LMFDB - Newform orbit 1456.2.cc.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(225,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1456.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.cc (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:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.58891012706304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 $ x^12 - 5x^10 - 2x^9 + 15x^8 + 2x^7 - 30x^6 + 4x^5 + 60x^4 - 16x^3 - 80*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{8} - \beta_{3}) q^{3} + (\beta_{10} - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{11} + \beta_{9} + \cdots + \beta_1) q^{9}+O(q^{10}) $ q + (b8 - b3) * q^3 + (b10 - b7 + b6 - b2 - b1) * q^5 - b4 * q^7 + (-b11 + b9 + b8 - b6 + b5 + b2 + b1) * q^9	$ q + (\beta_{8} - \beta_{3}) q^{3} + (\beta_{10} - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{11} + 2 \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) $ q + (b8 - b3) * q^3 + (b10 - b7 + b6 - b2 - b1) * q^5 - b4 * q^7 + (-b11 + b9 + b8 - b6 + b5 + b2 + b1) * q^9 + (b11 - b8 - 2b7 + b6 - b5 - 2b4 - b3 - b1) * q^11 + (-b11 + b10 + 2b9 + 2b8 - 2b6 + b5 - b4 + 2b2 - b1 - 1) * q^13 + (b11 - b9 - b7 - b5 - b4 - 1) * q^15 + (b10 + b8 - 2b7 + b5 - b4 + b3 + b2) q^17 + (b9 + b8 - b6 - b5 - 3b4 - b3 + b2 - b1 - 2) q^19 + (b2 + b1) * q^21 + (b11 - 3b10 - b9 + b8 + b7 - 2b6 + b5 - b4 - b3 + 3b2 + b1 + 1) q^23 + (2b8 - b7 - 2b5 - 2b4 + b2 - b1 - 1) q^25 + (2b11 + b10 + b7 - b6 - 2b5 + 2b4 - 2b3 - 1) * q^27 + (b10 - 3b9 + b8 + 2b6 - b3 - b2 + b1 + 3) * q^29 + (-2b11 - 2b10 + 3b8 + 2b7 - 2b6 + b5 - 2b3) * q^31 + (-2b10 + b8 + 2b6 - b5 - b3 + 2b1) q^33 + (b9 - b8 - b5 - b3) * q^35 + (2b10 - b8 - b7 - 3b6 - b4 - b3 + 2b2 + b1) q^37 + (-b11 + 2b10 + b9 - b7 - 3b6 - 2b5 + b4 - b3 - 2b1 - 3) * q^39 + (-3b11 + b10 - b9 + 4b8 - 2b7 - b6 + 3b5 - 2b4 + 4b3 + b2 - 1) * q^41 + (2b11 + b9 - b8 - 2b7 - b5 - b4 + b3) * q^43 + (b10 - b9 + b6 + 2b4 - 2b2 + b1 + 2) * q^45 + (-2b11 + b10 - 2b9 + 2b8 - b7 + b6 - 2b3 - 2b2 - 2b1 + 1) * q^47 + (-b9 + 1) * q^49 + (b10 + b8 - b6 - b5 + b2 - b1 - 2) * q^51 + (-2b8 + 2b5 + 3) * q^53 + (-2b11 - b10 + 5b9 + b8 - b7 - 2b5 + b4 - b3 + b2 + b1 - 5) q^55 + (-b11 - b10 - b6 - b5 - b3 + 2b2 + 2b1) * q^57 + (-3b11 + 2b10 + 2b8 - 2b6 - 2b5 - 2b4 + b3 - 2b1) q^59 + (b10 - b9 + b8 - b6 + b5 + b3 + 2b2 + b1) q^61 + (-b11 + b10 + b8 - b7 - b6 + b5 - b4 + b3 + b2) * q^63 + (-2b11 + b10 + 5b9 + b8 + b7 - 2b6 - 2b5 - 3b3 + 2b1 - 2) * q^65 + (b11 - b10 - 2b7 - b6 - b5 - 2b4 - b2 + 2b1) q^67 + (-4b11 + 2b9 + 2b8 + 2b5 - 2b3) q^69 + (-2b11 + 2b9 - 2b4 + 2b3 - 4) * q^71 + (3b11 - 2b10 + 2b9 - 5b8 - 2b7 - 2b6 - 2b5 + 3b3 + 3b2 + 3b1 - 1) * q^73 + (-b11 - 2b10 + 4b8 + 2b7 - 2b6 - b5 - 2b4 - 4b3 + 2b2) q^75 + (b11 - b10 - b8 + b6 - b3 + 2) * q^77 + (4b11 - 3b10 - b8 + 3b7 + 3b6 - 3b5 + 6b4 - 4b3 - 2b2 + 2b1 + 1) q^79 + (-4b10 - 2b9 + 3b8 - b7 - b6 + b4 - 3b3 + 4b2 + 3b1 + 2) * q^81 + (4b10 - b8 - 6b7 + 4b6 - b5 - 4b2 - 4b1) q^83 + (-3b11 + 3b9 + 3b8 - b6 - 3b5 - 4b4 + b2 - b1 - 6) q^85 + (-3b11 + 2b10 - 5b9 + 5b8 + 2b7 - 3b6 + 5b5 + b4 + 2b3 + 5b2 + 3b1) * q^87 + (5b11 + b10 - 3b9 - 2b8 - b7 - 2b6 - 5b5 - b4 - 2b3 + b2 + b1 - 3) * q^89 + (b10 - b9 - b8 - 2b7 - b6 + 2b5 - b4 + 2b3 + b2 + b1 + 1) q^91 + (-4b11 - b10 - b9 - b7 - b6 + 4b5 - b4 - b2 + 2b1 - 1) q^93 + (-b11 - 2b10 + 6b9 - 3b8 + 4b7 - 3b6 - 3b5 + 2b4 - 4b3 + b2 + 3b1) q^95 + (-4b11 + 4b10 + 2b9 + 5b8 - 4b6 - 5b5 - b3 - 4b1 - 4) q^97 + (-b11 + 2b10 - 2b9 + b8 - b7 + 2b6 - b3 + 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{3} + 4 q^{9}+O(q^{10}) $ 12 * q + 2 * q^3 + 4 * q^9	$ 12 q + 2 q^{3} + 4 q^{9} + 2 q^{13} - 12 q^{15} + 2 q^{17} - 12 q^{19} + 6 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 6 q^{33} + 4 q^{35} - 6 q^{37} - 26 q^{39} - 12 q^{41} + 12 q^{43} + 18 q^{45} + 6 q^{49} - 16 q^{51} + 20 q^{53} - 24 q^{55} + 12 q^{59} - 4 q^{61} + 8 q^{65} + 6 q^{67} - 36 q^{71} + 10 q^{75} + 20 q^{77} + 20 q^{79} + 18 q^{81} - 36 q^{85} - 32 q^{87} - 36 q^{89} - 2 q^{91} - 42 q^{93} + 26 q^{95} - 6 q^{97}+O(q^{100}) $ 12 * q + 2 * q^3 + 4 * q^9 + 2 * q^13 - 12 * q^15 + 2 * q^17 - 12 * q^19 + 6 * q^23 + 4 * q^25 - 4 * q^27 + 20 * q^29 + 6 * q^33 + 4 * q^35 - 6 * q^37 - 26 * q^39 - 12 * q^41 + 12 * q^43 + 18 * q^45 + 6 * q^49 - 16 * q^51 + 20 * q^53 - 24 * q^55 + 12 * q^59 - 4 * q^61 + 8 * q^65 + 6 * q^67 - 36 * q^71 + 10 * q^75 + 20 * q^77 + 20 * q^79 + 18 * q^81 - 36 * q^85 - 32 * q^87 - 36 * q^89 - 2 * q^91 - 42 * q^93 + 26 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 $ x^12 - 5*x^10 - 2*x^9 + 15*x^8 + 2*x^7 - 30*x^6 + 4*x^5 + 60*x^4 - 16*x^3 - 80*x^2 + 64 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 80\nu ) / 32 $ (v^11 - 5v^9 - 2v^8 + 15v^7 + 2v^6 - 30v^5 + 4v^4 + 60v^3 - 16v^2 - 80*v) / 32
$\beta_{3}$	$=$	$ ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} + \cdots + 96 ) / 32 $ (-3v^11 + 4v^10 + 7v^9 - 6v^8 - 13v^7 + 30v^6 - 6v^5 - 28v^4 - 4v^3 + 16v^2 - 48*v + 96) / 32
$\beta_{4}$	$=$	$ ( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + \cdots - 64 ) / 32 $ (3v^11 + 2v^10 - 11v^9 - 8v^8 + 21v^7 + 4v^6 - 42v^5 + 84v^3 + 24v^2 - 64v - 64) / 32
$\beta_{5}$	$=$	$ ( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + \cdots - 64 ) / 16 $ (v^11 + 3v^10 - 3v^9 - 13v^8 + 7v^7 + 23v^6 - 26v^5 - 38v^4 + 60v^3 + 68v^2 - 56v - 64) / 16
$\beta_{6}$	$=$	$ ( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} + \cdots + 64 ) / 32 $ (-5v^11 + 4v^10 + 13v^9 - 10v^8 - 23v^7 + 42v^6 + 10v^5 - 68v^4 - 20v^3 + 48v^2 - 32*v + 64) / 32
$\beta_{7}$	$=$	$ ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} + \cdots - 32 ) / 32 $ (-5v^11 + 2v^10 + 17v^9 - 8v^8 - 39v^7 + 44v^6 + 50v^5 - 88v^4 - 68v^3 + 120v^2 + 16*v - 32) / 32
$\beta_{8}$	$=$	$ ( - \nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} + \cdots - 32 ) / 8 $ (-v^11 + v^10 + 5v^9 - 3v^8 - 13v^7 + 13v^6 + 20v^5 - 34v^4 - 28v^3 + 52v^2 + 24*v - 32) / 8
$\beta_{9}$	$=$	$ ( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + \cdots - 80 ) / 16 $ (-v^11 + 3v^10 + 5v^9 - 13v^8 - 13v^7 + 35v^6 + 12v^5 - 70v^4 + 8v^3 + 108v^2 - 16v - 80) / 16
$\beta_{10}$	$=$	$ ( - 3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + \cdots - 64 ) / 16 $ (-3v^11 + 15v^9 - 2v^8 - 37v^7 + 18v^6 + 66v^5 - 68v^4 - 92v^3 + 96v^2 + 96v - 64) / 16
$\beta_{11}$	$=$	$ ( - \nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} + \cdots - 160 ) / 16 $ (-v^11 + 4v^10 + 9v^9 - 18v^8 - 27v^7 + 50v^6 + 34v^5 - 116v^4 - 20v^3 + 192v^2 + 16v - 160) / 16

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1 $ -b11 + b8 + b7 - b6 + b5 + 1
$\nu^{3}$	$=$	$ -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 $ -b11 + b10 + b9 - b6 + b4 + b3 + b2 + b1
$\nu^{4}$	$=$	$ -\beta_{11} + \beta_{9} + \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1 $ -b11 + b9 + b7 - 2*b6 + b5 - b4 + b3 - b2 - 1
$\nu^{5}$	$=$	$ -2\beta_{11} + 4\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta _1 + 1 $ -2b11 + 4b9 + b8 - b7 - b6 - b5 - b1 + 1
$\nu^{6}$	$=$	$ 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \cdots - 2 $ 2b11 - 2b10 + b9 - b8 - b6 - 2b5 - 3b4 + b3 + b2 + 4*b1 - 2
$\nu^{7}$	$=$	$ - 5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + \cdots + 1 $ -5b11 - 3b10 + 5b9 + 7b8 - b7 - 2b6 + 3b5 - 7b4 - 3b3 + b2 + 1
$\nu^{8}$	$=$	$ -3\beta_{11} + 7\beta_{8} + \beta_{7} - 5\beta_{6} - \beta_{5} + 2\beta_{4} + 4\beta_{3} + 10\beta_{2} + 8\beta _1 - 3 $ -3b11 + 7b8 + b7 - 5b6 - b5 + 2b4 + 4b3 + 10b2 + 8*b1 - 3
$\nu^{9}$	$=$	$ - 7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} + \cdots - 8 $ -7b11 + 3b10 + 3b9 + 6b8 - 4b7 - 11b6 + 12b5 - 17b4 + 7b3 + 7b2 + b1 - 8
$\nu^{10}$	$=$	$ - 17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + \cdots - 11 $ -17b11 + 4b10 + 13b9 + 18b8 - 9b7 - 8b6 + 3b5 + 7b4 + 9b3 + 3b2 - 6*b1 - 11
$\nu^{11}$	$=$	$ 18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} + \cdots - 7 $ 18b11 + 4b10 - 6b9 - 13b8 - 21b7 - 11b6 - b5 - 26b4 + 22b3 + 14b2 + 3b1 - 7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$\beta_{9}$	$1$	$1$	$1$

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} $

225.1

−1.12906	−	0.851598i
−1.30089	+	0.554694i
1.40744	+	0.138282i
1.34408	+	0.439820i
−1.08105	+	0.911778i
0.759479	−	1.19298i
−1.12906	+	0.851598i
−1.30089	−	0.554694i
1.40744	−	0.138282i
1.34408	−	0.439820i
−1.08105	−	0.911778i
0.759479	+	1.19298i

−0.851598

1.47501i

1.80719i

−0.866025

0.500000i

0.0495616

0.0858431i

225.2

−0.554694

0.960759i

2.69851i

0.866025

−

0.500000i

0.884629

1.53222i

225.3

−0.138282

0.239512i

−

1.57603i

0.866025

−

0.500000i

1.46176

2.53184i

225.4

0.439820

−

0.761791i

−

3.76784i

−0.866025

0.500000i

1.11312

1.92797i

225.5

0.911778

−

1.57925i

−

0.0393510i

−0.866025

0.500000i

−0.162678

−

0.281767i

225.6

1.19298

−

2.06630i

0.877521i

0.866025

−

0.500000i

−1.34638

−

2.33201i

673.1

−0.851598

−

1.47501i

−

1.80719i

−0.866025

−

0.500000i

0.0495616

−

0.0858431i

673.2

−0.554694

−

0.960759i

−

2.69851i

0.866025

0.500000i

0.884629

−

1.53222i

673.3

−0.138282

−

0.239512i

1.57603i

0.866025

0.500000i

1.46176

−

2.53184i

673.4

0.439820

0.761791i

3.76784i

−0.866025

−

0.500000i

1.11312

−

1.92797i

673.5

0.911778

1.57925i

0.0393510i

−0.866025

−

0.500000i

−0.162678

0.281767i

673.6

1.19298

2.06630i

−

0.877521i

0.866025

0.500000i

−1.34638

2.33201i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1456.2.cc.e		12
4.b	odd	2	1		728.2.bm.b	✓	12
13.e	even	6	1	inner	1456.2.cc.e		12
52.i	odd	6	1		728.2.bm.b	✓	12
52.l	even	12	1		9464.2.a.bf		6
52.l	even	12	1		9464.2.a.bg		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
728.2.bm.b	✓	12	4.b	odd	2	1
728.2.bm.b	✓	12	52.i	odd	6	1
1456.2.cc.e		12	1.a	even	1	1	trivial
1456.2.cc.e		12	13.e	even	6	1	inner
9464.2.a.bf		6	52.l	even	12	1
9464.2.a.bg		6	52.l	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{12} - 2 T_{3}^{11} + 9 T_{3}^{10} - 6 T_{3}^{9} + 34 T_{3}^{8} - 18 T_{3}^{7} + 83 T_{3}^{6} + \cdots + 4 $ T3^12 - 2*T3^11 + 9*T3^10 - 6*T3^9 + 34*T3^8 - 18*T3^7 + 83*T3^6 + 87*T3^4 - 10*T3^3 + 50*T3^2 + 12*T3 + 4 acting on $S_{2}^{\mathrm{new}}(1456, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} - 2 T^{11} + \cdots + 4 $ T^12 - 2T^11 + 9T^10 - 6T^9 + 34T^8 - 18T^7 + 83T^6 + 87T^4 - 10T^3 + 50T^2 + 12T + 4
$5$	$ T^{12} + 28 T^{10} + \cdots + 1 $ T^12 + 28T^10 + 256T^8 + 950T^6 + 1432T^4 + 648*T^2 + 1
$7$	$ (T^{4} - T^{2} + 1)^{3} $ (T^4 - T^2 + 1)^3
$11$	$ T^{12} - 21 T^{10} + \cdots + 5476 $ T^12 - 21T^10 + 334T^8 + 414T^7 - 1963T^6 - 3852T^5 + 8899T^4 + 36594T^3 + 46906T^2 + 25308*T + 5476
$13$	$ T^{12} - 2 T^{11} + \cdots + 4826809 $ T^12 - 2T^11 - 11T^10 - 2T^9 - 106T^8 - 98T^7 + 5417T^6 - 1274T^5 - 17914T^4 - 4394T^3 - 314171T^2 - 742586*T + 4826809
$17$	$ T^{12} - 2 T^{11} + \cdots + 2401 $ T^12 - 2T^11 + 25T^10 - 102T^9 + 622T^8 - 1730T^7 + 4365T^6 - 5702T^5 + 7438T^4 - 4466T^3 + 6713T^2 - 3430*T + 2401
$19$	$ T^{12} + 12 T^{11} + \cdots + 64 $ T^12 + 12T^11 + 35T^10 - 156T^9 - 376T^8 + 1458T^7 + 2199T^6 - 14070T^5 + 23361T^4 - 18084T^3 + 6904T^2 - 1056*T + 64
$23$	$ T^{12} - 6 T^{11} + \cdots + 52765696 $ T^12 - 6T^11 + 110T^10 - 388T^9 + 7236T^8 - 28128T^7 + 204992T^6 - 651456T^5 + 3648448T^4 - 10589184T^3 + 32796672T^2 - 44862464*T + 52765696
$29$	$ T^{12} - 20 T^{11} + \cdots + 1 $ T^12 - 20T^11 + 300T^10 - 2292T^9 + 14260T^8 - 39852T^7 + 138278T^6 - 25260T^5 + 1919892T^4 + 1141388T^3 + 727244T^2 - 852*T + 1
$31$	$ T^{12} + 152 T^{10} + \cdots + 13118884 $ T^12 + 152T^10 + 7154T^8 + 151248T^6 + 1572153T^4 + 7639576*T^2 + 13118884
$37$	$ T^{12} + 6 T^{11} + \cdots + 614656 $ T^12 + 6T^11 - 69T^10 - 486T^9 + 4909T^8 + 11400T^7 - 104644T^6 - 164736T^5 + 1997440T^4 - 2372160T^3 - 166208T^2 + 1317120*T + 614656
$41$	$ T^{12} + 12 T^{11} + \cdots + 32993536 $ T^12 + 12T^11 - 57T^10 - 1260T^9 + 6253T^8 + 73536T^7 + 3548T^6 - 1299456T^5 - 425984T^4 + 18313152T^3 + 57820864T^2 + 69755136*T + 32993536
$43$	$ T^{12} - 12 T^{11} + \cdots + 148996 $ T^12 - 12T^11 + 123T^10 - 500T^9 + 2070T^8 - 1386T^7 + 10293T^6 + 12600T^5 + 103131T^4 + 181174T^3 + 312810T^2 + 233916*T + 148996
$47$	$ T^{12} + \cdots + 150601984 $ T^12 + 184T^10 + 12438T^8 + 404616T^6 + 6708569T^4 + 53130080*T^2 + 150601984
$53$	$ (T^{6} - 10 T^{5} + \cdots - 383)^{2} $ (T^6 - 10T^5 - 17T^4 + 196T^3 + 463T^2 + 6*T - 383)^2
$59$	$ T^{12} - 12 T^{11} + \cdots + 964324 $ T^12 - 12T^11 - 24T^10 + 864T^9 + 2011T^8 - 47154T^7 + 115352T^6 + 300774T^5 - 233795T^4 - 766182T^3 + 468046T^2 + 1632084*T + 964324
$61$	$ T^{12} + 4 T^{11} + \cdots + 4096 $ T^12 + 4T^11 + 31T^10 + 36T^9 + 385T^8 + 592T^7 + 2400T^6 + 2560T^5 + 8128T^4 + 10240T^3 + 14336T^2 + 8192*T + 4096
$67$	$ T^{12} + \cdots + 131010916 $ T^12 - 6T^11 - 80T^10 + 552T^9 + 5451T^8 - 24114T^7 - 184908T^6 + 563592T^5 + 5320169T^4 + 2333370T^3 - 28429714T^2 - 10644780*T + 131010916
$71$	$ T^{12} + 36 T^{11} + \cdots + 65536 $ T^12 + 36T^11 + 568T^10 + 4896T^9 + 24048T^8 + 60672T^7 + 35712T^6 - 149760T^5 - 106240T^4 + 706560T^3 + 1323008T^2 + 491520*T + 65536
$73$	$ T^{12} + \cdots + 322751244544 $ T^12 + 610T^10 + 145225T^8 + 17148728T^6 + 1050041392T^4 + 30933985152*T^2 + 322751244544
$79$	$ (T^{6} - 10 T^{5} + \cdots + 55648)^{2} $ (T^6 - 10T^5 - 142T^4 + 1440T^3 + 3624T^2 - 44000*T + 55648)^2
$83$	$ T^{12} + \cdots + 1205895076 $ T^12 + 488T^10 + 77378T^8 + 4526688T^6 + 101760105T^4 + 685457080*T^2 + 1205895076
$89$	$ T^{12} + 36 T^{11} + \cdots + 87616 $ T^12 + 36T^11 + 427T^10 - 180T^9 - 24480T^8 + 3054T^7 + 1104231T^6 - 3459906T^5 - 697375T^4 + 11672700T^3 + 14176472T^2 + 1893216*T + 87616
$97$	$ T^{12} + \cdots + 17249144896 $ T^12 + 6T^11 - 289T^10 - 1806T^9 + 66152T^8 + 590592T^7 - 4138509T^6 - 51227220T^5 + 232321449T^4 + 3846968148T^3 + 15912261880T^2 + 26324462496*T + 17249144896
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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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.cc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{8} - \beta_{3}) q^{3} + (\beta_{10} - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{11} + \beta_{9} + \cdots + \beta_1) q^{9}+O(q^{10}) \) q + (b8 - b3) * q^3 + (b10 - b7 + b6 - b2 - b1) * q^5 - b4 * q^7 + (-b11 + b9 + b8 - b6 + b5 + b2 + b1) * q^9	\( q + (\beta_{8} - \beta_{3}) q^{3} + (\beta_{10} - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{11} + 2 \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) q + (b8 - b3) * q^3 + (b10 - b7 + b6 - b2 - b1) * q^5 - b4 * q^7 + (-b11 + b9 + b8 - b6 + b5 + b2 + b1) * q^9 + (b11 - b8 - 2b7 + b6 - b5 - 2b4 - b3 - b1) * q^11 + (-b11 + b10 + 2b9 + 2b8 - 2b6 + b5 - b4 + 2b2 - b1 - 1) * q^13 + (b11 - b9 - b7 - b5 - b4 - 1) * q^15 + (b10 + b8 - 2b7 + b5 - b4 + b3 + b2) q^17 + (b9 + b8 - b6 - b5 - 3b4 - b3 + b2 - b1 - 2) q^19 + (b2 + b1) * q^21 + (b11 - 3b10 - b9 + b8 + b7 - 2b6 + b5 - b4 - b3 + 3b2 + b1 + 1) q^23 + (2b8 - b7 - 2b5 - 2b4 + b2 - b1 - 1) q^25 + (2b11 + b10 + b7 - b6 - 2b5 + 2b4 - 2b3 - 1) * q^27 + (b10 - 3b9 + b8 + 2b6 - b3 - b2 + b1 + 3) * q^29 + (-2b11 - 2b10 + 3b8 + 2b7 - 2b6 + b5 - 2b3) * q^31 + (-2b10 + b8 + 2b6 - b5 - b3 + 2b1) q^33 + (b9 - b8 - b5 - b3) * q^35 + (2b10 - b8 - b7 - 3b6 - b4 - b3 + 2b2 + b1) q^37 + (-b11 + 2b10 + b9 - b7 - 3b6 - 2b5 + b4 - b3 - 2b1 - 3) * q^39 + (-3b11 + b10 - b9 + 4b8 - 2b7 - b6 + 3b5 - 2b4 + 4b3 + b2 - 1) * q^41 + (2b11 + b9 - b8 - 2b7 - b5 - b4 + b3) * q^43 + (b10 - b9 + b6 + 2b4 - 2b2 + b1 + 2) * q^45 + (-2b11 + b10 - 2b9 + 2b8 - b7 + b6 - 2b3 - 2b2 - 2b1 + 1) * q^47 + (-b9 + 1) * q^49 + (b10 + b8 - b6 - b5 + b2 - b1 - 2) * q^51 + (-2b8 + 2b5 + 3) * q^53 + (-2b11 - b10 + 5b9 + b8 - b7 - 2b5 + b4 - b3 + b2 + b1 - 5) q^55 + (-b11 - b10 - b6 - b5 - b3 + 2b2 + 2b1) * q^57 + (-3b11 + 2b10 + 2b8 - 2b6 - 2b5 - 2b4 + b3 - 2b1) q^59 + (b10 - b9 + b8 - b6 + b5 + b3 + 2b2 + b1) q^61 + (-b11 + b10 + b8 - b7 - b6 + b5 - b4 + b3 + b2) * q^63 + (-2b11 + b10 + 5b9 + b8 + b7 - 2b6 - 2b5 - 3b3 + 2b1 - 2) * q^65 + (b11 - b10 - 2b7 - b6 - b5 - 2b4 - b2 + 2b1) q^67 + (-4b11 + 2b9 + 2b8 + 2b5 - 2b3) q^69 + (-2b11 + 2b9 - 2b4 + 2b3 - 4) * q^71 + (3b11 - 2b10 + 2b9 - 5b8 - 2b7 - 2b6 - 2b5 + 3b3 + 3b2 + 3b1 - 1) * q^73 + (-b11 - 2b10 + 4b8 + 2b7 - 2b6 - b5 - 2b4 - 4b3 + 2b2) q^75 + (b11 - b10 - b8 + b6 - b3 + 2) * q^77 + (4b11 - 3b10 - b8 + 3b7 + 3b6 - 3b5 + 6b4 - 4b3 - 2b2 + 2b1 + 1) q^79 + (-4b10 - 2b9 + 3b8 - b7 - b6 + b4 - 3b3 + 4b2 + 3b1 + 2) * q^81 + (4b10 - b8 - 6b7 + 4b6 - b5 - 4b2 - 4b1) q^83 + (-3b11 + 3b9 + 3b8 - b6 - 3b5 - 4b4 + b2 - b1 - 6) q^85 + (-3b11 + 2b10 - 5b9 + 5b8 + 2b7 - 3b6 + 5b5 + b4 + 2b3 + 5b2 + 3b1) * q^87 + (5b11 + b10 - 3b9 - 2b8 - b7 - 2b6 - 5b5 - b4 - 2b3 + b2 + b1 - 3) * q^89 + (b10 - b9 - b8 - 2b7 - b6 + 2b5 - b4 + 2b3 + b2 + b1 + 1) q^91 + (-4b11 - b10 - b9 - b7 - b6 + 4b5 - b4 - b2 + 2b1 - 1) q^93 + (-b11 - 2b10 + 6b9 - 3b8 + 4b7 - 3b6 - 3b5 + 2b4 - 4b3 + b2 + 3b1) q^95 + (-4b11 + 4b10 + 2b9 + 5b8 - 4b6 - 5b5 - b3 - 4b1 - 4) q^97 + (-b11 + 2b10 - 2b9 + b8 - b7 + 2b6 - b3 + 1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{3} + 4 q^{9}+O(q^{10}) \) 12 * q + 2 * q^3 + 4 * q^9	\( 12 q + 2 q^{3} + 4 q^{9} + 2 q^{13} - 12 q^{15} + 2 q^{17} - 12 q^{19} + 6 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 6 q^{33} + 4 q^{35} - 6 q^{37} - 26 q^{39} - 12 q^{41} + 12 q^{43} + 18 q^{45} + 6 q^{49} - 16 q^{51} + 20 q^{53} - 24 q^{55} + 12 q^{59} - 4 q^{61} + 8 q^{65} + 6 q^{67} - 36 q^{71} + 10 q^{75} + 20 q^{77} + 20 q^{79} + 18 q^{81} - 36 q^{85} - 32 q^{87} - 36 q^{89} - 2 q^{91} - 42 q^{93} + 26 q^{95} - 6 q^{97}+O(q^{100}) \) 12 * q + 2 * q^3 + 4 * q^9 + 2 * q^13 - 12 * q^15 + 2 * q^17 - 12 * q^19 + 6 * q^23 + 4 * q^25 - 4 * q^27 + 20 * q^29 + 6 * q^33 + 4 * q^35 - 6 * q^37 - 26 * q^39 - 12 * q^41 + 12 * q^43 + 18 * q^45 + 6 * q^49 - 16 * q^51 + 20 * q^53 - 24 * q^55 + 12 * q^59 - 4 * q^61 + 8 * q^65 + 6 * q^67 - 36 * q^71 + 10 * q^75 + 20 * q^77 + 20 * q^79 + 18 * q^81 - 36 * q^85 - 32 * q^87 - 36 * q^89 - 2 * q^91 - 42 * q^93 + 26 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1456.2.cc.e

Level	$1456$
Weight	$2$
Character orbit	1456.cc
Analytic conductor	$11.626$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(11.6262185343\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	12.0.58891012706304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) x^12 - 5x^10 - 2x^9 + 15x^8 + 2x^7 - 30x^6 + 4x^5 + 60x^4 - 16x^3 - 80*x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 80\nu ) / 32 \) (v^11 - 5v^9 - 2v^8 + 15v^7 + 2v^6 - 30v^5 + 4v^4 + 60v^3 - 16v^2 - 80*v) / 32
\(\beta_{3}\)	\(=\)	\( ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} + \cdots + 96 ) / 32 \) (-3v^11 + 4v^10 + 7v^9 - 6v^8 - 13v^7 + 30v^6 - 6v^5 - 28v^4 - 4v^3 + 16v^2 - 48*v + 96) / 32
\(\beta_{4}\)	\(=\)	\( ( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + \cdots - 64 ) / 32 \) (3v^11 + 2v^10 - 11v^9 - 8v^8 + 21v^7 + 4v^6 - 42v^5 + 84v^3 + 24v^2 - 64v - 64) / 32
\(\beta_{5}\)	\(=\)	\( ( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + \cdots - 64 ) / 16 \) (v^11 + 3v^10 - 3v^9 - 13v^8 + 7v^7 + 23v^6 - 26v^5 - 38v^4 + 60v^3 + 68v^2 - 56v - 64) / 16
\(\beta_{6}\)	\(=\)	\( ( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} + \cdots + 64 ) / 32 \) (-5v^11 + 4v^10 + 13v^9 - 10v^8 - 23v^7 + 42v^6 + 10v^5 - 68v^4 - 20v^3 + 48v^2 - 32*v + 64) / 32
\(\beta_{7}\)	\(=\)	\( ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} + \cdots - 32 ) / 32 \) (-5v^11 + 2v^10 + 17v^9 - 8v^8 - 39v^7 + 44v^6 + 50v^5 - 88v^4 - 68v^3 + 120v^2 + 16*v - 32) / 32
\(\beta_{8}\)	\(=\)	\( ( - \nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} + \cdots - 32 ) / 8 \) (-v^11 + v^10 + 5v^9 - 3v^8 - 13v^7 + 13v^6 + 20v^5 - 34v^4 - 28v^3 + 52v^2 + 24*v - 32) / 8
\(\beta_{9}\)	\(=\)	\( ( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + \cdots - 80 ) / 16 \) (-v^11 + 3v^10 + 5v^9 - 13v^8 - 13v^7 + 35v^6 + 12v^5 - 70v^4 + 8v^3 + 108v^2 - 16v - 80) / 16
\(\beta_{10}\)	\(=\)	\( ( - 3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + \cdots - 64 ) / 16 \) (-3v^11 + 15v^9 - 2v^8 - 37v^7 + 18v^6 + 66v^5 - 68v^4 - 92v^3 + 96v^2 + 96v - 64) / 16
\(\beta_{11}\)	\(=\)	\( ( - \nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} + \cdots - 160 ) / 16 \) (-v^11 + 4v^10 + 9v^9 - 18v^8 - 27v^7 + 50v^6 + 34v^5 - 116v^4 - 20v^3 + 192v^2 + 16v - 160) / 16

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1 \) -b11 + b8 + b7 - b6 + b5 + 1
\(\nu^{3}\)	\(=\)	\( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 \) -b11 + b10 + b9 - b6 + b4 + b3 + b2 + b1
\(\nu^{4}\)	\(=\)	\( -\beta_{11} + \beta_{9} + \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1 \) -b11 + b9 + b7 - 2*b6 + b5 - b4 + b3 - b2 - 1
\(\nu^{5}\)	\(=\)	\( -2\beta_{11} + 4\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta _1 + 1 \) -2b11 + 4b9 + b8 - b7 - b6 - b5 - b1 + 1
\(\nu^{6}\)	\(=\)	\( 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \cdots - 2 \) 2b11 - 2b10 + b9 - b8 - b6 - 2b5 - 3b4 + b3 + b2 + 4*b1 - 2
\(\nu^{7}\)	\(=\)	\( - 5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + \cdots + 1 \) -5b11 - 3b10 + 5b9 + 7b8 - b7 - 2b6 + 3b5 - 7b4 - 3b3 + b2 + 1
\(\nu^{8}\)	\(=\)	\( -3\beta_{11} + 7\beta_{8} + \beta_{7} - 5\beta_{6} - \beta_{5} + 2\beta_{4} + 4\beta_{3} + 10\beta_{2} + 8\beta _1 - 3 \) -3b11 + 7b8 + b7 - 5b6 - b5 + 2b4 + 4b3 + 10b2 + 8*b1 - 3
\(\nu^{9}\)	\(=\)	\( - 7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} + \cdots - 8 \) -7b11 + 3b10 + 3b9 + 6b8 - 4b7 - 11b6 + 12b5 - 17b4 + 7b3 + 7b2 + b1 - 8
\(\nu^{10}\)	\(=\)	\( - 17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + \cdots - 11 \) -17b11 + 4b10 + 13b9 + 18b8 - 9b7 - 8b6 + 3b5 + 7b4 + 9b3 + 3b2 - 6*b1 - 11
\(\nu^{11}\)	\(=\)	\( 18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} + \cdots - 7 \) 18b11 + 4b10 - 6b9 - 13b8 - 21b7 - 11b6 - b5 - 26b4 + 22b3 + 14b2 + 3b1 - 7

\(n\)	\(561\)	\(911\)	\(1093\)	\(1249\)
\(\chi(n)\)	\(\beta_{9}\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 225.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} - 2 T^{11} + \cdots + 4 \) T^12 - 2T^11 + 9T^10 - 6T^9 + 34T^8 - 18T^7 + 83T^6 + 87T^4 - 10T^3 + 50T^2 + 12T + 4
$5$	\( T^{12} + 28 T^{10} + \cdots + 1 \) T^12 + 28T^10 + 256T^8 + 950T^6 + 1432T^4 + 648*T^2 + 1
$7$	\( (T^{4} - T^{2} + 1)^{3} \) (T^4 - T^2 + 1)^3
$11$	\( T^{12} - 21 T^{10} + \cdots + 5476 \) T^12 - 21T^10 + 334T^8 + 414T^7 - 1963T^6 - 3852T^5 + 8899T^4 + 36594T^3 + 46906T^2 + 25308*T + 5476
$13$	\( T^{12} - 2 T^{11} + \cdots + 4826809 \) T^12 - 2T^11 - 11T^10 - 2T^9 - 106T^8 - 98T^7 + 5417T^6 - 1274T^5 - 17914T^4 - 4394T^3 - 314171T^2 - 742586*T + 4826809
$17$	\( T^{12} - 2 T^{11} + \cdots + 2401 \) T^12 - 2T^11 + 25T^10 - 102T^9 + 622T^8 - 1730T^7 + 4365T^6 - 5702T^5 + 7438T^4 - 4466T^3 + 6713T^2 - 3430*T + 2401
$19$	\( T^{12} + 12 T^{11} + \cdots + 64 \) T^12 + 12T^11 + 35T^10 - 156T^9 - 376T^8 + 1458T^7 + 2199T^6 - 14070T^5 + 23361T^4 - 18084T^3 + 6904T^2 - 1056*T + 64
$23$	\( T^{12} - 6 T^{11} + \cdots + 52765696 \) T^12 - 6T^11 + 110T^10 - 388T^9 + 7236T^8 - 28128T^7 + 204992T^6 - 651456T^5 + 3648448T^4 - 10589184T^3 + 32796672T^2 - 44862464*T + 52765696
$29$	\( T^{12} - 20 T^{11} + \cdots + 1 \) T^12 - 20T^11 + 300T^10 - 2292T^9 + 14260T^8 - 39852T^7 + 138278T^6 - 25260T^5 + 1919892T^4 + 1141388T^3 + 727244T^2 - 852*T + 1
$31$	\( T^{12} + 152 T^{10} + \cdots + 13118884 \) T^12 + 152T^10 + 7154T^8 + 151248T^6 + 1572153T^4 + 7639576*T^2 + 13118884
$37$	\( T^{12} + 6 T^{11} + \cdots + 614656 \) T^12 + 6T^11 - 69T^10 - 486T^9 + 4909T^8 + 11400T^7 - 104644T^6 - 164736T^5 + 1997440T^4 - 2372160T^3 - 166208T^2 + 1317120*T + 614656
$41$	\( T^{12} + 12 T^{11} + \cdots + 32993536 \) T^12 + 12T^11 - 57T^10 - 1260T^9 + 6253T^8 + 73536T^7 + 3548T^6 - 1299456T^5 - 425984T^4 + 18313152T^3 + 57820864T^2 + 69755136*T + 32993536
$43$	\( T^{12} - 12 T^{11} + \cdots + 148996 \) T^12 - 12T^11 + 123T^10 - 500T^9 + 2070T^8 - 1386T^7 + 10293T^6 + 12600T^5 + 103131T^4 + 181174T^3 + 312810T^2 + 233916*T + 148996
$47$	\( T^{12} + \cdots + 150601984 \) T^12 + 184T^10 + 12438T^8 + 404616T^6 + 6708569T^4 + 53130080*T^2 + 150601984
$53$	\( (T^{6} - 10 T^{5} + \cdots - 383)^{2} \) (T^6 - 10T^5 - 17T^4 + 196T^3 + 463T^2 + 6*T - 383)^2
$59$	\( T^{12} - 12 T^{11} + \cdots + 964324 \) T^12 - 12T^11 - 24T^10 + 864T^9 + 2011T^8 - 47154T^7 + 115352T^6 + 300774T^5 - 233795T^4 - 766182T^3 + 468046T^2 + 1632084*T + 964324
$61$	\( T^{12} + 4 T^{11} + \cdots + 4096 \) T^12 + 4T^11 + 31T^10 + 36T^9 + 385T^8 + 592T^7 + 2400T^6 + 2560T^5 + 8128T^4 + 10240T^3 + 14336T^2 + 8192*T + 4096
$67$	\( T^{12} + \cdots + 131010916 \) T^12 - 6T^11 - 80T^10 + 552T^9 + 5451T^8 - 24114T^7 - 184908T^6 + 563592T^5 + 5320169T^4 + 2333370T^3 - 28429714T^2 - 10644780*T + 131010916
$71$	\( T^{12} + 36 T^{11} + \cdots + 65536 \) T^12 + 36T^11 + 568T^10 + 4896T^9 + 24048T^8 + 60672T^7 + 35712T^6 - 149760T^5 - 106240T^4 + 706560T^3 + 1323008T^2 + 491520*T + 65536
$73$	\( T^{12} + \cdots + 322751244544 \) T^12 + 610T^10 + 145225T^8 + 17148728T^6 + 1050041392T^4 + 30933985152*T^2 + 322751244544
$79$	\( (T^{6} - 10 T^{5} + \cdots + 55648)^{2} \) (T^6 - 10T^5 - 142T^4 + 1440T^3 + 3624T^2 - 44000*T + 55648)^2
$83$	\( T^{12} + \cdots + 1205895076 \) T^12 + 488T^10 + 77378T^8 + 4526688T^6 + 101760105T^4 + 685457080*T^2 + 1205895076
$89$	\( T^{12} + 36 T^{11} + \cdots + 87616 \) T^12 + 36T^11 + 427T^10 - 180T^9 - 24480T^8 + 3054T^7 + 1104231T^6 - 3459906T^5 - 697375T^4 + 11672700T^3 + 14176472T^2 + 1893216*T + 87616
$97$	\( T^{12} + \cdots + 17249144896 \) T^12 + 6T^11 - 289T^10 - 1806T^9 + 66152T^8 + 590592T^7 - 4138509T^6 - 51227220T^5 + 232321449T^4 + 3846968148T^3 + 15912261880T^2 + 26324462496*T + 17249144896
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