LMFDB - Newform orbit 864.2.r.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(145,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("864.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.r (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 $ x^16 - x^15 + x^14 + 2x^12 - 4x^11 - 8x^9 + 4x^8 - 16x^7 - 32x^5 + 32x^4 + 64x^2 - 128*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + ( - \beta_{11} - \beta_{5} + \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{3} + \beta_1 - 1) q^{7}+O(q^{10}) $ q + (-b11 - b5 + b4) * q^5 + (-b6 - b3 + b1 - 1) * q^7	$ q + ( - \beta_{11} - \beta_{5} + \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{13} - \beta_{8}) q^{11} + (\beta_{8} - \beta_{5}) q^{13} + ( - \beta_{7} + 2) q^{17} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{11}) q^{19}+ \cdots + (\beta_{15} - \beta_{10} - 3 \beta_{6} + \cdots - 1) q^{97}+O(q^{100}) $ q + (-b11 - b5 + b4) * q^5 + (-b6 - b3 + b1 - 1) * q^7 + (b13 - b8) * q^11 + (b8 - b5) * q^13 + (-b7 + 2) * q^17 + (-b14 + b13 + 2b12 + b11) q^19 + (-b15 - b10 - b7 - b1) * q^23 + (-b15 - b6 - b3 + b2) * q^25 + (b13 + b12 - 2b9 - b8 - b5) q^29 + (-b15 - b10 - b7 + 2b6 + b1) q^31 + (-b14 + b13) * q^35 + (b13 + 2b12 - b11) q^37 + (-b15 + b10 + b7 - b6 + b1) * q^41 + b4 * q^43 + (2b10 - b6 - b3 - b1 + 1) q^47 + (-b10 - b7 + 2b6 + 2b1) * q^49 + (2b14 + b13 - b11) q^53 + (-b7 + 2b3 + b2 + 1) q^55 + (2b14 - b11 - 2b9 - 2b5 + b4) q^59 + (b13 - b12 - 2b9 - b8 + b5 + 2b4) * q^61 + (-b10 - 2b6 - 2b3 - b1 + 1) * q^65 + (-2b11 + b8 - 4b5 + 2b4) q^67 + (b7 - b3 - b2 + 4) * q^71 + (-b7 + 2b3 - 2) q^73 + (-2b14 + 3b11 + 2b9 + b5 - 3b4) * q^77 + (-b15 - b10 - 2b6 - 2b3 + b2 - 3b1 + 3) q^79 + (-b13 + 2b12 + b9 + b8 - 2b5) * q^83 + (-2b14 - b11 + 2b9 - b8 - 2b5 + b4) q^85 + (-b7 - b3 - b2 - 4) * q^89 + (b14 - 3b11) q^91 + (2b15 - 2b6 + 6b1) q^95 + (b15 - b10 - 3b6 - 3b3 - b2 + b1 - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{7}+O(q^{10}) $ 16 * q - 6 * q^7	$ 16 q - 6 q^{7} + 28 q^{17} - 10 q^{23} + 2 q^{25} + 10 q^{31} + 8 q^{41} + 6 q^{47} + 18 q^{49} + 4 q^{55} + 14 q^{65} + 72 q^{71} - 44 q^{73} + 30 q^{79} - 64 q^{89} + 44 q^{95}+O(q^{100}) $ 16 * q - 6 * q^7 + 28 * q^17 - 10 * q^23 + 2 * q^25 + 10 * q^31 + 8 * q^41 + 6 * q^47 + 18 * q^49 + 4 * q^55 + 14 * q^65 + 72 * q^71 - 44 * q^73 + 30 * q^79 - 64 * q^89 + 44 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 $ x^16 - x^15 + x^14 + 2*x^12 - 4*x^11 - 8*x^9 + 4*x^8 - 16*x^7 - 32*x^5 + 32*x^4 + 64*x^2 - 128*x + 256 :

$\beta_{1}$	$=$	$ ( - \nu^{15} - \nu^{14} - 3 \nu^{13} - 4 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} - 8 \nu^{9} - 4 \nu^{8} + \cdots + 256 ) / 192 $ (-v^15 - v^14 - 3v^13 - 4v^12 - 8v^11 - 6v^10 - 8v^9 - 4v^8 + 12v^7 + 24v^6 + 32v^5 + 72v^4 + 96v^3 + 128v^2 + 96*v + 256) / 192
$\beta_{2}$	$=$	$ ( - \nu^{14} - 3 \nu^{13} + 3 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} + 8 \nu^{8} + 60 \nu^{6} + \cdots + 128 \nu^{2} ) / 192 $ (-v^14 - 3v^13 + 3v^12 - 8v^11 - 6v^10 + 8v^8 + 60v^6 + 32v^5 + 96v^4 + 240v^3 + 128v^2) / 192
$\beta_{3}$	$=$	$ ( -\nu^{14} + \nu^{11} + 8\nu^{8} + 12\nu^{7} - 12\nu^{6} - 4\nu^{5} + 24\nu^{3} - 16\nu^{2} - 96\nu - 96 ) / 96 $ (-v^14 + v^11 + 8v^8 + 12v^7 - 12v^6 - 4v^5 + 24v^3 - 16v^2 - 96*v - 96) / 96
$\beta_{4}$	$=$	$ ( \nu^{14} + \nu^{13} + 2 \nu^{12} - \nu^{11} + 5 \nu^{10} + 4 \nu^{9} + 10 \nu^{8} - 8 \nu^{7} + \cdots + 32 ) / 96 $ (v^14 + v^13 + 2v^12 - v^11 + 5v^10 + 4v^9 + 10v^8 - 8v^7 - 4v^6 - 32v^5 - 20v^4 - 64v^3 - 80v^2 - 80*v + 32) / 96
$\beta_{5}$	$=$	$ ( \nu^{15} + 3 \nu^{14} + 5 \nu^{13} + 12 \nu^{12} + 18 \nu^{11} + 28 \nu^{10} + 24 \nu^{9} + 24 \nu^{8} + \cdots - 512 ) / 384 $ (v^15 + 3v^14 + 5v^13 + 12v^12 + 18v^11 + 28v^10 + 24v^9 + 24v^8 + 20v^7 - 64v^6 - 96v^5 - 160v^4 - 256v^3 - 384v^2 - 64v - 512) / 384
$\beta_{6}$	$=$	$ ( - \nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} + \cdots + 512 ) / 384 $ (-v^15 - 3v^14 - 5v^13 - 12v^12 - 18v^11 - 28v^10 - 24v^9 - 24v^8 - 20v^7 + 64v^6 + 96v^5 + 160v^4 + 256v^3 + 384v^2 + 832v + 512) / 384
$\beta_{7}$	$=$	$ ( 2 \nu^{15} + 3 \nu^{14} + \nu^{13} + \nu^{12} + 2 \nu^{10} - 28 \nu^{9} - 24 \nu^{8} - 32 \nu^{7} + \cdots + 192 ) / 192 $ (2v^15 + 3v^14 + v^13 + v^12 + 2v^10 - 28v^9 - 24v^8 - 32v^7 - 28v^6 - 32v^4 - 64v^3 + 256v + 192) / 192
$\beta_{8}$	$=$	$ ( - \nu^{15} + 2 \nu^{14} + 6 \nu^{13} + 7 \nu^{12} + 4 \nu^{11} + 8 \nu^{9} + 20 \nu^{8} + \cdots - 288 \nu ) / 192 $ (-v^15 + 2v^14 + 6v^13 + 7v^12 + 4v^11 + 8v^9 + 20v^8 + 12v^7 - 28v^6 - 64v^5 + 24v^4 - 112v^3 - 160v^2 - 288*v) / 192
$\beta_{9}$	$=$	$ ( 7 \nu^{15} + \nu^{14} + 7 \nu^{13} + 16 \nu^{12} + 38 \nu^{11} + 20 \nu^{10} + 32 \nu^{9} + \cdots - 1408 ) / 384 $ (7v^15 + v^14 + 7v^13 + 16v^12 + 38v^11 + 20v^10 + 32v^9 - 8v^8 + 28v^7 - 48v^6 - 224v^5 - 416v^4 - 288v^3 - 704v^2 - 320v - 1408) / 384
$\beta_{10}$	$=$	$ ( 7 \nu^{15} + \nu^{14} + 7 \nu^{13} + 16 \nu^{12} + 38 \nu^{11} + 20 \nu^{10} + 32 \nu^{9} + \cdots - 1408 ) / 384 $ (7v^15 + v^14 + 7v^13 + 16v^12 + 38v^11 + 20v^10 + 32v^9 - 8v^8 + 28v^7 - 48v^6 - 224v^5 - 416v^4 - 288v^3 + 64v^2 - 320v - 1408) / 384
$\beta_{11}$	$=$	$ ( 4 \nu^{15} + \nu^{14} + 3 \nu^{13} + \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} - 8 \nu^{8} + \cdots - 448 ) / 192 $ (4v^15 + v^14 + 3v^13 + v^12 + 8v^11 + 6v^10 + 8v^9 - 8v^8 - 12v^6 - 32v^5 - 96v^4 - 144v^3 - 128*v^2 - 448) / 192
$\beta_{12}$	$=$	$ ( 2 \nu^{15} + \nu^{14} + 4 \nu^{13} + 6 \nu^{12} + 11 \nu^{11} + 8 \nu^{10} + 12 \nu^{9} - 8 \nu^{8} + \cdots - 352 ) / 96 $ (2v^15 + v^14 + 4v^13 + 6v^12 + 11v^11 + 8v^10 + 12v^9 - 8v^8 + 4v^7 - 44v^6 - 44v^5 - 128v^4 - 104v^3 - 176v^2 - 32v - 352) / 96
$\beta_{13}$	$=$	$ ( 2 \nu^{15} - 3 \nu^{14} + 5 \nu^{13} + 13 \nu^{12} + 18 \nu^{11} + 10 \nu^{10} + 20 \nu^{9} + \cdots - 960 ) / 192 $ (2v^15 - 3v^14 + 5v^13 + 13v^12 + 18v^11 + 10v^10 + 20v^9 + 24v^8 + 56v^7 - 4v^6 - 72v^5 - 160v^4 - 208v^3 - 288v^2 - 448*v - 960) / 192
$\beta_{14}$	$=$	$ ( 6 \nu^{15} + \nu^{14} + 11 \nu^{13} + 11 \nu^{12} + 32 \nu^{11} + 22 \nu^{10} + 28 \nu^{9} + \cdots - 1216 ) / 192 $ (6v^15 + v^14 + 11v^13 + 11v^12 + 32v^11 + 22v^10 + 28v^9 - 8v^8 + 32v^7 - 52v^6 - 128v^5 - 352v^4 - 256v^3 - 512v^2 - 256v - 1216) / 192
$\beta_{15}$	$=$	$ ( - 12 \nu^{15} - 5 \nu^{14} - 15 \nu^{13} - 21 \nu^{12} - 64 \nu^{11} - 30 \nu^{10} - 48 \nu^{9} + \cdots + 2304 ) / 192 $ (-12v^15 - 5v^14 - 15v^13 - 21v^12 - 64v^11 - 30v^10 - 48v^9 + 16v^8 - 48v^7 + 156v^6 + 160v^5 + 768v^4 + 624v^3 + 736v^2 + 384*v + 2304) / 192

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

$\nu$	$=$	$ ( \beta_{6} + \beta_{5} ) / 2 $ (b6 + b5) / 2
$\nu^{2}$	$=$	$ ( \beta_{10} - \beta_{9} ) / 2 $ (b10 - b9) / 2
$\nu^{3}$	$=$	$ ( \beta_{14} - \beta_{13} - \beta_{11} + \beta_{3} + \beta_{2} ) / 2 $ (b14 - b13 - b11 + b3 + b2) / 2
$\nu^{4}$	$=$	$ ( \beta_{15} - \beta_{14} + 3\beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_1 ) / 2 $ (b15 - b14 + 3b11 + b9 + b8 - b6 + 2b5 - 3b4 - 2b1) / 2
$\nu^{5}$	$=$	$ ( - \beta_{15} + \beta_{13} - 2 \beta_{10} - 3 \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 2 ) / 2 $ (-b15 + b13 - 2b10 - 3b9 - b8 - b6 - b4 - b3 + b2 - 2*b1 + 2) / 2
$\nu^{6}$	$=$	$ ( 3\beta_{14} + \beta_{13} - 4\beta_{12} - \beta_{11} - 5\beta_{3} + \beta_{2} + 2 ) / 2 $ (3b14 + b13 - 4b12 - b11 - 5*b3 + b2 + 2) / 2
$\nu^{7}$	$=$	$ ( -\beta_{15} + \beta_{14} + \beta_{11} - \beta_{9} + 3\beta_{8} - \beta_{6} + 4\beta_{5} - \beta_{4} + 18\beta_1 ) / 2 $ (-b15 + b14 + b11 - b9 + 3b8 - b6 + 4b5 - b4 + 18*b1) / 2
$\nu^{8}$	$=$	$ ( - 3 \beta_{15} - \beta_{13} - 8 \beta_{12} - 3 \beta_{9} + \beta_{8} + 9 \beta_{6} + 8 \beta_{5} + \cdots + 10 ) / 2 $ (-3b15 - b13 - 8b12 - 3b9 + b8 + 9b6 + 8b5 + 5b4 + 9b3 + 3b2 - 10*b1 + 10) / 2
$\nu^{9}$	$=$	$ ( -5\beta_{14} - 3\beta_{13} + 8\beta_{12} + 7\beta_{11} - 12\beta_{7} - 5\beta_{3} + \beta_{2} + 6 ) / 2 $ (-5b14 - 3b13 + 8b12 + 7b11 - 12b7 - 5b3 + b2 + 6) / 2
$\nu^{10}$	$=$	$ ( 7 \beta_{15} + 13 \beta_{14} + \beta_{11} + 4 \beta_{10} - 13 \beta_{9} - 13 \beta_{8} + 4 \beta_{7} + \cdots + 2 \beta_1 ) / 2 $ (7b15 + 13b14 + b11 + 4b10 - 13b9 - 13b8 + 4b7 - 13b6 + 16b5 - b4 + 2*b1) / 2
$\nu^{11}$	$=$	$ ( \beta_{15} - 5 \beta_{13} - 24 \beta_{12} + 4 \beta_{10} + 13 \beta_{9} + 5 \beta_{8} - 3 \beta_{6} + \cdots + 50 ) / 2 $ (b15 - 5b13 - 24b12 + 4b10 + 13b9 + 5b8 - 3b6 + 24b5 - 39b4 - 3b3 - b2 - 50b1 + 50) / 2
$\nu^{12}$	$=$	$ ( -21\beta_{14} + 21\beta_{13} + 24\beta_{12} - 9\beta_{11} + 12\beta_{7} - 13\beta_{3} + 17\beta_{2} + 14 ) / 2 $ (-21b14 + 21b13 + 24b12 - 9b11 + 12b7 - 13b3 + 17*b2 + 14) / 2
$\nu^{13}$	$=$	$ ( 7 \beta_{15} + 61 \beta_{14} - 15 \beta_{11} + 4 \beta_{10} - 61 \beta_{9} + 19 \beta_{8} + \cdots - 14 \beta_1 ) / 2 $ (7b15 + 61b14 - 15b11 + 4b10 - 61b9 + 19b8 + 4b7 + 27b6 - 24b5 + 15b4 - 14*b1) / 2
$\nu^{14}$	$=$	$ ( - 31 \beta_{15} - 53 \beta_{13} - 40 \beta_{12} - 4 \beta_{10} + 5 \beta_{9} + 53 \beta_{8} - 35 \beta_{6} + \cdots - 94 ) / 2 $ (-31b15 - 53b13 - 40b12 - 4b10 + 5b9 + 53b8 - 35b6 + 40b5 - 7b4 - 35b3 + 31b2 + 94b1 - 94) / 2
$\nu^{15}$	$=$	$ ( -29\beta_{14} - 3\beta_{13} + 8\beta_{12} + 127\beta_{11} + 12\beta_{7} + 59\beta_{3} + 41\beta_{2} + 174 ) / 2 $ (-29b14 - 3b13 + 8b12 + 127b11 + 12b7 + 59b3 + 41*b2 + 174) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-\beta_{1}$	$1$

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} $

145.1

1.41411	+	0.0174668i
0.587625	−	1.28635i
−0.179748	+	1.40274i
−1.34532	−	0.436011i
1.05026	+	0.947078i
−1.12494	+	0.857038i
0.820200	−	1.15207i
−0.722180	−	1.21592i
1.41411	−	0.0174668i
0.587625	+	1.28635i
−0.179748	−	1.40274i
−1.34532	+	0.436011i
1.05026	−	0.947078i
−1.12494	−	0.857038i
0.820200	+	1.15207i
−0.722180	+	1.21592i

−3.17262

−

1.83171i

0.191926

0.332426i

145.2

−1.97542

−

1.14051i

0.907824

1.57240i

145.3

−1.19115

−

0.687709i

−1.80469

−

3.12581i

145.4

−0.602794

−

0.348023i

−0.795065

−

1.37709i

145.5

0.602794

0.348023i

−0.795065

−

1.37709i

145.6

1.19115

0.687709i

−1.80469

−

3.12581i

145.7

1.97542

1.14051i

0.907824

1.57240i

145.8

3.17262

1.83171i

0.191926

0.332426i

721.1

−3.17262

1.83171i

0.191926

−

0.332426i

721.2

−1.97542

1.14051i

0.907824

−

1.57240i

721.3

−1.19115

0.687709i

−1.80469

3.12581i

721.4

−0.602794

0.348023i

−0.795065

1.37709i

721.5

0.602794

−

0.348023i

−0.795065

1.37709i

721.6

1.19115

−

0.687709i

−1.80469

3.12581i

721.7

1.97542

−

1.14051i

0.907824

−

1.57240i

721.8

3.17262

−

1.83171i

0.191926

−

0.332426i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
9.c	even	3	1	inner
72.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	864.2.r.b		16
3.b	odd	2	1		288.2.r.b		16
4.b	odd	2	1		216.2.n.b		16
8.b	even	2	1	inner	864.2.r.b		16
8.d	odd	2	1		216.2.n.b		16
9.c	even	3	1	inner	864.2.r.b		16
9.c	even	3	1		2592.2.d.k		8
9.d	odd	6	1		288.2.r.b		16
9.d	odd	6	1		2592.2.d.j		8
12.b	even	2	1		72.2.n.b	✓	16
24.f	even	2	1		72.2.n.b	✓	16
24.h	odd	2	1		288.2.r.b		16
36.f	odd	6	1		216.2.n.b		16
36.f	odd	6	1		648.2.d.k		8
36.h	even	6	1		72.2.n.b	✓	16
36.h	even	6	1		648.2.d.j		8
72.j	odd	6	1		288.2.r.b		16
72.j	odd	6	1		2592.2.d.j		8
72.l	even	6	1		72.2.n.b	✓	16
72.l	even	6	1		648.2.d.j		8
72.n	even	6	1	inner	864.2.r.b		16
72.n	even	6	1		2592.2.d.k		8
72.p	odd	6	1		216.2.n.b		16
72.p	odd	6	1		648.2.d.k		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.2.n.b	✓	16	12.b	even	2	1
72.2.n.b	✓	16	24.f	even	2	1
72.2.n.b	✓	16	36.h	even	6	1
72.2.n.b	✓	16	72.l	even	6	1
216.2.n.b		16	4.b	odd	2	1
216.2.n.b		16	8.d	odd	2	1
216.2.n.b		16	36.f	odd	6	1
216.2.n.b		16	72.p	odd	6	1
288.2.r.b		16	3.b	odd	2	1
288.2.r.b		16	9.d	odd	6	1
288.2.r.b		16	24.h	odd	2	1
288.2.r.b		16	72.j	odd	6	1
648.2.d.j		8	36.h	even	6	1
648.2.d.j		8	72.l	even	6	1
648.2.d.k		8	36.f	odd	6	1
648.2.d.k		8	72.p	odd	6	1
864.2.r.b		16	1.a	even	1	1	trivial
864.2.r.b		16	8.b	even	2	1	inner
864.2.r.b		16	9.c	even	3	1	inner
864.2.r.b		16	72.n	even	6	1	inner
2592.2.d.j		8	9.d	odd	6	1
2592.2.d.j		8	72.j	odd	6	1
2592.2.d.k		8	9.c	even	3	1
2592.2.d.k		8	72.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{16} - 21T_{5}^{14} + 326T_{5}^{12} - 2049T_{5}^{10} + 9318T_{5}^{8} - 18357T_{5}^{6} + 26129T_{5}^{4} - 11712T_{5}^{2} + 4096 $ T5^16 - 21*T5^14 + 326*T5^12 - 2049*T5^10 + 9318*T5^8 - 18357*T5^6 + 26129*T5^4 - 11712*T5^2 + 4096 acting on $S_{2}^{\mathrm{new}}(864, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ T^{16} - 21 T^{14} + \cdots + 4096 $ T^16 - 21T^14 + 326T^12 - 2049T^10 + 9318T^8 - 18357T^6 + 26129T^4 - 11712*T^2 + 4096
$7$	$ (T^{8} + 3 T^{7} + 14 T^{6} + \cdots + 16)^{2} $ (T^8 + 3T^7 + 14T^6 + 3T^5 + 48T^4 + 21T^3 + 101T^2 - 36*T + 16)^2
$11$	$ T^{16} - 40 T^{14} + \cdots + 10556001 $ T^16 - 40T^14 + 1134T^12 - 14440T^10 + 129907T^8 - 718680T^6 + 2895966T^4 - 6822900*T^2 + 10556001
$13$	$ T^{16} - 53 T^{14} + \cdots + 20736 $ T^16 - 53T^14 + 2094T^12 - 36233T^10 + 467038T^8 - 578901T^6 + 587601T^4 - 119664*T^2 + 20736
$17$	$ (T^{4} - 7 T^{3} - 2 T^{2} + \cdots + 36)^{4} $ (T^4 - 7T^3 - 2T^2 + 48*T + 36)^4
$19$	$ (T^{8} + 83 T^{6} + \cdots + 5184)^{2} $ (T^8 + 83T^6 + 1660T^4 + 7056*T^2 + 5184)^2
$23$	$ (T^{8} + 5 T^{7} + \cdots + 90000)^{2} $ (T^8 + 5T^7 + 60T^6 + 275T^5 + 2650T^4 + 10875T^3 + 40125T^2 + 67500*T + 90000)^2
$29$	$ T^{16} + \cdots + 4032758016 $ T^16 - 109T^14 + 9246T^12 - 241441T^10 + 4385038T^8 - 46463373T^6 + 356481729T^4 - 1453416048*T^2 + 4032758016
$31$	$ (T^{8} - 5 T^{7} + \cdots + 92416)^{2} $ (T^8 - 5T^7 + 76T^6 - 155T^5 + 3322T^4 - 7415T^3 + 57529T^2 + 62320*T + 92416)^2
$37$	$ (T^{8} + 152 T^{6} + \cdots + 1327104)^{2} $ (T^8 + 152T^6 + 8080T^4 + 177264*T^2 + 1327104)^2
$41$	$ (T^{8} - 4 T^{7} + \cdots + 335241)^{2} $ (T^8 - 4T^7 + 90T^6 - 616T^5 + 7879T^4 - 38376T^3 + 165090T^2 - 264024*T + 335241)^2
$43$	$ T^{16} - 20 T^{14} + \cdots + 81 $ T^16 - 20T^14 + 294T^12 - 1880T^10 + 8827T^8 - 12360T^6 + 13446T^4 - 1080*T^2 + 81
$47$	$ (T^{8} - 3 T^{7} + \cdots + 2178576)^{2} $ (T^8 - 3T^7 + 90T^6 - 63T^5 + 5544T^4 - 3537T^3 + 142965T^2 + 225828*T + 2178576)^2
$53$	$ (T^{8} + 160 T^{6} + \cdots + 451584)^{2} $ (T^8 + 160T^6 + 7744T^4 + 113520*T^2 + 451584)^2
$59$	$ T^{16} + \cdots + 131079601 $ T^16 - 108T^14 + 8534T^12 - 287928T^10 + 7079403T^8 - 75952296T^6 + 591967766T^4 - 286866144*T^2 + 131079601
$61$	$ T^{16} + \cdots + 176319369216 $ T^16 - 165T^14 + 18846T^12 - 1107945T^10 + 47134062T^8 - 1011826485T^6 + 15331541409T^4 - 57650719680*T^2 + 176319369216
$67$	$ T^{16} + \cdots + 9881774573841 $ T^16 - 240T^14 + 38646T^12 - 3485160T^10 + 228454587T^8 - 8572738680T^6 + 223335161334T^4 - 1672043075100*T^2 + 9881774573841
$71$	$ (T^{4} - 18 T^{3} + \cdots - 864)^{4} $ (T^4 - 18T^3 + 48T^2 + 252*T - 864)^4
$73$	$ (T^{4} + 11 T^{3} + \cdots + 36)^{4} $ (T^4 + 11T^3 - 14T^2 - 84*T + 36)^4
$79$	$ (T^{8} - 15 T^{7} + \cdots + 3136)^{2} $ (T^8 - 15T^7 + 236T^6 - 1065T^5 + 9402T^4 - 8445T^3 + 377609T^2 - 34440*T + 3136)^2
$83$	$ T^{16} + \cdots + 31713911056 $ T^16 - 105T^14 + 7406T^12 - 289605T^10 + 8173602T^8 - 126163065T^6 + 1398102029T^4 - 8048506380*T^2 + 31713911056
$89$	$ (T^{4} + 16 T^{3} + \cdots - 504)^{4} $ (T^4 + 16T^3 + 52T^2 - 156*T - 504)^4
$97$	$ (T^{8} + 134 T^{6} + \cdots + 1324801)^{2} $ (T^8 + 134T^6 + 1560T^5 + 19107T^4 + 104520T^3 + 454166T^2 + 897780T + 1324801)^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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{11} - \beta_{5} + \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{3} + \beta_1 - 1) q^{7}+O(q^{10}) \) q + (-b11 - b5 + b4) * q^5 + (-b6 - b3 + b1 - 1) * q^7	\( q + ( - \beta_{11} - \beta_{5} + \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{13} - \beta_{8}) q^{11} + (\beta_{8} - \beta_{5}) q^{13} + ( - \beta_{7} + 2) q^{17} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{11}) q^{19}+ \cdots + (\beta_{15} - \beta_{10} - 3 \beta_{6} + \cdots - 1) q^{97}+O(q^{100}) \) q + (-b11 - b5 + b4) * q^5 + (-b6 - b3 + b1 - 1) * q^7 + (b13 - b8) * q^11 + (b8 - b5) * q^13 + (-b7 + 2) * q^17 + (-b14 + b13 + 2b12 + b11) q^19 + (-b15 - b10 - b7 - b1) * q^23 + (-b15 - b6 - b3 + b2) * q^25 + (b13 + b12 - 2b9 - b8 - b5) q^29 + (-b15 - b10 - b7 + 2b6 + b1) q^31 + (-b14 + b13) * q^35 + (b13 + 2b12 - b11) q^37 + (-b15 + b10 + b7 - b6 + b1) * q^41 + b4 * q^43 + (2b10 - b6 - b3 - b1 + 1) q^47 + (-b10 - b7 + 2b6 + 2b1) * q^49 + (2b14 + b13 - b11) q^53 + (-b7 + 2b3 + b2 + 1) q^55 + (2b14 - b11 - 2b9 - 2b5 + b4) q^59 + (b13 - b12 - 2b9 - b8 + b5 + 2b4) * q^61 + (-b10 - 2b6 - 2b3 - b1 + 1) * q^65 + (-2b11 + b8 - 4b5 + 2b4) q^67 + (b7 - b3 - b2 + 4) * q^71 + (-b7 + 2b3 - 2) q^73 + (-2b14 + 3b11 + 2b9 + b5 - 3b4) * q^77 + (-b15 - b10 - 2b6 - 2b3 + b2 - 3b1 + 3) q^79 + (-b13 + 2b12 + b9 + b8 - 2b5) * q^83 + (-2b14 - b11 + 2b9 - b8 - 2b5 + b4) q^85 + (-b7 - b3 - b2 - 4) * q^89 + (b14 - 3b11) q^91 + (2b15 - 2b6 + 6b1) q^95 + (b15 - b10 - 3b6 - 3b3 - b2 + b1 - 1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{7}+O(q^{10}) \) 16 * q - 6 * q^7	\( 16 q - 6 q^{7} + 28 q^{17} - 10 q^{23} + 2 q^{25} + 10 q^{31} + 8 q^{41} + 6 q^{47} + 18 q^{49} + 4 q^{55} + 14 q^{65} + 72 q^{71} - 44 q^{73} + 30 q^{79} - 64 q^{89} + 44 q^{95}+O(q^{100}) \) 16 * q - 6 * q^7 + 28 * q^17 - 10 * q^23 + 2 * q^25 + 10 * q^31 + 8 * q^41 + 6 * q^47 + 18 * q^49 + 4 * q^55 + 14 * q^65 + 72 * q^71 - 44 * q^73 + 30 * q^79 - 64 * q^89 + 44 * q^95

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	864.2.r.b

Level	$864$
Weight	$2$
Character orbit	864.r
Analytic conductor	$6.899$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.89907473464\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 \) x^16 - x^15 + x^14 + 2x^12 - 4x^11 - 8x^9 + 4x^8 - 16x^7 - 32x^5 + 32x^4 + 64x^2 - 128*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{8}\cdot 3^{4} \)
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - \nu^{15} - \nu^{14} - 3 \nu^{13} - 4 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} - 8 \nu^{9} - 4 \nu^{8} + \cdots + 256 ) / 192 \) (-v^15 - v^14 - 3v^13 - 4v^12 - 8v^11 - 6v^10 - 8v^9 - 4v^8 + 12v^7 + 24v^6 + 32v^5 + 72v^4 + 96v^3 + 128v^2 + 96*v + 256) / 192
\(\beta_{2}\)	\(=\)	\( ( - \nu^{14} - 3 \nu^{13} + 3 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} + 8 \nu^{8} + 60 \nu^{6} + \cdots + 128 \nu^{2} ) / 192 \) (-v^14 - 3v^13 + 3v^12 - 8v^11 - 6v^10 + 8v^8 + 60v^6 + 32v^5 + 96v^4 + 240v^3 + 128v^2) / 192
\(\beta_{3}\)	\(=\)	\( ( -\nu^{14} + \nu^{11} + 8\nu^{8} + 12\nu^{7} - 12\nu^{6} - 4\nu^{5} + 24\nu^{3} - 16\nu^{2} - 96\nu - 96 ) / 96 \) (-v^14 + v^11 + 8v^8 + 12v^7 - 12v^6 - 4v^5 + 24v^3 - 16v^2 - 96*v - 96) / 96
\(\beta_{4}\)	\(=\)	\( ( \nu^{14} + \nu^{13} + 2 \nu^{12} - \nu^{11} + 5 \nu^{10} + 4 \nu^{9} + 10 \nu^{8} - 8 \nu^{7} + \cdots + 32 ) / 96 \) (v^14 + v^13 + 2v^12 - v^11 + 5v^10 + 4v^9 + 10v^8 - 8v^7 - 4v^6 - 32v^5 - 20v^4 - 64v^3 - 80v^2 - 80*v + 32) / 96
\(\beta_{5}\)	\(=\)	\( ( \nu^{15} + 3 \nu^{14} + 5 \nu^{13} + 12 \nu^{12} + 18 \nu^{11} + 28 \nu^{10} + 24 \nu^{9} + 24 \nu^{8} + \cdots - 512 ) / 384 \) (v^15 + 3v^14 + 5v^13 + 12v^12 + 18v^11 + 28v^10 + 24v^9 + 24v^8 + 20v^7 - 64v^6 - 96v^5 - 160v^4 - 256v^3 - 384v^2 - 64v - 512) / 384
\(\beta_{6}\)	\(=\)	\( ( - \nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} + \cdots + 512 ) / 384 \) (-v^15 - 3v^14 - 5v^13 - 12v^12 - 18v^11 - 28v^10 - 24v^9 - 24v^8 - 20v^7 + 64v^6 + 96v^5 + 160v^4 + 256v^3 + 384v^2 + 832v + 512) / 384
\(\beta_{7}\)	\(=\)	\( ( 2 \nu^{15} + 3 \nu^{14} + \nu^{13} + \nu^{12} + 2 \nu^{10} - 28 \nu^{9} - 24 \nu^{8} - 32 \nu^{7} + \cdots + 192 ) / 192 \) (2v^15 + 3v^14 + v^13 + v^12 + 2v^10 - 28v^9 - 24v^8 - 32v^7 - 28v^6 - 32v^4 - 64v^3 + 256v + 192) / 192
\(\beta_{8}\)	\(=\)	\( ( - \nu^{15} + 2 \nu^{14} + 6 \nu^{13} + 7 \nu^{12} + 4 \nu^{11} + 8 \nu^{9} + 20 \nu^{8} + \cdots - 288 \nu ) / 192 \) (-v^15 + 2v^14 + 6v^13 + 7v^12 + 4v^11 + 8v^9 + 20v^8 + 12v^7 - 28v^6 - 64v^5 + 24v^4 - 112v^3 - 160v^2 - 288*v) / 192
\(\beta_{9}\)	\(=\)	\( ( 7 \nu^{15} + \nu^{14} + 7 \nu^{13} + 16 \nu^{12} + 38 \nu^{11} + 20 \nu^{10} + 32 \nu^{9} + \cdots - 1408 ) / 384 \) (7v^15 + v^14 + 7v^13 + 16v^12 + 38v^11 + 20v^10 + 32v^9 - 8v^8 + 28v^7 - 48v^6 - 224v^5 - 416v^4 - 288v^3 - 704v^2 - 320v - 1408) / 384
\(\beta_{10}\)	\(=\)	\( ( 7 \nu^{15} + \nu^{14} + 7 \nu^{13} + 16 \nu^{12} + 38 \nu^{11} + 20 \nu^{10} + 32 \nu^{9} + \cdots - 1408 ) / 384 \) (7v^15 + v^14 + 7v^13 + 16v^12 + 38v^11 + 20v^10 + 32v^9 - 8v^8 + 28v^7 - 48v^6 - 224v^5 - 416v^4 - 288v^3 + 64v^2 - 320v - 1408) / 384
\(\beta_{11}\)	\(=\)	\( ( 4 \nu^{15} + \nu^{14} + 3 \nu^{13} + \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} - 8 \nu^{8} + \cdots - 448 ) / 192 \) (4v^15 + v^14 + 3v^13 + v^12 + 8v^11 + 6v^10 + 8v^9 - 8v^8 - 12v^6 - 32v^5 - 96v^4 - 144v^3 - 128*v^2 - 448) / 192
\(\beta_{12}\)	\(=\)	\( ( 2 \nu^{15} + \nu^{14} + 4 \nu^{13} + 6 \nu^{12} + 11 \nu^{11} + 8 \nu^{10} + 12 \nu^{9} - 8 \nu^{8} + \cdots - 352 ) / 96 \) (2v^15 + v^14 + 4v^13 + 6v^12 + 11v^11 + 8v^10 + 12v^9 - 8v^8 + 4v^7 - 44v^6 - 44v^5 - 128v^4 - 104v^3 - 176v^2 - 32v - 352) / 96
\(\beta_{13}\)	\(=\)	\( ( 2 \nu^{15} - 3 \nu^{14} + 5 \nu^{13} + 13 \nu^{12} + 18 \nu^{11} + 10 \nu^{10} + 20 \nu^{9} + \cdots - 960 ) / 192 \) (2v^15 - 3v^14 + 5v^13 + 13v^12 + 18v^11 + 10v^10 + 20v^9 + 24v^8 + 56v^7 - 4v^6 - 72v^5 - 160v^4 - 208v^3 - 288v^2 - 448*v - 960) / 192
\(\beta_{14}\)	\(=\)	\( ( 6 \nu^{15} + \nu^{14} + 11 \nu^{13} + 11 \nu^{12} + 32 \nu^{11} + 22 \nu^{10} + 28 \nu^{9} + \cdots - 1216 ) / 192 \) (6v^15 + v^14 + 11v^13 + 11v^12 + 32v^11 + 22v^10 + 28v^9 - 8v^8 + 32v^7 - 52v^6 - 128v^5 - 352v^4 - 256v^3 - 512v^2 - 256v - 1216) / 192
\(\beta_{15}\)	\(=\)	\( ( - 12 \nu^{15} - 5 \nu^{14} - 15 \nu^{13} - 21 \nu^{12} - 64 \nu^{11} - 30 \nu^{10} - 48 \nu^{9} + \cdots + 2304 ) / 192 \) (-12v^15 - 5v^14 - 15v^13 - 21v^12 - 64v^11 - 30v^10 - 48v^9 + 16v^8 - 48v^7 + 156v^6 + 160v^5 + 768v^4 + 624v^3 + 736v^2 + 384*v + 2304) / 192

\(\nu\)	\(=\)	\( ( \beta_{6} + \beta_{5} ) / 2 \) (b6 + b5) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{10} - \beta_{9} ) / 2 \) (b10 - b9) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{14} - \beta_{13} - \beta_{11} + \beta_{3} + \beta_{2} ) / 2 \) (b14 - b13 - b11 + b3 + b2) / 2
\(\nu^{4}\)	\(=\)	\( ( \beta_{15} - \beta_{14} + 3\beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_1 ) / 2 \) (b15 - b14 + 3b11 + b9 + b8 - b6 + 2b5 - 3b4 - 2b1) / 2
\(\nu^{5}\)	\(=\)	\( ( - \beta_{15} + \beta_{13} - 2 \beta_{10} - 3 \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 2 ) / 2 \) (-b15 + b13 - 2b10 - 3b9 - b8 - b6 - b4 - b3 + b2 - 2*b1 + 2) / 2
\(\nu^{6}\)	\(=\)	\( ( 3\beta_{14} + \beta_{13} - 4\beta_{12} - \beta_{11} - 5\beta_{3} + \beta_{2} + 2 ) / 2 \) (3b14 + b13 - 4b12 - b11 - 5*b3 + b2 + 2) / 2
\(\nu^{7}\)	\(=\)	\( ( -\beta_{15} + \beta_{14} + \beta_{11} - \beta_{9} + 3\beta_{8} - \beta_{6} + 4\beta_{5} - \beta_{4} + 18\beta_1 ) / 2 \) (-b15 + b14 + b11 - b9 + 3b8 - b6 + 4b5 - b4 + 18*b1) / 2
\(\nu^{8}\)	\(=\)	\( ( - 3 \beta_{15} - \beta_{13} - 8 \beta_{12} - 3 \beta_{9} + \beta_{8} + 9 \beta_{6} + 8 \beta_{5} + \cdots + 10 ) / 2 \) (-3b15 - b13 - 8b12 - 3b9 + b8 + 9b6 + 8b5 + 5b4 + 9b3 + 3b2 - 10*b1 + 10) / 2
\(\nu^{9}\)	\(=\)	\( ( -5\beta_{14} - 3\beta_{13} + 8\beta_{12} + 7\beta_{11} - 12\beta_{7} - 5\beta_{3} + \beta_{2} + 6 ) / 2 \) (-5b14 - 3b13 + 8b12 + 7b11 - 12b7 - 5b3 + b2 + 6) / 2
\(\nu^{10}\)	\(=\)	\( ( 7 \beta_{15} + 13 \beta_{14} + \beta_{11} + 4 \beta_{10} - 13 \beta_{9} - 13 \beta_{8} + 4 \beta_{7} + \cdots + 2 \beta_1 ) / 2 \) (7b15 + 13b14 + b11 + 4b10 - 13b9 - 13b8 + 4b7 - 13b6 + 16b5 - b4 + 2*b1) / 2
\(\nu^{11}\)	\(=\)	\( ( \beta_{15} - 5 \beta_{13} - 24 \beta_{12} + 4 \beta_{10} + 13 \beta_{9} + 5 \beta_{8} - 3 \beta_{6} + \cdots + 50 ) / 2 \) (b15 - 5b13 - 24b12 + 4b10 + 13b9 + 5b8 - 3b6 + 24b5 - 39b4 - 3b3 - b2 - 50b1 + 50) / 2
\(\nu^{12}\)	\(=\)	\( ( -21\beta_{14} + 21\beta_{13} + 24\beta_{12} - 9\beta_{11} + 12\beta_{7} - 13\beta_{3} + 17\beta_{2} + 14 ) / 2 \) (-21b14 + 21b13 + 24b12 - 9b11 + 12b7 - 13b3 + 17*b2 + 14) / 2
\(\nu^{13}\)	\(=\)	\( ( 7 \beta_{15} + 61 \beta_{14} - 15 \beta_{11} + 4 \beta_{10} - 61 \beta_{9} + 19 \beta_{8} + \cdots - 14 \beta_1 ) / 2 \) (7b15 + 61b14 - 15b11 + 4b10 - 61b9 + 19b8 + 4b7 + 27b6 - 24b5 + 15b4 - 14*b1) / 2
\(\nu^{14}\)	\(=\)	\( ( - 31 \beta_{15} - 53 \beta_{13} - 40 \beta_{12} - 4 \beta_{10} + 5 \beta_{9} + 53 \beta_{8} - 35 \beta_{6} + \cdots - 94 ) / 2 \) (-31b15 - 53b13 - 40b12 - 4b10 + 5b9 + 53b8 - 35b6 + 40b5 - 7b4 - 35b3 + 31b2 + 94b1 - 94) / 2
\(\nu^{15}\)	\(=\)	\( ( -29\beta_{14} - 3\beta_{13} + 8\beta_{12} + 127\beta_{11} + 12\beta_{7} + 59\beta_{3} + 41\beta_{2} + 174 ) / 2 \) (-29b14 - 3b13 + 8b12 + 127b11 + 12b7 + 59b3 + 41*b2 + 174) / 2

\(n\)	\(325\)	\(353\)	\(703\)
\(\chi(n)\)	\(-1\)	\(-\beta_{1}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 145.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( T^{16} - 21 T^{14} + \cdots + 4096 \) T^16 - 21T^14 + 326T^12 - 2049T^10 + 9318T^8 - 18357T^6 + 26129T^4 - 11712*T^2 + 4096
$7$	\( (T^{8} + 3 T^{7} + 14 T^{6} + \cdots + 16)^{2} \) (T^8 + 3T^7 + 14T^6 + 3T^5 + 48T^4 + 21T^3 + 101T^2 - 36*T + 16)^2
$11$	\( T^{16} - 40 T^{14} + \cdots + 10556001 \) T^16 - 40T^14 + 1134T^12 - 14440T^10 + 129907T^8 - 718680T^6 + 2895966T^4 - 6822900*T^2 + 10556001
$13$	\( T^{16} - 53 T^{14} + \cdots + 20736 \) T^16 - 53T^14 + 2094T^12 - 36233T^10 + 467038T^8 - 578901T^6 + 587601T^4 - 119664*T^2 + 20736
$17$	\( (T^{4} - 7 T^{3} - 2 T^{2} + \cdots + 36)^{4} \) (T^4 - 7T^3 - 2T^2 + 48*T + 36)^4
$19$	\( (T^{8} + 83 T^{6} + \cdots + 5184)^{2} \) (T^8 + 83T^6 + 1660T^4 + 7056*T^2 + 5184)^2
$23$	\( (T^{8} + 5 T^{7} + \cdots + 90000)^{2} \) (T^8 + 5T^7 + 60T^6 + 275T^5 + 2650T^4 + 10875T^3 + 40125T^2 + 67500*T + 90000)^2
$29$	\( T^{16} + \cdots + 4032758016 \) T^16 - 109T^14 + 9246T^12 - 241441T^10 + 4385038T^8 - 46463373T^6 + 356481729T^4 - 1453416048*T^2 + 4032758016
$31$	\( (T^{8} - 5 T^{7} + \cdots + 92416)^{2} \) (T^8 - 5T^7 + 76T^6 - 155T^5 + 3322T^4 - 7415T^3 + 57529T^2 + 62320*T + 92416)^2
$37$	\( (T^{8} + 152 T^{6} + \cdots + 1327104)^{2} \) (T^8 + 152T^6 + 8080T^4 + 177264*T^2 + 1327104)^2
$41$	\( (T^{8} - 4 T^{7} + \cdots + 335241)^{2} \) (T^8 - 4T^7 + 90T^6 - 616T^5 + 7879T^4 - 38376T^3 + 165090T^2 - 264024*T + 335241)^2
$43$	\( T^{16} - 20 T^{14} + \cdots + 81 \) T^16 - 20T^14 + 294T^12 - 1880T^10 + 8827T^8 - 12360T^6 + 13446T^4 - 1080*T^2 + 81
$47$	\( (T^{8} - 3 T^{7} + \cdots + 2178576)^{2} \) (T^8 - 3T^7 + 90T^6 - 63T^5 + 5544T^4 - 3537T^3 + 142965T^2 + 225828*T + 2178576)^2
$53$	\( (T^{8} + 160 T^{6} + \cdots + 451584)^{2} \) (T^8 + 160T^6 + 7744T^4 + 113520*T^2 + 451584)^2
$59$	\( T^{16} + \cdots + 131079601 \) T^16 - 108T^14 + 8534T^12 - 287928T^10 + 7079403T^8 - 75952296T^6 + 591967766T^4 - 286866144*T^2 + 131079601
$61$	\( T^{16} + \cdots + 176319369216 \) T^16 - 165T^14 + 18846T^12 - 1107945T^10 + 47134062T^8 - 1011826485T^6 + 15331541409T^4 - 57650719680*T^2 + 176319369216
$67$	\( T^{16} + \cdots + 9881774573841 \) T^16 - 240T^14 + 38646T^12 - 3485160T^10 + 228454587T^8 - 8572738680T^6 + 223335161334T^4 - 1672043075100*T^2 + 9881774573841
$71$	\( (T^{4} - 18 T^{3} + \cdots - 864)^{4} \) (T^4 - 18T^3 + 48T^2 + 252*T - 864)^4
$73$	\( (T^{4} + 11 T^{3} + \cdots + 36)^{4} \) (T^4 + 11T^3 - 14T^2 - 84*T + 36)^4
$79$	\( (T^{8} - 15 T^{7} + \cdots + 3136)^{2} \) (T^8 - 15T^7 + 236T^6 - 1065T^5 + 9402T^4 - 8445T^3 + 377609T^2 - 34440*T + 3136)^2
$83$	\( T^{16} + \cdots + 31713911056 \) T^16 - 105T^14 + 7406T^12 - 289605T^10 + 8173602T^8 - 126163065T^6 + 1398102029T^4 - 8048506380*T^2 + 31713911056
$89$	\( (T^{4} + 16 T^{3} + \cdots - 504)^{4} \) (T^4 + 16T^3 + 52T^2 - 156*T - 504)^4
$97$	\( (T^{8} + 134 T^{6} + \cdots + 1324801)^{2} \) (T^8 + 134T^6 + 1560T^5 + 19107T^4 + 104520T^3 + 454166T^2 + 897780T + 1324801)^2
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