LMFDB - Newform orbit 2646.2.t.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(1979,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 5]))

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

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

chi := DirichletCharacter("2646.1979");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.t (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:	$21.1284163748$
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} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 $ x^16 - 6x^14 + 9x^12 + 54x^10 - 288x^8 + 486x^6 + 729x^4 - 4374*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 126)
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_{7} - \beta_{5}) q^{2} + ( - \beta_{8} + 1) q^{4} + ( - \beta_{6} + \beta_{3}) q^{5} - \beta_{5} q^{8}+O(q^{10}) $ q + (b7 - b5) * q^2 + (-b8 + 1) * q^4 + (-b6 + b3) * q^5 - b5 * q^8	$ q + (\beta_{7} - \beta_{5}) q^{2} + ( - \beta_{8} + 1) q^{4} + ( - \beta_{6} + \beta_{3}) q^{5} - \beta_{5} q^{8} + ( - \beta_{15} + \beta_{6} - \beta_{3}) q^{10} + (\beta_{14} - \beta_{9} - 2 \beta_{8} + \cdots + 2) q^{11}+ \cdots + ( - 2 \beta_{15} + \beta_{13} + \cdots - 4 \beta_{2}) q^{97}+O(q^{100}) $ q + (b7 - b5) * q^2 + (-b8 + 1) * q^4 + (-b6 + b3) * q^5 - b5 * q^8 + (-b15 + b6 - b3) * q^10 + (b14 - b9 - 2b8 + b5 - b1 + 2) q^11 + (-b11 - b6 + b3) * q^13 - b8 * q^16 + (-b12 - b2) * q^17 + (-b15 + b13 - b11 + b6 + b3 - b2) * q^19 + (b15 - b11 - b6 + b3) * q^20 + (-b14 + b9 + b8 - 2b5 - b4 + b1 - 2) q^22 + (-b9 + 4b8 + b7 - 2b5 - b4 - 2) * q^23 + (-b9 - 5b7 + 3b5 + b4 + 2) * q^25 + (-b15 - b3) * q^26 + (b14 - b9 + b7 + b5 + b4 + 2) * q^29 + (2b13 + b12 - b10 - b2) q^31 - b7 * q^32 + (b13 - b2) * q^34 + (b14 + b9 - 2b8 - 3b7 + 5b5 + b4 - 2b1 + 2) * q^37 + (-b15 + b13 - b11 + b10 + b3 - b2) * q^38 + (-b15 + b11 - b3) * q^40 + (-2b13 - b12 + 2b11 + 2b10 - 2b6 - 2b3 - b2) q^41 + (b9 - b8 - 2b7 + 3b5 + b4 - b1 + 1) * q^43 + (-b9 - b8 + b5 + b4 - b1 + 1) * q^44 + (-b14 - b8 + 3b7 + b5 - b4 - 1) q^46 + (-b13 - 2b12 - 2b11 + b10 + 2b6 + 2b3 - 2b2) q^47 + (b14 + 2b8 + b7 - b5 - b4 - 5) q^50 + (b15 - b11 - b6) * q^52 + (-2b15 - 3b13 - b12 + 2b11 + 2b10 - 2b3 - 3b2) * q^55 + (b14 + b7 - b5 - b4 + b1 + 1) * q^58 + (-b15 + b13 + b12 - 2b11 + b10 - 2b6 - b3) * q^59 + (-b15 - b13 - 2b12 - b11 - b10 + 2b2) * q^61 + (b13 - b12 + 2b10 - b2) q^62 - q^64 + (-b9 - 3b8 - b7 + b5 + b1 + 7) q^65 + (b9 + 3b8 + 4b7 - 9b5 + b4 - b1 - 3) q^67 + (b13 + b10 - b2) * q^68 + (-b9 - 2b8 + b7 + b5 - b4 + 1) q^71 + (-2b15 + b12 - 2b11 - b2) * q^73 + (-b14 + 2b9 + 4b8 - b7 - b5 + b4 + b1 - 3) * q^74 + (-b15 + b13 + b12 - b11 + b10 - b2) * q^76 + (-b14 + 2b9 + b8 - 3b7 - 5b5 - b4 + 2b1 - 3) * q^79 + (b15 - b11) * q^80 + (2b15 + b13 + 2b12 - 2b10 - 2b3 + b2) * q^82 + (3b15 + 2b13 + 4b12 - 2b11 + 4b10 - 2b6 + 3b3 - 2b2) * q^83 + (-b9 - 2b8 - b1 + 1) q^85 + (b9 + 2b8 - b7 + b4 - 1) q^86 + (b9 - b7 - b4 - 1) * q^88 + (-2b15 - 2b13 - 2b12 + 2b11 - 2b10 + 2b6 - 2b3) q^89 + (-b14 + 2b8 - b7 - b1 + 2) q^92 + (-2b15 + 2b13 + b12 - b10 + 2b3 - b2) q^94 + (b9 + b8 + 3b7 - b5 - b4 + b1 - 1) q^95 + (-2b15 + b13 - 3b12 + 2b11 + 3b10 - 2b6 + 2b3 - 4b2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4}+O(q^{10}) $ 16 * q + 8 * q^4	$ 16 q + 8 q^{4} - 8 q^{16} + 16 q^{25} + 12 q^{29} + 4 q^{37} + 4 q^{43} - 12 q^{44} - 12 q^{46} - 60 q^{50} + 24 q^{58} - 16 q^{64} + 84 q^{65} - 28 q^{67} - 4 q^{79} - 12 q^{85} + 48 q^{92} + 12 q^{95}+O(q^{100}) $ 16 * q + 8 * q^4 - 8 * q^16 + 16 * q^25 + 12 * q^29 + 4 * q^37 + 4 * q^43 - 12 * q^44 - 12 * q^46 - 60 * q^50 + 24 * q^58 - 16 * q^64 + 84 * q^65 - 28 * q^67 - 4 * q^79 - 12 * q^85 + 48 * q^92 + 12 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 $ x^16 - 6*x^14 + 9*x^12 + 54*x^10 - 288*x^8 + 486*x^6 + 729*x^4 - 4374*x^2 + 6561 :

$\beta_{1}$	$=$	$ ( -\nu^{14} - \nu^{12} + 6\nu^{10} - 36\nu^{8} + 72\nu^{6} + 234\nu^{4} + 729\nu^{2} - 243 ) / 1944 $ (-v^14 - v^12 + 6v^10 - 36v^8 + 72v^6 + 234v^4 + 729*v^2 - 243) / 1944
$\beta_{2}$	$=$	$ ( \nu^{15} + 3\nu^{13} - 45\nu^{11} + 135\nu^{9} + 198\nu^{7} - 1377\nu^{5} + 2916\nu^{3} + 2187\nu ) / 4374 $ (v^15 + 3v^13 - 45v^11 + 135v^9 + 198v^7 - 1377v^5 + 2916v^3 + 2187*v) / 4374
$\beta_{3}$	$=$	$ ( 2\nu^{15} + 9\nu^{13} - 18\nu^{11} + 396\nu^{7} - 216\nu^{5} + 324\nu^{3} + 9477\nu ) / 5832 $ (2v^15 + 9v^13 - 18v^11 + 396v^7 - 216v^5 + 324v^3 + 9477*v) / 5832
$\beta_{4}$	$=$	$ ( 2\nu^{14} - \nu^{12} + 6\nu^{10} - 36\nu^{8} + 180\nu^{6} + 396\nu^{4} - 972\nu^{2} + 4131 ) / 1944 $ (2v^14 - v^12 + 6v^10 - 36v^8 + 180v^6 + 396v^4 - 972v^2 + 4131) / 1944
$\beta_{5}$	$=$	$ ( \nu^{14} - 3\nu^{12} - 9\nu^{10} + 81\nu^{8} - 126\nu^{6} - 135\nu^{4} + 1458\nu^{2} - 2187 ) / 1458 $ (v^14 - 3v^12 - 9v^10 + 81v^8 - 126v^6 - 135v^4 + 1458v^2 - 2187) / 1458
$\beta_{6}$	$=$	$ ( 7\nu^{15} - 24\nu^{13} + 36\nu^{11} + 540\nu^{9} - 1044\nu^{7} + 1134\nu^{5} + 8019\nu^{3} - 13122\nu ) / 17496 $ (7v^15 - 24v^13 + 36v^11 + 540v^9 - 1044v^7 + 1134v^5 + 8019v^3 - 13122v) / 17496
$\beta_{7}$	$=$	$ ( -5\nu^{14} + 18\nu^{12} - 216\nu^{8} + 792\nu^{6} + 54\nu^{4} - 4617\nu^{2} + 8748 ) / 5832 $ (-5v^14 + 18v^12 - 216v^8 + 792v^6 + 54v^4 - 4617v^2 + 8748) / 5832
$\beta_{8}$	$=$	$ ( -2\nu^{14} + 21\nu^{12} - 18\nu^{10} - 108\nu^{8} + 576\nu^{6} - 648\nu^{4} - 972\nu^{2} + 9477 ) / 5832 $ (-2v^14 + 21v^12 - 18v^10 - 108v^8 + 576v^6 - 648v^4 - 972*v^2 + 9477) / 5832
$\beta_{9}$	$=$	$ ( \nu^{14} - 21\nu^{12} + 126\nu^{10} - 612\nu^{6} + 2862\nu^{4} - 2673\nu^{2} + 729 ) / 5832 $ (v^14 - 21v^12 + 126v^10 - 612v^6 + 2862v^4 - 2673*v^2 + 729) / 5832
$\beta_{10}$	$=$	$ ( 7\nu^{15} - 15\nu^{13} - 18\nu^{11} + 378\nu^{9} - 558\nu^{7} - 1458\nu^{5} + 8019\nu^{3} - 2187\nu ) / 8748 $ (7v^15 - 15v^13 - 18v^11 + 378v^9 - 558v^7 - 1458v^5 + 8019v^3 - 2187v) / 8748
$\beta_{11}$	$=$	$ ( \nu^{15} - 6\nu^{11} + 36\nu^{9} - 18\nu^{7} - 108\nu^{5} + 513\nu^{3} ) / 972 $ (v^15 - 6v^11 + 36v^9 - 18v^7 - 108v^5 + 513*v^3) / 972
$\beta_{12}$	$=$	$ ( -\nu^{15} + 5\nu^{13} - 54\nu^{9} + 180\nu^{7} - 36\nu^{5} - 621\nu^{3} + 1215\nu ) / 972 $ (-v^15 + 5v^13 - 54v^9 + 180v^7 - 36v^5 - 621v^3 + 1215v) / 972
$\beta_{13}$	$=$	$ ( 5\nu^{15} - 21\nu^{13} + 216\nu^{9} - 630\nu^{7} + 324\nu^{5} + 2511\nu^{3} - 6561\nu ) / 2916 $ (5v^15 - 21v^13 + 216v^9 - 630v^7 + 324v^5 + 2511v^3 - 6561*v) / 2916
$\beta_{14}$	$=$	$ ( -3\nu^{14} + 13\nu^{12} + 12\nu^{10} - 180\nu^{8} + 594\nu^{6} - 180\nu^{4} - 3321\nu^{2} + 8505 ) / 972 $ (-3v^14 + 13v^12 + 12v^10 - 180v^8 + 594v^6 - 180v^4 - 3321*v^2 + 8505) / 972
$\beta_{15}$	$=$	$ ( 35\nu^{15} - 138\nu^{13} - 36\nu^{11} + 2052\nu^{9} - 6192\nu^{7} + 2106\nu^{5} + 37179\nu^{3} - 87480\nu ) / 17496 $ (35v^15 - 138v^13 - 36v^11 + 2052v^9 - 6192v^7 + 2106v^5 + 37179v^3 - 87480v) / 17496

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

$\nu$	$=$	$ ( -\beta_{13} - 2\beta_{12} - \beta_{11} + 2\beta_{3} ) / 3 $ (-b13 - 2b12 - b11 + 2b3) / 3
$\nu^{2}$	$=$	$ \beta_{8} - \beta_{7} + \beta_1 $ b8 - b7 + b1
$\nu^{3}$	$=$	$ \beta_{15} + \beta_{12} - 2\beta_{11} + \beta_{10} - \beta_{6} + 2\beta_{3} $ b15 + b12 - 2b11 + b10 - b6 + 2b3
$\nu^{4}$	$=$	$ -2\beta_{14} + \beta_{9} + 3\beta_{8} + 6\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 $ -2b14 + b9 + 3b8 + 6*b7 - b5 + b4 + b1
$\nu^{5}$	$=$	$ 2\beta_{15} - 3\beta_{11} - 5\beta_{10} + 2\beta_{6} + 6\beta_{3} + \beta_{2} $ 2b15 - 3b11 - 5b10 + 2b6 + 6*b3 + b2
$\nu^{6}$	$=$	$ 3\beta_{14} - 6\beta_{8} + 3\beta_{7} + 12\beta_{5} + 3\beta_{4} + 3\beta _1 - 9 $ 3b14 - 6b8 + 3b7 + 12b5 + 3b4 + 3b1 - 9
$\nu^{7}$	$=$	$ -3\beta_{15} + 15\beta_{13} + 18\beta_{12} - 6\beta_{11} + 9\beta_{6} + 6\beta_{2} $ -3b15 + 15b13 + 18b12 - 6b11 + 9b6 + 6b2
$\nu^{8}$	$=$	$ 3\beta_{14} + 12\beta_{9} - 3\beta_{8} + 30\beta_{7} + 42\beta_{5} - 3\beta_{4} - 12\beta_1 $ 3b14 + 12b9 - 3b8 + 30b7 + 42b5 - 3b4 - 12*b1
$\nu^{9}$	$=$	$ -18\beta_{15} - 6\beta_{13} - 21\beta_{12} + 21\beta_{11} - 27\beta_{10} + 72\beta_{6} - 24\beta_{3} + 18\beta_{2} $ -18b15 - 6b13 - 21b12 + 21b11 - 27b10 + 72b6 - 24b3 + 18b2
$\nu^{10}$	$=$	$ 45\beta_{14} + 45\beta_{9} + 27\beta_{8} - 135\beta_{7} + 63\beta_{5} - 18\beta_{4} - 108 $ 45b14 + 45b9 + 27b8 - 135b7 + 63b5 - 18b4 - 108
$\nu^{11}$	$=$	$ - 63 \beta_{15} + 18 \beta_{13} + 54 \beta_{12} + 36 \beta_{11} + 153 \beta_{10} + 117 \beta_{6} + \cdots - 81 \beta_{2} $ -63b15 + 18b13 + 54b12 + 36b11 + 153b10 + 117b6 - 108b3 - 81b2
$\nu^{12}$	$=$	$ -90\beta_{14} + 126\beta_{9} + 648\beta_{8} - 126\beta_{5} - 36\beta_{4} - 90\beta _1 - 405 $ -90b14 + 126b9 + 648b8 - 126b5 - 36b4 - 90b1 - 405
$\nu^{13}$	$=$	$ 36 \beta_{15} - 405 \beta_{13} - 216 \beta_{12} + 459 \beta_{11} - 36 \beta_{10} + 144 \beta_{6} + \cdots - 198 \beta_{2} $ 36b15 - 405b13 - 216b12 + 459b11 - 36b10 + 144b6 - 162b3 - 198b2
$\nu^{14}$	$=$	$ -54\beta_{9} + 621\beta_{8} - 999\beta_{7} - 378\beta_{5} + 486\beta_{4} - 243\beta _1 - 1134 $ -54b9 + 621b8 - 999b7 - 378b5 + 486b4 - 243b1 - 1134
$\nu^{15}$	$=$	$ - 81 \beta_{15} + 594 \beta_{13} + 891 \beta_{12} + 1026 \beta_{11} + 837 \beta_{10} + \cdots - 918 \beta_{2} $ -81b15 + 594b13 + 891b12 + 1026b11 + 837b10 - 999b6 - 162b3 - 918b2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$\beta_{8}$	$1 - \beta_{8}$

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} $

1979.1

−1.62181	+	0.608059i
1.40917	+	1.00709i
−1.40917	−	1.00709i
1.62181	−	0.608059i
−1.69547	−	0.354107i
0.0967785	−	1.72934i
−0.0967785	+	1.72934i
1.69547	+	0.354107i
−1.62181	−	0.608059i
1.40917	−	1.00709i
−1.40917	+	1.00709i
1.62181	+	0.608059i
−1.69547	+	0.354107i
0.0967785	+	1.72934i
−0.0967785	−	1.72934i
1.69547	−	0.354107i

−0.866025

0.500000i

0.500000

−

0.866025i

−3.89111

1.00000i

3.36980

−

1.94556i

1979.2

−0.866025

0.500000i

0.500000

−

0.866025i

−2.34936

1.00000i

2.03460

−

1.17468i

1979.3

−0.866025

0.500000i

0.500000

−

0.866025i

2.34936

1.00000i

−2.03460

1.17468i

1979.4

−0.866025

0.500000i

0.500000

−

0.866025i

3.89111

1.00000i

−3.36980

1.94556i

1979.5

0.866025

−

0.500000i

0.500000

−

0.866025i

−1.79035

−

1.00000i

−1.55049

0.895175i

1979.6

0.866025

−

0.500000i

0.500000

−

0.866025i

−0.366598

−

1.00000i

−0.317483

0.183299i

1979.7

0.866025

−

0.500000i

0.500000

−

0.866025i

0.366598

−

1.00000i

0.317483

−

0.183299i

1979.8

0.866025

−

0.500000i

0.500000

−

0.866025i

1.79035

−

1.00000i

1.55049

−

0.895175i

2285.1

−0.866025

−

0.500000i

0.500000

0.866025i

−3.89111

−

1.00000i

3.36980

1.94556i

2285.2

−0.866025

−

0.500000i

0.500000

0.866025i

−2.34936

−

1.00000i

2.03460

1.17468i

2285.3

−0.866025

−

0.500000i

0.500000

0.866025i

2.34936

−

1.00000i

−2.03460

−

1.17468i

2285.4

−0.866025

−

0.500000i

0.500000

0.866025i

3.89111

−

1.00000i

−3.36980

−

1.94556i

2285.5

0.866025

0.500000i

0.500000

0.866025i

−1.79035

1.00000i

−1.55049

−

0.895175i

2285.6

0.866025

0.500000i

0.500000

0.866025i

−0.366598

1.00000i

−0.317483

−

0.183299i

2285.7

0.866025

0.500000i

0.500000

0.866025i

0.366598

1.00000i

0.317483

0.183299i

2285.8

0.866025

0.500000i

0.500000

0.866025i

1.79035

1.00000i

1.55049

0.895175i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
63.n	odd	6	1	inner
63.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2646.2.t.a		16
3.b	odd	2	1		882.2.t.b		16
7.b	odd	2	1	inner	2646.2.t.a		16
7.c	even	3	1		378.2.m.a		16
7.c	even	3	1		2646.2.l.b		16
7.d	odd	6	1		378.2.m.a		16
7.d	odd	6	1		2646.2.l.b		16
9.c	even	3	1		882.2.l.a		16
9.d	odd	6	1		2646.2.l.b		16
21.c	even	2	1		882.2.t.b		16
21.g	even	6	1		126.2.m.a	✓	16
21.g	even	6	1		882.2.l.a		16
21.h	odd	6	1		126.2.m.a	✓	16
21.h	odd	6	1		882.2.l.a		16
28.f	even	6	1		3024.2.cc.b		16
28.g	odd	6	1		3024.2.cc.b		16
63.g	even	3	1		882.2.t.b		16
63.g	even	3	1		1134.2.d.a		16
63.h	even	3	1		126.2.m.a	✓	16
63.i	even	6	1		378.2.m.a		16
63.j	odd	6	1		378.2.m.a		16
63.k	odd	6	1		882.2.t.b		16
63.k	odd	6	1		1134.2.d.a		16
63.l	odd	6	1		882.2.l.a		16
63.n	odd	6	1		1134.2.d.a		16
63.n	odd	6	1	inner	2646.2.t.a		16
63.o	even	6	1		2646.2.l.b		16
63.s	even	6	1		1134.2.d.a		16
63.s	even	6	1	inner	2646.2.t.a		16
63.t	odd	6	1		126.2.m.a	✓	16
84.j	odd	6	1		1008.2.cc.b		16
84.n	even	6	1		1008.2.cc.b		16
252.r	odd	6	1		3024.2.cc.b		16
252.u	odd	6	1		1008.2.cc.b		16
252.bb	even	6	1		3024.2.cc.b		16
252.bj	even	6	1		1008.2.cc.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.2.m.a	✓	16	21.g	even	6	1
126.2.m.a	✓	16	21.h	odd	6	1
126.2.m.a	✓	16	63.h	even	3	1
126.2.m.a	✓	16	63.t	odd	6	1
378.2.m.a		16	7.c	even	3	1
378.2.m.a		16	7.d	odd	6	1
378.2.m.a		16	63.i	even	6	1
378.2.m.a		16	63.j	odd	6	1
882.2.l.a		16	9.c	even	3	1
882.2.l.a		16	21.g	even	6	1
882.2.l.a		16	21.h	odd	6	1
882.2.l.a		16	63.l	odd	6	1
882.2.t.b		16	3.b	odd	2	1
882.2.t.b		16	21.c	even	2	1
882.2.t.b		16	63.g	even	3	1
882.2.t.b		16	63.k	odd	6	1
1008.2.cc.b		16	84.j	odd	6	1
1008.2.cc.b		16	84.n	even	6	1
1008.2.cc.b		16	252.u	odd	6	1
1008.2.cc.b		16	252.bj	even	6	1
1134.2.d.a		16	63.g	even	3	1
1134.2.d.a		16	63.k	odd	6	1
1134.2.d.a		16	63.n	odd	6	1
1134.2.d.a		16	63.s	even	6	1
2646.2.l.b		16	7.c	even	3	1
2646.2.l.b		16	7.d	odd	6	1
2646.2.l.b		16	9.d	odd	6	1
2646.2.l.b		16	63.o	even	6	1
2646.2.t.a		16	1.a	even	1	1	trivial
2646.2.t.a		16	7.b	odd	2	1	inner
2646.2.t.a		16	63.n	odd	6	1	inner
2646.2.t.a		16	63.s	even	6	1	inner
3024.2.cc.b		16	28.f	even	6	1
3024.2.cc.b		16	28.g	odd	6	1
3024.2.cc.b		16	252.r	odd	6	1
3024.2.cc.b		16	252.bb	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 24T_{5}^{6} + 153T_{5}^{4} - 288T_{5}^{2} + 36 $ T5^8 - 24*T5^6 + 153*T5^4 - 288*T5^2 + 36 acting on $S_{2}^{\mathrm{new}}(2646, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - T^{2} + 1)^{4} $ (T^4 - T^2 + 1)^4
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 24 T^{6} + \cdots + 36)^{2} $ (T^8 - 24T^6 + 153T^4 - 288*T^2 + 36)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} + 54 T^{6} + \cdots + 1296)^{2} $ (T^8 + 54T^6 + 801T^4 + 3240*T^2 + 1296)^2
$13$	$ T^{16} - 36 T^{14} + \cdots + 331776 $ T^16 - 36T^14 + 972T^12 - 9936T^10 + 73296T^8 - 238464T^6 + 559872T^4 - 497664*T^2 + 331776
$17$	$ T^{16} + 42 T^{14} + \cdots + 331776 $ T^16 + 42T^14 + 1287T^12 + 17442T^10 + 172521T^8 + 569808T^6 + 1404864T^4 + 746496*T^2 + 331776
$19$	$ T^{16} - 54 T^{14} + \cdots + 2313441 $ T^16 - 54T^14 + 2322T^12 - 28368T^10 + 251199T^8 - 937008T^6 + 2533842T^4 - 2819934*T^2 + 2313441
$23$	$ (T^{8} + 126 T^{6} + \cdots + 443556)^{2} $ (T^8 + 126T^6 + 5229T^4 + 82944*T^2 + 443556)^2
$29$	$ (T^{8} - 6 T^{7} + \cdots + 20736)^{2} $ (T^8 - 6T^7 - 18T^6 + 180T^5 + 612T^4 - 2160T^3 - 2592T^2 + 10368*T + 20736)^2
$31$	$ T^{16} + \cdots + 557256278016 $ T^16 - 144T^14 + 13932T^12 - 730944T^10 + 27632016T^8 - 631535616T^6 + 10400182272T^4 - 92876046336*T^2 + 557256278016
$37$	$ (T^{8} - 2 T^{7} + \cdots + 1784896)^{2} $ (T^8 - 2T^7 + 106T^6 - 164T^5 + 9436T^4 - 13424T^3 + 170128T^2 + 245824*T + 1784896)^2
$41$	$ T^{16} + \cdots + 73499483897856 $ T^16 + 258T^14 + 43695T^12 + 4294314T^10 + 307258425T^8 + 13938763392T^6 + 448658922240T^4 + 6883786653696*T^2 + 73499483897856
$43$	$ (T^{8} - 2 T^{7} + \cdots + 10816)^{2} $ (T^8 - 2T^7 + 43T^6 - 218T^5 + 1921T^4 - 6188T^3 + 17848T^2 - 15392*T + 10816)^2
$47$	$ T^{16} + \cdots + 1485512441856 $ T^16 + 240T^14 + 41292T^12 + 3258432T^10 + 186073488T^8 + 4759817472T^6 + 87539678208T^4 + 399459631104*T^2 + 1485512441856
$53$	$ T^{16} $ T^16
$59$	$ T^{16} + 294 T^{14} + \cdots + 1296 $ T^16 + 294T^14 + 68787T^12 + 5027598T^10 + 287789589T^8 + 1422558828T^6 + 6496369452T^4 + 2901744*T^2 + 1296
$61$	$ T^{16} + \cdots + 2425818710016 $ T^16 - 240T^14 + 40635T^12 - 3371184T^10 + 202203801T^8 - 5193676800T^6 + 96222587904T^4 - 545450360832*T^2 + 2425818710016
$67$	$ (T^{8} + 14 T^{7} + \cdots + 824464)^{2} $ (T^8 + 14T^7 + 307T^6 + 1766T^5 + 36469T^4 + 209684T^3 + 2654812T^2 + 1507280*T + 824464)^2
$71$	$ (T^{8} + 90 T^{6} + \cdots + 82944)^{2} $ (T^8 + 90T^6 + 2745T^4 + 31104*T^2 + 82944)^2
$73$	$ T^{16} + \cdots + 2927055626496 $ T^16 - 222T^14 + 37215T^12 - 2185686T^10 + 89156745T^8 - 2219198688T^6 + 40269720240T^4 - 422268609024*T^2 + 2927055626496
$79$	$ (T^{8} + 2 T^{7} + \cdots + 1444804)^{2} $ (T^8 + 2T^7 + 133T^6 + 1298T^5 + 19399T^4 + 105170T^3 + 450226T^2 + 935156*T + 1444804)^2
$83$	$ T^{16} + \cdots + 33\!\cdots\!56 $ T^16 + 708T^14 + 326268T^12 + 88712784T^10 + 17587710864T^8 + 2256409253376T^6 + 207879529033728T^4 + 10214329542377472*T^2 + 337116351515590656
$89$	$ T^{16} + \cdots + 34828517376 $ T^16 + 216T^14 + 33696T^12 + 2488320T^10 + 134182656T^8 + 1934917632T^6 + 21767823360T^4 + 29023764480*T^2 + 34828517376
$97$	$ T^{16} + \cdots + 45\!\cdots\!56 $ T^16 - 702T^14 + 353727T^12 - 85999734T^10 + 15192293193T^8 - 714581204256T^6 + 24485300891568T^4 - 390697151362560*T^2 + 4512402164941056
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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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{7} - \beta_{5}) q^{2} + ( - \beta_{8} + 1) q^{4} + ( - \beta_{6} + \beta_{3}) q^{5} - \beta_{5} q^{8}+O(q^{10}) \) q + (b7 - b5) * q^2 + (-b8 + 1) * q^4 + (-b6 + b3) * q^5 - b5 * q^8	\( q + (\beta_{7} - \beta_{5}) q^{2} + ( - \beta_{8} + 1) q^{4} + ( - \beta_{6} + \beta_{3}) q^{5} - \beta_{5} q^{8} + ( - \beta_{15} + \beta_{6} - \beta_{3}) q^{10} + (\beta_{14} - \beta_{9} - 2 \beta_{8} + \cdots + 2) q^{11}+ \cdots + ( - 2 \beta_{15} + \beta_{13} + \cdots - 4 \beta_{2}) q^{97}+O(q^{100}) \) q + (b7 - b5) * q^2 + (-b8 + 1) * q^4 + (-b6 + b3) * q^5 - b5 * q^8 + (-b15 + b6 - b3) * q^10 + (b14 - b9 - 2b8 + b5 - b1 + 2) q^11 + (-b11 - b6 + b3) * q^13 - b8 * q^16 + (-b12 - b2) * q^17 + (-b15 + b13 - b11 + b6 + b3 - b2) * q^19 + (b15 - b11 - b6 + b3) * q^20 + (-b14 + b9 + b8 - 2b5 - b4 + b1 - 2) q^22 + (-b9 + 4b8 + b7 - 2b5 - b4 - 2) * q^23 + (-b9 - 5b7 + 3b5 + b4 + 2) * q^25 + (-b15 - b3) * q^26 + (b14 - b9 + b7 + b5 + b4 + 2) * q^29 + (2b13 + b12 - b10 - b2) q^31 - b7 * q^32 + (b13 - b2) * q^34 + (b14 + b9 - 2b8 - 3b7 + 5b5 + b4 - 2b1 + 2) * q^37 + (-b15 + b13 - b11 + b10 + b3 - b2) * q^38 + (-b15 + b11 - b3) * q^40 + (-2b13 - b12 + 2b11 + 2b10 - 2b6 - 2b3 - b2) q^41 + (b9 - b8 - 2b7 + 3b5 + b4 - b1 + 1) * q^43 + (-b9 - b8 + b5 + b4 - b1 + 1) * q^44 + (-b14 - b8 + 3b7 + b5 - b4 - 1) q^46 + (-b13 - 2b12 - 2b11 + b10 + 2b6 + 2b3 - 2b2) q^47 + (b14 + 2b8 + b7 - b5 - b4 - 5) q^50 + (b15 - b11 - b6) * q^52 + (-2b15 - 3b13 - b12 + 2b11 + 2b10 - 2b3 - 3b2) * q^55 + (b14 + b7 - b5 - b4 + b1 + 1) * q^58 + (-b15 + b13 + b12 - 2b11 + b10 - 2b6 - b3) * q^59 + (-b15 - b13 - 2b12 - b11 - b10 + 2b2) * q^61 + (b13 - b12 + 2b10 - b2) q^62 - q^64 + (-b9 - 3b8 - b7 + b5 + b1 + 7) q^65 + (b9 + 3b8 + 4b7 - 9b5 + b4 - b1 - 3) q^67 + (b13 + b10 - b2) * q^68 + (-b9 - 2b8 + b7 + b5 - b4 + 1) q^71 + (-2b15 + b12 - 2b11 - b2) * q^73 + (-b14 + 2b9 + 4b8 - b7 - b5 + b4 + b1 - 3) * q^74 + (-b15 + b13 + b12 - b11 + b10 - b2) * q^76 + (-b14 + 2b9 + b8 - 3b7 - 5b5 - b4 + 2b1 - 3) * q^79 + (b15 - b11) * q^80 + (2b15 + b13 + 2b12 - 2b10 - 2b3 + b2) * q^82 + (3b15 + 2b13 + 4b12 - 2b11 + 4b10 - 2b6 + 3b3 - 2b2) * q^83 + (-b9 - 2b8 - b1 + 1) q^85 + (b9 + 2b8 - b7 + b4 - 1) q^86 + (b9 - b7 - b4 - 1) * q^88 + (-2b15 - 2b13 - 2b12 + 2b11 - 2b10 + 2b6 - 2b3) q^89 + (-b14 + 2b8 - b7 - b1 + 2) q^92 + (-2b15 + 2b13 + b12 - b10 + 2b3 - b2) q^94 + (b9 + b8 + 3b7 - b5 - b4 + b1 - 1) q^95 + (-2b15 + b13 - 3b12 + 2b11 + 3b10 - 2b6 + 2b3 - 4b2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4}+O(q^{10}) \) 16 * q + 8 * q^4	\( 16 q + 8 q^{4} - 8 q^{16} + 16 q^{25} + 12 q^{29} + 4 q^{37} + 4 q^{43} - 12 q^{44} - 12 q^{46} - 60 q^{50} + 24 q^{58} - 16 q^{64} + 84 q^{65} - 28 q^{67} - 4 q^{79} - 12 q^{85} + 48 q^{92} + 12 q^{95}+O(q^{100}) \) 16 * q + 8 * q^4 - 8 * q^16 + 16 * q^25 + 12 * q^29 + 4 * q^37 + 4 * q^43 - 12 * q^44 - 12 * q^46 - 60 * q^50 + 24 * q^58 - 16 * q^64 + 84 * q^65 - 28 * q^67 - 4 * q^79 - 12 * q^85 + 48 * q^92 + 12 * 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	2646.2.t.a

Level	$2646$
Weight	$2$
Character orbit	2646.t
Analytic conductor	$21.128$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(21.1284163748\)
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:	\( x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 \) x^16 - 6x^14 + 9x^12 + 54x^10 - 288x^8 + 486x^6 + 729x^4 - 4374*x^2 + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{4} \)
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{14} - \nu^{12} + 6\nu^{10} - 36\nu^{8} + 72\nu^{6} + 234\nu^{4} + 729\nu^{2} - 243 ) / 1944 \) (-v^14 - v^12 + 6v^10 - 36v^8 + 72v^6 + 234v^4 + 729*v^2 - 243) / 1944
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} + 3\nu^{13} - 45\nu^{11} + 135\nu^{9} + 198\nu^{7} - 1377\nu^{5} + 2916\nu^{3} + 2187\nu ) / 4374 \) (v^15 + 3v^13 - 45v^11 + 135v^9 + 198v^7 - 1377v^5 + 2916v^3 + 2187*v) / 4374
\(\beta_{3}\)	\(=\)	\( ( 2\nu^{15} + 9\nu^{13} - 18\nu^{11} + 396\nu^{7} - 216\nu^{5} + 324\nu^{3} + 9477\nu ) / 5832 \) (2v^15 + 9v^13 - 18v^11 + 396v^7 - 216v^5 + 324v^3 + 9477*v) / 5832
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{14} - \nu^{12} + 6\nu^{10} - 36\nu^{8} + 180\nu^{6} + 396\nu^{4} - 972\nu^{2} + 4131 ) / 1944 \) (2v^14 - v^12 + 6v^10 - 36v^8 + 180v^6 + 396v^4 - 972v^2 + 4131) / 1944
\(\beta_{5}\)	\(=\)	\( ( \nu^{14} - 3\nu^{12} - 9\nu^{10} + 81\nu^{8} - 126\nu^{6} - 135\nu^{4} + 1458\nu^{2} - 2187 ) / 1458 \) (v^14 - 3v^12 - 9v^10 + 81v^8 - 126v^6 - 135v^4 + 1458v^2 - 2187) / 1458
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{15} - 24\nu^{13} + 36\nu^{11} + 540\nu^{9} - 1044\nu^{7} + 1134\nu^{5} + 8019\nu^{3} - 13122\nu ) / 17496 \) (7v^15 - 24v^13 + 36v^11 + 540v^9 - 1044v^7 + 1134v^5 + 8019v^3 - 13122v) / 17496
\(\beta_{7}\)	\(=\)	\( ( -5\nu^{14} + 18\nu^{12} - 216\nu^{8} + 792\nu^{6} + 54\nu^{4} - 4617\nu^{2} + 8748 ) / 5832 \) (-5v^14 + 18v^12 - 216v^8 + 792v^6 + 54v^4 - 4617v^2 + 8748) / 5832
\(\beta_{8}\)	\(=\)	\( ( -2\nu^{14} + 21\nu^{12} - 18\nu^{10} - 108\nu^{8} + 576\nu^{6} - 648\nu^{4} - 972\nu^{2} + 9477 ) / 5832 \) (-2v^14 + 21v^12 - 18v^10 - 108v^8 + 576v^6 - 648v^4 - 972*v^2 + 9477) / 5832
\(\beta_{9}\)	\(=\)	\( ( \nu^{14} - 21\nu^{12} + 126\nu^{10} - 612\nu^{6} + 2862\nu^{4} - 2673\nu^{2} + 729 ) / 5832 \) (v^14 - 21v^12 + 126v^10 - 612v^6 + 2862v^4 - 2673*v^2 + 729) / 5832
\(\beta_{10}\)	\(=\)	\( ( 7\nu^{15} - 15\nu^{13} - 18\nu^{11} + 378\nu^{9} - 558\nu^{7} - 1458\nu^{5} + 8019\nu^{3} - 2187\nu ) / 8748 \) (7v^15 - 15v^13 - 18v^11 + 378v^9 - 558v^7 - 1458v^5 + 8019v^3 - 2187v) / 8748
\(\beta_{11}\)	\(=\)	\( ( \nu^{15} - 6\nu^{11} + 36\nu^{9} - 18\nu^{7} - 108\nu^{5} + 513\nu^{3} ) / 972 \) (v^15 - 6v^11 + 36v^9 - 18v^7 - 108v^5 + 513*v^3) / 972
\(\beta_{12}\)	\(=\)	\( ( -\nu^{15} + 5\nu^{13} - 54\nu^{9} + 180\nu^{7} - 36\nu^{5} - 621\nu^{3} + 1215\nu ) / 972 \) (-v^15 + 5v^13 - 54v^9 + 180v^7 - 36v^5 - 621v^3 + 1215v) / 972
\(\beta_{13}\)	\(=\)	\( ( 5\nu^{15} - 21\nu^{13} + 216\nu^{9} - 630\nu^{7} + 324\nu^{5} + 2511\nu^{3} - 6561\nu ) / 2916 \) (5v^15 - 21v^13 + 216v^9 - 630v^7 + 324v^5 + 2511v^3 - 6561*v) / 2916
\(\beta_{14}\)	\(=\)	\( ( -3\nu^{14} + 13\nu^{12} + 12\nu^{10} - 180\nu^{8} + 594\nu^{6} - 180\nu^{4} - 3321\nu^{2} + 8505 ) / 972 \) (-3v^14 + 13v^12 + 12v^10 - 180v^8 + 594v^6 - 180v^4 - 3321*v^2 + 8505) / 972
\(\beta_{15}\)	\(=\)	\( ( 35\nu^{15} - 138\nu^{13} - 36\nu^{11} + 2052\nu^{9} - 6192\nu^{7} + 2106\nu^{5} + 37179\nu^{3} - 87480\nu ) / 17496 \) (35v^15 - 138v^13 - 36v^11 + 2052v^9 - 6192v^7 + 2106v^5 + 37179v^3 - 87480v) / 17496

\(\nu\)	\(=\)	\( ( -\beta_{13} - 2\beta_{12} - \beta_{11} + 2\beta_{3} ) / 3 \) (-b13 - 2b12 - b11 + 2b3) / 3
\(\nu^{2}\)	\(=\)	\( \beta_{8} - \beta_{7} + \beta_1 \) b8 - b7 + b1
\(\nu^{3}\)	\(=\)	\( \beta_{15} + \beta_{12} - 2\beta_{11} + \beta_{10} - \beta_{6} + 2\beta_{3} \) b15 + b12 - 2b11 + b10 - b6 + 2b3
\(\nu^{4}\)	\(=\)	\( -2\beta_{14} + \beta_{9} + 3\beta_{8} + 6\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 \) -2b14 + b9 + 3b8 + 6*b7 - b5 + b4 + b1
\(\nu^{5}\)	\(=\)	\( 2\beta_{15} - 3\beta_{11} - 5\beta_{10} + 2\beta_{6} + 6\beta_{3} + \beta_{2} \) 2b15 - 3b11 - 5b10 + 2b6 + 6*b3 + b2
\(\nu^{6}\)	\(=\)	\( 3\beta_{14} - 6\beta_{8} + 3\beta_{7} + 12\beta_{5} + 3\beta_{4} + 3\beta _1 - 9 \) 3b14 - 6b8 + 3b7 + 12b5 + 3b4 + 3b1 - 9
\(\nu^{7}\)	\(=\)	\( -3\beta_{15} + 15\beta_{13} + 18\beta_{12} - 6\beta_{11} + 9\beta_{6} + 6\beta_{2} \) -3b15 + 15b13 + 18b12 - 6b11 + 9b6 + 6b2
\(\nu^{8}\)	\(=\)	\( 3\beta_{14} + 12\beta_{9} - 3\beta_{8} + 30\beta_{7} + 42\beta_{5} - 3\beta_{4} - 12\beta_1 \) 3b14 + 12b9 - 3b8 + 30b7 + 42b5 - 3b4 - 12*b1
\(\nu^{9}\)	\(=\)	\( -18\beta_{15} - 6\beta_{13} - 21\beta_{12} + 21\beta_{11} - 27\beta_{10} + 72\beta_{6} - 24\beta_{3} + 18\beta_{2} \) -18b15 - 6b13 - 21b12 + 21b11 - 27b10 + 72b6 - 24b3 + 18b2
\(\nu^{10}\)	\(=\)	\( 45\beta_{14} + 45\beta_{9} + 27\beta_{8} - 135\beta_{7} + 63\beta_{5} - 18\beta_{4} - 108 \) 45b14 + 45b9 + 27b8 - 135b7 + 63b5 - 18b4 - 108
\(\nu^{11}\)	\(=\)	\( - 63 \beta_{15} + 18 \beta_{13} + 54 \beta_{12} + 36 \beta_{11} + 153 \beta_{10} + 117 \beta_{6} + \cdots - 81 \beta_{2} \) -63b15 + 18b13 + 54b12 + 36b11 + 153b10 + 117b6 - 108b3 - 81b2
\(\nu^{12}\)	\(=\)	\( -90\beta_{14} + 126\beta_{9} + 648\beta_{8} - 126\beta_{5} - 36\beta_{4} - 90\beta _1 - 405 \) -90b14 + 126b9 + 648b8 - 126b5 - 36b4 - 90b1 - 405
\(\nu^{13}\)	\(=\)	\( 36 \beta_{15} - 405 \beta_{13} - 216 \beta_{12} + 459 \beta_{11} - 36 \beta_{10} + 144 \beta_{6} + \cdots - 198 \beta_{2} \) 36b15 - 405b13 - 216b12 + 459b11 - 36b10 + 144b6 - 162b3 - 198b2
\(\nu^{14}\)	\(=\)	\( -54\beta_{9} + 621\beta_{8} - 999\beta_{7} - 378\beta_{5} + 486\beta_{4} - 243\beta _1 - 1134 \) -54b9 + 621b8 - 999b7 - 378b5 + 486b4 - 243b1 - 1134
\(\nu^{15}\)	\(=\)	\( - 81 \beta_{15} + 594 \beta_{13} + 891 \beta_{12} + 1026 \beta_{11} + 837 \beta_{10} + \cdots - 918 \beta_{2} \) -81b15 + 594b13 + 891b12 + 1026b11 + 837b10 - 999b6 - 162b3 - 918b2

\(n\)	\(785\)	\(1081\)
\(\chi(n)\)	\(\beta_{8}\)	\(1 - \beta_{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1979.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 1)^{4} \) (T^4 - T^2 + 1)^4
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 24 T^{6} + \cdots + 36)^{2} \) (T^8 - 24T^6 + 153T^4 - 288*T^2 + 36)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} + 54 T^{6} + \cdots + 1296)^{2} \) (T^8 + 54T^6 + 801T^4 + 3240*T^2 + 1296)^2
$13$	\( T^{16} - 36 T^{14} + \cdots + 331776 \) T^16 - 36T^14 + 972T^12 - 9936T^10 + 73296T^8 - 238464T^6 + 559872T^4 - 497664*T^2 + 331776
$17$	\( T^{16} + 42 T^{14} + \cdots + 331776 \) T^16 + 42T^14 + 1287T^12 + 17442T^10 + 172521T^8 + 569808T^6 + 1404864T^4 + 746496*T^2 + 331776
$19$	\( T^{16} - 54 T^{14} + \cdots + 2313441 \) T^16 - 54T^14 + 2322T^12 - 28368T^10 + 251199T^8 - 937008T^6 + 2533842T^4 - 2819934*T^2 + 2313441
$23$	\( (T^{8} + 126 T^{6} + \cdots + 443556)^{2} \) (T^8 + 126T^6 + 5229T^4 + 82944*T^2 + 443556)^2
$29$	\( (T^{8} - 6 T^{7} + \cdots + 20736)^{2} \) (T^8 - 6T^7 - 18T^6 + 180T^5 + 612T^4 - 2160T^3 - 2592T^2 + 10368*T + 20736)^2
$31$	\( T^{16} + \cdots + 557256278016 \) T^16 - 144T^14 + 13932T^12 - 730944T^10 + 27632016T^8 - 631535616T^6 + 10400182272T^4 - 92876046336*T^2 + 557256278016
$37$	\( (T^{8} - 2 T^{7} + \cdots + 1784896)^{2} \) (T^8 - 2T^7 + 106T^6 - 164T^5 + 9436T^4 - 13424T^3 + 170128T^2 + 245824*T + 1784896)^2
$41$	\( T^{16} + \cdots + 73499483897856 \) T^16 + 258T^14 + 43695T^12 + 4294314T^10 + 307258425T^8 + 13938763392T^6 + 448658922240T^4 + 6883786653696*T^2 + 73499483897856
$43$	\( (T^{8} - 2 T^{7} + \cdots + 10816)^{2} \) (T^8 - 2T^7 + 43T^6 - 218T^5 + 1921T^4 - 6188T^3 + 17848T^2 - 15392*T + 10816)^2
$47$	\( T^{16} + \cdots + 1485512441856 \) T^16 + 240T^14 + 41292T^12 + 3258432T^10 + 186073488T^8 + 4759817472T^6 + 87539678208T^4 + 399459631104*T^2 + 1485512441856
$53$	\( T^{16} \) T^16
$59$	\( T^{16} + 294 T^{14} + \cdots + 1296 \) T^16 + 294T^14 + 68787T^12 + 5027598T^10 + 287789589T^8 + 1422558828T^6 + 6496369452T^4 + 2901744*T^2 + 1296
$61$	\( T^{16} + \cdots + 2425818710016 \) T^16 - 240T^14 + 40635T^12 - 3371184T^10 + 202203801T^8 - 5193676800T^6 + 96222587904T^4 - 545450360832*T^2 + 2425818710016
$67$	\( (T^{8} + 14 T^{7} + \cdots + 824464)^{2} \) (T^8 + 14T^7 + 307T^6 + 1766T^5 + 36469T^4 + 209684T^3 + 2654812T^2 + 1507280*T + 824464)^2
$71$	\( (T^{8} + 90 T^{6} + \cdots + 82944)^{2} \) (T^8 + 90T^6 + 2745T^4 + 31104*T^2 + 82944)^2
$73$	\( T^{16} + \cdots + 2927055626496 \) T^16 - 222T^14 + 37215T^12 - 2185686T^10 + 89156745T^8 - 2219198688T^6 + 40269720240T^4 - 422268609024*T^2 + 2927055626496
$79$	\( (T^{8} + 2 T^{7} + \cdots + 1444804)^{2} \) (T^8 + 2T^7 + 133T^6 + 1298T^5 + 19399T^4 + 105170T^3 + 450226T^2 + 935156*T + 1444804)^2
$83$	\( T^{16} + \cdots + 33\!\cdots\!56 \) T^16 + 708T^14 + 326268T^12 + 88712784T^10 + 17587710864T^8 + 2256409253376T^6 + 207879529033728T^4 + 10214329542377472*T^2 + 337116351515590656
$89$	\( T^{16} + \cdots + 34828517376 \) T^16 + 216T^14 + 33696T^12 + 2488320T^10 + 134182656T^8 + 1934917632T^6 + 21767823360T^4 + 29023764480*T^2 + 34828517376
$97$	\( T^{16} + \cdots + 45\!\cdots\!56 \) T^16 - 702T^14 + 353727T^12 - 85999734T^10 + 15192293193T^8 - 714581204256T^6 + 24485300891568T^4 - 390697151362560*T^2 + 4512402164941056
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