LMFDB - Newform orbit 3456.2.r.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(577,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("3456.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.r (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.9349208943630483456.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 $ x^16 - 8x^15 + 48x^14 - 196x^13 + 642x^12 - 1668x^11 + 3580x^10 - 6328x^9 + 9297x^8 - 11276x^7 + 11224x^6 - 9024x^5 + 5736x^4 - 2780x^3 + 972x^2 - 220*x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 1152)
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_{7} + \beta_{5}) q^{5} + \beta_{13} q^{7}+O(q^{10}) $ q + (-b7 + b5) * q^5 + b13 * q^7	$ q + ( - \beta_{7} + \beta_{5}) q^{5} + \beta_{13} q^{7} + \beta_{10} q^{11} - \beta_{7} q^{13} + (\beta_{3} - 4) q^{17} + (\beta_{15} - \beta_{13} + \beta_{2}) q^{23} + ( - 2 \beta_{14} + 2 \beta_1) q^{25} + (\beta_{8} + 2 \beta_{6}) q^{29} + (\beta_{15} + \beta_{13} - \beta_{2}) q^{31} + (\beta_{12} + \beta_{11} + \cdots + 3 \beta_{9}) q^{35}+ \cdots + (3 \beta_{14} - 9 \beta_1) q^{97}+O(q^{100}) $ q + (-b7 + b5) * q^5 + b13 * q^7 + b10 * q^11 - b7 * q^13 + (b3 - 4) * q^17 + (b15 - b13 + b2) * q^23 + (-2b14 + 2b1) * q^25 + (b8 + 2b6) q^29 + (b15 + b13 - b2) * q^31 + (b12 + b11 + 3b10 + 3b9) * q^35 + (-6b8 - 6b7 + b6 - b5) * q^37 + (3b14 - 3b3 - b1 + 1) * q^41 + (-b12 + 3b10) q^43 + b13 * q^47 + (-3b14 + 3b3 + 2b1 - 2) q^49 + (4b8 + 4b7 - 3b6 + 3b5) * q^53 + (b4 - 3b2) q^55 + (b11 - b9) * q^59 + (3b8 - 2b6) * q^61 + (-b14 + b1) * q^65 + (2b11 + 3b9) * q^67 + (3b4 - 2b2) * q^71 + (b3 + 6) * q^73 + (9b7 - 3b5) * q^77 + (3b15 + b13 - 3b4) * q^79 + (b12 - b10) * q^83 + (10b7 - 5b5) * q^85 + (b3 - 2) * q^89 + (b10 + b9) * q^91 + (3b14 - 9b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 64 q^{17} + 16 q^{25} + 8 q^{41} - 16 q^{49} + 8 q^{65} + 96 q^{73} - 32 q^{89} - 72 q^{97}+O(q^{100}) $ 16 * q - 64 * q^17 + 16 * q^25 + 8 * q^41 - 16 * q^49 + 8 * q^65 + 96 * q^73 - 32 * q^89 - 72 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 $ x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25 :

$\beta_{1}$	$=$	$ ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} + \cdots - 125460 ) / 17095 $ (3456v^15 - 25920v^14 + 151876v^13 - 594074v^12 + 1879372v^11 - 4666596v^10 + 9554736v^9 - 15945783v^8 + 21928484v^7 - 24527176v^6 + 22138664v^5 - 15739188v^4 + 8598668v^3 - 3418428v^2 + 929924*v - 125460) / 17095
$\beta_{2}$	$=$	$ ( 183 \nu^{14} - 1281 \nu^{13} + 7408 \nu^{12} - 27795 \nu^{11} + 85851 \nu^{10} - 204998 \nu^{9} + \cdots + 10130 ) / 65 $ (183v^14 - 1281v^13 + 7408v^12 - 27795v^11 + 85851v^10 - 204998v^9 + 405777v^8 - 646527v^7 + 846324v^6 - 886329v^5 + 737358v^4 - 466904v^3 + 215139v^2 - 64206v + 10130) / 65
$\beta_{3}$	$=$	$ ( - 54 \nu^{14} + 378 \nu^{13} - 2193 \nu^{12} + 8244 \nu^{11} - 25569 \nu^{10} + 61284 \nu^{9} + \cdots - 3308 ) / 13 $ (-54v^14 + 378v^13 - 2193v^12 + 8244v^11 - 25569v^10 + 61284v^9 - 122021v^8 + 195620v^7 - 258293v^6 + 273134v^5 - 230109v^4 + 147802v^3 - 69203v^2 + 20980v - 3308) / 13
$\beta_{4}$	$=$	$ ( - 447 \nu^{14} + 3129 \nu^{13} - 18112 \nu^{12} + 67995 \nu^{11} - 210309 \nu^{10} + 502832 \nu^{9} + \cdots - 25820 ) / 65 $ (-447v^14 + 3129v^13 - 18112v^12 + 67995v^11 - 210309v^10 + 502832v^9 - 997608v^8 + 1593543v^7 - 2094381v^6 + 2203926v^5 - 1846302v^4 + 1179041v^3 - 548766v^2 + 165459v - 25820) / 65
$\beta_{5}$	$=$	$ ( 22230 \nu^{15} - 144633 \nu^{14} + 820486 \nu^{13} - 2915351 \nu^{12} + 8665028 \nu^{11} + \cdots + 599315 ) / 17095 $ (22230v^15 - 144633v^14 + 820486v^13 - 2915351v^12 + 8665028v^11 - 19406977v^10 + 35943838v^9 - 51803400v^8 + 59433982v^7 - 50057927v^6 + 27990758v^5 - 4701824v^4 - 7461590v^3 + 7646662v^2 - 3312642*v + 599315) / 17095
$\beta_{6}$	$=$	$ ( - 22230 \nu^{15} + 188817 \nu^{14} - 1129774 \nu^{13} + 4704014 \nu^{12} - 15376262 \nu^{11} + \cdots + 1317955 ) / 17095 $ (-22230v^15 + 188817v^14 - 1129774v^13 + 4704014v^12 - 15376262v^11 + 40133218v^10 - 85426762v^9 + 149690685v^8 - 215272528v^7 + 253642808v^6 - 240583652v^5 + 180367151v^4 - 102603910v^3 + 41706077v^2 - 10736292*v + 1317955) / 17095
$\beta_{7}$	$=$	$ ( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} + \cdots + 218885 ) / 17095 $ (26630v^15 - 182630v^14 + 1056740v^13 - 3923413v^12 + 12097498v^11 - 28666706v^10 + 56590650v^9 - 89528958v^8 + 116711850v^7 - 121073088v^6 + 99711814v^5 - 61611091v^4 + 27082796v^3 - 6949514v^2 + 510382*v + 218885) / 17095
$\beta_{8}$	$=$	$ ( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} + \cdots - 2071845 ) / 17095 $ (26630v^15 - 216820v^14 + 1296070v^13 - 5311527v^12 + 17314892v^11 - 44845414v^10 + 95362110v^9 - 166733397v^8 + 240513840v^7 - 284726942v^6 + 273082466v^5 - 208344314v^4 + 122001074v^3 - 52073476v^2 + 14507768*v - 2071845) / 17095
$\beta_{9}$	$=$	$ ( - 63322 \nu^{15} + 450456 \nu^{14} - 2625131 \nu^{13} + 9970934 \nu^{12} - 31148391 \nu^{11} + \cdots - 31980 ) / 17095 $ (-63322v^15 + 450456v^14 - 2625131v^13 + 9970934v^12 - 31148391v^11 + 75404770v^10 - 151597035v^9 + 245879883v^8 - 328994304v^7 + 353294264v^6 - 303546609v^5 + 199496197v^4 - 96502001v^3 + 30387501v^2 - 5033857*v - 31980) / 17095
$\beta_{10}$	$=$	$ ( - 63322 \nu^{15} + 499374 \nu^{14} - 2967557 \nu^{13} + 11975783 \nu^{12} - 38725947 \nu^{11} + \cdots + 4658625 ) / 17095 $ (-63322v^15 + 499374v^14 - 2967557v^13 + 11975783v^12 - 38725947v^11 + 99182863v^10 - 209187723v^9 + 362452266v^8 - 519102276v^7 + 610524044v^6 - 583075896v^5 + 444109867v^4 - 260715782v^3 + 112213113v^2 - 31745452*v + 4658625) / 17095
$\beta_{11}$	$=$	$ ( 85667 \nu^{15} - 575043 \nu^{14} + 3300730 \nu^{13} - 12042490 \nu^{12} + 36603345 \nu^{11} + \cdots + 1386930 ) / 17095 $ (85667v^15 - 575043v^14 + 3300730v^13 - 12042490v^12 + 36603345v^11 - 84888272v^10 + 163693827v^9 - 250619850v^8 + 313409097v^7 - 306048004v^6 + 230440767v^5 - 121926812v^4 + 38225860v^3 - 549609v^2 - 4411243*v + 1386930) / 17095
$\beta_{12}$	$=$	$ ( 85667 \nu^{15} - 709962 \nu^{14} + 4245163 \nu^{13} - 17516572 \nu^{12} + 57170208 \nu^{11} + \cdots - 6084900 ) / 17095 $ (85667v^15 - 709962v^14 + 4245163v^13 - 17516572v^12 + 57170208v^11 - 148592921v^10 + 316196481v^9 - 553624254v^8 + 798059448v^7 - 943650739v^6 + 901764573v^5 - 683781602v^4 + 396137563v^3 - 165998175v^2 + 44913092*v - 6084900) / 17095
$\beta_{13}$	$=$	$ ( 93453 \nu^{15} - 676833 \nu^{14} + 3953628 \nu^{13} - 15189086 \nu^{12} + 47722461 \nu^{11} + \cdots - 1707325 ) / 17095 $ (93453v^15 - 676833v^14 + 3953628v^13 - 15189086v^12 + 47722461v^11 - 116877858v^10 + 237231160v^9 - 390812883v^8 + 531576918v^7 - 585725841v^6 + 519678225v^5 - 360178374v^4 + 188593489v^3 - 69527715v^2 + 16218096*v - 1707325) / 17095
$\beta_{14}$	$=$	$ ( - 162552 \nu^{15} + 1183635 \nu^{14} - 6913620 \nu^{13} + 26621820 \nu^{12} - 83662434 \nu^{11} + \cdots + 2721635 ) / 17095 $ (-162552v^15 + 1183635v^14 - 6913620v^13 + 26621820v^12 - 83662434v^11 + 205121880v^10 - 416302560v^9 + 685838345v^8 - 931919096v^7 + 1025248280v^6 - 906867386v^5 + 625568415v^4 - 324976348v^3 + 118357235v^2 - 26928904*v + 2721635) / 17095
$\beta_{15}$	$=$	$ ( 237171 \nu^{15} - 1837563 \nu^{14} + 10869000 \nu^{13} - 43377514 \nu^{12} + 139191957 \nu^{11} + \cdots - 10614560 ) / 17095 $ (237171v^15 - 1837563v^14 + 10869000v^13 - 43377514v^12 + 139191957v^11 - 352432410v^10 + 735109127v^9 - 1255047384v^8 + 1767900759v^7 - 2034532848v^6 + 1892170491v^5 - 1390869990v^4 + 780030104v^3 - 315037881v^2 + 82065441*v - 10614560) / 17095

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

$\nu$	$=$	$ ( 2 \beta_{14} + 2 \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + \cdots + 3 ) / 6 $ (2b14 + 2b13 + b12 + b11 - b10 - b9 - 3b8 - 3b7 - b3 - b2 + 3) / 6
$\nu^{2}$	$=$	$ ( 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{9} - 6 \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 12 ) / 6 $ (2b14 + 2b13 + 2b12 - 2b9 - 6b7 + b6 + b5 - b3 + 2b2 - 12) / 6
$\nu^{3}$	$=$	$ ( - 2 \beta_{15} - 10 \beta_{14} - 7 \beta_{13} - 3 \beta_{11} + 9 \beta_{10} + 6 \beta_{9} + 21 \beta_{8} + \cdots - 24 ) / 6 $ (-2b15 - 10b14 - 7b13 - 3b11 + 9b10 + 6b9 + 21b8 + 12b7 + 6b6 - 3b5 + b4 + 5b3 + 8b2 + 9*b1 - 24) / 6
$\nu^{4}$	$=$	$ ( - 2 \beta_{15} - 11 \beta_{14} - 8 \beta_{13} - 4 \beta_{12} + 4 \beta_{10} + 12 \beta_{9} + 9 \beta_{8} + \cdots + 18 ) / 3 $ (-2b15 - 11b14 - 8b13 - 4b12 + 4b10 + 12b9 + 9b8 + 27b7 - 9b5 + 2b4 + b3 - 4b2 + 9b1 + 18) / 3
$\nu^{5}$	$=$	$ ( 12 \beta_{15} + 43 \beta_{14} + 39 \beta_{13} - 17 \beta_{12} + 8 \beta_{11} - 60 \beta_{10} + \cdots + 183 ) / 6 $ (12b15 + 43b14 + 39b13 - 17b12 + 8b11 - 60b10 - 15b9 - 138b8 - 33b7 - 55b6 + 5b5 - b4 - 44b3 - 67b2 - 75b1 + 183) / 6
$\nu^{6}$	$=$	$ ( 46 \beta_{15} + 185 \beta_{14} + 158 \beta_{13} + 8 \beta_{12} - 14 \beta_{11} - 110 \beta_{10} + \cdots - 30 ) / 6 $ (46b15 + 185b14 + 158b13 + 8b12 - 14b11 - 110b10 - 196b9 - 258b8 - 438b7 - 70b6 + 155b5 - 38b4 - 25b3 - 4b2 - 270*b1 - 30) / 6
$\nu^{7}$	$=$	$ ( - 38 \beta_{15} - 128 \beta_{14} - 154 \beta_{13} + 128 \beta_{12} - 40 \beta_{11} + 361 \beta_{10} + \cdots - 1299 ) / 6 $ (-38b15 - 128b14 - 154b13 + 128b12 - 40b11 + 361b10 - 101b9 + 801b8 - 207b7 + 364b6 + 112b5 - 51b4 + 379b3 + 511b2 + 378*b1 - 1299) / 6
$\nu^{8}$	$=$	$ ( - 188 \beta_{15} - 714 \beta_{14} - 696 \beta_{13} + 92 \beta_{12} + 88 \beta_{11} + 584 \beta_{10} + \cdots - 606 ) / 3 $ (-188b15 - 714b14 - 696b13 + 92b12 + 88b11 + 584b10 + 688b9 + 1338b8 + 1560b7 + 486b6 - 564b5 + 120b4 + 252b3 + 272b2 + 1407*b1 - 606) / 3
$\nu^{9}$	$=$	$ ( - 102 \beta_{15} - 379 \beta_{14} - 177 \beta_{13} - 645 \beta_{12} + 438 \beta_{11} - 1713 \beta_{10} + \cdots + 8202 ) / 6 $ (-102b15 - 379b14 - 177b13 - 645b12 + 438b11 - 1713b10 + 2190b9 - 3675b8 + 4830b7 - 1806b6 - 1968b5 + 726b4 - 2740b3 - 3408b2 - 27*b1 + 8202) / 6
$\nu^{10}$	$=$	$ ( 2652 \beta_{15} + 10223 \beta_{14} + 10744 \beta_{13} - 2348 \beta_{12} - 1386 \beta_{11} - 10686 \beta_{10} + \cdots + 17628 ) / 6 $ (2652b15 + 10223b14 + 10744b13 - 2348b12 - 1386b11 - 10686b10 - 8314b9 - 24006b8 - 18930b7 - 9581b6 + 7024b5 - 1008b4 - 6664b3 - 7658b2 - 23220*b1 + 17628) / 6
$\nu^{11}$	$=$	$ ( 3307 \beta_{15} + 12443 \beta_{14} + 12181 \beta_{13} + 1771 \beta_{12} - 5005 \beta_{11} + 3052 \beta_{10} + \cdots - 43491 ) / 6 $ (3307b15 + 12443b14 + 12181b13 + 1771b12 - 5005b11 + 3052b10 - 25273b9 + 5778b8 - 56559b7 + 3993b6 + 22341b5 - 6917b4 + 15806b3 + 18709b2 - 24783*b1 - 43491) / 6
$\nu^{12}$	$=$	$ ( - 8306 \beta_{15} - 32900 \beta_{14} - 35908 \beta_{13} + 10104 \beta_{12} + 3696 \beta_{11} + \cdots - 89109 ) / 3 $ (-8306b15 - 32900b14 - 35908b13 + 10104b12 + 3696b11 + 43092b10 + 19368b9 + 95841b8 + 44046b7 + 39501b6 - 16731b5 + 4b4 + 34369b3 + 39128b2 + 79695*b1 - 89109) / 3
$\nu^{13}$	$=$	$ ( - 40749 \beta_{15} - 158381 \beta_{14} - 167394 \beta_{13} + 10088 \beta_{12} + 48685 \beta_{11} + \cdots + 151791 ) / 6 $ (-40749b15 - 158381b14 - 167394b13 + 10088b12 + 48685b11 + 60771b10 + 234477b9 + 140085b8 + 525717b7 + 50609b6 - 208546b5 + 54025b4 - 57875b3 - 66635b2 + 362934*b1 + 151791) / 6
$\nu^{14}$	$=$	$ ( 86986 \beta_{15} + 352742 \beta_{14} + 392546 \beta_{13} - 144326 \beta_{12} - 13142 \beta_{11} + \cdots + 1521186 ) / 6 $ (86986b15 + 352742b14 + 392546b13 - 144326b12 - 13142b11 - 611624b10 - 65226b9 - 1355046b8 - 149892b7 - 563485b6 + 56225b5 + 58400b4 - 600613b3 - 674730b2 - 881244*b1 + 1521186) / 6
$\nu^{15}$	$=$	$ ( 394827 \beta_{15} + 1567315 \beta_{14} + 1700247 \beta_{13} - 234662 \beta_{12} - 405664 \beta_{11} + \cdots + 356820 ) / 6 $ (394827b15 + 1567315b14 + 1700247b13 - 234662b12 - 405664b11 - 1079521b10 - 1884968b9 - 2418363b8 - 4219254b7 - 968762b6 + 1689964b5 - 358179b4 - 142790b3 - 158280b2 - 3753099*b1 + 356820) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$-1$	$1$	$-1 + \beta_{1}$

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} $

577.1

0.500000	−	2.74530i
0.500000	+	0.331082i
0.500000	+	2.00333i
0.500000	+	0.410882i
0.500000	−	1.33108i
0.500000	+	1.74530i
0.500000	−	1.00333i
0.500000	+	0.589118i
0.500000	+	2.74530i
0.500000	−	0.331082i
0.500000	−	2.00333i
0.500000	−	0.410882i
0.500000	+	1.33108i
0.500000	−	1.74530i
0.500000	+	1.00333i
0.500000	−	0.589118i

−2.98735

−

1.72474i

−2.02166

−

3.50162i

577.2

−2.98735

−

1.72474i

2.02166

3.50162i

577.3

−1.25529

−

0.724745i

−0.642559

−

1.11295i

577.4

−1.25529

−

0.724745i

0.642559

1.11295i

577.5

1.25529

0.724745i

−0.642559

−

1.11295i

577.6

1.25529

0.724745i

0.642559

1.11295i

577.7

2.98735

1.72474i

−2.02166

−

3.50162i

577.8

2.98735

1.72474i

2.02166

3.50162i

2881.1

−2.98735

1.72474i

−2.02166

3.50162i

2881.2

−2.98735

1.72474i

2.02166

−

3.50162i

2881.3

−1.25529

0.724745i

−0.642559

1.11295i

2881.4

−1.25529

0.724745i

0.642559

−

1.11295i

2881.5

1.25529

−

0.724745i

−0.642559

1.11295i

2881.6

1.25529

−

0.724745i

0.642559

−

1.11295i

2881.7

2.98735

−

1.72474i

−2.02166

3.50162i

2881.8

2.98735

−

1.72474i

2.02166

−

3.50162i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
9.c	even	3	1	inner
36.f	odd	6	1	inner
72.n	even	6	1	inner
72.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3456.2.r.e		16
3.b	odd	2	1		1152.2.r.f	✓	16
4.b	odd	2	1	inner	3456.2.r.e		16
8.b	even	2	1	inner	3456.2.r.e		16
8.d	odd	2	1	inner	3456.2.r.e		16
9.c	even	3	1	inner	3456.2.r.e		16
9.d	odd	6	1		1152.2.r.f	✓	16
12.b	even	2	1		1152.2.r.f	✓	16
24.f	even	2	1		1152.2.r.f	✓	16
24.h	odd	2	1		1152.2.r.f	✓	16
36.f	odd	6	1	inner	3456.2.r.e		16
36.h	even	6	1		1152.2.r.f	✓	16
72.j	odd	6	1		1152.2.r.f	✓	16
72.l	even	6	1		1152.2.r.f	✓	16
72.n	even	6	1	inner	3456.2.r.e		16
72.p	odd	6	1	inner	3456.2.r.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.2.r.f	✓	16	3.b	odd	2	1
1152.2.r.f	✓	16	9.d	odd	6	1
1152.2.r.f	✓	16	12.b	even	2	1
1152.2.r.f	✓	16	24.f	even	2	1
1152.2.r.f	✓	16	24.h	odd	2	1
1152.2.r.f	✓	16	36.h	even	6	1
1152.2.r.f	✓	16	72.j	odd	6	1
1152.2.r.f	✓	16	72.l	even	6	1
3456.2.r.e		16	1.a	even	1	1	trivial
3456.2.r.e		16	4.b	odd	2	1	inner
3456.2.r.e		16	8.b	even	2	1	inner
3456.2.r.e		16	8.d	odd	2	1	inner
3456.2.r.e		16	9.c	even	3	1	inner
3456.2.r.e		16	36.f	odd	6	1	inner
3456.2.r.e		16	72.n	even	6	1	inner
3456.2.r.e		16	72.p	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3456, [\chi])$:

$ T_{5}^{8} - 14T_{5}^{6} + 171T_{5}^{4} - 350T_{5}^{2} + 625 $

$ T_{7}^{8} + 18T_{7}^{6} + 297T_{7}^{4} + 486T_{7}^{2} + 729 $

$ T_{11}^{8} - 18T_{11}^{6} + 297T_{11}^{4} - 486T_{11}^{2} + 729 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 14 T^{6} + \cdots + 625)^{2} $ (T^8 - 14T^6 + 171T^4 - 350*T^2 + 625)^2
$7$	$ (T^{8} + 18 T^{6} + \cdots + 729)^{2} $ (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$11$	$ (T^{8} - 18 T^{6} + \cdots + 729)^{2} $ (T^8 - 18T^6 + 297T^4 - 486*T^2 + 729)^2
$13$	$ (T^{4} - T^{2} + 1)^{4} $ (T^4 - T^2 + 1)^4
$17$	$ (T^{2} + 8 T + 10)^{8} $ (T^2 + 8*T + 10)^8
$19$	$ T^{16} $ T^16
$23$	$ (T^{8} + 54 T^{6} + \cdots + 59049)^{2} $ (T^8 + 54T^6 + 2673T^4 + 13122*T^2 + 59049)^2
$29$	$ (T^{8} - 50 T^{6} + \cdots + 279841)^{2} $ (T^8 - 50T^6 + 1971T^4 - 26450*T^2 + 279841)^2
$31$	$ (T^{8} + 54 T^{6} + \cdots + 455625)^{2} $ (T^8 + 54T^6 + 2241T^4 + 36450*T^2 + 455625)^2
$37$	$ (T^{4} + 84 T^{2} + 900)^{4} $ (T^4 + 84*T^2 + 900)^4
$41$	$ (T^{4} - 2 T^{3} + \cdots + 2809)^{4} $ (T^4 - 2T^3 + 57T^2 + 106*T + 2809)^4
$43$	$ (T^{8} - 198 T^{6} + \cdots + 95004009)^{2} $ (T^8 - 198T^6 + 29457T^4 - 1929906*T^2 + 95004009)^2
$47$	$ (T^{8} + 18 T^{6} + \cdots + 729)^{2} $ (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$53$	$ (T^{4} + 140 T^{2} + 1444)^{4} $ (T^4 + 140*T^2 + 1444)^4
$59$	$ (T^{8} - 54 T^{6} + \cdots + 59049)^{2} $ (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$61$	$ (T^{8} - 66 T^{6} + \cdots + 50625)^{2} $ (T^8 - 66T^6 + 4131T^4 - 14850*T^2 + 50625)^2
$67$	$ (T^{8} - 306 T^{6} + \cdots + 204004089)^{2} $ (T^8 - 306T^6 + 79353T^4 - 4370598*T^2 + 204004089)^2
$71$	$ (T^{4} - 396 T^{2} + 38988)^{4} $ (T^4 - 396*T^2 + 38988)^4
$73$	$ (T^{2} - 12 T + 30)^{8} $ (T^2 - 12*T + 30)^8
$79$	$ (T^{8} + 342 T^{6} + \cdots + 515607849)^{2} $ (T^8 + 342T^6 + 94257T^4 + 7765794*T^2 + 515607849)^2
$83$	$ (T^{8} - 54 T^{6} + \cdots + 59049)^{2} $ (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$89$	$ (T^{2} + 4 T - 2)^{8} $ (T^2 + 4*T - 2)^8
$97$	$ (T^{4} + 18 T^{3} + \cdots + 729)^{4} $ (T^4 + 18T^3 + 297T^2 + 486*T + 729)^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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{7} + \beta_{5}) q^{5} + \beta_{13} q^{7}+O(q^{10}) \) q + (-b7 + b5) * q^5 + b13 * q^7	\( q + ( - \beta_{7} + \beta_{5}) q^{5} + \beta_{13} q^{7} + \beta_{10} q^{11} - \beta_{7} q^{13} + (\beta_{3} - 4) q^{17} + (\beta_{15} - \beta_{13} + \beta_{2}) q^{23} + ( - 2 \beta_{14} + 2 \beta_1) q^{25} + (\beta_{8} + 2 \beta_{6}) q^{29} + (\beta_{15} + \beta_{13} - \beta_{2}) q^{31} + (\beta_{12} + \beta_{11} + \cdots + 3 \beta_{9}) q^{35}+ \cdots + (3 \beta_{14} - 9 \beta_1) q^{97}+O(q^{100}) \) q + (-b7 + b5) * q^5 + b13 * q^7 + b10 * q^11 - b7 * q^13 + (b3 - 4) * q^17 + (b15 - b13 + b2) * q^23 + (-2b14 + 2b1) * q^25 + (b8 + 2b6) q^29 + (b15 + b13 - b2) * q^31 + (b12 + b11 + 3b10 + 3b9) * q^35 + (-6b8 - 6b7 + b6 - b5) * q^37 + (3b14 - 3b3 - b1 + 1) * q^41 + (-b12 + 3b10) q^43 + b13 * q^47 + (-3b14 + 3b3 + 2b1 - 2) q^49 + (4b8 + 4b7 - 3b6 + 3b5) * q^53 + (b4 - 3b2) q^55 + (b11 - b9) * q^59 + (3b8 - 2b6) * q^61 + (-b14 + b1) * q^65 + (2b11 + 3b9) * q^67 + (3b4 - 2b2) * q^71 + (b3 + 6) * q^73 + (9b7 - 3b5) * q^77 + (3b15 + b13 - 3b4) * q^79 + (b12 - b10) * q^83 + (10b7 - 5b5) * q^85 + (b3 - 2) * q^89 + (b10 + b9) * q^91 + (3b14 - 9b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 64 q^{17} + 16 q^{25} + 8 q^{41} - 16 q^{49} + 8 q^{65} + 96 q^{73} - 32 q^{89} - 72 q^{97}+O(q^{100}) \) 16 * q - 64 * q^17 + 16 * q^25 + 8 * q^41 - 16 * q^49 + 8 * q^65 + 96 * q^73 - 32 * q^89 - 72 * q^97

Introduction

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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	3456.2.r.e

Level	$3456$
Weight	$2$
Character orbit	3456.r
Analytic conductor	$27.596$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(27.5962989386\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	16.0.9349208943630483456.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) x^16 - 8x^15 + 48x^14 - 196x^13 + 642x^12 - 1668x^11 + 3580x^10 - 6328x^9 + 9297x^8 - 11276x^7 + 11224x^6 - 9024x^5 + 5736x^4 - 2780x^3 + 972x^2 - 220*x + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{8} \)
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} + \cdots - 125460 ) / 17095 \) (3456v^15 - 25920v^14 + 151876v^13 - 594074v^12 + 1879372v^11 - 4666596v^10 + 9554736v^9 - 15945783v^8 + 21928484v^7 - 24527176v^6 + 22138664v^5 - 15739188v^4 + 8598668v^3 - 3418428v^2 + 929924*v - 125460) / 17095
\(\beta_{2}\)	\(=\)	\( ( 183 \nu^{14} - 1281 \nu^{13} + 7408 \nu^{12} - 27795 \nu^{11} + 85851 \nu^{10} - 204998 \nu^{9} + \cdots + 10130 ) / 65 \) (183v^14 - 1281v^13 + 7408v^12 - 27795v^11 + 85851v^10 - 204998v^9 + 405777v^8 - 646527v^7 + 846324v^6 - 886329v^5 + 737358v^4 - 466904v^3 + 215139v^2 - 64206v + 10130) / 65
\(\beta_{3}\)	\(=\)	\( ( - 54 \nu^{14} + 378 \nu^{13} - 2193 \nu^{12} + 8244 \nu^{11} - 25569 \nu^{10} + 61284 \nu^{9} + \cdots - 3308 ) / 13 \) (-54v^14 + 378v^13 - 2193v^12 + 8244v^11 - 25569v^10 + 61284v^9 - 122021v^8 + 195620v^7 - 258293v^6 + 273134v^5 - 230109v^4 + 147802v^3 - 69203v^2 + 20980v - 3308) / 13
\(\beta_{4}\)	\(=\)	\( ( - 447 \nu^{14} + 3129 \nu^{13} - 18112 \nu^{12} + 67995 \nu^{11} - 210309 \nu^{10} + 502832 \nu^{9} + \cdots - 25820 ) / 65 \) (-447v^14 + 3129v^13 - 18112v^12 + 67995v^11 - 210309v^10 + 502832v^9 - 997608v^8 + 1593543v^7 - 2094381v^6 + 2203926v^5 - 1846302v^4 + 1179041v^3 - 548766v^2 + 165459v - 25820) / 65
\(\beta_{5}\)	\(=\)	\( ( 22230 \nu^{15} - 144633 \nu^{14} + 820486 \nu^{13} - 2915351 \nu^{12} + 8665028 \nu^{11} + \cdots + 599315 ) / 17095 \) (22230v^15 - 144633v^14 + 820486v^13 - 2915351v^12 + 8665028v^11 - 19406977v^10 + 35943838v^9 - 51803400v^8 + 59433982v^7 - 50057927v^6 + 27990758v^5 - 4701824v^4 - 7461590v^3 + 7646662v^2 - 3312642*v + 599315) / 17095
\(\beta_{6}\)	\(=\)	\( ( - 22230 \nu^{15} + 188817 \nu^{14} - 1129774 \nu^{13} + 4704014 \nu^{12} - 15376262 \nu^{11} + \cdots + 1317955 ) / 17095 \) (-22230v^15 + 188817v^14 - 1129774v^13 + 4704014v^12 - 15376262v^11 + 40133218v^10 - 85426762v^9 + 149690685v^8 - 215272528v^7 + 253642808v^6 - 240583652v^5 + 180367151v^4 - 102603910v^3 + 41706077v^2 - 10736292*v + 1317955) / 17095
\(\beta_{7}\)	\(=\)	\( ( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} + \cdots + 218885 ) / 17095 \) (26630v^15 - 182630v^14 + 1056740v^13 - 3923413v^12 + 12097498v^11 - 28666706v^10 + 56590650v^9 - 89528958v^8 + 116711850v^7 - 121073088v^6 + 99711814v^5 - 61611091v^4 + 27082796v^3 - 6949514v^2 + 510382*v + 218885) / 17095
\(\beta_{8}\)	\(=\)	\( ( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} + \cdots - 2071845 ) / 17095 \) (26630v^15 - 216820v^14 + 1296070v^13 - 5311527v^12 + 17314892v^11 - 44845414v^10 + 95362110v^9 - 166733397v^8 + 240513840v^7 - 284726942v^6 + 273082466v^5 - 208344314v^4 + 122001074v^3 - 52073476v^2 + 14507768*v - 2071845) / 17095
\(\beta_{9}\)	\(=\)	\( ( - 63322 \nu^{15} + 450456 \nu^{14} - 2625131 \nu^{13} + 9970934 \nu^{12} - 31148391 \nu^{11} + \cdots - 31980 ) / 17095 \) (-63322v^15 + 450456v^14 - 2625131v^13 + 9970934v^12 - 31148391v^11 + 75404770v^10 - 151597035v^9 + 245879883v^8 - 328994304v^7 + 353294264v^6 - 303546609v^5 + 199496197v^4 - 96502001v^3 + 30387501v^2 - 5033857*v - 31980) / 17095
\(\beta_{10}\)	\(=\)	\( ( - 63322 \nu^{15} + 499374 \nu^{14} - 2967557 \nu^{13} + 11975783 \nu^{12} - 38725947 \nu^{11} + \cdots + 4658625 ) / 17095 \) (-63322v^15 + 499374v^14 - 2967557v^13 + 11975783v^12 - 38725947v^11 + 99182863v^10 - 209187723v^9 + 362452266v^8 - 519102276v^7 + 610524044v^6 - 583075896v^5 + 444109867v^4 - 260715782v^3 + 112213113v^2 - 31745452*v + 4658625) / 17095
\(\beta_{11}\)	\(=\)	\( ( 85667 \nu^{15} - 575043 \nu^{14} + 3300730 \nu^{13} - 12042490 \nu^{12} + 36603345 \nu^{11} + \cdots + 1386930 ) / 17095 \) (85667v^15 - 575043v^14 + 3300730v^13 - 12042490v^12 + 36603345v^11 - 84888272v^10 + 163693827v^9 - 250619850v^8 + 313409097v^7 - 306048004v^6 + 230440767v^5 - 121926812v^4 + 38225860v^3 - 549609v^2 - 4411243*v + 1386930) / 17095
\(\beta_{12}\)	\(=\)	\( ( 85667 \nu^{15} - 709962 \nu^{14} + 4245163 \nu^{13} - 17516572 \nu^{12} + 57170208 \nu^{11} + \cdots - 6084900 ) / 17095 \) (85667v^15 - 709962v^14 + 4245163v^13 - 17516572v^12 + 57170208v^11 - 148592921v^10 + 316196481v^9 - 553624254v^8 + 798059448v^7 - 943650739v^6 + 901764573v^5 - 683781602v^4 + 396137563v^3 - 165998175v^2 + 44913092*v - 6084900) / 17095
\(\beta_{13}\)	\(=\)	\( ( 93453 \nu^{15} - 676833 \nu^{14} + 3953628 \nu^{13} - 15189086 \nu^{12} + 47722461 \nu^{11} + \cdots - 1707325 ) / 17095 \) (93453v^15 - 676833v^14 + 3953628v^13 - 15189086v^12 + 47722461v^11 - 116877858v^10 + 237231160v^9 - 390812883v^8 + 531576918v^7 - 585725841v^6 + 519678225v^5 - 360178374v^4 + 188593489v^3 - 69527715v^2 + 16218096*v - 1707325) / 17095
\(\beta_{14}\)	\(=\)	\( ( - 162552 \nu^{15} + 1183635 \nu^{14} - 6913620 \nu^{13} + 26621820 \nu^{12} - 83662434 \nu^{11} + \cdots + 2721635 ) / 17095 \) (-162552v^15 + 1183635v^14 - 6913620v^13 + 26621820v^12 - 83662434v^11 + 205121880v^10 - 416302560v^9 + 685838345v^8 - 931919096v^7 + 1025248280v^6 - 906867386v^5 + 625568415v^4 - 324976348v^3 + 118357235v^2 - 26928904*v + 2721635) / 17095
\(\beta_{15}\)	\(=\)	\( ( 237171 \nu^{15} - 1837563 \nu^{14} + 10869000 \nu^{13} - 43377514 \nu^{12} + 139191957 \nu^{11} + \cdots - 10614560 ) / 17095 \) (237171v^15 - 1837563v^14 + 10869000v^13 - 43377514v^12 + 139191957v^11 - 352432410v^10 + 735109127v^9 - 1255047384v^8 + 1767900759v^7 - 2034532848v^6 + 1892170491v^5 - 1390869990v^4 + 780030104v^3 - 315037881v^2 + 82065441*v - 10614560) / 17095

\(\nu\)	\(=\)	\( ( 2 \beta_{14} + 2 \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + \cdots + 3 ) / 6 \) (2b14 + 2b13 + b12 + b11 - b10 - b9 - 3b8 - 3b7 - b3 - b2 + 3) / 6
\(\nu^{2}\)	\(=\)	\( ( 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{9} - 6 \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 12 ) / 6 \) (2b14 + 2b13 + 2b12 - 2b9 - 6b7 + b6 + b5 - b3 + 2b2 - 12) / 6
\(\nu^{3}\)	\(=\)	\( ( - 2 \beta_{15} - 10 \beta_{14} - 7 \beta_{13} - 3 \beta_{11} + 9 \beta_{10} + 6 \beta_{9} + 21 \beta_{8} + \cdots - 24 ) / 6 \) (-2b15 - 10b14 - 7b13 - 3b11 + 9b10 + 6b9 + 21b8 + 12b7 + 6b6 - 3b5 + b4 + 5b3 + 8b2 + 9*b1 - 24) / 6
\(\nu^{4}\)	\(=\)	\( ( - 2 \beta_{15} - 11 \beta_{14} - 8 \beta_{13} - 4 \beta_{12} + 4 \beta_{10} + 12 \beta_{9} + 9 \beta_{8} + \cdots + 18 ) / 3 \) (-2b15 - 11b14 - 8b13 - 4b12 + 4b10 + 12b9 + 9b8 + 27b7 - 9b5 + 2b4 + b3 - 4b2 + 9b1 + 18) / 3
\(\nu^{5}\)	\(=\)	\( ( 12 \beta_{15} + 43 \beta_{14} + 39 \beta_{13} - 17 \beta_{12} + 8 \beta_{11} - 60 \beta_{10} + \cdots + 183 ) / 6 \) (12b15 + 43b14 + 39b13 - 17b12 + 8b11 - 60b10 - 15b9 - 138b8 - 33b7 - 55b6 + 5b5 - b4 - 44b3 - 67b2 - 75b1 + 183) / 6
\(\nu^{6}\)	\(=\)	\( ( 46 \beta_{15} + 185 \beta_{14} + 158 \beta_{13} + 8 \beta_{12} - 14 \beta_{11} - 110 \beta_{10} + \cdots - 30 ) / 6 \) (46b15 + 185b14 + 158b13 + 8b12 - 14b11 - 110b10 - 196b9 - 258b8 - 438b7 - 70b6 + 155b5 - 38b4 - 25b3 - 4b2 - 270*b1 - 30) / 6
\(\nu^{7}\)	\(=\)	\( ( - 38 \beta_{15} - 128 \beta_{14} - 154 \beta_{13} + 128 \beta_{12} - 40 \beta_{11} + 361 \beta_{10} + \cdots - 1299 ) / 6 \) (-38b15 - 128b14 - 154b13 + 128b12 - 40b11 + 361b10 - 101b9 + 801b8 - 207b7 + 364b6 + 112b5 - 51b4 + 379b3 + 511b2 + 378*b1 - 1299) / 6
\(\nu^{8}\)	\(=\)	\( ( - 188 \beta_{15} - 714 \beta_{14} - 696 \beta_{13} + 92 \beta_{12} + 88 \beta_{11} + 584 \beta_{10} + \cdots - 606 ) / 3 \) (-188b15 - 714b14 - 696b13 + 92b12 + 88b11 + 584b10 + 688b9 + 1338b8 + 1560b7 + 486b6 - 564b5 + 120b4 + 252b3 + 272b2 + 1407*b1 - 606) / 3
\(\nu^{9}\)	\(=\)	\( ( - 102 \beta_{15} - 379 \beta_{14} - 177 \beta_{13} - 645 \beta_{12} + 438 \beta_{11} - 1713 \beta_{10} + \cdots + 8202 ) / 6 \) (-102b15 - 379b14 - 177b13 - 645b12 + 438b11 - 1713b10 + 2190b9 - 3675b8 + 4830b7 - 1806b6 - 1968b5 + 726b4 - 2740b3 - 3408b2 - 27*b1 + 8202) / 6
\(\nu^{10}\)	\(=\)	\( ( 2652 \beta_{15} + 10223 \beta_{14} + 10744 \beta_{13} - 2348 \beta_{12} - 1386 \beta_{11} - 10686 \beta_{10} + \cdots + 17628 ) / 6 \) (2652b15 + 10223b14 + 10744b13 - 2348b12 - 1386b11 - 10686b10 - 8314b9 - 24006b8 - 18930b7 - 9581b6 + 7024b5 - 1008b4 - 6664b3 - 7658b2 - 23220*b1 + 17628) / 6
\(\nu^{11}\)	\(=\)	\( ( 3307 \beta_{15} + 12443 \beta_{14} + 12181 \beta_{13} + 1771 \beta_{12} - 5005 \beta_{11} + 3052 \beta_{10} + \cdots - 43491 ) / 6 \) (3307b15 + 12443b14 + 12181b13 + 1771b12 - 5005b11 + 3052b10 - 25273b9 + 5778b8 - 56559b7 + 3993b6 + 22341b5 - 6917b4 + 15806b3 + 18709b2 - 24783*b1 - 43491) / 6
\(\nu^{12}\)	\(=\)	\( ( - 8306 \beta_{15} - 32900 \beta_{14} - 35908 \beta_{13} + 10104 \beta_{12} + 3696 \beta_{11} + \cdots - 89109 ) / 3 \) (-8306b15 - 32900b14 - 35908b13 + 10104b12 + 3696b11 + 43092b10 + 19368b9 + 95841b8 + 44046b7 + 39501b6 - 16731b5 + 4b4 + 34369b3 + 39128b2 + 79695*b1 - 89109) / 3
\(\nu^{13}\)	\(=\)	\( ( - 40749 \beta_{15} - 158381 \beta_{14} - 167394 \beta_{13} + 10088 \beta_{12} + 48685 \beta_{11} + \cdots + 151791 ) / 6 \) (-40749b15 - 158381b14 - 167394b13 + 10088b12 + 48685b11 + 60771b10 + 234477b9 + 140085b8 + 525717b7 + 50609b6 - 208546b5 + 54025b4 - 57875b3 - 66635b2 + 362934*b1 + 151791) / 6
\(\nu^{14}\)	\(=\)	\( ( 86986 \beta_{15} + 352742 \beta_{14} + 392546 \beta_{13} - 144326 \beta_{12} - 13142 \beta_{11} + \cdots + 1521186 ) / 6 \) (86986b15 + 352742b14 + 392546b13 - 144326b12 - 13142b11 - 611624b10 - 65226b9 - 1355046b8 - 149892b7 - 563485b6 + 56225b5 + 58400b4 - 600613b3 - 674730b2 - 881244*b1 + 1521186) / 6
\(\nu^{15}\)	\(=\)	\( ( 394827 \beta_{15} + 1567315 \beta_{14} + 1700247 \beta_{13} - 234662 \beta_{12} - 405664 \beta_{11} + \cdots + 356820 ) / 6 \) (394827b15 + 1567315b14 + 1700247b13 - 234662b12 - 405664b11 - 1079521b10 - 1884968b9 - 2418363b8 - 4219254b7 - 968762b6 + 1689964b5 - 358179b4 - 142790b3 - 158280b2 - 3753099*b1 + 356820) / 6

\(n\)	\(2053\)	\(2431\)	\(2945\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1 + \beta_{1}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 577.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 14 T^{6} + \cdots + 625)^{2} \) (T^8 - 14T^6 + 171T^4 - 350*T^2 + 625)^2
$7$	\( (T^{8} + 18 T^{6} + \cdots + 729)^{2} \) (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$11$	\( (T^{8} - 18 T^{6} + \cdots + 729)^{2} \) (T^8 - 18T^6 + 297T^4 - 486*T^2 + 729)^2
$13$	\( (T^{4} - T^{2} + 1)^{4} \) (T^4 - T^2 + 1)^4
$17$	\( (T^{2} + 8 T + 10)^{8} \) (T^2 + 8*T + 10)^8
$19$	\( T^{16} \) T^16
$23$	\( (T^{8} + 54 T^{6} + \cdots + 59049)^{2} \) (T^8 + 54T^6 + 2673T^4 + 13122*T^2 + 59049)^2
$29$	\( (T^{8} - 50 T^{6} + \cdots + 279841)^{2} \) (T^8 - 50T^6 + 1971T^4 - 26450*T^2 + 279841)^2
$31$	\( (T^{8} + 54 T^{6} + \cdots + 455625)^{2} \) (T^8 + 54T^6 + 2241T^4 + 36450*T^2 + 455625)^2
$37$	\( (T^{4} + 84 T^{2} + 900)^{4} \) (T^4 + 84*T^2 + 900)^4
$41$	\( (T^{4} - 2 T^{3} + \cdots + 2809)^{4} \) (T^4 - 2T^3 + 57T^2 + 106*T + 2809)^4
$43$	\( (T^{8} - 198 T^{6} + \cdots + 95004009)^{2} \) (T^8 - 198T^6 + 29457T^4 - 1929906*T^2 + 95004009)^2
$47$	\( (T^{8} + 18 T^{6} + \cdots + 729)^{2} \) (T^8 + 18T^6 + 297T^4 + 486*T^2 + 729)^2
$53$	\( (T^{4} + 140 T^{2} + 1444)^{4} \) (T^4 + 140*T^2 + 1444)^4
$59$	\( (T^{8} - 54 T^{6} + \cdots + 59049)^{2} \) (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$61$	\( (T^{8} - 66 T^{6} + \cdots + 50625)^{2} \) (T^8 - 66T^6 + 4131T^4 - 14850*T^2 + 50625)^2
$67$	\( (T^{8} - 306 T^{6} + \cdots + 204004089)^{2} \) (T^8 - 306T^6 + 79353T^4 - 4370598*T^2 + 204004089)^2
$71$	\( (T^{4} - 396 T^{2} + 38988)^{4} \) (T^4 - 396*T^2 + 38988)^4
$73$	\( (T^{2} - 12 T + 30)^{8} \) (T^2 - 12*T + 30)^8
$79$	\( (T^{8} + 342 T^{6} + \cdots + 515607849)^{2} \) (T^8 + 342T^6 + 94257T^4 + 7765794*T^2 + 515607849)^2
$83$	\( (T^{8} - 54 T^{6} + \cdots + 59049)^{2} \) (T^8 - 54T^6 + 2673T^4 - 13122*T^2 + 59049)^2
$89$	\( (T^{2} + 4 T - 2)^{8} \) (T^2 + 4*T - 2)^8
$97$	\( (T^{4} + 18 T^{3} + \cdots + 729)^{4} \) (T^4 + 18T^3 + 297T^2 + 486*T + 729)^4
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