LMFDB - Newform orbit 2720.2.c.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2720,2,Mod(1121,2720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2720.1121"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2720 = 2^{5} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2720.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-12,0,0,0,-16,0,0,0,-4,0,0,0,24,0,0,0,-16,0, 0,0,0,0,0,0,8,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(35)] == traces)

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

Self dual:	no
Analytic conductor:	$21.7193093498$
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} + 30x^{14} + 337x^{12} + 1804x^{10} + 4944x^{8} + 6760x^{6} + 4048x^{4} + 880x^{2} + 16 $ x^16 + 30x^14 + 337x^12 + 1804x^10 + 4944x^8 + 6760x^6 + 4048x^4 + 880*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8} $
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_1 q^{3} + \beta_{3} q^{5} + (\beta_{3} - \beta_{2}) q^{7} + (\beta_{9} + \beta_{8} - 1) q^{9} + (\beta_{14} - \beta_{12} - \beta_{3}) q^{11} + (\beta_{7} - \beta_{5} - 1) q^{13} + \beta_{10} q^{15}+ \cdots + ( - 3 \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^3 + b3 * q^5 + (b3 - b2) * q^7 + (b9 + b8 - 1) * q^9 + (b14 - b12 - b3) * q^11 + (b7 - b5 - 1) * q^13 + b10 * q^15 + (b12 + b9 + b4 - 1) * q^17 - b8 * q^19 + (2b10 - b8 + b7 + b5 + 1) q^21 + (-b15 + b12 - b6 + 2b3 + b2 - b1) q^23 - q^25 + (b14 + b13 - 2b12 + 2b11 + 2b6 - b3 + b2 - b1 + 1) q^27 + (-b15 + b12 + b6 - b2 + 2b1) q^29 + (-b15 + b13 - b11 - b6 + b1 + 1) * q^31 + (b14 + b13 - b8 + b7 + b5 + 2b4) q^33 + (b5 - 1) * q^35 - 2b6 q^37 + (-b15 + b12 - b11 - b6 - b3 - b1) * q^39 + (-b14 + b13 + 1) * q^41 + (-b14 - b13 - 2b10 + 2b7 - b5 - b4) * q^43 + (-b15 + b12 - b11) * q^45 + (b9 + b7 - b4 - 2) * q^47 + (b14 + b13 + b8 - b7 - b5 - 1) * q^49 + (-b15 + b13 - b10 + b9 + b8 - b7 + b6 - b5 - b4 - b1) * q^51 + (b14 + b13 - 2b10 - b8 + 2b7 - 1) * q^53 - b4 * q^55 + (-b11 + 2b1) q^57 + (-2b14 - 2b13 + b8 - 2b7 - 2b4 + 2) * q^59 + (-b15 - b14 + b13 + b12 - b6 - b2 + 1) * q^61 + (-3b15 + 3b12 - 2b11 + b6 - 2b3 - b2 + b1) * q^63 + (b6 - b3 - b2) * q^65 + (-b14 - b13 + 2b10 + b5 - b4 - 2) q^67 + (-b14 - b13 + b8 - 3b7 - b5 - 2b4 + 2) * q^69 + (b13 - b12 + b11 - 2b6 + b3 + 1) q^71 + (-b14 - b13 + 2b12 - 3b11 - 2b6 - 2b2 - 1) * q^73 - b1 * q^75 + (2b10 + b8 - 3b7 + b5 - 2b4 - 1) q^77 + (-b15 + b13 + b6 - 2b3 - 2b2 + 3b1 + 1) q^79 + (2b10 - 2b9 - b8 + b7 - b5 + 2b4 + 4) q^81 + (-b14 - b13 - 2b9 - 2b7 + b5 - 3b4 + 2) q^83 + (-b15 + b14 + b13 + b4 + 1) * q^85 + (-b14 - b13 - 2b10 + b9 + 2b8 - b7 - b5 - 2b4 - 6) q^87 + (b14 + b13 + 2b10 + b9 - b8 - 1) q^89 + (-b15 + 2b14 - b12 + b6 + 3b3 + 2b2 - b1) q^91 + (-2b14 - 2b13 - 2b10 + 2b8 - b7 + b5 - 2b4 - 3) q^93 + b11 * q^95 + (2b15 - b14 - b13 + b11 - 2b6 + 2b3 - 1) q^97 + (-3b15 + b14 + 2b12 - 2b11 - b6 + b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 12 q^{9} - 16 q^{13} - 4 q^{17} + 24 q^{21} - 16 q^{25} + 8 q^{33} - 12 q^{35} + 4 q^{43} - 24 q^{47} - 24 q^{49} - 20 q^{51} - 8 q^{53} + 24 q^{59} - 28 q^{67} + 16 q^{69} - 24 q^{77} + 56 q^{81}+ \cdots - 48 q^{93}+O(q^{100}) $ 16 * q - 12 * q^9 - 16 * q^13 - 4 * q^17 + 24 * q^21 - 16 * q^25 + 8 * q^33 - 12 * q^35 + 4 * q^43 - 24 * q^47 - 24 * q^49 - 20 * q^51 - 8 * q^53 + 24 * q^59 - 28 * q^67 + 16 * q^69 - 24 * q^77 + 56 * q^81 + 20 * q^83 + 8 * q^85 - 100 * q^87 - 12 * q^89 - 48 * q^93

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 30x^{14} + 337x^{12} + 1804x^{10} + 4944x^{8} + 6760x^{6} + 4048x^{4} + 880x^{2} + 16 $ x^16 + 30*x^14 + 337*x^12 + 1804*x^10 + 4944*x^8 + 6760*x^6 + 4048*x^4 + 880*x^2 + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -3\nu^{15} - 80\nu^{13} - 743\nu^{11} - 2902\nu^{9} - 4888\nu^{7} - 3252\nu^{5} - 1288\nu^{3} - 352\nu ) / 16 $ (-3v^15 - 80v^13 - 743v^11 - 2902v^9 - 4888v^7 - 3252v^5 - 1288v^3 - 352v) / 16
$\beta_{3}$	$=$	$ ( 69 \nu^{15} + 1860 \nu^{13} + 17617 \nu^{11} + 71526 \nu^{9} + 129704 \nu^{7} + 96556 \nu^{5} + \cdots + 2000 \nu ) / 208 $ (69v^15 + 1860v^13 + 17617v^11 + 71526v^9 + 129704v^7 + 96556v^5 + 28312v^3 + 2000v) / 208
$\beta_{4}$	$=$	$ ( 23\nu^{14} + 607\nu^{12} + 5530\nu^{10} + 20735\nu^{8} + 31669\nu^{6} + 14800\nu^{4} + 1932\nu^{2} + 268 ) / 52 $ (23v^14 + 607v^12 + 5530v^10 + 20735v^8 + 31669v^6 + 14800v^4 + 1932*v^2 + 268) / 52
$\beta_{5}$	$=$	$ ( 61\nu^{14} + 1624\nu^{12} + 15039\nu^{10} + 58370\nu^{8} + 96418\nu^{6} + 56740\nu^{4} + 9752\nu^{2} - 128 ) / 104 $ (61v^14 + 1624v^12 + 15039v^10 + 58370v^8 + 96418v^6 + 56740v^4 + 9752*v^2 - 128) / 104
$\beta_{6}$	$=$	$ ( -27\nu^{15} - 738\nu^{13} - 7157\nu^{11} - 30303\nu^{9} - 58777\nu^{7} - 47812\nu^{5} - 12304\nu^{3} + 280\nu ) / 52 $ (-27v^15 - 738v^13 - 7157v^11 - 30303v^9 - 58777v^7 - 47812v^5 - 12304v^3 + 280v) / 52
$\beta_{7}$	$=$	$ ( 79\nu^{14} + 2142\nu^{12} + 20495\nu^{10} + 84786\nu^{8} + 158734\nu^{6} + 123420\nu^{4} + 33312\nu^{2} + 760 ) / 104 $ (79v^14 + 2142v^12 + 20495v^10 + 84786v^8 + 158734v^6 + 123420v^4 + 33312*v^2 + 760) / 104
$\beta_{8}$	$=$	$ ( 50\nu^{14} + 1345\nu^{12} + 12687\nu^{10} + 51038\nu^{8} + 90446\nu^{6} + 62612\nu^{4} + 14288\nu^{2} + 248 ) / 52 $ (50v^14 + 1345v^12 + 12687v^10 + 51038v^8 + 90446v^6 + 62612v^4 + 14288*v^2 + 248) / 52
$\beta_{9}$	$=$	$ ( -50\nu^{14} - 1345\nu^{12} - 12687\nu^{10} - 51038\nu^{8} - 90446\nu^{6} - 62612\nu^{4} - 14236\nu^{2} - 40 ) / 52 $ (-50v^14 - 1345v^12 - 12687v^10 - 51038v^8 - 90446v^6 - 62612v^4 - 14236*v^2 - 40) / 52
$\beta_{10}$	$=$	$ ( - 105 \nu^{14} - 2818 \nu^{12} - 26475 \nu^{10} - 105716 \nu^{8} - 184942 \nu^{6} - 125500 \nu^{4} + \cdots - 552 ) / 104 $ (-105v^14 - 2818v^12 - 26475v^10 - 105716v^8 - 184942v^6 - 125500v^4 - 29360*v^2 - 552) / 104
$\beta_{11}$	$=$	$ ( 50\nu^{15} + 1345\nu^{13} + 12687\nu^{11} + 51038\nu^{9} + 90446\nu^{7} + 62612\nu^{5} + 14288\nu^{3} + 352\nu ) / 52 $ (50v^15 + 1345v^13 + 12687v^11 + 51038v^9 + 90446v^7 + 62612v^5 + 14288v^3 + 352v) / 52
$\beta_{12}$	$=$	$ ( 19 \nu^{15} + 146 \nu^{14} + 489 \nu^{13} + 3930 \nu^{12} + 4241 \nu^{11} + 37110 \nu^{10} + \cdots + 408 ) / 104 $ (19v^15 + 146v^14 + 489v^13 + 3930v^12 + 4241v^11 + 37110v^10 + 14157v^9 + 149552v^8 + 15026v^7 + 265758v^6 - 5108v^5 + 185244v^4 - 7348v^3 + 43360v^2 - 744*v + 408) / 104
$\beta_{13}$	$=$	$ ( 133 \nu^{15} + 292 \nu^{14} + 3514 \nu^{13} + 7860 \nu^{12} + 32053 \nu^{11} + 74220 \nu^{10} + \cdots + 608 ) / 208 $ (133v^15 + 292v^14 + 3514v^13 + 7860v^12 + 32053v^11 + 74220v^10 + 120120v^9 + 299104v^8 + 180556v^7 + 531516v^6 + 71156v^5 + 370488v^4 - 11344v^3 + 86720v^2 - 7184*v + 608) / 208
$\beta_{14}$	$=$	$ ( - 133 \nu^{15} + 292 \nu^{14} - 3514 \nu^{13} + 7860 \nu^{12} - 32053 \nu^{11} + 74220 \nu^{10} + \cdots + 816 ) / 208 $ (-133v^15 + 292v^14 - 3514v^13 + 7860v^12 - 32053v^11 + 74220v^10 - 120120v^9 + 299104v^8 - 180556v^7 + 531516v^6 - 71156v^5 + 370488v^4 + 11344v^3 + 86720v^2 + 7184*v + 816) / 208
$\beta_{15}$	$=$	$ ( - 159 \nu^{15} + 292 \nu^{14} - 4346 \nu^{13} + 7860 \nu^{12} - 42167 \nu^{11} + 74220 \nu^{10} + \cdots + 816 ) / 208 $ (-159v^15 + 292v^14 - 4346v^13 + 7860v^12 - 42167v^11 + 74220v^10 - 178984v^9 + 299104v^8 - 350960v^7 + 531516v^6 - 299332v^5 + 370488v^4 - 98064v^3 + 86720v^2 - 7792*v + 816) / 208

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{9} + \beta_{8} - 4 $ b9 + b8 - 4
$\nu^{3}$	$=$	$ \beta_{14} + \beta_{13} - 2\beta_{12} + 2\beta_{11} + 2\beta_{6} - \beta_{3} + \beta_{2} - 7\beta _1 + 1 $ b14 + b13 - 2b12 + 2b11 + 2b6 - b3 + b2 - 7b1 + 1
$\nu^{4}$	$=$	$ 2\beta_{10} - 11\beta_{9} - 10\beta_{8} + \beta_{7} - \beta_{5} + 2\beta_{4} + 31 $ 2b10 - 11b9 - 10b8 + b7 - b5 + 2b4 + 31
$\nu^{5}$	$=$	$ - 3 \beta_{15} - 13 \beta_{14} - 13 \beta_{13} + 29 \beta_{12} - 26 \beta_{11} - 25 \beta_{6} + \cdots - 13 $ -3b15 - 13b14 - 13b13 + 29b12 - 26b11 - 25b6 + 6b3 - 13b2 + 62*b1 - 13
$\nu^{6}$	$=$	$ - 3 \beta_{14} - 3 \beta_{13} - 40 \beta_{10} + 113 \beta_{9} + 104 \beta_{8} - 19 \beta_{7} + \cdots - 286 $ -3b14 - 3b13 - 40b10 + 113b9 + 104b8 - 19b7 + 9b5 - 32b4 - 286
$\nu^{7}$	$=$	$ 65 \beta_{15} + 145 \beta_{14} + 139 \beta_{13} - 349 \beta_{12} + 298 \beta_{11} + 267 \beta_{6} + \cdots + 139 $ 65b15 + 145b14 + 139b13 - 349b12 + 298b11 + 267b6 - 16b3 + 135b2 - 600*b1 + 139
$\nu^{8}$	$=$	$ 71 \beta_{14} + 71 \beta_{13} + 592 \beta_{10} - 1145 \beta_{9} - 1112 \beta_{8} + 279 \beta_{7} + \cdots + 2810 $ 71b14 + 71b13 + 592b10 - 1145b9 - 1112b8 + 279b7 - 53b5 + 420b4 + 2810
$\nu^{9}$	$=$	$ - 1013 \beta_{15} - 1565 \beta_{14} - 1423 \beta_{13} + 4001 \beta_{12} - 3322 \beta_{11} - 2763 \beta_{6} + \cdots - 1423 $ -1013b15 - 1565b14 - 1423b13 + 4001b12 - 3322b11 - 2763b6 - 264b3 - 1339b2 + 6024*b1 - 1423
$\nu^{10}$	$=$	$ - 1155 \beta_{14} - 1155 \beta_{13} - 7820 \beta_{10} + 11633 \beta_{9} + 12008 \beta_{8} + \cdots - 28366 $ -1155b14 - 1155b13 - 7820b10 + 11633b9 + 12008b8 - 3675b7 + 101b5 - 5156b4 - 28366
$\nu^{11}$	$=$	$ 13805 \beta_{15} + 16789 \beta_{14} + 14479 \beta_{13} - 45073 \beta_{12} + 36718 \beta_{11} + \cdots + 14479 $ 13805b15 + 16789b14 + 14479b13 - 45073b12 + 36718b11 + 28523b6 + 6772b3 + 13215b2 - 61596*b1 + 14479
$\nu^{12}$	$=$	$ 16115 \beta_{14} + 16115 \beta_{13} + 97348 \beta_{10} - 119077 \beta_{9} - 130172 \beta_{8} + \cdots + 290650 $ 16115b14 + 16115b13 + 97348b10 - 119077b9 - 130172b8 + 45663b7 + 3335b5 + 61188b4 + 290650
$\nu^{13}$	$=$	$ - 175241 \beta_{15} - 180265 \beta_{14} - 148035 \beta_{13} + 503541 \beta_{12} - 404450 \beta_{11} + \cdots - 148035 $ -175241b15 - 180265b14 - 148035b13 + 503541b12 - 404450b11 - 296007b6 - 108444b3 - 131267b2 + 637188*b1 - 148035
$\nu^{14}$	$=$	$ - 207471 \beta_{14} - 207471 \beta_{13} - 1168852 \beta_{10} + 1229265 \beta_{9} + 1413912 \beta_{8} + \cdots - 3009562 $ -207471b14 - 207471b13 - 1168852b10 + 1229265b9 + 1413912b8 - 547515b7 - 76267b5 - 711012b4 - 3009562
$\nu^{15}$	$=$	$ 2131309 \beta_{15} + 1940277 \beta_{14} + 1525335 \beta_{13} - 5596921 \beta_{12} + 4446774 \beta_{11} + \cdots + 1525335 $ 2131309b15 + 1940277b14 + 1525335b13 - 5596921b12 + 4446774b11 + 3093275b6 + 1490012b3 + 1316495b2 - 6650340*b1 + 1525335

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2720\mathbb{Z}\right)^\times$.

$n$	$511$	$1601$	$1701$	$2177$
$\chi(n)$	$1$	$-1$	$1$	$1$

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} $

1121.1

	−	3.30605i
	−	2.97343i
	−	1.77785i
	−	1.77495i
	−	1.67348i
	−	0.778614i
	−	0.700592i
	−	0.141255i
		0.141255i
		0.700592i
		0.778614i
		1.67348i
		1.77495i
		1.77785i
		2.97343i
		3.30605i

−

3.30605i

1.00000i

2.92445i

−7.92996

1121.2

−

2.97343i

−

1.00000i

−

2.79457i

−5.84130

1121.3

−

1.77785i

1.00000i

2.74610i

−0.160765

1121.4

−

1.77495i

−

1.00000i

−

0.336810i

−0.150436

1121.5

−

1.67348i

−

1.00000i

4.87128i

0.199481

1121.6

−

0.778614i

1.00000i

0.401608i

2.39376

1121.7

−

0.700592i

1.00000i

−

2.22408i

2.50917

1121.8

−

0.141255i

−

1.00000i

−

3.89182i

2.98005

1121.9

0.141255i

1.00000i

3.89182i

2.98005

1121.10

0.700592i

−

1.00000i

2.22408i

2.50917

1121.11

0.778614i

−

1.00000i

−

0.401608i

2.39376

1121.12

1.67348i

1.00000i

−

4.87128i

0.199481

1121.13

1.77495i

1.00000i

0.336810i

−0.150436

1121.14

1.77785i

−

1.00000i

−

2.74610i

−0.160765

1121.15

2.97343i

1.00000i

2.79457i

−5.84130

1121.16

3.30605i

−

1.00000i

−

2.92445i

−7.92996

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2720.2.c.e	✓	16
4.b	odd	2	1		2720.2.c.f	yes	16
17.b	even	2	1	inner	2720.2.c.e	✓	16
68.d	odd	2	1		2720.2.c.f	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2720.2.c.e	✓	16	1.a	even	1	1	trivial
2720.2.c.e	✓	16	17.b	even	2	1	inner
2720.2.c.f	yes	16	4.b	odd	2	1
2720.2.c.f	yes	16	68.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2720, [\chi])$:

$ T_{3}^{16} + 30T_{3}^{14} + 337T_{3}^{12} + 1804T_{3}^{10} + 4944T_{3}^{8} + 6760T_{3}^{6} + 4048T_{3}^{4} + 880T_{3}^{2} + 16 $

T3^16 + 30*T3^14 + 337*T3^12 + 1804*T3^10 + 4944*T3^8 + 6760*T3^6 + 4048*T3^4 + 880*T3^2 + 16

$ T_{19}^{8} - 53T_{19}^{6} + 108T_{19}^{5} + 544T_{19}^{4} - 2272T_{19}^{3} + 3072T_{19}^{2} - 1664T_{19} + 256 $

T19^8 - 53*T19^6 + 108*T19^5 + 544*T19^4 - 2272*T19^3 + 3072*T19^2 - 1664*T19 + 256

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 30 T^{14} + \cdots + 16 $ T^16 + 30T^14 + 337T^12 + 1804T^10 + 4944T^8 + 6760T^6 + 4048T^4 + 880*T^2 + 16
$5$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$7$	$ T^{16} + 68 T^{14} + \cdots + 16384 $ T^16 + 68T^14 + 1808T^12 + 24296T^10 + 176064T^8 + 662992T^6 + 1067792T^4 + 257280*T^2 + 16384
$11$	$ T^{16} + 88 T^{14} + \cdots + 13897984 $ T^16 + 88T^14 + 3144T^12 + 59112T^10 + 633904T^8 + 3917360T^6 + 13333136T^4 + 22175616*T^2 + 13897984
$13$	$ (T^{8} + 8 T^{7} + \cdots - 2672)^{2} $ (T^8 + 8T^7 - 13T^6 - 172T^5 - 40T^4 + 1136T^3 + 960T^2 - 2320*T - 2672)^2
$17$	$ T^{16} + \cdots + 6975757441 $ T^16 + 4T^15 + 4T^14 + 28T^13 - 268T^12 - 300T^11 + 124T^10 - 4212T^9 + 96918T^8 - 71604T^7 + 35836T^6 - 1473900T^5 - 22383628T^4 + 39755996T^3 + 96550276T^2 + 1641354692*T + 6975757441
$19$	$ (T^{8} - 53 T^{6} + \cdots + 256)^{2} $ (T^8 - 53T^6 + 108T^5 + 544T^4 - 2272T^3 + 3072T^2 - 1664T + 256)^2
$23$	$ T^{16} + \cdots + 586027264 $ T^16 + 196T^14 + 14464T^12 + 527832T^10 + 10318592T^8 + 107978960T^6 + 561995152T^4 + 1207264640*T^2 + 586027264
$29$	$ T^{16} + 250 T^{14} + \cdots + 262144 $ T^16 + 250T^14 + 22729T^12 + 903312T^10 + 15036608T^8 + 101743616T^6 + 239644672T^4 + 27590656*T^2 + 262144
$31$	$ T^{16} + \cdots + 202435984 $ T^16 + 274T^14 + 29321T^12 + 1528248T^10 + 39212136T^8 + 442690600T^6 + 1944140800T^4 + 2807359120*T^2 + 202435984
$37$	$ T^{16} + 216 T^{14} + \cdots + 1048576 $ T^16 + 216T^14 + 18448T^12 + 801024T^10 + 18804736T^8 + 232562688T^6 + 1349451776T^4 + 2834038784*T^2 + 1048576
$41$	$ T^{16} + 248 T^{14} + \cdots + 4194304 $ T^16 + 248T^14 + 22640T^12 + 945664T^10 + 18677504T^8 + 159733760T^6 + 405114880T^4 + 109379584*T^2 + 4194304
$43$	$ (T^{8} - 2 T^{7} + \cdots + 1030144)^{2} $ (T^8 - 2T^7 - 228T^6 + 456T^5 + 13472T^4 - 12864T^3 - 219728T^2 + 103616*T + 1030144)^2
$47$	$ (T^{8} + 12 T^{7} + \cdots + 84592)^{2} $ (T^8 + 12T^7 - 87T^6 - 1014T^5 + 3732T^4 + 23288T^3 - 84000T^2 - 704*T + 84592)^2
$53$	$ (T^{8} + 4 T^{7} + \cdots - 297664)^{2} $ (T^8 + 4T^7 - 225T^6 - 1008T^5 + 8332T^4 + 38176T^3 - 68080T^2 - 392320*T - 297664)^2
$59$	$ (T^{8} - 12 T^{7} + \cdots - 24951040)^{2} $ (T^8 - 12T^7 - 269T^6 + 3944T^5 + 13664T^4 - 363936T^3 + 780864T^2 + 6883712*T - 24951040)^2
$61$	$ T^{16} + \cdots + 2671158759424 $ T^16 + 450T^14 + 74665T^12 + 6111192T^10 + 274309872T^8 + 7082172160T^6 + 104790243072T^4 + 824568223744*T^2 + 2671158759424
$67$	$ (T^{8} + 14 T^{7} + \cdots + 3008)^{2} $ (T^8 + 14T^7 - 40T^6 - 928T^5 - 992T^4 + 9504T^3 + 25008T^2 + 19008*T + 3008)^2
$71$	$ T^{16} + \cdots + 43207221168400 $ T^16 + 506T^14 + 105033T^12 + 11656416T^10 + 757344168T^8 + 29590683864T^6 + 680903856896T^4 + 8447505115216*T^2 + 43207221168400
$73$	$ T^{16} + \cdots + 1186670350336 $ T^16 + 714T^14 + 192609T^12 + 24648712T^10 + 1561243760T^8 + 46891614464T^6 + 631313719040T^4 + 3003152582656*T^2 + 1186670350336
$79$	$ T^{16} + \cdots + 1392777625600 $ T^16 + 496T^14 + 97416T^12 + 9711640T^10 + 523688400T^8 + 15083117168T^6 + 213611124112T^4 + 1198562169856*T^2 + 1392777625600
$83$	$ (T^{8} - 10 T^{7} + \cdots - 753728)^{2} $ (T^8 - 10T^7 - 248T^6 + 3624T^5 - 7776T^4 - 61184T^3 + 207728T^2 + 188736*T - 753728)^2
$89$	$ (T^{8} + 6 T^{7} + \cdots - 7236496)^{2} $ (T^8 + 6T^7 - 259T^6 - 1900T^5 + 16052T^4 + 155192T^3 - 81056T^2 - 3363856*T - 7236496)^2
$97$	$ T^{16} + \cdots + 9413656576 $ T^16 + 578T^14 + 105473T^12 + 8141120T^10 + 286128512T^8 + 4600359424T^6 + 28937736192T^4 + 32392773632*T^2 + 9413656576
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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:	\( N \)	\(=\)	\( 2720 = 2^{5} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2720.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{3} - \beta_{2}) q^{7} + (\beta_{9} + \beta_{8} - 1) q^{9} + (\beta_{14} - \beta_{12} - \beta_{3}) q^{11} + (\beta_{7} - \beta_{5} - 1) q^{13} + \beta_{10} q^{15}+ \cdots + ( - 3 \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 + b3 * q^5 + (b3 - b2) * q^7 + (b9 + b8 - 1) * q^9 + (b14 - b12 - b3) * q^11 + (b7 - b5 - 1) * q^13 + b10 * q^15 + (b12 + b9 + b4 - 1) * q^17 - b8 * q^19 + (2b10 - b8 + b7 + b5 + 1) q^21 + (-b15 + b12 - b6 + 2b3 + b2 - b1) q^23 - q^25 + (b14 + b13 - 2b12 + 2b11 + 2b6 - b3 + b2 - b1 + 1) q^27 + (-b15 + b12 + b6 - b2 + 2b1) q^29 + (-b15 + b13 - b11 - b6 + b1 + 1) * q^31 + (b14 + b13 - b8 + b7 + b5 + 2b4) q^33 + (b5 - 1) * q^35 - 2b6 q^37 + (-b15 + b12 - b11 - b6 - b3 - b1) * q^39 + (-b14 + b13 + 1) * q^41 + (-b14 - b13 - 2b10 + 2b7 - b5 - b4) * q^43 + (-b15 + b12 - b11) * q^45 + (b9 + b7 - b4 - 2) * q^47 + (b14 + b13 + b8 - b7 - b5 - 1) * q^49 + (-b15 + b13 - b10 + b9 + b8 - b7 + b6 - b5 - b4 - b1) * q^51 + (b14 + b13 - 2b10 - b8 + 2b7 - 1) * q^53 - b4 * q^55 + (-b11 + 2b1) q^57 + (-2b14 - 2b13 + b8 - 2b7 - 2b4 + 2) * q^59 + (-b15 - b14 + b13 + b12 - b6 - b2 + 1) * q^61 + (-3b15 + 3b12 - 2b11 + b6 - 2b3 - b2 + b1) * q^63 + (b6 - b3 - b2) * q^65 + (-b14 - b13 + 2b10 + b5 - b4 - 2) q^67 + (-b14 - b13 + b8 - 3b7 - b5 - 2b4 + 2) * q^69 + (b13 - b12 + b11 - 2b6 + b3 + 1) q^71 + (-b14 - b13 + 2b12 - 3b11 - 2b6 - 2b2 - 1) * q^73 - b1 * q^75 + (2b10 + b8 - 3b7 + b5 - 2b4 - 1) q^77 + (-b15 + b13 + b6 - 2b3 - 2b2 + 3b1 + 1) q^79 + (2b10 - 2b9 - b8 + b7 - b5 + 2b4 + 4) q^81 + (-b14 - b13 - 2b9 - 2b7 + b5 - 3b4 + 2) q^83 + (-b15 + b14 + b13 + b4 + 1) * q^85 + (-b14 - b13 - 2b10 + b9 + 2b8 - b7 - b5 - 2b4 - 6) q^87 + (b14 + b13 + 2b10 + b9 - b8 - 1) q^89 + (-b15 + 2b14 - b12 + b6 + 3b3 + 2b2 - b1) q^91 + (-2b14 - 2b13 - 2b10 + 2b8 - b7 + b5 - 2b4 - 3) q^93 + b11 * q^95 + (2b15 - b14 - b13 + b11 - 2b6 + 2b3 - 1) q^97 + (-3b15 + b14 + 2b12 - 2b11 - b6 + b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 12 q^{9} - 16 q^{13} - 4 q^{17} + 24 q^{21} - 16 q^{25} + 8 q^{33} - 12 q^{35} + 4 q^{43} - 24 q^{47} - 24 q^{49} - 20 q^{51} - 8 q^{53} + 24 q^{59} - 28 q^{67} + 16 q^{69} - 24 q^{77} + 56 q^{81}+ \cdots - 48 q^{93}+O(q^{100}) \) 16 * q - 12 * q^9 - 16 * q^13 - 4 * q^17 + 24 * q^21 - 16 * q^25 + 8 * q^33 - 12 * q^35 + 4 * q^43 - 24 * q^47 - 24 * q^49 - 20 * q^51 - 8 * q^53 + 24 * q^59 - 28 * q^67 + 16 * q^69 - 24 * q^77 + 56 * q^81 + 20 * q^83 + 8 * q^85 - 100 * q^87 - 12 * q^89 - 48 * q^93

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	2720.2.c.e

Level	$2720$
Weight	$2$
Character orbit	2720.c
Analytic conductor	$21.719$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(21.7193093498\)
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:	\( x^{16} + 30x^{14} + 337x^{12} + 1804x^{10} + 4944x^{8} + 6760x^{6} + 4048x^{4} + 880x^{2} + 16 \) x^16 + 30x^14 + 337x^12 + 1804x^10 + 4944x^8 + 6760x^6 + 4048x^4 + 880*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -3\nu^{15} - 80\nu^{13} - 743\nu^{11} - 2902\nu^{9} - 4888\nu^{7} - 3252\nu^{5} - 1288\nu^{3} - 352\nu ) / 16 \) (-3v^15 - 80v^13 - 743v^11 - 2902v^9 - 4888v^7 - 3252v^5 - 1288v^3 - 352v) / 16
\(\beta_{3}\)	\(=\)	\( ( 69 \nu^{15} + 1860 \nu^{13} + 17617 \nu^{11} + 71526 \nu^{9} + 129704 \nu^{7} + 96556 \nu^{5} + \cdots + 2000 \nu ) / 208 \) (69v^15 + 1860v^13 + 17617v^11 + 71526v^9 + 129704v^7 + 96556v^5 + 28312v^3 + 2000v) / 208
\(\beta_{4}\)	\(=\)	\( ( 23\nu^{14} + 607\nu^{12} + 5530\nu^{10} + 20735\nu^{8} + 31669\nu^{6} + 14800\nu^{4} + 1932\nu^{2} + 268 ) / 52 \) (23v^14 + 607v^12 + 5530v^10 + 20735v^8 + 31669v^6 + 14800v^4 + 1932*v^2 + 268) / 52
\(\beta_{5}\)	\(=\)	\( ( 61\nu^{14} + 1624\nu^{12} + 15039\nu^{10} + 58370\nu^{8} + 96418\nu^{6} + 56740\nu^{4} + 9752\nu^{2} - 128 ) / 104 \) (61v^14 + 1624v^12 + 15039v^10 + 58370v^8 + 96418v^6 + 56740v^4 + 9752*v^2 - 128) / 104
\(\beta_{6}\)	\(=\)	\( ( -27\nu^{15} - 738\nu^{13} - 7157\nu^{11} - 30303\nu^{9} - 58777\nu^{7} - 47812\nu^{5} - 12304\nu^{3} + 280\nu ) / 52 \) (-27v^15 - 738v^13 - 7157v^11 - 30303v^9 - 58777v^7 - 47812v^5 - 12304v^3 + 280v) / 52
\(\beta_{7}\)	\(=\)	\( ( 79\nu^{14} + 2142\nu^{12} + 20495\nu^{10} + 84786\nu^{8} + 158734\nu^{6} + 123420\nu^{4} + 33312\nu^{2} + 760 ) / 104 \) (79v^14 + 2142v^12 + 20495v^10 + 84786v^8 + 158734v^6 + 123420v^4 + 33312*v^2 + 760) / 104
\(\beta_{8}\)	\(=\)	\( ( 50\nu^{14} + 1345\nu^{12} + 12687\nu^{10} + 51038\nu^{8} + 90446\nu^{6} + 62612\nu^{4} + 14288\nu^{2} + 248 ) / 52 \) (50v^14 + 1345v^12 + 12687v^10 + 51038v^8 + 90446v^6 + 62612v^4 + 14288*v^2 + 248) / 52
\(\beta_{9}\)	\(=\)	\( ( -50\nu^{14} - 1345\nu^{12} - 12687\nu^{10} - 51038\nu^{8} - 90446\nu^{6} - 62612\nu^{4} - 14236\nu^{2} - 40 ) / 52 \) (-50v^14 - 1345v^12 - 12687v^10 - 51038v^8 - 90446v^6 - 62612v^4 - 14236*v^2 - 40) / 52
\(\beta_{10}\)	\(=\)	\( ( - 105 \nu^{14} - 2818 \nu^{12} - 26475 \nu^{10} - 105716 \nu^{8} - 184942 \nu^{6} - 125500 \nu^{4} + \cdots - 552 ) / 104 \) (-105v^14 - 2818v^12 - 26475v^10 - 105716v^8 - 184942v^6 - 125500v^4 - 29360*v^2 - 552) / 104
\(\beta_{11}\)	\(=\)	\( ( 50\nu^{15} + 1345\nu^{13} + 12687\nu^{11} + 51038\nu^{9} + 90446\nu^{7} + 62612\nu^{5} + 14288\nu^{3} + 352\nu ) / 52 \) (50v^15 + 1345v^13 + 12687v^11 + 51038v^9 + 90446v^7 + 62612v^5 + 14288v^3 + 352v) / 52
\(\beta_{12}\)	\(=\)	\( ( 19 \nu^{15} + 146 \nu^{14} + 489 \nu^{13} + 3930 \nu^{12} + 4241 \nu^{11} + 37110 \nu^{10} + \cdots + 408 ) / 104 \) (19v^15 + 146v^14 + 489v^13 + 3930v^12 + 4241v^11 + 37110v^10 + 14157v^9 + 149552v^8 + 15026v^7 + 265758v^6 - 5108v^5 + 185244v^4 - 7348v^3 + 43360v^2 - 744*v + 408) / 104
\(\beta_{13}\)	\(=\)	\( ( 133 \nu^{15} + 292 \nu^{14} + 3514 \nu^{13} + 7860 \nu^{12} + 32053 \nu^{11} + 74220 \nu^{10} + \cdots + 608 ) / 208 \) (133v^15 + 292v^14 + 3514v^13 + 7860v^12 + 32053v^11 + 74220v^10 + 120120v^9 + 299104v^8 + 180556v^7 + 531516v^6 + 71156v^5 + 370488v^4 - 11344v^3 + 86720v^2 - 7184*v + 608) / 208
\(\beta_{14}\)	\(=\)	\( ( - 133 \nu^{15} + 292 \nu^{14} - 3514 \nu^{13} + 7860 \nu^{12} - 32053 \nu^{11} + 74220 \nu^{10} + \cdots + 816 ) / 208 \) (-133v^15 + 292v^14 - 3514v^13 + 7860v^12 - 32053v^11 + 74220v^10 - 120120v^9 + 299104v^8 - 180556v^7 + 531516v^6 - 71156v^5 + 370488v^4 + 11344v^3 + 86720v^2 + 7184*v + 816) / 208
\(\beta_{15}\)	\(=\)	\( ( - 159 \nu^{15} + 292 \nu^{14} - 4346 \nu^{13} + 7860 \nu^{12} - 42167 \nu^{11} + 74220 \nu^{10} + \cdots + 816 ) / 208 \) (-159v^15 + 292v^14 - 4346v^13 + 7860v^12 - 42167v^11 + 74220v^10 - 178984v^9 + 299104v^8 - 350960v^7 + 531516v^6 - 299332v^5 + 370488v^4 - 98064v^3 + 86720v^2 - 7792*v + 816) / 208

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{9} + \beta_{8} - 4 \) b9 + b8 - 4
\(\nu^{3}\)	\(=\)	\( \beta_{14} + \beta_{13} - 2\beta_{12} + 2\beta_{11} + 2\beta_{6} - \beta_{3} + \beta_{2} - 7\beta _1 + 1 \) b14 + b13 - 2b12 + 2b11 + 2b6 - b3 + b2 - 7b1 + 1
\(\nu^{4}\)	\(=\)	\( 2\beta_{10} - 11\beta_{9} - 10\beta_{8} + \beta_{7} - \beta_{5} + 2\beta_{4} + 31 \) 2b10 - 11b9 - 10b8 + b7 - b5 + 2b4 + 31
\(\nu^{5}\)	\(=\)	\( - 3 \beta_{15} - 13 \beta_{14} - 13 \beta_{13} + 29 \beta_{12} - 26 \beta_{11} - 25 \beta_{6} + \cdots - 13 \) -3b15 - 13b14 - 13b13 + 29b12 - 26b11 - 25b6 + 6b3 - 13b2 + 62*b1 - 13
\(\nu^{6}\)	\(=\)	\( - 3 \beta_{14} - 3 \beta_{13} - 40 \beta_{10} + 113 \beta_{9} + 104 \beta_{8} - 19 \beta_{7} + \cdots - 286 \) -3b14 - 3b13 - 40b10 + 113b9 + 104b8 - 19b7 + 9b5 - 32b4 - 286
\(\nu^{7}\)	\(=\)	\( 65 \beta_{15} + 145 \beta_{14} + 139 \beta_{13} - 349 \beta_{12} + 298 \beta_{11} + 267 \beta_{6} + \cdots + 139 \) 65b15 + 145b14 + 139b13 - 349b12 + 298b11 + 267b6 - 16b3 + 135b2 - 600*b1 + 139
\(\nu^{8}\)	\(=\)	\( 71 \beta_{14} + 71 \beta_{13} + 592 \beta_{10} - 1145 \beta_{9} - 1112 \beta_{8} + 279 \beta_{7} + \cdots + 2810 \) 71b14 + 71b13 + 592b10 - 1145b9 - 1112b8 + 279b7 - 53b5 + 420b4 + 2810
\(\nu^{9}\)	\(=\)	\( - 1013 \beta_{15} - 1565 \beta_{14} - 1423 \beta_{13} + 4001 \beta_{12} - 3322 \beta_{11} - 2763 \beta_{6} + \cdots - 1423 \) -1013b15 - 1565b14 - 1423b13 + 4001b12 - 3322b11 - 2763b6 - 264b3 - 1339b2 + 6024*b1 - 1423
\(\nu^{10}\)	\(=\)	\( - 1155 \beta_{14} - 1155 \beta_{13} - 7820 \beta_{10} + 11633 \beta_{9} + 12008 \beta_{8} + \cdots - 28366 \) -1155b14 - 1155b13 - 7820b10 + 11633b9 + 12008b8 - 3675b7 + 101b5 - 5156b4 - 28366
\(\nu^{11}\)	\(=\)	\( 13805 \beta_{15} + 16789 \beta_{14} + 14479 \beta_{13} - 45073 \beta_{12} + 36718 \beta_{11} + \cdots + 14479 \) 13805b15 + 16789b14 + 14479b13 - 45073b12 + 36718b11 + 28523b6 + 6772b3 + 13215b2 - 61596*b1 + 14479
\(\nu^{12}\)	\(=\)	\( 16115 \beta_{14} + 16115 \beta_{13} + 97348 \beta_{10} - 119077 \beta_{9} - 130172 \beta_{8} + \cdots + 290650 \) 16115b14 + 16115b13 + 97348b10 - 119077b9 - 130172b8 + 45663b7 + 3335b5 + 61188b4 + 290650
\(\nu^{13}\)	\(=\)	\( - 175241 \beta_{15} - 180265 \beta_{14} - 148035 \beta_{13} + 503541 \beta_{12} - 404450 \beta_{11} + \cdots - 148035 \) -175241b15 - 180265b14 - 148035b13 + 503541b12 - 404450b11 - 296007b6 - 108444b3 - 131267b2 + 637188*b1 - 148035
\(\nu^{14}\)	\(=\)	\( - 207471 \beta_{14} - 207471 \beta_{13} - 1168852 \beta_{10} + 1229265 \beta_{9} + 1413912 \beta_{8} + \cdots - 3009562 \) -207471b14 - 207471b13 - 1168852b10 + 1229265b9 + 1413912b8 - 547515b7 - 76267b5 - 711012b4 - 3009562
\(\nu^{15}\)	\(=\)	\( 2131309 \beta_{15} + 1940277 \beta_{14} + 1525335 \beta_{13} - 5596921 \beta_{12} + 4446774 \beta_{11} + \cdots + 1525335 \) 2131309b15 + 1940277b14 + 1525335b13 - 5596921b12 + 4446774b11 + 3093275b6 + 1490012b3 + 1316495b2 - 6650340*b1 + 1525335

\(n\)	\(511\)	\(1601\)	\(1701\)	\(2177\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1121.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 30 T^{14} + \cdots + 16 \) T^16 + 30T^14 + 337T^12 + 1804T^10 + 4944T^8 + 6760T^6 + 4048T^4 + 880*T^2 + 16
$5$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$7$	\( T^{16} + 68 T^{14} + \cdots + 16384 \) T^16 + 68T^14 + 1808T^12 + 24296T^10 + 176064T^8 + 662992T^6 + 1067792T^4 + 257280*T^2 + 16384
$11$	\( T^{16} + 88 T^{14} + \cdots + 13897984 \) T^16 + 88T^14 + 3144T^12 + 59112T^10 + 633904T^8 + 3917360T^6 + 13333136T^4 + 22175616*T^2 + 13897984
$13$	\( (T^{8} + 8 T^{7} + \cdots - 2672)^{2} \) (T^8 + 8T^7 - 13T^6 - 172T^5 - 40T^4 + 1136T^3 + 960T^2 - 2320*T - 2672)^2
$17$	\( T^{16} + \cdots + 6975757441 \) T^16 + 4T^15 + 4T^14 + 28T^13 - 268T^12 - 300T^11 + 124T^10 - 4212T^9 + 96918T^8 - 71604T^7 + 35836T^6 - 1473900T^5 - 22383628T^4 + 39755996T^3 + 96550276T^2 + 1641354692*T + 6975757441
$19$	\( (T^{8} - 53 T^{6} + \cdots + 256)^{2} \) (T^8 - 53T^6 + 108T^5 + 544T^4 - 2272T^3 + 3072T^2 - 1664T + 256)^2
$23$	\( T^{16} + \cdots + 586027264 \) T^16 + 196T^14 + 14464T^12 + 527832T^10 + 10318592T^8 + 107978960T^6 + 561995152T^4 + 1207264640*T^2 + 586027264
$29$	\( T^{16} + 250 T^{14} + \cdots + 262144 \) T^16 + 250T^14 + 22729T^12 + 903312T^10 + 15036608T^8 + 101743616T^6 + 239644672T^4 + 27590656*T^2 + 262144
$31$	\( T^{16} + \cdots + 202435984 \) T^16 + 274T^14 + 29321T^12 + 1528248T^10 + 39212136T^8 + 442690600T^6 + 1944140800T^4 + 2807359120*T^2 + 202435984
$37$	\( T^{16} + 216 T^{14} + \cdots + 1048576 \) T^16 + 216T^14 + 18448T^12 + 801024T^10 + 18804736T^8 + 232562688T^6 + 1349451776T^4 + 2834038784*T^2 + 1048576
$41$	\( T^{16} + 248 T^{14} + \cdots + 4194304 \) T^16 + 248T^14 + 22640T^12 + 945664T^10 + 18677504T^8 + 159733760T^6 + 405114880T^4 + 109379584*T^2 + 4194304
$43$	\( (T^{8} - 2 T^{7} + \cdots + 1030144)^{2} \) (T^8 - 2T^7 - 228T^6 + 456T^5 + 13472T^4 - 12864T^3 - 219728T^2 + 103616*T + 1030144)^2
$47$	\( (T^{8} + 12 T^{7} + \cdots + 84592)^{2} \) (T^8 + 12T^7 - 87T^6 - 1014T^5 + 3732T^4 + 23288T^3 - 84000T^2 - 704*T + 84592)^2
$53$	\( (T^{8} + 4 T^{7} + \cdots - 297664)^{2} \) (T^8 + 4T^7 - 225T^6 - 1008T^5 + 8332T^4 + 38176T^3 - 68080T^2 - 392320*T - 297664)^2
$59$	\( (T^{8} - 12 T^{7} + \cdots - 24951040)^{2} \) (T^8 - 12T^7 - 269T^6 + 3944T^5 + 13664T^4 - 363936T^3 + 780864T^2 + 6883712*T - 24951040)^2
$61$	\( T^{16} + \cdots + 2671158759424 \) T^16 + 450T^14 + 74665T^12 + 6111192T^10 + 274309872T^8 + 7082172160T^6 + 104790243072T^4 + 824568223744*T^2 + 2671158759424
$67$	\( (T^{8} + 14 T^{7} + \cdots + 3008)^{2} \) (T^8 + 14T^7 - 40T^6 - 928T^5 - 992T^4 + 9504T^3 + 25008T^2 + 19008*T + 3008)^2
$71$	\( T^{16} + \cdots + 43207221168400 \) T^16 + 506T^14 + 105033T^12 + 11656416T^10 + 757344168T^8 + 29590683864T^6 + 680903856896T^4 + 8447505115216*T^2 + 43207221168400
$73$	\( T^{16} + \cdots + 1186670350336 \) T^16 + 714T^14 + 192609T^12 + 24648712T^10 + 1561243760T^8 + 46891614464T^6 + 631313719040T^4 + 3003152582656*T^2 + 1186670350336
$79$	\( T^{16} + \cdots + 1392777625600 \) T^16 + 496T^14 + 97416T^12 + 9711640T^10 + 523688400T^8 + 15083117168T^6 + 213611124112T^4 + 1198562169856*T^2 + 1392777625600
$83$	\( (T^{8} - 10 T^{7} + \cdots - 753728)^{2} \) (T^8 - 10T^7 - 248T^6 + 3624T^5 - 7776T^4 - 61184T^3 + 207728T^2 + 188736*T - 753728)^2
$89$	\( (T^{8} + 6 T^{7} + \cdots - 7236496)^{2} \) (T^8 + 6T^7 - 259T^6 - 1900T^5 + 16052T^4 + 155192T^3 - 81056T^2 - 3363856*T - 7236496)^2
$97$	\( T^{16} + \cdots + 9413656576 \) T^16 + 578T^14 + 105473T^12 + 8141120T^10 + 286128512T^8 + 4600359424T^6 + 28937736192T^4 + 32392773632*T^2 + 9413656576
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