LMFDB - Newform orbit 4320.2.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4320,2,Mod(431,4320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4320.431"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,-16,0,0,0,0,0,0,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + x^{14} + 2 x^{13} - 6 x^{12} + 8 x^{11} - 20 x^{9} + 44 x^{8} - 40 x^{7} + \cdots + 256 $ x^16 - 2x^15 + x^14 + 2x^13 - 6x^12 + 8x^11 - 20x^9 + 44x^8 - 40x^7 + 64x^5 - 96x^4 + 64x^3 + 64x^2 - 256x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

$f(q)$	$=$	$ q - q^{5} + \beta_1 q^{7} + \beta_{8} q^{11} - \beta_{13} q^{13} + \beta_{2} q^{17} - \beta_{6} q^{19} - \beta_{11} q^{23} + q^{25} + ( - \beta_{14} + \beta_{6}) q^{29} + ( - \beta_{15} - \beta_{7}) q^{31}+ \cdots + (\beta_{14} - \beta_{12} - \beta_{11} + \cdots + 2) q^{97}+O(q^{100}) $ q - q^5 + b1 * q^7 + b8 * q^11 - b13 * q^13 + b2 * q^17 - b6 * q^19 - b11 * q^23 + q^25 + (-b14 + b6) * q^29 + (-b15 - b7) * q^31 - b1 * q^35 + (b15 - b1) * q^37 + (b8 - b7 - b3 + b1) * q^41 + (-b14 - b12 + b10 + b6 + b4) * q^43 + (-b10 - b6 + b5 + 2) * q^47 + (b14 + b11 + b6 - b5 - 3) * q^49 + (b14 + b12 - b10 + b5 - b4 - 1) * q^53 - b8 * q^55 + (b15 + 2b13 + b8 - b1) q^59 + (b13 - b9 - b1) * q^61 + b13 * q^65 + (-b14 - b4 - 2) * q^67 + (-b14 - b11 - b10 - b5 + b4 + 4) * q^71 + (b14 + b12 + b10 + b5 - b4 - 2) * q^73 + (b12 - b10 + b6 - b5 + b4) * q^77 + (b13 + b9 + 2b7 + b3 - b2) q^79 + (-b15 + 2b13 + b8 + b7) q^83 - b2 * q^85 + (b13 + b9 + b2 - b1) * q^89 + (-b14 + b12 - b11 + b6 + b5 + b4 + 2) * q^91 + b6 * q^95 + (b14 - b12 - b11 - b10 - 2b6 - b4 + 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} - 4 q^{19} - 4 q^{23} + 16 q^{25} + 16 q^{47} - 32 q^{49} - 16 q^{53} - 32 q^{67} + 48 q^{71} - 16 q^{73} + 24 q^{91} + 4 q^{95} + 16 q^{97}+O(q^{100}) $ 16 * q - 16 * q^5 - 4 * q^19 - 4 * q^23 + 16 * q^25 + 16 * q^47 - 32 * q^49 - 16 * q^53 - 32 * q^67 + 48 * q^71 - 16 * q^73 + 24 * q^91 + 4 * q^95 + 16 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2 x^{15} + x^{14} + 2 x^{13} - 6 x^{12} + 8 x^{11} - 20 x^{9} + 44 x^{8} - 40 x^{7} + \cdots + 256 $ x^16 - 2*x^15 + x^14 + 2*x^13 - 6*x^12 + 8*x^11 - 20*x^9 + 44*x^8 - 40*x^7 + 64*x^5 - 96*x^4 + 64*x^3 + 64*x^2 - 256*x + 256 :

$\beta_{1}$	$=$	$ ( - \nu^{15} + 6 \nu^{14} + 3 \nu^{13} - 14 \nu^{12} + 34 \nu^{11} - 32 \nu^{10} - 96 \nu^{9} + \cdots + 384 ) / 896 $ (-v^15 + 6v^14 + 3v^13 - 14v^12 + 34v^11 - 32v^10 - 96v^9 + 68v^8 + 20v^7 + 72v^6 + 272v^5 - 480v^4 + 448v^3 - 64v^2 - 704v + 384) / 896
$\beta_{2}$	$=$	$ ( - \nu^{15} + 13 \nu^{14} - 11 \nu^{13} - 7 \nu^{12} + 20 \nu^{11} - 74 \nu^{10} + 44 \nu^{9} + \cdots - 960 ) / 448 $ (-v^15 + 13v^14 - 11v^13 - 7v^12 + 20v^11 - 74v^10 + 44v^9 - 44v^8 - 64v^7 + 268v^6 - 232v^5 + 304v^4 + 112v^3 - 736v^2 + 640v - 960) / 448
$\beta_{3}$	$=$	$ ( - \nu^{15} + 13 \nu^{14} + 3 \nu^{13} - 7 \nu^{12} + 34 \nu^{11} + 38 \nu^{10} - 40 \nu^{9} + \cdots + 1728 ) / 448 $ (-v^15 + 13v^14 + 3v^13 - 7v^12 + 34v^11 + 38v^10 - 40v^9 + 180v^8 - 64v^7 + 100v^6 + 272v^5 + 80v^4 + 336v^3 + 832v^2 - 704v + 1728) / 448
$\beta_{4}$	$=$	$ ( \nu^{14} - 3 \nu^{13} + 5 \nu^{12} - 3 \nu^{11} - 2 \nu^{10} + 10 \nu^{9} - 16 \nu^{8} + 4 \nu^{7} + \cdots - 96 ) / 32 $ (v^14 - 3v^13 + 5v^12 - 3v^11 - 2v^10 + 10v^9 - 16v^8 + 4v^7 + 24v^6 - 60v^5 + 80v^4 - 32v^3 - 48v^2 + 96*v - 96) / 32
$\beta_{5}$	$=$	$ ( - \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 6 \nu^{12} - 6 \nu^{11} - 16 \nu^{10} + 16 \nu^{9} + \cdots - 64 \nu ) / 128 $ (-v^15 + 2v^14 - 5v^13 + 6v^12 - 6v^11 - 16v^10 + 16v^9 + 20v^8 + 4v^7 + 24v^6 - 112v^5 + 64v^4 - 64v^2 - 64*v) / 128
$\beta_{6}$	$=$	$ ( - \nu^{15} - 2 \nu^{14} + 7 \nu^{13} - 6 \nu^{12} + 6 \nu^{11} - 24 \nu^{9} + 36 \nu^{8} - 12 \nu^{7} + \cdots + 256 ) / 128 $ (-v^15 - 2v^14 + 7v^13 - 6v^12 + 6v^11 - 24v^9 + 36v^8 - 12v^7 - 72v^6 + 96v^5 - 96v^4 + 128v^3 + 128v^2 - 320*v + 256) / 128
$\beta_{7}$	$=$	$ ( 9 \nu^{15} - 12 \nu^{14} + \nu^{13} - 54 \nu^{11} + 92 \nu^{10} - 88 \nu^{9} - 52 \nu^{8} + 212 \nu^{7} + \cdots - 768 ) / 896 $ (9v^15 - 12v^14 + v^13 - 54v^11 + 92v^10 - 88v^9 - 52v^8 + 212v^7 - 368v^6 + 128v^5 + 288v^4 - 672v^3 + 1024v^2 - 832*v - 768) / 896
$\beta_{8}$	$=$	$ ( 9 \nu^{15} + 2 \nu^{14} + \nu^{13} + 14 \nu^{12} - 54 \nu^{11} + 8 \nu^{10} + 24 \nu^{9} - 52 \nu^{8} + \cdots + 128 ) / 896 $ (9v^15 + 2v^14 + v^13 + 14v^12 - 54v^11 + 8v^10 + 24v^9 - 52v^8 + 44v^7 + 136v^6 + 128v^5 + 512v^4 + 128v^2 - 832*v + 128) / 896
$\beta_{9}$	$=$	$ ( - 13 \nu^{15} + 36 \nu^{14} - 17 \nu^{13} - 56 \nu^{12} + 162 \nu^{11} - 220 \nu^{10} + 320 \nu^{9} + \cdots + 512 ) / 896 $ (-13v^15 + 36v^14 - 17v^13 - 56v^12 + 162v^11 - 220v^10 + 320v^9 + 100v^8 - 580v^7 + 544v^6 - 48v^5 - 864v^4 + 2912v^3 - 3520v^2 + 2496*v + 512) / 896
$\beta_{10}$	$=$	$ ( - \nu^{15} + 2 \nu^{14} - \nu^{13} - 2 \nu^{12} + 6 \nu^{11} - 8 \nu^{10} + 20 \nu^{8} - 44 \nu^{7} + \cdots + 256 ) / 64 $ (-v^15 + 2v^14 - v^13 - 2v^12 + 6v^11 - 8v^10 + 20v^8 - 44v^7 + 40v^6 - 64v^4 + 96v^3 - 64v^2 + 64*v + 256) / 64
$\beta_{11}$	$=$	$ ( - \nu^{15} + \nu^{13} - 4 \nu^{12} + 4 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} - 4 \nu^{7} + 8 \nu^{6} + \cdots + 64 ) / 64 $ (-v^15 + v^13 - 4v^12 + 4v^10 - 4v^9 + 4v^8 - 4v^7 + 8v^6 + 8v^5 - 32v^4 - 32*v^2 + 64) / 64
$\beta_{12}$	$=$	$ ( - \nu^{15} - 2 \nu^{14} + \nu^{13} - 6 \nu^{12} + 8 \nu^{11} - 12 \nu^{9} + 20 \nu^{8} - 28 \nu^{7} + \cdots + 192 ) / 64 $ (-v^15 - 2v^14 + v^13 - 6v^12 + 8v^11 - 12v^9 + 20v^8 - 28v^7 + 72v^5 - 96v^4 + 64v^3 - 96v^2 - 128*v + 192) / 64
$\beta_{13}$	$=$	$ ( - 23 \nu^{15} + 26 \nu^{14} - 15 \nu^{13} - 42 \nu^{12} + 138 \nu^{11} - 120 \nu^{10} - 24 \nu^{9} + \cdots + 3456 ) / 896 $ (-23v^15 + 26v^14 - 15v^13 - 42v^12 + 138v^11 - 120v^10 - 24v^9 + 332v^8 - 660v^7 + 424v^6 - 128v^5 - 1408v^4 + 1344v^3 - 1024v^2 - 1856*v + 3456) / 896
$\beta_{14}$	$=$	$ ( - \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 2 \nu^{12} + \nu^{11} - 4 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + \cdots + 96 ) / 32 $ (-v^15 + 2v^14 - 2v^13 + 2v^12 + v^11 - 4v^10 + 2v^9 + 4v^8 - 20v^7 + 20v^6 - 20v^5 + 16v^4 + 48v^2 - 32v + 96) / 32
$\beta_{15}$	$=$	$ ( 27 \nu^{15} - 22 \nu^{14} - 25 \nu^{13} + 70 \nu^{12} - 78 \nu^{11} + 24 \nu^{10} + 128 \nu^{9} + \cdots - 4992 ) / 896 $ (27v^15 - 22v^14 - 25v^13 + 70v^12 - 78v^11 + 24v^10 + 128v^9 - 492v^8 + 692v^7 + 184v^6 - 848v^5 + 1312v^4 - 448v^3 + 832v^2 + 2880*v - 4992) / 896

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

$\nu$	$=$	$ ( -\beta_{13} + \beta_{10} - \beta_{8} ) / 4 $ (-b13 + b10 - b8) / 4
$\nu^{2}$	$=$	$ ( \beta_{15} + \beta_{14} - \beta_{4} - \beta_1 ) / 4 $ (b15 + b14 - b4 - b1) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 2\beta_{7} + 2\beta_{6} + \beta_{4} - \beta_{2} + \beta _1 - 2 ) / 4 $ (b15 + b11 + b10 + b9 + b8 + 2b7 + 2b6 + b4 - b2 + b1 - 2) / 4
$\nu^{4}$	$=$	$ ( -\beta_{13} + \beta_{12} + \beta_{7} + 2\beta_{6} + \beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 2 ) / 4 $ (-b13 + b12 + b7 + 2b6 + b4 + b3 + 2b2 - 2*b1 + 2) / 4
$\nu^{5}$	$=$	$ ( - \beta_{15} + 2 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 2 ) / 4 $ (-b15 + 2b14 - 4b13 + 2b12 - b11 - b10 + b9 + b8 - 2b7 - 2b6 - b4 - b2 + 3b1 - 2) / 4
$\nu^{6}$	$=$	$ ( 2 \beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{11} - 2 \beta_{9} + 4 \beta_{8} - 7 \beta_{7} + \cdots - 6 ) / 4 $ (2b15 + b13 + b12 + 2b11 - 2b9 + 4b8 - 7b7 - 2b6 - b4 + b3 - 6) / 4
$\nu^{7}$	$=$	$ ( \beta_{15} - 2 \beta_{13} + 3 \beta_{11} - 5 \beta_{10} + \beta_{9} - 3 \beta_{8} - 4 \beta_{7} + \cdots + 14 ) / 4 $ (b15 - 2b13 + 3b11 - 5b10 + b9 - 3b8 - 4b7 - 2b6 + 4b5 - b4 + 2b3 - b2 + 9*b1 + 14) / 4
$\nu^{8}$	$=$	$ ( - 2 \beta_{15} - 4 \beta_{14} - 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + \cdots - 22 ) / 4 $ (-2b15 - 4b14 - 3b13 + b12 - 2b11 + 4b10 - 2b9 - 4b8 - 3b7 - 2b6 + 8b5 - b4 + 5b3 - 4b1 - 22) / 4
$\nu^{9}$	$=$	$ ( \beta_{15} + 4 \beta_{14} + 2 \beta_{13} - \beta_{11} - 13 \beta_{10} + 5 \beta_{9} + 5 \beta_{8} + \cdots + 6 ) / 4 $ (b15 + 4b14 + 2b13 - b11 - 13b10 + 5b9 + 5b8 - 12b7 - 10b6 + 4b5 - 9b4 + 2b3 - 5b2 - 15b1 + 6) / 4
$\nu^{10}$	$=$	$ ( - 2 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} - 3 \beta_{12} + 10 \beta_{11} + 2 \beta_{9} + 9 \beta_{7} + \cdots - 38 ) / 4 $ (-2b15 - 8b14 + 9b13 - 3b12 + 10b11 + 2b9 + 9b7 - 2b6 - 8b5 + 11b4 + 9b3 - 12b2 - 4*b1 - 38) / 4
$\nu^{11}$	$=$	$ ( \beta_{15} - 4 \beta_{14} + 14 \beta_{13} + 4 \beta_{12} - 17 \beta_{11} - 21 \beta_{10} - 3 \beta_{9} + \cdots + 30 ) / 4 $ (b15 - 4b14 + 14b13 + 4b12 - 17b11 - 21b10 - 3b9 - 27b8 - 16b7 - 2b6 - 12b5 + 3b4 + 14b3 + 11b2 - 7b1 + 30) / 4
$\nu^{12}$	$=$	$ ( - 14 \beta_{15} - 8 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} - 26 \beta_{11} + 4 \beta_{10} - 10 \beta_{9} + \cdots - 78 ) / 4 $ (-14b15 - 8b14 - 7b13 - 3b12 - 26b11 + 4b10 - 10b9 - 4b8 - 55b7 + 6b6 - 8b5 + 23b4 + b3 - 32b2 + 32b1 - 78) / 4
$\nu^{13}$	$=$	$ ( - 15 \beta_{15} - 28 \beta_{14} + 18 \beta_{13} - 40 \beta_{12} + 7 \beta_{11} - 5 \beta_{10} + \cdots - 42 ) / 4 $ (-15b15 - 28b14 + 18b13 - 40b12 + 7b11 - 5b10 - 27b9 + 21b8 - 108b7 + 38b6 - 28b5 + 7b4 + 2b3 + 11b2 - 31*b1 - 42) / 4
$\nu^{14}$	$=$	$ ( - 50 \beta_{15} - 8 \beta_{14} + \beta_{13} - 67 \beta_{12} + 10 \beta_{11} - 8 \beta_{10} + 18 \beta_{9} + \cdots - 6 ) / 4 $ (-50b15 - 8b14 + b13 - 67b12 + 10b11 - 8b10 + 18b9 - 8b8 + 41b7 - 18b6 - 24b5 + 11b4 + 9b3 + 4b2 + 92b1 - 6) / 4
$\nu^{15}$	$=$	$ ( - 7 \beta_{15} - 76 \beta_{14} + 78 \beta_{13} - 44 \beta_{12} - 113 \beta_{11} + 59 \beta_{10} + \cdots + 78 ) / 4 $ (-7b15 - 76b14 + 78b13 - 44b12 - 113b11 + 59b10 - 19b9 + 53b8 + 96b7 - 50b6 - 28b5 - 21b4 + 14b3 + 43b2 - 47*b1 + 78) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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} $

431.1

−0.0685300	+	1.41255i
0.963606	+	1.03512i
0.527497	−	1.31215i
−1.34868	−	0.425523i
−1.31385	−	0.523269i
−0.427379	−	1.34809i
1.29231	−	0.574397i
1.37502	−	0.330644i
1.37502	+	0.330644i
1.29231	+	0.574397i
−0.427379	+	1.34809i
−1.31385	+	0.523269i
−1.34868	+	0.425523i
0.527497	+	1.31215i
0.963606	−	1.03512i
−0.0685300	−	1.41255i

−1.00000

−

5.03977i

431.2

−1.00000

−

4.74492i

431.3

−1.00000

−

2.90264i

431.4

−1.00000

−

2.46349i

431.5

−1.00000

−

2.25998i

431.6

−1.00000

−

1.58153i

431.7

−1.00000

−

1.39827i

431.8

−1.00000

−

0.168502i

431.9

−1.00000

0.168502i

431.10

−1.00000

1.39827i

431.11

−1.00000

1.58153i

431.12

−1.00000

2.25998i

431.13

−1.00000

2.46349i

431.14

−1.00000

2.90264i

431.15

−1.00000

4.74492i

431.16

−1.00000

5.03977i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4320.2.b.a		16
3.b	odd	2	1		4320.2.b.c		16
4.b	odd	2	1		1080.2.b.b	✓	16
8.b	even	2	1		1080.2.b.c	yes	16
8.d	odd	2	1		4320.2.b.c		16
12.b	even	2	1		1080.2.b.c	yes	16
24.f	even	2	1	inner	4320.2.b.a		16
24.h	odd	2	1		1080.2.b.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1080.2.b.b	✓	16	4.b	odd	2	1
1080.2.b.b	✓	16	24.h	odd	2	1
1080.2.b.c	yes	16	8.b	even	2	1
1080.2.b.c	yes	16	12.b	even	2	1
4320.2.b.a		16	1.a	even	1	1	trivial
4320.2.b.a		16	24.f	even	2	1	inner
4320.2.b.c		16	3.b	odd	2	1
4320.2.b.c		16	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4320, [\chi])$:

$ T_{7}^{16} + 72 T_{7}^{14} + 1944 T_{7}^{12} + 25144 T_{7}^{10} + 170640 T_{7}^{8} + 614304 T_{7}^{6} + \cdots + 20736 $

T7^16 + 72*T7^14 + 1944*T7^12 + 25144*T7^10 + 170640*T7^8 + 614304*T7^6 + 1094032*T7^4 + 760896*T7^2 + 20736

$ T_{23}^{8} + 2T_{23}^{7} - 90T_{23}^{6} - 78T_{23}^{5} + 1544T_{23}^{4} + 1126T_{23}^{3} - 7622T_{23}^{2} - 3642T_{23} + 10551 $

T23^8 + 2*T23^7 - 90*T23^6 - 78*T23^5 + 1544*T23^4 + 1126*T23^3 - 7622*T23^2 - 3642*T23 + 10551

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T + 1)^{16} $ (T + 1)^16
$7$	$ T^{16} + 72 T^{14} + \cdots + 20736 $ T^16 + 72T^14 + 1944T^12 + 25144T^10 + 170640T^8 + 614304T^6 + 1094032T^4 + 760896*T^2 + 20736
$11$	$ T^{16} + 96 T^{14} + \cdots + 135424 $ T^16 + 96T^14 + 3648T^12 + 69224T^10 + 678944T^8 + 3216192T^6 + 6138640T^4 + 1940288*T^2 + 135424
$13$	$ T^{16} + 104 T^{14} + \cdots + 2304 $ T^16 + 104T^14 + 3456T^12 + 40104T^10 + 162368T^8 + 257728T^6 + 157072T^4 + 36672*T^2 + 2304
$17$	$ T^{16} + \cdots + 121506529 $ T^16 + 152T^14 + 8884T^12 + 254064T^10 + 3779174T^8 + 29051864T^6 + 111898788T^4 + 197308032*T^2 + 121506529
$19$	$ (T^{8} + 2 T^{7} + \cdots + 1231)^{2} $ (T^8 + 2T^7 - 62T^6 - 210T^5 + 572T^4 + 3710T^3 + 6498T^2 + 4722*T + 1231)^2
$23$	$ (T^{8} + 2 T^{7} + \cdots + 10551)^{2} $ (T^8 + 2T^7 - 90T^6 - 78T^5 + 1544T^4 + 1126T^3 - 7622T^2 - 3642*T + 10551)^2
$29$	$ (T^{8} - 128 T^{6} + \cdots - 75408)^{2} $ (T^8 - 128T^6 - 152T^5 + 4348T^4 + 9392T^3 - 41876T^2 - 134280T - 75408)^2
$31$	$ T^{16} + \cdots + 1270851201 $ T^16 + 232T^14 + 20468T^12 + 869664T^10 + 18622086T^8 + 194547080T^6 + 907934884T^4 + 1814419152*T^2 + 1270851201
$37$	$ T^{16} + \cdots + 7398752256 $ T^16 + 216T^14 + 19216T^12 + 918016T^10 + 25538560T^8 + 415465472T^6 + 3692560384T^4 + 14407434240*T^2 + 7398752256
$41$	$ T^{16} + \cdots + 62148495616 $ T^16 + 392T^14 + 58744T^12 + 4239576T^10 + 153913424T^8 + 2776912160T^6 + 23979313680T^4 + 85387005504*T^2 + 62148495616
$43$	$ (T^{8} - 216 T^{6} + \cdots + 4117456)^{2} $ (T^8 - 216T^6 + 44T^5 + 15548T^4 - 2976T^3 - 437516T^2 + 39608T + 4117456)^2
$47$	$ (T^{8} - 8 T^{7} + \cdots + 106752)^{2} $ (T^8 - 8T^7 - 144T^6 + 960T^5 + 6560T^4 - 24160T^3 - 125072T^2 - 38592*T + 106752)^2
$53$	$ (T^{8} + 8 T^{7} + \cdots + 3362241)^{2} $ (T^8 + 8T^7 - 224T^6 - 1400T^5 + 16690T^4 + 73560T^3 - 473684T^2 - 1132728*T + 3362241)^2
$59$	$ T^{16} + 440 T^{14} + \cdots + 9339136 $ T^16 + 440T^14 + 71296T^12 + 5256456T^10 + 179703872T^8 + 2765050304T^6 + 16207649808T^4 + 17295146304*T^2 + 9339136
$61$	$ T^{16} + \cdots + 17225775009 $ T^16 + 392T^14 + 50268T^12 + 3017592T^10 + 93106022T^8 + 1450652344T^6 + 10205919004T^4 + 26702699976*T^2 + 17225775009
$67$	$ (T^{8} + 16 T^{7} + \cdots - 118784)^{2} $ (T^8 + 16T^7 - 124T^6 - 2704T^5 - 4608T^4 + 67072T^3 + 223744T^2 - 18432*T - 118784)^2
$71$	$ (T^{8} - 24 T^{7} + \cdots + 2680272)^{2} $ (T^8 - 24T^7 - 56T^6 + 4660T^5 - 23936T^4 - 131248T^3 + 1356508T^2 - 3476040*T + 2680272)^2
$73$	$ (T^{8} + 8 T^{7} + \cdots - 36272)^{2} $ (T^8 + 8T^7 - 220T^6 - 2720T^5 - 8232T^4 + 8384T^3 + 66052T^2 + 49656*T - 36272)^2
$79$	$ T^{16} + \cdots + 19\!\cdots\!29 $ T^16 + 960T^14 + 378964T^12 + 79613560T^10 + 9615632854T^8 + 671301521264T^6 + 25608502345156T^4 + 450645038441016*T^2 + 1992981645409329
$83$	$ T^{16} + \cdots + 18891677809 $ T^16 + 664T^14 + 157868T^12 + 16312808T^10 + 729982998T^8 + 12199988008T^6 + 52057753420T^4 + 62038269912*T^2 + 18891677809
$89$	$ T^{16} + \cdots + 51758070016 $ T^16 + 480T^14 + 79288T^12 + 5776600T^10 + 192538032T^8 + 2949145760T^6 + 20240694800T^4 + 58461394240*T^2 + 51758070016
$97$	$ (T^{8} - 8 T^{7} + \cdots + 8676352)^{2} $ (T^8 - 8T^7 - 420T^6 + 1568T^5 + 51760T^4 - 42240T^3 - 2089408T^2 - 3092992*T + 8676352)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - q^{5} + \beta_1 q^{7} + \beta_{8} q^{11} - \beta_{13} q^{13} + \beta_{2} q^{17} - \beta_{6} q^{19} - \beta_{11} q^{23} + q^{25} + ( - \beta_{14} + \beta_{6}) q^{29} + ( - \beta_{15} - \beta_{7}) q^{31}+ \cdots + (\beta_{14} - \beta_{12} - \beta_{11} + \cdots + 2) q^{97}+O(q^{100}) \) q - q^5 + b1 * q^7 + b8 * q^11 - b13 * q^13 + b2 * q^17 - b6 * q^19 - b11 * q^23 + q^25 + (-b14 + b6) * q^29 + (-b15 - b7) * q^31 - b1 * q^35 + (b15 - b1) * q^37 + (b8 - b7 - b3 + b1) * q^41 + (-b14 - b12 + b10 + b6 + b4) * q^43 + (-b10 - b6 + b5 + 2) * q^47 + (b14 + b11 + b6 - b5 - 3) * q^49 + (b14 + b12 - b10 + b5 - b4 - 1) * q^53 - b8 * q^55 + (b15 + 2b13 + b8 - b1) q^59 + (b13 - b9 - b1) * q^61 + b13 * q^65 + (-b14 - b4 - 2) * q^67 + (-b14 - b11 - b10 - b5 + b4 + 4) * q^71 + (b14 + b12 + b10 + b5 - b4 - 2) * q^73 + (b12 - b10 + b6 - b5 + b4) * q^77 + (b13 + b9 + 2b7 + b3 - b2) q^79 + (-b15 + 2b13 + b8 + b7) q^83 - b2 * q^85 + (b13 + b9 + b2 - b1) * q^89 + (-b14 + b12 - b11 + b6 + b5 + b4 + 2) * q^91 + b6 * q^95 + (b14 - b12 - b11 - b10 - 2b6 - b4 + 2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 4 q^{19} - 4 q^{23} + 16 q^{25} + 16 q^{47} - 32 q^{49} - 16 q^{53} - 32 q^{67} + 48 q^{71} - 16 q^{73} + 24 q^{91} + 4 q^{95} + 16 q^{97}+O(q^{100}) \) 16 * q - 16 * q^5 - 4 * q^19 - 4 * q^23 + 16 * q^25 + 16 * q^47 - 32 * q^49 - 16 * q^53 - 32 * q^67 + 48 * q^71 - 16 * q^73 + 24 * q^91 + 4 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	4320.2.b.a

Level	$4320$
Weight	$2$
Character orbit	4320.b
Analytic conductor	$34.495$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(34.4953736732\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 2 x^{15} + x^{14} + 2 x^{13} - 6 x^{12} + 8 x^{11} - 20 x^{9} + 44 x^{8} - 40 x^{7} + \cdots + 256 \) x^16 - 2x^15 + x^14 + 2x^13 - 6x^12 + 8x^11 - 20x^9 + 44x^8 - 40x^7 + 64x^5 - 96x^4 + 64x^3 + 64x^2 - 256x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{20} \)
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - \nu^{15} + 6 \nu^{14} + 3 \nu^{13} - 14 \nu^{12} + 34 \nu^{11} - 32 \nu^{10} - 96 \nu^{9} + \cdots + 384 ) / 896 \) (-v^15 + 6v^14 + 3v^13 - 14v^12 + 34v^11 - 32v^10 - 96v^9 + 68v^8 + 20v^7 + 72v^6 + 272v^5 - 480v^4 + 448v^3 - 64v^2 - 704v + 384) / 896
\(\beta_{2}\)	\(=\)	\( ( - \nu^{15} + 13 \nu^{14} - 11 \nu^{13} - 7 \nu^{12} + 20 \nu^{11} - 74 \nu^{10} + 44 \nu^{9} + \cdots - 960 ) / 448 \) (-v^15 + 13v^14 - 11v^13 - 7v^12 + 20v^11 - 74v^10 + 44v^9 - 44v^8 - 64v^7 + 268v^6 - 232v^5 + 304v^4 + 112v^3 - 736v^2 + 640v - 960) / 448
\(\beta_{3}\)	\(=\)	\( ( - \nu^{15} + 13 \nu^{14} + 3 \nu^{13} - 7 \nu^{12} + 34 \nu^{11} + 38 \nu^{10} - 40 \nu^{9} + \cdots + 1728 ) / 448 \) (-v^15 + 13v^14 + 3v^13 - 7v^12 + 34v^11 + 38v^10 - 40v^9 + 180v^8 - 64v^7 + 100v^6 + 272v^5 + 80v^4 + 336v^3 + 832v^2 - 704v + 1728) / 448
\(\beta_{4}\)	\(=\)	\( ( \nu^{14} - 3 \nu^{13} + 5 \nu^{12} - 3 \nu^{11} - 2 \nu^{10} + 10 \nu^{9} - 16 \nu^{8} + 4 \nu^{7} + \cdots - 96 ) / 32 \) (v^14 - 3v^13 + 5v^12 - 3v^11 - 2v^10 + 10v^9 - 16v^8 + 4v^7 + 24v^6 - 60v^5 + 80v^4 - 32v^3 - 48v^2 + 96*v - 96) / 32
\(\beta_{5}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 6 \nu^{12} - 6 \nu^{11} - 16 \nu^{10} + 16 \nu^{9} + \cdots - 64 \nu ) / 128 \) (-v^15 + 2v^14 - 5v^13 + 6v^12 - 6v^11 - 16v^10 + 16v^9 + 20v^8 + 4v^7 + 24v^6 - 112v^5 + 64v^4 - 64v^2 - 64*v) / 128
\(\beta_{6}\)	\(=\)	\( ( - \nu^{15} - 2 \nu^{14} + 7 \nu^{13} - 6 \nu^{12} + 6 \nu^{11} - 24 \nu^{9} + 36 \nu^{8} - 12 \nu^{7} + \cdots + 256 ) / 128 \) (-v^15 - 2v^14 + 7v^13 - 6v^12 + 6v^11 - 24v^9 + 36v^8 - 12v^7 - 72v^6 + 96v^5 - 96v^4 + 128v^3 + 128v^2 - 320*v + 256) / 128
\(\beta_{7}\)	\(=\)	\( ( 9 \nu^{15} - 12 \nu^{14} + \nu^{13} - 54 \nu^{11} + 92 \nu^{10} - 88 \nu^{9} - 52 \nu^{8} + 212 \nu^{7} + \cdots - 768 ) / 896 \) (9v^15 - 12v^14 + v^13 - 54v^11 + 92v^10 - 88v^9 - 52v^8 + 212v^7 - 368v^6 + 128v^5 + 288v^4 - 672v^3 + 1024v^2 - 832*v - 768) / 896
\(\beta_{8}\)	\(=\)	\( ( 9 \nu^{15} + 2 \nu^{14} + \nu^{13} + 14 \nu^{12} - 54 \nu^{11} + 8 \nu^{10} + 24 \nu^{9} - 52 \nu^{8} + \cdots + 128 ) / 896 \) (9v^15 + 2v^14 + v^13 + 14v^12 - 54v^11 + 8v^10 + 24v^9 - 52v^8 + 44v^7 + 136v^6 + 128v^5 + 512v^4 + 128v^2 - 832*v + 128) / 896
\(\beta_{9}\)	\(=\)	\( ( - 13 \nu^{15} + 36 \nu^{14} - 17 \nu^{13} - 56 \nu^{12} + 162 \nu^{11} - 220 \nu^{10} + 320 \nu^{9} + \cdots + 512 ) / 896 \) (-13v^15 + 36v^14 - 17v^13 - 56v^12 + 162v^11 - 220v^10 + 320v^9 + 100v^8 - 580v^7 + 544v^6 - 48v^5 - 864v^4 + 2912v^3 - 3520v^2 + 2496*v + 512) / 896
\(\beta_{10}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{14} - \nu^{13} - 2 \nu^{12} + 6 \nu^{11} - 8 \nu^{10} + 20 \nu^{8} - 44 \nu^{7} + \cdots + 256 ) / 64 \) (-v^15 + 2v^14 - v^13 - 2v^12 + 6v^11 - 8v^10 + 20v^8 - 44v^7 + 40v^6 - 64v^4 + 96v^3 - 64v^2 + 64*v + 256) / 64
\(\beta_{11}\)	\(=\)	\( ( - \nu^{15} + \nu^{13} - 4 \nu^{12} + 4 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} - 4 \nu^{7} + 8 \nu^{6} + \cdots + 64 ) / 64 \) (-v^15 + v^13 - 4v^12 + 4v^10 - 4v^9 + 4v^8 - 4v^7 + 8v^6 + 8v^5 - 32v^4 - 32*v^2 + 64) / 64
\(\beta_{12}\)	\(=\)	\( ( - \nu^{15} - 2 \nu^{14} + \nu^{13} - 6 \nu^{12} + 8 \nu^{11} - 12 \nu^{9} + 20 \nu^{8} - 28 \nu^{7} + \cdots + 192 ) / 64 \) (-v^15 - 2v^14 + v^13 - 6v^12 + 8v^11 - 12v^9 + 20v^8 - 28v^7 + 72v^5 - 96v^4 + 64v^3 - 96v^2 - 128*v + 192) / 64
\(\beta_{13}\)	\(=\)	\( ( - 23 \nu^{15} + 26 \nu^{14} - 15 \nu^{13} - 42 \nu^{12} + 138 \nu^{11} - 120 \nu^{10} - 24 \nu^{9} + \cdots + 3456 ) / 896 \) (-23v^15 + 26v^14 - 15v^13 - 42v^12 + 138v^11 - 120v^10 - 24v^9 + 332v^8 - 660v^7 + 424v^6 - 128v^5 - 1408v^4 + 1344v^3 - 1024v^2 - 1856*v + 3456) / 896
\(\beta_{14}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 2 \nu^{12} + \nu^{11} - 4 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + \cdots + 96 ) / 32 \) (-v^15 + 2v^14 - 2v^13 + 2v^12 + v^11 - 4v^10 + 2v^9 + 4v^8 - 20v^7 + 20v^6 - 20v^5 + 16v^4 + 48v^2 - 32v + 96) / 32
\(\beta_{15}\)	\(=\)	\( ( 27 \nu^{15} - 22 \nu^{14} - 25 \nu^{13} + 70 \nu^{12} - 78 \nu^{11} + 24 \nu^{10} + 128 \nu^{9} + \cdots - 4992 ) / 896 \) (27v^15 - 22v^14 - 25v^13 + 70v^12 - 78v^11 + 24v^10 + 128v^9 - 492v^8 + 692v^7 + 184v^6 - 848v^5 + 1312v^4 - 448v^3 + 832v^2 + 2880*v - 4992) / 896

\(\nu\)	\(=\)	\( ( -\beta_{13} + \beta_{10} - \beta_{8} ) / 4 \) (-b13 + b10 - b8) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} + \beta_{14} - \beta_{4} - \beta_1 ) / 4 \) (b15 + b14 - b4 - b1) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 2\beta_{7} + 2\beta_{6} + \beta_{4} - \beta_{2} + \beta _1 - 2 ) / 4 \) (b15 + b11 + b10 + b9 + b8 + 2b7 + 2b6 + b4 - b2 + b1 - 2) / 4
\(\nu^{4}\)	\(=\)	\( ( -\beta_{13} + \beta_{12} + \beta_{7} + 2\beta_{6} + \beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 2 ) / 4 \) (-b13 + b12 + b7 + 2b6 + b4 + b3 + 2b2 - 2*b1 + 2) / 4
\(\nu^{5}\)	\(=\)	\( ( - \beta_{15} + 2 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 2 ) / 4 \) (-b15 + 2b14 - 4b13 + 2b12 - b11 - b10 + b9 + b8 - 2b7 - 2b6 - b4 - b2 + 3b1 - 2) / 4
\(\nu^{6}\)	\(=\)	\( ( 2 \beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{11} - 2 \beta_{9} + 4 \beta_{8} - 7 \beta_{7} + \cdots - 6 ) / 4 \) (2b15 + b13 + b12 + 2b11 - 2b9 + 4b8 - 7b7 - 2b6 - b4 + b3 - 6) / 4
\(\nu^{7}\)	\(=\)	\( ( \beta_{15} - 2 \beta_{13} + 3 \beta_{11} - 5 \beta_{10} + \beta_{9} - 3 \beta_{8} - 4 \beta_{7} + \cdots + 14 ) / 4 \) (b15 - 2b13 + 3b11 - 5b10 + b9 - 3b8 - 4b7 - 2b6 + 4b5 - b4 + 2b3 - b2 + 9*b1 + 14) / 4
\(\nu^{8}\)	\(=\)	\( ( - 2 \beta_{15} - 4 \beta_{14} - 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + \cdots - 22 ) / 4 \) (-2b15 - 4b14 - 3b13 + b12 - 2b11 + 4b10 - 2b9 - 4b8 - 3b7 - 2b6 + 8b5 - b4 + 5b3 - 4b1 - 22) / 4
\(\nu^{9}\)	\(=\)	\( ( \beta_{15} + 4 \beta_{14} + 2 \beta_{13} - \beta_{11} - 13 \beta_{10} + 5 \beta_{9} + 5 \beta_{8} + \cdots + 6 ) / 4 \) (b15 + 4b14 + 2b13 - b11 - 13b10 + 5b9 + 5b8 - 12b7 - 10b6 + 4b5 - 9b4 + 2b3 - 5b2 - 15b1 + 6) / 4
\(\nu^{10}\)	\(=\)	\( ( - 2 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} - 3 \beta_{12} + 10 \beta_{11} + 2 \beta_{9} + 9 \beta_{7} + \cdots - 38 ) / 4 \) (-2b15 - 8b14 + 9b13 - 3b12 + 10b11 + 2b9 + 9b7 - 2b6 - 8b5 + 11b4 + 9b3 - 12b2 - 4*b1 - 38) / 4
\(\nu^{11}\)	\(=\)	\( ( \beta_{15} - 4 \beta_{14} + 14 \beta_{13} + 4 \beta_{12} - 17 \beta_{11} - 21 \beta_{10} - 3 \beta_{9} + \cdots + 30 ) / 4 \) (b15 - 4b14 + 14b13 + 4b12 - 17b11 - 21b10 - 3b9 - 27b8 - 16b7 - 2b6 - 12b5 + 3b4 + 14b3 + 11b2 - 7b1 + 30) / 4
\(\nu^{12}\)	\(=\)	\( ( - 14 \beta_{15} - 8 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} - 26 \beta_{11} + 4 \beta_{10} - 10 \beta_{9} + \cdots - 78 ) / 4 \) (-14b15 - 8b14 - 7b13 - 3b12 - 26b11 + 4b10 - 10b9 - 4b8 - 55b7 + 6b6 - 8b5 + 23b4 + b3 - 32b2 + 32b1 - 78) / 4
\(\nu^{13}\)	\(=\)	\( ( - 15 \beta_{15} - 28 \beta_{14} + 18 \beta_{13} - 40 \beta_{12} + 7 \beta_{11} - 5 \beta_{10} + \cdots - 42 ) / 4 \) (-15b15 - 28b14 + 18b13 - 40b12 + 7b11 - 5b10 - 27b9 + 21b8 - 108b7 + 38b6 - 28b5 + 7b4 + 2b3 + 11b2 - 31*b1 - 42) / 4
\(\nu^{14}\)	\(=\)	\( ( - 50 \beta_{15} - 8 \beta_{14} + \beta_{13} - 67 \beta_{12} + 10 \beta_{11} - 8 \beta_{10} + 18 \beta_{9} + \cdots - 6 ) / 4 \) (-50b15 - 8b14 + b13 - 67b12 + 10b11 - 8b10 + 18b9 - 8b8 + 41b7 - 18b6 - 24b5 + 11b4 + 9b3 + 4b2 + 92b1 - 6) / 4
\(\nu^{15}\)	\(=\)	\( ( - 7 \beta_{15} - 76 \beta_{14} + 78 \beta_{13} - 44 \beta_{12} - 113 \beta_{11} + 59 \beta_{10} + \cdots + 78 ) / 4 \) (-7b15 - 76b14 + 78b13 - 44b12 - 113b11 + 59b10 - 19b9 + 53b8 + 96b7 - 50b6 - 28b5 - 21b4 + 14b3 + 43b2 - 47*b1 + 78) / 4

\(n\)	\(2081\)	\(2431\)	\(3457\)	\(3781\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 431.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T + 1)^{16} \) (T + 1)^16
$7$	\( T^{16} + 72 T^{14} + \cdots + 20736 \) T^16 + 72T^14 + 1944T^12 + 25144T^10 + 170640T^8 + 614304T^6 + 1094032T^4 + 760896*T^2 + 20736
$11$	\( T^{16} + 96 T^{14} + \cdots + 135424 \) T^16 + 96T^14 + 3648T^12 + 69224T^10 + 678944T^8 + 3216192T^6 + 6138640T^4 + 1940288*T^2 + 135424
$13$	\( T^{16} + 104 T^{14} + \cdots + 2304 \) T^16 + 104T^14 + 3456T^12 + 40104T^10 + 162368T^8 + 257728T^6 + 157072T^4 + 36672*T^2 + 2304
$17$	\( T^{16} + \cdots + 121506529 \) T^16 + 152T^14 + 8884T^12 + 254064T^10 + 3779174T^8 + 29051864T^6 + 111898788T^4 + 197308032*T^2 + 121506529
$19$	\( (T^{8} + 2 T^{7} + \cdots + 1231)^{2} \) (T^8 + 2T^7 - 62T^6 - 210T^5 + 572T^4 + 3710T^3 + 6498T^2 + 4722*T + 1231)^2
$23$	\( (T^{8} + 2 T^{7} + \cdots + 10551)^{2} \) (T^8 + 2T^7 - 90T^6 - 78T^5 + 1544T^4 + 1126T^3 - 7622T^2 - 3642*T + 10551)^2
$29$	\( (T^{8} - 128 T^{6} + \cdots - 75408)^{2} \) (T^8 - 128T^6 - 152T^5 + 4348T^4 + 9392T^3 - 41876T^2 - 134280T - 75408)^2
$31$	\( T^{16} + \cdots + 1270851201 \) T^16 + 232T^14 + 20468T^12 + 869664T^10 + 18622086T^8 + 194547080T^6 + 907934884T^4 + 1814419152*T^2 + 1270851201
$37$	\( T^{16} + \cdots + 7398752256 \) T^16 + 216T^14 + 19216T^12 + 918016T^10 + 25538560T^8 + 415465472T^6 + 3692560384T^4 + 14407434240*T^2 + 7398752256
$41$	\( T^{16} + \cdots + 62148495616 \) T^16 + 392T^14 + 58744T^12 + 4239576T^10 + 153913424T^8 + 2776912160T^6 + 23979313680T^4 + 85387005504*T^2 + 62148495616
$43$	\( (T^{8} - 216 T^{6} + \cdots + 4117456)^{2} \) (T^8 - 216T^6 + 44T^5 + 15548T^4 - 2976T^3 - 437516T^2 + 39608T + 4117456)^2
$47$	\( (T^{8} - 8 T^{7} + \cdots + 106752)^{2} \) (T^8 - 8T^7 - 144T^6 + 960T^5 + 6560T^4 - 24160T^3 - 125072T^2 - 38592*T + 106752)^2
$53$	\( (T^{8} + 8 T^{7} + \cdots + 3362241)^{2} \) (T^8 + 8T^7 - 224T^6 - 1400T^5 + 16690T^4 + 73560T^3 - 473684T^2 - 1132728*T + 3362241)^2
$59$	\( T^{16} + 440 T^{14} + \cdots + 9339136 \) T^16 + 440T^14 + 71296T^12 + 5256456T^10 + 179703872T^8 + 2765050304T^6 + 16207649808T^4 + 17295146304*T^2 + 9339136
$61$	\( T^{16} + \cdots + 17225775009 \) T^16 + 392T^14 + 50268T^12 + 3017592T^10 + 93106022T^8 + 1450652344T^6 + 10205919004T^4 + 26702699976*T^2 + 17225775009
$67$	\( (T^{8} + 16 T^{7} + \cdots - 118784)^{2} \) (T^8 + 16T^7 - 124T^6 - 2704T^5 - 4608T^4 + 67072T^3 + 223744T^2 - 18432*T - 118784)^2
$71$	\( (T^{8} - 24 T^{7} + \cdots + 2680272)^{2} \) (T^8 - 24T^7 - 56T^6 + 4660T^5 - 23936T^4 - 131248T^3 + 1356508T^2 - 3476040*T + 2680272)^2
$73$	\( (T^{8} + 8 T^{7} + \cdots - 36272)^{2} \) (T^8 + 8T^7 - 220T^6 - 2720T^5 - 8232T^4 + 8384T^3 + 66052T^2 + 49656*T - 36272)^2
$79$	\( T^{16} + \cdots + 19\!\cdots\!29 \) T^16 + 960T^14 + 378964T^12 + 79613560T^10 + 9615632854T^8 + 671301521264T^6 + 25608502345156T^4 + 450645038441016*T^2 + 1992981645409329
$83$	\( T^{16} + \cdots + 18891677809 \) T^16 + 664T^14 + 157868T^12 + 16312808T^10 + 729982998T^8 + 12199988008T^6 + 52057753420T^4 + 62038269912*T^2 + 18891677809
$89$	\( T^{16} + \cdots + 51758070016 \) T^16 + 480T^14 + 79288T^12 + 5776600T^10 + 192538032T^8 + 2949145760T^6 + 20240694800T^4 + 58461394240*T^2 + 51758070016
$97$	\( (T^{8} - 8 T^{7} + \cdots + 8676352)^{2} \) (T^8 - 8T^7 - 420T^6 + 1568T^5 + 51760T^4 - 42240T^3 - 2089408T^2 - 3092992*T + 8676352)^2
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