LMFDB - Newform orbit 1800.2.b.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(251,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1800.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.b (of order $2$, degree $1$, 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:	$14.3730723638$
Analytic rank:	$0$
Dimension:	$16$
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} - 2x^{14} + 6x^{12} - 20x^{10} + 33x^{8} - 80x^{6} + 96x^{4} - 128x^{2} + 256 $ x^16 - 2x^14 + 6x^12 - 20x^10 + 33x^8 - 80x^6 + 96x^4 - 128*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{6} q^{2} + \beta_{5} q^{4} - \beta_{15} q^{7} + \beta_{13} q^{8}+O(q^{10}) $ q - b6 * q^2 + b5 * q^4 - b15 * q^7 + b13 * q^8	$ q - \beta_{6} q^{2} + \beta_{5} q^{4} - \beta_{15} q^{7} + \beta_{13} q^{8} + ( - \beta_{14} + \beta_{13} + \beta_{3}) q^{11} + (\beta_{15} - \beta_{10} - \beta_{8} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{14} - \beta_{12} + \cdots + \beta_1) q^{98}+O(q^{100}) $ q - b6 * q^2 + b5 * q^4 - b15 * q^7 + b13 * q^8 + (-b14 + b13 + b3) * q^11 + (b15 - b10 - b8 - b5 - 1) * q^13 + (-b13 - b7 - b3 + b1) * q^14 + (b15 - b10 - b8 - b5 + b4 - 1) * q^16 + (-b14 + b13 - b9 + b3 - b1) * q^17 + (-b11 - b8 + b5 + 1) * q^19 + (b15 + b11 - b5 + b4) * q^22 + (b14 + b13 - b12 + b9 + b7 - b6 - b1) * q^23 + (-b12 + b9 + b7 - b3 - b1) * q^26 + (b5 - b4 + b2 + 1) * q^28 + (-b14 - b13 + b12 - b7 + b6) * q^29 + (b10 + b8 + b5 + b4 + b2 + 1) * q^31 + (-b12 + b9 + b7 + b6 - b3 + b1) * q^32 + (2b15 - b8 - 2b5 + b4 - b2 + 1) * q^34 + (b15 - b11 + b8 - b5 + b4 + b2) * q^37 + (b13 - b12 + b9 - b6 + 2b3) q^38 + (-b14 + b13 + b9 + b1) * q^41 + (-b15 - b11 + b10 + 2b5 - b4 + b2) q^43 + (b12 + b7 + b6 - b3 + b1) * q^44 + (-b15 - b11 + b5 + b4 + 2) * q^46 + (2b9 + b7 + b6 - 2b1) * q^47 + (b15 + b11 - 2b10 + b8 - b5 + b4 - b2 - 2) q^49 + (-2b15 + 2b10 + b5 + 4) * q^52 + (b14 + b13 + b9 - b1) * q^53 + (-2b14 + b13 - 2b6 - 2b1) q^56 + (2b10 + b8 - b4 + b2 - 1) q^58 + (-b12 - b9 + 2b6 - b3 - b1) q^59 + (-2b15 + b11 + b10 + 2b5 + b4 + b2 + 1) * q^61 + (-2b14 + b13 + b12 - b9 + b6 + 2b3 + 2b1) q^62 + (-2b15 + 2b11 + 2) * q^64 + (-b11 - 2b10 + b8 + 3b5 + 1) * q^67 + (2b14 + b9 + 2b7 + 2b3) q^68 + (-b14 - b13 - b12 - b9 - 2b6 + b1) q^71 + (-b15 + b11 + 3b8 + b5 - b4 + b2 - 2) q^73 + (-2b14 - b12 - b9 + b7 + b6 + 3b3 + b1) * q^74 + (2b8 + b5 + b4 + b2 + 3) q^76 + (-b14 - b13 - b12 + 2b9 - b7 - 3b6 - 2b1) q^77 + (-2b15 - 2b11 + 2b10 + 4b8 + 2) * q^79 + (2b11 + b4 + b2 - 1) q^82 + (-b12 - b9 + 2b6 + b3 - b1) q^83 + (-2b14 + b13 - b7 + 3b3 - b1) * q^86 + (-b15 + 2b11 - b10 - b8 - b5 - b2 - 6) q^88 + (2b12 - 2b9 - 4b6 - 2b1) * q^89 + (2b15 + b11 - 3b8 - 5b5 + 2b4 - 2b2 + 3) q^91 + (-b12 - b7 - b6 + b3 + 3b1) q^92 + (-3b15 - b11 + 4b10 + 2b8 + b5 + b2 + 1) q^94 + (2b15 - 2b11 - 2b10 - 4b8 + 2b4 - 2b2 - 1) * q^97 + (2b14 - b12 - b9 + b7 + b6 - 5b3 + b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{4}+O(q^{10}) $ 16 * q - 4 * q^4	$ 16 q - 4 q^{4} - 8 q^{16} + 16 q^{19} - 4 q^{22} + 28 q^{28} + 12 q^{34} + 20 q^{46} - 32 q^{49} + 44 q^{52} - 20 q^{58} + 32 q^{64} + 16 q^{67} - 32 q^{73} + 36 q^{76} - 16 q^{82} - 88 q^{88} + 48 q^{91} - 20 q^{94} - 16 q^{97}+O(q^{100}) $ 16 * q - 4 * q^4 - 8 * q^16 + 16 * q^19 - 4 * q^22 + 28 * q^28 + 12 * q^34 + 20 * q^46 - 32 * q^49 + 44 * q^52 - 20 * q^58 + 32 * q^64 + 16 * q^67 - 32 * q^73 + 36 * q^76 - 16 * q^82 - 88 * q^88 + 48 * q^91 - 20 * q^94 - 16 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2x^{14} + 6x^{12} - 20x^{10} + 33x^{8} - 80x^{6} + 96x^{4} - 128x^{2} + 256 $ x^16 - 2*x^14 + 6*x^12 - 20*x^10 + 33*x^8 - 80*x^6 + 96*x^4 - 128*x^2 + 256 :

$\beta_{1}$	$=$	$ 2\nu $ 2*v
$\beta_{2}$	$=$	$ 2\nu^{2} $ 2*v^2
$\beta_{3}$	$=$	$ ( \nu^{15} - 2\nu^{13} - 10\nu^{11} + 12\nu^{9} - 63\nu^{7} + 112\nu^{5} - 176\nu^{3} + 512\nu ) / 128 $ (v^15 - 2v^13 - 10v^11 + 12v^9 - 63v^7 + 112v^5 - 176v^3 + 512*v) / 128
$\beta_{4}$	$=$	$ ( -\nu^{14} + 14\nu^{12} - 14\nu^{10} + 28\nu^{8} - 49\nu^{6} + 28\nu^{4} - 112\nu^{2} - 528 ) / 80 $ (-v^14 + 14v^12 - 14v^10 + 28v^8 - 49v^6 + 28v^4 - 112v^2 - 528) / 80
$\beta_{5}$	$=$	$ ( -\nu^{14} + 2\nu^{12} + 10\nu^{10} - 12\nu^{8} + 63\nu^{6} - 112\nu^{4} + 176\nu^{2} - 512 ) / 64 $ (-v^14 + 2v^12 + 10v^10 - 12v^8 + 63v^6 - 112v^4 + 176v^2 - 512) / 64
$\beta_{6}$	$=$	$ ( -7\nu^{15} + 38\nu^{13} - 58\nu^{11} + 156\nu^{9} - 263\nu^{7} + 456\nu^{5} - 384\nu^{3} - 576\nu ) / 640 $ (-7v^15 + 38v^13 - 58v^11 + 156v^9 - 263v^7 + 456v^5 - 384v^3 - 576v) / 640
$\beta_{7}$	$=$	$ ( 3\nu^{15} - 42\nu^{13} + 82\nu^{11} - 164\nu^{9} + 387\nu^{7} - 564\nu^{5} + 1176\nu^{3} + 224\nu ) / 320 $ (3v^15 - 42v^13 + 82v^11 - 164v^9 + 387v^7 - 564v^5 + 1176v^3 + 224v) / 320
$\beta_{8}$	$=$	$ ( 7\nu^{14} - 18\nu^{12} + 98\nu^{10} - 196\nu^{8} + 343\nu^{6} - 756\nu^{4} + 784\nu^{2} - 1344 ) / 320 $ (7v^14 - 18v^12 + 98v^10 - 196v^8 + 343v^6 - 756v^4 + 784*v^2 - 1344) / 320
$\beta_{9}$	$=$	$ ( \nu^{15} - 2\nu^{13} + 6\nu^{11} - 20\nu^{9} + 33\nu^{7} - 80\nu^{5} + 96\nu^{3} - 128\nu ) / 64 $ (v^15 - 2v^13 + 6v^11 - 20v^9 + 33v^7 - 80v^5 + 96v^3 - 128*v) / 64
$\beta_{10}$	$=$	$ ( 3\nu^{14} + 8\nu^{12} - 18\nu^{10} + 56\nu^{8} - 133\nu^{6} + 206\nu^{4} - 584\nu^{2} + 304 ) / 80 $ (3v^14 + 8v^12 - 18v^10 + 56v^8 - 133v^6 + 206v^4 - 584*v^2 + 304) / 80
$\beta_{11}$	$=$	$ ( -3\nu^{14} + 12\nu^{12} - 22\nu^{10} + 64\nu^{8} - 107\nu^{6} + 134\nu^{4} - 216\nu^{2} - 64 ) / 80 $ (-3v^14 + 12v^12 - 22v^10 + 64v^8 - 107v^6 + 134v^4 - 216*v^2 - 64) / 80
$\beta_{12}$	$=$	$ ( -7\nu^{15} + 18\nu^{13} - 98\nu^{11} + 196\nu^{9} - 343\nu^{7} + 756\nu^{5} - 784\nu^{3} + 1344\nu ) / 320 $ (-7v^15 + 18v^13 - 98v^11 + 196v^9 - 343v^7 + 756v^5 - 784v^3 + 1344v) / 320
$\beta_{13}$	$=$	$ ( -21\nu^{15} + 14\nu^{13} - 54\nu^{11} + 28\nu^{9} + 91\nu^{7} + 308\nu^{5} + 1008\nu^{3} - 448\nu ) / 640 $ (-21v^15 + 14v^13 - 54v^11 + 28v^9 + 91v^7 + 308v^5 + 1008v^3 - 448v) / 640
$\beta_{14}$	$=$	$ ( 3\nu^{15} - 2\nu^{13} + 2\nu^{11} - 4\nu^{9} - 13\nu^{7} + 36\nu^{5} - 184\nu^{3} + 224\nu ) / 80 $ (3v^15 - 2v^13 + 2v^11 - 4v^9 - 13v^7 + 36v^5 - 184v^3 + 224v) / 80
$\beta_{15}$	$=$	$ ( 7\nu^{14} - 13\nu^{12} + 48\nu^{10} - 86\nu^{8} + 163\nu^{6} - 351\nu^{4} + 184\nu^{2} - 384 ) / 80 $ (7v^14 - 13v^12 + 48v^10 - 86v^8 + 163v^6 - 351v^4 + 184*v^2 - 384) / 80

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

$\nu$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{2} ) / 2 $ (b2) / 2
$\nu^{3}$	$=$	$ ( \beta_{9} + \beta_{7} + 3\beta_{6} + \beta_{3} ) / 2 $ (b9 + b7 + 3*b6 + b3) / 2
$\nu^{4}$	$=$	$ ( -2\beta_{15} - 2\beta_{11} + 2\beta_{10} + 2\beta_{8} + 2\beta_{5} - \beta_{4} + \beta_{2} - 1 ) / 2 $ (-2b15 - 2b11 + 2b10 + 2b8 + 2*b5 - b4 + b2 - 1) / 2
$\nu^{5}$	$=$	$ ( 2\beta_{14} + 2\beta_{13} - \beta_{12} - \beta_{9} + \beta_{7} + 3\beta_{6} + \beta_{3} - 2\beta_1 ) / 2 $ (2b14 + 2b13 - b12 - b9 + b7 + 3b6 + b3 - 2b1) / 2
$\nu^{6}$	$=$	$ ( -3\beta_{11} + \beta_{10} - 4\beta_{8} + 4\beta_{5} - \beta_{2} + 9 ) / 2 $ (-3b11 + b10 - 4b8 + 4*b5 - b2 + 9) / 2
$\nu^{7}$	$=$	$ ( 4\beta_{14} + 2\beta_{13} + 3\beta_{12} - 4\beta_{6} - 8\beta_{3} + 3\beta_1 ) / 2 $ (4b14 + 2b13 + 3b12 - 4b6 - 8b3 + 3b1) / 2
$\nu^{8}$	$=$	$ ( 4\beta_{15} + 4\beta_{11} - 8\beta_{8} + 4\beta_{5} - 3\beta_{4} + 3\beta_{2} + 1 ) / 2 $ (4b15 + 4b11 - 8b8 + 4b5 - 3b4 + 3b2 + 1) / 2
$\nu^{9}$	$=$	$ ( -8\beta_{13} + 8\beta_{12} + 6\beta_{7} + 6\beta_{6} - 10\beta_{3} + \beta_1 ) / 2 $ (-8b13 + 8b12 + 6b7 + 6b6 - 10*b3 + b1) / 2
$\nu^{10}$	$=$	$ ( -12\beta_{15} + 2\beta_{11} + 14\beta_{10} + 28\beta_{8} + 12\beta_{5} - 14\beta_{4} + 7\beta_{2} + 12 ) / 2 $ (-12b15 + 2b11 + 14b10 + 28b8 + 12b5 - 14b4 + 7*b2 + 12) / 2
$\nu^{11}$	$=$	$ ( -4\beta_{13} - 14\beta_{12} - 21\beta_{9} + 7\beta_{7} + 21\beta_{6} + 7\beta_{3} ) / 2 $ (-4b13 - 14b12 - 21b9 + 7b7 + 21b6 + 7b3) / 2
$\nu^{12}$	$=$	$ ( -14\beta_{15} - 14\beta_{11} + 14\beta_{10} + 22\beta_{8} + 14\beta_{5} + 7\beta_{4} + 7\beta_{2} + 119 ) / 2 $ (-14b15 - 14b11 + 14b10 + 22b8 + 14b5 + 7b4 + 7*b2 + 119) / 2
$\nu^{13}$	$=$	$ ( 14\beta_{14} + 14\beta_{13} - 15\beta_{12} + 7\beta_{9} - 7\beta_{7} + 35\beta_{6} - 7\beta_{3} + 56\beta_1 ) / 2 $ (14b14 + 14b13 - 15b12 + 7b9 - 7b7 + 35b6 - 7b3 + 56b1) / 2
$\nu^{14}$	$=$	$ ( 28\beta_{15} - 21\beta_{11} + 7\beta_{10} - 56\beta_{8} + 22\beta_{4} + 49\beta_{2} + 1 ) / 2 $ (28b15 - 21b11 + 7b10 - 56b8 + 22b4 + 49b2 + 1) / 2
$\nu^{15}$	$=$	$ ( 56\beta_{14} - 14\beta_{13} + 35\beta_{12} + 92\beta_{9} + 48\beta_{7} + 148\beta_{6} - 8\beta_{3} + \beta_1 ) / 2 $ (56b14 - 14b13 + 35b12 + 92b9 + 48b7 + 148b6 - 8*b3 + b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

251.1

−0.725815	−	1.21375i
−0.725815	+	1.21375i
1.41315	+	0.0547972i
1.41315	−	0.0547972i
−1.16768	+	0.797831i
−1.16768	−	0.797831i
0.782771	−	1.17782i
0.782771	+	1.17782i
−0.782771	−	1.17782i
−0.782771	+	1.17782i
1.16768	+	0.797831i
1.16768	−	0.797831i
−1.41315	+	0.0547972i
−1.41315	−	0.0547972i
0.725815	−	1.21375i
0.725815	+	1.21375i

−1.36318

−

0.376495i

1.71650

1.02646i

1.04148i

−1.95344

−

2.04550i

251.2

−1.36318

0.376495i

1.71650

−

1.02646i

−

1.04148i

−1.95344

2.04550i

251.3

−1.01919

−

0.980435i

0.0774950

1.99850i

−

4.53014i

1.88042

−

2.11283i

251.4

−1.01919

0.980435i

0.0774950

−

1.99850i

4.53014i

1.88042

2.11283i

251.5

−0.660188

−

1.25066i

−1.12830

1.65134i

−

1.68313i

2.81016

0.320929i

251.6

−0.660188

1.25066i

−1.12830

−

1.65134i

1.68313i

2.81016

−

0.320929i

251.7

−0.408843

−

1.35383i

−1.66570

1.10700i

3.40004i

2.17970

1.80247i

251.8

−0.408843

1.35383i

−1.66570

−

1.10700i

−

3.40004i

2.17970

−

1.80247i

251.9

0.408843

−

1.35383i

−1.66570

−

1.10700i

−

3.40004i

−2.17970

1.80247i

251.10

0.408843

1.35383i

−1.66570

1.10700i

3.40004i

−2.17970

−

1.80247i

251.11

0.660188

−

1.25066i

−1.12830

−

1.65134i

1.68313i

−2.81016

0.320929i

251.12

0.660188

1.25066i

−1.12830

1.65134i

−

1.68313i

−2.81016

−

0.320929i

251.13

1.01919

−

0.980435i

0.0774950

−

1.99850i

4.53014i

−1.88042

−

2.11283i

251.14

1.01919

0.980435i

0.0774950

1.99850i

−

4.53014i

−1.88042

2.11283i

251.15

1.36318

−

0.376495i

1.71650

−

1.02646i

−

1.04148i

1.95344

−

2.04550i

251.16

1.36318

0.376495i

1.71650

1.02646i

1.04148i

1.95344

2.04550i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
24.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1800.2.b.h	✓	16
3.b	odd	2	1	inner	1800.2.b.h	✓	16
4.b	odd	2	1		7200.2.b.g		16
5.b	even	2	1		1800.2.b.i	yes	16
5.c	odd	4	2		1800.2.m.f		32
8.b	even	2	1		7200.2.b.g		16
8.d	odd	2	1	inner	1800.2.b.h	✓	16
12.b	even	2	1		7200.2.b.g		16
15.d	odd	2	1		1800.2.b.i	yes	16
15.e	even	4	2		1800.2.m.f		32
20.d	odd	2	1		7200.2.b.h		16
20.e	even	4	2		7200.2.m.f		32
24.f	even	2	1	inner	1800.2.b.h	✓	16
24.h	odd	2	1		7200.2.b.g		16
40.e	odd	2	1		1800.2.b.i	yes	16
40.f	even	2	1		7200.2.b.h		16
40.i	odd	4	2		7200.2.m.f		32
40.k	even	4	2		1800.2.m.f		32
60.h	even	2	1		7200.2.b.h		16
60.l	odd	4	2		7200.2.m.f		32
120.i	odd	2	1		7200.2.b.h		16
120.m	even	2	1		1800.2.b.i	yes	16
120.q	odd	4	2		1800.2.m.f		32
120.w	even	4	2		7200.2.m.f		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1800.2.b.h	✓	16	1.a	even	1	1	trivial
1800.2.b.h	✓	16	3.b	odd	2	1	inner
1800.2.b.h	✓	16	8.d	odd	2	1	inner
1800.2.b.h	✓	16	24.f	even	2	1	inner
1800.2.b.i	yes	16	5.b	even	2	1
1800.2.b.i	yes	16	15.d	odd	2	1
1800.2.b.i	yes	16	40.e	odd	2	1
1800.2.b.i	yes	16	120.m	even	2	1
1800.2.m.f		32	5.c	odd	4	2
1800.2.m.f		32	15.e	even	4	2
1800.2.m.f		32	40.k	even	4	2
1800.2.m.f		32	120.q	odd	4	2
7200.2.b.g		16	4.b	odd	2	1
7200.2.b.g		16	8.b	even	2	1
7200.2.b.g		16	12.b	even	2	1
7200.2.b.g		16	24.h	odd	2	1
7200.2.b.h		16	20.d	odd	2	1
7200.2.b.h		16	40.f	even	2	1
7200.2.b.h		16	60.h	even	2	1
7200.2.b.h		16	120.i	odd	2	1
7200.2.m.f		32	20.e	even	4	2
7200.2.m.f		32	40.i	odd	4	2
7200.2.m.f		32	60.l	odd	4	2
7200.2.m.f		32	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1800, [\chi])$:

$ T_{7}^{8} + 36T_{7}^{6} + 366T_{7}^{4} + 1028T_{7}^{2} + 729 $

$ T_{23}^{8} - 80T_{23}^{6} + 2136T_{23}^{4} - 19168T_{23}^{2} + 3600 $

$ T_{43}^{4} - 66T_{43}^{2} - 284T_{43} - 335 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 2 T^{14} + \cdots + 256 $ T^16 + 2T^14 + 4T^12 + 4T^8 + 64T^4 + 128*T^2 + 256
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + 36 T^{6} + \cdots + 729)^{2} $ (T^8 + 36T^6 + 366T^4 + 1028*T^2 + 729)^2
$11$	$ (T^{8} + 48 T^{6} + \cdots + 4624)^{2} $ (T^8 + 48T^6 + 600T^4 + 2848*T^2 + 4624)^2
$13$	$ (T^{8} + 52 T^{6} + 286 T^{4} + \cdots + 9)^{2} $ (T^8 + 52T^6 + 286T^4 + 164*T^2 + 9)^2
$17$	$ (T^{8} + 88 T^{6} + \cdots + 10000)^{2} $ (T^8 + 88T^6 + 2136T^4 + 10368*T^2 + 10000)^2
$19$	$ (T^{4} - 4 T^{3} - 18 T^{2} + \cdots - 95)^{4} $ (T^4 - 4T^3 - 18T^2 + 96*T - 95)^4
$23$	$ (T^{8} - 80 T^{6} + \cdots + 3600)^{2} $ (T^8 - 80T^6 + 2136T^4 - 19168*T^2 + 3600)^2
$29$	$ (T^{8} - 104 T^{6} + \cdots + 3600)^{2} $ (T^8 - 104T^6 + 2776T^4 - 11008*T^2 + 3600)^2
$31$	$ (T^{8} + 164 T^{6} + \cdots + 585225)^{2} $ (T^8 + 164T^6 + 7742T^4 + 126596*T^2 + 585225)^2
$37$	$ (T^{8} + 192 T^{6} + \cdots + 57600)^{2} $ (T^8 + 192T^6 + 9440T^4 + 134912*T^2 + 57600)^2
$41$	$ (T^{8} + 112 T^{6} + \cdots + 23104)^{2} $ (T^8 + 112T^6 + 2832T^4 + 22176*T^2 + 23104)^2
$43$	$ (T^{4} - 66 T^{2} + \cdots - 335)^{4} $ (T^4 - 66T^2 - 284T - 335)^4
$47$	$ (T^{8} - 248 T^{6} + \cdots + 7551504)^{2} $ (T^8 - 248T^6 + 20376T^4 - 674656*T^2 + 7551504)^2
$53$	$ (T^{8} - 112 T^{6} + \cdots + 576)^{2} $ (T^8 - 112T^6 + 656T^4 - 1120*T^2 + 576)^2
$59$	$ (T^{8} + 184 T^{6} + \cdots + 13456)^{2} $ (T^8 + 184T^6 + 7416T^4 + 86496*T^2 + 13456)^2
$61$	$ (T^{8} + 292 T^{6} + \cdots + 1265625)^{2} $ (T^8 + 292T^6 + 23950T^4 + 432500*T^2 + 1265625)^2
$67$	$ (T^{4} - 4 T^{3} - 154 T^{2} + \cdots + 25)^{4} $ (T^4 - 4T^3 - 154T^2 + 72*T + 25)^4
$71$	$ (T^{8} - 200 T^{6} + \cdots + 129600)^{2} $ (T^8 - 200T^6 + 5776T^4 - 50272*T^2 + 129600)^2
$73$	$ (T^{4} + 8 T^{3} + \cdots + 160)^{4} $ (T^4 + 8T^3 - 88T^2 - 336*T + 160)^4
$79$	$ (T^{8} + 432 T^{6} + \cdots + 16646400)^{2} $ (T^8 + 432T^6 + 62048T^4 + 3090176*T^2 + 16646400)^2
$83$	$ (T^{8} + 152 T^{6} + \cdots + 250000)^{2} $ (T^8 + 152T^6 + 6776T^4 + 79200*T^2 + 250000)^2
$89$	$ (T^{8} + 384 T^{6} + \cdots + 1638400)^{2} $ (T^8 + 384T^6 + 34304T^4 + 892928*T^2 + 1638400)^2
$97$	$ (T^{4} + 4 T^{3} + \cdots + 16265)^{4} $ (T^4 + 4T^3 - 290T^2 - 428*T + 16265)^4
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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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} + \beta_{5} q^{4} - \beta_{15} q^{7} + \beta_{13} q^{8}+O(q^{10}) \) q - b6 * q^2 + b5 * q^4 - b15 * q^7 + b13 * q^8	\( q - \beta_{6} q^{2} + \beta_{5} q^{4} - \beta_{15} q^{7} + \beta_{13} q^{8} + ( - \beta_{14} + \beta_{13} + \beta_{3}) q^{11} + (\beta_{15} - \beta_{10} - \beta_{8} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{14} - \beta_{12} + \cdots + \beta_1) q^{98}+O(q^{100}) \) q - b6 * q^2 + b5 * q^4 - b15 * q^7 + b13 * q^8 + (-b14 + b13 + b3) * q^11 + (b15 - b10 - b8 - b5 - 1) * q^13 + (-b13 - b7 - b3 + b1) * q^14 + (b15 - b10 - b8 - b5 + b4 - 1) * q^16 + (-b14 + b13 - b9 + b3 - b1) * q^17 + (-b11 - b8 + b5 + 1) * q^19 + (b15 + b11 - b5 + b4) * q^22 + (b14 + b13 - b12 + b9 + b7 - b6 - b1) * q^23 + (-b12 + b9 + b7 - b3 - b1) * q^26 + (b5 - b4 + b2 + 1) * q^28 + (-b14 - b13 + b12 - b7 + b6) * q^29 + (b10 + b8 + b5 + b4 + b2 + 1) * q^31 + (-b12 + b9 + b7 + b6 - b3 + b1) * q^32 + (2b15 - b8 - 2b5 + b4 - b2 + 1) * q^34 + (b15 - b11 + b8 - b5 + b4 + b2) * q^37 + (b13 - b12 + b9 - b6 + 2b3) q^38 + (-b14 + b13 + b9 + b1) * q^41 + (-b15 - b11 + b10 + 2b5 - b4 + b2) q^43 + (b12 + b7 + b6 - b3 + b1) * q^44 + (-b15 - b11 + b5 + b4 + 2) * q^46 + (2b9 + b7 + b6 - 2b1) * q^47 + (b15 + b11 - 2b10 + b8 - b5 + b4 - b2 - 2) q^49 + (-2b15 + 2b10 + b5 + 4) * q^52 + (b14 + b13 + b9 - b1) * q^53 + (-2b14 + b13 - 2b6 - 2b1) q^56 + (2b10 + b8 - b4 + b2 - 1) q^58 + (-b12 - b9 + 2b6 - b3 - b1) q^59 + (-2b15 + b11 + b10 + 2b5 + b4 + b2 + 1) * q^61 + (-2b14 + b13 + b12 - b9 + b6 + 2b3 + 2b1) q^62 + (-2b15 + 2b11 + 2) * q^64 + (-b11 - 2b10 + b8 + 3b5 + 1) * q^67 + (2b14 + b9 + 2b7 + 2b3) q^68 + (-b14 - b13 - b12 - b9 - 2b6 + b1) q^71 + (-b15 + b11 + 3b8 + b5 - b4 + b2 - 2) q^73 + (-2b14 - b12 - b9 + b7 + b6 + 3b3 + b1) * q^74 + (2b8 + b5 + b4 + b2 + 3) q^76 + (-b14 - b13 - b12 + 2b9 - b7 - 3b6 - 2b1) q^77 + (-2b15 - 2b11 + 2b10 + 4b8 + 2) * q^79 + (2b11 + b4 + b2 - 1) q^82 + (-b12 - b9 + 2b6 + b3 - b1) q^83 + (-2b14 + b13 - b7 + 3b3 - b1) * q^86 + (-b15 + 2b11 - b10 - b8 - b5 - b2 - 6) q^88 + (2b12 - 2b9 - 4b6 - 2b1) * q^89 + (2b15 + b11 - 3b8 - 5b5 + 2b4 - 2b2 + 3) q^91 + (-b12 - b7 - b6 + b3 + 3b1) q^92 + (-3b15 - b11 + 4b10 + 2b8 + b5 + b2 + 1) q^94 + (2b15 - 2b11 - 2b10 - 4b8 + 2b4 - 2b2 - 1) * q^97 + (2b14 - b12 - b9 + b7 + b6 - 5b3 + b1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{4}+O(q^{10}) \) 16 * q - 4 * q^4	\( 16 q - 4 q^{4} - 8 q^{16} + 16 q^{19} - 4 q^{22} + 28 q^{28} + 12 q^{34} + 20 q^{46} - 32 q^{49} + 44 q^{52} - 20 q^{58} + 32 q^{64} + 16 q^{67} - 32 q^{73} + 36 q^{76} - 16 q^{82} - 88 q^{88} + 48 q^{91} - 20 q^{94} - 16 q^{97}+O(q^{100}) \) 16 * q - 4 * q^4 - 8 * q^16 + 16 * q^19 - 4 * q^22 + 28 * q^28 + 12 * q^34 + 20 * q^46 - 32 * q^49 + 44 * q^52 - 20 * q^58 + 32 * q^64 + 16 * q^67 - 32 * q^73 + 36 * q^76 - 16 * q^82 - 88 * q^88 + 48 * q^91 - 20 * q^94 - 16 * 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	1800.2.b.h

Level	$1800$
Weight	$2$
Character orbit	1800.b
Analytic conductor	$14.373$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(14.3730723638\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 2x^{14} + 6x^{12} - 20x^{10} + 33x^{8} - 80x^{6} + 96x^{4} - 128x^{2} + 256 \) x^16 - 2x^14 + 6x^12 - 20x^10 + 33x^8 - 80x^6 + 96x^4 - 128*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index:	\( 2^{8}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( 2\nu \) 2*v
\(\beta_{2}\)	\(=\)	\( 2\nu^{2} \) 2*v^2
\(\beta_{3}\)	\(=\)	\( ( \nu^{15} - 2\nu^{13} - 10\nu^{11} + 12\nu^{9} - 63\nu^{7} + 112\nu^{5} - 176\nu^{3} + 512\nu ) / 128 \) (v^15 - 2v^13 - 10v^11 + 12v^9 - 63v^7 + 112v^5 - 176v^3 + 512*v) / 128
\(\beta_{4}\)	\(=\)	\( ( -\nu^{14} + 14\nu^{12} - 14\nu^{10} + 28\nu^{8} - 49\nu^{6} + 28\nu^{4} - 112\nu^{2} - 528 ) / 80 \) (-v^14 + 14v^12 - 14v^10 + 28v^8 - 49v^6 + 28v^4 - 112v^2 - 528) / 80
\(\beta_{5}\)	\(=\)	\( ( -\nu^{14} + 2\nu^{12} + 10\nu^{10} - 12\nu^{8} + 63\nu^{6} - 112\nu^{4} + 176\nu^{2} - 512 ) / 64 \) (-v^14 + 2v^12 + 10v^10 - 12v^8 + 63v^6 - 112v^4 + 176v^2 - 512) / 64
\(\beta_{6}\)	\(=\)	\( ( -7\nu^{15} + 38\nu^{13} - 58\nu^{11} + 156\nu^{9} - 263\nu^{7} + 456\nu^{5} - 384\nu^{3} - 576\nu ) / 640 \) (-7v^15 + 38v^13 - 58v^11 + 156v^9 - 263v^7 + 456v^5 - 384v^3 - 576v) / 640
\(\beta_{7}\)	\(=\)	\( ( 3\nu^{15} - 42\nu^{13} + 82\nu^{11} - 164\nu^{9} + 387\nu^{7} - 564\nu^{5} + 1176\nu^{3} + 224\nu ) / 320 \) (3v^15 - 42v^13 + 82v^11 - 164v^9 + 387v^7 - 564v^5 + 1176v^3 + 224v) / 320
\(\beta_{8}\)	\(=\)	\( ( 7\nu^{14} - 18\nu^{12} + 98\nu^{10} - 196\nu^{8} + 343\nu^{6} - 756\nu^{4} + 784\nu^{2} - 1344 ) / 320 \) (7v^14 - 18v^12 + 98v^10 - 196v^8 + 343v^6 - 756v^4 + 784*v^2 - 1344) / 320
\(\beta_{9}\)	\(=\)	\( ( \nu^{15} - 2\nu^{13} + 6\nu^{11} - 20\nu^{9} + 33\nu^{7} - 80\nu^{5} + 96\nu^{3} - 128\nu ) / 64 \) (v^15 - 2v^13 + 6v^11 - 20v^9 + 33v^7 - 80v^5 + 96v^3 - 128*v) / 64
\(\beta_{10}\)	\(=\)	\( ( 3\nu^{14} + 8\nu^{12} - 18\nu^{10} + 56\nu^{8} - 133\nu^{6} + 206\nu^{4} - 584\nu^{2} + 304 ) / 80 \) (3v^14 + 8v^12 - 18v^10 + 56v^8 - 133v^6 + 206v^4 - 584*v^2 + 304) / 80
\(\beta_{11}\)	\(=\)	\( ( -3\nu^{14} + 12\nu^{12} - 22\nu^{10} + 64\nu^{8} - 107\nu^{6} + 134\nu^{4} - 216\nu^{2} - 64 ) / 80 \) (-3v^14 + 12v^12 - 22v^10 + 64v^8 - 107v^6 + 134v^4 - 216*v^2 - 64) / 80
\(\beta_{12}\)	\(=\)	\( ( -7\nu^{15} + 18\nu^{13} - 98\nu^{11} + 196\nu^{9} - 343\nu^{7} + 756\nu^{5} - 784\nu^{3} + 1344\nu ) / 320 \) (-7v^15 + 18v^13 - 98v^11 + 196v^9 - 343v^7 + 756v^5 - 784v^3 + 1344v) / 320
\(\beta_{13}\)	\(=\)	\( ( -21\nu^{15} + 14\nu^{13} - 54\nu^{11} + 28\nu^{9} + 91\nu^{7} + 308\nu^{5} + 1008\nu^{3} - 448\nu ) / 640 \) (-21v^15 + 14v^13 - 54v^11 + 28v^9 + 91v^7 + 308v^5 + 1008v^3 - 448v) / 640
\(\beta_{14}\)	\(=\)	\( ( 3\nu^{15} - 2\nu^{13} + 2\nu^{11} - 4\nu^{9} - 13\nu^{7} + 36\nu^{5} - 184\nu^{3} + 224\nu ) / 80 \) (3v^15 - 2v^13 + 2v^11 - 4v^9 - 13v^7 + 36v^5 - 184v^3 + 224v) / 80
\(\beta_{15}\)	\(=\)	\( ( 7\nu^{14} - 13\nu^{12} + 48\nu^{10} - 86\nu^{8} + 163\nu^{6} - 351\nu^{4} + 184\nu^{2} - 384 ) / 80 \) (7v^14 - 13v^12 + 48v^10 - 86v^8 + 163v^6 - 351v^4 + 184*v^2 - 384) / 80

\(\nu\)	\(=\)	\( ( \beta_1 ) / 2 \) (b1) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{2} ) / 2 \) (b2) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{9} + \beta_{7} + 3\beta_{6} + \beta_{3} ) / 2 \) (b9 + b7 + 3*b6 + b3) / 2
\(\nu^{4}\)	\(=\)	\( ( -2\beta_{15} - 2\beta_{11} + 2\beta_{10} + 2\beta_{8} + 2\beta_{5} - \beta_{4} + \beta_{2} - 1 ) / 2 \) (-2b15 - 2b11 + 2b10 + 2b8 + 2*b5 - b4 + b2 - 1) / 2
\(\nu^{5}\)	\(=\)	\( ( 2\beta_{14} + 2\beta_{13} - \beta_{12} - \beta_{9} + \beta_{7} + 3\beta_{6} + \beta_{3} - 2\beta_1 ) / 2 \) (2b14 + 2b13 - b12 - b9 + b7 + 3b6 + b3 - 2b1) / 2
\(\nu^{6}\)	\(=\)	\( ( -3\beta_{11} + \beta_{10} - 4\beta_{8} + 4\beta_{5} - \beta_{2} + 9 ) / 2 \) (-3b11 + b10 - 4b8 + 4*b5 - b2 + 9) / 2
\(\nu^{7}\)	\(=\)	\( ( 4\beta_{14} + 2\beta_{13} + 3\beta_{12} - 4\beta_{6} - 8\beta_{3} + 3\beta_1 ) / 2 \) (4b14 + 2b13 + 3b12 - 4b6 - 8b3 + 3b1) / 2
\(\nu^{8}\)	\(=\)	\( ( 4\beta_{15} + 4\beta_{11} - 8\beta_{8} + 4\beta_{5} - 3\beta_{4} + 3\beta_{2} + 1 ) / 2 \) (4b15 + 4b11 - 8b8 + 4b5 - 3b4 + 3b2 + 1) / 2
\(\nu^{9}\)	\(=\)	\( ( -8\beta_{13} + 8\beta_{12} + 6\beta_{7} + 6\beta_{6} - 10\beta_{3} + \beta_1 ) / 2 \) (-8b13 + 8b12 + 6b7 + 6b6 - 10*b3 + b1) / 2
\(\nu^{10}\)	\(=\)	\( ( -12\beta_{15} + 2\beta_{11} + 14\beta_{10} + 28\beta_{8} + 12\beta_{5} - 14\beta_{4} + 7\beta_{2} + 12 ) / 2 \) (-12b15 + 2b11 + 14b10 + 28b8 + 12b5 - 14b4 + 7*b2 + 12) / 2
\(\nu^{11}\)	\(=\)	\( ( -4\beta_{13} - 14\beta_{12} - 21\beta_{9} + 7\beta_{7} + 21\beta_{6} + 7\beta_{3} ) / 2 \) (-4b13 - 14b12 - 21b9 + 7b7 + 21b6 + 7b3) / 2
\(\nu^{12}\)	\(=\)	\( ( -14\beta_{15} - 14\beta_{11} + 14\beta_{10} + 22\beta_{8} + 14\beta_{5} + 7\beta_{4} + 7\beta_{2} + 119 ) / 2 \) (-14b15 - 14b11 + 14b10 + 22b8 + 14b5 + 7b4 + 7*b2 + 119) / 2
\(\nu^{13}\)	\(=\)	\( ( 14\beta_{14} + 14\beta_{13} - 15\beta_{12} + 7\beta_{9} - 7\beta_{7} + 35\beta_{6} - 7\beta_{3} + 56\beta_1 ) / 2 \) (14b14 + 14b13 - 15b12 + 7b9 - 7b7 + 35b6 - 7b3 + 56b1) / 2
\(\nu^{14}\)	\(=\)	\( ( 28\beta_{15} - 21\beta_{11} + 7\beta_{10} - 56\beta_{8} + 22\beta_{4} + 49\beta_{2} + 1 ) / 2 \) (28b15 - 21b11 + 7b10 - 56b8 + 22b4 + 49b2 + 1) / 2
\(\nu^{15}\)	\(=\)	\( ( 56\beta_{14} - 14\beta_{13} + 35\beta_{12} + 92\beta_{9} + 48\beta_{7} + 148\beta_{6} - 8\beta_{3} + \beta_1 ) / 2 \) (56b14 - 14b13 + 35b12 + 92b9 + 48b7 + 148b6 - 8*b3 + b1) / 2

\(n\)	\(577\)	\(901\)	\(1001\)	\(1351\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 251.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 2 T^{14} + \cdots + 256 \) T^16 + 2T^14 + 4T^12 + 4T^8 + 64T^4 + 128*T^2 + 256
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + 36 T^{6} + \cdots + 729)^{2} \) (T^8 + 36T^6 + 366T^4 + 1028*T^2 + 729)^2
$11$	\( (T^{8} + 48 T^{6} + \cdots + 4624)^{2} \) (T^8 + 48T^6 + 600T^4 + 2848*T^2 + 4624)^2
$13$	\( (T^{8} + 52 T^{6} + 286 T^{4} + \cdots + 9)^{2} \) (T^8 + 52T^6 + 286T^4 + 164*T^2 + 9)^2
$17$	\( (T^{8} + 88 T^{6} + \cdots + 10000)^{2} \) (T^8 + 88T^6 + 2136T^4 + 10368*T^2 + 10000)^2
$19$	\( (T^{4} - 4 T^{3} - 18 T^{2} + \cdots - 95)^{4} \) (T^4 - 4T^3 - 18T^2 + 96*T - 95)^4
$23$	\( (T^{8} - 80 T^{6} + \cdots + 3600)^{2} \) (T^8 - 80T^6 + 2136T^4 - 19168*T^2 + 3600)^2
$29$	\( (T^{8} - 104 T^{6} + \cdots + 3600)^{2} \) (T^8 - 104T^6 + 2776T^4 - 11008*T^2 + 3600)^2
$31$	\( (T^{8} + 164 T^{6} + \cdots + 585225)^{2} \) (T^8 + 164T^6 + 7742T^4 + 126596*T^2 + 585225)^2
$37$	\( (T^{8} + 192 T^{6} + \cdots + 57600)^{2} \) (T^8 + 192T^6 + 9440T^4 + 134912*T^2 + 57600)^2
$41$	\( (T^{8} + 112 T^{6} + \cdots + 23104)^{2} \) (T^8 + 112T^6 + 2832T^4 + 22176*T^2 + 23104)^2
$43$	\( (T^{4} - 66 T^{2} + \cdots - 335)^{4} \) (T^4 - 66T^2 - 284T - 335)^4
$47$	\( (T^{8} - 248 T^{6} + \cdots + 7551504)^{2} \) (T^8 - 248T^6 + 20376T^4 - 674656*T^2 + 7551504)^2
$53$	\( (T^{8} - 112 T^{6} + \cdots + 576)^{2} \) (T^8 - 112T^6 + 656T^4 - 1120*T^2 + 576)^2
$59$	\( (T^{8} + 184 T^{6} + \cdots + 13456)^{2} \) (T^8 + 184T^6 + 7416T^4 + 86496*T^2 + 13456)^2
$61$	\( (T^{8} + 292 T^{6} + \cdots + 1265625)^{2} \) (T^8 + 292T^6 + 23950T^4 + 432500*T^2 + 1265625)^2
$67$	\( (T^{4} - 4 T^{3} - 154 T^{2} + \cdots + 25)^{4} \) (T^4 - 4T^3 - 154T^2 + 72*T + 25)^4
$71$	\( (T^{8} - 200 T^{6} + \cdots + 129600)^{2} \) (T^8 - 200T^6 + 5776T^4 - 50272*T^2 + 129600)^2
$73$	\( (T^{4} + 8 T^{3} + \cdots + 160)^{4} \) (T^4 + 8T^3 - 88T^2 - 336*T + 160)^4
$79$	\( (T^{8} + 432 T^{6} + \cdots + 16646400)^{2} \) (T^8 + 432T^6 + 62048T^4 + 3090176*T^2 + 16646400)^2
$83$	\( (T^{8} + 152 T^{6} + \cdots + 250000)^{2} \) (T^8 + 152T^6 + 6776T^4 + 79200*T^2 + 250000)^2
$89$	\( (T^{8} + 384 T^{6} + \cdots + 1638400)^{2} \) (T^8 + 384T^6 + 34304T^4 + 892928*T^2 + 1638400)^2
$97$	\( (T^{4} + 4 T^{3} + \cdots + 16265)^{4} \) (T^4 + 4T^3 - 290T^2 - 428*T + 16265)^4
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