LMFDB - Newform orbit 1428.2.cc.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1428,2,Mod(361,1428)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1428, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 0, 8, 9]))

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

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

chi := DirichletCharacter("1428.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1428.cc (of order $12$, degree $4$, 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:	$11.4026374086$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
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} - 4x^{14} + 8x^{12} - 88x^{10} + 127x^{8} + 616x^{6} + 392x^{4} + 1372x^{2} + 2401 $ x^16 - 4x^14 + 8x^12 - 88x^10 + 127x^8 + 616x^6 + 392x^4 + 1372*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-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_{14} - \beta_{5}) q^{3} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_{6}) q^{5}+ \cdots + ( - \beta_{11} - \beta_{9}) q^{9}+O(q^{10}) $ q + (b14 - b5) * q^3 + (-b12 - b11 + b10 - b9 + b6) * q^5 + (-b12 + b11 - b9 + b7 - b2 + 2) * q^7 + (-b11 - b9) * q^9	$ q + (\beta_{14} - \beta_{5}) q^{3} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_{6}) q^{5}+ \cdots + ( - \beta_{15} + \beta_{10} + \beta_{9} + \cdots + 1) q^{99}+O(q^{100}) $ q + (b14 - b5) * q^3 + (-b12 - b11 + b10 - b9 + b6) * q^5 + (-b12 + b11 - b9 + b7 - b2 + 2) * q^7 + (-b11 - b9) * q^9 + (-b13 + b12 - b11 - b8 + b3 + b1 - 1) * q^11 + (2b15 - 2b13 + b7 - 2b6 + b5 + 1) q^13 + (b10 - b7 + b5 - b4 + b3) * q^15 + (-b14 + b11 - 2b8 + b5 - b4 + 2b3 + 2b2) q^17 + (b14 - b11 - 2b10 - b9 - 2b8 - b7 + b2) * q^19 + (2b14 + b7 - b5 - 2b2 - 1) * q^21 + (3b12 - 3b11 - b10 - 3b9 + b6) q^23 + (-2b15 - b14 - 2b13 + b11 - 2b8 + b5 - 2b4 + 2b3 - b2 + 2b1) * q^25 - b7 * q^27 + (b13 - b9 - 4b5 + b3 - 1) q^29 + (5b14 - 2b13 - 2b12 - 2b11 + 2b8 - 5b5 - 2b3 + 2b1 + 2) * q^31 + (-b14 - b7 + b6 + b2 + b1) * q^33 + (-2b15 - b14 + 2b13 - 2b12 + 2b10 - b8 + 2b6 + b5 - b4 - b3 + b2 - 2) q^35 + (b14 - 2b12 - 2b11 - 2b9 - 2b8 - 2b1) q^37 + (2b15 + b14 + b12 + b11 - b5 + 2b4 - 1) * q^39 + (b15 + 5b13 + b10 - b6 - b4 - 5b3) * q^41 + (3b9 + 3b7 - 3b5) q^43 + (-b13 - b12 + b11 - b8 + b3 + b1 + 1) * q^45 + (2b14 - 2b12 + 2b7 - 2b6 - 2b2 - 2b1) * q^47 + (-2b14 + 5b11 + 4b7 - 2b5 - 2b2) q^49 + (2b12 + b11 + b9 - b8 + b7 - b2 + 2b1) * q^51 + (-3b14 + 2b11 - b8 + 3b5 - b4 + b3 - 3b2) * q^53 + (b7 + b5 - 2) * q^55 + (2b13 - b9 - b7 - 2b3 + 1) * q^57 + (-2b14 - 8b11 - 2b8 + 2b5 - 2b4 + 2b3 - 2b2) q^59 + (2b10 - 2b8 - 2b6 + 2b1) * q^61 + (-b14 - b12 - b11 - 2b9 + b5 - 1) q^63 + (6b14 - 3b12 - 3b11 + 2b10 - 3b9 + b8 + 2b6 + b1) * q^65 + (2b15 + b14 - 2b13 - 3b12 - b5 - b2 + 2b1 + 3) * q^67 + (b10 - 3b7 - 3b5 - b4 - b3) * q^69 + (-b15 - b13 + b10 - 2b9 + b6 - b4 - b3 - 2) q^71 + (-3b14 - 2b13 - b12 - b11 + 2b8 + 3b5 - 2b3 + 2b1 + 1) * q^73 + (-b12 + b11 - 2b10 + b9 - 2b8 + b7 + 2b6 - b2 + 2b1) * q^75 + (-b15 - b13 - 4b11 - 2b10 - 2b9 - 2b8 - b7 - b6 + b5 + 2b1) q^77 + (b12 - b11 - b9 + 2b8 + 7b7 - 7b2 - 2b1) * q^79 + (-b12 + 1) * q^81 + (-4b10 + 2b9 + 4b7 - 4b5 + 4b4 - 4b3) * q^83 + (-2b10 + b9 + 5b7 - 4b5 + 2b4 - 4b3 - 4) q^85 + (-b15 - b14 - b13 - 4b11 + b5 - b2 + b1) q^87 + (-2b14 + 2b10 - 2b8 - 2b7 + 2b6 + 2b2 + 2b1) q^89 + (4b15 - 4b13 + 2b10 - 2b9 - 4b8 + 3b7 - 2b6 - 4b4 + 4b3 + b2 + 4b1 + 2) * q^91 + (2b14 - 5b11 - 5b9 - 2b7 + 2b6 + 2b2 - 2b1) q^93 + (-b15 + b12 - b11 + b4 + 6b2 - 1) q^95 + (2b15 - 2b13 - 2b10 - 7b9 - 2b6 - 4b5 + 2b4 - 2b3 - 7) * q^97 + (-b15 + b10 + b9 + b6 - b4 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{5} + 24 q^{7}+O(q^{10}) $ 16 * q - 8 * q^5 + 24 * q^7	$ 16 q - 8 q^{5} + 24 q^{7} - 8 q^{11} + 16 q^{13} - 16 q^{21} + 24 q^{23} - 16 q^{29} + 16 q^{31} - 48 q^{35} - 16 q^{37} - 8 q^{39} + 8 q^{45} - 16 q^{47} + 16 q^{51} - 32 q^{55} + 16 q^{57} - 24 q^{63} - 24 q^{65} + 24 q^{67} - 32 q^{71} + 8 q^{73} - 8 q^{75} + 8 q^{79} + 8 q^{81} - 64 q^{85} + 32 q^{91} - 8 q^{95} - 112 q^{97} + 16 q^{99}+O(q^{100}) $ 16 * q - 8 * q^5 + 24 * q^7 - 8 * q^11 + 16 * q^13 - 16 * q^21 + 24 * q^23 - 16 * q^29 + 16 * q^31 - 48 * q^35 - 16 * q^37 - 8 * q^39 + 8 * q^45 - 16 * q^47 + 16 * q^51 - 32 * q^55 + 16 * q^57 - 24 * q^63 - 24 * q^65 + 24 * q^67 - 32 * q^71 + 8 * q^73 - 8 * q^75 + 8 * q^79 + 8 * q^81 - 64 * q^85 + 32 * q^91 - 8 * q^95 - 112 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4x^{14} + 8x^{12} - 88x^{10} + 127x^{8} + 616x^{6} + 392x^{4} + 1372x^{2} + 2401 $ x^16 - 4*x^14 + 8*x^12 - 88*x^10 + 127*x^8 + 616*x^6 + 392*x^4 + 1372*x^2 + 2401 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 30\nu^{14} + 161\nu^{12} - 420\nu^{10} - 4215\nu^{8} - 7868\nu^{6} - 11025\nu^{4} + 714765\nu^{2} - 19208 ) / 1167705 $ (30v^14 + 161v^12 - 420v^10 - 4215v^8 - 7868v^6 - 11025v^4 + 714765*v^2 - 19208) / 1167705
$\beta_{3}$	$=$	$ ( 30 \nu^{15} + 161 \nu^{13} - 420 \nu^{11} - 4215 \nu^{9} - 7868 \nu^{7} - 11025 \nu^{5} + \cdots - 19208 \nu ) / 1167705 $ (30v^15 + 161v^13 - 420v^11 - 4215v^9 - 7868v^7 - 11025v^5 + 714765v^3 - 19208v) / 1167705
$\beta_{4}$	$=$	$ ( 562 \nu^{15} - 1320 \nu^{13} - 3150 \nu^{11} - 23356 \nu^{9} - 59010 \nu^{7} + 1016775 \nu^{5} + \cdots - 144060 \nu ) / 2724645 $ (562v^15 - 1320v^13 - 3150v^11 - 23356v^9 - 59010v^7 + 1016775v^5 - 120736v^3 - 144060v) / 2724645
$\beta_{5}$	$=$	$ ( 13456 \nu^{14} - 110867 \nu^{12} + 378450 \nu^{10} - 1890568 \nu^{8} + 8361131 \nu^{6} + \cdots - 19565749 ) / 57217545 $ (13456v^14 - 110867v^12 + 378450v^10 - 1890568v^8 + 8361131v^6 - 4945080v^4 - 13269298*v^2 - 19565749) / 57217545
$\beta_{6}$	$=$	$ ( - 13456 \nu^{15} + 110867 \nu^{13} - 378450 \nu^{11} + 1890568 \nu^{9} - 8361131 \nu^{7} + \cdots + 19565749 \nu ) / 57217545 $ (-13456v^15 + 110867v^13 - 378450v^11 + 1890568v^9 - 8361131v^7 + 4945080v^5 + 13269298v^3 + 19565749v) / 57217545
$\beta_{7}$	$=$	$ ( - 16200 \nu^{14} + 111553 \nu^{12} - 455625 \nu^{10} + 2276100 \nu^{8} - 7263874 \nu^{6} + \cdots - 57710779 ) / 57217545 $ (-16200v^14 + 111553v^12 - 455625v^10 + 2276100v^8 - 7263874v^6 + 5953500v^4 + 15975225*v^2 - 57710779) / 57217545
$\beta_{8}$	$=$	$ ( - 16200 \nu^{15} + 111553 \nu^{13} - 455625 \nu^{11} + 2276100 \nu^{9} - 7263874 \nu^{7} + \cdots - 57710779 \nu ) / 57217545 $ (-16200v^15 + 111553v^13 - 455625v^11 + 2276100v^9 - 7263874v^7 + 5953500v^5 + 15975225v^3 - 57710779v) / 57217545
$\beta_{9}$	$=$	$ ( - 464 \nu^{14} + 3823 \nu^{12} - 13050 \nu^{10} + 65192 \nu^{8} - 244470 \nu^{6} + 170520 \nu^{4} + \cdots - 596820 ) / 1271501 $ (-464v^14 + 3823v^12 - 13050v^10 + 65192v^8 - 244470v^6 + 170520v^4 + 457562*v^2 - 596820) / 1271501
$\beta_{10}$	$=$	$ ( - 464 \nu^{15} + 3823 \nu^{13} - 13050 \nu^{11} + 65192 \nu^{9} - 244470 \nu^{7} + \cdots - 596820 \nu ) / 1271501 $ (-464v^15 + 3823v^13 - 13050v^11 + 65192v^9 - 244470v^7 + 170520v^5 + 457562v^3 - 596820v) / 1271501
$\beta_{11}$	$=$	$ ( 10894 \nu^{14} - 66585 \nu^{12} + 173700 \nu^{10} - 1141372 \nu^{8} + 3253980 \nu^{6} + \cdots + 7943880 ) / 19072515 $ (10894v^14 - 66585v^12 + 173700v^10 - 1141372v^8 + 3253980v^6 + 4559625v^4 - 7708582*v^2 + 7943880) / 19072515
$\beta_{12}$	$=$	$ ( 1124 \nu^{14} - 5336 \nu^{12} + 13920 \nu^{10} - 122537 \nu^{8} + 260768 \nu^{6} + 365400 \nu^{4} + \cdots + 2370473 ) / 1733865 $ (1124v^14 - 5336v^12 + 13920v^10 - 122537v^8 + 260768v^6 + 365400v^4 + 749308*v^2 + 2370473) / 1733865
$\beta_{13}$	$=$	$ ( - 1124 \nu^{15} + 5336 \nu^{13} - 13920 \nu^{11} + 122537 \nu^{9} - 260768 \nu^{7} + \cdots - 636608 \nu ) / 1733865 $ (-1124v^15 + 5336v^13 - 13920v^11 + 122537v^9 - 260768v^7 - 365400v^5 - 749308v^3 - 636608v) / 1733865
$\beta_{14}$	$=$	$ ( - 39307 \nu^{14} + 154484 \nu^{12} - 313770 \nu^{10} + 3381841 \nu^{8} - 4606457 \nu^{6} + \cdots - 51223277 ) / 57217545 $ (-39307v^14 + 154484v^12 - 313770v^10 + 3381841v^8 - 4606457v^6 - 23115855v^4 - 14399924*v^2 - 51223277) / 57217545
$\beta_{15}$	$=$	$ ( 39307 \nu^{15} - 154484 \nu^{13} + 313770 \nu^{11} - 3381841 \nu^{9} + 4606457 \nu^{7} + \cdots + 51223277 \nu ) / 57217545 $ (39307v^15 - 154484v^13 + 313770v^11 - 3381841v^9 + 4606457v^7 + 23115855v^5 + 14399924v^3 + 51223277v) / 57217545

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{12} + \beta_{11} + 3\beta_{2} + 1 $ -b12 + b11 + 3*b2 + 1
$\nu^{3}$	$=$	$ \beta_{13} - \beta_{10} + \beta_{4} + 3\beta_{3} $ b13 - b10 + b4 + 3*b3
$\nu^{4}$	$=$	$ 6\beta_{14} + 11\beta_{11} + 11\beta_{9} - 6\beta_{7} + 6\beta_{2} $ 6b14 + 11b11 + 11b9 - 6b7 + 6*b2
$\nu^{5}$	$=$	$ -6\beta_{15} - 6\beta_{8} + 11\beta_{4} + 6\beta_{3} $ -6b15 - 6b8 + 11b4 + 6b3
$\nu^{6}$	$=$	$ 29\beta_{9} + 45\beta_{5} + 29 $ 29b9 + 45b5 + 29
$\nu^{7}$	$=$	$ 29\beta_{10} - 45\beta_{6} + 29\beta_1 $ 29b10 - 45b6 + 29*b1
$\nu^{8}$	$=$	$ -132\beta_{14} - 193\beta_{12} + 132\beta_{5} + 132\beta_{2} + 193 $ -132b14 - 193b12 + 132b5 + 132b2 + 193
$\nu^{9}$	$=$	$ 132\beta_{15} + 193\beta_{13} - 132\beta_{6} + 132\beta_{3} $ 132b15 + 193b13 - 132b6 + 132b3
$\nu^{10}$	$=$	$ -589\beta_{12} + 589\beta_{11} + 589\beta_{9} - 843\beta_{7} + 843\beta_{2} $ -589b12 + 589b11 + 589b9 - 843b7 + 843*b2
$\nu^{11}$	$=$	$ 589\beta_{13} - 843\beta_{8} + 589\beta_{4} + 843\beta_{3} - 589\beta_1 $ 589b13 - 843b8 + 589b4 + 843b3 - 589*b1
$\nu^{12}$	$=$	$ 3707\beta_{9} - 2610\beta_{7} + 2610\beta_{5} $ 3707b9 - 2610b7 + 2610*b5
$\nu^{13}$	$=$	$ 3707\beta_{10} - 2610\beta_{8} - 2610\beta_{6} $ 3707b10 - 2610b8 - 2610*b6
$\nu^{14}$	$=$	$ -16341\beta_{14} - 11537\beta_{12} - 11537\beta_{11} + 16341\beta_{5} + 11537 $ -16341b14 - 11537b12 - 11537b11 + 16341b5 + 11537
$\nu^{15}$	$=$	$ 16341\beta_{15} + 11537\beta_{13} + 11537\beta_{10} - 16341\beta_{6} - 11537\beta_{4} $ 16341b15 + 11537b13 + 11537b10 - 16341b6 - 11537*b4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times$.

$n$	$409$	$715$	$953$	$1261$
$\chi(n)$	$-\beta_{12}$	$1$	$1$	$\beta_{9}$

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} $

361.1

−0.164369	−	1.24851i
0.164369	+	1.24851i
2.08303	−	0.274236i
−2.08303	+	0.274236i
−0.164369	+	1.24851i
0.164369	−	1.24851i
2.08303	+	0.274236i
−2.08303	−	0.274236i
1.27901	−	1.66684i
−1.27901	+	1.66684i
0.999054	+	0.766601i
−0.999054	−	0.766601i
1.27901	+	1.66684i
−1.27901	−	1.66684i
0.999054	−	0.766601i
−0.999054	+	0.766601i

−0.258819

−

0.965926i

−3.61359

−

0.968258i

2.62484

−

0.331951i

−0.866025

0.500000i

361.2

−0.258819

−

0.965926i

0.881536

0.236207i

2.62484

−

0.331951i

−0.866025

0.500000i

361.3

0.258819

0.965926i

−2.91927

−

0.782216i

2.10721

1.59990i

−0.866025

0.500000i

361.4

0.258819

0.965926i

0.187220

0.0501655i

2.10721

1.59990i

−0.866025

0.500000i

625.1

−0.258819

0.965926i

−3.61359

0.968258i

2.62484

0.331951i

−0.866025

−

0.500000i

625.2

−0.258819

0.965926i

0.881536

−

0.236207i

2.62484

0.331951i

−0.866025

−

0.500000i

625.3

0.258819

−

0.965926i

−2.91927

0.782216i

2.10721

−

1.59990i

−0.866025

−

0.500000i

625.4

0.258819

−

0.965926i

0.187220

−

0.0501655i

2.10721

−

1.59990i

−0.866025

−

0.500000i

1033.1

−0.965926

0.258819i

−0.0501655

0.187220i

1.59990

−

2.10721i

0.866025

−

0.500000i

1033.2

−0.965926

0.258819i

0.782216

−

2.91927i

1.59990

−

2.10721i

0.866025

−

0.500000i

1033.3

0.965926

−

0.258819i

−0.236207

0.881536i

−0.331951

−

2.62484i

0.866025

−

0.500000i

1033.4

0.965926

−

0.258819i

0.968258

−

3.61359i

−0.331951

−

2.62484i

0.866025

−

0.500000i

1381.1

−0.965926

−

0.258819i

−0.0501655

−

0.187220i

1.59990

2.10721i

0.866025

0.500000i

1381.2

−0.965926

−

0.258819i

0.782216

2.91927i

1.59990

2.10721i

0.866025

0.500000i

1381.3

0.965926

0.258819i

−0.236207

−

0.881536i

−0.331951

2.62484i

0.866025

0.500000i

1381.4

0.965926

0.258819i

0.968258

3.61359i

−0.331951

2.62484i

0.866025

0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
17.c	even	4	1	inner
119.n	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1428.2.cc.b	✓	16
7.c	even	3	1	inner	1428.2.cc.b	✓	16
17.c	even	4	1	inner	1428.2.cc.b	✓	16
119.n	even	12	1	inner	1428.2.cc.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1428.2.cc.b	✓	16	1.a	even	1	1	trivial
1428.2.cc.b	✓	16	7.c	even	3	1	inner
1428.2.cc.b	✓	16	17.c	even	4	1	inner
1428.2.cc.b	✓	16	119.n	even	12	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{16} + 8 T_{5}^{15} + 32 T_{5}^{14} + 160 T_{5}^{13} + 628 T_{5}^{12} + 1264 T_{5}^{11} + \cdots + 16 $ T5^16 + 8*T5^15 + 32*T5^14 + 160*T5^13 + 628*T5^12 + 1264*T5^11 + 2816*T5^10 + 6688*T5^9 - 628*T5^8 - 14240*T5^7 + 6656*T5^6 - 9280*T5^5 + 13776*T5^4 - 3456*T5^3 + 512*T5^2 - 128*T5 + 16 acting on $S_{2}^{\mathrm{new}}(1428, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - T^{4} + 1)^{2} $ (T^8 - T^4 + 1)^2
$5$	$ T^{16} + 8 T^{15} + \cdots + 16 $ T^16 + 8T^15 + 32T^14 + 160T^13 + 628T^12 + 1264T^11 + 2816T^10 + 6688T^9 - 628T^8 - 14240T^7 + 6656T^6 - 9280T^5 + 13776T^4 - 3456T^3 + 512T^2 - 128*T + 16
$7$	$ (T^{8} - 12 T^{7} + \cdots + 2401)^{2} $ (T^8 - 12T^7 + 72T^6 - 288T^5 + 863T^4 - 2016T^3 + 3528T^2 - 4116*T + 2401)^2
$11$	$ T^{16} + 8 T^{15} + \cdots + 16 $ T^16 + 8T^15 + 32T^14 + 160T^13 + 628T^12 + 1264T^11 + 2816T^10 + 6688T^9 - 628T^8 - 14240T^7 + 6656T^6 - 9280T^5 + 13776T^4 - 3456T^3 + 512T^2 - 128*T + 16
$13$	$ (T^{4} - 4 T^{3} - 30 T^{2} + \cdots + 97)^{4} $ (T^4 - 4T^3 - 30T^2 + 100*T + 97)^4
$17$	$ T^{16} + \cdots + 6975757441 $ T^16 - 16T^14 + 240T^13 + 240T^12 - 3960T^11 + 23392T^10 + 63360T^9 - 402049T^8 + 1077120T^7 + 6760288T^6 - 19455480T^5 + 20045040T^4 + 340765680T^3 - 386201104*T^2 + 6975757441
$19$	$ T^{16} + \cdots + 671898241 $ T^16 - 76T^14 + 4266T^12 - 92848T^10 + 1421523T^8 - 12603568T^6 + 80893226T^4 - 283990476*T^2 + 671898241
$23$	$ T^{16} - 24 T^{15} + \cdots + 6250000 $ T^16 - 24T^15 + 288T^14 - 2976T^13 + 27348T^12 - 186192T^11 + 1020672T^10 - 5136672T^9 + 21143756T^8 - 64799520T^7 + 172915200T^6 - 405528000T^5 + 557970000T^4 - 241200000T^3 + 72000000T^2 - 30000000*T + 6250000
$29$	$ (T^{8} + 8 T^{7} + \cdots + 6724)^{2} $ (T^8 + 8T^7 + 32T^6 - 272T^5 + 1132T^4 - 80T^3 + 128T^2 - 1312*T + 6724)^2
$31$	$ T^{16} + \cdots + 104726589427201 $ T^16 - 16T^15 + 128T^14 - 992T^13 - 2818T^12 + 120816T^11 - 1080320T^10 + 9048592T^9 + 24761859T^8 - 1162258608T^7 + 10704327680T^6 - 91782795984T^5 + 674680944382T^4 - 3169647920992T^3 + 12838338995328T^2 - 51855866724816*T + 104726589427201
$37$	$ T^{16} + \cdots + 150644120641 $ T^16 + 16T^15 + 128T^14 + 1504T^13 + 10462T^12 + 4624T^11 - 134144T^10 - 1847568T^9 - 12282525T^8 - 2957136T^7 + 160211968T^6 + 2153614288T^5 + 16982531230T^4 + 39141963616T^3 + 60858627200T^2 + 135410445520*T + 150644120641
$41$	$ (T^{8} + 33168 T^{4} + 261921856)^{2} $ (T^8 + 33168*T^4 + 261921856)^2
$43$	$ (T^{4} + 54 T^{2} + 81)^{4} $ (T^4 + 54*T^2 + 81)^4
$47$	$ (T^{8} + 8 T^{7} + \cdots + 256)^{2} $ (T^8 + 8T^7 + 88T^6 + 1360T^4 + 2560T^3 + 8832T^2 + 1536T + 256)^2
$53$	$ T^{16} - 104 T^{14} + \cdots + 45212176 $ T^16 - 104T^14 + 8332T^12 - 222976T^10 + 4324812T^8 - 42518528T^6 + 295879984T^4 - 118880320*T^2 + 45212176
$59$	$ T^{16} + \cdots + 87578116096 $ T^16 - 352T^14 + 87744T^12 - 10502144T^10 + 915442688T^8 - 40040923136T^6 + 1228263849984T^4 - 329402810368*T^2 + 87578116096
$61$	$ T^{16} + \cdots + 644513529856 $ T^16 - 14592T^12 + 212123648T^8 - 11714691072*T^4 + 644513529856
$67$	$ (T^{8} - 12 T^{7} + \cdots + 289)^{2} $ (T^8 - 12T^7 + 126T^6 - 368T^5 + 1219T^4 + 1776T^3 + 5470T^2 + 1292*T + 289)^2
$71$	$ (T^{8} + 16 T^{7} + \cdots + 18496)^{2} $ (T^8 + 16T^7 + 128T^6 + 384T^5 + 528T^4 + 128T^3 + 8192T^2 + 17408*T + 18496)^2
$73$	$ T^{16} - 8 T^{15} + \cdots + 5764801 $ T^16 - 8T^15 + 32T^14 - 1712T^13 + 2046T^12 + 112280T^11 + 501760T^10 + 2597000T^9 + 13890275T^8 + 37524200T^7 + 86051840T^6 + 252297080T^5 + 287654206T^4 - 174523888T^3 + 33882912T^2 - 19765032*T + 5764801
$79$	$ T^{16} + \cdots + 110841719041 $ T^16 - 8T^15 + 32T^14 - 688T^13 - 10018T^12 + 129944T^11 - 482304T^10 + 6260872T^9 + 165097539T^8 - 1141745944T^7 + 3154366464T^6 - 32196987272T^5 + 100114768862T^4 + 165437744656T^3 + 88223521568T^2 + 139848824024*T + 110841719041
$83$	$ (T^{8} + 400 T^{6} + \cdots + 614656)^{2} $ (T^8 + 400T^6 + 33376T^4 + 313600*T^2 + 614656)^2
$89$	$ (T^{8} + 80 T^{6} + \cdots + 200704)^{2} $ (T^8 + 80T^6 + 256T^5 + 5952T^4 + 10240T^3 + 52224T^2 - 57344T + 200704)^2
$97$	$ (T^{8} + 56 T^{7} + \cdots + 2155024)^{2} $ (T^8 + 56T^7 + 1568T^6 + 25008T^5 + 245000T^4 + 1333856T^3 + 3235968T^2 - 3734592*T + 2155024)^2
show more
show less

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:	\( N \)	\(=\)	\( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1428.cc (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{14} - \beta_{5}) q^{3} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_{6}) q^{5}+ \cdots + ( - \beta_{11} - \beta_{9}) q^{9}+O(q^{10}) \) q + (b14 - b5) * q^3 + (-b12 - b11 + b10 - b9 + b6) * q^5 + (-b12 + b11 - b9 + b7 - b2 + 2) * q^7 + (-b11 - b9) * q^9	\( q + (\beta_{14} - \beta_{5}) q^{3} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_{6}) q^{5}+ \cdots + ( - \beta_{15} + \beta_{10} + \beta_{9} + \cdots + 1) q^{99}+O(q^{100}) \) q + (b14 - b5) * q^3 + (-b12 - b11 + b10 - b9 + b6) * q^5 + (-b12 + b11 - b9 + b7 - b2 + 2) * q^7 + (-b11 - b9) * q^9 + (-b13 + b12 - b11 - b8 + b3 + b1 - 1) * q^11 + (2b15 - 2b13 + b7 - 2b6 + b5 + 1) q^13 + (b10 - b7 + b5 - b4 + b3) * q^15 + (-b14 + b11 - 2b8 + b5 - b4 + 2b3 + 2b2) q^17 + (b14 - b11 - 2b10 - b9 - 2b8 - b7 + b2) * q^19 + (2b14 + b7 - b5 - 2b2 - 1) * q^21 + (3b12 - 3b11 - b10 - 3b9 + b6) q^23 + (-2b15 - b14 - 2b13 + b11 - 2b8 + b5 - 2b4 + 2b3 - b2 + 2b1) * q^25 - b7 * q^27 + (b13 - b9 - 4b5 + b3 - 1) q^29 + (5b14 - 2b13 - 2b12 - 2b11 + 2b8 - 5b5 - 2b3 + 2b1 + 2) * q^31 + (-b14 - b7 + b6 + b2 + b1) * q^33 + (-2b15 - b14 + 2b13 - 2b12 + 2b10 - b8 + 2b6 + b5 - b4 - b3 + b2 - 2) q^35 + (b14 - 2b12 - 2b11 - 2b9 - 2b8 - 2b1) q^37 + (2b15 + b14 + b12 + b11 - b5 + 2b4 - 1) * q^39 + (b15 + 5b13 + b10 - b6 - b4 - 5b3) * q^41 + (3b9 + 3b7 - 3b5) q^43 + (-b13 - b12 + b11 - b8 + b3 + b1 + 1) * q^45 + (2b14 - 2b12 + 2b7 - 2b6 - 2b2 - 2b1) * q^47 + (-2b14 + 5b11 + 4b7 - 2b5 - 2b2) q^49 + (2b12 + b11 + b9 - b8 + b7 - b2 + 2b1) * q^51 + (-3b14 + 2b11 - b8 + 3b5 - b4 + b3 - 3b2) * q^53 + (b7 + b5 - 2) * q^55 + (2b13 - b9 - b7 - 2b3 + 1) * q^57 + (-2b14 - 8b11 - 2b8 + 2b5 - 2b4 + 2b3 - 2b2) q^59 + (2b10 - 2b8 - 2b6 + 2b1) * q^61 + (-b14 - b12 - b11 - 2b9 + b5 - 1) q^63 + (6b14 - 3b12 - 3b11 + 2b10 - 3b9 + b8 + 2b6 + b1) * q^65 + (2b15 + b14 - 2b13 - 3b12 - b5 - b2 + 2b1 + 3) * q^67 + (b10 - 3b7 - 3b5 - b4 - b3) * q^69 + (-b15 - b13 + b10 - 2b9 + b6 - b4 - b3 - 2) q^71 + (-3b14 - 2b13 - b12 - b11 + 2b8 + 3b5 - 2b3 + 2b1 + 1) * q^73 + (-b12 + b11 - 2b10 + b9 - 2b8 + b7 + 2b6 - b2 + 2b1) * q^75 + (-b15 - b13 - 4b11 - 2b10 - 2b9 - 2b8 - b7 - b6 + b5 + 2b1) q^77 + (b12 - b11 - b9 + 2b8 + 7b7 - 7b2 - 2b1) * q^79 + (-b12 + 1) * q^81 + (-4b10 + 2b9 + 4b7 - 4b5 + 4b4 - 4b3) * q^83 + (-2b10 + b9 + 5b7 - 4b5 + 2b4 - 4b3 - 4) q^85 + (-b15 - b14 - b13 - 4b11 + b5 - b2 + b1) q^87 + (-2b14 + 2b10 - 2b8 - 2b7 + 2b6 + 2b2 + 2b1) q^89 + (4b15 - 4b13 + 2b10 - 2b9 - 4b8 + 3b7 - 2b6 - 4b4 + 4b3 + b2 + 4b1 + 2) * q^91 + (2b14 - 5b11 - 5b9 - 2b7 + 2b6 + 2b2 - 2b1) q^93 + (-b15 + b12 - b11 + b4 + 6b2 - 1) q^95 + (2b15 - 2b13 - 2b10 - 7b9 - 2b6 - 4b5 + 2b4 - 2b3 - 7) * q^97 + (-b15 + b10 + b9 + b6 - b4 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{5} + 24 q^{7}+O(q^{10}) \) 16 * q - 8 * q^5 + 24 * q^7	\( 16 q - 8 q^{5} + 24 q^{7} - 8 q^{11} + 16 q^{13} - 16 q^{21} + 24 q^{23} - 16 q^{29} + 16 q^{31} - 48 q^{35} - 16 q^{37} - 8 q^{39} + 8 q^{45} - 16 q^{47} + 16 q^{51} - 32 q^{55} + 16 q^{57} - 24 q^{63} - 24 q^{65} + 24 q^{67} - 32 q^{71} + 8 q^{73} - 8 q^{75} + 8 q^{79} + 8 q^{81} - 64 q^{85} + 32 q^{91} - 8 q^{95} - 112 q^{97} + 16 q^{99}+O(q^{100}) \) 16 * q - 8 * q^5 + 24 * q^7 - 8 * q^11 + 16 * q^13 - 16 * q^21 + 24 * q^23 - 16 * q^29 + 16 * q^31 - 48 * q^35 - 16 * q^37 - 8 * q^39 + 8 * q^45 - 16 * q^47 + 16 * q^51 - 32 * q^55 + 16 * q^57 - 24 * q^63 - 24 * q^65 + 24 * q^67 - 32 * q^71 + 8 * q^73 - 8 * q^75 + 8 * q^79 + 8 * q^81 - 64 * q^85 + 32 * q^91 - 8 * q^95 - 112 * q^97 + 16 * q^99

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	1428.2.cc.b

Level	$1428$
Weight	$2$
Character orbit	1428.cc
Analytic conductor	$11.403$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(11.4026374086\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4x^{14} + 8x^{12} - 88x^{10} + 127x^{8} + 616x^{6} + 392x^{4} + 1372x^{2} + 2401 \) x^16 - 4x^14 + 8x^12 - 88x^10 + 127x^8 + 616x^6 + 392x^4 + 1372*x^2 + 2401
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 30\nu^{14} + 161\nu^{12} - 420\nu^{10} - 4215\nu^{8} - 7868\nu^{6} - 11025\nu^{4} + 714765\nu^{2} - 19208 ) / 1167705 \) (30v^14 + 161v^12 - 420v^10 - 4215v^8 - 7868v^6 - 11025v^4 + 714765*v^2 - 19208) / 1167705
\(\beta_{3}\)	\(=\)	\( ( 30 \nu^{15} + 161 \nu^{13} - 420 \nu^{11} - 4215 \nu^{9} - 7868 \nu^{7} - 11025 \nu^{5} + \cdots - 19208 \nu ) / 1167705 \) (30v^15 + 161v^13 - 420v^11 - 4215v^9 - 7868v^7 - 11025v^5 + 714765v^3 - 19208v) / 1167705
\(\beta_{4}\)	\(=\)	\( ( 562 \nu^{15} - 1320 \nu^{13} - 3150 \nu^{11} - 23356 \nu^{9} - 59010 \nu^{7} + 1016775 \nu^{5} + \cdots - 144060 \nu ) / 2724645 \) (562v^15 - 1320v^13 - 3150v^11 - 23356v^9 - 59010v^7 + 1016775v^5 - 120736v^3 - 144060v) / 2724645
\(\beta_{5}\)	\(=\)	\( ( 13456 \nu^{14} - 110867 \nu^{12} + 378450 \nu^{10} - 1890568 \nu^{8} + 8361131 \nu^{6} + \cdots - 19565749 ) / 57217545 \) (13456v^14 - 110867v^12 + 378450v^10 - 1890568v^8 + 8361131v^6 - 4945080v^4 - 13269298*v^2 - 19565749) / 57217545
\(\beta_{6}\)	\(=\)	\( ( - 13456 \nu^{15} + 110867 \nu^{13} - 378450 \nu^{11} + 1890568 \nu^{9} - 8361131 \nu^{7} + \cdots + 19565749 \nu ) / 57217545 \) (-13456v^15 + 110867v^13 - 378450v^11 + 1890568v^9 - 8361131v^7 + 4945080v^5 + 13269298v^3 + 19565749v) / 57217545
\(\beta_{7}\)	\(=\)	\( ( - 16200 \nu^{14} + 111553 \nu^{12} - 455625 \nu^{10} + 2276100 \nu^{8} - 7263874 \nu^{6} + \cdots - 57710779 ) / 57217545 \) (-16200v^14 + 111553v^12 - 455625v^10 + 2276100v^8 - 7263874v^6 + 5953500v^4 + 15975225*v^2 - 57710779) / 57217545
\(\beta_{8}\)	\(=\)	\( ( - 16200 \nu^{15} + 111553 \nu^{13} - 455625 \nu^{11} + 2276100 \nu^{9} - 7263874 \nu^{7} + \cdots - 57710779 \nu ) / 57217545 \) (-16200v^15 + 111553v^13 - 455625v^11 + 2276100v^9 - 7263874v^7 + 5953500v^5 + 15975225v^3 - 57710779v) / 57217545
\(\beta_{9}\)	\(=\)	\( ( - 464 \nu^{14} + 3823 \nu^{12} - 13050 \nu^{10} + 65192 \nu^{8} - 244470 \nu^{6} + 170520 \nu^{4} + \cdots - 596820 ) / 1271501 \) (-464v^14 + 3823v^12 - 13050v^10 + 65192v^8 - 244470v^6 + 170520v^4 + 457562*v^2 - 596820) / 1271501
\(\beta_{10}\)	\(=\)	\( ( - 464 \nu^{15} + 3823 \nu^{13} - 13050 \nu^{11} + 65192 \nu^{9} - 244470 \nu^{7} + \cdots - 596820 \nu ) / 1271501 \) (-464v^15 + 3823v^13 - 13050v^11 + 65192v^9 - 244470v^7 + 170520v^5 + 457562v^3 - 596820v) / 1271501
\(\beta_{11}\)	\(=\)	\( ( 10894 \nu^{14} - 66585 \nu^{12} + 173700 \nu^{10} - 1141372 \nu^{8} + 3253980 \nu^{6} + \cdots + 7943880 ) / 19072515 \) (10894v^14 - 66585v^12 + 173700v^10 - 1141372v^8 + 3253980v^6 + 4559625v^4 - 7708582*v^2 + 7943880) / 19072515
\(\beta_{12}\)	\(=\)	\( ( 1124 \nu^{14} - 5336 \nu^{12} + 13920 \nu^{10} - 122537 \nu^{8} + 260768 \nu^{6} + 365400 \nu^{4} + \cdots + 2370473 ) / 1733865 \) (1124v^14 - 5336v^12 + 13920v^10 - 122537v^8 + 260768v^6 + 365400v^4 + 749308*v^2 + 2370473) / 1733865
\(\beta_{13}\)	\(=\)	\( ( - 1124 \nu^{15} + 5336 \nu^{13} - 13920 \nu^{11} + 122537 \nu^{9} - 260768 \nu^{7} + \cdots - 636608 \nu ) / 1733865 \) (-1124v^15 + 5336v^13 - 13920v^11 + 122537v^9 - 260768v^7 - 365400v^5 - 749308v^3 - 636608v) / 1733865
\(\beta_{14}\)	\(=\)	\( ( - 39307 \nu^{14} + 154484 \nu^{12} - 313770 \nu^{10} + 3381841 \nu^{8} - 4606457 \nu^{6} + \cdots - 51223277 ) / 57217545 \) (-39307v^14 + 154484v^12 - 313770v^10 + 3381841v^8 - 4606457v^6 - 23115855v^4 - 14399924*v^2 - 51223277) / 57217545
\(\beta_{15}\)	\(=\)	\( ( 39307 \nu^{15} - 154484 \nu^{13} + 313770 \nu^{11} - 3381841 \nu^{9} + 4606457 \nu^{7} + \cdots + 51223277 \nu ) / 57217545 \) (39307v^15 - 154484v^13 + 313770v^11 - 3381841v^9 + 4606457v^7 + 23115855v^5 + 14399924v^3 + 51223277v) / 57217545

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{12} + \beta_{11} + 3\beta_{2} + 1 \) -b12 + b11 + 3*b2 + 1
\(\nu^{3}\)	\(=\)	\( \beta_{13} - \beta_{10} + \beta_{4} + 3\beta_{3} \) b13 - b10 + b4 + 3*b3
\(\nu^{4}\)	\(=\)	\( 6\beta_{14} + 11\beta_{11} + 11\beta_{9} - 6\beta_{7} + 6\beta_{2} \) 6b14 + 11b11 + 11b9 - 6b7 + 6*b2
\(\nu^{5}\)	\(=\)	\( -6\beta_{15} - 6\beta_{8} + 11\beta_{4} + 6\beta_{3} \) -6b15 - 6b8 + 11b4 + 6b3
\(\nu^{6}\)	\(=\)	\( 29\beta_{9} + 45\beta_{5} + 29 \) 29b9 + 45b5 + 29
\(\nu^{7}\)	\(=\)	\( 29\beta_{10} - 45\beta_{6} + 29\beta_1 \) 29b10 - 45b6 + 29*b1
\(\nu^{8}\)	\(=\)	\( -132\beta_{14} - 193\beta_{12} + 132\beta_{5} + 132\beta_{2} + 193 \) -132b14 - 193b12 + 132b5 + 132b2 + 193
\(\nu^{9}\)	\(=\)	\( 132\beta_{15} + 193\beta_{13} - 132\beta_{6} + 132\beta_{3} \) 132b15 + 193b13 - 132b6 + 132b3
\(\nu^{10}\)	\(=\)	\( -589\beta_{12} + 589\beta_{11} + 589\beta_{9} - 843\beta_{7} + 843\beta_{2} \) -589b12 + 589b11 + 589b9 - 843b7 + 843*b2
\(\nu^{11}\)	\(=\)	\( 589\beta_{13} - 843\beta_{8} + 589\beta_{4} + 843\beta_{3} - 589\beta_1 \) 589b13 - 843b8 + 589b4 + 843b3 - 589*b1
\(\nu^{12}\)	\(=\)	\( 3707\beta_{9} - 2610\beta_{7} + 2610\beta_{5} \) 3707b9 - 2610b7 + 2610*b5
\(\nu^{13}\)	\(=\)	\( 3707\beta_{10} - 2610\beta_{8} - 2610\beta_{6} \) 3707b10 - 2610b8 - 2610*b6
\(\nu^{14}\)	\(=\)	\( -16341\beta_{14} - 11537\beta_{12} - 11537\beta_{11} + 16341\beta_{5} + 11537 \) -16341b14 - 11537b12 - 11537b11 + 16341b5 + 11537
\(\nu^{15}\)	\(=\)	\( 16341\beta_{15} + 11537\beta_{13} + 11537\beta_{10} - 16341\beta_{6} - 11537\beta_{4} \) 16341b15 + 11537b13 + 11537b10 - 16341b6 - 11537*b4

\(n\)	\(409\)	\(715\)	\(953\)	\(1261\)
\(\chi(n)\)	\(-\beta_{12}\)	\(1\)	\(1\)	\(\beta_{9}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 361.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - T^{4} + 1)^{2} \) (T^8 - T^4 + 1)^2
$5$	\( T^{16} + 8 T^{15} + \cdots + 16 \) T^16 + 8T^15 + 32T^14 + 160T^13 + 628T^12 + 1264T^11 + 2816T^10 + 6688T^9 - 628T^8 - 14240T^7 + 6656T^6 - 9280T^5 + 13776T^4 - 3456T^3 + 512T^2 - 128*T + 16
$7$	\( (T^{8} - 12 T^{7} + \cdots + 2401)^{2} \) (T^8 - 12T^7 + 72T^6 - 288T^5 + 863T^4 - 2016T^3 + 3528T^2 - 4116*T + 2401)^2
$11$	\( T^{16} + 8 T^{15} + \cdots + 16 \) T^16 + 8T^15 + 32T^14 + 160T^13 + 628T^12 + 1264T^11 + 2816T^10 + 6688T^9 - 628T^8 - 14240T^7 + 6656T^6 - 9280T^5 + 13776T^4 - 3456T^3 + 512T^2 - 128*T + 16
$13$	\( (T^{4} - 4 T^{3} - 30 T^{2} + \cdots + 97)^{4} \) (T^4 - 4T^3 - 30T^2 + 100*T + 97)^4
$17$	\( T^{16} + \cdots + 6975757441 \) T^16 - 16T^14 + 240T^13 + 240T^12 - 3960T^11 + 23392T^10 + 63360T^9 - 402049T^8 + 1077120T^7 + 6760288T^6 - 19455480T^5 + 20045040T^4 + 340765680T^3 - 386201104*T^2 + 6975757441
$19$	\( T^{16} + \cdots + 671898241 \) T^16 - 76T^14 + 4266T^12 - 92848T^10 + 1421523T^8 - 12603568T^6 + 80893226T^4 - 283990476*T^2 + 671898241
$23$	\( T^{16} - 24 T^{15} + \cdots + 6250000 \) T^16 - 24T^15 + 288T^14 - 2976T^13 + 27348T^12 - 186192T^11 + 1020672T^10 - 5136672T^9 + 21143756T^8 - 64799520T^7 + 172915200T^6 - 405528000T^5 + 557970000T^4 - 241200000T^3 + 72000000T^2 - 30000000*T + 6250000
$29$	\( (T^{8} + 8 T^{7} + \cdots + 6724)^{2} \) (T^8 + 8T^7 + 32T^6 - 272T^5 + 1132T^4 - 80T^3 + 128T^2 - 1312*T + 6724)^2
$31$	\( T^{16} + \cdots + 104726589427201 \) T^16 - 16T^15 + 128T^14 - 992T^13 - 2818T^12 + 120816T^11 - 1080320T^10 + 9048592T^9 + 24761859T^8 - 1162258608T^7 + 10704327680T^6 - 91782795984T^5 + 674680944382T^4 - 3169647920992T^3 + 12838338995328T^2 - 51855866724816*T + 104726589427201
$37$	\( T^{16} + \cdots + 150644120641 \) T^16 + 16T^15 + 128T^14 + 1504T^13 + 10462T^12 + 4624T^11 - 134144T^10 - 1847568T^9 - 12282525T^8 - 2957136T^7 + 160211968T^6 + 2153614288T^5 + 16982531230T^4 + 39141963616T^3 + 60858627200T^2 + 135410445520*T + 150644120641
$41$	\( (T^{8} + 33168 T^{4} + 261921856)^{2} \) (T^8 + 33168*T^4 + 261921856)^2
$43$	\( (T^{4} + 54 T^{2} + 81)^{4} \) (T^4 + 54*T^2 + 81)^4
$47$	\( (T^{8} + 8 T^{7} + \cdots + 256)^{2} \) (T^8 + 8T^7 + 88T^6 + 1360T^4 + 2560T^3 + 8832T^2 + 1536T + 256)^2
$53$	\( T^{16} - 104 T^{14} + \cdots + 45212176 \) T^16 - 104T^14 + 8332T^12 - 222976T^10 + 4324812T^8 - 42518528T^6 + 295879984T^4 - 118880320*T^2 + 45212176
$59$	\( T^{16} + \cdots + 87578116096 \) T^16 - 352T^14 + 87744T^12 - 10502144T^10 + 915442688T^8 - 40040923136T^6 + 1228263849984T^4 - 329402810368*T^2 + 87578116096
$61$	\( T^{16} + \cdots + 644513529856 \) T^16 - 14592T^12 + 212123648T^8 - 11714691072*T^4 + 644513529856
$67$	\( (T^{8} - 12 T^{7} + \cdots + 289)^{2} \) (T^8 - 12T^7 + 126T^6 - 368T^5 + 1219T^4 + 1776T^3 + 5470T^2 + 1292*T + 289)^2
$71$	\( (T^{8} + 16 T^{7} + \cdots + 18496)^{2} \) (T^8 + 16T^7 + 128T^6 + 384T^5 + 528T^4 + 128T^3 + 8192T^2 + 17408*T + 18496)^2
$73$	\( T^{16} - 8 T^{15} + \cdots + 5764801 \) T^16 - 8T^15 + 32T^14 - 1712T^13 + 2046T^12 + 112280T^11 + 501760T^10 + 2597000T^9 + 13890275T^8 + 37524200T^7 + 86051840T^6 + 252297080T^5 + 287654206T^4 - 174523888T^3 + 33882912T^2 - 19765032*T + 5764801
$79$	\( T^{16} + \cdots + 110841719041 \) T^16 - 8T^15 + 32T^14 - 688T^13 - 10018T^12 + 129944T^11 - 482304T^10 + 6260872T^9 + 165097539T^8 - 1141745944T^7 + 3154366464T^6 - 32196987272T^5 + 100114768862T^4 + 165437744656T^3 + 88223521568T^2 + 139848824024*T + 110841719041
$83$	\( (T^{8} + 400 T^{6} + \cdots + 614656)^{2} \) (T^8 + 400T^6 + 33376T^4 + 313600*T^2 + 614656)^2
$89$	\( (T^{8} + 80 T^{6} + \cdots + 200704)^{2} \) (T^8 + 80T^6 + 256T^5 + 5952T^4 + 10240T^3 + 52224T^2 - 57344T + 200704)^2
$97$	\( (T^{8} + 56 T^{7} + \cdots + 2155024)^{2} \) (T^8 + 56T^7 + 1568T^6 + 25008T^5 + 245000T^4 + 1333856T^3 + 3235968T^2 - 3734592*T + 2155024)^2
show more
show less