LMFDB - Newform orbit 1728.2.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(287,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

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

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

chi := DirichletCharacter("1728.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.p (of order $6$, degree $2$, not 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 $ x^16 + 11x^14 + 85x^12 + 332x^10 + 940x^8 + 1064x^6 + 880x^4 + 128*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
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_{5} + \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{10}) q^{7}+O(q^{10}) $ q + (b5 + b3 + b2) * q^5 + (-b15 + b14 + b12 + b10) * q^7	$ q + (\beta_{5} + \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{10}) q^{7}+ \cdots + (\beta_{8} - 2 \beta_{6} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) $ q + (b5 + b3 + b2) * q^5 + (-b15 + b14 + b12 + b10) * q^7 + (b15 - b14 + b13 - b12 - b11 - b10 + b9 - b7) * q^11 + (-b6 - 2b1) q^13 + (-2b6 - 2b3 - b1 - 1) * q^17 + (2b14 + b10) q^19 + (-b15 + 2b12 - b11 - 2b7) * q^23 + (b8 + b6 - b5 - 2b3 + b1 - 2) q^25 + (-b8 - 2b3 - 2) q^29 + (-b15 - 2b14 - 3b11) * q^31 + (b12 + b10 - b9 - 2b7) q^35 + (-2b6 - 2b5 - 2b3 - b2 - b1 - 1) q^37 + (b8 + b5 + b4 - b2) * q^41 + (b15 + b12) * q^43 + (b15 + b14 - b13 + b12 - b10 + b9) * q^47 + (b8 - 2b6 + 2b5 - b4 - b3 + 2b2) q^49 + (b4 - b2 + 3) * q^53 + (b12 - 2b10 - b9 + 3b7) * q^55 + (b15 - 3b13 + 2b11) * q^59 + (-b8 + 2b4 + 2b3 - 2) * q^61 + (-b8 - 2b6 + 2b4 + 2b3 + 2b1 - 2) * q^65 + (-b15 - b14 - b13 + 2b12 - b11 - 2b10 - 2b9 - 2b7) * q^67 + (-2b14 + 2b13 - 6b11 - b10 + b9 - 3b7) * q^71 + (b4 - 2b2 + 4b1 + 3) * q^73 + (-b8 + 3b6 - b5 + b4 + 9b3 - b2) * q^77 + (-3b15 + 2b13 + 3b12 - 3b11 + 2b9 - 3b7) * q^79 + (b15 + b14 + 2b13 - b12 + 2b11 + b10 + 2b9 + 2b7) * q^83 + (b8 + 2b6 + b5 + b4 - b3 - b2 + 4b1 - 2) * q^85 + (-2b6 - 2b5 - 10b3 - b2 - b1 - 5) q^89 + (2b15 - 4b14 - b12 - 2b11 - 2b10 - b7) * q^91 + (-2b15 - 3b13 + 4b12 - 6b9) * q^95 + (b8 - 2b6 - b5 - 2b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{5}+O(q^{10}) $ 16 * q - 6 * q^5	$ 16 q - 6 q^{5} + 6 q^{13} - 14 q^{25} - 18 q^{29} + 6 q^{49} + 48 q^{53} - 42 q^{61} - 54 q^{65} + 28 q^{73} - 66 q^{77} - 36 q^{85} + 8 q^{97}+O(q^{100}) $ 16 * q - 6 * q^5 + 6 * q^13 - 14 * q^25 - 18 * q^29 + 6 * q^49 + 48 * q^53 - 42 * q^61 - 54 * q^65 + 28 * q^73 - 66 * q^77 - 36 * q^85 + 8 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 $ x^16 + 11*x^14 + 85*x^12 + 332*x^10 + 940*x^8 + 1064*x^6 + 880*x^4 + 128*x^2 + 16 :

$\beta_{1}$	$=$	$ ( 678 \nu^{14} + 6789 \nu^{12} + 51076 \nu^{10} + 174698 \nu^{8} + 464941 \nu^{6} + 179896 \nu^{4} + \cdots - 1117832 ) / 392916 $ (678v^14 + 6789v^12 + 51076v^10 + 174698v^8 + 464941v^6 + 179896v^4 + 26216*v^2 - 1117832) / 392916
$\beta_{2}$	$=$	$ ( 3399 \nu^{14} + 34180 \nu^{12} + 256058 \nu^{10} + 875809 \nu^{8} + 2248150 \nu^{6} + \cdots - 1217016 ) / 392916 $ (3399v^14 + 34180v^12 + 256058v^10 + 875809v^8 + 2248150v^6 + 901868v^4 + 131428*v^2 - 1217016) / 392916
$\beta_{3}$	$=$	$ ( 15229 \nu^{14} + 164807 \nu^{12} + 1267309 \nu^{10} + 4851724 \nu^{8} + 13616468 \nu^{6} + \cdots + 272784 ) / 1571664 $ (15229v^14 + 164807v^12 + 1267309v^10 + 4851724v^8 + 13616468v^6 + 14343892v^4 + 12681936*v^2 + 272784) / 1571664
$\beta_{4}$	$=$	$ ( 2739 \nu^{14} + 27861 \nu^{12} + 206338 \nu^{10} + 705749 \nu^{8} + 1761071 \nu^{6} + 726748 \nu^{4} + \cdots - 522936 ) / 130972 $ (2739v^14 + 27861v^12 + 206338v^10 + 705749v^8 + 1761071v^6 + 726748v^4 + 105908*v^2 - 522936) / 130972
$\beta_{5}$	$=$	$ ( 11161 \nu^{14} + 124073 \nu^{12} + 960853 \nu^{10} + 3803536 \nu^{8} + 10826822 \nu^{6} + \cdots + 1478952 ) / 392916 $ (11161v^14 + 124073v^12 + 960853v^10 + 3803536v^8 + 10826822v^6 + 12871600v^4 + 10167144*v^2 + 1478952) / 392916
$\beta_{6}$	$=$	$ ( - 15229 \nu^{14} - 164807 \nu^{12} - 1267309 \nu^{10} - 4851724 \nu^{8} - 13616468 \nu^{6} + \cdots - 272784 ) / 523888 $ (-15229v^14 - 164807v^12 - 1267309v^10 - 4851724v^8 - 13616468v^6 - 14343892v^4 - 12158048*v^2 - 272784) / 523888
$\beta_{7}$	$=$	$ ( - 16809 \nu^{15} - 187872 \nu^{13} - 1462736 \nu^{11} - 5837297 \nu^{9} - 16795704 \nu^{7} + \cdots - 4754976 \nu ) / 785832 $ (-16809v^15 - 187872v^13 - 1462736v^11 - 5837297v^9 - 16795704v^7 - 20569544v^5 - 17676308v^3 - 4754976v) / 785832
$\beta_{8}$	$=$	$ ( - 52871 \nu^{14} - 609241 \nu^{12} - 4779695 \nu^{10} - 19636892 \nu^{8} - 56825800 \nu^{6} + \cdots - 7837008 ) / 1571664 $ (-52871v^14 - 609241v^12 - 4779695v^10 - 19636892v^8 - 56825800v^6 - 73402412v^4 - 53865600*v^2 - 7837008) / 1571664
$\beta_{9}$	$=$	$ ( 427 \nu^{15} + 4723 \nu^{13} + 36549 \nu^{11} + 143882 \nu^{9} + 409832 \nu^{7} + 483588 \nu^{5} + \cdots + 112384 \nu ) / 17208 $ (427v^15 + 4723v^13 + 36549v^11 + 143882v^9 + 409832v^7 + 483588v^5 + 417712v^3 + 112384v) / 17208
$\beta_{10}$	$=$	$ ( 76115 \nu^{15} + 866909 \nu^{13} + 6792687 \nu^{11} + 27721648 \nu^{9} + 80801692 \nu^{7} + \cdots + 24160784 \nu ) / 2357496 $ (76115v^15 + 866909v^13 + 6792687v^11 + 27721648v^9 + 80801692v^7 + 105764220v^5 + 89864864v^3 + 24160784v) / 2357496
$\beta_{11}$	$=$	$ ( - 13530 \nu^{15} - 145911 \nu^{13} - 1117489 \nu^{11} - 4239319 \nu^{9} - 11736870 \nu^{7} + \cdots + 869496 \nu ) / 392916 $ (-13530v^15 - 145911v^13 - 1117489v^11 - 4239319v^9 - 11736870v^7 - 11644738v^5 - 9036340v^3 + 869496v) / 392916
$\beta_{12}$	$=$	$ ( - 27507 \nu^{15} - 314408 \nu^{13} - 2465110 \nu^{11} - 10099993 \nu^{9} - 29483494 \nu^{7} + \cdots - 8911640 \nu ) / 785832 $ (-27507v^15 - 314408v^13 - 2465110v^11 - 10099993v^9 - 29483494v^7 - 39124720v^5 - 33151744v^3 - 8911640v) / 785832
$\beta_{13}$	$=$	$ ( 82961 \nu^{15} + 892865 \nu^{13} + 6850017 \nu^{11} + 26014876 \nu^{9} + 72425530 \nu^{7} + \cdots - 5684968 \nu ) / 2357496 $ (82961v^15 + 892865v^13 + 6850017v^11 + 26014876v^9 + 72425530v^7 + 72742140v^5 + 58172408v^3 - 5684968v) / 2357496
$\beta_{14}$	$=$	$ ( 149879 \nu^{15} + 1614221 \nu^{13} + 12349575 \nu^{11} + 46728172 \nu^{9} + 128779516 \nu^{7} + \cdots - 8691136 \nu ) / 2357496 $ (149879v^15 + 1614221v^13 + 12349575v^11 + 46728172v^9 + 128779516v^7 + 125336268v^5 + 92717072v^3 - 8691136v) / 2357496
$\beta_{15}$	$=$	$ ( 56172 \nu^{15} + 604481 \nu^{13} + 4624540 \nu^{11} + 17486008 \nu^{9} + 48189193 \nu^{7} + \cdots - 3195100 \nu ) / 785832 $ (56172v^15 + 604481v^13 + 4624540v^11 + 17486008v^9 + 48189193v^7 + 46730500v^5 + 34260124v^3 - 3195100v) / 785832

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

$\nu$	$=$	$ ( 2\beta_{15} - 2\beta_{14} - \beta_{13} + 2\beta_{12} + 2\beta_{10} + \beta_{9} ) / 6 $ (2b15 - 2b14 - b13 + 2b12 + 2b10 + b9) / 6
$\nu^{2}$	$=$	$ \beta_{6} + 3\beta_{3} $ b6 + 3*b3
$\nu^{3}$	$=$	$ ( -8\beta_{15} + 10\beta_{14} + 2\beta_{13} + 4\beta_{12} + 4\beta_{11} + 5\beta_{10} + \beta_{9} + 2\beta_{7} ) / 3 $ (-8b15 + 10b14 + 2b13 + 4b12 + 4b11 + 5b10 + b9 + 2*b7) / 3
$\nu^{4}$	$=$	$ -6\beta_{6} - \beta_{5} - 14\beta_{3} - 6\beta _1 - 14 $ -6b6 - b5 - 14b3 - 6*b1 - 14
$\nu^{5}$	$=$	$ ( 18 \beta_{15} - 25 \beta_{14} - 5 \beta_{13} - 36 \beta_{12} - 14 \beta_{11} - 50 \beta_{10} + \cdots - 28 \beta_{7} ) / 3 $ (18b15 - 25b14 - 5b13 - 36b12 - 14b11 - 50b10 - 10b9 - 28b7) / 3
$\nu^{6}$	$=$	$ \beta_{4} - 9\beta_{2} + 33\beta _1 + 70 $ b4 - 9b2 + 33b1 + 70
$\nu^{7}$	$=$	$ ( 86 \beta_{15} - 125 \beta_{14} - 31 \beta_{13} + 86 \beta_{12} - 84 \beta_{11} + 125 \beta_{10} + \cdots + 84 \beta_{7} ) / 3 $ (86b15 - 125b14 - 31b13 + 86b12 - 84b11 + 125b10 + 31b9 + 84b7) / 3
$\nu^{8}$	$=$	$ 11\beta_{8} + 179\beta_{6} + 63\beta_{5} - 11\beta_{4} + 358\beta_{3} + 63\beta_{2} $ 11b8 + 179b6 + 63b5 - 11b4 + 358b3 + 63b2
$\nu^{9}$	$=$	$ ( - 844 \beta_{15} + 1262 \beta_{14} + 382 \beta_{13} + 422 \beta_{12} + 968 \beta_{11} + \cdots + 484 \beta_{7} ) / 3 $ (-844b15 + 1262b14 + 382b13 + 422b12 + 968b11 + 631b10 + 191b9 + 484b7) / 3
$\nu^{10}$	$=$	$ -85\beta_{8} - 969\beta_{6} - 401\beta_{5} - 1854\beta_{3} - 969\beta _1 - 1854 $ -85b8 - 969b6 - 401b5 - 1854b3 - 969*b1 - 1854
$\nu^{11}$	$=$	$ ( 2106 \beta_{15} - 3221 \beta_{14} - 1141 \beta_{13} - 4212 \beta_{12} - 2740 \beta_{11} + \cdots - 5480 \beta_{7} ) / 3 $ (2106b15 - 3221b14 - 1141b13 - 4212b12 - 2740b11 - 6442b10 - 2282b9 - 5480b7) / 3
$\nu^{12}$	$=$	$ 571\beta_{4} - 2427\beta_{2} + 5247\beta _1 + 9690 $ 571b4 - 2427b2 + 5247*b1 + 9690
$\nu^{13}$	$=$	$ ( 10654 \beta_{15} - 16615 \beta_{14} - 6647 \beta_{13} + 10654 \beta_{12} - 15348 \beta_{11} + \cdots + 15348 \beta_{7} ) / 3 $ (10654b15 - 16615b14 - 6647b13 + 10654b12 - 15348b11 + 16615b10 + 6647b9 + 15348b7) / 3
$\nu^{14}$	$=$	$ 3569\beta_{8} + 28429\beta_{6} + 14241\beta_{5} - 3569\beta_{4} + 51022\beta_{3} + 14241\beta_{2} $ 3569b8 + 28429b6 + 14241b5 - 3569b4 + 51022b3 + 14241b2
$\nu^{15}$	$=$	$ ( - 109076 \beta_{15} + 173002 \beta_{14} + 76058 \beta_{13} + 54538 \beta_{12} + 170680 \beta_{11} + \cdots + 85340 \beta_{7} ) / 3 $ (-109076b15 + 173002b14 + 76058b13 + 54538b12 + 170680b11 + 86501b10 + 38029b9 + 85340b7) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$-1$	$-\beta_{3}$

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} $

287.1

−0.192865	−	0.334053i
0.192865	+	0.334053i
1.16543	+	2.01859i
−1.16543	−	2.01859i
0.539169	+	0.933868i
−0.539169	−	0.933868i
1.03144	+	1.78651i
−1.03144	−	1.78651i
−0.192865	+	0.334053i
0.192865	−	0.334053i
1.16543	−	2.01859i
−1.16543	+	2.01859i
0.539169	−	0.933868i
−0.539169	+	0.933868i
1.03144	−	1.78651i
−1.03144	+	1.78651i

−2.06470

3.57617i

−0.287429

0.165947i

287.2

−2.06470

3.57617i

0.287429

−

0.165947i

287.3

−0.959555

1.66200i

−2.63027

1.51859i

287.4

−0.959555

1.66200i

2.63027

−

1.51859i

287.5

0.312371

−

0.541042i

−0.751481

0.433868i

287.6

0.312371

−

0.541042i

0.751481

−

0.433868i

287.7

1.21189

−

2.09905i

−3.96035

2.28651i

287.8

1.21189

−

2.09905i

3.96035

−

2.28651i

1439.1

−2.06470

−

3.57617i

−0.287429

−

0.165947i

1439.2

−2.06470

−

3.57617i

0.287429

0.165947i

1439.3

−0.959555

−

1.66200i

−2.63027

−

1.51859i

1439.4

−0.959555

−

1.66200i

2.63027

1.51859i

1439.5

0.312371

0.541042i

−0.751481

−

0.433868i

1439.6

0.312371

0.541042i

0.751481

0.433868i

1439.7

1.21189

2.09905i

−3.96035

−

2.28651i

1439.8

1.21189

2.09905i

3.96035

2.28651i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
72.j	odd	6	1	inner
72.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1728.2.p.a		16
3.b	odd	2	1		576.2.p.c	yes	16
4.b	odd	2	1	inner	1728.2.p.a		16
8.b	even	2	1		1728.2.p.c		16
8.d	odd	2	1		1728.2.p.c		16
9.c	even	3	1		576.2.p.a	✓	16
9.c	even	3	1		5184.2.f.f		16
9.d	odd	6	1		1728.2.p.c		16
9.d	odd	6	1		5184.2.f.a		16
12.b	even	2	1		576.2.p.c	yes	16
24.f	even	2	1		576.2.p.a	✓	16
24.h	odd	2	1		576.2.p.a	✓	16
36.f	odd	6	1		576.2.p.a	✓	16
36.f	odd	6	1		5184.2.f.f		16
36.h	even	6	1		1728.2.p.c		16
36.h	even	6	1		5184.2.f.a		16
72.j	odd	6	1	inner	1728.2.p.a		16
72.j	odd	6	1		5184.2.f.f		16
72.l	even	6	1	inner	1728.2.p.a		16
72.l	even	6	1		5184.2.f.f		16
72.n	even	6	1		576.2.p.c	yes	16
72.n	even	6	1		5184.2.f.a		16
72.p	odd	6	1		576.2.p.c	yes	16
72.p	odd	6	1		5184.2.f.a		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
576.2.p.a	✓	16	9.c	even	3	1
576.2.p.a	✓	16	24.f	even	2	1
576.2.p.a	✓	16	24.h	odd	2	1
576.2.p.a	✓	16	36.f	odd	6	1
576.2.p.c	yes	16	3.b	odd	2	1
576.2.p.c	yes	16	12.b	even	2	1
576.2.p.c	yes	16	72.n	even	6	1
576.2.p.c	yes	16	72.p	odd	6	1
1728.2.p.a		16	1.a	even	1	1	trivial
1728.2.p.a		16	4.b	odd	2	1	inner
1728.2.p.a		16	72.j	odd	6	1	inner
1728.2.p.a		16	72.l	even	6	1	inner
1728.2.p.c		16	8.b	even	2	1
1728.2.p.c		16	8.d	odd	2	1
1728.2.p.c		16	9.d	odd	6	1
1728.2.p.c		16	36.h	even	6	1
5184.2.f.a		16	9.d	odd	6	1
5184.2.f.a		16	36.h	even	6	1
5184.2.f.a		16	72.n	even	6	1
5184.2.f.a		16	72.p	odd	6	1
5184.2.f.f		16	9.c	even	3	1
5184.2.f.f		16	36.f	odd	6	1
5184.2.f.f		16	72.j	odd	6	1
5184.2.f.f		16	72.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} + 3T_{5}^{7} + 18T_{5}^{6} + 3T_{5}^{5} + 114T_{5}^{4} + 63T_{5}^{3} + 333T_{5}^{2} - 180T_{5} + 144 $ T5^8 + 3*T5^7 + 18*T5^6 + 3*T5^5 + 114*T5^4 + 63*T5^3 + 333*T5^2 - 180*T5 + 144 acting on $S_{2}^{\mathrm{new}}(1728, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} + 3 T^{7} + \cdots + 144)^{2} $ (T^8 + 3T^7 + 18T^6 + 3T^5 + 114T^4 + 63T^3 + 333T^2 - 180*T + 144)^2
$7$	$ T^{16} - 31 T^{14} + \cdots + 256 $ T^16 - 31T^14 + 742T^12 - 6451T^10 + 42706T^8 - 36019T^6 + 25057T^4 - 2704*T^2 + 256
$11$	$ T^{16} + \cdots + 639128961 $ T^16 - 60T^14 + 2334T^12 - 54792T^10 + 942435T^8 - 10365624T^6 + 80015310T^4 - 267574104*T^2 + 639128961
$13$	$ (T^{8} - 3 T^{7} + \cdots + 20736)^{2} $ (T^8 - 3T^7 - 24T^6 + 81T^5 + 540T^4 - 1215T^3 - 3213T^2 + 6480*T + 20736)^2
$17$	$ (T^{8} + 63 T^{6} + \cdots + 11664)^{2} $ (T^8 + 63T^6 + 1188T^4 + 7776*T^2 + 11664)^2
$19$	$ (T^{8} - 75 T^{6} + \cdots + 20736)^{2} $ (T^8 - 75T^6 + 1728T^4 - 12096*T^2 + 20736)^2
$23$	$ T^{16} + \cdots + 2176782336 $ T^16 + 81T^14 + 4482T^12 + 132921T^10 + 2838726T^8 + 29321109T^6 + 217674297T^4 + 827630784*T^2 + 2176782336
$29$	$ (T^{8} + 9 T^{7} + \cdots + 11664)^{2} $ (T^8 + 9T^7 + 90T^6 + 81T^5 + 702T^4 - 1215T^3 + 7533T^2 - 8748*T + 11664)^2
$31$	$ T^{16} + \cdots + 18339659776 $ T^16 - 187T^14 + 24994T^12 - 1605643T^10 + 75084934T^8 - 1244515399T^6 + 15507830881T^4 - 17583587584*T^2 + 18339659776
$37$	$ (T^{8} + 84 T^{6} + \cdots + 20736)^{2} $ (T^8 + 84T^6 + 1728T^4 + 10800*T^2 + 20736)^2
$41$	$ (T^{8} - 126 T^{6} + \cdots + 700569)^{2} $ (T^8 - 126T^6 + 15039T^4 - 68040T^3 - 8262T^2 + 451980*T + 700569)^2
$43$	$ T^{16} + \cdots + 1275989841 $ T^16 + 72T^14 + 3510T^12 + 91368T^10 + 1716795T^8 + 19263096T^6 + 152779446T^4 + 520812180*T^2 + 1275989841
$47$	$ T^{16} + \cdots + 142657607172096 $ T^16 + 261T^14 + 44334T^12 + 4403889T^10 + 318387834T^8 + 15227300241T^6 + 529960897449T^4 + 10776523751424*T^2 + 142657607172096
$53$	$ (T^{4} - 12 T^{3} + \cdots - 384)^{4} $ (T^4 - 12T^3 + 12T^2 + 204*T - 384)^4
$59$	$ T^{16} - 144 T^{14} + \cdots + 531441 $ T^16 - 144T^14 + 16470T^12 - 600696T^10 + 17218251T^8 - 28815912T^6 + 43184502T^4 - 4960116*T^2 + 531441
$61$	$ (T^{8} + 21 T^{7} + \cdots + 82944)^{2} $ (T^8 + 21T^7 + 84T^6 - 1323T^5 - 6264T^4 + 94689T^3 + 771147T^2 + 432864*T + 82944)^2
$67$	$ T^{16} + \cdots + 5103121662081 $ T^16 + 228T^14 + 37134T^12 + 2712312T^10 + 141485859T^8 + 3970540296T^6 + 79850237886T^4 + 760707726696*T^2 + 5103121662081
$71$	$ (T^{8} - 432 T^{6} + \cdots + 9144576)^{2} $ (T^8 - 432T^6 + 52704T^4 - 1963440*T^2 + 9144576)^2
$73$	$ (T^{4} - 7 T^{3} + \cdots - 188)^{4} $ (T^4 - 7T^3 - 132T^2 - 340*T - 188)^4
$79$	$ T^{16} - 235 T^{14} + \cdots + 256 $ T^16 - 235T^14 + 47506T^12 - 1803499T^10 + 58353190T^8 - 40386007T^6 + 27260785T^4 - 83728*T^2 + 256
$83$	$ T^{16} + \cdots + 306402103296 $ T^16 - 183T^14 + 22998T^12 - 1511307T^10 + 72125586T^8 - 1940433867T^6 + 35920312353T^4 - 113072459328*T^2 + 306402103296
$89$	$ (T^{8} + 324 T^{6} + \cdots + 2985984)^{2} $ (T^8 + 324T^6 + 26784T^4 + 594864*T^2 + 2985984)^2
$97$	$ (T^{8} - 4 T^{7} + \cdots + 36481)^{2} $ (T^8 - 4T^7 + 94T^6 + 776T^5 + 5347T^4 + 16568T^3 + 38926T^2 + 44312*T + 36481)^2
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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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{5} + \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{10}) q^{7}+O(q^{10}) \) q + (b5 + b3 + b2) * q^5 + (-b15 + b14 + b12 + b10) * q^7	\( q + (\beta_{5} + \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{10}) q^{7}+ \cdots + (\beta_{8} - 2 \beta_{6} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) q + (b5 + b3 + b2) * q^5 + (-b15 + b14 + b12 + b10) * q^7 + (b15 - b14 + b13 - b12 - b11 - b10 + b9 - b7) * q^11 + (-b6 - 2b1) q^13 + (-2b6 - 2b3 - b1 - 1) * q^17 + (2b14 + b10) q^19 + (-b15 + 2b12 - b11 - 2b7) * q^23 + (b8 + b6 - b5 - 2b3 + b1 - 2) q^25 + (-b8 - 2b3 - 2) q^29 + (-b15 - 2b14 - 3b11) * q^31 + (b12 + b10 - b9 - 2b7) q^35 + (-2b6 - 2b5 - 2b3 - b2 - b1 - 1) q^37 + (b8 + b5 + b4 - b2) * q^41 + (b15 + b12) * q^43 + (b15 + b14 - b13 + b12 - b10 + b9) * q^47 + (b8 - 2b6 + 2b5 - b4 - b3 + 2b2) q^49 + (b4 - b2 + 3) * q^53 + (b12 - 2b10 - b9 + 3b7) * q^55 + (b15 - 3b13 + 2b11) * q^59 + (-b8 + 2b4 + 2b3 - 2) * q^61 + (-b8 - 2b6 + 2b4 + 2b3 + 2b1 - 2) * q^65 + (-b15 - b14 - b13 + 2b12 - b11 - 2b10 - 2b9 - 2b7) * q^67 + (-2b14 + 2b13 - 6b11 - b10 + b9 - 3b7) * q^71 + (b4 - 2b2 + 4b1 + 3) * q^73 + (-b8 + 3b6 - b5 + b4 + 9b3 - b2) * q^77 + (-3b15 + 2b13 + 3b12 - 3b11 + 2b9 - 3b7) * q^79 + (b15 + b14 + 2b13 - b12 + 2b11 + b10 + 2b9 + 2b7) * q^83 + (b8 + 2b6 + b5 + b4 - b3 - b2 + 4b1 - 2) * q^85 + (-2b6 - 2b5 - 10b3 - b2 - b1 - 5) q^89 + (2b15 - 4b14 - b12 - 2b11 - 2b10 - b7) * q^91 + (-2b15 - 3b13 + 4b12 - 6b9) * q^95 + (b8 - 2b6 - b5 - 2b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{5}+O(q^{10}) \) 16 * q - 6 * q^5	\( 16 q - 6 q^{5} + 6 q^{13} - 14 q^{25} - 18 q^{29} + 6 q^{49} + 48 q^{53} - 42 q^{61} - 54 q^{65} + 28 q^{73} - 66 q^{77} - 36 q^{85} + 8 q^{97}+O(q^{100}) \) 16 * q - 6 * q^5 + 6 * q^13 - 14 * q^25 - 18 * q^29 + 6 * q^49 + 48 * q^53 - 42 * q^61 - 54 * q^65 + 28 * q^73 - 66 * q^77 - 36 * q^85 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1728.2.p.a

Level	$1728$
Weight	$2$
Character orbit	1728.p
Analytic conductor	$13.798$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(13.7981494693\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 \) x^16 + 11x^14 + 85x^12 + 332x^10 + 940x^8 + 1064x^6 + 880x^4 + 128*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{4} \)
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 678 \nu^{14} + 6789 \nu^{12} + 51076 \nu^{10} + 174698 \nu^{8} + 464941 \nu^{6} + 179896 \nu^{4} + \cdots - 1117832 ) / 392916 \) (678v^14 + 6789v^12 + 51076v^10 + 174698v^8 + 464941v^6 + 179896v^4 + 26216*v^2 - 1117832) / 392916
\(\beta_{2}\)	\(=\)	\( ( 3399 \nu^{14} + 34180 \nu^{12} + 256058 \nu^{10} + 875809 \nu^{8} + 2248150 \nu^{6} + \cdots - 1217016 ) / 392916 \) (3399v^14 + 34180v^12 + 256058v^10 + 875809v^8 + 2248150v^6 + 901868v^4 + 131428*v^2 - 1217016) / 392916
\(\beta_{3}\)	\(=\)	\( ( 15229 \nu^{14} + 164807 \nu^{12} + 1267309 \nu^{10} + 4851724 \nu^{8} + 13616468 \nu^{6} + \cdots + 272784 ) / 1571664 \) (15229v^14 + 164807v^12 + 1267309v^10 + 4851724v^8 + 13616468v^6 + 14343892v^4 + 12681936*v^2 + 272784) / 1571664
\(\beta_{4}\)	\(=\)	\( ( 2739 \nu^{14} + 27861 \nu^{12} + 206338 \nu^{10} + 705749 \nu^{8} + 1761071 \nu^{6} + 726748 \nu^{4} + \cdots - 522936 ) / 130972 \) (2739v^14 + 27861v^12 + 206338v^10 + 705749v^8 + 1761071v^6 + 726748v^4 + 105908*v^2 - 522936) / 130972
\(\beta_{5}\)	\(=\)	\( ( 11161 \nu^{14} + 124073 \nu^{12} + 960853 \nu^{10} + 3803536 \nu^{8} + 10826822 \nu^{6} + \cdots + 1478952 ) / 392916 \) (11161v^14 + 124073v^12 + 960853v^10 + 3803536v^8 + 10826822v^6 + 12871600v^4 + 10167144*v^2 + 1478952) / 392916
\(\beta_{6}\)	\(=\)	\( ( - 15229 \nu^{14} - 164807 \nu^{12} - 1267309 \nu^{10} - 4851724 \nu^{8} - 13616468 \nu^{6} + \cdots - 272784 ) / 523888 \) (-15229v^14 - 164807v^12 - 1267309v^10 - 4851724v^8 - 13616468v^6 - 14343892v^4 - 12158048*v^2 - 272784) / 523888
\(\beta_{7}\)	\(=\)	\( ( - 16809 \nu^{15} - 187872 \nu^{13} - 1462736 \nu^{11} - 5837297 \nu^{9} - 16795704 \nu^{7} + \cdots - 4754976 \nu ) / 785832 \) (-16809v^15 - 187872v^13 - 1462736v^11 - 5837297v^9 - 16795704v^7 - 20569544v^5 - 17676308v^3 - 4754976v) / 785832
\(\beta_{8}\)	\(=\)	\( ( - 52871 \nu^{14} - 609241 \nu^{12} - 4779695 \nu^{10} - 19636892 \nu^{8} - 56825800 \nu^{6} + \cdots - 7837008 ) / 1571664 \) (-52871v^14 - 609241v^12 - 4779695v^10 - 19636892v^8 - 56825800v^6 - 73402412v^4 - 53865600*v^2 - 7837008) / 1571664
\(\beta_{9}\)	\(=\)	\( ( 427 \nu^{15} + 4723 \nu^{13} + 36549 \nu^{11} + 143882 \nu^{9} + 409832 \nu^{7} + 483588 \nu^{5} + \cdots + 112384 \nu ) / 17208 \) (427v^15 + 4723v^13 + 36549v^11 + 143882v^9 + 409832v^7 + 483588v^5 + 417712v^3 + 112384v) / 17208
\(\beta_{10}\)	\(=\)	\( ( 76115 \nu^{15} + 866909 \nu^{13} + 6792687 \nu^{11} + 27721648 \nu^{9} + 80801692 \nu^{7} + \cdots + 24160784 \nu ) / 2357496 \) (76115v^15 + 866909v^13 + 6792687v^11 + 27721648v^9 + 80801692v^7 + 105764220v^5 + 89864864v^3 + 24160784v) / 2357496
\(\beta_{11}\)	\(=\)	\( ( - 13530 \nu^{15} - 145911 \nu^{13} - 1117489 \nu^{11} - 4239319 \nu^{9} - 11736870 \nu^{7} + \cdots + 869496 \nu ) / 392916 \) (-13530v^15 - 145911v^13 - 1117489v^11 - 4239319v^9 - 11736870v^7 - 11644738v^5 - 9036340v^3 + 869496v) / 392916
\(\beta_{12}\)	\(=\)	\( ( - 27507 \nu^{15} - 314408 \nu^{13} - 2465110 \nu^{11} - 10099993 \nu^{9} - 29483494 \nu^{7} + \cdots - 8911640 \nu ) / 785832 \) (-27507v^15 - 314408v^13 - 2465110v^11 - 10099993v^9 - 29483494v^7 - 39124720v^5 - 33151744v^3 - 8911640v) / 785832
\(\beta_{13}\)	\(=\)	\( ( 82961 \nu^{15} + 892865 \nu^{13} + 6850017 \nu^{11} + 26014876 \nu^{9} + 72425530 \nu^{7} + \cdots - 5684968 \nu ) / 2357496 \) (82961v^15 + 892865v^13 + 6850017v^11 + 26014876v^9 + 72425530v^7 + 72742140v^5 + 58172408v^3 - 5684968v) / 2357496
\(\beta_{14}\)	\(=\)	\( ( 149879 \nu^{15} + 1614221 \nu^{13} + 12349575 \nu^{11} + 46728172 \nu^{9} + 128779516 \nu^{7} + \cdots - 8691136 \nu ) / 2357496 \) (149879v^15 + 1614221v^13 + 12349575v^11 + 46728172v^9 + 128779516v^7 + 125336268v^5 + 92717072v^3 - 8691136v) / 2357496
\(\beta_{15}\)	\(=\)	\( ( 56172 \nu^{15} + 604481 \nu^{13} + 4624540 \nu^{11} + 17486008 \nu^{9} + 48189193 \nu^{7} + \cdots - 3195100 \nu ) / 785832 \) (56172v^15 + 604481v^13 + 4624540v^11 + 17486008v^9 + 48189193v^7 + 46730500v^5 + 34260124v^3 - 3195100v) / 785832

\(\nu\)	\(=\)	\( ( 2\beta_{15} - 2\beta_{14} - \beta_{13} + 2\beta_{12} + 2\beta_{10} + \beta_{9} ) / 6 \) (2b15 - 2b14 - b13 + 2b12 + 2b10 + b9) / 6
\(\nu^{2}\)	\(=\)	\( \beta_{6} + 3\beta_{3} \) b6 + 3*b3
\(\nu^{3}\)	\(=\)	\( ( -8\beta_{15} + 10\beta_{14} + 2\beta_{13} + 4\beta_{12} + 4\beta_{11} + 5\beta_{10} + \beta_{9} + 2\beta_{7} ) / 3 \) (-8b15 + 10b14 + 2b13 + 4b12 + 4b11 + 5b10 + b9 + 2*b7) / 3
\(\nu^{4}\)	\(=\)	\( -6\beta_{6} - \beta_{5} - 14\beta_{3} - 6\beta _1 - 14 \) -6b6 - b5 - 14b3 - 6*b1 - 14
\(\nu^{5}\)	\(=\)	\( ( 18 \beta_{15} - 25 \beta_{14} - 5 \beta_{13} - 36 \beta_{12} - 14 \beta_{11} - 50 \beta_{10} + \cdots - 28 \beta_{7} ) / 3 \) (18b15 - 25b14 - 5b13 - 36b12 - 14b11 - 50b10 - 10b9 - 28b7) / 3
\(\nu^{6}\)	\(=\)	\( \beta_{4} - 9\beta_{2} + 33\beta _1 + 70 \) b4 - 9b2 + 33b1 + 70
\(\nu^{7}\)	\(=\)	\( ( 86 \beta_{15} - 125 \beta_{14} - 31 \beta_{13} + 86 \beta_{12} - 84 \beta_{11} + 125 \beta_{10} + \cdots + 84 \beta_{7} ) / 3 \) (86b15 - 125b14 - 31b13 + 86b12 - 84b11 + 125b10 + 31b9 + 84b7) / 3
\(\nu^{8}\)	\(=\)	\( 11\beta_{8} + 179\beta_{6} + 63\beta_{5} - 11\beta_{4} + 358\beta_{3} + 63\beta_{2} \) 11b8 + 179b6 + 63b5 - 11b4 + 358b3 + 63b2
\(\nu^{9}\)	\(=\)	\( ( - 844 \beta_{15} + 1262 \beta_{14} + 382 \beta_{13} + 422 \beta_{12} + 968 \beta_{11} + \cdots + 484 \beta_{7} ) / 3 \) (-844b15 + 1262b14 + 382b13 + 422b12 + 968b11 + 631b10 + 191b9 + 484b7) / 3
\(\nu^{10}\)	\(=\)	\( -85\beta_{8} - 969\beta_{6} - 401\beta_{5} - 1854\beta_{3} - 969\beta _1 - 1854 \) -85b8 - 969b6 - 401b5 - 1854b3 - 969*b1 - 1854
\(\nu^{11}\)	\(=\)	\( ( 2106 \beta_{15} - 3221 \beta_{14} - 1141 \beta_{13} - 4212 \beta_{12} - 2740 \beta_{11} + \cdots - 5480 \beta_{7} ) / 3 \) (2106b15 - 3221b14 - 1141b13 - 4212b12 - 2740b11 - 6442b10 - 2282b9 - 5480b7) / 3
\(\nu^{12}\)	\(=\)	\( 571\beta_{4} - 2427\beta_{2} + 5247\beta _1 + 9690 \) 571b4 - 2427b2 + 5247*b1 + 9690
\(\nu^{13}\)	\(=\)	\( ( 10654 \beta_{15} - 16615 \beta_{14} - 6647 \beta_{13} + 10654 \beta_{12} - 15348 \beta_{11} + \cdots + 15348 \beta_{7} ) / 3 \) (10654b15 - 16615b14 - 6647b13 + 10654b12 - 15348b11 + 16615b10 + 6647b9 + 15348b7) / 3
\(\nu^{14}\)	\(=\)	\( 3569\beta_{8} + 28429\beta_{6} + 14241\beta_{5} - 3569\beta_{4} + 51022\beta_{3} + 14241\beta_{2} \) 3569b8 + 28429b6 + 14241b5 - 3569b4 + 51022b3 + 14241b2
\(\nu^{15}\)	\(=\)	\( ( - 109076 \beta_{15} + 173002 \beta_{14} + 76058 \beta_{13} + 54538 \beta_{12} + 170680 \beta_{11} + \cdots + 85340 \beta_{7} ) / 3 \) (-109076b15 + 173002b14 + 76058b13 + 54538b12 + 170680b11 + 86501b10 + 38029b9 + 85340b7) / 3

\(n\)	\(325\)	\(703\)	\(1217\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\beta_{3}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 287.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} + 3 T^{7} + \cdots + 144)^{2} \) (T^8 + 3T^7 + 18T^6 + 3T^5 + 114T^4 + 63T^3 + 333T^2 - 180*T + 144)^2
$7$	\( T^{16} - 31 T^{14} + \cdots + 256 \) T^16 - 31T^14 + 742T^12 - 6451T^10 + 42706T^8 - 36019T^6 + 25057T^4 - 2704*T^2 + 256
$11$	\( T^{16} + \cdots + 639128961 \) T^16 - 60T^14 + 2334T^12 - 54792T^10 + 942435T^8 - 10365624T^6 + 80015310T^4 - 267574104*T^2 + 639128961
$13$	\( (T^{8} - 3 T^{7} + \cdots + 20736)^{2} \) (T^8 - 3T^7 - 24T^6 + 81T^5 + 540T^4 - 1215T^3 - 3213T^2 + 6480*T + 20736)^2
$17$	\( (T^{8} + 63 T^{6} + \cdots + 11664)^{2} \) (T^8 + 63T^6 + 1188T^4 + 7776*T^2 + 11664)^2
$19$	\( (T^{8} - 75 T^{6} + \cdots + 20736)^{2} \) (T^8 - 75T^6 + 1728T^4 - 12096*T^2 + 20736)^2
$23$	\( T^{16} + \cdots + 2176782336 \) T^16 + 81T^14 + 4482T^12 + 132921T^10 + 2838726T^8 + 29321109T^6 + 217674297T^4 + 827630784*T^2 + 2176782336
$29$	\( (T^{8} + 9 T^{7} + \cdots + 11664)^{2} \) (T^8 + 9T^7 + 90T^6 + 81T^5 + 702T^4 - 1215T^3 + 7533T^2 - 8748*T + 11664)^2
$31$	\( T^{16} + \cdots + 18339659776 \) T^16 - 187T^14 + 24994T^12 - 1605643T^10 + 75084934T^8 - 1244515399T^6 + 15507830881T^4 - 17583587584*T^2 + 18339659776
$37$	\( (T^{8} + 84 T^{6} + \cdots + 20736)^{2} \) (T^8 + 84T^6 + 1728T^4 + 10800*T^2 + 20736)^2
$41$	\( (T^{8} - 126 T^{6} + \cdots + 700569)^{2} \) (T^8 - 126T^6 + 15039T^4 - 68040T^3 - 8262T^2 + 451980*T + 700569)^2
$43$	\( T^{16} + \cdots + 1275989841 \) T^16 + 72T^14 + 3510T^12 + 91368T^10 + 1716795T^8 + 19263096T^6 + 152779446T^4 + 520812180*T^2 + 1275989841
$47$	\( T^{16} + \cdots + 142657607172096 \) T^16 + 261T^14 + 44334T^12 + 4403889T^10 + 318387834T^8 + 15227300241T^6 + 529960897449T^4 + 10776523751424*T^2 + 142657607172096
$53$	\( (T^{4} - 12 T^{3} + \cdots - 384)^{4} \) (T^4 - 12T^3 + 12T^2 + 204*T - 384)^4
$59$	\( T^{16} - 144 T^{14} + \cdots + 531441 \) T^16 - 144T^14 + 16470T^12 - 600696T^10 + 17218251T^8 - 28815912T^6 + 43184502T^4 - 4960116*T^2 + 531441
$61$	\( (T^{8} + 21 T^{7} + \cdots + 82944)^{2} \) (T^8 + 21T^7 + 84T^6 - 1323T^5 - 6264T^4 + 94689T^3 + 771147T^2 + 432864*T + 82944)^2
$67$	\( T^{16} + \cdots + 5103121662081 \) T^16 + 228T^14 + 37134T^12 + 2712312T^10 + 141485859T^8 + 3970540296T^6 + 79850237886T^4 + 760707726696*T^2 + 5103121662081
$71$	\( (T^{8} - 432 T^{6} + \cdots + 9144576)^{2} \) (T^8 - 432T^6 + 52704T^4 - 1963440*T^2 + 9144576)^2
$73$	\( (T^{4} - 7 T^{3} + \cdots - 188)^{4} \) (T^4 - 7T^3 - 132T^2 - 340*T - 188)^4
$79$	\( T^{16} - 235 T^{14} + \cdots + 256 \) T^16 - 235T^14 + 47506T^12 - 1803499T^10 + 58353190T^8 - 40386007T^6 + 27260785T^4 - 83728*T^2 + 256
$83$	\( T^{16} + \cdots + 306402103296 \) T^16 - 183T^14 + 22998T^12 - 1511307T^10 + 72125586T^8 - 1940433867T^6 + 35920312353T^4 - 113072459328*T^2 + 306402103296
$89$	\( (T^{8} + 324 T^{6} + \cdots + 2985984)^{2} \) (T^8 + 324T^6 + 26784T^4 + 594864*T^2 + 2985984)^2
$97$	\( (T^{8} - 4 T^{7} + \cdots + 36481)^{2} \) (T^8 - 4T^7 + 94T^6 + 776T^5 + 5347T^4 + 16568T^3 + 38926T^2 + 44312*T + 36481)^2
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