LMFDB - Newform orbit 4704.2.b.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(1567,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4704.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.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:	$37.5616291108$
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} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 $ x^16 + 24x^14 + 218x^12 + 968x^10 + 2241x^8 + 2672x^6 + 1512x^4 + 320*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 672)
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 + q^{3} - \beta_{9} q^{5} + q^{9}+O(q^{10}) $ q + q^3 - b9 * q^5 + q^9	$ q + q^{3} - \beta_{9} q^{5} + q^{9} + ( - \beta_{14} + \beta_{13} - \beta_1) q^{11} + (\beta_{8} - \beta_1) q^{13} - \beta_{9} q^{15} + ( - \beta_{13} - \beta_{11}) q^{17} + (\beta_{7} - \beta_{6} + \beta_{5}) q^{19} - \beta_{15} q^{23} + (\beta_{7} - \beta_{2}) q^{25} + q^{27} + ( - \beta_{10} - \beta_{6} - \beta_{3} + 1) q^{29} + (\beta_{10} - \beta_{5} - \beta_{2}) q^{31} + ( - \beta_{14} + \beta_{13} - \beta_1) q^{33} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{37} + (\beta_{8} - \beta_1) q^{39} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_1) q^{41}+ \cdots + ( - \beta_{14} + \beta_{13} - \beta_1) q^{99}+O(q^{100}) $ q + q^3 - b9 * q^5 + q^9 + (-b14 + b13 - b1) * q^11 + (b8 - b1) * q^13 - b9 * q^15 + (-b13 - b11) * q^17 + (b7 - b6 + b5) * q^19 - b15 * q^23 + (b7 - b2) * q^25 + q^27 + (-b10 - b6 - b3 + 1) * q^29 + (b10 - b5 - b2) * q^31 + (-b14 + b13 - b1) * q^33 + (-b5 + b4 - b3) * q^37 + (b8 - b1) * q^39 + (-b15 - b14 - b12 + b8 - b1) * q^41 + (-b14 - b13 + b11 + b8 + b1) * q^43 - b9 * q^45 + (-b10 + b3 - b2 - 1) * q^47 + (-b13 - b11) * q^51 + (-b10 - 2b7 - 2b5 - b4 - b3 - b2) * q^53 + (2b5 - b4 - 1) q^55 + (b7 - b6 + b5) * q^57 + (-b7 + b6 + b5 - 1) * q^59 + (b13 - 2b12 + b8 + b1) q^61 + (b10 + b5 - b4 - b3 - b2) * q^65 + (-b15 - b13 + b12 - b11 - 2b9) q^67 - b15 * q^69 + (b15 + b13 - 2b9 - b8 + b1) q^71 + (-b15 - 2b14 + 2b13 + b12 + 2b9 - 2b1) * q^73 + (b7 - b2) * q^75 + (-b15 + b14 + b13 + b12 + b9 + b8 - b1) * q^79 + q^81 + (b10 + b7 - b2 + 1) * q^83 + (b10 - 2b7 + b6 - b5 + 2b3 + b2) * q^85 + (-b10 - b6 - b3 + 1) * q^87 + (-2b11 + 2b9 + 2b8 - 2b1) * q^89 + (b10 - b5 - b2) * q^93 + (-b15 + 2b14 - b13 - 2b11 + 2b9 + b8 + b1) q^95 + (b15 + b12 - 2b9 + b1) q^97 + (-b14 + b13 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 16 * q^9	$ 16 q + 16 q^{3} + 16 q^{9} - 8 q^{19} - 8 q^{25} + 16 q^{27} - 8 q^{31} - 8 q^{37} - 16 q^{47} - 16 q^{53} - 16 q^{55} - 8 q^{57} - 8 q^{59} - 16 q^{65} - 8 q^{75} + 16 q^{81} + 8 q^{83} + 32 q^{85} - 8 q^{93}+O(q^{100}) $ 16 * q + 16 * q^3 + 16 * q^9 - 8 * q^19 - 8 * q^25 + 16 * q^27 - 8 * q^31 - 8 * q^37 - 16 * q^47 - 16 * q^53 - 16 * q^55 - 8 * q^57 - 8 * q^59 - 16 * q^65 - 8 * q^75 + 16 * q^81 + 8 * q^83 + 32 * q^85 - 8 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 $ x^16 + 24*x^14 + 218*x^12 + 968*x^10 + 2241*x^8 + 2672*x^6 + 1512*x^4 + 320*x^2 + 16 :

$\beta_{1}$	$=$	$ ( \nu^{15} + 22\nu^{13} + 174\nu^{11} + 612\nu^{9} + 865\nu^{7} - 26\nu^{5} - 1020\nu^{3} - 488\nu ) / 64 $ (v^15 + 22v^13 + 174v^11 + 612v^9 + 865v^7 - 26v^5 - 1020v^3 - 488*v) / 64
$\beta_{2}$	$=$	$ ( \nu^{14} + 22\nu^{12} + 174\nu^{10} + 616\nu^{8} + 937\nu^{6} + 386\nu^{4} - 148\nu^{2} ) / 8 $ (v^14 + 22v^12 + 174v^10 + 616v^8 + 937v^6 + 386v^4 - 148v^2) / 8
$\beta_{3}$	$=$	$ ( \nu^{14} + 23\nu^{12} + 194\nu^{10} + 754\nu^{8} + 1349\nu^{6} + 911\nu^{4} + 68\nu^{2} - 20 ) / 8 $ (v^14 + 23v^12 + 194v^10 + 754v^8 + 1349v^6 + 911v^4 + 68v^2 - 20) / 8
$\beta_{4}$	$=$	$ ( -3\nu^{14} - 68\nu^{12} - 566\nu^{10} - 2200\nu^{8} - 4111\nu^{6} - 3380\nu^{4} - 924\nu^{2} ) / 16 $ (-3v^14 - 68v^12 - 566v^10 - 2200v^8 - 4111v^6 - 3380v^4 - 924*v^2) / 16
$\beta_{5}$	$=$	$ ( 3\nu^{14} + 70\nu^{12} + 606\nu^{10} + 2476\nu^{8} + 4935\nu^{6} + 4430\nu^{4} + 1388\nu^{2} + 56 ) / 16 $ (3v^14 + 70v^12 + 606v^10 + 2476v^8 + 4935v^6 + 4430v^4 + 1388*v^2 + 56) / 16
$\beta_{6}$	$=$	$ ( 7\nu^{14} + 160\nu^{12} + 1346\nu^{10} + 5288\nu^{8} + 9927\nu^{6} + 8008\nu^{4} + 2044\nu^{2} + 32 ) / 32 $ (7v^14 + 160v^12 + 1346v^10 + 5288v^8 + 9927v^6 + 8008v^4 + 2044*v^2 + 32) / 32
$\beta_{7}$	$=$	$ ( -3\nu^{14} - 68\nu^{12} - 562\nu^{10} - 2128\nu^{8} - 3699\nu^{6} - 2492\nu^{4} - 324\nu^{2} + 16 ) / 8 $ (-3v^14 - 68v^12 - 562v^10 - 2128v^8 - 3699v^6 - 2492v^4 - 324*v^2 + 16) / 8
$\beta_{8}$	$=$	$ ( 5\nu^{15} + 110\nu^{13} + 866\nu^{11} + 3008\nu^{9} + 4273\nu^{7} + 1034\nu^{5} - 1452\nu^{3} - 248\nu ) / 32 $ (5v^15 + 110v^13 + 866v^11 + 3008v^9 + 4273v^7 + 1034v^5 - 1452v^3 - 248v) / 32
$\beta_{9}$	$=$	$ ( -11\nu^{15} - 242\nu^{13} - 1898\nu^{11} - 6492\nu^{9} - 8715\nu^{7} - 850\nu^{5} + 4324\nu^{3} + 728\nu ) / 64 $ (-11v^15 - 242v^13 - 1898v^11 - 6492v^9 - 8715v^7 - 850v^5 + 4324v^3 + 728v) / 64
$\beta_{10}$	$=$	$ ( 17\nu^{14} + 388\nu^{12} + 3254\nu^{10} + 12720\nu^{8} + 23801\nu^{6} + 19580\nu^{4} + 5828\nu^{2} + 464 ) / 32 $ (17v^14 + 388v^12 + 3254v^10 + 12720v^8 + 23801v^6 + 19580v^4 + 5828*v^2 + 464) / 32
$\beta_{11}$	$=$	$ ( 3\nu^{15} + 74\nu^{13} + 698\nu^{11} + 3252\nu^{9} + 7963\nu^{7} + 10018\nu^{5} + 5876\nu^{3} + 1304\nu ) / 32 $ (3v^15 + 74v^13 + 698v^11 + 3252v^9 + 7963v^7 + 10018v^5 + 5876v^3 + 1304v) / 32
$\beta_{12}$	$=$	$ ( -5\nu^{15} - 118\nu^{13} - 1046\nu^{11} - 4492\nu^{9} - 9965\nu^{7} - 11358\nu^{5} - 6220\nu^{3} - 1256\nu ) / 32 $ (-5v^15 - 118v^13 - 1046v^11 - 4492v^9 - 9965v^7 - 11358v^5 - 6220v^3 - 1256v) / 32
$\beta_{13}$	$=$	$ ( -11\nu^{15} - 258\nu^{13} - 2258\nu^{11} - 9444\nu^{9} - 19843\nu^{7} - 20330\nu^{5} - 9356\nu^{3} - 1448\nu ) / 64 $ (-11v^15 - 258v^13 - 2258v^11 - 9444v^9 - 19843v^7 - 20330v^5 - 9356v^3 - 1448v) / 64
$\beta_{14}$	$=$	$ ( 11\nu^{15} + 254\nu^{13} + 2170\nu^{11} + 8740\nu^{9} + 17227\nu^{7} + 15646\nu^{5} + 5644\nu^{3} + 568\nu ) / 32 $ (11v^15 + 254v^13 + 2170v^11 + 8740v^9 + 17227v^7 + 15646v^5 + 5644v^3 + 568v) / 32
$\beta_{15}$	$=$	$ ( -23\nu^{15} - 530\nu^{13} - 4506\nu^{11} - 17940\nu^{9} - 34367\nu^{7} - 28906\nu^{5} - 8204\nu^{3} - 328\nu ) / 64 $ (-23v^15 - 530v^13 - 4506v^11 - 17940v^9 - 34367v^7 - 28906v^5 - 8204v^3 - 328v) / 64

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

$\nu$	$=$	$ ( -\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} ) / 4 $ (-b15 - b14 + b13 - b12) / 4
$\nu^{2}$	$=$	$ ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 6 ) / 2 $ (b5 + b4 - b3 + b2 - 6) / 2
$\nu^{3}$	$=$	$ ( 7\beta_{15} + 3\beta_{14} - 9\beta_{13} + 5\beta_{12} + 2\beta_{11} - 4\beta_{9} - 2\beta_{8} + 10\beta_1 ) / 4 $ (7b15 + 3b14 - 9b13 + 5b12 + 2b11 - 4b9 - 2b8 + 10b1) / 4
$\nu^{4}$	$=$	$ \beta_{10} + \beta_{7} - \beta_{6} - 5\beta_{5} - 4\beta_{4} + 6\beta_{3} - 4\beta_{2} + 17 $ b10 + b7 - b6 - 5b5 - 4b4 + 6b3 - 4b2 + 17
$\nu^{5}$	$=$	$ ( -53\beta_{15} - 5\beta_{14} + 89\beta_{13} - 31\beta_{12} - 14\beta_{11} + 40\beta_{9} + 20\beta_{8} - 116\beta_1 ) / 4 $ (-53b15 - 5b14 + 89b13 - 31b12 - 14b11 + 40b9 + 20b8 - 116b1) / 4
$\nu^{6}$	$=$	$ ( -24\beta_{10} - 26\beta_{7} + 32\beta_{6} + 89\beta_{5} + 73\beta_{4} - 121\beta_{3} + 65\beta_{2} - 246 ) / 2 $ (-24b10 - 26b7 + 32b6 + 89b5 + 73b4 - 121b3 + 65*b2 - 246) / 2
$\nu^{7}$	$=$	$ ( 433 \beta_{15} - 59 \beta_{14} - 863 \beta_{13} + 223 \beta_{12} + 98 \beta_{11} - 364 \beta_{9} + \cdots + 1142 \beta_1 ) / 4 $ (433b15 - 59b14 - 863b13 + 223b12 + 98b11 - 364b9 - 174b8 + 1142b1) / 4
$\nu^{8}$	$=$	$ 121\beta_{10} + 135\beta_{7} - 185\beta_{6} - 399\beta_{5} - 342\beta_{4} + 580\beta_{3} - 280\beta_{2} + 1007 $ 121b10 + 135b7 - 185b6 - 399b5 - 342b4 + 580b3 - 280*b2 + 1007
$\nu^{9}$	$=$	$ ( - 3735 \beta_{15} + 1049 \beta_{14} + 8211 \beta_{13} - 1789 \beta_{12} - 754 \beta_{11} + \cdots - 10844 \beta_1 ) / 4 $ (-3735b15 + 1049b14 + 8211b13 - 1789b12 - 754b11 + 3304b9 + 1516b8 - 10844b1) / 4
$\nu^{10}$	$=$	$ ( - 2328 \beta_{10} - 2622 \beta_{7} + 3808 \beta_{6} + 7267 \beta_{5} + 6411 \beta_{4} - 10931 \beta_{3} + \cdots - 17570 ) / 2 $ (-2328b10 - 2622b7 + 3808b6 + 7267b5 + 6411b4 - 10931b3 + 5011*b2 - 17570) / 2
$\nu^{11}$	$=$	$ ( 33339 \beta_{15} - 12137 \beta_{14} - 77285 \beta_{13} + 15349 \beta_{12} + 6278 \beta_{11} + \cdots + 101810 \beta_1 ) / 4 $ (33339b15 - 12137b14 - 77285b13 + 15349b12 + 6278b11 - 30228b9 - 13482b8 + 101810b1) / 4
$\nu^{12}$	$=$	$ 11001 \beta_{10} + 12421 \beta_{7} - 18617 \beta_{6} - 33425 \beta_{5} - 29960 \beta_{4} + 51162 \beta_{3} + \cdots + 79153 $ 11001b10 + 12421b7 - 18617b6 - 33425b5 - 29960b4 + 51162b3 - 22876*b2 + 79153
$\nu^{13}$	$=$	$ ( - 303493 \beta_{15} + 124235 \beta_{14} + 723385 \beta_{13} - 136775 \beta_{12} - 54966 \beta_{11} + \cdots - 951588 \beta_1 ) / 4 $ (-303493b15 + 124235b14 + 723385b13 - 136775b12 - 54966b11 + 278440b9 + 122020b8 - 951588b1) / 4
$\nu^{14}$	$=$	$ ( - 206328 \beta_{10} - 233026 \beta_{7} + 355264 \beta_{6} + 618425 \beta_{5} + 558905 \beta_{4} + \cdots - 1449686 ) / 2 $ (-206328b10 - 233026b7 + 355264b6 + 618425b5 + 558905b4 - 955097b3 + 421937*b2 - 1449686) / 2
$\nu^{15}$	$=$	$ ( 2792409 \beta_{15} - 1209843 \beta_{14} - 6751895 \beta_{13} + 1244103 \beta_{12} + 495234 \beta_{11} + \cdots + 8876134 \beta_1 ) / 4 $ (2792409b15 - 1209843b14 - 6751895b13 + 1244103b12 + 495234b11 - 2576236b9 - 1117374b8 + 8876134b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times$.

$n$	$1471$	$1765$	$3137$	$4609$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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} $

1567.1

	−	0.549125i
	−	1.06676i
	−	1.60698i
		0.266470i
		1.11907i
		2.19832i
	−	3.05093i
	−	2.12462i
		2.12462i
		3.05093i
	−	2.19832i
	−	1.11907i
	−	0.266470i
		1.60698i
		1.06676i
		0.549125i

1.00000

−

3.97528i

1.00000

1567.2

1.00000

−

3.56107i

1.00000

1567.3

1.00000

−

2.70012i

1.00000

1567.4

1.00000

−

1.74722i

1.00000

1567.5

1.00000

−

1.65416i

1.00000

1567.6

1.00000

−

1.33300i

1.00000

1567.7

1.00000

−

0.760056i

1.00000

1567.8

1.00000

−

0.285903i

1.00000

1567.9

1.00000

0.285903i

1.00000

1567.10

1.00000

0.760056i

1.00000

1567.11

1.00000

1.33300i

1.00000

1567.12

1.00000

1.65416i

1.00000

1567.13

1.00000

1.74722i

1.00000

1567.14

1.00000

2.70012i

1.00000

1567.15

1.00000

3.56107i

1.00000

1567.16

1.00000

3.97528i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4704.2.b.e		16
4.b	odd	2	1		4704.2.b.d		16
7.b	odd	2	1		4704.2.b.d		16
7.c	even	3	1		672.2.bl.a	✓	16
7.d	odd	6	1		672.2.bl.b	yes	16
21.g	even	6	1		2016.2.cs.c		16
21.h	odd	6	1		2016.2.cs.a		16
28.d	even	2	1	inner	4704.2.b.e		16
28.f	even	6	1		672.2.bl.a	✓	16
28.g	odd	6	1		672.2.bl.b	yes	16
56.j	odd	6	1		1344.2.bl.k		16
56.k	odd	6	1		1344.2.bl.k		16
56.m	even	6	1		1344.2.bl.l		16
56.p	even	6	1		1344.2.bl.l		16
84.j	odd	6	1		2016.2.cs.a		16
84.n	even	6	1		2016.2.cs.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.2.bl.a	✓	16	7.c	even	3	1
672.2.bl.a	✓	16	28.f	even	6	1
672.2.bl.b	yes	16	7.d	odd	6	1
672.2.bl.b	yes	16	28.g	odd	6	1
1344.2.bl.k		16	56.j	odd	6	1
1344.2.bl.k		16	56.k	odd	6	1
1344.2.bl.l		16	56.m	even	6	1
1344.2.bl.l		16	56.p	even	6	1
2016.2.cs.a		16	21.h	odd	6	1
2016.2.cs.a		16	84.j	odd	6	1
2016.2.cs.c		16	21.g	even	6	1
2016.2.cs.c		16	84.n	even	6	1
4704.2.b.d		16	4.b	odd	2	1
4704.2.b.d		16	7.b	odd	2	1
4704.2.b.e		16	1.a	even	1	1	trivial
4704.2.b.e		16	28.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4704, [\chi])$:

$ T_{5}^{16} + 44T_{5}^{14} + 726T_{5}^{12} + 5692T_{5}^{10} + 22673T_{5}^{8} + 46192T_{5}^{6} + 44544T_{5}^{4} + 15872T_{5}^{2} + 1024 $

$ T_{19}^{8} + 4T_{19}^{7} - 90T_{19}^{6} - 260T_{19}^{5} + 1977T_{19}^{4} + 3400T_{19}^{3} - 10360T_{19}^{2} - 2400T_{19} + 5392 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T - 1)^{16} $ (T - 1)^16
$5$	$ T^{16} + 44 T^{14} + \cdots + 1024 $ T^16 + 44T^14 + 726T^12 + 5692T^10 + 22673T^8 + 46192T^6 + 44544T^4 + 15872*T^2 + 1024
$7$	$ T^{16} $ T^16
$11$	$ T^{16} + 76 T^{14} + \cdots + 135424 $ T^16 + 76T^14 + 2214T^12 + 31804T^10 + 238017T^8 + 890032T^6 + 1432928T^4 + 833280*T^2 + 135424
$13$	$ T^{16} + 124 T^{14} + \cdots + 16384 $ T^16 + 124T^14 + 5638T^12 + 114460T^10 + 977089T^8 + 2357920T^6 + 1275904T^4 + 249856*T^2 + 16384
$17$	$ T^{16} + \cdots + 3743481856 $ T^16 + 176T^14 + 12672T^12 + 485632T^10 + 10718720T^8 + 136744960T^6 + 950894592T^4 + 3121283072*T^2 + 3743481856
$19$	$ (T^{8} + 4 T^{7} + \cdots + 5392)^{2} $ (T^8 + 4T^7 - 90T^6 - 260T^5 + 1977T^4 + 3400T^3 - 10360T^2 - 2400*T + 5392)^2
$23$	$ T^{16} + 192 T^{14} + \cdots + 4194304 $ T^16 + 192T^14 + 13856T^12 + 464384T^10 + 7123200T^8 + 40804352T^6 + 64487424T^4 + 30408704*T^2 + 4194304
$29$	$ (T^{8} - 106 T^{6} + \cdots + 4096)^{2} $ (T^8 - 106T^6 + 96T^5 + 3425T^4 - 5280T^3 - 28928T^2 + 43008T + 4096)^2
$31$	$ (T^{8} + 4 T^{7} + \cdots - 143303)^{2} $ (T^8 + 4T^7 - 140T^6 - 148T^5 + 6414T^4 - 8708T^3 - 77948T^2 + 223700*T - 143303)^2
$37$	$ (T^{8} + 4 T^{7} + \cdots - 368)^{2} $ (T^8 + 4T^7 - 122T^6 - 316T^5 + 3369T^4 + 8848T^3 - 9176T^2 - 5248*T - 368)^2
$41$	$ T^{16} + \cdots + 57538576384 $ T^16 + 320T^14 + 38592T^12 + 2327552T^10 + 76199424T^8 + 1350287360T^6 + 12027478016T^4 + 45745963008*T^2 + 57538576384
$43$	$ T^{16} + \cdots + 17573803020544 $ T^16 + 460T^14 + 86918T^12 + 8782204T^10 + 518098081T^8 + 18236031536T^6 + 372907834208T^4 + 4034711396096*T^2 + 17573803020544
$47$	$ (T^{8} + 8 T^{7} + \cdots - 198656)^{2} $ (T^8 + 8T^7 - 168T^6 - 1568T^5 + 3104T^4 + 46336T^3 + 56832T^2 - 126976*T - 198656)^2
$53$	$ (T^{8} + 8 T^{7} + \cdots + 1276816)^{2} $ (T^8 + 8T^7 - 282T^6 - 1808T^5 + 24881T^4 + 84520T^3 - 826032T^2 + 279968*T + 1276816)^2
$59$	$ (T^{8} + 4 T^{7} + \cdots - 218864)^{2} $ (T^8 + 4T^7 - 178T^6 - 692T^5 + 8857T^4 + 32888T^3 - 98320T^2 - 356128*T - 218864)^2
$61$	$ T^{16} + \cdots + 644513529856 $ T^16 + 512T^14 + 103040T^12 + 10661888T^10 + 611848192T^8 + 19285147648T^6 + 303313190912T^4 + 1798718881792*T^2 + 644513529856
$67$	$ T^{16} + 508 T^{14} + \cdots + 21827584 $ T^16 + 508T^14 + 88998T^12 + 6697436T^10 + 208201697T^8 + 1861235744T^6 + 6460800384T^4 + 7763200000*T^2 + 21827584
$71$	$ T^{16} + \cdots + 2218786816 $ T^16 + 432T^14 + 64640T^12 + 4232704T^10 + 129537024T^8 + 1756168192T^6 + 8302755840T^4 + 12492734464*T^2 + 2218786816
$73$	$ T^{16} + \cdots + 362073308594176 $ T^16 + 844T^14 + 282166T^12 + 47810924T^10 + 4384833873T^8 + 218045404864T^6 + 5745889535488T^4 + 73848185077760*T^2 + 362073308594176
$79$	$ T^{16} + \cdots + 157558981969 $ T^16 + 504T^14 + 99500T^12 + 9933032T^10 + 541144950T^8 + 15913509512T^6 + 225426174732T^4 + 1052310328856*T^2 + 157558981969
$83$	$ (T^{8} - 4 T^{7} + \cdots - 314144)^{2} $ (T^8 - 4T^7 - 154T^6 + 956T^5 + 3561T^4 - 32016T^3 + 19800T^2 + 197984*T - 314144)^2
$89$	$ T^{16} + \cdots + 16211639271424 $ T^16 + 880T^14 + 314976T^12 + 59073280T^10 + 6236070144T^8 + 369420845056T^6 + 11389044654080T^4 + 141914271645696*T^2 + 16211639271424
$97$	$ T^{16} + \cdots + 135706771456 $ T^16 + 492T^14 + 93974T^12 + 8790668T^10 + 417570801T^8 + 9376033664T^6 + 81916494336T^4 + 206125629440*T^2 + 135706771456
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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:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + q^{3} - \beta_{9} q^{5} + q^{9}+O(q^{10}) \) q + q^3 - b9 * q^5 + q^9	\( q + q^{3} - \beta_{9} q^{5} + q^{9} + ( - \beta_{14} + \beta_{13} - \beta_1) q^{11} + (\beta_{8} - \beta_1) q^{13} - \beta_{9} q^{15} + ( - \beta_{13} - \beta_{11}) q^{17} + (\beta_{7} - \beta_{6} + \beta_{5}) q^{19} - \beta_{15} q^{23} + (\beta_{7} - \beta_{2}) q^{25} + q^{27} + ( - \beta_{10} - \beta_{6} - \beta_{3} + 1) q^{29} + (\beta_{10} - \beta_{5} - \beta_{2}) q^{31} + ( - \beta_{14} + \beta_{13} - \beta_1) q^{33} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{37} + (\beta_{8} - \beta_1) q^{39} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_1) q^{41}+ \cdots + ( - \beta_{14} + \beta_{13} - \beta_1) q^{99}+O(q^{100}) \) q + q^3 - b9 * q^5 + q^9 + (-b14 + b13 - b1) * q^11 + (b8 - b1) * q^13 - b9 * q^15 + (-b13 - b11) * q^17 + (b7 - b6 + b5) * q^19 - b15 * q^23 + (b7 - b2) * q^25 + q^27 + (-b10 - b6 - b3 + 1) * q^29 + (b10 - b5 - b2) * q^31 + (-b14 + b13 - b1) * q^33 + (-b5 + b4 - b3) * q^37 + (b8 - b1) * q^39 + (-b15 - b14 - b12 + b8 - b1) * q^41 + (-b14 - b13 + b11 + b8 + b1) * q^43 - b9 * q^45 + (-b10 + b3 - b2 - 1) * q^47 + (-b13 - b11) * q^51 + (-b10 - 2b7 - 2b5 - b4 - b3 - b2) * q^53 + (2b5 - b4 - 1) q^55 + (b7 - b6 + b5) * q^57 + (-b7 + b6 + b5 - 1) * q^59 + (b13 - 2b12 + b8 + b1) q^61 + (b10 + b5 - b4 - b3 - b2) * q^65 + (-b15 - b13 + b12 - b11 - 2b9) q^67 - b15 * q^69 + (b15 + b13 - 2b9 - b8 + b1) q^71 + (-b15 - 2b14 + 2b13 + b12 + 2b9 - 2b1) * q^73 + (b7 - b2) * q^75 + (-b15 + b14 + b13 + b12 + b9 + b8 - b1) * q^79 + q^81 + (b10 + b7 - b2 + 1) * q^83 + (b10 - 2b7 + b6 - b5 + 2b3 + b2) * q^85 + (-b10 - b6 - b3 + 1) * q^87 + (-2b11 + 2b9 + 2b8 - 2b1) * q^89 + (b10 - b5 - b2) * q^93 + (-b15 + 2b14 - b13 - 2b11 + 2b9 + b8 + b1) q^95 + (b15 + b12 - 2b9 + b1) q^97 + (-b14 + b13 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 16 * q^9	\( 16 q + 16 q^{3} + 16 q^{9} - 8 q^{19} - 8 q^{25} + 16 q^{27} - 8 q^{31} - 8 q^{37} - 16 q^{47} - 16 q^{53} - 16 q^{55} - 8 q^{57} - 8 q^{59} - 16 q^{65} - 8 q^{75} + 16 q^{81} + 8 q^{83} + 32 q^{85} - 8 q^{93}+O(q^{100}) \) 16 * q + 16 * q^3 + 16 * q^9 - 8 * q^19 - 8 * q^25 + 16 * q^27 - 8 * q^31 - 8 * q^37 - 16 * q^47 - 16 * q^53 - 16 * q^55 - 8 * q^57 - 8 * q^59 - 16 * q^65 - 8 * q^75 + 16 * q^81 + 8 * q^83 + 32 * q^85 - 8 * 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	4704.2.b.e

Level	$4704$
Weight	$2$
Character orbit	4704.b
Analytic conductor	$37.562$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(37.5616291108\)
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:	\( x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 \) x^16 + 24x^14 + 218x^12 + 968x^10 + 2241x^8 + 2672x^6 + 1512x^4 + 320*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index:	\( 2^{22} \)
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{15} + 22\nu^{13} + 174\nu^{11} + 612\nu^{9} + 865\nu^{7} - 26\nu^{5} - 1020\nu^{3} - 488\nu ) / 64 \) (v^15 + 22v^13 + 174v^11 + 612v^9 + 865v^7 - 26v^5 - 1020v^3 - 488*v) / 64
\(\beta_{2}\)	\(=\)	\( ( \nu^{14} + 22\nu^{12} + 174\nu^{10} + 616\nu^{8} + 937\nu^{6} + 386\nu^{4} - 148\nu^{2} ) / 8 \) (v^14 + 22v^12 + 174v^10 + 616v^8 + 937v^6 + 386v^4 - 148v^2) / 8
\(\beta_{3}\)	\(=\)	\( ( \nu^{14} + 23\nu^{12} + 194\nu^{10} + 754\nu^{8} + 1349\nu^{6} + 911\nu^{4} + 68\nu^{2} - 20 ) / 8 \) (v^14 + 23v^12 + 194v^10 + 754v^8 + 1349v^6 + 911v^4 + 68v^2 - 20) / 8
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{14} - 68\nu^{12} - 566\nu^{10} - 2200\nu^{8} - 4111\nu^{6} - 3380\nu^{4} - 924\nu^{2} ) / 16 \) (-3v^14 - 68v^12 - 566v^10 - 2200v^8 - 4111v^6 - 3380v^4 - 924*v^2) / 16
\(\beta_{5}\)	\(=\)	\( ( 3\nu^{14} + 70\nu^{12} + 606\nu^{10} + 2476\nu^{8} + 4935\nu^{6} + 4430\nu^{4} + 1388\nu^{2} + 56 ) / 16 \) (3v^14 + 70v^12 + 606v^10 + 2476v^8 + 4935v^6 + 4430v^4 + 1388*v^2 + 56) / 16
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{14} + 160\nu^{12} + 1346\nu^{10} + 5288\nu^{8} + 9927\nu^{6} + 8008\nu^{4} + 2044\nu^{2} + 32 ) / 32 \) (7v^14 + 160v^12 + 1346v^10 + 5288v^8 + 9927v^6 + 8008v^4 + 2044*v^2 + 32) / 32
\(\beta_{7}\)	\(=\)	\( ( -3\nu^{14} - 68\nu^{12} - 562\nu^{10} - 2128\nu^{8} - 3699\nu^{6} - 2492\nu^{4} - 324\nu^{2} + 16 ) / 8 \) (-3v^14 - 68v^12 - 562v^10 - 2128v^8 - 3699v^6 - 2492v^4 - 324*v^2 + 16) / 8
\(\beta_{8}\)	\(=\)	\( ( 5\nu^{15} + 110\nu^{13} + 866\nu^{11} + 3008\nu^{9} + 4273\nu^{7} + 1034\nu^{5} - 1452\nu^{3} - 248\nu ) / 32 \) (5v^15 + 110v^13 + 866v^11 + 3008v^9 + 4273v^7 + 1034v^5 - 1452v^3 - 248v) / 32
\(\beta_{9}\)	\(=\)	\( ( -11\nu^{15} - 242\nu^{13} - 1898\nu^{11} - 6492\nu^{9} - 8715\nu^{7} - 850\nu^{5} + 4324\nu^{3} + 728\nu ) / 64 \) (-11v^15 - 242v^13 - 1898v^11 - 6492v^9 - 8715v^7 - 850v^5 + 4324v^3 + 728v) / 64
\(\beta_{10}\)	\(=\)	\( ( 17\nu^{14} + 388\nu^{12} + 3254\nu^{10} + 12720\nu^{8} + 23801\nu^{6} + 19580\nu^{4} + 5828\nu^{2} + 464 ) / 32 \) (17v^14 + 388v^12 + 3254v^10 + 12720v^8 + 23801v^6 + 19580v^4 + 5828*v^2 + 464) / 32
\(\beta_{11}\)	\(=\)	\( ( 3\nu^{15} + 74\nu^{13} + 698\nu^{11} + 3252\nu^{9} + 7963\nu^{7} + 10018\nu^{5} + 5876\nu^{3} + 1304\nu ) / 32 \) (3v^15 + 74v^13 + 698v^11 + 3252v^9 + 7963v^7 + 10018v^5 + 5876v^3 + 1304v) / 32
\(\beta_{12}\)	\(=\)	\( ( -5\nu^{15} - 118\nu^{13} - 1046\nu^{11} - 4492\nu^{9} - 9965\nu^{7} - 11358\nu^{5} - 6220\nu^{3} - 1256\nu ) / 32 \) (-5v^15 - 118v^13 - 1046v^11 - 4492v^9 - 9965v^7 - 11358v^5 - 6220v^3 - 1256v) / 32
\(\beta_{13}\)	\(=\)	\( ( -11\nu^{15} - 258\nu^{13} - 2258\nu^{11} - 9444\nu^{9} - 19843\nu^{7} - 20330\nu^{5} - 9356\nu^{3} - 1448\nu ) / 64 \) (-11v^15 - 258v^13 - 2258v^11 - 9444v^9 - 19843v^7 - 20330v^5 - 9356v^3 - 1448v) / 64
\(\beta_{14}\)	\(=\)	\( ( 11\nu^{15} + 254\nu^{13} + 2170\nu^{11} + 8740\nu^{9} + 17227\nu^{7} + 15646\nu^{5} + 5644\nu^{3} + 568\nu ) / 32 \) (11v^15 + 254v^13 + 2170v^11 + 8740v^9 + 17227v^7 + 15646v^5 + 5644v^3 + 568v) / 32
\(\beta_{15}\)	\(=\)	\( ( -23\nu^{15} - 530\nu^{13} - 4506\nu^{11} - 17940\nu^{9} - 34367\nu^{7} - 28906\nu^{5} - 8204\nu^{3} - 328\nu ) / 64 \) (-23v^15 - 530v^13 - 4506v^11 - 17940v^9 - 34367v^7 - 28906v^5 - 8204v^3 - 328v) / 64

\(\nu\)	\(=\)	\( ( -\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} ) / 4 \) (-b15 - b14 + b13 - b12) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 6 ) / 2 \) (b5 + b4 - b3 + b2 - 6) / 2
\(\nu^{3}\)	\(=\)	\( ( 7\beta_{15} + 3\beta_{14} - 9\beta_{13} + 5\beta_{12} + 2\beta_{11} - 4\beta_{9} - 2\beta_{8} + 10\beta_1 ) / 4 \) (7b15 + 3b14 - 9b13 + 5b12 + 2b11 - 4b9 - 2b8 + 10b1) / 4
\(\nu^{4}\)	\(=\)	\( \beta_{10} + \beta_{7} - \beta_{6} - 5\beta_{5} - 4\beta_{4} + 6\beta_{3} - 4\beta_{2} + 17 \) b10 + b7 - b6 - 5b5 - 4b4 + 6b3 - 4b2 + 17
\(\nu^{5}\)	\(=\)	\( ( -53\beta_{15} - 5\beta_{14} + 89\beta_{13} - 31\beta_{12} - 14\beta_{11} + 40\beta_{9} + 20\beta_{8} - 116\beta_1 ) / 4 \) (-53b15 - 5b14 + 89b13 - 31b12 - 14b11 + 40b9 + 20b8 - 116b1) / 4
\(\nu^{6}\)	\(=\)	\( ( -24\beta_{10} - 26\beta_{7} + 32\beta_{6} + 89\beta_{5} + 73\beta_{4} - 121\beta_{3} + 65\beta_{2} - 246 ) / 2 \) (-24b10 - 26b7 + 32b6 + 89b5 + 73b4 - 121b3 + 65*b2 - 246) / 2
\(\nu^{7}\)	\(=\)	\( ( 433 \beta_{15} - 59 \beta_{14} - 863 \beta_{13} + 223 \beta_{12} + 98 \beta_{11} - 364 \beta_{9} + \cdots + 1142 \beta_1 ) / 4 \) (433b15 - 59b14 - 863b13 + 223b12 + 98b11 - 364b9 - 174b8 + 1142b1) / 4
\(\nu^{8}\)	\(=\)	\( 121\beta_{10} + 135\beta_{7} - 185\beta_{6} - 399\beta_{5} - 342\beta_{4} + 580\beta_{3} - 280\beta_{2} + 1007 \) 121b10 + 135b7 - 185b6 - 399b5 - 342b4 + 580b3 - 280*b2 + 1007
\(\nu^{9}\)	\(=\)	\( ( - 3735 \beta_{15} + 1049 \beta_{14} + 8211 \beta_{13} - 1789 \beta_{12} - 754 \beta_{11} + \cdots - 10844 \beta_1 ) / 4 \) (-3735b15 + 1049b14 + 8211b13 - 1789b12 - 754b11 + 3304b9 + 1516b8 - 10844b1) / 4
\(\nu^{10}\)	\(=\)	\( ( - 2328 \beta_{10} - 2622 \beta_{7} + 3808 \beta_{6} + 7267 \beta_{5} + 6411 \beta_{4} - 10931 \beta_{3} + \cdots - 17570 ) / 2 \) (-2328b10 - 2622b7 + 3808b6 + 7267b5 + 6411b4 - 10931b3 + 5011*b2 - 17570) / 2
\(\nu^{11}\)	\(=\)	\( ( 33339 \beta_{15} - 12137 \beta_{14} - 77285 \beta_{13} + 15349 \beta_{12} + 6278 \beta_{11} + \cdots + 101810 \beta_1 ) / 4 \) (33339b15 - 12137b14 - 77285b13 + 15349b12 + 6278b11 - 30228b9 - 13482b8 + 101810b1) / 4
\(\nu^{12}\)	\(=\)	\( 11001 \beta_{10} + 12421 \beta_{7} - 18617 \beta_{6} - 33425 \beta_{5} - 29960 \beta_{4} + 51162 \beta_{3} + \cdots + 79153 \) 11001b10 + 12421b7 - 18617b6 - 33425b5 - 29960b4 + 51162b3 - 22876*b2 + 79153
\(\nu^{13}\)	\(=\)	\( ( - 303493 \beta_{15} + 124235 \beta_{14} + 723385 \beta_{13} - 136775 \beta_{12} - 54966 \beta_{11} + \cdots - 951588 \beta_1 ) / 4 \) (-303493b15 + 124235b14 + 723385b13 - 136775b12 - 54966b11 + 278440b9 + 122020b8 - 951588b1) / 4
\(\nu^{14}\)	\(=\)	\( ( - 206328 \beta_{10} - 233026 \beta_{7} + 355264 \beta_{6} + 618425 \beta_{5} + 558905 \beta_{4} + \cdots - 1449686 ) / 2 \) (-206328b10 - 233026b7 + 355264b6 + 618425b5 + 558905b4 - 955097b3 + 421937*b2 - 1449686) / 2
\(\nu^{15}\)	\(=\)	\( ( 2792409 \beta_{15} - 1209843 \beta_{14} - 6751895 \beta_{13} + 1244103 \beta_{12} + 495234 \beta_{11} + \cdots + 8876134 \beta_1 ) / 4 \) (2792409b15 - 1209843b14 - 6751895b13 + 1244103b12 + 495234b11 - 2576236b9 - 1117374b8 + 8876134b1) / 4

\(n\)	\(1471\)	\(1765\)	\(3137\)	\(4609\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1567.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T - 1)^{16} \) (T - 1)^16
$5$	\( T^{16} + 44 T^{14} + \cdots + 1024 \) T^16 + 44T^14 + 726T^12 + 5692T^10 + 22673T^8 + 46192T^6 + 44544T^4 + 15872*T^2 + 1024
$7$	\( T^{16} \) T^16
$11$	\( T^{16} + 76 T^{14} + \cdots + 135424 \) T^16 + 76T^14 + 2214T^12 + 31804T^10 + 238017T^8 + 890032T^6 + 1432928T^4 + 833280*T^2 + 135424
$13$	\( T^{16} + 124 T^{14} + \cdots + 16384 \) T^16 + 124T^14 + 5638T^12 + 114460T^10 + 977089T^8 + 2357920T^6 + 1275904T^4 + 249856*T^2 + 16384
$17$	\( T^{16} + \cdots + 3743481856 \) T^16 + 176T^14 + 12672T^12 + 485632T^10 + 10718720T^8 + 136744960T^6 + 950894592T^4 + 3121283072*T^2 + 3743481856
$19$	\( (T^{8} + 4 T^{7} + \cdots + 5392)^{2} \) (T^8 + 4T^7 - 90T^6 - 260T^5 + 1977T^4 + 3400T^3 - 10360T^2 - 2400*T + 5392)^2
$23$	\( T^{16} + 192 T^{14} + \cdots + 4194304 \) T^16 + 192T^14 + 13856T^12 + 464384T^10 + 7123200T^8 + 40804352T^6 + 64487424T^4 + 30408704*T^2 + 4194304
$29$	\( (T^{8} - 106 T^{6} + \cdots + 4096)^{2} \) (T^8 - 106T^6 + 96T^5 + 3425T^4 - 5280T^3 - 28928T^2 + 43008T + 4096)^2
$31$	\( (T^{8} + 4 T^{7} + \cdots - 143303)^{2} \) (T^8 + 4T^7 - 140T^6 - 148T^5 + 6414T^4 - 8708T^3 - 77948T^2 + 223700*T - 143303)^2
$37$	\( (T^{8} + 4 T^{7} + \cdots - 368)^{2} \) (T^8 + 4T^7 - 122T^6 - 316T^5 + 3369T^4 + 8848T^3 - 9176T^2 - 5248*T - 368)^2
$41$	\( T^{16} + \cdots + 57538576384 \) T^16 + 320T^14 + 38592T^12 + 2327552T^10 + 76199424T^8 + 1350287360T^6 + 12027478016T^4 + 45745963008*T^2 + 57538576384
$43$	\( T^{16} + \cdots + 17573803020544 \) T^16 + 460T^14 + 86918T^12 + 8782204T^10 + 518098081T^8 + 18236031536T^6 + 372907834208T^4 + 4034711396096*T^2 + 17573803020544
$47$	\( (T^{8} + 8 T^{7} + \cdots - 198656)^{2} \) (T^8 + 8T^7 - 168T^6 - 1568T^5 + 3104T^4 + 46336T^3 + 56832T^2 - 126976*T - 198656)^2
$53$	\( (T^{8} + 8 T^{7} + \cdots + 1276816)^{2} \) (T^8 + 8T^7 - 282T^6 - 1808T^5 + 24881T^4 + 84520T^3 - 826032T^2 + 279968*T + 1276816)^2
$59$	\( (T^{8} + 4 T^{7} + \cdots - 218864)^{2} \) (T^8 + 4T^7 - 178T^6 - 692T^5 + 8857T^4 + 32888T^3 - 98320T^2 - 356128*T - 218864)^2
$61$	\( T^{16} + \cdots + 644513529856 \) T^16 + 512T^14 + 103040T^12 + 10661888T^10 + 611848192T^8 + 19285147648T^6 + 303313190912T^4 + 1798718881792*T^2 + 644513529856
$67$	\( T^{16} + 508 T^{14} + \cdots + 21827584 \) T^16 + 508T^14 + 88998T^12 + 6697436T^10 + 208201697T^8 + 1861235744T^6 + 6460800384T^4 + 7763200000*T^2 + 21827584
$71$	\( T^{16} + \cdots + 2218786816 \) T^16 + 432T^14 + 64640T^12 + 4232704T^10 + 129537024T^8 + 1756168192T^6 + 8302755840T^4 + 12492734464*T^2 + 2218786816
$73$	\( T^{16} + \cdots + 362073308594176 \) T^16 + 844T^14 + 282166T^12 + 47810924T^10 + 4384833873T^8 + 218045404864T^6 + 5745889535488T^4 + 73848185077760*T^2 + 362073308594176
$79$	\( T^{16} + \cdots + 157558981969 \) T^16 + 504T^14 + 99500T^12 + 9933032T^10 + 541144950T^8 + 15913509512T^6 + 225426174732T^4 + 1052310328856*T^2 + 157558981969
$83$	\( (T^{8} - 4 T^{7} + \cdots - 314144)^{2} \) (T^8 - 4T^7 - 154T^6 + 956T^5 + 3561T^4 - 32016T^3 + 19800T^2 + 197984*T - 314144)^2
$89$	\( T^{16} + \cdots + 16211639271424 \) T^16 + 880T^14 + 314976T^12 + 59073280T^10 + 6236070144T^8 + 369420845056T^6 + 11389044654080T^4 + 141914271645696*T^2 + 16211639271424
$97$	\( T^{16} + \cdots + 135706771456 \) T^16 + 492T^14 + 93974T^12 + 8790668T^10 + 417570801T^8 + 9376033664T^6 + 81916494336T^4 + 206125629440*T^2 + 135706771456
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