T^12 - 27744T^10 + 344499T^8 - 1616736T^6 + 5565298T^4 - 4357152*T^2 + 2825761
$29$	$ (T^{4} + 4 T^{3} + \cdots + 112)^{4} $ (T^4 + 4T^3 - 28T^2 - 16*T + 112)^4
$31$	$ T^{16} + \cdots + 12897917761 $ T^16 + 128T^14 + 11810T^12 + 500768T^10 + 15386851T^8 + 164644384T^6 + 1274227298T^4 + 4809874288*T^2 + 12897917761
$37$	$ (T^{8} + 4 T^{7} + \cdots + 113569)^{2} $ (T^8 + 4T^7 + 74T^6 + 208T^5 + 3907T^4 + 10064T^3 + 67946T^2 - 74140*T + 113569)^2
$41$	$ (T^{8} + 56 T^{6} + \cdots + 12544)^{2} $ (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$43$	$ (T^{8} + 192 T^{6} + \cdots + 12544)^{2} $ (T^8 + 192T^6 + 6368T^4 + 18432*T^2 + 12544)^2
$47$	$ T^{16} + \cdots + 9597924961 $ T^16 + 176T^14 + 22130T^12 + 1341344T^10 + 59185171T^8 + 918901408T^6 + 10749032402T^4 + 10558706944*T^2 + 9597924961
$53$	$ (T^{8} - 4 T^{7} + \cdots + 2401)^{2} $ (T^8 - 4T^7 + 106T^6 + 560T^5 + 7651T^4 + 9392T^3 + 14410T^2 - 4900*T + 2401)^2
$59$	$ T^{16} + \cdots + 43617904801 $ T^16 + 256T^14 + 51026T^12 + 3233056T^10 + 148698739T^8 + 3386380832T^6 + 54931126514T^4 + 50280814448*T^2 + 43617904801
$61$	$ (T^{8} + 12 T^{7} + \cdots + 4844401)^{2} $ (T^8 + 12T^7 - 62T^6 - 1320T^5 + 6587T^4 + 91080T^3 - 13582T^2 - 1822428*T + 4844401)^2
$67$	$ T^{16} + \cdots + 15527402881 $ T^16 - 288T^14 + 61570T^12 - 5603424T^10 + 377193795T^8 - 5830527072T^6 + 73592115970T^4 - 34410027696*T^2 + 15527402881
$71$	$ (T^{8} + 320 T^{6} + \cdots + 6635776)^{2} $ (T^8 + 320T^6 + 29664T^4 + 806912*T^2 + 6635776)^2
$73$	$ (T^{8} + 12 T^{7} + \cdots + 564001)^{2} $ (T^8 + 12T^7 - 22T^6 - 840T^5 + 2147T^4 + 61320T^3 + 308362T^2 + 657876*T + 564001)^2
$79$	$ T^{16} + \cdots + 16748793615841 $ T^16 - 336T^14 + 89266T^12 - 6829920T^10 + 367844691T^8 - 10361634912T^6 + 211185354130T^4 - 2270862491520*T^2 + 16748793615841
$83$	$ (T^{8} - 416 T^{6} + \cdots + 3211264)^{2} $ (T^8 - 416T^6 + 39680T^4 - 827392*T^2 + 3211264)^2
$89$	$ (T^{8} - 36 T^{7} + \cdots + 27952369)^{2} $ (T^8 - 36T^7 + 466T^6 - 1224T^5 - 18757T^4 + 71400T^3 + 1290242T^2 - 11102700*T + 27952369)^2
$97$	$ (T^{8} + 472 T^{6} + \cdots + 118026496)^{2} $ (T^8 + 472T^6 + 78896T^4 + 5378432*T^2 + 118026496)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.cs (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{10} q^{5} + \beta_{15} q^{7}+O(q^{10}) \) q + b10 * q^5 + b15 * q^7	\( q + \beta_{10} q^{5} + \beta_{15} q^{7} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - 2 \beta_{12} - 2 \beta_{10} + \cdots + 3) q^{97}+O(q^{100}) \) q + b10 * q^5 + b15 * q^7 + (-b15 + b13 - b7 + b3 + b2) * q^11 + (b14 - b8 + 2b6 - 1) q^13 - b8 * q^17 + (-b15 - b7 - b5 + b2) * q^19 + (-b13 + b2) * q^23 + (-b14 - b9 + b8 - 2b6 + 2) q^25 + (-b4 - 1) * q^29 + (-b13 - 2b11 - b5 - b3 + b2 - b1) q^31 + (-3b13 - 3b7 - b5 + b2 - 2b1) q^35 + (b14 - 2b12 - b10 - b9 - b6 + b4) q^37 + (-b14 + b8 - 2b6 + 1) q^41 + (-2b11 + 2b7) * q^43 + (-b15 + b11 + b7 - b5 - b3 + b2 + b1) * q^47 + (-2b12 - b9 - 2) q^49 + (2b14 - b12 - 2b10 - b8 - b6 + 1) * q^53 + (-3b15 + 3b13 - b5 - b2 + 6b1) q^55 + (2b15 + 2b13 + 3b7 - 2b2 + 3b1) q^59 + (b14 + b10 - b9 + b6 - b4 - 2) * q^61 + (-b14 - 4b12 - 2b10 + 2b8 - b6) q^65 + (-b15 + 2b13 + b7 + b5 - b3 + 2b2) * q^67 + (b15 + b13 - b11 + b7 + 3b5 - 3b2) * q^71 + (b14 - b9 + b8 - b6 + 2b4 - 1) q^73 + (3b14 + b12 - 2b9 + b8 + 2b6 + 2b4 - 5) * q^77 + (-b13 + 2b7 + b2 + 4b1) * q^79 + (-b15 + b13 - b11 + b7 + b5 - 2b3 + b2 + 4b1) * q^83 + (b14 - b12 + b10 + b8 - 1) * q^85 + (-2b14 + 2b10 - 3b6 + 6) q^89 + (b15 - b11 + b7 - 3b3 - 3b2 + 3b1) q^91 + (-4b15 + 2b13 - 2b7 - 2b5 + b3 + 2b2 + b1) q^95 + (-2b12 - 2b10 + 2b9 - 2b8 - 6b6 - b4 + 3) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{25} - 16 q^{29} - 8 q^{37} - 32 q^{49} + 8 q^{53} - 24 q^{61} - 8 q^{65} - 24 q^{73} - 64 q^{77} - 16 q^{85} + 72 q^{89}+O(q^{100}) \) 16 * q + 16 * q^25 - 16 * q^29 - 8 * q^37 - 32 * q^49 + 8 * q^53 - 24 * q^61 - 8 * q^65 - 24 * q^73 - 64 * q^77 - 16 * q^85 + 72 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2016.2.cs.b

Level	$2016$
Weight	$2$
Character orbit	2016.cs
Analytic conductor	$16.098$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(16.0978410475\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	16.0.2353561680715186176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 \) x^16 - 4x^15 + 2x^14 + 41x^12 - 92x^11 + 66x^10 - 104x^9 + 291x^8 - 388x^7 + 366x^6 - 344x^5 + 286x^4 - 184x^3 + 84x^2 - 24x + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{20} \)
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 508009675 \nu^{15} + 17835297016 \nu^{14} - 56517662852 \nu^{13} + 582270848 \nu^{12} + \cdots + 242647661018 ) / 20109322702 \) (-508009675v^15 + 17835297016v^14 - 56517662852v^13 + 582270848v^12 - 7088890831v^11 + 702274874462v^10 - 1170090037720v^9 + 377376368526v^8 - 1364027322083v^7 + 4137630712600v^6 - 4084528421066v^5 + 3083496853086v^4 - 3236291968028v^3 + 2419000098258v^2 - 1055996592624*v + 242647661018) / 20109322702
\(\beta_{2}\)	\(=\)	\( ( 2306431404 \nu^{15} - 37444380512 \nu^{14} + 95714273392 \nu^{13} + 18886793845 \nu^{12} + \cdots - 394015237814 ) / 20109322702 \) (2306431404v^15 - 37444380512v^14 + 95714273392v^13 + 18886793845v^12 + 90029143982v^11 - 1367786602618v^10 + 1863018589666v^9 - 575541348219v^8 + 2921911891676v^7 - 7182761898450v^6 + 6570657118770v^5 - 5342350769881v^4 + 5286105515374v^3 - 3757872246264v^2 + 1675584716836*v - 394015237814) / 20109322702
\(\beta_{3}\)	\(=\)	\( ( 2358518199 \nu^{15} + 7744007704 \nu^{14} - 40220309508 \nu^{13} - 37698238811 \nu^{12} + \cdots + 6869804700 ) / 20109322702 \) (2358518199v^15 + 7744007704v^14 - 40220309508v^13 - 37698238811v^12 + 72348596101v^11 + 470758796196v^10 - 465557954766v^9 - 370921530677v^8 - 836020034165v^7 + 2010052783442v^6 - 927678153432v^5 + 1113778618237v^4 - 1251623532730v^3 + 664629352550v^2 - 193341101940*v + 6869804700) / 20109322702
\(\beta_{4}\)	\(=\)	\( ( 32131571 \nu^{15} - 338443886 \nu^{14} + 801340954 \nu^{13} - 7179161 \nu^{12} + \cdots - 4131081209 ) / 137735087 \) (32131571v^15 - 338443886v^14 + 801340954v^13 - 7179161v^12 + 1133962371v^11 - 11663672814v^10 + 17217925190v^9 - 7634184353v^8 + 25474337063v^7 - 64850621098v^6 + 63560694880v^5 - 49589483157v^4 + 50404045920v^3 - 36993277864v^2 + 15863090132*v - 4131081209) / 137735087
\(\beta_{5}\)	\(=\)	\( ( 10020520327 \nu^{15} - 45461900870 \nu^{14} + 44547235448 \nu^{13} - 12981941615 \nu^{12} + \cdots - 298001022024 ) / 20109322702 \) (10020520327v^15 - 45461900870v^14 + 44547235448v^13 - 12981941615v^12 + 385912971961v^11 - 1152439387714v^10 + 1281760102330v^9 - 1264840720461v^8 + 3117754162047v^7 - 5736727389604v^6 + 5819705362052v^5 - 4432128787003v^4 + 4137228095470v^3 - 2995942660194v^2 + 1162324265412*v - 298001022024) / 20109322702
\(\beta_{6}\)	\(=\)	\( ( 11175608166 \nu^{15} - 37005033535 \nu^{14} - 2381595084 \nu^{13} - 3936897701 \nu^{12} + \cdots - 67263229296 ) / 20109322702 \) (11175608166v^15 - 37005033535v^14 - 2381595084v^13 - 3936897701v^12 + 454687961278v^11 - 715591444243v^10 + 275560834120v^9 - 1011875313139v^8 + 2563606006170v^7 - 2641228538245v^6 + 2425489864376v^5 - 2311962665959v^4 + 1743124162188v^3 - 985628049642v^2 + 363876573956*v - 67263229296) / 20109322702
\(\beta_{7}\)	\(=\)	\( ( - 23323681491 \nu^{15} + 83201471534 \nu^{14} - 1607368764 \nu^{13} - 31036989629 \nu^{12} + \cdots - 57823456800 ) / 20109322702 \) (-23323681491v^15 + 83201471534v^14 - 1607368764v^13 - 31036989629v^12 - 972730040921v^11 + 1728562328958v^10 - 418212216102v^9 + 1650986994057v^8 - 5926250882147v^7 + 5756424025160v^6 - 3904479730020v^5 + 4320814620631v^4 - 3222266417878v^3 + 1215012220410v^2 - 194003175156*v - 57823456800) / 20109322702
\(\beta_{8}\)	\(=\)	\( ( 26335970486 \nu^{15} - 70391872069 \nu^{14} - 65246694428 \nu^{13} - 2526173993 \nu^{12} + \cdots + 188442633362 ) / 20109322702 \) (26335970486v^15 - 70391872069v^14 - 65246694428v^13 - 2526173993v^12 + 1078964554130v^11 - 1005388669869v^10 - 601570882444v^9 - 1845135723059v^8 + 4716559684762v^7 - 2071430507171v^6 + 1009660171492v^5 - 1797402444403v^4 + 670277831416v^3 + 568901432862v^2 - 555412302964*v + 188442633362) / 20109322702
\(\beta_{9}\)	\(=\)	\( ( 20583835651 \nu^{15} - 78924515133 \nu^{14} + 24486137978 \nu^{13} + 16944509126 \nu^{12} + \cdots - 137133576807 ) / 10054661351 \) (20583835651v^15 - 78924515133v^14 + 24486137978v^13 + 16944509126v^12 + 846855475659v^11 - 1758926814389v^10 + 913477887258v^9 - 1721634118436v^8 + 5659512709199v^7 - 6784243698523v^6 + 5445073635924v^5 - 5329410473936v^4 + 4424303976488v^3 - 2300351404182v^2 + 807236148784*v - 137133576807) / 10054661351
\(\beta_{10}\)	\(=\)	\( ( - 46688945564 \nu^{15} + 219887457941 \nu^{14} - 194424781504 \nu^{13} - 50488884423 \nu^{12} + \cdots + 828605565416 ) / 20109322702 \) (-46688945564v^15 + 219887457941v^14 - 194424781504v^13 - 50488884423v^12 - 1890422689576v^11 + 5678839565545v^10 - 4832040026616v^9 + 4520765331063v^8 - 15902360689892v^7 + 25178802231927v^6 - 21652899765320v^5 + 18807008211767v^4 - 17224972534876v^3 + 10719181869982v^2 - 4138992241108*v + 828605565416) / 20109322702
\(\beta_{11}\)	\(=\)	\( ( - 48745618686 \nu^{15} + 142220971434 \nu^{14} + 68706464252 \nu^{13} + 38916237257 \nu^{12} + \cdots + 132752818922 ) / 20109322702 \) (-48745618686v^15 + 142220971434v^14 + 68706464252v^13 + 38916237257v^12 - 1976323335252v^11 + 2339384761444v^10 - 183551886138v^9 + 4297225454217v^8 - 9571930476062v^7 + 7523047520356v^6 - 7276603158718v^5 + 7211236725035v^4 - 4647700997594v^3 + 2259841492452v^2 - 613753670388*v + 132752818922) / 20109322702
\(\beta_{12}\)	\(=\)	\( ( - 51918966616 \nu^{15} + 156637786453 \nu^{14} + 91292310268 \nu^{13} - 53280838773 \nu^{12} + \cdots - 435752795150 ) / 20109322702 \) (-51918966616v^15 + 156637786453v^14 + 91292310268v^13 - 53280838773v^12 - 2183806181780v^11 + 2664420234629v^10 + 895817893144v^9 + 3375808124601v^8 - 11013464257600v^7 + 6196852490403v^6 - 2797053807220v^5 + 5190337377625v^4 - 2414367784536v^3 - 578312257270v^2 + 892787498524*v - 435752795150) / 20109322702
\(\beta_{13}\)	\(=\)	\( ( - 52362709317 \nu^{15} + 193624330268 \nu^{14} - 47845583320 \nu^{13} - 16301340676 \nu^{12} + \cdots + 388645444070 ) / 20109322702 \) (-52362709317v^15 + 193624330268v^14 - 47845583320v^13 - 16301340676v^12 - 2131250452977v^11 + 4188622968602v^10 - 2252824563400v^9 + 4571066636438v^8 - 13594781396441v^7 + 16438115098564v^6 - 13965682547094v^5 + 12831594190630v^4 - 10686788900436v^3 + 6015340435046v^2 - 2040124934368*v + 388645444070) / 20109322702
\(\beta_{14}\)	\(=\)	\( ( - 62373751860 \nu^{15} + 229249674509 \nu^{14} - 55369811856 \nu^{13} - 7335930805 \nu^{12} + \cdots + 492025369408 ) / 20109322702 \) (-62373751860v^15 + 229249674509v^14 - 55369811856v^13 - 7335930805v^12 - 2539366718576v^11 + 4921777999277v^10 - 2727452903728v^9 + 5672835053989v^8 - 16079561020912v^7 + 19425764769699v^6 - 17215318352204v^5 + 15709615329357v^4 - 12784129629776v^3 + 7555243431778v^2 - 2624472816964*v + 492025369408) / 20109322702
\(\beta_{15}\)	\(=\)	\( ( - 65140863369 \nu^{15} + 213460867466 \nu^{14} + 34856893632 \nu^{13} - 15203334586 \nu^{12} + \cdots + 26610324654 ) / 20109322702 \) (-65140863369v^15 + 213460867466v^14 + 34856893632v^13 - 15203334586v^12 - 2674354185421v^11 + 4073707574976v^10 - 905327150412v^9 + 5240912244508v^8 - 14837104039053v^7 + 13747516506566v^6 - 11020742479438v^5 + 11296537134384v^4 - 8172670591536v^3 + 3778071443838v^2 - 923061465008*v + 26610324654) / 20109322702

\(\nu\)	\(=\)	\( ( 2\beta_{14} - \beta_{13} - \beta_{11} - \beta_{9} - \beta_{7} + 4\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 3\beta_1 ) / 8 \) (2b14 - b13 - b11 - b9 - b7 + 4b6 - b4 - b3 + b2 - 3*b1) / 8
\(\nu^{2}\)	\(=\)	\( ( -\beta_{13} - \beta_{11} + \beta_{9} + 3\beta_{7} - 6\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 6 ) / 4 \) (-b13 - b11 + b9 + 3b7 - 6b6 - b4 - b3 + b2 + b1 + 6) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + 9 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 8 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + \cdots + 20 ) / 8 \) (b15 + 9b14 - 2b13 - 2b12 - 8b11 - 2b10 - 5b9 - b8 - 6b7 + b5 + 4b4 - 3b3 + 2b2 - b1 + 20) / 8
\(\nu^{4}\)	\(=\)	\( ( - 2 \beta_{15} + 6 \beta_{14} - 4 \beta_{13} - 8 \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 10 \beta_1 ) / 4 \) (-2b15 + 6b14 - 4b13 - 8b11 - 2b10 - b9 - 4b8 + 4b7 - 14b6 - 2b5 - 5b4 - 6b3 - 10b1) / 4
\(\nu^{5}\)	\(=\)	\( ( - 14 \beta_{15} + 18 \beta_{14} - 15 \beta_{13} - 10 \beta_{12} - 15 \beta_{11} + 7 \beta_{9} + \cdots + 104 ) / 8 \) (-14b15 + 18b14 - 15b13 - 10b12 - 15b11 + 7b9 - 18b8 + 27b7 - 104b6 + 6b5 + 11b4 + 9b3 + 15b2 + 29b1 + 104) / 8
\(\nu^{6}\)	\(=\)	\( ( 7 \beta_{15} + 44 \beta_{14} - 2 \beta_{13} - 20 \beta_{12} - 42 \beta_{11} - 20 \beta_{10} - 32 \beta_{9} + \cdots - 12 ) / 4 \) (7b15 + 44b14 - 2b13 - 20b12 - 42b11 - 20b10 - 32b9 - 20b8 - 38b7 - 3b5 + 12b4 + b3 - 8b2 - 47*b1 - 12) / 4
\(\nu^{7}\)	\(=\)	\( ( - 63 \beta_{15} + 11 \beta_{14} - 111 \beta_{13} - 41 \beta_{11} + 6 \beta_{10} + 68 \beta_{9} + \cdots - 60 \beta_1 ) / 8 \) (-63b15 + 11b14 - 111b13 - 41b11 + 6b10 + 68b9 - 147b8 + 155b7 - 472b6 - 63b5 - 79b4 + 22b3 - 29b2 - 60b1) / 8
\(\nu^{8}\)	\(=\)	\( ( 91 \beta_{14} - 24 \beta_{13} - 110 \beta_{12} - 24 \beta_{11} - 51 \beta_{9} - 91 \beta_{8} + \cdots + 166 ) / 4 \) (91b14 - 24b13 - 110b12 - 24b11 - 51b9 - 91b8 - 104b7 - 166b6 + 32b5 + 142b4 + 184b3 + 24b2 + 24*b1 + 166) / 4
\(\nu^{9}\)	\(=\)	\( ( 241 \beta_{15} + 177 \beta_{14} - 164 \beta_{13} - 222 \beta_{12} - 326 \beta_{11} - 222 \beta_{10} + \cdots - 1816 ) / 8 \) (241b15 + 177b14 - 164b13 - 222b12 - 326b11 - 222b10 - 329b9 - 481b8 - 480b7 - 395b5 - 152b4 + 231b3 - 472b2 - 1183b1 - 1816) / 8
\(\nu^{10}\)	\(=\)	\( ( - 213 \beta_{15} - 498 \beta_{14} - 487 \beta_{13} + 593 \beta_{11} + 420 \beta_{10} + 512 \beta_{9} + \cdots + 342 \beta_1 ) / 4 \) (-213b15 - 498b14 - 487b13 + 593b11 + 420b10 + 512b9 - 526b8 + 449b7 - 1624b6 - 213b5 - 14b4 + 806b3 - 23b2 + 342b1) / 4
\(\nu^{11}\)	\(=\)	\( ( 1836 \beta_{15} + 296 \beta_{14} + 649 \beta_{13} - 1982 \beta_{12} + 649 \beta_{11} - 2221 \beta_{9} + \cdots - 5232 ) / 8 \) (1836b15 + 296b14 + 649b13 - 1982b12 + 649b11 - 2221b9 - 296b8 - 5249b7 + 5232b6 - 144b5 + 2517b4 + 3807b3 - 649b2 - 2485b1 - 5232) / 8
\(\nu^{12}\)	\(=\)	\( ( 788 \beta_{15} - 3359 \beta_{14} - 1416 \beta_{13} + 962 \beta_{12} + 2176 \beta_{11} + 962 \beta_{10} + \cdots - 10066 ) / 4 \) (788b15 - 3359b14 - 1416b13 + 962b12 + 2176b11 + 962b10 + 1078b9 - 1203b8 + 264b7 - 2700b5 - 2281b4 + 1284b3 - 2072b2 - 3092b1 - 10066) / 4
\(\nu^{13}\)	\(=\)	\( ( 533 \beta_{15} - 13907 \beta_{14} - 1923 \beta_{13} + 19683 \beta_{11} + 11418 \beta_{10} + \cdots + 13172 \beta_1 ) / 8 \) (533b15 - 13907b14 - 1923b13 + 19683b11 + 11418b10 + 5972b9 + 1963b8 - 5121b7 + 4632b6 + 533b5 + 7935b4 + 19150b3 + 4055b2 + 13172b1) / 8
\(\nu^{14}\)	\(=\)	\( ( 12180 \beta_{15} - 7760 \beta_{14} + 6223 \beta_{13} - 1340 \beta_{12} + 6223 \beta_{11} - 9634 \beta_{9} + \cdots - 49232 ) / 4 \) (12180b15 - 7760b14 + 6223b13 - 1340b12 + 6223b11 - 9634b9 + 7760b8 - 23497b7 + 49232b6 - 5398b5 + 1874b4 + 5653b3 - 6223b2 - 18403b1 - 49232) / 4
\(\nu^{15}\)	\(=\)	\( ( - 6773 \beta_{15} - 105043 \beta_{14} - 22984 \beta_{13} + 51026 \beta_{12} + 90770 \beta_{11} + \cdots - 75680 ) / 8 \) (-6773b15 - 105043b14 - 22984b13 + 51026b12 + 90770b11 + 51026b10 + 61371b9 + 17699b8 + 51324b7 - 32673b5 - 43672b4 + 9689b3 - 2916b2 + 38719b1 - 75680) / 8

\(n\)	\(127\)	\(577\)	\(1765\)	\(1793\)
\(\chi(n)\)	\(-1\)	\(\beta_{6}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 703.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 14 T^{6} + \cdots + 289)^{2} \) (T^8 - 14T^6 + 179T^4 - 238*T^2 + 289)^2
$7$	\( T^{16} + 16 T^{14} + \cdots + 5764801 \) T^16 + 16T^14 + 60T^12 - 784T^10 - 10426T^8 - 38416T^6 + 144060T^4 + 1882384*T^2 + 5764801
$11$	\( T^{16} + \cdots + 352275361 \) T^16 - 64T^14 + 2770T^12 - 65248T^10 + 1111795T^8 - 10602976T^6 + 71309170T^4 - 184086352*T^2 + 352275361
$13$	\( (T^{8} + 56 T^{6} + \cdots + 12544)^{2} \) (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$17$	\( (T^{8} - 22 T^{6} + \cdots + 49)^{2} \) (T^8 - 22T^6 + 491T^4 - 1056T^3 + 922T^2 - 336*T + 49)^2
$19$	\( T^{16} + 32 T^{14} + \cdots + 1 \) T^16 + 32T^14 + 962T^12 + 1952T^10 + 3331T^8 + 928T^6 + 194T^4 + 16*T^2 + 1
$23$	\( T^{16} - 48 T^{14} + \cdots + 2825761 \) T^16 - 48T^14 + 1618T^12 - 27744T^10 + 344499T^8 - 1616736T^6 + 5565298T^4 - 4357152*T^2 + 2825761
$29$	\( (T^{4} + 4 T^{3} + \cdots + 112)^{4} \) (T^4 + 4T^3 - 28T^2 - 16*T + 112)^4
$31$	\( T^{16} + \cdots + 12897917761 \) T^16 + 128T^14 + 11810T^12 + 500768T^10 + 15386851T^8 + 164644384T^6 + 1274227298T^4 + 4809874288*T^2 + 12897917761
$37$	\( (T^{8} + 4 T^{7} + \cdots + 113569)^{2} \) (T^8 + 4T^7 + 74T^6 + 208T^5 + 3907T^4 + 10064T^3 + 67946T^2 - 74140*T + 113569)^2
$41$	\( (T^{8} + 56 T^{6} + \cdots + 12544)^{2} \) (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$43$	\( (T^{8} + 192 T^{6} + \cdots + 12544)^{2} \) (T^8 + 192T^6 + 6368T^4 + 18432*T^2 + 12544)^2
$47$	\( T^{16} + \cdots + 9597924961 \) T^16 + 176T^14 + 22130T^12 + 1341344T^10 + 59185171T^8 + 918901408T^6 + 10749032402T^4 + 10558706944*T^2 + 9597924961
$53$	\( (T^{8} - 4 T^{7} + \cdots + 2401)^{2} \) (T^8 - 4T^7 + 106T^6 + 560T^5 + 7651T^4 + 9392T^3 + 14410T^2 - 4900*T + 2401)^2
$59$	\( T^{16} + \cdots + 43617904801 \) T^16 + 256T^14 + 51026T^12 + 3233056T^10 + 148698739T^8 + 3386380832T^6 + 54931126514T^4 + 50280814448*T^2 + 43617904801
$61$	\( (T^{8} + 12 T^{7} + \cdots + 4844401)^{2} \) (T^8 + 12T^7 - 62T^6 - 1320T^5 + 6587T^4 + 91080T^3 - 13582T^2 - 1822428*T + 4844401)^2
$67$	\( T^{16} + \cdots + 15527402881 \) T^16 - 288T^14 + 61570T^12 - 5603424T^10 + 377193795T^8 - 5830527072T^6 + 73592115970T^4 - 34410027696*T^2 + 15527402881
$71$	\( (T^{8} + 320 T^{6} + \cdots + 6635776)^{2} \) (T^8 + 320T^6 + 29664T^4 + 806912*T^2 + 6635776)^2
$73$	\( (T^{8} + 12 T^{7} + \cdots + 564001)^{2} \) (T^8 + 12T^7 - 22T^6 - 840T^5 + 2147T^4 + 61320T^3 + 308362T^2 + 657876*T + 564001)^2
$79$	\( T^{16} + \cdots + 16748793615841 \) T^16 - 336T^14 + 89266T^12 - 6829920T^10 + 367844691T^8 - 10361634912T^6 + 211185354130T^4 - 2270862491520*T^2 + 16748793615841
$83$	\( (T^{8} - 416 T^{6} + \cdots + 3211264)^{2} \) (T^8 - 416T^6 + 39680T^4 - 827392*T^2 + 3211264)^2
$89$	\( (T^{8} - 36 T^{7} + \cdots + 27952369)^{2} \) (T^8 - 36T^7 + 466T^6 - 1224T^5 - 18757T^4 + 71400T^3 + 1290242T^2 - 11102700*T + 27952369)^2
$97$	\( (T^{8} + 472 T^{6} + \cdots + 118026496)^{2} \) (T^8 + 472T^6 + 78896T^4 + 5378432*T^2 + 118026496)^2
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