LMFDB - Newform orbit 3360.2.g.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(1681,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, 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("3360.1681");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.g (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:	$26.8297350792$
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} + 2 x^{14} - 2 x^{13} + x^{12} - 8 x^{10} + 24 x^{9} - 32 x^{8} + 48 x^{7} + \cdots + 256 $ x^16 - 2x^15 + 2x^14 - 2x^13 + x^12 - 8x^10 + 24x^9 - 32x^8 + 48x^7 - 32x^6 + 16x^4 - 64x^3 + 128x^2 - 256x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q - \beta_1 q^{3} - \beta_1 q^{5} + q^{7} - q^{9}+O(q^{10}) $ q - b1 * q^3 - b1 * q^5 + q^7 - q^9	$ q - \beta_1 q^{3} - \beta_1 q^{5} + q^{7} - q^{9} + \beta_{8} q^{11} - \beta_{2} q^{13} - q^{15} - \beta_{4} q^{17} - \beta_{7} q^{19} - \beta_1 q^{21} + \beta_{12} q^{23} - q^{25} + \beta_1 q^{27} + ( - \beta_{10} + \beta_{8} + \cdots + \beta_1) q^{29}+ \cdots - \beta_{8} q^{99}+O(q^{100}) $ q - b1 * q^3 - b1 * q^5 + q^7 - q^9 + b8 * q^11 - b2 * q^13 - q^15 - b4 * q^17 - b7 * q^19 - b1 * q^21 + b12 * q^23 - q^25 + b1 * q^27 + (-b10 + b8 + b7 + b2 + b1) * q^29 + (-b15 + b14 + b12 + b4) * q^31 + b15 * q^33 - b1 * q^35 + b5 * q^37 + b13 * q^39 + (b14 - b13 - b3) * q^41 + (-b10 - b8 - b5 + b1) * q^43 + b1 * q^45 + (b14 + b12 + b4 - 2) * q^47 + q^49 + b6 * q^51 + (-b10 - b6 + b5 - b1) * q^53 + b15 * q^55 + b3 * q^57 + (-b10 + b8 + b6 - b5 + b1) * q^59 + (-b10 + b8 + b7 - b6 + b5 + b2 + b1) * q^61 - q^63 + b13 * q^65 + (-b9 + b7 - b6 + b2 - 2b1) q^67 - b9 * q^69 + (-2b15 + 2b14 + b12 + b3 + 4) * q^71 + (-b15 + b14 + b13 - b4 + b3) * q^73 + b1 * q^75 + b8 * q^77 + (b15 - b14 - b13 + b11 - b3 - 3) * q^79 + q^81 + (-b10 + b9 + b8 - b7 + b6 - b5 + b2 - b1) * q^83 + b6 * q^85 + (b15 - b13 + b11 - b3 + 1) * q^87 + (2b15 + b14 + b4) q^89 - b2 * q^91 + (-b9 + b8 - b6 + b5) * q^93 + b3 * q^95 + (b15 - b14 - b13 - b4 + b3) * q^97 - b8 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7} - 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^7 - 16 * q^9	$ 16 q + 16 q^{7} - 16 q^{9} - 16 q^{15} - 16 q^{25} - 32 q^{47} + 16 q^{49} - 16 q^{63} + 64 q^{71} - 48 q^{79} + 16 q^{81} + 16 q^{87}+O(q^{100}) $ 16 * q + 16 * q^7 - 16 * q^9 - 16 * q^15 - 16 * q^25 - 32 * q^47 + 16 * q^49 - 16 * q^63 + 64 * q^71 - 48 * q^79 + 16 * q^81 + 16 * q^87

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2 x^{15} + 2 x^{14} - 2 x^{13} + x^{12} - 8 x^{10} + 24 x^{9} - 32 x^{8} + 48 x^{7} + \cdots + 256 $ x^16 - 2*x^15 + 2*x^14 - 2*x^13 + x^12 - 8*x^10 + 24*x^9 - 32*x^8 + 48*x^7 - 32*x^6 + 16*x^4 - 64*x^3 + 128*x^2 - 256*x + 256 :

$\beta_{1}$	$=$	$ ( 3 \nu^{15} - 8 \nu^{14} + 6 \nu^{13} - 10 \nu^{12} - \nu^{11} - 10 \nu^{10} - 44 \nu^{9} + 48 \nu^{8} + \cdots - 896 ) / 512 $ (3v^15 - 8v^14 + 6v^13 - 10v^12 - v^11 - 10v^10 - 44v^9 + 48v^8 - 64v^7 + 144v^6 - 64v^5 + 48v^3 - 224v^2 - 64*v - 896) / 512
$\beta_{2}$	$=$	$ ( \nu^{15} + 2 \nu^{13} + 2 \nu^{12} - 11 \nu^{11} + 10 \nu^{10} - 20 \nu^{9} + 16 \nu^{8} - 16 \nu^{7} + \cdots - 128 ) / 128 $ (v^15 + 2v^13 + 2v^12 - 11v^11 + 10v^10 - 20v^9 + 16v^8 - 16v^7 + 16v^6 + 128v^5 + 144v^3 - 160v^2 - 64v - 128) / 128
$\beta_{3}$	$=$	$ ( - \nu^{15} - 2 \nu^{14} + 2 \nu^{13} + 2 \nu^{12} - \nu^{11} + 4 \nu^{10} + 4 \nu^{9} + 8 \nu^{8} + \cdots - 64 \nu ) / 128 $ (-v^15 - 2v^14 + 2v^13 + 2v^12 - v^11 + 4v^10 + 4v^9 + 8v^8 - 32v^7 - 16v^6 - 32v^5 - 64v^4 + 240v^3 - 64v) / 128
$\beta_{4}$	$=$	$ ( 3 \nu^{15} - 4 \nu^{14} + 14 \nu^{13} - 18 \nu^{12} + 23 \nu^{11} - 6 \nu^{10} - 60 \nu^{9} + \cdots - 1152 ) / 256 $ (3v^15 - 4v^14 + 14v^13 - 18v^12 + 23v^11 - 6v^10 - 60v^9 + 96v^8 - 192v^7 + 208v^6 - 192v^5 + 256v^4 + 176v^3 - 160v^2 + 704*v - 1152) / 256
$\beta_{5}$	$=$	$ ( - 3 \nu^{15} + 4 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} + 9 \nu^{11} - 26 \nu^{10} + 76 \nu^{9} + \cdots + 1152 ) / 256 $ (-3v^15 + 4v^14 + 2v^13 - 14v^12 + 9v^11 - 26v^10 + 76v^9 - 96v^8 + 64v^7 - 80v^6 - 64v^5 + 256v^4 - 432v^3 + 160v^2 - 960*v + 1152) / 256
$\beta_{6}$	$=$	$ ( 5 \nu^{15} - 4 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} + \nu^{11} - 18 \nu^{10} + 12 \nu^{9} + 32 \nu^{8} + \cdots + 128 ) / 256 $ (5v^15 - 4v^14 + 2v^13 - 14v^12 + v^11 - 18v^10 + 12v^9 + 32v^8 + 48v^6 + 64v^5 - 304v^3 + 288v^2 - 448v + 128) / 256
$\beta_{7}$	$=$	$ ( - 5 \nu^{15} + 4 \nu^{14} - 2 \nu^{13} + 14 \nu^{12} + 31 \nu^{11} - 46 \nu^{10} + 52 \nu^{9} + \cdots - 128 ) / 256 $ (-5v^15 + 4v^14 - 2v^13 + 14v^12 + 31v^11 - 46v^10 + 52v^9 - 96v^8 + 32v^7 - 48v^6 - 64v^5 + 256v^4 - 208v^3 + 736v^2 - 320*v - 128) / 256
$\beta_{8}$	$=$	$ ( 3 \nu^{15} - 2 \nu^{14} + 2 \nu^{13} - 6 \nu^{12} + 3 \nu^{11} - 4 \nu^{10} - 20 \nu^{9} + 40 \nu^{8} + \cdots - 512 ) / 128 $ (3v^15 - 2v^14 + 2v^13 - 6v^12 + 3v^11 - 4v^10 - 20v^9 + 40v^8 - 32v^7 + 112v^6 - 32v^5 + 64v^4 + 48v^3 - 128v^2 + 320*v - 512) / 128
$\beta_{9}$	$=$	$ ( 3 \nu^{15} - 6 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} - 5 \nu^{11} - 24 \nu^{10} - 28 \nu^{9} + \cdots - 1280 ) / 128 $ (3v^15 - 6v^14 + 2v^13 - 14v^12 - 5v^11 - 24v^10 - 28v^9 + 56v^8 - 96v^7 + 176v^6 - 64v^5 + 64v^4 - 144v^3 - 448v^2 + 192*v - 1280) / 128
$\beta_{10}$	$=$	$ ( 11 \nu^{15} - 26 \nu^{13} - 10 \nu^{12} - 41 \nu^{11} - 50 \nu^{10} - 172 \nu^{9} + 144 \nu^{8} + \cdots - 4480 ) / 512 $ (11v^15 - 26v^13 - 10v^12 - 41v^11 - 50v^10 - 172v^9 + 144v^8 - 64v^7 + 400v^6 + 320v^5 + 176v^3 - 1376v^2 - 64*v - 4480) / 512
$\beta_{11}$	$=$	$ ( - 7 \nu^{15} - 4 \nu^{14} - 6 \nu^{13} - 6 \nu^{12} + 5 \nu^{11} + 22 \nu^{10} + 76 \nu^{9} + \cdots + 896 ) / 256 $ (-7v^15 - 4v^14 - 6v^13 - 6v^12 + 5v^11 + 22v^10 + 76v^9 - 32v^8 + 32v^7 - 16v^6 - 192v^5 - 256v^4 - 240v^3 + 160v^2 - 704*v + 896) / 256
$\beta_{12}$	$=$	$ ( - 11 \nu^{15} + 12 \nu^{14} - 14 \nu^{13} + 18 \nu^{12} - 15 \nu^{11} - 2 \nu^{10} + 124 \nu^{9} + \cdots + 2176 ) / 256 $ (-11v^15 + 12v^14 - 14v^13 + 18v^12 - 15v^11 - 2v^10 + 124v^9 - 224v^8 + 256v^7 - 336v^6 + 64v^5 - 304v^3 + 1056v^2 - 1216v + 2176) / 256
$\beta_{13}$	$=$	$ ( - 11 \nu^{15} - 22 \nu^{13} + 10 \nu^{12} + 9 \nu^{11} + 18 \nu^{10} + 124 \nu^{9} - 112 \nu^{8} + \cdots + 1920 ) / 256 $ (-11v^15 - 22v^13 + 10v^12 + 9v^11 + 18v^10 + 124v^9 - 112v^8 + 192v^7 - 336v^6 - 128v^5 - 128v^4 - 560v^3 + 608v^2 - 704v + 1920) / 256
$\beta_{14}$	$=$	$ ( 4 \nu^{15} - 3 \nu^{14} + 2 \nu^{13} - 2 \nu^{12} + 2 \nu^{11} + 5 \nu^{10} - 36 \nu^{9} + 44 \nu^{8} + \cdots - 576 ) / 64 $ (4v^15 - 3v^14 + 2v^13 - 2v^12 + 2v^11 + 5v^10 - 36v^9 + 44v^8 - 64v^7 + 80v^6 + 32v^5 - 32v^4 + 96v^3 - 176v^2 + 448*v - 576) / 64
$\beta_{15}$	$=$	$ ( 5 \nu^{15} - 3 \nu^{14} + 4 \nu^{13} - \nu^{11} - \nu^{10} - 40 \nu^{9} + 44 \nu^{8} - 72 \nu^{7} + \cdots - 704 ) / 64 $ (5v^15 - 3v^14 + 4v^13 - v^11 - v^10 - 40v^9 + 44v^8 - 72v^7 + 96v^6 + 32v^5 + 32v^4 + 112v^3 - 80v^2 + 320v - 704) / 64

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

$\nu$	$=$	$ ( -\beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta _1 + 1 ) / 8 $ (-b15 + b14 + b12 - b11 - b10 + b9 + b8 - b5 - b1 + 1) / 8
$\nu^{2}$	$=$	$ ( \beta_{12} + \beta_{6} - \beta_{5} + \beta_{4} ) / 4 $ (b12 + b6 - b5 + b4) / 4
$\nu^{3}$	$=$	$ ( - \beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{5} + \cdots + 1 ) / 8 $ (-b15 + b14 + b12 - b11 + b10 - b9 + 3b8 + b5 + 4b3 + b1 + 1) / 8
$\nu^{4}$	$=$	$ ( \beta_{15} - \beta_{14} + 2\beta_{13} - \beta_{11} + 2\beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{2} + 1 ) / 4 $ (b15 - b14 + 2b13 - b11 + 2b8 - b6 + b5 + b4 + 2*b2 + 1) / 4
$\nu^{5}$	$=$	$ ( - 5 \beta_{15} + 5 \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 4 \beta_{7} + \cdots + 1 ) / 8 $ (-5b15 + 5b14 + b12 - b11 - b10 + b9 + b8 + 4b7 + 4b6 - b5 + 8*b2 - b1 + 1) / 8
$\nu^{6}$	$=$	$ ( - 2 \beta_{13} + \beta_{12} + 4 \beta_{11} + 4 \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 12 ) / 4 $ (-2b13 + b12 + 4b11 + 4b8 + 2b7 - b6 - b5 - b4 - 2b3 + 2b2 + 12) / 4
$\nu^{7}$	$=$	$ ( - 5 \beta_{15} - 3 \beta_{14} - 3 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} - 5 \beta_{9} + 15 \beta_{8} + \cdots - 27 ) / 8 $ (-5b15 - 3b14 - 3b12 - 5b11 - 3b10 - 5b9 + 15b8 - 3b5 - 8b4 - 4b3 + 29*b1 - 27) / 8
$\nu^{8}$	$=$	$ ( \beta_{15} - 9 \beta_{14} + 2 \beta_{13} - 8 \beta_{12} - \beta_{11} + 2 \beta_{8} + 7 \beta_{6} + \cdots + 1 ) / 4 $ (b15 - 9b14 + 2b13 - 8b12 - b11 + 2b8 + 7b6 - 7b5 + b4 + 2b2 - 16b1 + 1) / 4
$\nu^{9}$	$=$	$ ( 3 \beta_{15} - 3 \beta_{14} - 7 \beta_{12} + 7 \beta_{11} - 9 \beta_{10} + 9 \beta_{9} + 25 \beta_{8} + \cdots - 71 ) / 8 $ (3b15 - 3b14 - 7b12 + 7b11 - 9b10 + 9b9 + 25b8 + 4b7 + 4b6 + 7b5 - 16b4 + 16b3 + 8b2 - 73b1 - 71) / 8
$\nu^{10}$	$=$	$ ( 8 \beta_{15} + 6 \beta_{13} + \beta_{12} + 12 \beta_{11} - 8 \beta_{9} + 4 \beta_{8} - 6 \beta_{7} + \cdots - 60 ) / 4 $ (8b15 + 6b13 + b12 + 12b11 - 8b9 + 4b8 - 6b7 - 9b6 + 7b5 + 7b4 - 10b3 + 10*b2 - 60) / 4
$\nu^{11}$	$=$	$ ( - 21 \beta_{15} + 29 \beta_{14} - 35 \beta_{12} + 11 \beta_{11} + 13 \beta_{10} - 37 \beta_{9} + \cdots - 43 ) / 8 $ (-21b15 + 29b14 - 35b12 + 11b11 + 13b10 - 37b9 - b8 + 32b7 + 29b5 - 8b4 - 4b3 + 45*b1 - 43) / 8
$\nu^{12}$	$=$	$ ( 33 \beta_{15} - 33 \beta_{14} + 2 \beta_{13} - 32 \beta_{12} + 15 \beta_{11} - 8 \beta_{10} - 6 \beta_{8} + \cdots + 49 ) / 4 $ (33b15 - 33b14 + 2b13 - 32b12 + 15b11 - 8b10 - 6b8 + 16b7 - 25b6 - 7b5 - 31b4 - 16b3 + 2b2 - 24b1 + 49) / 4
$\nu^{13}$	$=$	$ ( - 13 \beta_{15} - 35 \beta_{14} - 64 \beta_{13} - 39 \beta_{12} - 9 \beta_{11} - 89 \beta_{10} + \cdots - 311 ) / 8 $ (-13b15 - 35b14 - 64b13 - 39b12 - 9b11 - 89b10 + 9b9 - 23b8 + 4b7 + 4b6 - 25b5 - 48b4 - 48b3 + 8b2 + 103*b1 - 311) / 8
$\nu^{14}$	$=$	$ ( 16 \beta_{15} - 32 \beta_{14} - 58 \beta_{13} - 15 \beta_{12} + 36 \beta_{11} + 32 \beta_{10} + \cdots - 84 ) / 4 $ (16b15 - 32b14 - 58b13 - 15b12 + 36b11 + 32b10 - 16b9 - 44b8 - 22b7 + 7b6 - b5 + 23b4 - 26b3 - 54b2 - 224b1 - 84) / 4
$\nu^{15}$	$=$	$ ( 171 \beta_{15} - 19 \beta_{14} - 51 \beta_{12} + 11 \beta_{11} + 45 \beta_{10} - 85 \beta_{9} + \cdots - 299 ) / 8 $ (171b15 - 19b14 - 51b12 + 11b11 + 45b10 - 85b9 + 31b8 - 96b7 - 32b6 + 205b5 - 136b4 + 60b3 - 192b2 - 179b1 - 299) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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} $

1681.1

−0.908085	−	1.08415i
1.37218	−	0.342246i
0.787917	−	1.17439i
0.582664	+	1.28861i
−1.41228	+	0.0739393i
−0.727187	+	1.21293i
−0.0395670	−	1.41366i
1.34436	+	0.438970i
1.34436	−	0.438970i
−0.0395670	+	1.41366i
−0.727187	−	1.21293i
−1.41228	−	0.0739393i
0.582664	−	1.28861i
0.787917	+	1.17439i
1.37218	+	0.342246i
−0.908085	+	1.08415i

−

1.00000i

−

1.00000i

1.00000

−1.00000

1681.2

−

1.00000i

−

1.00000i

1.00000

−1.00000

1681.3

−

1.00000i

−

1.00000i

1.00000

−1.00000

1681.4

−

1.00000i

−

1.00000i

1.00000

−1.00000

1681.5

−

1.00000i

−

1.00000i

1.00000

−1.00000

1681.6

−

1.00000i

−

1.00000i

1.00000

−1.00000

1681.7

−

1.00000i

−

1.00000i

1.00000

−1.00000

1681.8

−

1.00000i

−

1.00000i

1.00000

−1.00000

1681.9

1.00000i

1.00000

−1.00000

1681.10

1.00000i

1.00000

−1.00000

1681.11

1.00000i

1.00000

−1.00000

1681.12

1.00000i

1.00000

−1.00000

1681.13

1.00000i

1.00000

−1.00000

1681.14

1.00000i

1.00000

−1.00000

1681.15

1.00000i

1.00000

−1.00000

1681.16

1.00000i

1.00000

−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	3360.2.g.d		16
4.b	odd	2	1		840.2.g.d	✓	16
8.b	even	2	1	inner	3360.2.g.d		16
8.d	odd	2	1		840.2.g.d	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
840.2.g.d	✓	16	4.b	odd	2	1
840.2.g.d	✓	16	8.d	odd	2	1
3360.2.g.d		16	1.a	even	1	1	trivial
3360.2.g.d		16	8.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{16} + 104 T_{11}^{14} + 4320 T_{11}^{12} + 91904 T_{11}^{10} + 1068352 T_{11}^{8} + \cdots + 9437184 $ T11^16 + 104*T11^14 + 4320*T11^12 + 91904*T11^10 + 1068352*T11^8 + 6695424*T11^6 + 20595712*T11^4 + 24379392*T11^2 + 9437184 acting on $S_{2}^{\mathrm{new}}(3360, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$5$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$7$	$ (T - 1)^{16} $ (T - 1)^16
$11$	$ T^{16} + 104 T^{14} + \cdots + 9437184 $ T^16 + 104T^14 + 4320T^12 + 91904T^10 + 1068352T^8 + 6695424T^6 + 20595712T^4 + 24379392*T^2 + 9437184
$13$	$ T^{16} + \cdots + 256000000 $ T^16 + 160T^14 + 10608T^12 + 376960T^10 + 7742784T^8 + 91922432T^6 + 585893888T^4 + 1575714816*T^2 + 256000000
$17$	$ (T^{8} - 84 T^{6} + \cdots - 3072)^{2} $ (T^8 - 84T^6 + 80T^5 + 1920T^4 - 3456T^3 - 6784T^2 + 10752T - 3072)^2
$19$	$ T^{16} + 200 T^{14} + \cdots + 9437184 $ T^16 + 200T^14 + 15456T^12 + 568576T^10 + 9733440T^8 + 62114304T^6 + 120374272T^4 + 62128128*T^2 + 9437184
$23$	$ (T^{8} - 108 T^{6} + \cdots - 24576)^{2} $ (T^8 - 108T^6 + 160T^5 + 2912T^4 - 9088T^3 - 2432T^2 + 30720T - 24576)^2
$29$	$ T^{16} + \cdots + 12572688384 $ T^16 + 312T^14 + 38128T^12 + 2347776T^10 + 77137664T^8 + 1305188352T^6 + 10014871552T^4 + 28452716544*T^2 + 12572688384
$31$	$ (T^{8} - 128 T^{6} + \cdots + 10240)^{2} $ (T^8 - 128T^6 + 176T^5 + 4232T^4 - 11232T^3 - 22464T^2 + 66048T + 10240)^2
$37$	$ T^{16} + 168 T^{14} + \cdots + 1048576 $ T^16 + 168T^14 + 8848T^12 + 176640T^10 + 1588224T^8 + 6463488T^6 + 10240000T^4 + 5767168*T^2 + 1048576
$41$	$ (T^{8} - 216 T^{6} + \cdots + 76800)^{2} $ (T^8 - 216T^6 - 224T^5 + 15120T^4 + 28544T^3 - 334592T^2 - 807936T + 76800)^2
$43$	$ T^{16} + \cdots + 4498653184 $ T^16 + 472T^14 + 82928T^12 + 6620928T^10 + 232425216T^8 + 2848077824T^6 + 10779217920T^4 + 13369999360*T^2 + 4498653184
$47$	$ (T^{8} + 16 T^{7} + \cdots - 35328)^{2} $ (T^8 + 16T^7 + 4T^6 - 944T^5 - 3280T^4 + 6656T^3 + 24128T^2 - 16128*T - 35328)^2
$53$	$ T^{16} + \cdots + 267413094400 $ T^16 + 488T^14 + 86368T^12 + 6939904T^10 + 270793024T^8 + 5317351936T^6 + 51873006592T^4 + 222808244224*T^2 + 267413094400
$59$	$ T^{16} + \cdots + 4244786642944 $ T^16 + 528T^14 + 109408T^12 + 11422976T^10 + 638497024T^8 + 18678300672T^6 + 263933263872T^4 + 1730926346240*T^2 + 4244786642944
$61$	$ T^{16} + \cdots + 260573495296 $ T^16 + 536T^14 + 111984T^12 + 11913984T^10 + 694114048T^8 + 21671573504T^6 + 315132694528T^4 + 1286669205504*T^2 + 260573495296
$67$	$ T^{16} + \cdots + 171568267264 $ T^16 + 568T^14 + 115696T^12 + 10280704T^10 + 406079232T^8 + 7601068032T^6 + 64707743744T^4 + 201723346944*T^2 + 171568267264
$71$	$ (T^{8} - 32 T^{7} + \cdots - 2977792)^{2} $ (T^8 - 32T^7 + 96T^6 + 5952T^5 - 57848T^4 - 39008T^3 + 1798464T^2 - 3383296*T - 2977792)^2
$73$	$ (T^{8} - 304 T^{6} + \cdots + 2407040)^{2} $ (T^8 - 304T^6 + 384T^5 + 27704T^4 - 49376T^3 - 812096T^2 + 1615488T + 2407040)^2
$79$	$ (T^{8} + 24 T^{7} + \cdots + 294912)^{2} $ (T^8 + 24T^7 + 24T^6 - 3136T^5 - 20912T^4 + 34432T^3 + 593408T^2 + 1302528*T + 294912)^2
$83$	$ T^{16} + \cdots + 87214679654400 $ T^16 + 896T^14 + 323200T^12 + 60291072T^10 + 6210424832T^8 + 347410923520T^6 + 9594635026432T^4 + 104166781353984*T^2 + 87214679654400
$89$	$ (T^{8} - 360 T^{6} + \cdots + 575488)^{2} $ (T^8 - 360T^6 + 160T^5 + 31312T^4 - 64896T^3 - 378368T^2 + 36864T + 575488)^2
$97$	$ (T^{8} - 336 T^{6} + \cdots + 640)^{2} $ (T^8 - 336T^6 + 32T^5 + 10232T^4 + 9248T^3 - 49728T^2 - 28032T + 640)^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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} - \beta_1 q^{5} + q^{7} - q^{9}+O(q^{10}) \) q - b1 * q^3 - b1 * q^5 + q^7 - q^9	\( q - \beta_1 q^{3} - \beta_1 q^{5} + q^{7} - q^{9} + \beta_{8} q^{11} - \beta_{2} q^{13} - q^{15} - \beta_{4} q^{17} - \beta_{7} q^{19} - \beta_1 q^{21} + \beta_{12} q^{23} - q^{25} + \beta_1 q^{27} + ( - \beta_{10} + \beta_{8} + \cdots + \beta_1) q^{29}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) q - b1 * q^3 - b1 * q^5 + q^7 - q^9 + b8 * q^11 - b2 * q^13 - q^15 - b4 * q^17 - b7 * q^19 - b1 * q^21 + b12 * q^23 - q^25 + b1 * q^27 + (-b10 + b8 + b7 + b2 + b1) * q^29 + (-b15 + b14 + b12 + b4) * q^31 + b15 * q^33 - b1 * q^35 + b5 * q^37 + b13 * q^39 + (b14 - b13 - b3) * q^41 + (-b10 - b8 - b5 + b1) * q^43 + b1 * q^45 + (b14 + b12 + b4 - 2) * q^47 + q^49 + b6 * q^51 + (-b10 - b6 + b5 - b1) * q^53 + b15 * q^55 + b3 * q^57 + (-b10 + b8 + b6 - b5 + b1) * q^59 + (-b10 + b8 + b7 - b6 + b5 + b2 + b1) * q^61 - q^63 + b13 * q^65 + (-b9 + b7 - b6 + b2 - 2b1) q^67 - b9 * q^69 + (-2b15 + 2b14 + b12 + b3 + 4) * q^71 + (-b15 + b14 + b13 - b4 + b3) * q^73 + b1 * q^75 + b8 * q^77 + (b15 - b14 - b13 + b11 - b3 - 3) * q^79 + q^81 + (-b10 + b9 + b8 - b7 + b6 - b5 + b2 - b1) * q^83 + b6 * q^85 + (b15 - b13 + b11 - b3 + 1) * q^87 + (2b15 + b14 + b4) q^89 - b2 * q^91 + (-b9 + b8 - b6 + b5) * q^93 + b3 * q^95 + (b15 - b14 - b13 - b4 + b3) * q^97 - b8 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7} - 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^7 - 16 * q^9	\( 16 q + 16 q^{7} - 16 q^{9} - 16 q^{15} - 16 q^{25} - 32 q^{47} + 16 q^{49} - 16 q^{63} + 64 q^{71} - 48 q^{79} + 16 q^{81} + 16 q^{87}+O(q^{100}) \) 16 * q + 16 * q^7 - 16 * q^9 - 16 * q^15 - 16 * q^25 - 32 * q^47 + 16 * q^49 - 16 * q^63 + 64 * q^71 - 48 * q^79 + 16 * q^81 + 16 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	3360.2.g.d

Level	$3360$
Weight	$2$
Character orbit	3360.g
Analytic conductor	$26.830$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(26.8297350792\)
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:	\( x^{16} - 2 x^{15} + 2 x^{14} - 2 x^{13} + x^{12} - 8 x^{10} + 24 x^{9} - 32 x^{8} + 48 x^{7} + \cdots + 256 \) x^16 - 2x^15 + 2x^14 - 2x^13 + x^12 - 8x^10 + 24x^9 - 32x^8 + 48x^7 - 32x^6 + 16x^4 - 64x^3 + 128x^2 - 256x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index:	\( 2^{21} \)
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 3 \nu^{15} - 8 \nu^{14} + 6 \nu^{13} - 10 \nu^{12} - \nu^{11} - 10 \nu^{10} - 44 \nu^{9} + 48 \nu^{8} + \cdots - 896 ) / 512 \) (3v^15 - 8v^14 + 6v^13 - 10v^12 - v^11 - 10v^10 - 44v^9 + 48v^8 - 64v^7 + 144v^6 - 64v^5 + 48v^3 - 224v^2 - 64*v - 896) / 512
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} + 2 \nu^{13} + 2 \nu^{12} - 11 \nu^{11} + 10 \nu^{10} - 20 \nu^{9} + 16 \nu^{8} - 16 \nu^{7} + \cdots - 128 ) / 128 \) (v^15 + 2v^13 + 2v^12 - 11v^11 + 10v^10 - 20v^9 + 16v^8 - 16v^7 + 16v^6 + 128v^5 + 144v^3 - 160v^2 - 64v - 128) / 128
\(\beta_{3}\)	\(=\)	\( ( - \nu^{15} - 2 \nu^{14} + 2 \nu^{13} + 2 \nu^{12} - \nu^{11} + 4 \nu^{10} + 4 \nu^{9} + 8 \nu^{8} + \cdots - 64 \nu ) / 128 \) (-v^15 - 2v^14 + 2v^13 + 2v^12 - v^11 + 4v^10 + 4v^9 + 8v^8 - 32v^7 - 16v^6 - 32v^5 - 64v^4 + 240v^3 - 64v) / 128
\(\beta_{4}\)	\(=\)	\( ( 3 \nu^{15} - 4 \nu^{14} + 14 \nu^{13} - 18 \nu^{12} + 23 \nu^{11} - 6 \nu^{10} - 60 \nu^{9} + \cdots - 1152 ) / 256 \) (3v^15 - 4v^14 + 14v^13 - 18v^12 + 23v^11 - 6v^10 - 60v^9 + 96v^8 - 192v^7 + 208v^6 - 192v^5 + 256v^4 + 176v^3 - 160v^2 + 704*v - 1152) / 256
\(\beta_{5}\)	\(=\)	\( ( - 3 \nu^{15} + 4 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} + 9 \nu^{11} - 26 \nu^{10} + 76 \nu^{9} + \cdots + 1152 ) / 256 \) (-3v^15 + 4v^14 + 2v^13 - 14v^12 + 9v^11 - 26v^10 + 76v^9 - 96v^8 + 64v^7 - 80v^6 - 64v^5 + 256v^4 - 432v^3 + 160v^2 - 960*v + 1152) / 256
\(\beta_{6}\)	\(=\)	\( ( 5 \nu^{15} - 4 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} + \nu^{11} - 18 \nu^{10} + 12 \nu^{9} + 32 \nu^{8} + \cdots + 128 ) / 256 \) (5v^15 - 4v^14 + 2v^13 - 14v^12 + v^11 - 18v^10 + 12v^9 + 32v^8 + 48v^6 + 64v^5 - 304v^3 + 288v^2 - 448v + 128) / 256
\(\beta_{7}\)	\(=\)	\( ( - 5 \nu^{15} + 4 \nu^{14} - 2 \nu^{13} + 14 \nu^{12} + 31 \nu^{11} - 46 \nu^{10} + 52 \nu^{9} + \cdots - 128 ) / 256 \) (-5v^15 + 4v^14 - 2v^13 + 14v^12 + 31v^11 - 46v^10 + 52v^9 - 96v^8 + 32v^7 - 48v^6 - 64v^5 + 256v^4 - 208v^3 + 736v^2 - 320*v - 128) / 256
\(\beta_{8}\)	\(=\)	\( ( 3 \nu^{15} - 2 \nu^{14} + 2 \nu^{13} - 6 \nu^{12} + 3 \nu^{11} - 4 \nu^{10} - 20 \nu^{9} + 40 \nu^{8} + \cdots - 512 ) / 128 \) (3v^15 - 2v^14 + 2v^13 - 6v^12 + 3v^11 - 4v^10 - 20v^9 + 40v^8 - 32v^7 + 112v^6 - 32v^5 + 64v^4 + 48v^3 - 128v^2 + 320*v - 512) / 128
\(\beta_{9}\)	\(=\)	\( ( 3 \nu^{15} - 6 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} - 5 \nu^{11} - 24 \nu^{10} - 28 \nu^{9} + \cdots - 1280 ) / 128 \) (3v^15 - 6v^14 + 2v^13 - 14v^12 - 5v^11 - 24v^10 - 28v^9 + 56v^8 - 96v^7 + 176v^6 - 64v^5 + 64v^4 - 144v^3 - 448v^2 + 192*v - 1280) / 128
\(\beta_{10}\)	\(=\)	\( ( 11 \nu^{15} - 26 \nu^{13} - 10 \nu^{12} - 41 \nu^{11} - 50 \nu^{10} - 172 \nu^{9} + 144 \nu^{8} + \cdots - 4480 ) / 512 \) (11v^15 - 26v^13 - 10v^12 - 41v^11 - 50v^10 - 172v^9 + 144v^8 - 64v^7 + 400v^6 + 320v^5 + 176v^3 - 1376v^2 - 64*v - 4480) / 512
\(\beta_{11}\)	\(=\)	\( ( - 7 \nu^{15} - 4 \nu^{14} - 6 \nu^{13} - 6 \nu^{12} + 5 \nu^{11} + 22 \nu^{10} + 76 \nu^{9} + \cdots + 896 ) / 256 \) (-7v^15 - 4v^14 - 6v^13 - 6v^12 + 5v^11 + 22v^10 + 76v^9 - 32v^8 + 32v^7 - 16v^6 - 192v^5 - 256v^4 - 240v^3 + 160v^2 - 704*v + 896) / 256
\(\beta_{12}\)	\(=\)	\( ( - 11 \nu^{15} + 12 \nu^{14} - 14 \nu^{13} + 18 \nu^{12} - 15 \nu^{11} - 2 \nu^{10} + 124 \nu^{9} + \cdots + 2176 ) / 256 \) (-11v^15 + 12v^14 - 14v^13 + 18v^12 - 15v^11 - 2v^10 + 124v^9 - 224v^8 + 256v^7 - 336v^6 + 64v^5 - 304v^3 + 1056v^2 - 1216v + 2176) / 256
\(\beta_{13}\)	\(=\)	\( ( - 11 \nu^{15} - 22 \nu^{13} + 10 \nu^{12} + 9 \nu^{11} + 18 \nu^{10} + 124 \nu^{9} - 112 \nu^{8} + \cdots + 1920 ) / 256 \) (-11v^15 - 22v^13 + 10v^12 + 9v^11 + 18v^10 + 124v^9 - 112v^8 + 192v^7 - 336v^6 - 128v^5 - 128v^4 - 560v^3 + 608v^2 - 704v + 1920) / 256
\(\beta_{14}\)	\(=\)	\( ( 4 \nu^{15} - 3 \nu^{14} + 2 \nu^{13} - 2 \nu^{12} + 2 \nu^{11} + 5 \nu^{10} - 36 \nu^{9} + 44 \nu^{8} + \cdots - 576 ) / 64 \) (4v^15 - 3v^14 + 2v^13 - 2v^12 + 2v^11 + 5v^10 - 36v^9 + 44v^8 - 64v^7 + 80v^6 + 32v^5 - 32v^4 + 96v^3 - 176v^2 + 448*v - 576) / 64
\(\beta_{15}\)	\(=\)	\( ( 5 \nu^{15} - 3 \nu^{14} + 4 \nu^{13} - \nu^{11} - \nu^{10} - 40 \nu^{9} + 44 \nu^{8} - 72 \nu^{7} + \cdots - 704 ) / 64 \) (5v^15 - 3v^14 + 4v^13 - v^11 - v^10 - 40v^9 + 44v^8 - 72v^7 + 96v^6 + 32v^5 + 32v^4 + 112v^3 - 80v^2 + 320v - 704) / 64

\(\nu\)	\(=\)	\( ( -\beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta _1 + 1 ) / 8 \) (-b15 + b14 + b12 - b11 - b10 + b9 + b8 - b5 - b1 + 1) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{12} + \beta_{6} - \beta_{5} + \beta_{4} ) / 4 \) (b12 + b6 - b5 + b4) / 4
\(\nu^{3}\)	\(=\)	\( ( - \beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{5} + \cdots + 1 ) / 8 \) (-b15 + b14 + b12 - b11 + b10 - b9 + 3b8 + b5 + 4b3 + b1 + 1) / 8
\(\nu^{4}\)	\(=\)	\( ( \beta_{15} - \beta_{14} + 2\beta_{13} - \beta_{11} + 2\beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{2} + 1 ) / 4 \) (b15 - b14 + 2b13 - b11 + 2b8 - b6 + b5 + b4 + 2*b2 + 1) / 4
\(\nu^{5}\)	\(=\)	\( ( - 5 \beta_{15} + 5 \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 4 \beta_{7} + \cdots + 1 ) / 8 \) (-5b15 + 5b14 + b12 - b11 - b10 + b9 + b8 + 4b7 + 4b6 - b5 + 8*b2 - b1 + 1) / 8
\(\nu^{6}\)	\(=\)	\( ( - 2 \beta_{13} + \beta_{12} + 4 \beta_{11} + 4 \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 12 ) / 4 \) (-2b13 + b12 + 4b11 + 4b8 + 2b7 - b6 - b5 - b4 - 2b3 + 2b2 + 12) / 4
\(\nu^{7}\)	\(=\)	\( ( - 5 \beta_{15} - 3 \beta_{14} - 3 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} - 5 \beta_{9} + 15 \beta_{8} + \cdots - 27 ) / 8 \) (-5b15 - 3b14 - 3b12 - 5b11 - 3b10 - 5b9 + 15b8 - 3b5 - 8b4 - 4b3 + 29*b1 - 27) / 8
\(\nu^{8}\)	\(=\)	\( ( \beta_{15} - 9 \beta_{14} + 2 \beta_{13} - 8 \beta_{12} - \beta_{11} + 2 \beta_{8} + 7 \beta_{6} + \cdots + 1 ) / 4 \) (b15 - 9b14 + 2b13 - 8b12 - b11 + 2b8 + 7b6 - 7b5 + b4 + 2b2 - 16b1 + 1) / 4
\(\nu^{9}\)	\(=\)	\( ( 3 \beta_{15} - 3 \beta_{14} - 7 \beta_{12} + 7 \beta_{11} - 9 \beta_{10} + 9 \beta_{9} + 25 \beta_{8} + \cdots - 71 ) / 8 \) (3b15 - 3b14 - 7b12 + 7b11 - 9b10 + 9b9 + 25b8 + 4b7 + 4b6 + 7b5 - 16b4 + 16b3 + 8b2 - 73b1 - 71) / 8
\(\nu^{10}\)	\(=\)	\( ( 8 \beta_{15} + 6 \beta_{13} + \beta_{12} + 12 \beta_{11} - 8 \beta_{9} + 4 \beta_{8} - 6 \beta_{7} + \cdots - 60 ) / 4 \) (8b15 + 6b13 + b12 + 12b11 - 8b9 + 4b8 - 6b7 - 9b6 + 7b5 + 7b4 - 10b3 + 10*b2 - 60) / 4
\(\nu^{11}\)	\(=\)	\( ( - 21 \beta_{15} + 29 \beta_{14} - 35 \beta_{12} + 11 \beta_{11} + 13 \beta_{10} - 37 \beta_{9} + \cdots - 43 ) / 8 \) (-21b15 + 29b14 - 35b12 + 11b11 + 13b10 - 37b9 - b8 + 32b7 + 29b5 - 8b4 - 4b3 + 45*b1 - 43) / 8
\(\nu^{12}\)	\(=\)	\( ( 33 \beta_{15} - 33 \beta_{14} + 2 \beta_{13} - 32 \beta_{12} + 15 \beta_{11} - 8 \beta_{10} - 6 \beta_{8} + \cdots + 49 ) / 4 \) (33b15 - 33b14 + 2b13 - 32b12 + 15b11 - 8b10 - 6b8 + 16b7 - 25b6 - 7b5 - 31b4 - 16b3 + 2b2 - 24b1 + 49) / 4
\(\nu^{13}\)	\(=\)	\( ( - 13 \beta_{15} - 35 \beta_{14} - 64 \beta_{13} - 39 \beta_{12} - 9 \beta_{11} - 89 \beta_{10} + \cdots - 311 ) / 8 \) (-13b15 - 35b14 - 64b13 - 39b12 - 9b11 - 89b10 + 9b9 - 23b8 + 4b7 + 4b6 - 25b5 - 48b4 - 48b3 + 8b2 + 103*b1 - 311) / 8
\(\nu^{14}\)	\(=\)	\( ( 16 \beta_{15} - 32 \beta_{14} - 58 \beta_{13} - 15 \beta_{12} + 36 \beta_{11} + 32 \beta_{10} + \cdots - 84 ) / 4 \) (16b15 - 32b14 - 58b13 - 15b12 + 36b11 + 32b10 - 16b9 - 44b8 - 22b7 + 7b6 - b5 + 23b4 - 26b3 - 54b2 - 224b1 - 84) / 4
\(\nu^{15}\)	\(=\)	\( ( 171 \beta_{15} - 19 \beta_{14} - 51 \beta_{12} + 11 \beta_{11} + 45 \beta_{10} - 85 \beta_{9} + \cdots - 299 ) / 8 \) (171b15 - 19b14 - 51b12 + 11b11 + 45b10 - 85b9 + 31b8 - 96b7 - 32b6 + 205b5 - 136b4 + 60b3 - 192b2 - 179b1 - 299) / 8

\(n\)	\(421\)	\(1121\)	\(1471\)	\(1921\)	\(2017\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1681.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$5$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$7$	\( (T - 1)^{16} \) (T - 1)^16
$11$	\( T^{16} + 104 T^{14} + \cdots + 9437184 \) T^16 + 104T^14 + 4320T^12 + 91904T^10 + 1068352T^8 + 6695424T^6 + 20595712T^4 + 24379392*T^2 + 9437184
$13$	\( T^{16} + \cdots + 256000000 \) T^16 + 160T^14 + 10608T^12 + 376960T^10 + 7742784T^8 + 91922432T^6 + 585893888T^4 + 1575714816*T^2 + 256000000
$17$	\( (T^{8} - 84 T^{6} + \cdots - 3072)^{2} \) (T^8 - 84T^6 + 80T^5 + 1920T^4 - 3456T^3 - 6784T^2 + 10752T - 3072)^2
$19$	\( T^{16} + 200 T^{14} + \cdots + 9437184 \) T^16 + 200T^14 + 15456T^12 + 568576T^10 + 9733440T^8 + 62114304T^6 + 120374272T^4 + 62128128*T^2 + 9437184
$23$	\( (T^{8} - 108 T^{6} + \cdots - 24576)^{2} \) (T^8 - 108T^6 + 160T^5 + 2912T^4 - 9088T^3 - 2432T^2 + 30720T - 24576)^2
$29$	\( T^{16} + \cdots + 12572688384 \) T^16 + 312T^14 + 38128T^12 + 2347776T^10 + 77137664T^8 + 1305188352T^6 + 10014871552T^4 + 28452716544*T^2 + 12572688384
$31$	\( (T^{8} - 128 T^{6} + \cdots + 10240)^{2} \) (T^8 - 128T^6 + 176T^5 + 4232T^4 - 11232T^3 - 22464T^2 + 66048T + 10240)^2
$37$	\( T^{16} + 168 T^{14} + \cdots + 1048576 \) T^16 + 168T^14 + 8848T^12 + 176640T^10 + 1588224T^8 + 6463488T^6 + 10240000T^4 + 5767168*T^2 + 1048576
$41$	\( (T^{8} - 216 T^{6} + \cdots + 76800)^{2} \) (T^8 - 216T^6 - 224T^5 + 15120T^4 + 28544T^3 - 334592T^2 - 807936T + 76800)^2
$43$	\( T^{16} + \cdots + 4498653184 \) T^16 + 472T^14 + 82928T^12 + 6620928T^10 + 232425216T^8 + 2848077824T^6 + 10779217920T^4 + 13369999360*T^2 + 4498653184
$47$	\( (T^{8} + 16 T^{7} + \cdots - 35328)^{2} \) (T^8 + 16T^7 + 4T^6 - 944T^5 - 3280T^4 + 6656T^3 + 24128T^2 - 16128*T - 35328)^2
$53$	\( T^{16} + \cdots + 267413094400 \) T^16 + 488T^14 + 86368T^12 + 6939904T^10 + 270793024T^8 + 5317351936T^6 + 51873006592T^4 + 222808244224*T^2 + 267413094400
$59$	\( T^{16} + \cdots + 4244786642944 \) T^16 + 528T^14 + 109408T^12 + 11422976T^10 + 638497024T^8 + 18678300672T^6 + 263933263872T^4 + 1730926346240*T^2 + 4244786642944
$61$	\( T^{16} + \cdots + 260573495296 \) T^16 + 536T^14 + 111984T^12 + 11913984T^10 + 694114048T^8 + 21671573504T^6 + 315132694528T^4 + 1286669205504*T^2 + 260573495296
$67$	\( T^{16} + \cdots + 171568267264 \) T^16 + 568T^14 + 115696T^12 + 10280704T^10 + 406079232T^8 + 7601068032T^6 + 64707743744T^4 + 201723346944*T^2 + 171568267264
$71$	\( (T^{8} - 32 T^{7} + \cdots - 2977792)^{2} \) (T^8 - 32T^7 + 96T^6 + 5952T^5 - 57848T^4 - 39008T^3 + 1798464T^2 - 3383296*T - 2977792)^2
$73$	\( (T^{8} - 304 T^{6} + \cdots + 2407040)^{2} \) (T^8 - 304T^6 + 384T^5 + 27704T^4 - 49376T^3 - 812096T^2 + 1615488T + 2407040)^2
$79$	\( (T^{8} + 24 T^{7} + \cdots + 294912)^{2} \) (T^8 + 24T^7 + 24T^6 - 3136T^5 - 20912T^4 + 34432T^3 + 593408T^2 + 1302528*T + 294912)^2
$83$	\( T^{16} + \cdots + 87214679654400 \) T^16 + 896T^14 + 323200T^12 + 60291072T^10 + 6210424832T^8 + 347410923520T^6 + 9594635026432T^4 + 104166781353984*T^2 + 87214679654400
$89$	\( (T^{8} - 360 T^{6} + \cdots + 575488)^{2} \) (T^8 - 360T^6 + 160T^5 + 31312T^4 - 64896T^3 - 378368T^2 + 36864T + 575488)^2
$97$	\( (T^{8} - 336 T^{6} + \cdots + 640)^{2} \) (T^8 - 336T^6 + 32T^5 + 10232T^4 + 9248T^3 - 49728T^2 - 28032T + 640)^2
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