LMFDB - Newform orbit 1280.2.j.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(63,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 3]))

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

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

chi := DirichletCharacter("1280.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.j (of order $4$, degree $2$, 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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 34x^{12} + 20x^{10} + 75x^{8} - 228x^{6} + 342x^{4} + 48x^{2} + 25 $ x^16 - 4x^14 + 34x^12 + 20x^10 + 75x^8 - 228x^6 + 342x^4 + 48*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{12} - \beta_{9}) q^{3} + (\beta_{6} - \beta_{5}) q^{5} - \beta_{2} q^{7} + (\beta_{7} + \beta_{4} + \beta_{3} - 1) q^{9}+O(q^{10}) $ q + (-b12 - b9) * q^3 + (b6 - b5) * q^5 - b2 * q^7 + (b7 + b4 + b3 - 1) * q^9	$ q + ( - \beta_{12} - \beta_{9}) q^{3} + (\beta_{6} - \beta_{5}) q^{5} - \beta_{2} q^{7} + (\beta_{7} + \beta_{4} + \beta_{3} - 1) q^{9} + ( - \beta_{12} + \beta_{11} + \cdots - \beta_1) q^{11}+ \cdots + (3 \beta_{12} - 3 \beta_{11} + \cdots - \beta_1) q^{99}+O(q^{100}) $ q + (-b12 - b9) * q^3 + (b6 - b5) * q^5 - b2 * q^7 + (b7 + b4 + b3 - 1) * q^9 + (-b12 + b11 - b10 - b9 - b2 - b1) * q^11 + (-b8 + b7 - b5 + 1) * q^13 + (-2b11 + b10 - b2) q^15 + (b8 - b4) * q^17 + (b10 + b9 + b2 + b1) * q^19 + (-b13 - b7 - b6 + b5 - b3) * q^21 + (-b15 + 2b14 + b12 + b11) q^23 + (-b13 + 2b3) q^25 + (2b15 + b14 + 2b12 + 2b9 - 2b2 + b1) * q^27 + (b13 - b8 + b7 + b5 - b4 - b3 - 1) * q^29 + (-b14 + 2b12 + 2b9 - b1) * q^31 + (b13 + b8 + b7 + 4b6 + b4 - 4) q^33 + (b14 + b11 - 3b10 - b2) q^35 + (b8 + 2b7 + b5) q^37 + (2b15 + b14 + 2b9 - 2b2 + b1) q^39 + (b13 + b8 + 3b6 - b5 - 2b4 + 2b3 + 2) q^41 + (b15 + 2b14 - b11 + 3b10 + b2 - 2b1) q^43 + (-2b13 - 2b8 - b7 - 7b6 - b3) q^45 + (b15 + 2b14 + b10 - b9) q^47 + (b13 - 2b6 + b4 - b3 - 1) q^49 + (-2b15 - b12 - b11) q^51 + (b13 - b6 + b4 - b3 - 1) * q^53 + (-2b15 + b14 - b12 - b11 - 2b10 - 3b9 - 2b2 + b1) * q^55 + (-b13 - b7 - b5 + b3 + 1) * q^57 + (-b15 + 3b14 - 2b12 - 2b11 + b10 - b9) q^59 + (2b13 - 2b8 - 2b7 - b6 - b5 + 2b4 - b3 - 2) * q^61 + (-3b12 + 3b11 - 4b10 - 4b9 + b2 - 2b1) q^63 + (-3b13 + b6 - b5 + b3 + 5) q^65 + (-b15 + b11 + 3b10 - b2) q^67 + (b13 - b7 + 3b6 - b5 - b3 + 2) q^69 + (-b15 + 2b14 - 2b11 - b2 - 2b1) q^71 + (b8 - 4b6 + 2b5 - b4 + 2b3 - 2) q^73 + (b15 + b14 + 4b12 + b11 - 3b10 + 2b9 - b2 - b1) q^75 + (-b13 - 2b8 + b6 + 2b5 + 2b4 - 2b3 - 2) * q^77 + (b15 + b14 + 4b11 - 2b10 + b2 - b1) * q^79 + (2b8 - 5b7 + 2b5 - b4 - b3 + 7) q^81 + (b14 + 5b12 - 3b9 + b1) * q^83 + (2b8 + b7 + b6 + b5 - b4 - 4) q^85 + (-b12 + b11 + 3b10 + 3b9 + 2b1) q^87 + (b8 - 2b7 + b5 + b4 + b3 + 1) q^89 + (-3b12 + 3b11 - 4b10 - 4b9 - 2b2) q^91 + (-2b4 - 2b3 + 8) * q^93 + (2b15 - b12 - b11 + 3b10 + 2b9 + 2b2) * q^95 + (-2b13 + 2b7 - 4b6 - b5 - b3 - 5) q^97 + (3b12 - 3b11 + 4b10 + 4b9 - 5b2 - b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} + 16 q^{13} - 8 q^{17} + 8 q^{21} - 16 q^{25} - 16 q^{29} - 56 q^{33} + 8 q^{45} + 8 q^{57} - 8 q^{61} + 72 q^{65} + 40 q^{69} - 56 q^{73} + 112 q^{81} - 72 q^{85} + 16 q^{89} + 128 q^{93} - 72 q^{97}+O(q^{100}) $ 16 * q - 16 * q^9 + 16 * q^13 - 8 * q^17 + 8 * q^21 - 16 * q^25 - 16 * q^29 - 56 * q^33 + 8 * q^45 + 8 * q^57 - 8 * q^61 + 72 * q^65 + 40 * q^69 - 56 * q^73 + 112 * q^81 - 72 * q^85 + 16 * q^89 + 128 * q^93 - 72 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4x^{14} + 34x^{12} + 20x^{10} + 75x^{8} - 228x^{6} + 342x^{4} + 48x^{2} + 25 $ x^16 - 4*x^14 + 34*x^12 + 20*x^10 + 75*x^8 - 228*x^6 + 342*x^4 + 48*x^2 + 25 :

$\beta_{1}$	$=$	$ ( - 69 \nu^{15} + 531 \nu^{13} - 4197 \nu^{11} + 10529 \nu^{9} - 28993 \nu^{7} + 20103 \nu^{5} + \cdots + 247570 \nu ) / 85400 $ (-69v^15 + 531v^13 - 4197v^11 + 10529v^9 - 28993v^7 + 20103v^5 - 179820v^3 + 247570v) / 85400
$\beta_{2}$	$=$	$ ( 121 \nu^{15} - 1451 \nu^{13} + 8511 \nu^{11} - 31967 \nu^{9} + 5259 \nu^{7} - 71381 \nu^{5} + \cdots - 433340 \nu ) / 85400 $ (121v^15 - 1451v^13 + 8511v^11 - 31967v^9 + 5259v^7 - 71381v^5 + 305794v^3 - 433340v) / 85400
$\beta_{3}$	$=$	$ ( 106\nu^{14} - 463\nu^{12} + 3867\nu^{10} + 379\nu^{8} + 10151\nu^{6} - 21953\nu^{4} + 37775\nu^{2} - 34950 ) / 17080 $ (106v^14 - 463v^12 + 3867v^10 + 379v^8 + 10151v^6 - 21953v^4 + 37775*v^2 - 34950) / 17080
$\beta_{4}$	$=$	$ ( - 551 \nu^{14} + 2161 \nu^{12} - 18236 \nu^{10} - 14768 \nu^{8} - 27424 \nu^{6} + 90746 \nu^{4} + \cdots - 109975 ) / 85400 $ (-551v^14 + 2161v^12 - 18236v^10 - 14768v^8 - 27424v^6 + 90746v^4 - 190989*v^2 - 109975) / 85400
$\beta_{5}$	$=$	$ ( - 832 \nu^{14} + 3618 \nu^{12} - 28886 \nu^{10} - 8413 \nu^{8} - 40424 \nu^{6} + 249549 \nu^{4} + \cdots + 32425 ) / 85400 $ (-832v^14 + 3618v^12 - 28886v^10 - 8413v^8 - 40424v^6 + 249549v^4 - 366510*v^2 + 32425) / 85400
$\beta_{6}$	$=$	$ ( 15\nu^{14} - 64\nu^{12} + 521\nu^{10} + 183\nu^{8} + 869\nu^{6} - 3823\nu^{4} + 5148\nu^{2} + 225 ) / 1400 $ (15v^14 - 64v^12 + 521v^10 + 183v^8 + 869v^6 - 3823v^4 + 5148*v^2 + 225) / 1400
$\beta_{7}$	$=$	$ ( -141\nu^{14} + 369\nu^{12} - 4028\nu^{10} - 9329\nu^{8} - 14942\nu^{6} + 19067\nu^{4} + 2825\nu^{2} - 43600 ) / 12200 $ (-141v^14 + 369v^12 - 4028v^10 - 9329v^8 - 14942v^6 + 19067v^4 + 2825*v^2 - 43600) / 12200
$\beta_{8}$	$=$	$ ( 1133 \nu^{14} - 4542 \nu^{12} + 38469 \nu^{10} + 21272 \nu^{8} + 87751 \nu^{6} - 303456 \nu^{4} + \cdots + 63300 ) / 85400 $ (1133v^14 - 4542v^12 + 38469v^10 + 21272v^8 + 87751v^6 - 303456v^4 + 358950*v^2 + 63300) / 85400
$\beta_{9}$	$=$	$ ( 127 \nu^{15} - 492 \nu^{13} + 4232 \nu^{11} + 3126 \nu^{9} + 9268 \nu^{7} - 29652 \nu^{5} + \cdots + 15230 \nu ) / 6100 $ (127v^15 - 492v^13 + 4232v^11 + 3126v^9 + 9268v^7 - 29652v^5 + 35863v^3 + 15230v) / 6100
$\beta_{10}$	$=$	$ ( - 957 \nu^{15} + 4023 \nu^{13} - 33426 \nu^{11} - 12143 \nu^{9} - 70824 \nu^{7} + 225909 \nu^{5} + \cdots + 50920 \nu ) / 42700 $ (-957v^15 + 4023v^13 - 33426v^11 - 12143v^9 - 70824v^7 + 225909v^5 - 363945v^3 + 50920v) / 42700
$\beta_{11}$	$=$	$ ( 2029 \nu^{15} - 9358 \nu^{13} + 73420 \nu^{11} + 475 \nu^{9} + 110690 \nu^{7} - 571577 \nu^{5} + \cdots - 133715 \nu ) / 85400 $ (2029v^15 - 9358v^13 + 73420v^11 + 475v^9 + 110690v^7 - 571577v^5 + 983609v^3 - 133715v) / 85400
$\beta_{12}$	$=$	$ ( 304 \nu^{15} - 1146 \nu^{13} + 10097 \nu^{11} + 8476 \nu^{9} + 25023 \nu^{7} - 57778 \nu^{5} + \cdots + 42460 \nu ) / 12200 $ (304v^15 - 1146v^13 + 10097v^11 + 8476v^9 + 25023v^7 - 57778v^5 + 100285v^3 + 42460v) / 12200
$\beta_{13}$	$=$	$ ( 3112 \nu^{14} - 12457 \nu^{12} + 106657 \nu^{10} + 59966 \nu^{8} + 257163 \nu^{6} - 657752 \nu^{4} + \cdots + 63375 ) / 85400 $ (3112v^14 - 12457v^12 + 106657v^10 + 59966v^8 + 257163v^6 - 657752v^4 + 1195843*v^2 + 63375) / 85400
$\beta_{14}$	$=$	$ ( 3016 \nu^{15} - 11941 \nu^{13} + 101616 \nu^{11} + 65778 \nu^{9} + 215284 \nu^{7} - 705046 \nu^{5} + \cdots + 86085 \nu ) / 85400 $ (3016v^15 - 11941v^13 + 101616v^11 + 65778v^9 + 215284v^7 - 705046v^5 + 963834v^3 + 86085v) / 85400
$\beta_{15}$	$=$	$ ( 4930 \nu^{15} - 19987 \nu^{13} + 168468 \nu^{11} + 90064 \nu^{9} + 356932 \nu^{7} + \cdots + 155045 \nu ) / 85400 $ (4930v^15 - 19987v^13 + 168468v^11 + 90064v^9 + 356932v^7 - 1156864v^5 + 1691724v^3 + 155045v) / 85400

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{14} + \beta_{10} ) / 2 $ (b15 - b14 + b10) / 2
$\nu^{2}$	$=$	$ ( -3\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} + 2 ) / 2 $ (-3b6 - 2b5 - b4 + b3 + 2) / 2
$\nu^{3}$	$=$	$ ( 3\beta_{15} - \beta_{12} + \beta_{11} - 8\beta_{9} - 5\beta_{2} - 2\beta_1 ) / 2 $ (3b15 - b12 + b11 - 8b9 - 5b2 - 2b1) / 2
$\nu^{4}$	$=$	$ ( -2\beta_{13} + 9\beta_{8} + 2\beta_{7} - 9\beta_{6} + 3\beta_{5} - 12\beta_{4} + 4\beta_{3} - 5 ) / 2 $ (-2b13 + 9b8 + 2b7 - 9b6 + 3b5 - 12b4 + 4*b3 - 5) / 2
$\nu^{5}$	$=$	$ ( -19\beta_{15} + 4\beta_{14} - \beta_{12} + 9\beta_{11} - 48\beta_{10} - 13\beta_{9} - 30\beta_{2} - 5\beta_1 ) / 2 $ (-19b15 + 4b14 - b12 + 9b11 - 48b10 - 13b9 - 30b2 - 5*b1) / 2
$\nu^{6}$	$=$	$ ( 7\beta_{13} + 44\beta_{8} + 17\beta_{7} + 23\beta_{6} + 66\beta_{5} - 10\beta_{4} - 50\beta_{3} - 121 ) / 2 $ (7b13 + 44b8 + 17b7 + 23b6 + 66b5 - 10b4 - 50*b3 - 121) / 2
$\nu^{7}$	$=$	$ ( - 217 \beta_{15} + 49 \beta_{14} + 48 \beta_{12} + 36 \beta_{11} - 171 \beta_{10} + 234 \beta_{9} + \cdots + 2 \beta_1 ) / 2 $ (-217b15 + 49b14 + 48b12 + 36b11 - 171b10 + 234b9 + 30b2 + 2b1) / 2
$\nu^{8}$	$=$	$ ( 108\beta_{13} - 187\beta_{8} + 4\beta_{7} + 469\beta_{6} + 219\beta_{5} + 324\beta_{4} - 356\beta_{3} - 397 ) / 2 $ (108b13 - 187b8 + 4b7 + 469b6 + 219b5 + 324b4 - 356*b3 - 397) / 2
$\nu^{9}$	$=$	$ ( - 358 \beta_{15} + 103 \beta_{14} + 327 \beta_{12} - 183 \beta_{11} + 865 \beta_{10} + 1491 \beta_{9} + \cdots + 315 \beta_1 ) / 2 $ (-358b15 + 103b14 + 327b12 - 183b11 + 865b10 + 1491b9 + 1290b2 + 315b1) / 2
$\nu^{10}$	$=$	$ ( 291 \beta_{13} - 2398 \beta_{8} - 579 \beta_{7} + 1332 \beta_{6} - 1250 \beta_{5} + 2031 \beta_{4} + \cdots + 2755 ) / 2 $ (291b13 - 2398b8 - 579b7 + 1332b6 - 1250b5 + 2031b4 + 189*b3 + 2755) / 2
$\nu^{11}$	$=$	$ ( 6336 \beta_{15} - 1573 \beta_{14} - 245 \beta_{12} - 2263 \beta_{11} + 10387 \beta_{10} + \cdots + 1278 \beta_1 ) / 2 $ (6336b15 - 1573b14 - 245b12 - 2263b11 + 10387b10 - 1140b9 + 5095b2 + 1278b1) / 2
$\nu^{12}$	$=$	$ - 1252 \beta_{13} - 2390 \beta_{8} - 1520 \beta_{7} - 5577 \beta_{6} - 7208 \beta_{5} - 1033 \beta_{4} + \cdots + 13627 $ -1252b13 - 2390b8 - 1520b7 - 5577b6 - 7208b5 - 1033b4 + 6869*b3 + 13627
$\nu^{13}$	$=$	$ ( 43625 \beta_{15} - 10735 \beta_{14} - 13158 \beta_{12} - 4058 \beta_{11} + 18863 \beta_{10} + \cdots - 5720 \beta_1 ) / 2 $ (43625b15 - 10735b14 - 13158b12 - 4058b11 + 18863b10 - 60724b9 - 23010b2 - 5720b1) / 2
$\nu^{14}$	$=$	$ ( - 23024 \beta_{13} + 64922 \beta_{8} + 6616 \beta_{7} - 101987 \beta_{6} - 23136 \beta_{5} + \cdots + 30456 ) / 2 $ (-23024b13 + 64922b8 + 6616b7 - 101987b6 - 23136b5 - 85447b4 + 59967*b3 + 30456) / 2
$\nu^{15}$	$=$	$ ( - 22885 \beta_{15} + 5772 \beta_{14} - 54313 \beta_{12} + 63385 \beta_{11} - 293188 \beta_{10} + \cdots - 73248 \beta_1 ) / 2 $ (-22885b15 + 5772b14 - 54313b12 + 63385b11 - 293188b10 - 251714b9 - 298455b2 - 73248b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$\beta_{6}$	$-\beta_{6}$	$-1$

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} $

63.1

−0.293774	+	0.413333i
1.12044	+	0.413333i
−2.05957	−	1.35246i
0.645356	+	1.35246i
−0.645356	−	1.35246i
2.05957	+	1.35246i
−1.12044	−	0.413333i
0.293774	−	0.413333i
0.293774	+	0.413333i
−1.12044	+	0.413333i
2.05957	−	1.35246i
−0.645356	+	1.35246i
0.645356	−	1.35246i
−2.05957	+	1.35246i
1.12044	−	0.413333i
−0.293774	−	0.413333i

−

3.31797i

−0.584541

−

2.15831i

−1.93949

1.93949i

−8.00895

63.2

−

1.90376i

0.584541

−

2.15831i

1.11282

−

1.11282i

−0.624302

63.3

−

1.13533i

1.91267

1.15831i

2.17151

−

2.17151i

1.71103

63.4

−

0.278883i

−1.91267

1.15831i

−0.533412

0.533412i

2.92222

63.5

0.278883i

−1.91267

1.15831i

0.533412

−

0.533412i

2.92222

63.6

1.13533i

1.91267

1.15831i

−2.17151

2.17151i

1.71103

63.7

1.90376i

0.584541

−

2.15831i

−1.11282

1.11282i

−0.624302

63.8

3.31797i

−0.584541

−

2.15831i

1.93949

−

1.93949i

−8.00895

447.1

−

3.31797i

−0.584541

2.15831i

1.93949

1.93949i

−8.00895

447.2

−

1.90376i

0.584541

2.15831i

−1.11282

−

1.11282i

−0.624302

447.3

−

1.13533i

1.91267

−

1.15831i

−2.17151

−

2.17151i

1.71103

447.4

−

0.278883i

−1.91267

−

1.15831i

0.533412

0.533412i

2.92222

447.5

0.278883i

−1.91267

−

1.15831i

−0.533412

−

0.533412i

2.92222

447.6

1.13533i

1.91267

−

1.15831i

2.17151

2.17151i

1.71103

447.7

1.90376i

0.584541

2.15831i

1.11282

1.11282i

−0.624302

447.8

3.31797i

−0.584541

2.15831i

−1.93949

−

1.93949i

−8.00895

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
80.j	even	4	1	inner
80.t	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1280.2.j.b	yes	16
4.b	odd	2	1	inner	1280.2.j.b	yes	16
5.c	odd	4	1		1280.2.s.a	yes	16
8.b	even	2	1		1280.2.j.a	✓	16
8.d	odd	2	1		1280.2.j.a	✓	16
16.e	even	4	1		1280.2.s.a	yes	16
16.e	even	4	1		1280.2.s.b	yes	16
16.f	odd	4	1		1280.2.s.a	yes	16
16.f	odd	4	1		1280.2.s.b	yes	16
20.e	even	4	1		1280.2.s.a	yes	16
40.i	odd	4	1		1280.2.s.b	yes	16
40.k	even	4	1		1280.2.s.b	yes	16
80.i	odd	4	1		1280.2.j.a	✓	16
80.j	even	4	1	inner	1280.2.j.b	yes	16
80.s	even	4	1		1280.2.j.a	✓	16
80.t	odd	4	1	inner	1280.2.j.b	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1280.2.j.a	✓	16	8.b	even	2	1
1280.2.j.a	✓	16	8.d	odd	2	1
1280.2.j.a	✓	16	80.i	odd	4	1
1280.2.j.a	✓	16	80.s	even	4	1
1280.2.j.b	yes	16	1.a	even	1	1	trivial
1280.2.j.b	yes	16	4.b	odd	2	1	inner
1280.2.j.b	yes	16	80.j	even	4	1	inner
1280.2.j.b	yes	16	80.t	odd	4	1	inner
1280.2.s.a	yes	16	5.c	odd	4	1
1280.2.s.a	yes	16	16.e	even	4	1
1280.2.s.a	yes	16	16.f	odd	4	1
1280.2.s.a	yes	16	20.e	even	4	1
1280.2.s.b	yes	16	16.e	even	4	1
1280.2.s.b	yes	16	16.f	odd	4	1
1280.2.s.b	yes	16	40.i	odd	4	1
1280.2.s.b	yes	16	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1280, [\chi])$:

$ T_{3}^{8} + 16T_{3}^{6} + 60T_{3}^{4} + 56T_{3}^{2} + 4 $

$ T_{13}^{4} - 4T_{13}^{3} - 28T_{13}^{2} + 64T_{13} + 212 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 16 T^{6} + 60 T^{4} + \cdots + 4)^{2} $ (T^8 + 16T^6 + 60T^4 + 56*T^2 + 4)^2
$5$	$ (T^{8} + 4 T^{6} + \cdots + 625)^{2} $ (T^8 + 4T^6 + 10T^4 + 100*T^2 + 625)^2
$7$	$ T^{16} + 152 T^{12} + \cdots + 10000 $ T^16 + 152T^12 + 5976T^8 + 32800*T^4 + 10000
$11$	$ T^{16} + \cdots + 100000000 $ T^16 + 1168T^12 + 282208T^8 + 20060416*T^4 + 100000000
$13$	$ (T^{4} - 4 T^{3} + \cdots + 212)^{4} $ (T^4 - 4T^3 - 28T^2 + 64*T + 212)^4
$17$	$ (T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 16)^{2} $ (T^8 + 4T^7 + 8T^6 - 72T^5 + 248T^4 - 144T^3 + 32T^2 + 32*T + 16)^2
$19$	$ T^{16} + 1264 T^{12} + \cdots + 614656 $ T^16 + 1264T^12 + 130656T^8 + 1351424*T^4 + 614656
$23$	$ T^{16} + 5624 T^{12} + \cdots + 54700816 $ T^16 + 5624T^12 + 2114136T^8 + 148268704*T^4 + 54700816
$29$	$ (T^{8} + 8 T^{7} + \cdots + 364816)^{2} $ (T^8 + 8T^7 + 32T^6 + 144T^5 + 4808T^4 + 42272T^3 + 194688T^2 + 376896*T + 364816)^2
$31$	$ (T^{8} + 104 T^{6} + \cdots + 40000)^{2} $ (T^8 + 104T^6 + 3184T^4 + 32000*T^2 + 40000)^2
$37$	$ (T^{4} - 64 T^{2} + 980)^{4} $ (T^4 - 64*T^2 + 980)^4
$41$	$ (T^{8} + 232 T^{6} + \cdots + 150544)^{2} $ (T^8 + 232T^6 + 15640T^4 + 303328*T^2 + 150544)^2
$43$	$ (T^{8} - 240 T^{6} + \cdots + 1716100)^{2} $ (T^8 - 240T^6 + 18076T^4 - 440680*T^2 + 1716100)^2
$47$	$ T^{16} + \cdots + 71997768976 $ T^16 + 4024T^12 + 3406296T^8 + 895454624*T^4 + 71997768976
$53$	$ (T^{8} + 56 T^{6} + \cdots + 10000)^{2} $ (T^8 + 56T^6 + 984T^4 + 6304*T^2 + 10000)^2
$59$	$ T^{16} + \cdots + 56364057760000 $ T^16 + 45872T^12 + 279750496T^8 + 272100780800*T^4 + 56364057760000
$61$	$ (T^{8} + 4 T^{7} + \cdots + 25806400)^{2} $ (T^8 + 4T^7 + 8T^6 - 112T^5 + 18064T^4 + 73760T^3 + 156800T^2 - 2844800*T + 25806400)^2
$67$	$ (T^{8} - 176 T^{6} + \cdots + 4900)^{2} $ (T^8 - 176T^6 + 9292T^4 - 154088*T^2 + 4900)^2
$71$	$ (T^{8} - 264 T^{6} + \cdots + 4293184)^{2} $ (T^8 - 264T^6 + 21040T^4 - 602624*T^2 + 4293184)^2
$73$	$ (T^{8} + 28 T^{7} + \cdots + 7728400)^{2} $ (T^8 + 28T^7 + 392T^6 + 2664T^5 + 9656T^4 + 22032T^3 + 380192T^2 + 2424160*T + 7728400)^2
$79$	$ (T^{8} - 320 T^{6} + \cdots + 313600)^{2} $ (T^8 - 320T^6 + 13216T^4 - 158720*T^2 + 313600)^2
$83$	$ (T^{8} + 568 T^{6} + \cdots + 68724100)^{2} $ (T^8 + 568T^6 + 97516T^4 + 5088408*T^2 + 68724100)^2
$89$	$ (T^{4} - 4 T^{3} + \cdots - 800)^{4} $ (T^4 - 4T^3 - 96T^2 + 640*T - 800)^4
$97$	$ (T^{8} + 36 T^{7} + \cdots + 68890000)^{2} $ (T^8 + 36T^7 + 648T^6 + 5624T^5 + 27416T^4 + 103280T^3 + 1767200T^2 + 15604000*T + 68890000)^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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{12} - \beta_{9}) q^{3} + (\beta_{6} - \beta_{5}) q^{5} - \beta_{2} q^{7} + (\beta_{7} + \beta_{4} + \beta_{3} - 1) q^{9}+O(q^{10}) \) q + (-b12 - b9) * q^3 + (b6 - b5) * q^5 - b2 * q^7 + (b7 + b4 + b3 - 1) * q^9	\( q + ( - \beta_{12} - \beta_{9}) q^{3} + (\beta_{6} - \beta_{5}) q^{5} - \beta_{2} q^{7} + (\beta_{7} + \beta_{4} + \beta_{3} - 1) q^{9} + ( - \beta_{12} + \beta_{11} + \cdots - \beta_1) q^{11}+ \cdots + (3 \beta_{12} - 3 \beta_{11} + \cdots - \beta_1) q^{99}+O(q^{100}) \) q + (-b12 - b9) * q^3 + (b6 - b5) * q^5 - b2 * q^7 + (b7 + b4 + b3 - 1) * q^9 + (-b12 + b11 - b10 - b9 - b2 - b1) * q^11 + (-b8 + b7 - b5 + 1) * q^13 + (-2b11 + b10 - b2) q^15 + (b8 - b4) * q^17 + (b10 + b9 + b2 + b1) * q^19 + (-b13 - b7 - b6 + b5 - b3) * q^21 + (-b15 + 2b14 + b12 + b11) q^23 + (-b13 + 2b3) q^25 + (2b15 + b14 + 2b12 + 2b9 - 2b2 + b1) * q^27 + (b13 - b8 + b7 + b5 - b4 - b3 - 1) * q^29 + (-b14 + 2b12 + 2b9 - b1) * q^31 + (b13 + b8 + b7 + 4b6 + b4 - 4) q^33 + (b14 + b11 - 3b10 - b2) q^35 + (b8 + 2b7 + b5) q^37 + (2b15 + b14 + 2b9 - 2b2 + b1) q^39 + (b13 + b8 + 3b6 - b5 - 2b4 + 2b3 + 2) q^41 + (b15 + 2b14 - b11 + 3b10 + b2 - 2b1) q^43 + (-2b13 - 2b8 - b7 - 7b6 - b3) q^45 + (b15 + 2b14 + b10 - b9) q^47 + (b13 - 2b6 + b4 - b3 - 1) q^49 + (-2b15 - b12 - b11) q^51 + (b13 - b6 + b4 - b3 - 1) * q^53 + (-2b15 + b14 - b12 - b11 - 2b10 - 3b9 - 2b2 + b1) * q^55 + (-b13 - b7 - b5 + b3 + 1) * q^57 + (-b15 + 3b14 - 2b12 - 2b11 + b10 - b9) q^59 + (2b13 - 2b8 - 2b7 - b6 - b5 + 2b4 - b3 - 2) * q^61 + (-3b12 + 3b11 - 4b10 - 4b9 + b2 - 2b1) q^63 + (-3b13 + b6 - b5 + b3 + 5) q^65 + (-b15 + b11 + 3b10 - b2) q^67 + (b13 - b7 + 3b6 - b5 - b3 + 2) q^69 + (-b15 + 2b14 - 2b11 - b2 - 2b1) q^71 + (b8 - 4b6 + 2b5 - b4 + 2b3 - 2) q^73 + (b15 + b14 + 4b12 + b11 - 3b10 + 2b9 - b2 - b1) q^75 + (-b13 - 2b8 + b6 + 2b5 + 2b4 - 2b3 - 2) * q^77 + (b15 + b14 + 4b11 - 2b10 + b2 - b1) * q^79 + (2b8 - 5b7 + 2b5 - b4 - b3 + 7) q^81 + (b14 + 5b12 - 3b9 + b1) * q^83 + (2b8 + b7 + b6 + b5 - b4 - 4) q^85 + (-b12 + b11 + 3b10 + 3b9 + 2b1) q^87 + (b8 - 2b7 + b5 + b4 + b3 + 1) q^89 + (-3b12 + 3b11 - 4b10 - 4b9 - 2b2) q^91 + (-2b4 - 2b3 + 8) * q^93 + (2b15 - b12 - b11 + 3b10 + 2b9 + 2b2) * q^95 + (-2b13 + 2b7 - 4b6 - b5 - b3 - 5) q^97 + (3b12 - 3b11 + 4b10 + 4b9 - 5b2 - b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} + 16 q^{13} - 8 q^{17} + 8 q^{21} - 16 q^{25} - 16 q^{29} - 56 q^{33} + 8 q^{45} + 8 q^{57} - 8 q^{61} + 72 q^{65} + 40 q^{69} - 56 q^{73} + 112 q^{81} - 72 q^{85} + 16 q^{89} + 128 q^{93} - 72 q^{97}+O(q^{100}) \) 16 * q - 16 * q^9 + 16 * q^13 - 8 * q^17 + 8 * q^21 - 16 * q^25 - 16 * q^29 - 56 * q^33 + 8 * q^45 + 8 * q^57 - 8 * q^61 + 72 * q^65 + 40 * q^69 - 56 * q^73 + 112 * q^81 - 72 * q^85 + 16 * q^89 + 128 * q^93 - 72 * 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	1280.2.j.b

Level	$1280$
Weight	$2$
Character orbit	1280.j
Analytic conductor	$10.221$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(10.2208514587\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4x^{14} + 34x^{12} + 20x^{10} + 75x^{8} - 228x^{6} + 342x^{4} + 48x^{2} + 25 \) x^16 - 4x^14 + 34x^12 + 20x^10 + 75x^8 - 228x^6 + 342x^4 + 48*x^2 + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{10} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( - 69 \nu^{15} + 531 \nu^{13} - 4197 \nu^{11} + 10529 \nu^{9} - 28993 \nu^{7} + 20103 \nu^{5} + \cdots + 247570 \nu ) / 85400 \) (-69v^15 + 531v^13 - 4197v^11 + 10529v^9 - 28993v^7 + 20103v^5 - 179820v^3 + 247570v) / 85400
\(\beta_{2}\)	\(=\)	\( ( 121 \nu^{15} - 1451 \nu^{13} + 8511 \nu^{11} - 31967 \nu^{9} + 5259 \nu^{7} - 71381 \nu^{5} + \cdots - 433340 \nu ) / 85400 \) (121v^15 - 1451v^13 + 8511v^11 - 31967v^9 + 5259v^7 - 71381v^5 + 305794v^3 - 433340v) / 85400
\(\beta_{3}\)	\(=\)	\( ( 106\nu^{14} - 463\nu^{12} + 3867\nu^{10} + 379\nu^{8} + 10151\nu^{6} - 21953\nu^{4} + 37775\nu^{2} - 34950 ) / 17080 \) (106v^14 - 463v^12 + 3867v^10 + 379v^8 + 10151v^6 - 21953v^4 + 37775*v^2 - 34950) / 17080
\(\beta_{4}\)	\(=\)	\( ( - 551 \nu^{14} + 2161 \nu^{12} - 18236 \nu^{10} - 14768 \nu^{8} - 27424 \nu^{6} + 90746 \nu^{4} + \cdots - 109975 ) / 85400 \) (-551v^14 + 2161v^12 - 18236v^10 - 14768v^8 - 27424v^6 + 90746v^4 - 190989*v^2 - 109975) / 85400
\(\beta_{5}\)	\(=\)	\( ( - 832 \nu^{14} + 3618 \nu^{12} - 28886 \nu^{10} - 8413 \nu^{8} - 40424 \nu^{6} + 249549 \nu^{4} + \cdots + 32425 ) / 85400 \) (-832v^14 + 3618v^12 - 28886v^10 - 8413v^8 - 40424v^6 + 249549v^4 - 366510*v^2 + 32425) / 85400
\(\beta_{6}\)	\(=\)	\( ( 15\nu^{14} - 64\nu^{12} + 521\nu^{10} + 183\nu^{8} + 869\nu^{6} - 3823\nu^{4} + 5148\nu^{2} + 225 ) / 1400 \) (15v^14 - 64v^12 + 521v^10 + 183v^8 + 869v^6 - 3823v^4 + 5148*v^2 + 225) / 1400
\(\beta_{7}\)	\(=\)	\( ( -141\nu^{14} + 369\nu^{12} - 4028\nu^{10} - 9329\nu^{8} - 14942\nu^{6} + 19067\nu^{4} + 2825\nu^{2} - 43600 ) / 12200 \) (-141v^14 + 369v^12 - 4028v^10 - 9329v^8 - 14942v^6 + 19067v^4 + 2825*v^2 - 43600) / 12200
\(\beta_{8}\)	\(=\)	\( ( 1133 \nu^{14} - 4542 \nu^{12} + 38469 \nu^{10} + 21272 \nu^{8} + 87751 \nu^{6} - 303456 \nu^{4} + \cdots + 63300 ) / 85400 \) (1133v^14 - 4542v^12 + 38469v^10 + 21272v^8 + 87751v^6 - 303456v^4 + 358950*v^2 + 63300) / 85400
\(\beta_{9}\)	\(=\)	\( ( 127 \nu^{15} - 492 \nu^{13} + 4232 \nu^{11} + 3126 \nu^{9} + 9268 \nu^{7} - 29652 \nu^{5} + \cdots + 15230 \nu ) / 6100 \) (127v^15 - 492v^13 + 4232v^11 + 3126v^9 + 9268v^7 - 29652v^5 + 35863v^3 + 15230v) / 6100
\(\beta_{10}\)	\(=\)	\( ( - 957 \nu^{15} + 4023 \nu^{13} - 33426 \nu^{11} - 12143 \nu^{9} - 70824 \nu^{7} + 225909 \nu^{5} + \cdots + 50920 \nu ) / 42700 \) (-957v^15 + 4023v^13 - 33426v^11 - 12143v^9 - 70824v^7 + 225909v^5 - 363945v^3 + 50920v) / 42700
\(\beta_{11}\)	\(=\)	\( ( 2029 \nu^{15} - 9358 \nu^{13} + 73420 \nu^{11} + 475 \nu^{9} + 110690 \nu^{7} - 571577 \nu^{5} + \cdots - 133715 \nu ) / 85400 \) (2029v^15 - 9358v^13 + 73420v^11 + 475v^9 + 110690v^7 - 571577v^5 + 983609v^3 - 133715v) / 85400
\(\beta_{12}\)	\(=\)	\( ( 304 \nu^{15} - 1146 \nu^{13} + 10097 \nu^{11} + 8476 \nu^{9} + 25023 \nu^{7} - 57778 \nu^{5} + \cdots + 42460 \nu ) / 12200 \) (304v^15 - 1146v^13 + 10097v^11 + 8476v^9 + 25023v^7 - 57778v^5 + 100285v^3 + 42460v) / 12200
\(\beta_{13}\)	\(=\)	\( ( 3112 \nu^{14} - 12457 \nu^{12} + 106657 \nu^{10} + 59966 \nu^{8} + 257163 \nu^{6} - 657752 \nu^{4} + \cdots + 63375 ) / 85400 \) (3112v^14 - 12457v^12 + 106657v^10 + 59966v^8 + 257163v^6 - 657752v^4 + 1195843*v^2 + 63375) / 85400
\(\beta_{14}\)	\(=\)	\( ( 3016 \nu^{15} - 11941 \nu^{13} + 101616 \nu^{11} + 65778 \nu^{9} + 215284 \nu^{7} - 705046 \nu^{5} + \cdots + 86085 \nu ) / 85400 \) (3016v^15 - 11941v^13 + 101616v^11 + 65778v^9 + 215284v^7 - 705046v^5 + 963834v^3 + 86085v) / 85400
\(\beta_{15}\)	\(=\)	\( ( 4930 \nu^{15} - 19987 \nu^{13} + 168468 \nu^{11} + 90064 \nu^{9} + 356932 \nu^{7} + \cdots + 155045 \nu ) / 85400 \) (4930v^15 - 19987v^13 + 168468v^11 + 90064v^9 + 356932v^7 - 1156864v^5 + 1691724v^3 + 155045v) / 85400

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{14} + \beta_{10} ) / 2 \) (b15 - b14 + b10) / 2
\(\nu^{2}\)	\(=\)	\( ( -3\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} + 2 ) / 2 \) (-3b6 - 2b5 - b4 + b3 + 2) / 2
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{15} - \beta_{12} + \beta_{11} - 8\beta_{9} - 5\beta_{2} - 2\beta_1 ) / 2 \) (3b15 - b12 + b11 - 8b9 - 5b2 - 2b1) / 2
\(\nu^{4}\)	\(=\)	\( ( -2\beta_{13} + 9\beta_{8} + 2\beta_{7} - 9\beta_{6} + 3\beta_{5} - 12\beta_{4} + 4\beta_{3} - 5 ) / 2 \) (-2b13 + 9b8 + 2b7 - 9b6 + 3b5 - 12b4 + 4*b3 - 5) / 2
\(\nu^{5}\)	\(=\)	\( ( -19\beta_{15} + 4\beta_{14} - \beta_{12} + 9\beta_{11} - 48\beta_{10} - 13\beta_{9} - 30\beta_{2} - 5\beta_1 ) / 2 \) (-19b15 + 4b14 - b12 + 9b11 - 48b10 - 13b9 - 30b2 - 5*b1) / 2
\(\nu^{6}\)	\(=\)	\( ( 7\beta_{13} + 44\beta_{8} + 17\beta_{7} + 23\beta_{6} + 66\beta_{5} - 10\beta_{4} - 50\beta_{3} - 121 ) / 2 \) (7b13 + 44b8 + 17b7 + 23b6 + 66b5 - 10b4 - 50*b3 - 121) / 2
\(\nu^{7}\)	\(=\)	\( ( - 217 \beta_{15} + 49 \beta_{14} + 48 \beta_{12} + 36 \beta_{11} - 171 \beta_{10} + 234 \beta_{9} + \cdots + 2 \beta_1 ) / 2 \) (-217b15 + 49b14 + 48b12 + 36b11 - 171b10 + 234b9 + 30b2 + 2b1) / 2
\(\nu^{8}\)	\(=\)	\( ( 108\beta_{13} - 187\beta_{8} + 4\beta_{7} + 469\beta_{6} + 219\beta_{5} + 324\beta_{4} - 356\beta_{3} - 397 ) / 2 \) (108b13 - 187b8 + 4b7 + 469b6 + 219b5 + 324b4 - 356*b3 - 397) / 2
\(\nu^{9}\)	\(=\)	\( ( - 358 \beta_{15} + 103 \beta_{14} + 327 \beta_{12} - 183 \beta_{11} + 865 \beta_{10} + 1491 \beta_{9} + \cdots + 315 \beta_1 ) / 2 \) (-358b15 + 103b14 + 327b12 - 183b11 + 865b10 + 1491b9 + 1290b2 + 315b1) / 2
\(\nu^{10}\)	\(=\)	\( ( 291 \beta_{13} - 2398 \beta_{8} - 579 \beta_{7} + 1332 \beta_{6} - 1250 \beta_{5} + 2031 \beta_{4} + \cdots + 2755 ) / 2 \) (291b13 - 2398b8 - 579b7 + 1332b6 - 1250b5 + 2031b4 + 189*b3 + 2755) / 2
\(\nu^{11}\)	\(=\)	\( ( 6336 \beta_{15} - 1573 \beta_{14} - 245 \beta_{12} - 2263 \beta_{11} + 10387 \beta_{10} + \cdots + 1278 \beta_1 ) / 2 \) (6336b15 - 1573b14 - 245b12 - 2263b11 + 10387b10 - 1140b9 + 5095b2 + 1278b1) / 2
\(\nu^{12}\)	\(=\)	\( - 1252 \beta_{13} - 2390 \beta_{8} - 1520 \beta_{7} - 5577 \beta_{6} - 7208 \beta_{5} - 1033 \beta_{4} + \cdots + 13627 \) -1252b13 - 2390b8 - 1520b7 - 5577b6 - 7208b5 - 1033b4 + 6869*b3 + 13627
\(\nu^{13}\)	\(=\)	\( ( 43625 \beta_{15} - 10735 \beta_{14} - 13158 \beta_{12} - 4058 \beta_{11} + 18863 \beta_{10} + \cdots - 5720 \beta_1 ) / 2 \) (43625b15 - 10735b14 - 13158b12 - 4058b11 + 18863b10 - 60724b9 - 23010b2 - 5720b1) / 2
\(\nu^{14}\)	\(=\)	\( ( - 23024 \beta_{13} + 64922 \beta_{8} + 6616 \beta_{7} - 101987 \beta_{6} - 23136 \beta_{5} + \cdots + 30456 ) / 2 \) (-23024b13 + 64922b8 + 6616b7 - 101987b6 - 23136b5 - 85447b4 + 59967*b3 + 30456) / 2
\(\nu^{15}\)	\(=\)	\( ( - 22885 \beta_{15} + 5772 \beta_{14} - 54313 \beta_{12} + 63385 \beta_{11} - 293188 \beta_{10} + \cdots - 73248 \beta_1 ) / 2 \) (-22885b15 + 5772b14 - 54313b12 + 63385b11 - 293188b10 - 251714b9 - 298455b2 - 73248b1) / 2

\(n\)	\(257\)	\(261\)	\(511\)
\(\chi(n)\)	\(\beta_{6}\)	\(-\beta_{6}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 63.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 16 T^{6} + 60 T^{4} + \cdots + 4)^{2} \) (T^8 + 16T^6 + 60T^4 + 56*T^2 + 4)^2
$5$	\( (T^{8} + 4 T^{6} + \cdots + 625)^{2} \) (T^8 + 4T^6 + 10T^4 + 100*T^2 + 625)^2
$7$	\( T^{16} + 152 T^{12} + \cdots + 10000 \) T^16 + 152T^12 + 5976T^8 + 32800*T^4 + 10000
$11$	\( T^{16} + \cdots + 100000000 \) T^16 + 1168T^12 + 282208T^8 + 20060416*T^4 + 100000000
$13$	\( (T^{4} - 4 T^{3} + \cdots + 212)^{4} \) (T^4 - 4T^3 - 28T^2 + 64*T + 212)^4
$17$	\( (T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 16)^{2} \) (T^8 + 4T^7 + 8T^6 - 72T^5 + 248T^4 - 144T^3 + 32T^2 + 32*T + 16)^2
$19$	\( T^{16} + 1264 T^{12} + \cdots + 614656 \) T^16 + 1264T^12 + 130656T^8 + 1351424*T^4 + 614656
$23$	\( T^{16} + 5624 T^{12} + \cdots + 54700816 \) T^16 + 5624T^12 + 2114136T^8 + 148268704*T^4 + 54700816
$29$	\( (T^{8} + 8 T^{7} + \cdots + 364816)^{2} \) (T^8 + 8T^7 + 32T^6 + 144T^5 + 4808T^4 + 42272T^3 + 194688T^2 + 376896*T + 364816)^2
$31$	\( (T^{8} + 104 T^{6} + \cdots + 40000)^{2} \) (T^8 + 104T^6 + 3184T^4 + 32000*T^2 + 40000)^2
$37$	\( (T^{4} - 64 T^{2} + 980)^{4} \) (T^4 - 64*T^2 + 980)^4
$41$	\( (T^{8} + 232 T^{6} + \cdots + 150544)^{2} \) (T^8 + 232T^6 + 15640T^4 + 303328*T^2 + 150544)^2
$43$	\( (T^{8} - 240 T^{6} + \cdots + 1716100)^{2} \) (T^8 - 240T^6 + 18076T^4 - 440680*T^2 + 1716100)^2
$47$	\( T^{16} + \cdots + 71997768976 \) T^16 + 4024T^12 + 3406296T^8 + 895454624*T^4 + 71997768976
$53$	\( (T^{8} + 56 T^{6} + \cdots + 10000)^{2} \) (T^8 + 56T^6 + 984T^4 + 6304*T^2 + 10000)^2
$59$	\( T^{16} + \cdots + 56364057760000 \) T^16 + 45872T^12 + 279750496T^8 + 272100780800*T^4 + 56364057760000
$61$	\( (T^{8} + 4 T^{7} + \cdots + 25806400)^{2} \) (T^8 + 4T^7 + 8T^6 - 112T^5 + 18064T^4 + 73760T^3 + 156800T^2 - 2844800*T + 25806400)^2
$67$	\( (T^{8} - 176 T^{6} + \cdots + 4900)^{2} \) (T^8 - 176T^6 + 9292T^4 - 154088*T^2 + 4900)^2
$71$	\( (T^{8} - 264 T^{6} + \cdots + 4293184)^{2} \) (T^8 - 264T^6 + 21040T^4 - 602624*T^2 + 4293184)^2
$73$	\( (T^{8} + 28 T^{7} + \cdots + 7728400)^{2} \) (T^8 + 28T^7 + 392T^6 + 2664T^5 + 9656T^4 + 22032T^3 + 380192T^2 + 2424160*T + 7728400)^2
$79$	\( (T^{8} - 320 T^{6} + \cdots + 313600)^{2} \) (T^8 - 320T^6 + 13216T^4 - 158720*T^2 + 313600)^2
$83$	\( (T^{8} + 568 T^{6} + \cdots + 68724100)^{2} \) (T^8 + 568T^6 + 97516T^4 + 5088408*T^2 + 68724100)^2
$89$	\( (T^{4} - 4 T^{3} + \cdots - 800)^{4} \) (T^4 - 4T^3 - 96T^2 + 640*T - 800)^4
$97$	\( (T^{8} + 36 T^{7} + \cdots + 68890000)^{2} \) (T^8 + 36T^7 + 648T^6 + 5624T^5 + 27416T^4 + 103280T^3 + 1767200T^2 + 15604000*T + 68890000)^2
show more
show less