LMFDB - Newform orbit 1232.2.j.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(111,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1232.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.j (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:	$9.83756952902$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2x^{14} - 11x^{12} + 48x^{10} + 768x^{6} - 2816x^{4} - 8192x^{2} + 65536 $ x^16 - 2x^14 - 11x^12 + 48x^10 + 768x^6 - 2816x^4 - 8192x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
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_{15} q^{3} - \beta_{12} q^{5} - \beta_{13} q^{7} + (\beta_{10} + \beta_{9} + \beta_{6} + 1) q^{9}+O(q^{10}) $ q + b15 * q^3 - b12 * q^5 - b13 * q^7 + (b10 + b9 + b6 + 1) * q^9	$ q + \beta_{15} q^{3} - \beta_{12} q^{5} - \beta_{13} q^{7} + (\beta_{10} + \beta_{9} + \beta_{6} + 1) q^{9} - \beta_{2} q^{11} + (\beta_{12} - \beta_{10} + \cdots + \beta_{4}) q^{13}+ \cdots + ( - \beta_{14} + \beta_{13} + \cdots - \beta_{2}) q^{99}+O(q^{100}) $ q + b15 * q^3 - b12 * q^5 - b13 * q^7 + (b10 + b9 + b6 + 1) * q^9 - b2 * q^11 + (b12 - b10 + b9 + b4) * q^13 + (b14 - b13 + b5 + b3 + 2b2) q^15 + b7 * q^17 + (-b14 - b13 + 2b11) q^19 + (b9 - b4 + b1 + 1) * q^21 + (-b14 + b13 - b5 + b3 + 2b2) q^23 + (-b6 - b1 - 3) * q^25 + (b15 - 2b11 + b8) q^27 + (b10 + b9) * q^29 + (b14 + b13 - b11 - b8) * q^31 + b4 * q^33 + (-b15 + b14 - b11 - b8 + b5 + 3b2) q^35 + (b10 + b9 - b6 - 2) * q^37 + (-b14 + b13 - 2b5 + b3 - 4b2) * q^39 + (-b12 + 3b4) q^41 + (-b3 - 2b2) q^43 + (-2b10 + 2b9 - 2b7 - 4b4) * q^45 + (2b11 + b8) q^47 + (b12 + 2b9 + b6 - 2b4 - b1 - 1) * q^49 + (-b14 + b13 - 2b5 - b3 - 2b2) * q^51 + (b10 + b9 - b1 - 2) * q^53 - b11 * q^55 + (-b10 - b9 - 2b6 - 2) q^57 + (-2b15 + b14 + b13 + b11 + b8) q^59 + (2b12 - 3b10 + 3b9 - b7) q^61 + (b15 - b14 - b13 - b11 + b8 + b5 + b3 + 5b2) q^63 + (2b1 + 4) q^65 + (b14 - b13 + 3b5 - b3 - 2b2) * q^67 + (b12 - 2b10 + 2b9 + 2b4) q^69 + (b14 - b13 + b5 - b3 - 2b2) q^71 + (-b10 + b9 - b7 - 2b4) q^73 + (-3b15 + b14 + b13 + 3b11 - 2b8) q^75 + b10 * q^77 + (-b14 + b13 - b3 + 4b2) q^79 + (b10 + b9 + 2b6 + 3b1 + 7) * q^81 + (-2b15 + b14 + b13 - 2b11 - 2b8) q^83 + (4b10 + 4b9 + 2b6) q^85 + (2b15 - b14 - b13) q^87 + (b12 + 2b7) q^89 + (b8 - 2b3 + 2b2) * q^91 + (-b10 - b9 - b6 - 2b1 - 2) q^93 + (b14 - b13 - 2b3 - 10b2) * q^95 + (-5b12 + 2b10 - 2b9 + 4b4) * q^97 + (-b14 + b13 - b5 - b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{9}+O(q^{10}) $ 16 * q + 24 * q^9	$ 16 q + 24 q^{9} + 12 q^{21} - 40 q^{25} + 8 q^{29} - 24 q^{37} - 16 q^{53} - 40 q^{57} + 48 q^{65} + 4 q^{77} + 96 q^{81} + 32 q^{85} - 24 q^{93}+O(q^{100}) $ 16 * q + 24 * q^9 + 12 * q^21 - 40 * q^25 + 8 * q^29 - 24 * q^37 - 16 * q^53 - 40 * q^57 + 48 * q^65 + 4 * q^77 + 96 * q^81 + 32 * q^85 - 24 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2x^{14} - 11x^{12} + 48x^{10} + 768x^{6} - 2816x^{4} - 8192x^{2} + 65536 $ x^16 - 2*x^14 - 11*x^12 + 48*x^10 + 768*x^6 - 2816*x^4 - 8192*x^2 + 65536 :

$\beta_{1}$	$=$	$ ( -\nu^{14} - 46\nu^{12} - 21\nu^{10} + 736\nu^{8} + 1152\nu^{6} + 1280\nu^{4} - 77056\nu^{2} + 12288 ) / 73728 $ (-v^14 - 46v^12 - 21v^10 + 736v^8 + 1152v^6 + 1280v^4 - 77056v^2 + 12288) / 73728
$\beta_{2}$	$=$	$ ( 3\nu^{14} - 22\nu^{12} - \nu^{10} - 192\nu^{8} + 256\nu^{6} + 3840\nu^{4} - 24832\nu^{2} + 20480 ) / 73728 $ (3v^14 - 22v^12 - v^10 - 192v^8 + 256v^6 + 3840v^4 - 24832v^2 + 20480) / 73728
$\beta_{3}$	$=$	$ ( \nu^{14} - 18\nu^{12} + 149\nu^{10} - 32\nu^{8} - 128\nu^{6} - 1280\nu^{4} - 25344\nu^{2} + 135168 ) / 24576 $ (v^14 - 18v^12 + 149v^10 - 32v^8 - 128v^6 - 1280v^4 - 25344v^2 + 135168) / 24576
$\beta_{4}$	$=$	$ ( -\nu^{15} - 14\nu^{13} + 43\nu^{11} + 128\nu^{9} - 768\nu^{7} - 768\nu^{5} - 25856\nu^{3} + 69632\nu ) / 98304 $ (-v^15 - 14v^13 + 43v^11 + 128v^9 - 768v^7 - 768v^5 - 25856v^3 + 69632*v) / 98304
$\beta_{5}$	$=$	$ ( 3\nu^{14} - 6\nu^{12} + 31\nu^{10} + 16\nu^{8} + 320\nu^{6} + 3328\nu^{4} - 7424\nu^{2} + 24576 ) / 12288 $ (3v^14 - 6v^12 + 31v^10 + 16v^8 + 320v^6 + 3328v^4 - 7424*v^2 + 24576) / 12288
$\beta_{6}$	$=$	$ ( 19\nu^{14} + 10\nu^{12} - 177\nu^{10} + 128\nu^{8} - 1152\nu^{6} + 12544\nu^{4} - 47360\nu^{2} - 159744 ) / 73728 $ (19v^14 + 10v^12 - 177v^10 + 128v^8 - 1152v^6 + 12544v^4 - 47360*v^2 - 159744) / 73728
$\beta_{7}$	$=$	$ ( 5\nu^{15} + 6\nu^{13} + 169\nu^{11} + 64\nu^{9} - 1024\nu^{7} + 22784\nu^{5} - 16128\nu^{3} + 77824\nu ) / 147456 $ (5v^15 + 6v^13 + 169v^11 + 64v^9 - 1024v^7 + 22784v^5 - 16128v^3 + 77824v) / 147456
$\beta_{8}$	$=$	$ ( \nu^{15} - 34\nu^{13} + 85\nu^{11} + 80\nu^{9} - 352\nu^{7} + 1024\nu^{5} - 14080\nu^{3} + 96256\nu ) / 36864 $ (v^15 - 34v^13 + 85v^11 + 80v^9 - 352v^7 + 1024v^5 - 14080v^3 + 96256*v) / 36864
$\beta_{9}$	$=$	$ ( - 25 \nu^{15} - 36 \nu^{14} + 66 \nu^{13} - 120 \nu^{12} + 499 \nu^{11} + 780 \nu^{10} + \cdots + 638976 ) / 589824 $ (-25v^15 - 36v^14 + 66v^13 - 120v^12 + 499v^11 + 780v^10 + 160v^9 + 384v^8 + 2048v^7 - 9216v^6 - 29440v^5 + 21504v^4 + 74496v^3 + 3072v^2 + 126976*v + 638976) / 589824
$\beta_{10}$	$=$	$ ( 25 \nu^{15} - 36 \nu^{14} - 66 \nu^{13} - 120 \nu^{12} - 499 \nu^{11} + 780 \nu^{10} + \cdots + 638976 ) / 589824 $ (25v^15 - 36v^14 - 66v^13 - 120v^12 - 499v^11 + 780v^10 - 160v^9 + 384v^8 - 2048v^7 - 9216v^6 + 29440v^5 + 21504v^4 - 74496v^3 + 3072v^2 - 126976*v + 638976) / 589824
$\beta_{11}$	$=$	$ ( - 17 \nu^{15} + 146 \nu^{13} - 293 \nu^{11} + 512 \nu^{9} - 4096 \nu^{7} - 15104 \nu^{5} + \cdots - 438272 \nu ) / 294912 $ (-17v^15 + 146v^13 - 293v^11 + 512v^9 - 4096v^7 - 15104v^5 + 174848v^3 - 438272v) / 294912
$\beta_{12}$	$=$	$ ( \nu^{15} - 2\nu^{13} - 11\nu^{11} + 48\nu^{9} + 768\nu^{5} - 2816\nu^{3} + 8192\nu ) / 16384 $ (v^15 - 2v^13 - 11v^11 + 48v^9 + 768v^5 - 2816v^3 + 8192v) / 16384
$\beta_{13}$	$=$	$ ( - 35 \nu^{15} - 36 \nu^{14} + 182 \nu^{13} + 392 \nu^{12} - 95 \nu^{11} - 244 \nu^{10} + \cdots - 606208 ) / 589824 $ (-35v^15 - 36v^14 + 182v^13 + 392v^12 - 95v^11 - 244v^10 - 352v^9 - 1152v^8 - 4096v^7 + 7168v^6 - 28928v^5 + 9216v^4 + 225536v^3 + 232448v^2 + 4096*v - 606208) / 589824
$\beta_{14}$	$=$	$ ( - 35 \nu^{15} + 36 \nu^{14} + 182 \nu^{13} - 392 \nu^{12} - 95 \nu^{11} + 244 \nu^{10} + \cdots + 606208 ) / 589824 $ (-35v^15 + 36v^14 + 182v^13 - 392v^12 - 95v^11 + 244v^10 - 352v^9 + 1152v^8 - 4096v^7 - 7168v^6 - 28928v^5 - 9216v^4 + 225536v^3 - 232448v^2 + 4096*v + 606208) / 589824
$\beta_{15}$	$=$	$ ( 7\nu^{15} - 22\nu^{13} + 67\nu^{11} - 88\nu^{9} - 256\nu^{7} + 4096\nu^{5} - 24832\nu^{3} + 71680\nu ) / 73728 $ (7v^15 - 22v^13 + 67v^11 - 88v^9 - 256v^7 + 4096v^5 - 24832v^3 + 71680v) / 73728

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

$\nu$	$=$	$ ( \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} ) / 2 $ (b14 + b13 + b12 - b11) / 2
$\nu^{2}$	$=$	$ ( \beta_{14} - \beta_{13} - \beta_{6} + \beta_{5} - \beta_{3} - 2\beta_{2} - \beta_1 ) / 2 $ (b14 - b13 - b6 + b5 - b3 - 2*b2 - b1) / 2
$\nu^{3}$	$=$	$ ( \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + 2\beta_{8} - 6\beta_{4} ) / 2 $ (b14 + b13 + b12 + b11 + 2b8 - 6b4) / 2
$\nu^{4}$	$=$	$ ( - 3 \beta_{14} + 3 \beta_{13} + 6 \beta_{10} + 6 \beta_{9} + \beta_{6} + 3 \beta_{5} - 3 \beta_{3} + \cdots + 4 ) / 2 $ (-3b14 + 3b13 + 6b10 + 6b9 + b6 + 3b5 - 3b3 + 6*b2 + b1 + 4) / 2
$\nu^{5}$	$=$	$ ( - 12 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - 5 \beta_{11} + 6 \beta_{10} + \cdots - 6 \beta_{4} ) / 2 $ (-12b15 + b14 + b13 - b12 - 5b11 + 6b10 - 6b9 + 2b8 + 12b7 - 6*b4) / 2
$\nu^{6}$	$=$	$ ( - 7 \beta_{14} + 7 \beta_{13} - 24 \beta_{10} - 24 \beta_{9} - 25 \beta_{6} + 17 \beta_{5} + \cdots + 11 \beta_1 ) / 2 $ (-7b14 + 7b13 - 24b10 - 24b9 - 25b6 + 17b5 - 5b3 + 14b2 + 11*b1) / 2
$\nu^{7}$	$=$	$ ( - 48 \beta_{15} - 11 \beta_{14} - 11 \beta_{13} + 41 \beta_{12} - 55 \beta_{11} - 36 \beta_{10} + \cdots - 78 \beta_{4} ) / 2 $ (-48b15 - 11b14 - 11b13 + 41b12 - 55b11 - 36b10 + 36b9 - 6b8 - 78*b4) / 2
$\nu^{8}$	$=$	$ ( 33 \beta_{14} - 33 \beta_{13} + 18 \beta_{10} + 18 \beta_{9} + 9 \beta_{6} + 51 \beta_{5} - 27 \beta_{3} + \cdots + 44 ) / 2 $ (33b14 - 33b13 + 18b10 + 18b9 + 9b6 + 51b5 - 27b3 - 354b2 + 81*b1 + 44) / 2
$\nu^{9}$	$=$	$ ( - 132 \beta_{15} - 155 \beta_{14} - 155 \beta_{13} + 343 \beta_{12} + 203 \beta_{11} + \cdots + 66 \beta_{4} ) / 2 $ (-132b15 - 155b14 - 155b13 + 343b12 + 203b11 - 198b10 + 198b9 + 42b8 + 36b7 + 66b4) / 2
$\nu^{10}$	$=$	$ ( - 59 \beta_{14} + 59 \beta_{13} + 60 \beta_{10} + 60 \beta_{9} - 49 \beta_{6} + 145 \beta_{5} + \cdots - 1512 ) / 2 $ (-59b14 + 59b13 + 60b10 + 60b9 - 49b6 + 145b5 + 227b3 - 458b2 - 85*b1 - 1512) / 2
$\nu^{11}$	$=$	$ ( 552 \beta_{15} - 479 \beta_{14} - 479 \beta_{13} - 479 \beta_{12} + 553 \beta_{11} - 696 \beta_{10} + \cdots - 54 \beta_{4} ) / 2 $ (552b15 - 479b14 - 479b13 - 479b12 + 553b11 - 696b10 + 696b9 + 50b8 + 264b7 - 54b4) / 2
$\nu^{12}$	$=$	$ ( - 1467 \beta_{14} + 1467 \beta_{13} - 66 \beta_{10} - 66 \beta_{9} + 1201 \beta_{6} - 477 \beta_{5} + \cdots + 2068 ) / 2 $ (-1467b14 + 1467b13 - 66b10 - 66b9 + 1201b6 - 477b5 + 909b3 - 1386b2 + 193*b1 + 2068) / 2
$\nu^{13}$	$=$	$ ( 1956 \beta_{15} + 937 \beta_{14} + 937 \beta_{13} + 1871 \beta_{12} - 677 \beta_{11} + \cdots + 2826 \beta_{4} ) / 2 $ (1956b15 + 937b14 + 937b13 + 1871b12 - 677b11 - 1842b10 + 1842b9 - 2734b8 + 924b7 + 2826b4) / 2
$\nu^{14}$	$=$	$ ( 4049 \beta_{14} - 4049 \beta_{13} - 4944 \beta_{10} - 4944 \beta_{9} + 1943 \beta_{6} + 2801 \beta_{5} + \cdots - 1296 ) / 2 $ (4049b14 - 4049b13 - 4944b10 - 4944b9 + 1943b6 + 2801b5 + 1003b3 - 9250b2 - 3925*b1 - 1296) / 2
$\nu^{15}$	$=$	$ ( 25536 \beta_{15} - 2099 \beta_{14} - 2099 \beta_{13} + 10169 \beta_{12} + 9833 \beta_{11} + \cdots - 10398 \beta_{4} ) / 2 $ (25536b15 - 2099b14 - 2099b13 + 10169b12 + 9833b11 - 6444b10 + 6444b9 - 2838b8 - 6192b7 - 10398b4) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$.

$n$	$309$	$353$	$463$	$673$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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} $

111.1

0.0861690	+	1.99814i
0.0861690	−	1.99814i
−1.82710	+	0.813447i
−1.82710	−	0.813447i
1.95998	+	0.398075i
1.95998	−	0.398075i
1.14575	+	1.63929i
1.14575	−	1.63929i
−1.14575	+	1.63929i
−1.14575	−	1.63929i
−1.95998	+	0.398075i
−1.95998	−	0.398075i
1.82710	+	0.813447i
1.82710	−	0.813447i
−0.0861690	+	1.99814i
−0.0861690	−	1.99814i

−3.31046

−

3.99629i

−2.08431

1.62961i

7.95912

111.2

−3.31046

3.99629i

−2.08431

−

1.62961i

7.95912

111.3

−2.26497

−

1.62689i

2.64055

0.165823i

2.13007

111.4

−2.26497

1.62689i

2.64055

−

0.165823i

2.13007

111.5

−1.37550

−

0.796150i

−1.56191

−

2.13552i

−1.10800

111.6

−1.37550

0.796150i

−1.56191

2.13552i

−1.10800

111.7

−0.137121

−

3.27858i

0.493541

2.59931i

−2.98120

111.8

−0.137121

3.27858i

0.493541

−

2.59931i

−2.98120

111.9

0.137121

−

3.27858i

−0.493541

−

2.59931i

−2.98120

111.10

0.137121

3.27858i

−0.493541

2.59931i

−2.98120

111.11

1.37550

−

0.796150i

1.56191

2.13552i

−1.10800

111.12

1.37550

0.796150i

1.56191

−

2.13552i

−1.10800

111.13

2.26497

−

1.62689i

−2.64055

−

0.165823i

2.13007

111.14

2.26497

1.62689i

−2.64055

0.165823i

2.13007

111.15

3.31046

−

3.99629i

2.08431

−

1.62961i

7.95912

111.16

3.31046

3.99629i

2.08431

1.62961i

7.95912

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
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	1232.2.j.a	✓	16
4.b	odd	2	1	inner	1232.2.j.a	✓	16
7.b	odd	2	1	inner	1232.2.j.a	✓	16
28.d	even	2	1	inner	1232.2.j.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1232.2.j.a	✓	16	1.a	even	1	1	trivial
1232.2.j.a	✓	16	4.b	odd	2	1	inner
1232.2.j.a	✓	16	7.b	odd	2	1	inner
1232.2.j.a	✓	16	28.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 18T_{3}^{6} + 87T_{3}^{4} - 108T_{3}^{2} + 2 $ T3^8 - 18*T3^6 + 87*T3^4 - 108*T3^2 + 2 acting on $S_{2}^{\mathrm{new}}(1232, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 18 T^{6} + 87 T^{4} + \cdots + 2)^{2} $ (T^8 - 18T^6 + 87T^4 - 108*T^2 + 2)^2
$5$	$ (T^{8} + 30 T^{6} + \cdots + 288)^{2} $ (T^8 + 30T^6 + 261T^4 + 608*T^2 + 288)^2
$7$	$ T^{16} - 144 T^{10} + \cdots + 5764801 $ T^16 - 144T^10 - 2210T^8 - 7056*T^6 + 5764801
$11$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$13$	$ (T^{8} + 56 T^{6} + \cdots + 10368)^{2} $ (T^8 + 56T^6 + 874T^4 + 5184*T^2 + 10368)^2
$17$	$ (T^{8} + 88 T^{6} + \cdots + 23328)^{2} $ (T^8 + 88T^6 + 2602T^4 + 26640*T^2 + 23328)^2
$19$	$ (T^{8} - 72 T^{6} + \cdots + 41472)^{2} $ (T^8 - 72T^6 + 1748T^4 - 16128*T^2 + 41472)^2
$23$	$ (T^{8} + 102 T^{6} + \cdots + 9216)^{2} $ (T^8 + 102T^6 + 2937T^4 + 20800*T^2 + 9216)^2
$29$	$ (T^{4} - 2 T^{3} - 26 T^{2} + \cdots + 24)^{4} $ (T^4 - 2T^3 - 26T^2 + 64*T + 24)^4
$31$	$ (T^{8} - 114 T^{6} + \cdots + 85698)^{2} $ (T^8 - 114T^6 + 3935T^4 - 45828*T^2 + 85698)^2
$37$	$ (T^{4} + 6 T^{3} + \cdots - 674)^{4} $ (T^4 + 6T^3 - 65T^2 - 456*T - 674)^4
$41$	$ (T^{8} + 144 T^{6} + \cdots + 824328)^{2} $ (T^8 + 144T^6 + 6846T^4 + 129584*T^2 + 824328)^2
$43$	$ (T^{8} + 76 T^{6} + \cdots + 11664)^{2} $ (T^8 + 76T^6 + 1768T^4 + 13104*T^2 + 11664)^2
$47$	$ (T^{8} - 176 T^{6} + \cdots + 93312)^{2} $ (T^8 - 176T^6 + 4410T^4 - 36288*T^2 + 93312)^2
$53$	$ (T^{4} + 4 T^{3} + \cdots + 384)^{4} $ (T^4 + 4T^3 - 50T^2 - 32*T + 384)^4
$59$	$ (T^{8} - 298 T^{6} + \cdots + 1854738)^{2} $ (T^8 - 298T^6 + 29687T^4 - 1007780*T^2 + 1854738)^2
$61$	$ (T^{8} + 392 T^{6} + \cdots + 44142408)^{2} $ (T^8 + 392T^6 + 51534T^4 + 2681424*T^2 + 44142408)^2
$67$	$ (T^{8} + 422 T^{6} + \cdots + 33454656)^{2} $ (T^8 + 422T^6 + 51689T^4 + 2366800*T^2 + 33454656)^2
$71$	$ (T^{8} + 102 T^{6} + \cdots + 9216)^{2} $ (T^8 + 102T^6 + 2937T^4 + 20800*T^2 + 9216)^2
$73$	$ (T^{8} + 280 T^{6} + \cdots + 3528)^{2} $ (T^8 + 280T^6 + 6926T^4 + 19472*T^2 + 3528)^2
$79$	$ (T^{8} + 196 T^{6} + \cdots + 46656)^{2} $ (T^8 + 196T^6 + 9684T^4 + 95904*T^2 + 46656)^2
$83$	$ (T^{8} - 472 T^{6} + \cdots + 165888)^{2} $ (T^8 - 472T^6 + 54740T^4 - 585728*T^2 + 165888)^2
$89$	$ (T^{8} + 414 T^{6} + \cdots + 22257792)^{2} $ (T^8 + 414T^6 + 53925T^4 + 2355392*T^2 + 22257792)^2
$97$	$ (T^{8} + 518 T^{6} + \cdots + 6223392)^{2} $ (T^8 + 518T^6 + 77485T^4 + 2632608*T^2 + 6223392)^2
show more
show less

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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{15} q^{3} - \beta_{12} q^{5} - \beta_{13} q^{7} + (\beta_{10} + \beta_{9} + \beta_{6} + 1) q^{9}+O(q^{10}) \) q + b15 * q^3 - b12 * q^5 - b13 * q^7 + (b10 + b9 + b6 + 1) * q^9	\( q + \beta_{15} q^{3} - \beta_{12} q^{5} - \beta_{13} q^{7} + (\beta_{10} + \beta_{9} + \beta_{6} + 1) q^{9} - \beta_{2} q^{11} + (\beta_{12} - \beta_{10} + \cdots + \beta_{4}) q^{13}+ \cdots + ( - \beta_{14} + \beta_{13} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) q + b15 * q^3 - b12 * q^5 - b13 * q^7 + (b10 + b9 + b6 + 1) * q^9 - b2 * q^11 + (b12 - b10 + b9 + b4) * q^13 + (b14 - b13 + b5 + b3 + 2b2) q^15 + b7 * q^17 + (-b14 - b13 + 2b11) q^19 + (b9 - b4 + b1 + 1) * q^21 + (-b14 + b13 - b5 + b3 + 2b2) q^23 + (-b6 - b1 - 3) * q^25 + (b15 - 2b11 + b8) q^27 + (b10 + b9) * q^29 + (b14 + b13 - b11 - b8) * q^31 + b4 * q^33 + (-b15 + b14 - b11 - b8 + b5 + 3b2) q^35 + (b10 + b9 - b6 - 2) * q^37 + (-b14 + b13 - 2b5 + b3 - 4b2) * q^39 + (-b12 + 3b4) q^41 + (-b3 - 2b2) q^43 + (-2b10 + 2b9 - 2b7 - 4b4) * q^45 + (2b11 + b8) q^47 + (b12 + 2b9 + b6 - 2b4 - b1 - 1) * q^49 + (-b14 + b13 - 2b5 - b3 - 2b2) * q^51 + (b10 + b9 - b1 - 2) * q^53 - b11 * q^55 + (-b10 - b9 - 2b6 - 2) q^57 + (-2b15 + b14 + b13 + b11 + b8) q^59 + (2b12 - 3b10 + 3b9 - b7) q^61 + (b15 - b14 - b13 - b11 + b8 + b5 + b3 + 5b2) q^63 + (2b1 + 4) q^65 + (b14 - b13 + 3b5 - b3 - 2b2) * q^67 + (b12 - 2b10 + 2b9 + 2b4) q^69 + (b14 - b13 + b5 - b3 - 2b2) q^71 + (-b10 + b9 - b7 - 2b4) q^73 + (-3b15 + b14 + b13 + 3b11 - 2b8) q^75 + b10 * q^77 + (-b14 + b13 - b3 + 4b2) q^79 + (b10 + b9 + 2b6 + 3b1 + 7) * q^81 + (-2b15 + b14 + b13 - 2b11 - 2b8) q^83 + (4b10 + 4b9 + 2b6) q^85 + (2b15 - b14 - b13) q^87 + (b12 + 2b7) q^89 + (b8 - 2b3 + 2b2) * q^91 + (-b10 - b9 - b6 - 2b1 - 2) q^93 + (b14 - b13 - 2b3 - 10b2) * q^95 + (-5b12 + 2b10 - 2b9 + 4b4) * q^97 + (-b14 + b13 - b5 - b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{9}+O(q^{10}) \) 16 * q + 24 * q^9	\( 16 q + 24 q^{9} + 12 q^{21} - 40 q^{25} + 8 q^{29} - 24 q^{37} - 16 q^{53} - 40 q^{57} + 48 q^{65} + 4 q^{77} + 96 q^{81} + 32 q^{85} - 24 q^{93}+O(q^{100}) \) 16 * q + 24 * q^9 + 12 * q^21 - 40 * q^25 + 8 * q^29 - 24 * q^37 - 16 * q^53 - 40 * q^57 + 48 * q^65 + 4 * q^77 + 96 * q^81 + 32 * q^85 - 24 * 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	1232.2.j.a

Level	$1232$
Weight	$2$
Character orbit	1232.j
Analytic conductor	$9.838$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(9.83756952902\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 2x^{14} - 11x^{12} + 48x^{10} + 768x^{6} - 2816x^{4} - 8192x^{2} + 65536 \) x^16 - 2x^14 - 11x^12 + 48x^10 + 768x^6 - 2816x^4 - 8192x^2 + 65536
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{14} - 46\nu^{12} - 21\nu^{10} + 736\nu^{8} + 1152\nu^{6} + 1280\nu^{4} - 77056\nu^{2} + 12288 ) / 73728 \) (-v^14 - 46v^12 - 21v^10 + 736v^8 + 1152v^6 + 1280v^4 - 77056v^2 + 12288) / 73728
\(\beta_{2}\)	\(=\)	\( ( 3\nu^{14} - 22\nu^{12} - \nu^{10} - 192\nu^{8} + 256\nu^{6} + 3840\nu^{4} - 24832\nu^{2} + 20480 ) / 73728 \) (3v^14 - 22v^12 - v^10 - 192v^8 + 256v^6 + 3840v^4 - 24832v^2 + 20480) / 73728
\(\beta_{3}\)	\(=\)	\( ( \nu^{14} - 18\nu^{12} + 149\nu^{10} - 32\nu^{8} - 128\nu^{6} - 1280\nu^{4} - 25344\nu^{2} + 135168 ) / 24576 \) (v^14 - 18v^12 + 149v^10 - 32v^8 - 128v^6 - 1280v^4 - 25344v^2 + 135168) / 24576
\(\beta_{4}\)	\(=\)	\( ( -\nu^{15} - 14\nu^{13} + 43\nu^{11} + 128\nu^{9} - 768\nu^{7} - 768\nu^{5} - 25856\nu^{3} + 69632\nu ) / 98304 \) (-v^15 - 14v^13 + 43v^11 + 128v^9 - 768v^7 - 768v^5 - 25856v^3 + 69632*v) / 98304
\(\beta_{5}\)	\(=\)	\( ( 3\nu^{14} - 6\nu^{12} + 31\nu^{10} + 16\nu^{8} + 320\nu^{6} + 3328\nu^{4} - 7424\nu^{2} + 24576 ) / 12288 \) (3v^14 - 6v^12 + 31v^10 + 16v^8 + 320v^6 + 3328v^4 - 7424*v^2 + 24576) / 12288
\(\beta_{6}\)	\(=\)	\( ( 19\nu^{14} + 10\nu^{12} - 177\nu^{10} + 128\nu^{8} - 1152\nu^{6} + 12544\nu^{4} - 47360\nu^{2} - 159744 ) / 73728 \) (19v^14 + 10v^12 - 177v^10 + 128v^8 - 1152v^6 + 12544v^4 - 47360*v^2 - 159744) / 73728
\(\beta_{7}\)	\(=\)	\( ( 5\nu^{15} + 6\nu^{13} + 169\nu^{11} + 64\nu^{9} - 1024\nu^{7} + 22784\nu^{5} - 16128\nu^{3} + 77824\nu ) / 147456 \) (5v^15 + 6v^13 + 169v^11 + 64v^9 - 1024v^7 + 22784v^5 - 16128v^3 + 77824v) / 147456
\(\beta_{8}\)	\(=\)	\( ( \nu^{15} - 34\nu^{13} + 85\nu^{11} + 80\nu^{9} - 352\nu^{7} + 1024\nu^{5} - 14080\nu^{3} + 96256\nu ) / 36864 \) (v^15 - 34v^13 + 85v^11 + 80v^9 - 352v^7 + 1024v^5 - 14080v^3 + 96256*v) / 36864
\(\beta_{9}\)	\(=\)	\( ( - 25 \nu^{15} - 36 \nu^{14} + 66 \nu^{13} - 120 \nu^{12} + 499 \nu^{11} + 780 \nu^{10} + \cdots + 638976 ) / 589824 \) (-25v^15 - 36v^14 + 66v^13 - 120v^12 + 499v^11 + 780v^10 + 160v^9 + 384v^8 + 2048v^7 - 9216v^6 - 29440v^5 + 21504v^4 + 74496v^3 + 3072v^2 + 126976*v + 638976) / 589824
\(\beta_{10}\)	\(=\)	\( ( 25 \nu^{15} - 36 \nu^{14} - 66 \nu^{13} - 120 \nu^{12} - 499 \nu^{11} + 780 \nu^{10} + \cdots + 638976 ) / 589824 \) (25v^15 - 36v^14 - 66v^13 - 120v^12 - 499v^11 + 780v^10 - 160v^9 + 384v^8 - 2048v^7 - 9216v^6 + 29440v^5 + 21504v^4 - 74496v^3 + 3072v^2 - 126976*v + 638976) / 589824
\(\beta_{11}\)	\(=\)	\( ( - 17 \nu^{15} + 146 \nu^{13} - 293 \nu^{11} + 512 \nu^{9} - 4096 \nu^{7} - 15104 \nu^{5} + \cdots - 438272 \nu ) / 294912 \) (-17v^15 + 146v^13 - 293v^11 + 512v^9 - 4096v^7 - 15104v^5 + 174848v^3 - 438272v) / 294912
\(\beta_{12}\)	\(=\)	\( ( \nu^{15} - 2\nu^{13} - 11\nu^{11} + 48\nu^{9} + 768\nu^{5} - 2816\nu^{3} + 8192\nu ) / 16384 \) (v^15 - 2v^13 - 11v^11 + 48v^9 + 768v^5 - 2816v^3 + 8192v) / 16384
\(\beta_{13}\)	\(=\)	\( ( - 35 \nu^{15} - 36 \nu^{14} + 182 \nu^{13} + 392 \nu^{12} - 95 \nu^{11} - 244 \nu^{10} + \cdots - 606208 ) / 589824 \) (-35v^15 - 36v^14 + 182v^13 + 392v^12 - 95v^11 - 244v^10 - 352v^9 - 1152v^8 - 4096v^7 + 7168v^6 - 28928v^5 + 9216v^4 + 225536v^3 + 232448v^2 + 4096*v - 606208) / 589824
\(\beta_{14}\)	\(=\)	\( ( - 35 \nu^{15} + 36 \nu^{14} + 182 \nu^{13} - 392 \nu^{12} - 95 \nu^{11} + 244 \nu^{10} + \cdots + 606208 ) / 589824 \) (-35v^15 + 36v^14 + 182v^13 - 392v^12 - 95v^11 + 244v^10 - 352v^9 + 1152v^8 - 4096v^7 - 7168v^6 - 28928v^5 - 9216v^4 + 225536v^3 - 232448v^2 + 4096*v + 606208) / 589824
\(\beta_{15}\)	\(=\)	\( ( 7\nu^{15} - 22\nu^{13} + 67\nu^{11} - 88\nu^{9} - 256\nu^{7} + 4096\nu^{5} - 24832\nu^{3} + 71680\nu ) / 73728 \) (7v^15 - 22v^13 + 67v^11 - 88v^9 - 256v^7 + 4096v^5 - 24832v^3 + 71680v) / 73728

\(\nu\)	\(=\)	\( ( \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} ) / 2 \) (b14 + b13 + b12 - b11) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{14} - \beta_{13} - \beta_{6} + \beta_{5} - \beta_{3} - 2\beta_{2} - \beta_1 ) / 2 \) (b14 - b13 - b6 + b5 - b3 - 2*b2 - b1) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + 2\beta_{8} - 6\beta_{4} ) / 2 \) (b14 + b13 + b12 + b11 + 2b8 - 6b4) / 2
\(\nu^{4}\)	\(=\)	\( ( - 3 \beta_{14} + 3 \beta_{13} + 6 \beta_{10} + 6 \beta_{9} + \beta_{6} + 3 \beta_{5} - 3 \beta_{3} + \cdots + 4 ) / 2 \) (-3b14 + 3b13 + 6b10 + 6b9 + b6 + 3b5 - 3b3 + 6*b2 + b1 + 4) / 2
\(\nu^{5}\)	\(=\)	\( ( - 12 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - 5 \beta_{11} + 6 \beta_{10} + \cdots - 6 \beta_{4} ) / 2 \) (-12b15 + b14 + b13 - b12 - 5b11 + 6b10 - 6b9 + 2b8 + 12b7 - 6*b4) / 2
\(\nu^{6}\)	\(=\)	\( ( - 7 \beta_{14} + 7 \beta_{13} - 24 \beta_{10} - 24 \beta_{9} - 25 \beta_{6} + 17 \beta_{5} + \cdots + 11 \beta_1 ) / 2 \) (-7b14 + 7b13 - 24b10 - 24b9 - 25b6 + 17b5 - 5b3 + 14b2 + 11*b1) / 2
\(\nu^{7}\)	\(=\)	\( ( - 48 \beta_{15} - 11 \beta_{14} - 11 \beta_{13} + 41 \beta_{12} - 55 \beta_{11} - 36 \beta_{10} + \cdots - 78 \beta_{4} ) / 2 \) (-48b15 - 11b14 - 11b13 + 41b12 - 55b11 - 36b10 + 36b9 - 6b8 - 78*b4) / 2
\(\nu^{8}\)	\(=\)	\( ( 33 \beta_{14} - 33 \beta_{13} + 18 \beta_{10} + 18 \beta_{9} + 9 \beta_{6} + 51 \beta_{5} - 27 \beta_{3} + \cdots + 44 ) / 2 \) (33b14 - 33b13 + 18b10 + 18b9 + 9b6 + 51b5 - 27b3 - 354b2 + 81*b1 + 44) / 2
\(\nu^{9}\)	\(=\)	\( ( - 132 \beta_{15} - 155 \beta_{14} - 155 \beta_{13} + 343 \beta_{12} + 203 \beta_{11} + \cdots + 66 \beta_{4} ) / 2 \) (-132b15 - 155b14 - 155b13 + 343b12 + 203b11 - 198b10 + 198b9 + 42b8 + 36b7 + 66b4) / 2
\(\nu^{10}\)	\(=\)	\( ( - 59 \beta_{14} + 59 \beta_{13} + 60 \beta_{10} + 60 \beta_{9} - 49 \beta_{6} + 145 \beta_{5} + \cdots - 1512 ) / 2 \) (-59b14 + 59b13 + 60b10 + 60b9 - 49b6 + 145b5 + 227b3 - 458b2 - 85*b1 - 1512) / 2
\(\nu^{11}\)	\(=\)	\( ( 552 \beta_{15} - 479 \beta_{14} - 479 \beta_{13} - 479 \beta_{12} + 553 \beta_{11} - 696 \beta_{10} + \cdots - 54 \beta_{4} ) / 2 \) (552b15 - 479b14 - 479b13 - 479b12 + 553b11 - 696b10 + 696b9 + 50b8 + 264b7 - 54b4) / 2
\(\nu^{12}\)	\(=\)	\( ( - 1467 \beta_{14} + 1467 \beta_{13} - 66 \beta_{10} - 66 \beta_{9} + 1201 \beta_{6} - 477 \beta_{5} + \cdots + 2068 ) / 2 \) (-1467b14 + 1467b13 - 66b10 - 66b9 + 1201b6 - 477b5 + 909b3 - 1386b2 + 193*b1 + 2068) / 2
\(\nu^{13}\)	\(=\)	\( ( 1956 \beta_{15} + 937 \beta_{14} + 937 \beta_{13} + 1871 \beta_{12} - 677 \beta_{11} + \cdots + 2826 \beta_{4} ) / 2 \) (1956b15 + 937b14 + 937b13 + 1871b12 - 677b11 - 1842b10 + 1842b9 - 2734b8 + 924b7 + 2826b4) / 2
\(\nu^{14}\)	\(=\)	\( ( 4049 \beta_{14} - 4049 \beta_{13} - 4944 \beta_{10} - 4944 \beta_{9} + 1943 \beta_{6} + 2801 \beta_{5} + \cdots - 1296 ) / 2 \) (4049b14 - 4049b13 - 4944b10 - 4944b9 + 1943b6 + 2801b5 + 1003b3 - 9250b2 - 3925*b1 - 1296) / 2
\(\nu^{15}\)	\(=\)	\( ( 25536 \beta_{15} - 2099 \beta_{14} - 2099 \beta_{13} + 10169 \beta_{12} + 9833 \beta_{11} + \cdots - 10398 \beta_{4} ) / 2 \) (25536b15 - 2099b14 - 2099b13 + 10169b12 + 9833b11 - 6444b10 + 6444b9 - 2838b8 - 6192b7 - 10398b4) / 2

\(n\)	\(309\)	\(353\)	\(463\)	\(673\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 111.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 18 T^{6} + 87 T^{4} + \cdots + 2)^{2} \) (T^8 - 18T^6 + 87T^4 - 108*T^2 + 2)^2
$5$	\( (T^{8} + 30 T^{6} + \cdots + 288)^{2} \) (T^8 + 30T^6 + 261T^4 + 608*T^2 + 288)^2
$7$	\( T^{16} - 144 T^{10} + \cdots + 5764801 \) T^16 - 144T^10 - 2210T^8 - 7056*T^6 + 5764801
$11$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$13$	\( (T^{8} + 56 T^{6} + \cdots + 10368)^{2} \) (T^8 + 56T^6 + 874T^4 + 5184*T^2 + 10368)^2
$17$	\( (T^{8} + 88 T^{6} + \cdots + 23328)^{2} \) (T^8 + 88T^6 + 2602T^4 + 26640*T^2 + 23328)^2
$19$	\( (T^{8} - 72 T^{6} + \cdots + 41472)^{2} \) (T^8 - 72T^6 + 1748T^4 - 16128*T^2 + 41472)^2
$23$	\( (T^{8} + 102 T^{6} + \cdots + 9216)^{2} \) (T^8 + 102T^6 + 2937T^4 + 20800*T^2 + 9216)^2
$29$	\( (T^{4} - 2 T^{3} - 26 T^{2} + \cdots + 24)^{4} \) (T^4 - 2T^3 - 26T^2 + 64*T + 24)^4
$31$	\( (T^{8} - 114 T^{6} + \cdots + 85698)^{2} \) (T^8 - 114T^6 + 3935T^4 - 45828*T^2 + 85698)^2
$37$	\( (T^{4} + 6 T^{3} + \cdots - 674)^{4} \) (T^4 + 6T^3 - 65T^2 - 456*T - 674)^4
$41$	\( (T^{8} + 144 T^{6} + \cdots + 824328)^{2} \) (T^8 + 144T^6 + 6846T^4 + 129584*T^2 + 824328)^2
$43$	\( (T^{8} + 76 T^{6} + \cdots + 11664)^{2} \) (T^8 + 76T^6 + 1768T^4 + 13104*T^2 + 11664)^2
$47$	\( (T^{8} - 176 T^{6} + \cdots + 93312)^{2} \) (T^8 - 176T^6 + 4410T^4 - 36288*T^2 + 93312)^2
$53$	\( (T^{4} + 4 T^{3} + \cdots + 384)^{4} \) (T^4 + 4T^3 - 50T^2 - 32*T + 384)^4
$59$	\( (T^{8} - 298 T^{6} + \cdots + 1854738)^{2} \) (T^8 - 298T^6 + 29687T^4 - 1007780*T^2 + 1854738)^2
$61$	\( (T^{8} + 392 T^{6} + \cdots + 44142408)^{2} \) (T^8 + 392T^6 + 51534T^4 + 2681424*T^2 + 44142408)^2
$67$	\( (T^{8} + 422 T^{6} + \cdots + 33454656)^{2} \) (T^8 + 422T^6 + 51689T^4 + 2366800*T^2 + 33454656)^2
$71$	\( (T^{8} + 102 T^{6} + \cdots + 9216)^{2} \) (T^8 + 102T^6 + 2937T^4 + 20800*T^2 + 9216)^2
$73$	\( (T^{8} + 280 T^{6} + \cdots + 3528)^{2} \) (T^8 + 280T^6 + 6926T^4 + 19472*T^2 + 3528)^2
$79$	\( (T^{8} + 196 T^{6} + \cdots + 46656)^{2} \) (T^8 + 196T^6 + 9684T^4 + 95904*T^2 + 46656)^2
$83$	\( (T^{8} - 472 T^{6} + \cdots + 165888)^{2} \) (T^8 - 472T^6 + 54740T^4 - 585728*T^2 + 165888)^2
$89$	\( (T^{8} + 414 T^{6} + \cdots + 22257792)^{2} \) (T^8 + 414T^6 + 53925T^4 + 2355392*T^2 + 22257792)^2
$97$	\( (T^{8} + 518 T^{6} + \cdots + 6223392)^{2} \) (T^8 + 518T^6 + 77485T^4 + 2632608*T^2 + 6223392)^2
show more
show less