LMFDB - Newform orbit 900.2.s.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,2,Mod(49,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.49"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	900.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.18653618192$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.1333317747165888577536.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 $ x^16 + 3x^14 + 5x^12 + 15x^10 + 45x^8 + 60x^6 + 80x^4 + 192*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{15} + \beta_{3}) q^{3} + ( - \beta_{15} - \beta_{9} + \cdots + \beta_{3}) q^{7} + ( - \beta_{11} + \beta_{4} + \beta_{2} - 1) q^{9} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{11} + ( - \beta_{15} - \beta_{13} - \beta_{5}) q^{13}+ \cdots + ( - \beta_{14} - \beta_{11} + \beta_{6} + \cdots - 3) q^{99}+O(q^{100}) $ q + (-b15 + b3) * q^3 + (-b15 - b9 + b8 + b3) * q^7 + (-b11 + b4 + b2 - 1) * q^9 + (-b6 + b4 + b1) * q^11 + (-b15 - b13 - b5) * q^13 + (2b10 + b9 + 2b5 - b3) * q^17 + (-b14 + b11 + b6 - 2b4 - b2 + b1 + 2) q^19 + (b6 + b4 + b2 + b1 + 2) * q^21 + (-b13 + b10 - b8 + b7 + 2b3) q^23 + (b15 + b13 + 3b10 - b9 - b8 + 2b5 + b3) * q^27 + (2b14 - 2b12 - b11 - 2b6 - 3b4 + 2b2 - b1 - 1) q^29 + (-b12 - b11 - b6 + b2 - 1) * q^31 + (-b10 + b9 - 2b8 - 2b5) * q^33 + (b15 - b13 - b10 + b9 + b8 - b7 - b5 - 3b3) q^37 + (-b14 - b11 - b6 + b4 - b2 + b1 + 2) * q^39 + (-b14 + b12 - b11 + 2b6 - b4 - b2 + b1 + 2) q^41 + (-2b15 - 2b13 - 3b10 + 2b9 - 2b7 - 2b5) * q^43 + (3b15 + b13 - 2b10 - 2b9 + b8 + 2b7 + 2b5) q^47 + (-2b14 + 5b12 + b11 + 2b6 + 3b4 + 2b1 + 2) q^49 + (-b12 - 2b6 - 3b4 + b2 - 2b1 - 1) q^51 + (-b15 + b13 - 2b10 + b9 - b8 + 2b7 - 2b5) q^53 + (-b15 + b13 - 5b10 + b9 - b7 - 6b5) * q^57 + (2b14 - 5b12 - b11 + 3b6 + b4 + 4b2 - 2b1 + 3) q^59 + (2b14 - 2b11 - b6 - b4 + 2b2 - b1 - 2) q^61 + (b13 - 4b10 - 3b8 - 3b5 + 3b3) * q^63 + (-2b15 - b13 - 2b10 + b8 - 2b7 - 4b3) * q^67 + (b12 - 3b11 - 7b6 + 3b4 - b2 + 2b1 - 5) * q^69 + (-2b14 - b12 + b11 + b6 - 2b4 + b2 - b1 - 2) * q^71 + (b15 - b13 - 3b10 - b9 + b8 - 3b5 - 2b3) q^73 + (3b15 + b13 + 2b10 - 2b8 + 2b7 - 6b5 + 4b3) * q^77 + (-2b14 - 2b12 + 3b11 + 3b6 + b4 - 2b2 + 3b1 + 3) * q^79 + (b11 + 5b6 - 3b4 - 2b2 + 2b1 - 1) * q^81 + (5b15 - b13 + 3b10 + b9 + b7 + b5 - 2b3) q^83 + (3b13 - 4b10 + b9 - 2b8 + 7b5) * q^87 + (3b14 + 3b12 + 3b2 - 3b1 - 3) * q^89 + (-b12 - b11 - b6 + 2b4 - 3) q^91 + (b13 + 2b10 - b8 - b7 + 2b5) * q^93 + (-3b13 + 3b9 - 2b7 - 2b5 - b3) * q^97 + (-b14 - b11 + b6 + b4 - 2b2 - 3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 10 q^{9} + 6 q^{11} + 16 q^{19} + 26 q^{21} + 18 q^{29} - 4 q^{31} + 34 q^{39} + 18 q^{41} + 18 q^{49} + 6 q^{51} + 30 q^{59} + 2 q^{61} - 18 q^{69} - 48 q^{71} - 14 q^{79} - 62 q^{81} - 12 q^{89}+ \cdots - 66 q^{99}+O(q^{100}) $ 16 * q - 10 * q^9 + 6 * q^11 + 16 * q^19 + 26 * q^21 + 18 * q^29 - 4 * q^31 + 34 * q^39 + 18 * q^41 + 18 * q^49 + 6 * q^51 + 30 * q^59 + 2 * q^61 - 18 * q^69 - 48 * q^71 - 14 * q^79 - 62 * q^81 - 12 * q^89 - 44 * q^91 - 66 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 $ x^16 + 3*x^14 + 5*x^12 + 15*x^10 + 45*x^8 + 60*x^6 + 80*x^4 + 192*x^2 + 256 :

$\beta_{1}$	$=$	$ ( -\nu^{14} - 35\nu^{12} - 165\nu^{10} - 495\nu^{8} - 525\nu^{6} - 540\nu^{4} + 880\nu^{2} - 1792 ) / 2880 $ (-v^14 - 35v^12 - 165v^10 - 495v^8 - 525v^6 - 540v^4 + 880v^2 - 1792) / 2880
$\beta_{2}$	$=$	$ ( -\nu^{14} + 25\nu^{12} + 255\nu^{10} + 525\nu^{8} + 615\nu^{6} + 1920\nu^{4} + 7600\nu^{2} + 4928 ) / 2880 $ (-v^14 + 25v^12 + 255v^10 + 525v^8 + 615v^6 + 1920v^4 + 7600v^2 + 4928) / 2880
$\beta_{3}$	$=$	$ ( \nu^{15} + 5\nu^{13} + 15\nu^{11} - 75\nu^{9} + 15\nu^{7} - 30\nu^{5} + 140\nu^{3} + 352\nu ) / 1440 $ (v^15 + 5v^13 + 15v^11 - 75v^9 + 15v^7 - 30v^5 + 140v^3 + 352*v) / 1440
$\beta_{4}$	$=$	$ ( 7\nu^{14} + 65\nu^{12} + 135\nu^{10} + 165\nu^{8} + 495\nu^{6} + 1920\nu^{4} - 160\nu^{2} + 64 ) / 2880 $ (7v^14 + 65v^12 + 135v^10 + 165v^8 + 495v^6 + 1920v^4 - 160*v^2 + 64) / 2880
$\beta_{5}$	$=$	$ ( -7\nu^{15} - 5\nu^{13} + 45\nu^{11} + 135\nu^{9} + 405\nu^{7} - 180\nu^{5} + 880\nu^{3} + 896\nu ) / 5760 $ (-7v^15 - 5v^13 + 45v^11 + 135v^9 + 405v^7 - 180v^5 + 880v^3 + 896v) / 5760
$\beta_{6}$	$=$	$ ( 3\nu^{14} + 25\nu^{12} + 15\nu^{10} + 45\nu^{8} + 135\nu^{6} + 180\nu^{4} - 64 ) / 960 $ (3v^14 + 25v^12 + 15v^10 + 45v^8 + 135v^6 + 180v^4 - 64) / 960
$\beta_{7}$	$=$	$ ( -3\nu^{15} - 45\nu^{13} - 75\nu^{11} - 145\nu^{9} - 435\nu^{7} - 760\nu^{5} - 240\nu^{3} - 2176\nu ) / 960 $ (-3v^15 - 45v^13 - 75v^11 - 145v^9 - 435v^7 - 760v^5 - 240v^3 - 2176v) / 960
$\beta_{8}$	$=$	$ ( 31\nu^{15} - 115\nu^{13} - 405\nu^{11} - 255\nu^{9} - 765\nu^{7} - 4620\nu^{5} - 3280\nu^{3} + 6592\nu ) / 5760 $ (31v^15 - 115v^13 - 405v^11 - 255v^9 - 765v^7 - 4620v^5 - 3280v^3 + 6592v) / 5760
$\beta_{9}$	$=$	$ ( -37\nu^{15} - 275\nu^{13} - 645\nu^{11} - 1215\nu^{9} - 1725\nu^{7} - 4320\nu^{5} - 4640\nu^{3} - 1984\nu ) / 5760 $ (-37v^15 - 275v^13 - 645v^11 - 1215v^9 - 1725v^7 - 4320v^5 - 4640v^3 - 1984v) / 5760
$\beta_{10}$	$=$	$ ( -37\nu^{15} - 35\nu^{13} - 165\nu^{11} - 735\nu^{9} - 1245\nu^{7} - 960\nu^{5} - 4880\nu^{3} - 6784\nu ) / 5760 $ (-37v^15 - 35v^13 - 165v^11 - 735v^9 - 1245v^7 - 960v^5 - 4880v^3 - 6784v) / 5760
$\beta_{11}$	$=$	$ ( -11\nu^{14} - 25\nu^{12} - 15\nu^{10} - 45\nu^{8} - 375\nu^{6} + 180\nu^{4} + 320\nu^{2} - 992 ) / 720 $ (-11v^14 - 25v^12 - 15v^10 - 45v^8 - 375v^6 + 180v^4 + 320*v^2 - 992) / 720
$\beta_{12}$	$=$	$ ( 49\nu^{14} + 35\nu^{12} + 165\nu^{10} + 495\nu^{8} + 525\nu^{6} - 180\nu^{4} + 2960\nu^{2} + 2368 ) / 2880 $ (49v^14 + 35v^12 + 165v^10 + 495v^8 + 525v^6 - 180v^4 + 2960*v^2 + 2368) / 2880
$\beta_{13}$	$=$	$ ( -9\nu^{15} - 5\nu^{13} - 75\nu^{11} - 25\nu^{9} - 75\nu^{7} - 190\nu^{5} - 360\nu^{3} + 1792\nu ) / 960 $ (-9v^15 - 5v^13 - 75v^11 - 25v^9 - 75v^7 - 190v^5 - 360v^3 + 1792v) / 960
$\beta_{14}$	$=$	$ ( 43\nu^{14} + 35\nu^{12} - 75\nu^{10} + 375\nu^{8} + 645\nu^{6} + 150\nu^{4} + 800\nu^{2} + 5056 ) / 1440 $ (43v^14 + 35v^12 - 75v^10 + 375v^8 + 645v^6 + 150v^4 + 800*v^2 + 5056) / 1440
$\beta_{15}$	$=$	$ ( 26\nu^{15} + 55\nu^{13} + 45\nu^{11} + 195\nu^{9} + 585\nu^{7} + 285\nu^{5} + 460\nu^{3} + 2192\nu ) / 1440 $ (26v^15 + 55v^13 + 45v^11 + 195v^9 + 585v^7 + 285v^5 + 460v^3 + 2192v) / 1440

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

$\nu$	$=$	$ ( -\beta_{15} - \beta_{10} + \beta_{8} - \beta_{7} - 2\beta_{5} + \beta_{3} ) / 3 $ (-b15 - b10 + b8 - b7 - 2*b5 + b3) / 3
$\nu^{2}$	$=$	$ ( \beta_{6} - \beta_{4} + \beta_{2} + \beta _1 - 1 ) / 3 $ (b6 - b4 + b2 + b1 - 1) / 3
$\nu^{3}$	$=$	$ ( \beta_{15} + 2\beta_{13} - 2\beta_{10} - \beta_{9} - 2\beta_{8} + 3\beta_{7} + \beta_{5} + 4\beta_{3} ) / 3 $ (b15 + 2b13 - 2b10 - b9 - 2b8 + 3b7 + b5 + 4*b3) / 3
$\nu^{4}$	$=$	$ ( \beta_{14} - \beta_{12} + \beta_{11} - 3\beta_{6} + 5\beta_{4} + \beta _1 - 1 ) / 3 $ (b14 - b12 + b11 - 3b6 + 5b4 + b1 - 1) / 3
$\nu^{5}$	$=$	$ ( -\beta_{15} - \beta_{13} - 4\beta_{10} + 4\beta_{9} - 3\beta_{8} - 3\beta_{7} - 15\beta_{5} - 4\beta_{3} ) / 3 $ (-b15 - b13 - 4b10 + 4b9 - 3b8 - 3b7 - 15b5 - 4b3) / 3
$\nu^{6}$	$=$	$ ( \beta_{14} - 7\beta_{12} - 8\beta_{11} - 11\beta_{6} + \beta_{4} + 4\beta_{2} - \beta _1 - 17 ) / 3 $ (b14 - 7b12 - 8b11 - 11b6 + b4 + 4b2 - b1 - 17) / 3
$\nu^{7}$	$=$	$ ( 7\beta_{15} - 2\beta_{13} + 4\beta_{10} + 9\beta_{9} - 6\beta_{8} - 2\beta_{7} + 27\beta_{5} - 4\beta_{3} ) / 3 $ (7b15 - 2b13 + 4b10 + 9b9 - 6b8 - 2b7 + 27b5 - 4b3) / 3
$\nu^{8}$	$=$	$ ( 5\beta_{14} - 2\beta_{12} + 8\beta_{11} + 9\beta_{6} - 11\beta_{4} - 16\beta _1 - 14 ) / 3 $ (5b14 - 2b12 + 8b11 + 9b6 - 11b4 - 16b1 - 14) / 3
$\nu^{9}$	$=$	$ ( -4\beta_{15} + \beta_{13} - \beta_{10} - 10\beta_{9} + 8\beta_{8} + \beta_{7} + 5\beta_{5} - 33\beta_{3} ) / 3 $ (-4b15 + b13 - b10 - 10b9 + 8b8 + b7 + 5b5 - 33*b3) / 3
$\nu^{10}$	$=$	$ ( -19\beta_{14} + 28\beta_{12} - 7\beta_{11} - 20\beta_{6} + 18\beta_{4} - 5\beta_{2} - 3\beta _1 + 39 ) / 3 $ (-19b14 + 28b12 - 7b11 - 20b6 + 18b4 - 5b2 - 3*b1 + 39) / 3
$\nu^{11}$	$=$	$ ( -41\beta_{15} - 36\beta_{13} + \beta_{10} - 14\beta_{9} + 41\beta_{8} - 31\beta_{7} + 14\beta_{5} + 27\beta_{3} ) / 3 $ (-41b15 - 36b13 + b10 - 14b9 + 41b8 - 31b7 + 14b5 + 27*b3) / 3
$\nu^{12}$	$=$	$ -4\beta_{14} + 4\beta_{12} + 8\beta_{11} + 69\beta_{6} - 17\beta_{4} - 3\beta_{2} + 5\beta _1 + 35 $ -4b14 + 4b12 + 8b11 + 69b6 - 17b4 - 3b2 + 5*b1 + 35
$\nu^{13}$	$=$	$ ( 71\beta_{15} + 90\beta_{13} + 98\beta_{10} - 99\beta_{9} - 26\beta_{8} + 89\beta_{7} + 79\beta_{5} + 100\beta_{3} ) / 3 $ (71b15 + 90b13 + 98b10 - 99b9 - 26b8 + 89b7 + 79b5 + 100b3) / 3
$\nu^{14}$	$=$	$ ( 15\beta_{14} + 165\beta_{12} + 15\beta_{11} - 125\beta_{6} + 155\beta_{4} - 80\beta_{2} + 115\beta _1 + 29 ) / 3 $ (15b14 + 165b12 + 15b11 - 125b6 + 155b4 - 80b2 + 115*b1 + 29) / 3
$\nu^{15}$	$=$	$ ( 37 \beta_{15} - 115 \beta_{13} - 128 \beta_{10} + 80 \beta_{9} + 43 \beta_{8} - 33 \beta_{7} + \cdots - 32 \beta_{3} ) / 3 $ (37b15 - 115b13 - 128b10 + 80b9 + 43b8 - 33b7 - 521b5 - 32b3) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1 - \beta_{6}$	$1$	$-1$

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} $

49.1

−0.485097	+	1.32841i
−1.15347	+	0.818235i
0.263711	+	1.38941i
−1.27069	−	0.620769i
1.27069	+	0.620769i
−0.263711	−	1.38941i
1.15347	−	0.818235i
0.485097	−	1.32841i
−0.485097	−	1.32841i
−1.15347	−	0.818235i
0.263711	−	1.38941i
−1.27069	+	0.620769i
1.27069	−	0.620769i
−0.263711	+	1.38941i
1.15347	+	0.818235i
0.485097	+	1.32841i

−1.39299

1.02936i

−0.589266

0.340213i

0.880830

−

2.86778i

49.2

−1.28535

1.16098i

−4.32643

2.49787i

0.304233

−

2.98453i

49.3

−1.07141

−

1.36091i

−0.0748933

0.0432397i

−0.704170

2.91619i

49.4

−0.0977414

−

1.72929i

−2.94604

1.70089i

−2.98089

0.338047i

49.5

0.0977414

1.72929i

2.94604

−

1.70089i

−2.98089

0.338047i

49.6

1.07141

1.36091i

0.0748933

−

0.0432397i

−0.704170

2.91619i

49.7

1.28535

−

1.16098i

4.32643

−

2.49787i

0.304233

−

2.98453i

49.8

1.39299

−

1.02936i

0.589266

−

0.340213i

0.880830

−

2.86778i

349.1

−1.39299

−

1.02936i

−0.589266

−

0.340213i

0.880830

2.86778i

349.2

−1.28535

−

1.16098i

−4.32643

−

2.49787i

0.304233

2.98453i

349.3

−1.07141

1.36091i

−0.0748933

−

0.0432397i

−0.704170

−

2.91619i

349.4

−0.0977414

1.72929i

−2.94604

−

1.70089i

−2.98089

−

0.338047i

349.5

0.0977414

−

1.72929i

2.94604

1.70089i

−2.98089

−

0.338047i

349.6

1.07141

−

1.36091i

0.0748933

0.0432397i

−0.704170

−

2.91619i

349.7

1.28535

1.16098i

4.32643

2.49787i

0.304233

2.98453i

349.8

1.39299

1.02936i

0.589266

0.340213i

0.880830

2.86778i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.s.d		16
3.b	odd	2	1		2700.2.s.d		16
5.b	even	2	1	inner	900.2.s.d		16
5.c	odd	4	1		900.2.i.d	✓	8
5.c	odd	4	1		900.2.i.e	yes	8
9.c	even	3	1	inner	900.2.s.d		16
9.c	even	3	1		8100.2.d.q		8
9.d	odd	6	1		2700.2.s.d		16
9.d	odd	6	1		8100.2.d.s		8
15.d	odd	2	1		2700.2.s.d		16
15.e	even	4	1		2700.2.i.d		8
15.e	even	4	1		2700.2.i.e		8
45.h	odd	6	1		2700.2.s.d		16
45.h	odd	6	1		8100.2.d.s		8
45.j	even	6	1	inner	900.2.s.d		16
45.j	even	6	1		8100.2.d.q		8
45.k	odd	12	1		900.2.i.d	✓	8
45.k	odd	12	1		900.2.i.e	yes	8
45.k	odd	12	1		8100.2.a.x		4
45.k	odd	12	1		8100.2.a.z		4
45.l	even	12	1		2700.2.i.d		8
45.l	even	12	1		2700.2.i.e		8
45.l	even	12	1		8100.2.a.y		4
45.l	even	12	1		8100.2.a.ba		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.i.d	✓	8	5.c	odd	4	1
900.2.i.d	✓	8	45.k	odd	12	1
900.2.i.e	yes	8	5.c	odd	4	1
900.2.i.e	yes	8	45.k	odd	12	1
900.2.s.d		16	1.a	even	1	1	trivial
900.2.s.d		16	5.b	even	2	1	inner
900.2.s.d		16	9.c	even	3	1	inner
900.2.s.d		16	45.j	even	6	1	inner
2700.2.i.d		8	15.e	even	4	1
2700.2.i.d		8	45.l	even	12	1
2700.2.i.e		8	15.e	even	4	1
2700.2.i.e		8	45.l	even	12	1
2700.2.s.d		16	3.b	odd	2	1
2700.2.s.d		16	9.d	odd	6	1
2700.2.s.d		16	15.d	odd	2	1
2700.2.s.d		16	45.h	odd	6	1
8100.2.a.x		4	45.k	odd	12	1
8100.2.a.y		4	45.l	even	12	1
8100.2.a.z		4	45.k	odd	12	1
8100.2.a.ba		4	45.l	even	12	1
8100.2.d.q		8	9.c	even	3	1
8100.2.d.q		8	45.j	even	6	1
8100.2.d.s		8	9.d	odd	6	1
8100.2.d.s		8	45.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(900, [\chi])$:

$ T_{7}^{16} - 37T_{7}^{14} + 1063T_{7}^{12} - 11050T_{7}^{10} + 88603T_{7}^{8} - 41542T_{7}^{6} + 18190T_{7}^{4} - 136T_{7}^{2} + 1 $

T7^16 - 37*T7^14 + 1063*T7^12 - 11050*T7^10 + 88603*T7^8 - 41542*T7^6 + 18190*T7^4 - 136*T7^2 + 1

$ T_{11}^{8} - 3T_{11}^{7} + 24T_{11}^{6} - 45T_{11}^{5} + 387T_{11}^{4} - 837T_{11}^{3} + 1620T_{11}^{2} - 1215T_{11} + 729 $

T11^8 - 3*T11^7 + 24*T11^6 - 45*T11^5 + 387*T11^4 - 837*T11^3 + 1620*T11^2 - 1215*T11 + 729

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 5 T^{14} + \cdots + 6561 $ T^16 + 5T^14 + 28T^12 + 123T^10 + 351T^8 + 1107T^6 + 2268T^4 + 3645*T^2 + 6561
$5$	$ T^{16} $ T^16
$7$	$ T^{16} - 37 T^{14} + \cdots + 1 $ T^16 - 37T^14 + 1063T^12 - 11050T^10 + 88603T^8 - 41542T^6 + 18190T^4 - 136*T^2 + 1
$11$	$ (T^{8} - 3 T^{7} + \cdots + 729)^{2} $ (T^8 - 3T^7 + 24T^6 - 45T^5 + 387T^4 - 837T^3 + 1620T^2 - 1215*T + 729)^2
$13$	$ T^{16} - 40 T^{14} + \cdots + 3418801 $ T^16 - 40T^14 + 1186T^12 - 13462T^10 + 107587T^8 - 493366T^6 + 1633915T^4 - 2864101*T^2 + 3418801
$17$	$ (T^{8} + 57 T^{6} + \cdots + 729)^{2} $ (T^8 + 57T^6 + 414T^4 + 972*T^2 + 729)^2
$19$	$ (T^{4} - 4 T^{3} - 27 T^{2} + \cdots - 23)^{4} $ (T^4 - 4T^3 - 27T^2 + 80*T - 23)^4
$23$	$ T^{16} - 147 T^{14} + \cdots + 531441 $ T^16 - 147T^14 + 15336T^12 - 796581T^10 + 30121875T^8 - 393573249T^6 + 3936127608T^4 - 45762975*T^2 + 531441
$29$	$ (T^{8} - 9 T^{7} + \cdots + 1347921)^{2} $ (T^8 - 9T^7 + 141T^6 - 792T^5 + 10755T^4 - 60858T^3 + 373896T^2 - 773226*T + 1347921)^2
$31$	$ (T^{8} + 2 T^{7} + \cdots + 625)^{2} $ (T^8 + 2T^7 + 22T^6 + 38T^5 + 373T^4 + 566T^3 + 1819T^2 - 925*T + 625)^2
$37$	$ (T^{8} + 79 T^{6} + \cdots + 9409)^{2} $ (T^8 + 79T^6 + 1761T^4 + 8095*T^2 + 9409)^2
$41$	$ (T^{8} - 9 T^{7} + \cdots + 6561)^{2} $ (T^8 - 9T^7 + 126T^6 - 405T^5 + 5589T^4 - 16767T^3 + 167670T^2 + 32805*T + 6561)^2
$43$	$ T^{16} - 172 T^{14} + \cdots + 1874161 $ T^16 - 172T^14 + 19810T^12 - 1313584T^10 + 63920923T^8 - 1795716592T^6 + 33758767378T^4 - 251583868*T^2 + 1874161
$47$	$ T^{16} + \cdots + 996005996001 $ T^16 - 246T^14 + 46089T^12 - 3089286T^10 + 150590340T^8 - 2825433414T^6 + 38445734457T^4 - 229418473878*T^2 + 996005996001
$53$	$ (T^{8} + 300 T^{6} + \cdots + 7733961)^{2} $ (T^8 + 300T^6 + 26550T^4 + 819477*T^2 + 7733961)^2
$59$	$ (T^{8} - 15 T^{7} + \cdots + 101425041)^{2} $ (T^8 - 15T^7 + 339T^6 - 3546T^5 + 62487T^4 - 601722T^3 + 5758290T^2 - 26466588*T + 101425041)^2
$61$	$ (T^{8} - T^{7} + \cdots + 100489)^{2} $ (T^8 - T^7 + 76T^6 - 547T^5 + 6253T^4 - 23959T^3 + 72946T^2 - 98587T + 100489)^2
$67$	$ T^{16} + \cdots + 66625573677601 $ T^16 - 289T^14 + 57595T^12 - 5817454T^10 + 421934407T^8 - 16997203558T^6 + 489920603626T^4 - 6836704033420*T^2 + 66625573677601
$71$	$ (T^{4} + 12 T^{3} + \cdots - 729)^{4} $ (T^4 + 12T^3 - 72T^2 - 891*T - 729)^4
$73$	$ (T^{8} + 124 T^{6} + \cdots + 265225)^{2} $ (T^8 + 124T^6 + 4974T^4 + 73489*T^2 + 265225)^2
$79$	$ (T^{8} + 7 T^{7} + \cdots + 240033049)^{2} $ (T^8 + 7T^7 + 286T^6 + 79T^5 + 46759T^4 - 10949T^3 + 4427002T^2 - 13463417*T + 240033049)^2
$83$	$ T^{16} + \cdots + 187213570125201 $ T^16 - 336T^14 + 79650T^12 - 8789742T^10 + 691620363T^8 - 30383225550T^6 + 962296116003T^4 - 16288548138657*T^2 + 187213570125201
$89$	$ (T^{4} + 3 T^{3} + \cdots + 5913)^{4} $ (T^4 + 3T^3 - 243T^2 + 513*T + 5913)^4
$97$	$ T^{16} + \cdots + 16354914662641 $ T^16 - 295T^14 + 58108T^12 - 6554797T^10 + 540730363T^8 - 26179887313T^6 + 858921556924T^4 - 3995021326939*T^2 + 16354914662641
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{15} + \beta_{3}) q^{3} + ( - \beta_{15} - \beta_{9} + \cdots + \beta_{3}) q^{7} + ( - \beta_{11} + \beta_{4} + \beta_{2} - 1) q^{9} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{11} + ( - \beta_{15} - \beta_{13} - \beta_{5}) q^{13}+ \cdots + ( - \beta_{14} - \beta_{11} + \beta_{6} + \cdots - 3) q^{99}+O(q^{100}) \) q + (-b15 + b3) * q^3 + (-b15 - b9 + b8 + b3) * q^7 + (-b11 + b4 + b2 - 1) * q^9 + (-b6 + b4 + b1) * q^11 + (-b15 - b13 - b5) * q^13 + (2b10 + b9 + 2b5 - b3) * q^17 + (-b14 + b11 + b6 - 2b4 - b2 + b1 + 2) q^19 + (b6 + b4 + b2 + b1 + 2) * q^21 + (-b13 + b10 - b8 + b7 + 2b3) q^23 + (b15 + b13 + 3b10 - b9 - b8 + 2b5 + b3) * q^27 + (2b14 - 2b12 - b11 - 2b6 - 3b4 + 2b2 - b1 - 1) q^29 + (-b12 - b11 - b6 + b2 - 1) * q^31 + (-b10 + b9 - 2b8 - 2b5) * q^33 + (b15 - b13 - b10 + b9 + b8 - b7 - b5 - 3b3) q^37 + (-b14 - b11 - b6 + b4 - b2 + b1 + 2) * q^39 + (-b14 + b12 - b11 + 2b6 - b4 - b2 + b1 + 2) q^41 + (-2b15 - 2b13 - 3b10 + 2b9 - 2b7 - 2b5) * q^43 + (3b15 + b13 - 2b10 - 2b9 + b8 + 2b7 + 2b5) q^47 + (-2b14 + 5b12 + b11 + 2b6 + 3b4 + 2b1 + 2) q^49 + (-b12 - 2b6 - 3b4 + b2 - 2b1 - 1) q^51 + (-b15 + b13 - 2b10 + b9 - b8 + 2b7 - 2b5) q^53 + (-b15 + b13 - 5b10 + b9 - b7 - 6b5) * q^57 + (2b14 - 5b12 - b11 + 3b6 + b4 + 4b2 - 2b1 + 3) q^59 + (2b14 - 2b11 - b6 - b4 + 2b2 - b1 - 2) q^61 + (b13 - 4b10 - 3b8 - 3b5 + 3b3) * q^63 + (-2b15 - b13 - 2b10 + b8 - 2b7 - 4b3) * q^67 + (b12 - 3b11 - 7b6 + 3b4 - b2 + 2b1 - 5) * q^69 + (-2b14 - b12 + b11 + b6 - 2b4 + b2 - b1 - 2) * q^71 + (b15 - b13 - 3b10 - b9 + b8 - 3b5 - 2b3) q^73 + (3b15 + b13 + 2b10 - 2b8 + 2b7 - 6b5 + 4b3) * q^77 + (-2b14 - 2b12 + 3b11 + 3b6 + b4 - 2b2 + 3b1 + 3) * q^79 + (b11 + 5b6 - 3b4 - 2b2 + 2b1 - 1) * q^81 + (5b15 - b13 + 3b10 + b9 + b7 + b5 - 2b3) q^83 + (3b13 - 4b10 + b9 - 2b8 + 7b5) * q^87 + (3b14 + 3b12 + 3b2 - 3b1 - 3) * q^89 + (-b12 - b11 - b6 + 2b4 - 3) q^91 + (b13 + 2b10 - b8 - b7 + 2b5) * q^93 + (-3b13 + 3b9 - 2b7 - 2b5 - b3) * q^97 + (-b14 - b11 + b6 + b4 - 2b2 - 3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{9} + 6 q^{11} + 16 q^{19} + 26 q^{21} + 18 q^{29} - 4 q^{31} + 34 q^{39} + 18 q^{41} + 18 q^{49} + 6 q^{51} + 30 q^{59} + 2 q^{61} - 18 q^{69} - 48 q^{71} - 14 q^{79} - 62 q^{81} - 12 q^{89}+ \cdots - 66 q^{99}+O(q^{100}) \) 16 * q - 10 * q^9 + 6 * q^11 + 16 * q^19 + 26 * q^21 + 18 * q^29 - 4 * q^31 + 34 * q^39 + 18 * q^41 + 18 * q^49 + 6 * q^51 + 30 * q^59 + 2 * q^61 - 18 * q^69 - 48 * q^71 - 14 * q^79 - 62 * q^81 - 12 * q^89 - 44 * q^91 - 66 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	900.2.s.d

Level	$900$
Weight	$2$
Character orbit	900.s
Analytic conductor	$7.187$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	16.0.1333317747165888577536.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 \) x^16 + 3x^14 + 5x^12 + 15x^10 + 45x^8 + 60x^6 + 80x^4 + 192*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{14} - 35\nu^{12} - 165\nu^{10} - 495\nu^{8} - 525\nu^{6} - 540\nu^{4} + 880\nu^{2} - 1792 ) / 2880 \) (-v^14 - 35v^12 - 165v^10 - 495v^8 - 525v^6 - 540v^4 + 880v^2 - 1792) / 2880
\(\beta_{2}\)	\(=\)	\( ( -\nu^{14} + 25\nu^{12} + 255\nu^{10} + 525\nu^{8} + 615\nu^{6} + 1920\nu^{4} + 7600\nu^{2} + 4928 ) / 2880 \) (-v^14 + 25v^12 + 255v^10 + 525v^8 + 615v^6 + 1920v^4 + 7600v^2 + 4928) / 2880
\(\beta_{3}\)	\(=\)	\( ( \nu^{15} + 5\nu^{13} + 15\nu^{11} - 75\nu^{9} + 15\nu^{7} - 30\nu^{5} + 140\nu^{3} + 352\nu ) / 1440 \) (v^15 + 5v^13 + 15v^11 - 75v^9 + 15v^7 - 30v^5 + 140v^3 + 352*v) / 1440
\(\beta_{4}\)	\(=\)	\( ( 7\nu^{14} + 65\nu^{12} + 135\nu^{10} + 165\nu^{8} + 495\nu^{6} + 1920\nu^{4} - 160\nu^{2} + 64 ) / 2880 \) (7v^14 + 65v^12 + 135v^10 + 165v^8 + 495v^6 + 1920v^4 - 160*v^2 + 64) / 2880
\(\beta_{5}\)	\(=\)	\( ( -7\nu^{15} - 5\nu^{13} + 45\nu^{11} + 135\nu^{9} + 405\nu^{7} - 180\nu^{5} + 880\nu^{3} + 896\nu ) / 5760 \) (-7v^15 - 5v^13 + 45v^11 + 135v^9 + 405v^7 - 180v^5 + 880v^3 + 896v) / 5760
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{14} + 25\nu^{12} + 15\nu^{10} + 45\nu^{8} + 135\nu^{6} + 180\nu^{4} - 64 ) / 960 \) (3v^14 + 25v^12 + 15v^10 + 45v^8 + 135v^6 + 180v^4 - 64) / 960
\(\beta_{7}\)	\(=\)	\( ( -3\nu^{15} - 45\nu^{13} - 75\nu^{11} - 145\nu^{9} - 435\nu^{7} - 760\nu^{5} - 240\nu^{3} - 2176\nu ) / 960 \) (-3v^15 - 45v^13 - 75v^11 - 145v^9 - 435v^7 - 760v^5 - 240v^3 - 2176v) / 960
\(\beta_{8}\)	\(=\)	\( ( 31\nu^{15} - 115\nu^{13} - 405\nu^{11} - 255\nu^{9} - 765\nu^{7} - 4620\nu^{5} - 3280\nu^{3} + 6592\nu ) / 5760 \) (31v^15 - 115v^13 - 405v^11 - 255v^9 - 765v^7 - 4620v^5 - 3280v^3 + 6592v) / 5760
\(\beta_{9}\)	\(=\)	\( ( -37\nu^{15} - 275\nu^{13} - 645\nu^{11} - 1215\nu^{9} - 1725\nu^{7} - 4320\nu^{5} - 4640\nu^{3} - 1984\nu ) / 5760 \) (-37v^15 - 275v^13 - 645v^11 - 1215v^9 - 1725v^7 - 4320v^5 - 4640v^3 - 1984v) / 5760
\(\beta_{10}\)	\(=\)	\( ( -37\nu^{15} - 35\nu^{13} - 165\nu^{11} - 735\nu^{9} - 1245\nu^{7} - 960\nu^{5} - 4880\nu^{3} - 6784\nu ) / 5760 \) (-37v^15 - 35v^13 - 165v^11 - 735v^9 - 1245v^7 - 960v^5 - 4880v^3 - 6784v) / 5760
\(\beta_{11}\)	\(=\)	\( ( -11\nu^{14} - 25\nu^{12} - 15\nu^{10} - 45\nu^{8} - 375\nu^{6} + 180\nu^{4} + 320\nu^{2} - 992 ) / 720 \) (-11v^14 - 25v^12 - 15v^10 - 45v^8 - 375v^6 + 180v^4 + 320*v^2 - 992) / 720
\(\beta_{12}\)	\(=\)	\( ( 49\nu^{14} + 35\nu^{12} + 165\nu^{10} + 495\nu^{8} + 525\nu^{6} - 180\nu^{4} + 2960\nu^{2} + 2368 ) / 2880 \) (49v^14 + 35v^12 + 165v^10 + 495v^8 + 525v^6 - 180v^4 + 2960*v^2 + 2368) / 2880
\(\beta_{13}\)	\(=\)	\( ( -9\nu^{15} - 5\nu^{13} - 75\nu^{11} - 25\nu^{9} - 75\nu^{7} - 190\nu^{5} - 360\nu^{3} + 1792\nu ) / 960 \) (-9v^15 - 5v^13 - 75v^11 - 25v^9 - 75v^7 - 190v^5 - 360v^3 + 1792v) / 960
\(\beta_{14}\)	\(=\)	\( ( 43\nu^{14} + 35\nu^{12} - 75\nu^{10} + 375\nu^{8} + 645\nu^{6} + 150\nu^{4} + 800\nu^{2} + 5056 ) / 1440 \) (43v^14 + 35v^12 - 75v^10 + 375v^8 + 645v^6 + 150v^4 + 800*v^2 + 5056) / 1440
\(\beta_{15}\)	\(=\)	\( ( 26\nu^{15} + 55\nu^{13} + 45\nu^{11} + 195\nu^{9} + 585\nu^{7} + 285\nu^{5} + 460\nu^{3} + 2192\nu ) / 1440 \) (26v^15 + 55v^13 + 45v^11 + 195v^9 + 585v^7 + 285v^5 + 460v^3 + 2192v) / 1440

\(\nu\)	\(=\)	\( ( -\beta_{15} - \beta_{10} + \beta_{8} - \beta_{7} - 2\beta_{5} + \beta_{3} ) / 3 \) (-b15 - b10 + b8 - b7 - 2*b5 + b3) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{6} - \beta_{4} + \beta_{2} + \beta _1 - 1 ) / 3 \) (b6 - b4 + b2 + b1 - 1) / 3
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + 2\beta_{13} - 2\beta_{10} - \beta_{9} - 2\beta_{8} + 3\beta_{7} + \beta_{5} + 4\beta_{3} ) / 3 \) (b15 + 2b13 - 2b10 - b9 - 2b8 + 3b7 + b5 + 4*b3) / 3
\(\nu^{4}\)	\(=\)	\( ( \beta_{14} - \beta_{12} + \beta_{11} - 3\beta_{6} + 5\beta_{4} + \beta _1 - 1 ) / 3 \) (b14 - b12 + b11 - 3b6 + 5b4 + b1 - 1) / 3
\(\nu^{5}\)	\(=\)	\( ( -\beta_{15} - \beta_{13} - 4\beta_{10} + 4\beta_{9} - 3\beta_{8} - 3\beta_{7} - 15\beta_{5} - 4\beta_{3} ) / 3 \) (-b15 - b13 - 4b10 + 4b9 - 3b8 - 3b7 - 15b5 - 4b3) / 3
\(\nu^{6}\)	\(=\)	\( ( \beta_{14} - 7\beta_{12} - 8\beta_{11} - 11\beta_{6} + \beta_{4} + 4\beta_{2} - \beta _1 - 17 ) / 3 \) (b14 - 7b12 - 8b11 - 11b6 + b4 + 4b2 - b1 - 17) / 3
\(\nu^{7}\)	\(=\)	\( ( 7\beta_{15} - 2\beta_{13} + 4\beta_{10} + 9\beta_{9} - 6\beta_{8} - 2\beta_{7} + 27\beta_{5} - 4\beta_{3} ) / 3 \) (7b15 - 2b13 + 4b10 + 9b9 - 6b8 - 2b7 + 27b5 - 4b3) / 3
\(\nu^{8}\)	\(=\)	\( ( 5\beta_{14} - 2\beta_{12} + 8\beta_{11} + 9\beta_{6} - 11\beta_{4} - 16\beta _1 - 14 ) / 3 \) (5b14 - 2b12 + 8b11 + 9b6 - 11b4 - 16b1 - 14) / 3
\(\nu^{9}\)	\(=\)	\( ( -4\beta_{15} + \beta_{13} - \beta_{10} - 10\beta_{9} + 8\beta_{8} + \beta_{7} + 5\beta_{5} - 33\beta_{3} ) / 3 \) (-4b15 + b13 - b10 - 10b9 + 8b8 + b7 + 5b5 - 33*b3) / 3
\(\nu^{10}\)	\(=\)	\( ( -19\beta_{14} + 28\beta_{12} - 7\beta_{11} - 20\beta_{6} + 18\beta_{4} - 5\beta_{2} - 3\beta _1 + 39 ) / 3 \) (-19b14 + 28b12 - 7b11 - 20b6 + 18b4 - 5b2 - 3*b1 + 39) / 3
\(\nu^{11}\)	\(=\)	\( ( -41\beta_{15} - 36\beta_{13} + \beta_{10} - 14\beta_{9} + 41\beta_{8} - 31\beta_{7} + 14\beta_{5} + 27\beta_{3} ) / 3 \) (-41b15 - 36b13 + b10 - 14b9 + 41b8 - 31b7 + 14b5 + 27*b3) / 3
\(\nu^{12}\)	\(=\)	\( -4\beta_{14} + 4\beta_{12} + 8\beta_{11} + 69\beta_{6} - 17\beta_{4} - 3\beta_{2} + 5\beta _1 + 35 \) -4b14 + 4b12 + 8b11 + 69b6 - 17b4 - 3b2 + 5*b1 + 35
\(\nu^{13}\)	\(=\)	\( ( 71\beta_{15} + 90\beta_{13} + 98\beta_{10} - 99\beta_{9} - 26\beta_{8} + 89\beta_{7} + 79\beta_{5} + 100\beta_{3} ) / 3 \) (71b15 + 90b13 + 98b10 - 99b9 - 26b8 + 89b7 + 79b5 + 100b3) / 3
\(\nu^{14}\)	\(=\)	\( ( 15\beta_{14} + 165\beta_{12} + 15\beta_{11} - 125\beta_{6} + 155\beta_{4} - 80\beta_{2} + 115\beta _1 + 29 ) / 3 \) (15b14 + 165b12 + 15b11 - 125b6 + 155b4 - 80b2 + 115*b1 + 29) / 3
\(\nu^{15}\)	\(=\)	\( ( 37 \beta_{15} - 115 \beta_{13} - 128 \beta_{10} + 80 \beta_{9} + 43 \beta_{8} - 33 \beta_{7} + \cdots - 32 \beta_{3} ) / 3 \) (37b15 - 115b13 - 128b10 + 80b9 + 43b8 - 33b7 - 521b5 - 32b3) / 3

\(n\)	\(101\)	\(451\)	\(577\)
\(\chi(n)\)	\(-1 - \beta_{6}\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 5 T^{14} + \cdots + 6561 \) T^16 + 5T^14 + 28T^12 + 123T^10 + 351T^8 + 1107T^6 + 2268T^4 + 3645*T^2 + 6561
$5$	\( T^{16} \) T^16
$7$	\( T^{16} - 37 T^{14} + \cdots + 1 \) T^16 - 37T^14 + 1063T^12 - 11050T^10 + 88603T^8 - 41542T^6 + 18190T^4 - 136*T^2 + 1
$11$	\( (T^{8} - 3 T^{7} + \cdots + 729)^{2} \) (T^8 - 3T^7 + 24T^6 - 45T^5 + 387T^4 - 837T^3 + 1620T^2 - 1215*T + 729)^2
$13$	\( T^{16} - 40 T^{14} + \cdots + 3418801 \) T^16 - 40T^14 + 1186T^12 - 13462T^10 + 107587T^8 - 493366T^6 + 1633915T^4 - 2864101*T^2 + 3418801
$17$	\( (T^{8} + 57 T^{6} + \cdots + 729)^{2} \) (T^8 + 57T^6 + 414T^4 + 972*T^2 + 729)^2
$19$	\( (T^{4} - 4 T^{3} - 27 T^{2} + \cdots - 23)^{4} \) (T^4 - 4T^3 - 27T^2 + 80*T - 23)^4
$23$	\( T^{16} - 147 T^{14} + \cdots + 531441 \) T^16 - 147T^14 + 15336T^12 - 796581T^10 + 30121875T^8 - 393573249T^6 + 3936127608T^4 - 45762975*T^2 + 531441
$29$	\( (T^{8} - 9 T^{7} + \cdots + 1347921)^{2} \) (T^8 - 9T^7 + 141T^6 - 792T^5 + 10755T^4 - 60858T^3 + 373896T^2 - 773226*T + 1347921)^2
$31$	\( (T^{8} + 2 T^{7} + \cdots + 625)^{2} \) (T^8 + 2T^7 + 22T^6 + 38T^5 + 373T^4 + 566T^3 + 1819T^2 - 925*T + 625)^2
$37$	\( (T^{8} + 79 T^{6} + \cdots + 9409)^{2} \) (T^8 + 79T^6 + 1761T^4 + 8095*T^2 + 9409)^2
$41$	\( (T^{8} - 9 T^{7} + \cdots + 6561)^{2} \) (T^8 - 9T^7 + 126T^6 - 405T^5 + 5589T^4 - 16767T^3 + 167670T^2 + 32805*T + 6561)^2
$43$	\( T^{16} - 172 T^{14} + \cdots + 1874161 \) T^16 - 172T^14 + 19810T^12 - 1313584T^10 + 63920923T^8 - 1795716592T^6 + 33758767378T^4 - 251583868*T^2 + 1874161
$47$	\( T^{16} + \cdots + 996005996001 \) T^16 - 246T^14 + 46089T^12 - 3089286T^10 + 150590340T^8 - 2825433414T^6 + 38445734457T^4 - 229418473878*T^2 + 996005996001
$53$	\( (T^{8} + 300 T^{6} + \cdots + 7733961)^{2} \) (T^8 + 300T^6 + 26550T^4 + 819477*T^2 + 7733961)^2
$59$	\( (T^{8} - 15 T^{7} + \cdots + 101425041)^{2} \) (T^8 - 15T^7 + 339T^6 - 3546T^5 + 62487T^4 - 601722T^3 + 5758290T^2 - 26466588*T + 101425041)^2
$61$	\( (T^{8} - T^{7} + \cdots + 100489)^{2} \) (T^8 - T^7 + 76T^6 - 547T^5 + 6253T^4 - 23959T^3 + 72946T^2 - 98587T + 100489)^2
$67$	\( T^{16} + \cdots + 66625573677601 \) T^16 - 289T^14 + 57595T^12 - 5817454T^10 + 421934407T^8 - 16997203558T^6 + 489920603626T^4 - 6836704033420*T^2 + 66625573677601
$71$	\( (T^{4} + 12 T^{3} + \cdots - 729)^{4} \) (T^4 + 12T^3 - 72T^2 - 891*T - 729)^4
$73$	\( (T^{8} + 124 T^{6} + \cdots + 265225)^{2} \) (T^8 + 124T^6 + 4974T^4 + 73489*T^2 + 265225)^2
$79$	\( (T^{8} + 7 T^{7} + \cdots + 240033049)^{2} \) (T^8 + 7T^7 + 286T^6 + 79T^5 + 46759T^4 - 10949T^3 + 4427002T^2 - 13463417*T + 240033049)^2
$83$	\( T^{16} + \cdots + 187213570125201 \) T^16 - 336T^14 + 79650T^12 - 8789742T^10 + 691620363T^8 - 30383225550T^6 + 962296116003T^4 - 16288548138657*T^2 + 187213570125201
$89$	\( (T^{4} + 3 T^{3} + \cdots + 5913)^{4} \) (T^4 + 3T^3 - 243T^2 + 513*T + 5913)^4
$97$	\( T^{16} + \cdots + 16354914662641 \) T^16 - 295T^14 + 58108T^12 - 6554797T^10 + 540730363T^8 - 26179887313T^6 + 858921556924T^4 - 3995021326939*T^2 + 16354914662641
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