LMFDB - Newform orbit 6240.2.w.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(3121,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6240.3121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.w (of order $2$, degree $1$, 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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 5 x^{16} + 10 x^{15} - 12 x^{14} + 16 x^{13} - 2 x^{12} - 40 x^{11} + \cdots + 1024 $ x^20 - 2x^19 + 2x^18 - 5x^16 + 10x^15 - 12x^14 + 16x^13 - 2x^12 - 40x^11 + 72x^10 - 80x^9 - 8x^8 + 128x^7 - 192x^6 + 320x^5 - 320x^4 + 512x^2 - 1024*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{23} $
Twist minimal:	no (minimal twist has level 1560)
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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{2} q^{3} - \beta_{2} q^{5} + (\beta_{10} + 1) q^{7} - q^{9}+O(q^{10}) $ q - b2 * q^3 - b2 * q^5 + (b10 + 1) * q^7 - q^9	$ q - \beta_{2} q^{3} - \beta_{2} q^{5} + (\beta_{10} + 1) q^{7} - q^{9} + (\beta_{13} - \beta_{2}) q^{11} - \beta_{2} q^{13} - q^{15} + (\beta_{17} - \beta_{10} + 1) q^{17} + ( - \beta_{11} + \beta_{2}) q^{19} + (\beta_{15} - \beta_{2}) q^{21} + (\beta_{17} - \beta_{12} - \beta_{10} + \cdots - 2) q^{23}+ \cdots + ( - \beta_{13} + \beta_{2}) q^{99}+O(q^{100}) $ q - b2 * q^3 - b2 * q^5 + (b10 + 1) * q^7 - q^9 + (b13 - b2) * q^11 - b2 * q^13 - q^15 + (b17 - b10 + 1) * q^17 + (-b11 + b2) * q^19 + (b15 - b2) * q^21 + (b17 - b12 - b10 + b9 - 2) * q^23 - q^25 + b2 * q^27 + (-b8 - b1) * q^29 + (b18 + b17 + b9 + b7 + 1) * q^31 + (b12 - 1) * q^33 + (b15 - b2) * q^35 + (b16 - b15 + b11 - b3 + b2) * q^37 - q^39 + (b12 - b7 + b4 + 1) * q^41 + (-b16 + b2) * q^43 + b2 * q^45 + (b17 + b10 - b9 + b7 + 2b6) q^47 + (-b19 - b18 - b17 + b10 - b9 + 1) * q^49 + (-b15 - b5 - b2) * q^51 + (b15 + b14 - b13 - b8 + b3 + 3b2) q^53 + (b12 - 1) * q^55 + (-b7 + 1) * q^57 + (b16 - b14 + b11 - 2b3 - b2) q^59 + (b14 - b13 - b8 - b5 + b3 - b1) * q^61 + (-b10 - 1) * q^63 - q^65 + (-b14 - b8 - b3 - b2 + b1) * q^67 + (-b15 + b13 + b8 - b5 + 2b2) q^69 + (b18 - b10 - b9 - b7 + b4 + 1) * q^71 + (b19 + b18 + b9 - b6 + 3) * q^73 + b2 * q^75 + (-b15 + b14 + b13 + b8 - 2b5 - b3 - 3b2) * q^77 + (b19 + b18 + 3b12 + b10 - b6 + 1) q^79 + q^81 + (b15 - 2b8 + b5 + b3 + 3b2 + b1) * q^83 + (-b15 - b5 - b2) * q^85 + (b18 + b9) * q^87 + (-b19 + b17 - b12 + 2b7 + b6 - 2b4 + 2) * q^89 + (b15 - b2) * q^91 + (-b11 + b8 - b5 - b2 + b1) * q^93 + (-b7 + 1) * q^95 + (-2b18 - b12 - b10 - b9 + 2b6 - 4) * q^97 + (-b13 + b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7} - 20 q^{9}+O(q^{10}) $ 20 * q + 16 * q^7 - 20 * q^9	$ 20 q + 16 q^{7} - 20 q^{9} - 20 q^{15} + 16 q^{17} - 48 q^{23} - 20 q^{25} + 8 q^{31} - 16 q^{33} - 20 q^{39} + 16 q^{41} + 8 q^{47} + 36 q^{49} - 16 q^{55} + 16 q^{57} - 16 q^{63} - 20 q^{65} + 8 q^{71} + 40 q^{73} + 8 q^{79} + 20 q^{81} - 8 q^{87} + 56 q^{89} + 16 q^{95} - 48 q^{97}+O(q^{100}) $ 20 * q + 16 * q^7 - 20 * q^9 - 20 * q^15 + 16 * q^17 - 48 * q^23 - 20 * q^25 + 8 * q^31 - 16 * q^33 - 20 * q^39 + 16 * q^41 + 8 * q^47 + 36 * q^49 - 16 * q^55 + 16 * q^57 - 16 * q^63 - 20 * q^65 + 8 * q^71 + 40 * q^73 + 8 * q^79 + 20 * q^81 - 8 * q^87 + 56 * q^89 + 16 * q^95 - 48 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 2 x^{19} + 2 x^{18} - 5 x^{16} + 10 x^{15} - 12 x^{14} + 16 x^{13} - 2 x^{12} - 40 x^{11} + \cdots + 1024 $ x^20 - 2*x^19 + 2*x^18 - 5*x^16 + 10*x^15 - 12*x^14 + 16*x^13 - 2*x^12 - 40*x^11 + 72*x^10 - 80*x^9 - 8*x^8 + 128*x^7 - 192*x^6 + 320*x^5 - 320*x^4 + 512*x^2 - 1024*x + 1024 :

$\beta_{1}$	$=$	$ ( \nu^{19} - 10 \nu^{18} - 2 \nu^{17} - 8 \nu^{16} + 3 \nu^{15} + 18 \nu^{14} - 24 \nu^{13} + \cdots - 1024 ) / 1024 $ (v^19 - 10v^18 - 2v^17 - 8v^16 + 3v^15 + 18v^14 - 24v^13 + 72v^12 - 2v^11 - 120v^10 + 112v^9 - 48v^8 + 376v^7 + 128v^6 - 416v^5 + 1344v^4 - 1728v^3 - 2304v^2 + 256v - 1024) / 1024
$\beta_{2}$	$=$	$ ( 7 \nu^{19} - 4 \nu^{18} + 6 \nu^{17} + 36 \nu^{16} + 21 \nu^{15} - 28 \nu^{14} - 76 \nu^{13} + \cdots + 7680 ) / 4096 $ (7v^19 - 4v^18 + 6v^17 + 36v^16 + 21v^15 - 28v^14 - 76v^13 - 24v^12 - 30v^11 - 204v^10 + 144v^9 + 368v^8 - 280v^7 - 656v^6 - 544v^5 - 704v^3 - 128v^2 + 5888v + 7680) / 4096
$\beta_{3}$	$=$	$ ( 5 \nu^{19} - 36 \nu^{18} + 2 \nu^{17} - 68 \nu^{16} - 49 \nu^{15} + 100 \nu^{14} + 12 \nu^{13} + \cdots - 19968 ) / 2048 $ (5v^19 - 36v^18 + 2v^17 - 68v^16 - 49v^15 + 100v^14 + 12v^13 + 408v^12 - 58v^11 - 180v^10 + 560v^9 - 1264v^8 + 696v^7 + 1680v^6 + 672v^5 + 4352v^4 - 4672v^3 - 896v^2 - 6912*v - 19968) / 2048
$\beta_{4}$	$=$	$ ( \nu^{19} - 2 \nu^{18} + 2 \nu^{17} + 4 \nu^{16} - 9 \nu^{15} + 10 \nu^{14} - 12 \nu^{13} + 12 \nu^{12} + \cdots - 1536 ) / 256 $ (v^19 - 2v^18 + 2v^17 + 4v^16 - 9v^15 + 10v^14 - 12v^13 + 12v^12 + 18v^11 - 80v^10 + 128v^9 - 40v^8 - 160v^7 + 160v^6 - 240v^5 + 224v^4 - 128v^3 - 640v^2 + 1664v - 1536) / 256
$\beta_{5}$	$=$	$ ( \nu^{19} - 2 \nu^{18} + 2 \nu^{17} - 5 \nu^{15} + 10 \nu^{14} - 12 \nu^{13} + 16 \nu^{12} + \cdots - 1024 ) / 256 $ (v^19 - 2v^18 + 2v^17 - 5v^15 + 10v^14 - 12v^13 + 16v^12 - 2v^11 - 40v^10 + 72v^9 - 80v^8 - 8v^7 + 128v^6 - 192v^5 + 320v^4 - 320v^3 + 1024v - 1024) / 256
$\beta_{6}$	$=$	$ ( - \nu^{19} + 2 \nu^{18} - 2 \nu^{17} + 5 \nu^{15} - 10 \nu^{14} + 12 \nu^{13} - 16 \nu^{12} + \cdots + 1024 ) / 256 $ (-v^19 + 2v^18 - 2v^17 + 5v^15 - 10v^14 + 12v^13 - 16v^12 + 2v^11 + 40v^10 - 72v^9 + 80v^8 + 8v^7 - 128v^6 + 192v^5 - 320v^4 + 320*v^3 + 1024) / 256
$\beta_{7}$	$=$	$ ( - 9 \nu^{19} - 20 \nu^{18} + 38 \nu^{17} + 4 \nu^{16} - 59 \nu^{15} + 52 \nu^{14} - 76 \nu^{13} + \cdots - 12800 ) / 2048 $ (-9v^19 - 20v^18 + 38v^17 + 4v^16 - 59v^15 + 52v^14 - 76v^13 + 104v^12 - 158v^11 - 236v^10 + 1040v^9 - 784v^8 - 88v^7 + 1904v^6 - 1184v^5 + 768v^4 - 2240v^3 + 896v^2 + 6912*v - 12800) / 2048
$\beta_{8}$	$=$	$ ( 7 \nu^{19} - 52 \nu^{18} + 6 \nu^{17} - 28 \nu^{16} - 107 \nu^{15} + 148 \nu^{14} - 76 \nu^{13} + \cdots - 29184 ) / 2048 $ (7v^19 - 52v^18 + 6v^17 - 28v^16 - 107v^15 + 148v^14 - 76v^13 + 392v^12 + 34v^11 - 492v^10 + 1232v^9 - 1488v^8 - 536v^7 + 3056v^6 - 288v^5 + 5376v^4 - 5824v^3 - 2176v^2 + 1792*v - 29184) / 2048
$\beta_{9}$	$=$	$ ( 3 \nu^{19} - 28 \nu^{18} + 46 \nu^{17} + 20 \nu^{16} - 87 \nu^{15} + 60 \nu^{14} - 140 \nu^{13} + \cdots - 16896 ) / 1024 $ (3v^19 - 28v^18 + 46v^17 + 20v^16 - 87v^15 + 60v^14 - 140v^13 + 184v^12 - 118v^11 - 364v^10 + 1424v^9 - 752v^8 - 888v^7 + 1136v^6 - 2208v^5 + 2688v^4 - 4032v^3 + 3968v^2 + 13056*v - 16896) / 1024
$\beta_{10}$	$=$	$ ( - 11 \nu^{19} - 4 \nu^{18} + 2 \nu^{17} - 4 \nu^{16} + 63 \nu^{15} + 4 \nu^{14} + 44 \nu^{13} + \cdots + 4608 ) / 2048 $ (-11v^19 - 4v^18 + 2v^17 - 4v^16 + 63v^15 + 4v^14 + 44v^13 - 40v^12 - 250v^11 + 140v^10 - 80v^9 + 400v^8 + 504v^7 - 368v^6 + 288v^5 - 1792v^4 - 2112v^3 + 128v^2 - 1792*v + 4608) / 2048
$\beta_{11}$	$=$	$ ( - 21 \nu^{19} - 12 \nu^{18} - 66 \nu^{17} - 124 \nu^{16} - 63 \nu^{15} + 12 \nu^{14} + 212 \nu^{13} + \cdots - 37376 ) / 4096 $ (-21v^19 - 12v^18 - 66v^17 - 124v^16 - 63v^15 + 12v^14 + 212v^13 + 456v^12 - 102v^11 + 404v^10 - 304v^9 - 1744v^8 + 968v^7 + 1648v^6 + 3424v^5 + 4864v^4 - 1472v^3 - 5248v^2 - 21760*v - 37376) / 4096
$\beta_{12}$	$=$	$ ( 11 \nu^{19} - 20 \nu^{18} + 14 \nu^{17} + 20 \nu^{16} - 95 \nu^{15} + 52 \nu^{14} - 124 \nu^{13} + \cdots - 12800 ) / 2048 $ (11v^19 - 20v^18 + 14v^17 + 20v^16 - 95v^15 + 52v^14 - 124v^13 + 136v^12 + 90v^11 - 540v^10 + 1040v^9 - 464v^8 - 824v^7 + 1072v^6 - 928v^5 + 3840v^4 - 1984v^3 + 384v^2 + 11008*v - 12800) / 2048
$\beta_{13}$	$=$	$ ( 29 \nu^{19} - 76 \nu^{18} - 30 \nu^{17} + 76 \nu^{16} - 169 \nu^{15} + 236 \nu^{14} - 260 \nu^{13} + \cdots - 13824 ) / 4096 $ (29v^19 - 76v^18 - 30v^17 + 76v^16 - 169v^15 + 236v^14 - 260v^13 + 312v^12 + 406v^11 - 1860v^10 + 1200v^9 + 336v^8 - 1672v^7 + 3536v^6 - 2912v^5 + 6656v^4 - 4672v^3 - 17792v^2 + 19712*v - 13824) / 4096
$\beta_{14}$	$=$	$ ( 41 \nu^{19} - 36 \nu^{18} - 54 \nu^{17} + 236 \nu^{16} - 133 \nu^{15} + 164 \nu^{14} - 292 \nu^{13} + \cdots - 1536 ) / 4096 $ (41v^19 - 36v^18 - 54v^17 + 236v^16 - 133v^15 + 164v^14 - 292v^13 + 88v^12 + 766v^11 - 2212v^10 + 1392v^9 + 2192v^8 - 3304v^7 + 3024v^6 - 5344v^5 + 6400v^4 - 832v^3 - 22912v^2 + 43264*v - 1536) / 4096
$\beta_{15}$	$=$	$ ( - 43 \nu^{19} + 76 \nu^{18} + 2 \nu^{17} - 68 \nu^{16} + 127 \nu^{15} - 204 \nu^{14} + 236 \nu^{13} + \cdots + 12800 ) / 4096 $ (-43v^19 + 76v^18 + 2v^17 - 68v^16 + 127v^15 - 204v^14 + 236v^13 - 392v^12 - 410v^11 + 2028v^10 - 1616v^9 + 80v^8 + 1592v^7 - 3696v^6 + 4256v^5 - 9472v^4 + 7616v^3 + 11392v^2 - 23296*v + 12800) / 4096
$\beta_{16}$	$=$	$ ( 47 \nu^{19} - 52 \nu^{18} + 54 \nu^{17} + 100 \nu^{16} - 115 \nu^{15} + 212 \nu^{14} - 396 \nu^{13} + \cdots - 512 ) / 4096 $ (47v^19 - 52v^18 + 54v^17 + 100v^16 - 115v^15 + 212v^14 - 396v^13 + 232v^12 + 530v^11 - 1484v^10 + 1424v^9 - 144v^8 - 2136v^7 + 1392v^6 - 4384v^5 + 8704v^4 - 192v^3 - 6272v^2 + 24320*v - 512) / 4096
$\beta_{17}$	$=$	$ ( - 25 \nu^{19} + 52 \nu^{18} - 74 \nu^{17} - 44 \nu^{16} + 245 \nu^{15} - 116 \nu^{14} + 260 \nu^{13} + \cdots + 34304 ) / 2048 $ (-25v^19 + 52v^18 - 74v^17 - 44v^16 + 245v^15 - 116v^14 + 260v^13 - 376v^12 - 94v^11 + 900v^10 - 2416v^9 + 1712v^8 + 2408v^7 - 3280v^6 + 3808v^5 - 7424v^4 + 4928v^3 - 3712v^2 - 22784*v + 34304) / 2048
$\beta_{18}$	$=$	$ ( - 27 \nu^{19} + 32 \nu^{18} - 38 \nu^{17} - 12 \nu^{16} + 159 \nu^{15} - 112 \nu^{14} + 276 \nu^{13} + \cdots + 23040 ) / 1024 $ (-27v^19 + 32v^18 - 38v^17 - 12v^16 + 159v^15 - 112v^14 + 276v^13 - 328v^12 - 234v^11 + 868v^10 - 1616v^9 + 1168v^8 + 1688v^7 - 2192v^6 + 3296v^5 - 7552v^4 + 2496v^3 - 128v^2 - 15616*v + 23040) / 1024
$\beta_{19}$	$=$	$ ( 33 \nu^{19} - 20 \nu^{18} + 10 \nu^{17} + 12 \nu^{16} - 157 \nu^{15} + 84 \nu^{14} - 212 \nu^{13} + \cdots - 16896 ) / 1024 $ (33v^19 - 20v^18 + 10v^17 + 12v^16 - 157v^15 + 84v^14 - 212v^13 + 184v^12 + 430v^11 - 740v^10 + 976v^9 - 752v^8 - 1768v^7 + 2128v^6 - 2784v^5 + 6656v^4 + 2752v^3 - 2432v^2 + 11520*v - 16896) / 1024

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

$\nu$	$=$	$ ( \beta_{6} + \beta_{5} ) / 4 $ (b6 + b5) / 4
$\nu^{2}$	$=$	$ ( \beta_{17} - \beta_{15} - \beta_{10} + \beta_{9} - \beta_{4} - \beta_{2} - \beta_1 ) / 4 $ (b17 - b15 - b10 + b9 - b4 - b2 - b1) / 4
$\nu^{3}$	$=$	$ ( \beta_{19} + \beta_{18} + \beta_{17} + \beta_{16} + \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + \cdots - \beta_1 ) / 4 $ (b19 + b18 + b17 + b16 + b11 + b9 - b8 + b7 + b3 + 2*b2 - b1) / 4
$\nu^{4}$	$=$	$ ( - \beta_{17} + 2 \beta_{16} - \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + \cdots + 2 ) / 4 $ (-b17 + 2b16 - b15 - 2b13 + 2b12 + 2b11 + 3b10 - b9 - b4 - 2b3 - 3*b2 + b1 + 2) / 4
$\nu^{5}$	$=$	$ ( - \beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} + 2 \beta_{15} + 4 \beta_{12} - \beta_{11} + \cdots - 4 ) / 4 $ (-b19 - b18 - b17 + b16 + 2b15 + 4b12 - b11 + 2b10 - 3b9 + b8 + b7 + 2b6 + b3 + 2b2 - b1 - 4) / 4
$\nu^{6}$	$=$	$ ( - 2 \beta_{18} - \beta_{17} - 2 \beta_{16} - \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \cdots - \beta_1 ) / 4 $ (-2b18 - b17 - 2b16 - b15 + 2b14 - 2b13 - 2b12 - 2b11 + 3b10 - 3b9 + 2b8 + 4b7 + 2b6 - 3b4 - b2 - b1) / 4
$\nu^{7}$	$=$	$ ( \beta_{19} + 3 \beta_{18} + \beta_{17} - \beta_{16} + 2 \beta_{15} + 2 \beta_{14} + 4 \beta_{12} + \cdots - 14 ) / 4 $ (b19 + 3b18 + b17 - b16 + 2b15 + 2b14 + 4b12 - 3b11 - 6b10 + b9 - 3b8 + 3b7 - 2b6 + 2b5 - 2b4 + 5b3 + 3*b1 - 14) / 4
$\nu^{8}$	$=$	$ ( - 4 \beta_{18} - \beta_{17} - 2 \beta_{16} + 3 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + \cdots - 18 ) / 4 $ (-4b18 - b17 - 2b16 + 3b15 + 4b14 + 2b13 + 2b12 - 6b11 + 11b10 - b9 - 4b8 + 4b6 - 8b5 - 5b4 + 6b3 - 19b2 + b1 - 18) / 4
$\nu^{9}$	$=$	$ ( 5 \beta_{19} + 5 \beta_{18} + 5 \beta_{17} - 5 \beta_{16} + 6 \beta_{15} + 4 \beta_{14} - 8 \beta_{13} + \cdots + 36 ) / 4 $ (5b19 + 5b18 + 5b17 - 5b16 + 6b15 + 4b14 - 8b13 + 12b12 - 7b11 + 2b10 + 3b9 + 3b8 - b7 - 8b6 - 10b5 + 8b4 + 7b3 + 6b2 + 5b1 + 36) / 4
$\nu^{10}$	$=$	$ ( 4 \beta_{19} + 2 \beta_{18} - 9 \beta_{17} - 2 \beta_{16} + 23 \beta_{15} + 10 \beta_{14} - 6 \beta_{13} + \cdots + 28 ) / 4 $ (4b19 + 2b18 - 9b17 - 2b16 + 23b15 + 10b14 - 6b13 - 6b12 - 10b11 + 11b10 - 3b9 + 14b8 - 8b7 + 6b6 + 4b5 - 3b4 - 5b2 + 15b1 + 28) / 4
$\nu^{11}$	$=$	$ ( - 7 \beta_{19} - 5 \beta_{18} + 17 \beta_{17} + 3 \beta_{16} - 10 \beta_{15} + 2 \beta_{14} - 8 \beta_{13} + \cdots - 14 ) / 4 $ (-7b19 - 5b18 + 17b17 + 3b16 - 10b15 + 2b14 - 8b13 - 12b12 - 19b11 - 50b10 + 13b9 + 21b8 - b7 + 14b6 - 18b5 - 2b4 - 7b3 - 48b2 + 15b1 - 14) / 4
$\nu^{12}$	$=$	$ ( - 8 \beta_{19} - 16 \beta_{18} + 15 \beta_{17} - 2 \beta_{16} + 11 \beta_{15} + 24 \beta_{14} + \cdots + 66 ) / 4 $ (-8b19 - 16b18 + 15b17 - 2b16 + 11b15 + 24b14 + 10b13 - 46b12 + 2b11 - 29b10 + 19b9 - 24b8 + 8b7 + 16b6 - 8b5 - 9b4 + 50b3 + 25b2 - 27*b1 + 66) / 4
$\nu^{13}$	$=$	$ ( 9 \beta_{19} + 37 \beta_{18} - 23 \beta_{17} + 15 \beta_{16} - 50 \beta_{15} - 40 \beta_{13} + 12 \beta_{12} + \cdots + 88 ) / 4 $ (9b19 + 37b18 - 23b17 + 15b16 - 50b15 - 40b13 + 12b12 + 21b11 + 10b10 + 11b9 - 29b8 - 21b7 - 8b6 + 2b5 + 28b4 + 15b3 - 70b2 - 3b1 + 88) / 4
$\nu^{14}$	$=$	$ ( - 4 \beta_{19} + 6 \beta_{18} + 15 \beta_{17} + 38 \beta_{16} + 47 \beta_{15} + 14 \beta_{14} + \cdots + 104 ) / 4 $ (-4b19 + 6b18 + 15b17 + 38b16 + 47b15 + 14b14 + 18b13 + 34b12 - 10b11 + 43b10 + 9b9 - 6b8 - 40b7 - 30b6 + 44b5 + 9b4 + 20b3 - 153b2 - 37*b1 + 104) / 4
$\nu^{15}$	$=$	$ ( 9 \beta_{19} - 9 \beta_{18} + 65 \beta_{17} + 35 \beta_{16} - 66 \beta_{15} + 6 \beta_{14} - 104 \beta_{13} + \cdots - 202 ) / 4 $ (9b19 - 9b18 + 65b17 + 35b16 - 66b15 + 6b14 - 104b13 - 44b12 + 13b11 + 14b10 + 9b9 + 73b8 + 31b7 + 26b6 - 2b5 + 18b4 - 27b3 + 124b2 - 21*b1 - 202) / 4
$\nu^{16}$	$=$	$ ( - 16 \beta_{19} + 60 \beta_{18} - 33 \beta_{17} + 78 \beta_{16} + 35 \beta_{15} + 84 \beta_{14} + \cdots - 130 ) / 4 $ (-16b19 + 60b18 - 33b17 + 78b16 + 35b15 + 84b14 - 54b13 - 14b12 + 58b11 - 109b10 + 15b9 + 12b8 + 24b7 - 124b6 - 53b4 + 6b3 + 149b2 - 31b1 - 130) / 4
$\nu^{17}$	$=$	$ ( - 79 \beta_{19} - 79 \beta_{18} - 175 \beta_{17} + 143 \beta_{16} - 18 \beta_{15} - 36 \beta_{14} + \cdots - 620 ) / 4 $ (-79b19 - 79b18 - 175b17 + 143b16 - 18b15 - 36b14 - 40b13 - 36b12 - 35b11 + 130b10 - 161b9 - 89b8 + 91b7 + 100b6 - 94b5 + 24b4 + 35b3 - 258b2 + 49*b1 - 620) / 4
$\nu^{18}$	$=$	$ ( - 124 \beta_{19} - 206 \beta_{18} - 65 \beta_{17} - 122 \beta_{16} + 119 \beta_{15} + 154 \beta_{14} + \cdots + 196 ) / 4 $ (-124b19 - 206b18 - 65b17 - 122b16 + 119b15 + 154b14 - 110b13 + 146b12 - 50b11 + 155b10 - 219b9 - 162b8 + 24b7 - 138b6 + 36b5 + 77b4 + 136b3 - 269b2 - 105*b1 + 196) / 4
$\nu^{19}$	$=$	$ ( 153 \beta_{19} + 139 \beta_{18} - 303 \beta_{17} - 277 \beta_{16} + 254 \beta_{15} + 82 \beta_{14} + \cdots - 782 ) / 4 $ (153b19 + 139b18 - 303b17 - 277b16 + 254b15 + 82b14 - 104b13 + 132b12 - 339b11 + 342b10 - 267b9 - 27b8 - 185b7 - 146b6 + 182b5 + 174b4 + 401b3 + 752b2 + 23*b1 - 782) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6240\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$5761$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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} $

3121.1

0.635342	−	1.26346i
−1.31746	+	0.514103i
1.20431	−	0.741368i
−0.524636	−	1.31330i
1.40238	−	0.182570i
1.22562	+	0.705594i
−0.732592	+	1.20967i
0.288202	+	1.38454i
−1.41183	+	0.0820714i
0.230662	−	1.39528i
0.635342	+	1.26346i
−1.31746	−	0.514103i
1.20431	+	0.741368i
−0.524636	+	1.31330i
1.40238	+	0.182570i
1.22562	−	0.705594i
−0.732592	−	1.20967i
0.288202	−	1.38454i
−1.41183	−	0.0820714i
0.230662	+	1.39528i

−

1.00000i

−

1.00000i

−4.21598

−1.00000

3121.2

−

1.00000i

−

1.00000i

−3.05712

−1.00000

3121.3

−

1.00000i

−

1.00000i

−1.16629

−1.00000

3121.4

−

1.00000i

−

1.00000i

−0.433192

−1.00000

3121.5

−

1.00000i

−

1.00000i

0.184370

−1.00000

3121.6

−

1.00000i

−

1.00000i

2.07003

−1.00000

3121.7

−

1.00000i

−

1.00000i

2.74234

−1.00000

3121.8

−

1.00000i

−

1.00000i

3.39492

−1.00000

3121.9

−

1.00000i

−

1.00000i

4.19666

−1.00000

3121.10

−

1.00000i

−

1.00000i

4.28425

−1.00000

3121.11

1.00000i

−4.21598

−1.00000

3121.12

1.00000i

−3.05712

−1.00000

3121.13

1.00000i

−1.16629

−1.00000

3121.14

1.00000i

−0.433192

−1.00000

3121.15

1.00000i

0.184370

−1.00000

3121.16

1.00000i

2.07003

−1.00000

3121.17

1.00000i

2.74234

−1.00000

3121.18

1.00000i

3.39492

−1.00000

3121.19

1.00000i

4.19666

−1.00000

3121.20

1.00000i

4.28425

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	6240.2.w.f		20
4.b	odd	2	1		1560.2.w.f	✓	20
8.b	even	2	1	inner	6240.2.w.f		20
8.d	odd	2	1		1560.2.w.f	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1560.2.w.f	✓	20	4.b	odd	2	1
1560.2.w.f	✓	20	8.d	odd	2	1
6240.2.w.f		20	1.a	even	1	1	trivial
6240.2.w.f		20	8.b	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 $S_{2}^{\mathrm{new}}(6240, [\chi])$:

$ T_{7}^{10} - 8 T_{7}^{9} - 12 T_{7}^{8} + 228 T_{7}^{7} - 275 T_{7}^{6} - 1636 T_{7}^{5} + 3358 T_{7}^{4} + \cdots + 416 $

$ T_{11}^{20} + 124 T_{11}^{18} + 6590 T_{11}^{16} + 195972 T_{11}^{14} + 3566297 T_{11}^{12} + \cdots + 198246400 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ (T^{2} + 1)^{10} $ (T^2 + 1)^10
$5$	$ (T^{2} + 1)^{10} $ (T^2 + 1)^10
$7$	$ (T^{10} - 8 T^{9} + \cdots + 416)^{2} $ (T^10 - 8T^9 - 12T^8 + 228T^7 - 275T^6 - 1636T^5 + 3358T^4 + 2048T^3 - 5256T^2 - 1376*T + 416)^2
$11$	$ T^{20} + \cdots + 198246400 $ T^20 + 124T^18 + 6590T^16 + 195972T^14 + 3566297T^12 + 40705880T^10 + 285958160T^8 + 1154361344T^6 + 2294396928T^4 + 1593081856*T^2 + 198246400
$13$	$ (T^{2} + 1)^{10} $ (T^2 + 1)^10
$17$	$ (T^{10} - 8 T^{9} + \cdots - 3488)^{2} $ (T^10 - 8T^9 - 52T^8 + 584T^7 - 743T^6 - 5080T^5 + 12746T^4 + 4512T^3 - 31848T^2 + 23488*T - 3488)^2
$19$	$ T^{20} + \cdots + 7487094784 $ T^20 + 236T^18 + 22868T^16 + 1178144T^14 + 34850368T^12 + 596692480T^10 + 5716189184T^8 + 28793151488T^6 + 65434238976T^4 + 41580363776*T^2 + 7487094784
$23$	$ (T^{10} + 24 T^{9} + \cdots - 3400352)^{2} $ (T^10 + 24T^9 + 144T^8 - 744T^7 - 10551T^6 - 17888T^5 + 156666T^4 + 571552T^3 - 351208T^2 - 3481280*T - 3400352)^2
$29$	$ T^{20} + \cdots + 1701250662400 $ T^20 + 292T^18 + 33284T^16 + 1966176T^14 + 67179584T^12 + 1394441216T^10 + 17890430208T^8 + 140832331776T^6 + 655736352768T^4 + 1644537217024*T^2 + 1701250662400
$31$	$ (T^{10} - 4 T^{9} + \cdots - 16384)^{2} $ (T^10 - 4T^9 - 152T^8 + 424T^7 + 8136T^6 - 15168T^5 - 179776T^4 + 226304T^3 + 1363968T^2 - 1376256*T - 16384)^2
$37$	$ T^{20} + \cdots + 251442073600 $ T^20 + 412T^18 + 67206T^16 + 5649252T^14 + 269839185T^12 + 7598719496T^10 + 126545781616T^8 + 1217185260288T^6 + 6235284012800T^4 + 13184465012736*T^2 + 251442073600
$41$	$ (T^{10} - 8 T^{9} + \cdots - 50990720)^{2} $ (T^10 - 8T^9 - 234T^8 + 1600T^7 + 19245T^6 - 99832T^5 - 673056T^4 + 2066016T^3 + 10018768T^2 - 13075968*T - 50990720)^2
$43$	$ T^{20} + 296 T^{18} + \cdots + 67108864 $ T^20 + 296T^18 + 36704T^16 + 2481088T^14 + 99666752T^12 + 2435392000T^10 + 35701392384T^8 + 299085889536T^6 + 1277051011072T^4 + 2030387068928*T^2 + 67108864
$47$	$ (T^{10} - 4 T^{9} + \cdots - 497096704)^{2} $ (T^10 - 4T^9 - 368T^8 + 1144T^7 + 48888T^6 - 105024T^5 - 2831296T^4 + 3503488T^3 + 67217024T^2 - 43328512*T - 497096704)^2
$53$	$ T^{20} + \cdots + 91447449849856 $ T^20 + 524T^18 + 110318T^16 + 12264884T^14 + 792514057T^12 + 30786746328T^10 + 718112310640T^8 + 9718317236480T^6 + 69842501019392T^4 + 212846527469568*T^2 + 91447449849856
$59$	$ T^{20} + \cdots + 261369043615744 $ T^20 + 680T^18 + 188784T^16 + 27661056T^14 + 2317690624T^12 + 113266599936T^10 + 3188745146368T^8 + 49441579466752T^6 + 383236351459328T^4 + 1146807597924352*T^2 + 261369043615744
$61$	$ T^{20} + \cdots + 318928165273600 $ T^20 + 700T^18 + 198278T^16 + 29428604T^14 + 2487578753T^12 + 122221770432T^10 + 3413678279680T^8 + 50174489395200T^6 + 328190517051392T^4 + 799345154719744*T^2 + 318928165273600
$67$	$ T^{20} + \cdots + 16\!\cdots\!00 $ T^20 + 768T^18 + 243616T^16 + 41233152T^14 + 4027226368T^12 + 230813331456T^10 + 7638310928384T^8 + 138139767668736T^6 + 1213395432898560T^4 + 4179949816643584*T^2 + 1691170216345600
$71$	$ (T^{10} - 4 T^{9} + \cdots + 104160256)^{2} $ (T^10 - 4T^9 - 458T^8 + 1344T^7 + 73173T^6 - 211540T^5 - 5000852T^4 + 16685376T^3 + 124314240T^2 - 506847232*T + 104160256)^2
$73$	$ (T^{10} - 20 T^{9} + \cdots + 4096)^{2} $ (T^10 - 20T^9 - 34T^8 + 1984T^7 - 7720T^6 - 19936T^5 + 172832T^4 - 354816T^3 + 284672T^2 - 83968*T + 4096)^2
$79$	$ (T^{10} - 4 T^{9} + \cdots - 1678716928)^{2} $ (T^10 - 4T^9 - 486T^8 + 1684T^7 + 80225T^6 - 246088T^5 - 5490064T^4 + 13203712T^3 + 159068928T^2 - 208398336*T - 1678716928)^2
$83$	$ T^{20} + \cdots + 35\!\cdots\!24 $ T^20 + 1256T^18 + 673936T^16 + 201685504T^14 + 36801587200T^12 + 4192680083456T^10 + 291809364869120T^8 + 11456284758900736T^6 + 205514705039523840T^4 + 694519625777414144*T^2 + 35204854787866624
$89$	$ (T^{10} - 28 T^{9} + \cdots - 2344371904)^{2} $ (T^10 - 28T^9 - 238T^8 + 12520T^7 - 44507T^6 - 1518604T^5 + 12821260T^4 + 25549152T^3 - 601358416T^2 + 2145673536*T - 2344371904)^2
$97$	$ (T^{10} + 24 T^{9} + \cdots + 176269088)^{2} $ (T^10 + 24T^9 - 168T^8 - 7380T^7 - 16251T^6 + 653732T^5 + 3276846T^4 - 16266912T^3 - 106951848T^2 - 11190496*T + 176269088)^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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.w (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{3} - \beta_{2} q^{5} + (\beta_{10} + 1) q^{7} - q^{9}+O(q^{10}) \) q - b2 * q^3 - b2 * q^5 + (b10 + 1) * q^7 - q^9	\( q - \beta_{2} q^{3} - \beta_{2} q^{5} + (\beta_{10} + 1) q^{7} - q^{9} + (\beta_{13} - \beta_{2}) q^{11} - \beta_{2} q^{13} - q^{15} + (\beta_{17} - \beta_{10} + 1) q^{17} + ( - \beta_{11} + \beta_{2}) q^{19} + (\beta_{15} - \beta_{2}) q^{21} + (\beta_{17} - \beta_{12} - \beta_{10} + \cdots - 2) q^{23}+ \cdots + ( - \beta_{13} + \beta_{2}) q^{99}+O(q^{100}) \) q - b2 * q^3 - b2 * q^5 + (b10 + 1) * q^7 - q^9 + (b13 - b2) * q^11 - b2 * q^13 - q^15 + (b17 - b10 + 1) * q^17 + (-b11 + b2) * q^19 + (b15 - b2) * q^21 + (b17 - b12 - b10 + b9 - 2) * q^23 - q^25 + b2 * q^27 + (-b8 - b1) * q^29 + (b18 + b17 + b9 + b7 + 1) * q^31 + (b12 - 1) * q^33 + (b15 - b2) * q^35 + (b16 - b15 + b11 - b3 + b2) * q^37 - q^39 + (b12 - b7 + b4 + 1) * q^41 + (-b16 + b2) * q^43 + b2 * q^45 + (b17 + b10 - b9 + b7 + 2b6) q^47 + (-b19 - b18 - b17 + b10 - b9 + 1) * q^49 + (-b15 - b5 - b2) * q^51 + (b15 + b14 - b13 - b8 + b3 + 3b2) q^53 + (b12 - 1) * q^55 + (-b7 + 1) * q^57 + (b16 - b14 + b11 - 2b3 - b2) q^59 + (b14 - b13 - b8 - b5 + b3 - b1) * q^61 + (-b10 - 1) * q^63 - q^65 + (-b14 - b8 - b3 - b2 + b1) * q^67 + (-b15 + b13 + b8 - b5 + 2b2) q^69 + (b18 - b10 - b9 - b7 + b4 + 1) * q^71 + (b19 + b18 + b9 - b6 + 3) * q^73 + b2 * q^75 + (-b15 + b14 + b13 + b8 - 2b5 - b3 - 3b2) * q^77 + (b19 + b18 + 3b12 + b10 - b6 + 1) q^79 + q^81 + (b15 - 2b8 + b5 + b3 + 3b2 + b1) * q^83 + (-b15 - b5 - b2) * q^85 + (b18 + b9) * q^87 + (-b19 + b17 - b12 + 2b7 + b6 - 2b4 + 2) * q^89 + (b15 - b2) * q^91 + (-b11 + b8 - b5 - b2 + b1) * q^93 + (-b7 + 1) * q^95 + (-2b18 - b12 - b10 - b9 + 2b6 - 4) * q^97 + (-b13 + b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7} - 20 q^{9}+O(q^{10}) \) 20 * q + 16 * q^7 - 20 * q^9	\( 20 q + 16 q^{7} - 20 q^{9} - 20 q^{15} + 16 q^{17} - 48 q^{23} - 20 q^{25} + 8 q^{31} - 16 q^{33} - 20 q^{39} + 16 q^{41} + 8 q^{47} + 36 q^{49} - 16 q^{55} + 16 q^{57} - 16 q^{63} - 20 q^{65} + 8 q^{71} + 40 q^{73} + 8 q^{79} + 20 q^{81} - 8 q^{87} + 56 q^{89} + 16 q^{95} - 48 q^{97}+O(q^{100}) \) 20 * q + 16 * q^7 - 20 * q^9 - 20 * q^15 + 16 * q^17 - 48 * q^23 - 20 * q^25 + 8 * q^31 - 16 * q^33 - 20 * q^39 + 16 * q^41 + 8 * q^47 + 36 * q^49 - 16 * q^55 + 16 * q^57 - 16 * q^63 - 20 * q^65 + 8 * q^71 + 40 * q^73 + 8 * q^79 + 20 * q^81 - 8 * q^87 + 56 * q^89 + 16 * q^95 - 48 * q^97

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	6240.2.w.f

Level	$6240$
Weight	$2$
Character orbit	6240.w
Analytic conductor	$49.827$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(49.8266508613\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 2 x^{19} + 2 x^{18} - 5 x^{16} + 10 x^{15} - 12 x^{14} + 16 x^{13} - 2 x^{12} - 40 x^{11} + \cdots + 1024 \) x^20 - 2x^19 + 2x^18 - 5x^16 + 10x^15 - 12x^14 + 16x^13 - 2x^12 - 40x^11 + 72x^10 - 80x^9 - 8x^8 + 128x^7 - 192x^6 + 320x^5 - 320x^4 + 512x^2 - 1024*x + 1024
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 2^{23} \)
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{19} - 10 \nu^{18} - 2 \nu^{17} - 8 \nu^{16} + 3 \nu^{15} + 18 \nu^{14} - 24 \nu^{13} + \cdots - 1024 ) / 1024 \) (v^19 - 10v^18 - 2v^17 - 8v^16 + 3v^15 + 18v^14 - 24v^13 + 72v^12 - 2v^11 - 120v^10 + 112v^9 - 48v^8 + 376v^7 + 128v^6 - 416v^5 + 1344v^4 - 1728v^3 - 2304v^2 + 256v - 1024) / 1024
\(\beta_{2}\)	\(=\)	\( ( 7 \nu^{19} - 4 \nu^{18} + 6 \nu^{17} + 36 \nu^{16} + 21 \nu^{15} - 28 \nu^{14} - 76 \nu^{13} + \cdots + 7680 ) / 4096 \) (7v^19 - 4v^18 + 6v^17 + 36v^16 + 21v^15 - 28v^14 - 76v^13 - 24v^12 - 30v^11 - 204v^10 + 144v^9 + 368v^8 - 280v^7 - 656v^6 - 544v^5 - 704v^3 - 128v^2 + 5888v + 7680) / 4096
\(\beta_{3}\)	\(=\)	\( ( 5 \nu^{19} - 36 \nu^{18} + 2 \nu^{17} - 68 \nu^{16} - 49 \nu^{15} + 100 \nu^{14} + 12 \nu^{13} + \cdots - 19968 ) / 2048 \) (5v^19 - 36v^18 + 2v^17 - 68v^16 - 49v^15 + 100v^14 + 12v^13 + 408v^12 - 58v^11 - 180v^10 + 560v^9 - 1264v^8 + 696v^7 + 1680v^6 + 672v^5 + 4352v^4 - 4672v^3 - 896v^2 - 6912*v - 19968) / 2048
\(\beta_{4}\)	\(=\)	\( ( \nu^{19} - 2 \nu^{18} + 2 \nu^{17} + 4 \nu^{16} - 9 \nu^{15} + 10 \nu^{14} - 12 \nu^{13} + 12 \nu^{12} + \cdots - 1536 ) / 256 \) (v^19 - 2v^18 + 2v^17 + 4v^16 - 9v^15 + 10v^14 - 12v^13 + 12v^12 + 18v^11 - 80v^10 + 128v^9 - 40v^8 - 160v^7 + 160v^6 - 240v^5 + 224v^4 - 128v^3 - 640v^2 + 1664v - 1536) / 256
\(\beta_{5}\)	\(=\)	\( ( \nu^{19} - 2 \nu^{18} + 2 \nu^{17} - 5 \nu^{15} + 10 \nu^{14} - 12 \nu^{13} + 16 \nu^{12} + \cdots - 1024 ) / 256 \) (v^19 - 2v^18 + 2v^17 - 5v^15 + 10v^14 - 12v^13 + 16v^12 - 2v^11 - 40v^10 + 72v^9 - 80v^8 - 8v^7 + 128v^6 - 192v^5 + 320v^4 - 320v^3 + 1024v - 1024) / 256
\(\beta_{6}\)	\(=\)	\( ( - \nu^{19} + 2 \nu^{18} - 2 \nu^{17} + 5 \nu^{15} - 10 \nu^{14} + 12 \nu^{13} - 16 \nu^{12} + \cdots + 1024 ) / 256 \) (-v^19 + 2v^18 - 2v^17 + 5v^15 - 10v^14 + 12v^13 - 16v^12 + 2v^11 + 40v^10 - 72v^9 + 80v^8 + 8v^7 - 128v^6 + 192v^5 - 320v^4 + 320*v^3 + 1024) / 256
\(\beta_{7}\)	\(=\)	\( ( - 9 \nu^{19} - 20 \nu^{18} + 38 \nu^{17} + 4 \nu^{16} - 59 \nu^{15} + 52 \nu^{14} - 76 \nu^{13} + \cdots - 12800 ) / 2048 \) (-9v^19 - 20v^18 + 38v^17 + 4v^16 - 59v^15 + 52v^14 - 76v^13 + 104v^12 - 158v^11 - 236v^10 + 1040v^9 - 784v^8 - 88v^7 + 1904v^6 - 1184v^5 + 768v^4 - 2240v^3 + 896v^2 + 6912*v - 12800) / 2048
\(\beta_{8}\)	\(=\)	\( ( 7 \nu^{19} - 52 \nu^{18} + 6 \nu^{17} - 28 \nu^{16} - 107 \nu^{15} + 148 \nu^{14} - 76 \nu^{13} + \cdots - 29184 ) / 2048 \) (7v^19 - 52v^18 + 6v^17 - 28v^16 - 107v^15 + 148v^14 - 76v^13 + 392v^12 + 34v^11 - 492v^10 + 1232v^9 - 1488v^8 - 536v^7 + 3056v^6 - 288v^5 + 5376v^4 - 5824v^3 - 2176v^2 + 1792*v - 29184) / 2048
\(\beta_{9}\)	\(=\)	\( ( 3 \nu^{19} - 28 \nu^{18} + 46 \nu^{17} + 20 \nu^{16} - 87 \nu^{15} + 60 \nu^{14} - 140 \nu^{13} + \cdots - 16896 ) / 1024 \) (3v^19 - 28v^18 + 46v^17 + 20v^16 - 87v^15 + 60v^14 - 140v^13 + 184v^12 - 118v^11 - 364v^10 + 1424v^9 - 752v^8 - 888v^7 + 1136v^6 - 2208v^5 + 2688v^4 - 4032v^3 + 3968v^2 + 13056*v - 16896) / 1024
\(\beta_{10}\)	\(=\)	\( ( - 11 \nu^{19} - 4 \nu^{18} + 2 \nu^{17} - 4 \nu^{16} + 63 \nu^{15} + 4 \nu^{14} + 44 \nu^{13} + \cdots + 4608 ) / 2048 \) (-11v^19 - 4v^18 + 2v^17 - 4v^16 + 63v^15 + 4v^14 + 44v^13 - 40v^12 - 250v^11 + 140v^10 - 80v^9 + 400v^8 + 504v^7 - 368v^6 + 288v^5 - 1792v^4 - 2112v^3 + 128v^2 - 1792*v + 4608) / 2048
\(\beta_{11}\)	\(=\)	\( ( - 21 \nu^{19} - 12 \nu^{18} - 66 \nu^{17} - 124 \nu^{16} - 63 \nu^{15} + 12 \nu^{14} + 212 \nu^{13} + \cdots - 37376 ) / 4096 \) (-21v^19 - 12v^18 - 66v^17 - 124v^16 - 63v^15 + 12v^14 + 212v^13 + 456v^12 - 102v^11 + 404v^10 - 304v^9 - 1744v^8 + 968v^7 + 1648v^6 + 3424v^5 + 4864v^4 - 1472v^3 - 5248v^2 - 21760*v - 37376) / 4096
\(\beta_{12}\)	\(=\)	\( ( 11 \nu^{19} - 20 \nu^{18} + 14 \nu^{17} + 20 \nu^{16} - 95 \nu^{15} + 52 \nu^{14} - 124 \nu^{13} + \cdots - 12800 ) / 2048 \) (11v^19 - 20v^18 + 14v^17 + 20v^16 - 95v^15 + 52v^14 - 124v^13 + 136v^12 + 90v^11 - 540v^10 + 1040v^9 - 464v^8 - 824v^7 + 1072v^6 - 928v^5 + 3840v^4 - 1984v^3 + 384v^2 + 11008*v - 12800) / 2048
\(\beta_{13}\)	\(=\)	\( ( 29 \nu^{19} - 76 \nu^{18} - 30 \nu^{17} + 76 \nu^{16} - 169 \nu^{15} + 236 \nu^{14} - 260 \nu^{13} + \cdots - 13824 ) / 4096 \) (29v^19 - 76v^18 - 30v^17 + 76v^16 - 169v^15 + 236v^14 - 260v^13 + 312v^12 + 406v^11 - 1860v^10 + 1200v^9 + 336v^8 - 1672v^7 + 3536v^6 - 2912v^5 + 6656v^4 - 4672v^3 - 17792v^2 + 19712*v - 13824) / 4096
\(\beta_{14}\)	\(=\)	\( ( 41 \nu^{19} - 36 \nu^{18} - 54 \nu^{17} + 236 \nu^{16} - 133 \nu^{15} + 164 \nu^{14} - 292 \nu^{13} + \cdots - 1536 ) / 4096 \) (41v^19 - 36v^18 - 54v^17 + 236v^16 - 133v^15 + 164v^14 - 292v^13 + 88v^12 + 766v^11 - 2212v^10 + 1392v^9 + 2192v^8 - 3304v^7 + 3024v^6 - 5344v^5 + 6400v^4 - 832v^3 - 22912v^2 + 43264*v - 1536) / 4096
\(\beta_{15}\)	\(=\)	\( ( - 43 \nu^{19} + 76 \nu^{18} + 2 \nu^{17} - 68 \nu^{16} + 127 \nu^{15} - 204 \nu^{14} + 236 \nu^{13} + \cdots + 12800 ) / 4096 \) (-43v^19 + 76v^18 + 2v^17 - 68v^16 + 127v^15 - 204v^14 + 236v^13 - 392v^12 - 410v^11 + 2028v^10 - 1616v^9 + 80v^8 + 1592v^7 - 3696v^6 + 4256v^5 - 9472v^4 + 7616v^3 + 11392v^2 - 23296*v + 12800) / 4096
\(\beta_{16}\)	\(=\)	\( ( 47 \nu^{19} - 52 \nu^{18} + 54 \nu^{17} + 100 \nu^{16} - 115 \nu^{15} + 212 \nu^{14} - 396 \nu^{13} + \cdots - 512 ) / 4096 \) (47v^19 - 52v^18 + 54v^17 + 100v^16 - 115v^15 + 212v^14 - 396v^13 + 232v^12 + 530v^11 - 1484v^10 + 1424v^9 - 144v^8 - 2136v^7 + 1392v^6 - 4384v^5 + 8704v^4 - 192v^3 - 6272v^2 + 24320*v - 512) / 4096
\(\beta_{17}\)	\(=\)	\( ( - 25 \nu^{19} + 52 \nu^{18} - 74 \nu^{17} - 44 \nu^{16} + 245 \nu^{15} - 116 \nu^{14} + 260 \nu^{13} + \cdots + 34304 ) / 2048 \) (-25v^19 + 52v^18 - 74v^17 - 44v^16 + 245v^15 - 116v^14 + 260v^13 - 376v^12 - 94v^11 + 900v^10 - 2416v^9 + 1712v^8 + 2408v^7 - 3280v^6 + 3808v^5 - 7424v^4 + 4928v^3 - 3712v^2 - 22784*v + 34304) / 2048
\(\beta_{18}\)	\(=\)	\( ( - 27 \nu^{19} + 32 \nu^{18} - 38 \nu^{17} - 12 \nu^{16} + 159 \nu^{15} - 112 \nu^{14} + 276 \nu^{13} + \cdots + 23040 ) / 1024 \) (-27v^19 + 32v^18 - 38v^17 - 12v^16 + 159v^15 - 112v^14 + 276v^13 - 328v^12 - 234v^11 + 868v^10 - 1616v^9 + 1168v^8 + 1688v^7 - 2192v^6 + 3296v^5 - 7552v^4 + 2496v^3 - 128v^2 - 15616*v + 23040) / 1024
\(\beta_{19}\)	\(=\)	\( ( 33 \nu^{19} - 20 \nu^{18} + 10 \nu^{17} + 12 \nu^{16} - 157 \nu^{15} + 84 \nu^{14} - 212 \nu^{13} + \cdots - 16896 ) / 1024 \) (33v^19 - 20v^18 + 10v^17 + 12v^16 - 157v^15 + 84v^14 - 212v^13 + 184v^12 + 430v^11 - 740v^10 + 976v^9 - 752v^8 - 1768v^7 + 2128v^6 - 2784v^5 + 6656v^4 + 2752v^3 - 2432v^2 + 11520*v - 16896) / 1024

\(\nu\)	\(=\)	\( ( \beta_{6} + \beta_{5} ) / 4 \) (b6 + b5) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{17} - \beta_{15} - \beta_{10} + \beta_{9} - \beta_{4} - \beta_{2} - \beta_1 ) / 4 \) (b17 - b15 - b10 + b9 - b4 - b2 - b1) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{19} + \beta_{18} + \beta_{17} + \beta_{16} + \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + \cdots - \beta_1 ) / 4 \) (b19 + b18 + b17 + b16 + b11 + b9 - b8 + b7 + b3 + 2*b2 - b1) / 4
\(\nu^{4}\)	\(=\)	\( ( - \beta_{17} + 2 \beta_{16} - \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + \cdots + 2 ) / 4 \) (-b17 + 2b16 - b15 - 2b13 + 2b12 + 2b11 + 3b10 - b9 - b4 - 2b3 - 3*b2 + b1 + 2) / 4
\(\nu^{5}\)	\(=\)	\( ( - \beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} + 2 \beta_{15} + 4 \beta_{12} - \beta_{11} + \cdots - 4 ) / 4 \) (-b19 - b18 - b17 + b16 + 2b15 + 4b12 - b11 + 2b10 - 3b9 + b8 + b7 + 2b6 + b3 + 2b2 - b1 - 4) / 4
\(\nu^{6}\)	\(=\)	\( ( - 2 \beta_{18} - \beta_{17} - 2 \beta_{16} - \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \cdots - \beta_1 ) / 4 \) (-2b18 - b17 - 2b16 - b15 + 2b14 - 2b13 - 2b12 - 2b11 + 3b10 - 3b9 + 2b8 + 4b7 + 2b6 - 3b4 - b2 - b1) / 4
\(\nu^{7}\)	\(=\)	\( ( \beta_{19} + 3 \beta_{18} + \beta_{17} - \beta_{16} + 2 \beta_{15} + 2 \beta_{14} + 4 \beta_{12} + \cdots - 14 ) / 4 \) (b19 + 3b18 + b17 - b16 + 2b15 + 2b14 + 4b12 - 3b11 - 6b10 + b9 - 3b8 + 3b7 - 2b6 + 2b5 - 2b4 + 5b3 + 3*b1 - 14) / 4
\(\nu^{8}\)	\(=\)	\( ( - 4 \beta_{18} - \beta_{17} - 2 \beta_{16} + 3 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + \cdots - 18 ) / 4 \) (-4b18 - b17 - 2b16 + 3b15 + 4b14 + 2b13 + 2b12 - 6b11 + 11b10 - b9 - 4b8 + 4b6 - 8b5 - 5b4 + 6b3 - 19b2 + b1 - 18) / 4
\(\nu^{9}\)	\(=\)	\( ( 5 \beta_{19} + 5 \beta_{18} + 5 \beta_{17} - 5 \beta_{16} + 6 \beta_{15} + 4 \beta_{14} - 8 \beta_{13} + \cdots + 36 ) / 4 \) (5b19 + 5b18 + 5b17 - 5b16 + 6b15 + 4b14 - 8b13 + 12b12 - 7b11 + 2b10 + 3b9 + 3b8 - b7 - 8b6 - 10b5 + 8b4 + 7b3 + 6b2 + 5b1 + 36) / 4
\(\nu^{10}\)	\(=\)	\( ( 4 \beta_{19} + 2 \beta_{18} - 9 \beta_{17} - 2 \beta_{16} + 23 \beta_{15} + 10 \beta_{14} - 6 \beta_{13} + \cdots + 28 ) / 4 \) (4b19 + 2b18 - 9b17 - 2b16 + 23b15 + 10b14 - 6b13 - 6b12 - 10b11 + 11b10 - 3b9 + 14b8 - 8b7 + 6b6 + 4b5 - 3b4 - 5b2 + 15b1 + 28) / 4
\(\nu^{11}\)	\(=\)	\( ( - 7 \beta_{19} - 5 \beta_{18} + 17 \beta_{17} + 3 \beta_{16} - 10 \beta_{15} + 2 \beta_{14} - 8 \beta_{13} + \cdots - 14 ) / 4 \) (-7b19 - 5b18 + 17b17 + 3b16 - 10b15 + 2b14 - 8b13 - 12b12 - 19b11 - 50b10 + 13b9 + 21b8 - b7 + 14b6 - 18b5 - 2b4 - 7b3 - 48b2 + 15b1 - 14) / 4
\(\nu^{12}\)	\(=\)	\( ( - 8 \beta_{19} - 16 \beta_{18} + 15 \beta_{17} - 2 \beta_{16} + 11 \beta_{15} + 24 \beta_{14} + \cdots + 66 ) / 4 \) (-8b19 - 16b18 + 15b17 - 2b16 + 11b15 + 24b14 + 10b13 - 46b12 + 2b11 - 29b10 + 19b9 - 24b8 + 8b7 + 16b6 - 8b5 - 9b4 + 50b3 + 25b2 - 27*b1 + 66) / 4
\(\nu^{13}\)	\(=\)	\( ( 9 \beta_{19} + 37 \beta_{18} - 23 \beta_{17} + 15 \beta_{16} - 50 \beta_{15} - 40 \beta_{13} + 12 \beta_{12} + \cdots + 88 ) / 4 \) (9b19 + 37b18 - 23b17 + 15b16 - 50b15 - 40b13 + 12b12 + 21b11 + 10b10 + 11b9 - 29b8 - 21b7 - 8b6 + 2b5 + 28b4 + 15b3 - 70b2 - 3b1 + 88) / 4
\(\nu^{14}\)	\(=\)	\( ( - 4 \beta_{19} + 6 \beta_{18} + 15 \beta_{17} + 38 \beta_{16} + 47 \beta_{15} + 14 \beta_{14} + \cdots + 104 ) / 4 \) (-4b19 + 6b18 + 15b17 + 38b16 + 47b15 + 14b14 + 18b13 + 34b12 - 10b11 + 43b10 + 9b9 - 6b8 - 40b7 - 30b6 + 44b5 + 9b4 + 20b3 - 153b2 - 37*b1 + 104) / 4
\(\nu^{15}\)	\(=\)	\( ( 9 \beta_{19} - 9 \beta_{18} + 65 \beta_{17} + 35 \beta_{16} - 66 \beta_{15} + 6 \beta_{14} - 104 \beta_{13} + \cdots - 202 ) / 4 \) (9b19 - 9b18 + 65b17 + 35b16 - 66b15 + 6b14 - 104b13 - 44b12 + 13b11 + 14b10 + 9b9 + 73b8 + 31b7 + 26b6 - 2b5 + 18b4 - 27b3 + 124b2 - 21*b1 - 202) / 4
\(\nu^{16}\)	\(=\)	\( ( - 16 \beta_{19} + 60 \beta_{18} - 33 \beta_{17} + 78 \beta_{16} + 35 \beta_{15} + 84 \beta_{14} + \cdots - 130 ) / 4 \) (-16b19 + 60b18 - 33b17 + 78b16 + 35b15 + 84b14 - 54b13 - 14b12 + 58b11 - 109b10 + 15b9 + 12b8 + 24b7 - 124b6 - 53b4 + 6b3 + 149b2 - 31b1 - 130) / 4
\(\nu^{17}\)	\(=\)	\( ( - 79 \beta_{19} - 79 \beta_{18} - 175 \beta_{17} + 143 \beta_{16} - 18 \beta_{15} - 36 \beta_{14} + \cdots - 620 ) / 4 \) (-79b19 - 79b18 - 175b17 + 143b16 - 18b15 - 36b14 - 40b13 - 36b12 - 35b11 + 130b10 - 161b9 - 89b8 + 91b7 + 100b6 - 94b5 + 24b4 + 35b3 - 258b2 + 49*b1 - 620) / 4
\(\nu^{18}\)	\(=\)	\( ( - 124 \beta_{19} - 206 \beta_{18} - 65 \beta_{17} - 122 \beta_{16} + 119 \beta_{15} + 154 \beta_{14} + \cdots + 196 ) / 4 \) (-124b19 - 206b18 - 65b17 - 122b16 + 119b15 + 154b14 - 110b13 + 146b12 - 50b11 + 155b10 - 219b9 - 162b8 + 24b7 - 138b6 + 36b5 + 77b4 + 136b3 - 269b2 - 105*b1 + 196) / 4
\(\nu^{19}\)	\(=\)	\( ( 153 \beta_{19} + 139 \beta_{18} - 303 \beta_{17} - 277 \beta_{16} + 254 \beta_{15} + 82 \beta_{14} + \cdots - 782 ) / 4 \) (153b19 + 139b18 - 303b17 - 277b16 + 254b15 + 82b14 - 104b13 + 132b12 - 339b11 + 342b10 - 267b9 - 27b8 - 185b7 - 146b6 + 182b5 + 174b4 + 401b3 + 752b2 + 23*b1 - 782) / 4

\(n\)	\(1951\)	\(2081\)	\(2341\)	\(2497\)	\(5761\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3121.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( (T^{2} + 1)^{10} \) (T^2 + 1)^10
$5$	\( (T^{2} + 1)^{10} \) (T^2 + 1)^10
$7$	\( (T^{10} - 8 T^{9} + \cdots + 416)^{2} \) (T^10 - 8T^9 - 12T^8 + 228T^7 - 275T^6 - 1636T^5 + 3358T^4 + 2048T^3 - 5256T^2 - 1376*T + 416)^2
$11$	\( T^{20} + \cdots + 198246400 \) T^20 + 124T^18 + 6590T^16 + 195972T^14 + 3566297T^12 + 40705880T^10 + 285958160T^8 + 1154361344T^6 + 2294396928T^4 + 1593081856*T^2 + 198246400
$13$	\( (T^{2} + 1)^{10} \) (T^2 + 1)^10
$17$	\( (T^{10} - 8 T^{9} + \cdots - 3488)^{2} \) (T^10 - 8T^9 - 52T^8 + 584T^7 - 743T^6 - 5080T^5 + 12746T^4 + 4512T^3 - 31848T^2 + 23488*T - 3488)^2
$19$	\( T^{20} + \cdots + 7487094784 \) T^20 + 236T^18 + 22868T^16 + 1178144T^14 + 34850368T^12 + 596692480T^10 + 5716189184T^8 + 28793151488T^6 + 65434238976T^4 + 41580363776*T^2 + 7487094784
$23$	\( (T^{10} + 24 T^{9} + \cdots - 3400352)^{2} \) (T^10 + 24T^9 + 144T^8 - 744T^7 - 10551T^6 - 17888T^5 + 156666T^4 + 571552T^3 - 351208T^2 - 3481280*T - 3400352)^2
$29$	\( T^{20} + \cdots + 1701250662400 \) T^20 + 292T^18 + 33284T^16 + 1966176T^14 + 67179584T^12 + 1394441216T^10 + 17890430208T^8 + 140832331776T^6 + 655736352768T^4 + 1644537217024*T^2 + 1701250662400
$31$	\( (T^{10} - 4 T^{9} + \cdots - 16384)^{2} \) (T^10 - 4T^9 - 152T^8 + 424T^7 + 8136T^6 - 15168T^5 - 179776T^4 + 226304T^3 + 1363968T^2 - 1376256*T - 16384)^2
$37$	\( T^{20} + \cdots + 251442073600 \) T^20 + 412T^18 + 67206T^16 + 5649252T^14 + 269839185T^12 + 7598719496T^10 + 126545781616T^8 + 1217185260288T^6 + 6235284012800T^4 + 13184465012736*T^2 + 251442073600
$41$	\( (T^{10} - 8 T^{9} + \cdots - 50990720)^{2} \) (T^10 - 8T^9 - 234T^8 + 1600T^7 + 19245T^6 - 99832T^5 - 673056T^4 + 2066016T^3 + 10018768T^2 - 13075968*T - 50990720)^2
$43$	\( T^{20} + 296 T^{18} + \cdots + 67108864 \) T^20 + 296T^18 + 36704T^16 + 2481088T^14 + 99666752T^12 + 2435392000T^10 + 35701392384T^8 + 299085889536T^6 + 1277051011072T^4 + 2030387068928*T^2 + 67108864
$47$	\( (T^{10} - 4 T^{9} + \cdots - 497096704)^{2} \) (T^10 - 4T^9 - 368T^8 + 1144T^7 + 48888T^6 - 105024T^5 - 2831296T^4 + 3503488T^3 + 67217024T^2 - 43328512*T - 497096704)^2
$53$	\( T^{20} + \cdots + 91447449849856 \) T^20 + 524T^18 + 110318T^16 + 12264884T^14 + 792514057T^12 + 30786746328T^10 + 718112310640T^8 + 9718317236480T^6 + 69842501019392T^4 + 212846527469568*T^2 + 91447449849856
$59$	\( T^{20} + \cdots + 261369043615744 \) T^20 + 680T^18 + 188784T^16 + 27661056T^14 + 2317690624T^12 + 113266599936T^10 + 3188745146368T^8 + 49441579466752T^6 + 383236351459328T^4 + 1146807597924352*T^2 + 261369043615744
$61$	\( T^{20} + \cdots + 318928165273600 \) T^20 + 700T^18 + 198278T^16 + 29428604T^14 + 2487578753T^12 + 122221770432T^10 + 3413678279680T^8 + 50174489395200T^6 + 328190517051392T^4 + 799345154719744*T^2 + 318928165273600
$67$	\( T^{20} + \cdots + 16\!\cdots\!00 \) T^20 + 768T^18 + 243616T^16 + 41233152T^14 + 4027226368T^12 + 230813331456T^10 + 7638310928384T^8 + 138139767668736T^6 + 1213395432898560T^4 + 4179949816643584*T^2 + 1691170216345600
$71$	\( (T^{10} - 4 T^{9} + \cdots + 104160256)^{2} \) (T^10 - 4T^9 - 458T^8 + 1344T^7 + 73173T^6 - 211540T^5 - 5000852T^4 + 16685376T^3 + 124314240T^2 - 506847232*T + 104160256)^2
$73$	\( (T^{10} - 20 T^{9} + \cdots + 4096)^{2} \) (T^10 - 20T^9 - 34T^8 + 1984T^7 - 7720T^6 - 19936T^5 + 172832T^4 - 354816T^3 + 284672T^2 - 83968*T + 4096)^2
$79$	\( (T^{10} - 4 T^{9} + \cdots - 1678716928)^{2} \) (T^10 - 4T^9 - 486T^8 + 1684T^7 + 80225T^6 - 246088T^5 - 5490064T^4 + 13203712T^3 + 159068928T^2 - 208398336*T - 1678716928)^2
$83$	\( T^{20} + \cdots + 35\!\cdots\!24 \) T^20 + 1256T^18 + 673936T^16 + 201685504T^14 + 36801587200T^12 + 4192680083456T^10 + 291809364869120T^8 + 11456284758900736T^6 + 205514705039523840T^4 + 694519625777414144*T^2 + 35204854787866624
$89$	\( (T^{10} - 28 T^{9} + \cdots - 2344371904)^{2} \) (T^10 - 28T^9 - 238T^8 + 12520T^7 - 44507T^6 - 1518604T^5 + 12821260T^4 + 25549152T^3 - 601358416T^2 + 2145673536*T - 2344371904)^2
$97$	\( (T^{10} + 24 T^{9} + \cdots + 176269088)^{2} \) (T^10 + 24T^9 - 168T^8 - 7380T^7 - 16251T^6 + 653732T^5 + 3276846T^4 - 16266912T^3 - 106951848T^2 - 11190496*T + 176269088)^2
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