LMFDB - Newform orbit 2112.2.k.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(287,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("2112.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.k (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:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.968265199641600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 9x^{14} + 44x^{12} + 261x^{10} + 1029x^{8} + 1044x^{6} + 704x^{4} + 576x^{2} + 256 $ x^16 + 9x^14 + 44x^12 + 261x^10 + 1029x^8 + 1044x^6 + 704x^4 + 576*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
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_{9} q^{3} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{5} + (\beta_{13} + \beta_{7}) q^{7} + \beta_{2} q^{9}+O(q^{10}) $ q - b9 * q^3 + (b4 + b3 - b1 + 1) * q^5 + (b13 + b7) * q^7 + b2 * q^9	$ q - \beta_{9} q^{3} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{5} + (\beta_{13} + \beta_{7}) q^{7} + \beta_{2} q^{9} - \beta_{7} q^{11} + (\beta_{11} + \beta_{8} + \cdots + \beta_{3}) q^{13}+ \cdots - \beta_{6} q^{99}+O(q^{100}) $ q - b9 * q^3 + (b4 + b3 - b1 + 1) * q^5 + (b13 + b7) * q^7 + b2 * q^9 - b7 * q^11 + (b11 + b8 - b4 + b3) * q^13 + (b14 - b12 - b9 + 2b7 - b6) q^15 + (b10 - 2b8 + b4 - b3 + b2) q^17 + (-b14 + b9 + b6 + b5) * q^19 + (b10 + b8 - b3 + 1) * q^21 + (b15 + b14 + b12 + b9 - b6 + b5) * q^23 + (-b10 + 2b4 + 2b3 + b2 - 2b1 + 3) q^25 + (b15 + 2b13 - b9 - b5) q^27 + (-2b13 - b9 - 2b7 + b5) * q^31 + b3 * q^33 + (b14 + 2b13 - b9 + 4b7 + b6 + b5) * q^35 + (2b11 + b10 - 2b8 - 2b4 + 2b3 + b2) * q^37 + (-b15 - b14 + b13 - 2b12 - 2b7 + b5) * q^39 + (2b11 + b10 + b4 - b3 + b2) q^41 + (2b15 + 2b14 - 2b6) q^43 + (3b4 - b3 + b2 - 3b1) * q^45 + (b15 + b14 + 3b12 + 2b9 - b6 + 2b5) q^47 + (2b1 + 1) q^49 + (3b15 - b9 + 3b7 + b6) * q^51 + (b10 + b4 + b3 - b2 - b1 - 1) * q^53 + (-b13 + b9 - b7 - b5) * q^55 + (-b11 - 2b4 - b3 - b2 + 2b1 - 3) * q^57 + (-b9 - 8b7 + b5) q^59 + (3b11 - b10 - 3b8 - 2b4 + 2b3 - b2) * q^61 + (-b15 + b13 - b9 + b6 + 4b5) q^63 + (-b10 - 2b8 + b4 - b3 - b2) q^65 + (2b15 + 2b12 - b9 - b5) * q^67 + (2b11 - b10 + 2b8 + b4 + b3 - 2b2 - b1 - 4) q^69 + (-b15 + b14 + 3b12 - 3b9 - b6 - 3b5) q^71 + (b10 + 3b4 + 3b3 - b2 - 4) * q^73 + (b15 + 2b14 + 2b13 - 2b12 - 4b9 + 4b7 - 2b6 - 4b5) q^75 + (-b1 + 1) * q^77 + (-b14 - b13 - b9 + b7 - b6 + b5) * q^79 + (b10 + 4b8 + 4) q^81 + (-2b14 + 2b13 - 2b7 - 2b6) * q^83 + (2b11 + 3b10 + b4 - b3 + 3b2) q^85 + (-4b11 + b10 + 2b4 - 2b3 + b2) q^89 + (2b15 + 2b14 + 2b12 - 2b6) * q^91 + (-2b10 - 2b8 + 2b3 + b2 - 5) q^93 + (-3b14 + 2b12 + 5b9 + 3b6 + 5b5) q^95 + (b10 - b2 - 2b1 - 6) q^97 - b6 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5} - 4 q^{9}+O(q^{10}) $ 16 * q + 16 * q^5 - 4 * q^9	$ 16 q + 16 q^{5} - 4 q^{9} + 20 q^{21} + 40 q^{25} - 4 q^{45} + 16 q^{49} - 8 q^{53} - 44 q^{57} - 60 q^{69} - 56 q^{73} + 16 q^{77} + 68 q^{81} - 92 q^{93} - 88 q^{97}+O(q^{100}) $ 16 * q + 16 * q^5 - 4 * q^9 + 20 * q^21 + 40 * q^25 - 4 * q^45 + 16 * q^49 - 8 * q^53 - 44 * q^57 - 60 * q^69 - 56 * q^73 + 16 * q^77 + 68 * q^81 - 92 * q^93 - 88 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 9x^{14} + 44x^{12} + 261x^{10} + 1029x^{8} + 1044x^{6} + 704x^{4} + 576x^{2} + 256 $ x^16 + 9*x^14 + 44*x^12 + 261*x^10 + 1029*x^8 + 1044*x^6 + 704*x^4 + 576*x^2 + 256 :

$\beta_{1}$	$=$	$ ( -37\nu^{14} - 169\nu^{12} - 1320\nu^{10} - 8217\nu^{8} - 8709\nu^{6} - 5760\nu^{4} - 4928\nu^{2} + 1311424 ) / 587520 $ (-37v^14 - 169v^12 - 1320v^10 - 8217v^8 - 8709v^6 - 5760v^4 - 4928*v^2 + 1311424) / 587520
$\beta_{2}$	$=$	$ ( - 109 \nu^{14} - 1573 \nu^{12} - 9540 \nu^{10} - 49569 \nu^{8} - 243633 \nu^{6} - 573420 \nu^{4} + \cdots - 141632 ) / 81600 $ (-109v^14 - 1573v^12 - 9540v^10 - 49569v^8 - 243633v^6 - 573420v^4 - 168896*v^2 - 141632) / 81600
$\beta_{3}$	$=$	$ ( 2131 \nu^{14} + 19327 \nu^{12} + 90360 \nu^{10} + 520671 \nu^{8} + 2062467 \nu^{6} + 1325280 \nu^{4} + \cdots + 594368 ) / 1468800 $ (2131v^14 + 19327v^12 + 90360v^10 + 520671v^8 + 2062467v^6 + 1325280v^4 - 2434336*v^2 + 594368) / 1468800
$\beta_{4}$	$=$	$ ( 2143 \nu^{14} + 19731 \nu^{12} + 97680 \nu^{10} + 575163 \nu^{8} + 2303751 \nu^{6} + 2598840 \nu^{4} + \cdots + 1293504 ) / 489600 $ (2143v^14 + 19731v^12 + 97680v^10 + 575163v^8 + 2303751v^6 + 2598840v^4 + 1577792*v^2 + 1293504) / 489600
$\beta_{5}$	$=$	$ ( 6253 \nu^{15} + 50321 \nu^{13} + 223080 \nu^{11} + 1388673 \nu^{9} + 4996941 \nu^{7} + \cdots + 4934464 \nu ) / 1958400 $ (6253v^15 + 50321v^13 + 223080v^11 + 1388673v^9 + 4996941v^7 + 973440v^5 + 832832v^3 + 4934464v) / 1958400
$\beta_{6}$	$=$	$ ( 19129 \nu^{15} + 152653 \nu^{13} + 682440 \nu^{11} + 4248189 \nu^{9} + 15077913 \nu^{7} + \cdots - 15936448 \nu ) / 5875200 $ (19129v^15 + 152653v^13 + 682440v^11 + 4248189v^9 + 15077913v^7 + 2977920v^5 + 2547776v^3 - 15936448v) / 5875200
$\beta_{7}$	$=$	$ ( 1127 \nu^{15} + 9779 \nu^{13} + 45720 \nu^{11} + 273507 \nu^{9} + 1043559 \nu^{7} + \cdots + 300736 \nu ) / 345600 $ (1127v^15 + 9779v^13 + 45720v^11 + 273507v^9 + 1043559v^7 + 670560v^5 - 19712v^3 + 300736v) / 345600
$\beta_{8}$	$=$	$ ( - 2739 \nu^{14} - 23023 \nu^{12} - 107640 \nu^{10} - 656799 \nu^{8} - 2456883 \nu^{6} + \cdots - 1034432 ) / 326400 $ (-2739v^14 - 23023v^12 - 107640v^10 - 656799v^8 - 2456883v^6 - 1578720v^4 - 1674816*v^2 - 1034432) / 326400
$\beta_{9}$	$=$	$ ( - 2361 \nu^{15} - 22877 \nu^{13} - 116760 \nu^{11} - 674301 \nu^{9} - 2791017 \nu^{7} + \cdots - 1576768 \nu ) / 652800 $ (-2361v^15 - 22877v^13 - 116760v^11 - 674301v^9 - 2791017v^7 - 3745680v^5 - 1915584v^3 - 1576768v) / 652800
$\beta_{10}$	$=$	$ ( 473 \nu^{14} + 4081 \nu^{12} + 19080 \nu^{10} + 114693 \nu^{8} + 435501 \nu^{6} + 279840 \nu^{4} + \cdots + 125504 ) / 40800 $ (473v^14 + 4081v^12 + 19080v^10 + 114693v^8 + 435501v^6 + 279840v^4 + 58312*v^2 + 125504) / 40800
$\beta_{11}$	$=$	$ ( 11621 \nu^{14} + 95837 \nu^{12} + 439260 \nu^{10} + 2705961 \nu^{8} + 9941577 \nu^{6} + \cdots + 3563008 ) / 734400 $ (11621v^14 + 95837v^12 + 439260v^10 + 2705961v^8 + 9941577v^6 + 4710180v^4 + 4845424*v^2 + 3563008) / 734400
$\beta_{12}$	$=$	$ ( - 281 \nu^{15} - 2173 \nu^{13} - 9400 \nu^{11} - 59773 \nu^{9} - 206089 \nu^{7} + 12368 \nu^{5} + \cdots - 80448 \nu ) / 39168 $ (-281v^15 - 2173v^13 - 9400v^11 - 59773v^9 - 206089v^7 + 12368v^5 - 56672v^3 - 80448v) / 39168
$\beta_{13}$	$=$	$ ( - 1433 \nu^{15} - 12397 \nu^{13} - 57960 \nu^{11} - 347133 \nu^{9} - 1322937 \nu^{7} + \cdots - 381248 \nu ) / 195840 $ (-1433v^15 - 12397v^13 - 57960v^11 - 347133v^9 - 1322937v^7 - 850080v^5 + 24608v^3 - 381248v) / 195840
$\beta_{14}$	$=$	$ ( - 7839 \nu^{15} - 68923 \nu^{13} - 332040 \nu^{11} - 1987899 \nu^{9} - 7704783 \nu^{7} + \cdots - 4298432 \nu ) / 652800 $ (-7839v^15 - 68923v^13 - 332040v^11 - 1987899v^9 - 7704783v^7 - 6903120v^5 - 5265216v^3 - 4298432v) / 652800
$\beta_{15}$	$=$	$ ( 6439 \nu^{15} + 52163 \nu^{13} + 235560 \nu^{11} + 1462179 \nu^{9} + 5282103 \nu^{7} + \cdots + 1757632 \nu ) / 391680 $ (6439v^15 + 52163v^13 + 235560v^11 + 1462179v^9 + 5282103v^7 + 1775280v^5 + 2249216v^3 + 1757632v) / 391680

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{13} + \beta_{12} + \beta_{9} - \beta_{7} - 2\beta_{6} - \beta_{5} ) / 4 $ (b15 - b13 + b12 + b9 - b7 - 2*b6 - b5) / 4
$\nu^{2}$	$=$	$ ( -3\beta_{11} + 2\beta_{10} - 5\beta_{8} - 3\beta_{4} - 3\beta_{3} + 3\beta _1 - 5 ) / 4 $ (-3b11 + 2b10 - 5b8 - 3b4 - 3b3 + 3b1 - 5) / 4
$\nu^{3}$	$=$	$ ( -2\beta_{15} - 2\beta_{14} + 5\beta_{13} - \beta_{12} + 2\beta_{9} + 10\beta_{7} + 2\beta_{6} + 2\beta_{5} ) / 2 $ (-2b15 - 2b14 + 5b13 - b12 + 2b9 + 10b7 + 2b6 + 2*b5) / 2
$\nu^{4}$	$=$	$ ( -9\beta_{11} - \beta_{8} + 33\beta_{4} + 9\beta_{3} + 18\beta_{2} - 9\beta _1 + 1 ) / 4 $ (-9b11 - b8 + 33b4 + 9b3 + 18b2 - 9*b1 + 1) / 4
$\nu^{5}$	$=$	$ ( -33\beta_{15} - 10\beta_{14} - 33\beta_{13} - 63\beta_{12} - 11\beta_{9} - 63\beta_{7} - 33\beta_{5} ) / 4 $ (-33b15 - 10b14 - 33b13 - 63b12 - 11b9 - 63b7 - 33*b5) / 4
$\nu^{6}$	$=$	$ ( 45\beta_{11} - 40\beta_{10} + 20\beta_{8} - 45\beta_{4} + 45\beta_{3} - 40\beta_{2} + 36\beta _1 - 81 ) / 2 $ (45b11 - 40b10 + 20b8 - 45b4 + 45b3 - 40b2 + 36*b1 - 81) / 2
$\nu^{7}$	$=$	$ ( -17\beta_{15} + 17\beta_{13} + 143\beta_{12} - 17\beta_{9} - 143\beta_{7} + 182\beta_{6} + 389\beta_{5} ) / 4 $ (-17b15 + 17b13 + 143b12 - 17b9 - 143b7 + 182b6 + 389*b5) / 4
$\nu^{8}$	$=$	$ ( 255\beta_{11} + 126\beta_{10} + 697\beta_{8} + 255\beta_{4} - 537\beta_{3} - 255\beta _1 + 697 ) / 4 $ (255b11 + 126b10 + 697b8 + 255b4 - 537b3 - 255b1 + 697) / 4
$\nu^{9}$	$=$	$ ( 646 \beta_{15} + 578 \beta_{14} - 85 \beta_{13} + 289 \beta_{12} - 646 \beta_{9} - 190 \beta_{7} + \cdots - 646 \beta_{5} ) / 2 $ (646b15 + 578b14 - 85b13 + 289b12 - 646b9 - 190b7 - 578b6 - 646b5) / 2
$\nu^{10}$	$=$	$ ( -891\beta_{11} - 3203\beta_{8} - 3597\beta_{4} + 891\beta_{3} - 1210\beta_{2} - 891\beta _1 + 3203 ) / 4 $ (-891b11 - 3203b8 - 3597b4 + 891b3 - 1210b2 - 891b1 + 3203) / 4
$\nu^{11}$	$=$	$ ( 693\beta_{15} - 4094\beta_{14} + 693\beta_{13} + 5643\beta_{12} + 9847\beta_{9} + 5643\beta_{7} + 693\beta_{5} ) / 4 $ (693b15 - 4094b14 + 693b13 + 5643b12 + 9847b9 + 5643b7 + 693*b5) / 4
$\nu^{12}$	$=$	$ ( - 7245 \beta_{11} + 6480 \beta_{10} - 3240 \beta_{8} + 7245 \beta_{4} - 7245 \beta_{3} + 6480 \beta_{2} + \cdots - 7567 ) / 2 $ (-7245b11 + 6480b10 - 3240b8 + 7245b4 - 7245b3 + 6480b2 + 3384*b1 - 7567) / 2
$\nu^{13}$	$=$	$ ( - 21347 \beta_{15} + 21347 \beta_{13} - 47267 \beta_{12} - 21347 \beta_{9} + 47267 \beta_{7} + \cdots - 20305 \beta_{5} ) / 4 $ (-21347b15 + 21347b13 - 47267b12 - 21347b9 + 47267b7 + 466b6 - 20305*b5) / 4
$\nu^{14}$	$=$	$ ( 21957\beta_{11} - 68614\beta_{10} - 19517\beta_{8} + 21957\beta_{4} + 131469\beta_{3} - 21957\beta _1 - 19517 ) / 4 $ (21957b11 - 68614b10 - 19517b8 + 21957b4 + 131469b3 - 21957b1 - 19517) / 4
$\nu^{15}$	$=$	$ ( - 60698 \beta_{15} - 54290 \beta_{14} - 83875 \beta_{13} - 27145 \beta_{12} + 60698 \beta_{9} + \cdots + 60698 \beta_{5} ) / 2 $ (-60698b15 - 54290b14 - 83875b13 - 27145b12 + 60698b9 - 187550b7 + 54290b6 + 60698b5) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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} $

287.1

0.676408	−	0.553538i
−0.369600	+	2.25820i
0.676408	+	0.553538i
−0.369600	−	2.25820i
1.77086	+	1.44918i
−0.141174	−	0.862555i
1.77086	−	1.44918i
−0.141174	+	0.862555i
−1.77086	+	1.44918i
0.141174	−	0.862555i
−1.77086	−	1.44918i
0.141174	+	0.862555i
−0.676408	−	0.553538i
0.369600	+	2.25820i
−0.676408	+	0.553538i
0.369600	−	2.25820i

−1.62968

−

0.586627i

−2.40932

−

1.23607i

2.31174

1.91203i

287.2

−1.62968

−

0.586627i

4.40932

−

3.23607i

2.31174

1.91203i

287.3

−1.62968

0.586627i

−2.40932

1.23607i

2.31174

−

1.91203i

287.4

−1.62968

0.586627i

4.40932

3.23607i

2.31174

−

1.91203i

287.5

−0.306808

−

1.70466i

−0.173254

3.23607i

−2.81174

1.04601i

287.6

−0.306808

−

1.70466i

2.17325

1.23607i

−2.81174

1.04601i

287.7

−0.306808

1.70466i

−0.173254

−

3.23607i

−2.81174

−

1.04601i

287.8

−0.306808

1.70466i

2.17325

−

1.23607i

−2.81174

−

1.04601i

287.9

0.306808

−

1.70466i

−0.173254

3.23607i

−2.81174

−

1.04601i

287.10

0.306808

−

1.70466i

2.17325

1.23607i

−2.81174

−

1.04601i

287.11

0.306808

1.70466i

−0.173254

−

3.23607i

−2.81174

1.04601i

287.12

0.306808

1.70466i

2.17325

−

1.23607i

−2.81174

1.04601i

287.13

1.62968

−

0.586627i

−2.40932

−

1.23607i

2.31174

−

1.91203i

287.14

1.62968

−

0.586627i

4.40932

−

3.23607i

2.31174

−

1.91203i

287.15

1.62968

0.586627i

−2.40932

1.23607i

2.31174

1.91203i

287.16

1.62968

0.586627i

4.40932

3.23607i

2.31174

1.91203i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
24.f	even	2	1	inner
24.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2112.2.k.l	yes	16
3.b	odd	2	1		2112.2.k.k	✓	16
4.b	odd	2	1	inner	2112.2.k.l	yes	16
8.b	even	2	1		2112.2.k.k	✓	16
8.d	odd	2	1		2112.2.k.k	✓	16
12.b	even	2	1		2112.2.k.k	✓	16
24.f	even	2	1	inner	2112.2.k.l	yes	16
24.h	odd	2	1	inner	2112.2.k.l	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2112.2.k.k	✓	16	3.b	odd	2	1
2112.2.k.k	✓	16	8.b	even	2	1
2112.2.k.k	✓	16	8.d	odd	2	1
2112.2.k.k	✓	16	12.b	even	2	1
2112.2.k.l	yes	16	1.a	even	1	1	trivial
2112.2.k.l	yes	16	4.b	odd	2	1	inner
2112.2.k.l	yes	16	24.f	even	2	1	inner
2112.2.k.l	yes	16	24.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2112, [\chi])$:

$ T_{5}^{4} - 4T_{5}^{3} - 7T_{5}^{2} + 22T_{5} + 4 $

$ T_{19}^{8} - 60T_{19}^{6} + 512T_{19}^{4} - 960T_{19}^{2} + 256 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + T^{6} - 8 T^{4} + \cdots + 81)^{2} $ (T^8 + T^6 - 8T^4 + 9T^2 + 81)^2
$5$	$ (T^{4} - 4 T^{3} - 7 T^{2} + \cdots + 4)^{4} $ (T^4 - 4T^3 - 7T^2 + 22*T + 4)^4
$7$	$ (T^{4} + 12 T^{2} + 16)^{4} $ (T^4 + 12*T^2 + 16)^4
$11$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$13$	$ (T^{4} + 20 T^{2} + 16)^{4} $ (T^4 + 20*T^2 + 16)^4
$17$	$ (T^{8} + 84 T^{6} + \cdots + 102400)^{2} $ (T^8 + 84T^6 + 2384T^4 + 26880*T^2 + 102400)^2
$19$	$ (T^{8} - 60 T^{6} + \cdots + 256)^{2} $ (T^8 - 60T^6 + 512T^4 - 960*T^2 + 256)^2
$23$	$ (T^{8} - 98 T^{6} + \cdots + 16)^{2} $ (T^8 - 98T^6 + 2409T^4 - 6692*T^2 + 16)^2
$29$	$ T^{16} $ T^16
$31$	$ (T^{8} + 162 T^{6} + \cdots + 160000)^{2} $ (T^8 + 162T^6 + 7361T^4 + 106800*T^2 + 160000)^2
$37$	$ (T^{8} + 174 T^{6} + \cdots + 6400)^{2} $ (T^8 + 174T^6 + 7409T^4 + 91920*T^2 + 6400)^2
$41$	$ (T^{8} + 180 T^{6} + \cdots + 65536)^{2} $ (T^8 + 180T^6 + 9872T^4 + 153600*T^2 + 65536)^2
$43$	$ (T^{8} - 240 T^{6} + \cdots + 2027776)^{2} $ (T^8 - 240T^6 + 18272T^4 - 464640*T^2 + 2027776)^2
$47$	$ (T^{8} - 252 T^{6} + \cdots + 2611456)^{2} $ (T^8 - 252T^6 + 18944T^4 - 466368*T^2 + 2611456)^2
$53$	$ (T^{4} + 2 T^{3} + \cdots + 400)^{4} $ (T^4 + 2T^3 - 64T^2 + 40*T + 400)^4
$59$	$ (T^{8} + 282 T^{6} + \cdots + 10758400)^{2} $ (T^8 + 282T^6 + 26441T^4 + 951840*T^2 + 10758400)^2
$61$	$ (T^{8} + 372 T^{6} + \cdots + 14258176)^{2} $ (T^8 + 372T^6 + 41744T^4 + 1417728*T^2 + 14258176)^2
$67$	$ (T^{8} - 126 T^{6} + \cdots + 32400)^{2} $ (T^8 - 126T^6 + 3609T^4 - 22680*T^2 + 32400)^2
$71$	$ (T^{8} - 378 T^{6} + \cdots + 8271376)^{2} $ (T^8 - 378T^6 + 46769T^4 - 1943172*T^2 + 8271376)^2
$73$	$ (T^{4} + 14 T^{3} + \cdots - 6320)^{4} $ (T^4 + 14T^3 - 96T^2 - 1960*T - 6320)^4
$79$	$ (T^{8} + 180 T^{6} + \cdots + 20736)^{2} $ (T^8 + 180T^6 + 4608T^4 + 25920*T^2 + 20736)^2
$83$	$ (T^{8} + 504 T^{6} + \cdots + 48664576)^{2} $ (T^8 + 504T^6 + 77456T^4 + 3650304*T^2 + 48664576)^2
$89$	$ (T^{8} + 350 T^{6} + \cdots + 65536)^{2} $ (T^8 + 350T^6 + 36177T^4 + 1137920*T^2 + 65536)^2
$97$	$ (T^{4} + 22 T^{3} + \cdots - 1844)^{4} $ (T^4 + 22T^3 + 89T^2 - 352*T - 1844)^4
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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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{9} q^{3} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{5} + (\beta_{13} + \beta_{7}) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) q - b9 * q^3 + (b4 + b3 - b1 + 1) * q^5 + (b13 + b7) * q^7 + b2 * q^9	\( q - \beta_{9} q^{3} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{5} + (\beta_{13} + \beta_{7}) q^{7} + \beta_{2} q^{9} - \beta_{7} q^{11} + (\beta_{11} + \beta_{8} + \cdots + \beta_{3}) q^{13}+ \cdots - \beta_{6} q^{99}+O(q^{100}) \) q - b9 * q^3 + (b4 + b3 - b1 + 1) * q^5 + (b13 + b7) * q^7 + b2 * q^9 - b7 * q^11 + (b11 + b8 - b4 + b3) * q^13 + (b14 - b12 - b9 + 2b7 - b6) q^15 + (b10 - 2b8 + b4 - b3 + b2) q^17 + (-b14 + b9 + b6 + b5) * q^19 + (b10 + b8 - b3 + 1) * q^21 + (b15 + b14 + b12 + b9 - b6 + b5) * q^23 + (-b10 + 2b4 + 2b3 + b2 - 2b1 + 3) q^25 + (b15 + 2b13 - b9 - b5) q^27 + (-2b13 - b9 - 2b7 + b5) * q^31 + b3 * q^33 + (b14 + 2b13 - b9 + 4b7 + b6 + b5) * q^35 + (2b11 + b10 - 2b8 - 2b4 + 2b3 + b2) * q^37 + (-b15 - b14 + b13 - 2b12 - 2b7 + b5) * q^39 + (2b11 + b10 + b4 - b3 + b2) q^41 + (2b15 + 2b14 - 2b6) q^43 + (3b4 - b3 + b2 - 3b1) * q^45 + (b15 + b14 + 3b12 + 2b9 - b6 + 2b5) q^47 + (2b1 + 1) q^49 + (3b15 - b9 + 3b7 + b6) * q^51 + (b10 + b4 + b3 - b2 - b1 - 1) * q^53 + (-b13 + b9 - b7 - b5) * q^55 + (-b11 - 2b4 - b3 - b2 + 2b1 - 3) * q^57 + (-b9 - 8b7 + b5) q^59 + (3b11 - b10 - 3b8 - 2b4 + 2b3 - b2) * q^61 + (-b15 + b13 - b9 + b6 + 4b5) q^63 + (-b10 - 2b8 + b4 - b3 - b2) q^65 + (2b15 + 2b12 - b9 - b5) * q^67 + (2b11 - b10 + 2b8 + b4 + b3 - 2b2 - b1 - 4) q^69 + (-b15 + b14 + 3b12 - 3b9 - b6 - 3b5) q^71 + (b10 + 3b4 + 3b3 - b2 - 4) * q^73 + (b15 + 2b14 + 2b13 - 2b12 - 4b9 + 4b7 - 2b6 - 4b5) q^75 + (-b1 + 1) * q^77 + (-b14 - b13 - b9 + b7 - b6 + b5) * q^79 + (b10 + 4b8 + 4) q^81 + (-2b14 + 2b13 - 2b7 - 2b6) * q^83 + (2b11 + 3b10 + b4 - b3 + 3b2) q^85 + (-4b11 + b10 + 2b4 - 2b3 + b2) q^89 + (2b15 + 2b14 + 2b12 - 2b6) * q^91 + (-2b10 - 2b8 + 2b3 + b2 - 5) q^93 + (-3b14 + 2b12 + 5b9 + 3b6 + 5b5) q^95 + (b10 - b2 - 2b1 - 6) q^97 - b6 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} - 4 q^{9}+O(q^{10}) \) 16 * q + 16 * q^5 - 4 * q^9	\( 16 q + 16 q^{5} - 4 q^{9} + 20 q^{21} + 40 q^{25} - 4 q^{45} + 16 q^{49} - 8 q^{53} - 44 q^{57} - 60 q^{69} - 56 q^{73} + 16 q^{77} + 68 q^{81} - 92 q^{93} - 88 q^{97}+O(q^{100}) \) 16 * q + 16 * q^5 - 4 * q^9 + 20 * q^21 + 40 * q^25 - 4 * q^45 + 16 * q^49 - 8 * q^53 - 44 * q^57 - 60 * q^69 - 56 * q^73 + 16 * q^77 + 68 * q^81 - 92 * q^93 - 88 * q^97

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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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	2112.2.k.l

Level	$2112$
Weight	$2$
Character orbit	2112.k
Analytic conductor	$16.864$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(16.8644049069\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.968265199641600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 9x^{14} + 44x^{12} + 261x^{10} + 1029x^{8} + 1044x^{6} + 704x^{4} + 576x^{2} + 256 \) x^16 + 9x^14 + 44x^12 + 261x^10 + 1029x^8 + 1044x^6 + 704x^4 + 576*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -37\nu^{14} - 169\nu^{12} - 1320\nu^{10} - 8217\nu^{8} - 8709\nu^{6} - 5760\nu^{4} - 4928\nu^{2} + 1311424 ) / 587520 \) (-37v^14 - 169v^12 - 1320v^10 - 8217v^8 - 8709v^6 - 5760v^4 - 4928*v^2 + 1311424) / 587520
\(\beta_{2}\)	\(=\)	\( ( - 109 \nu^{14} - 1573 \nu^{12} - 9540 \nu^{10} - 49569 \nu^{8} - 243633 \nu^{6} - 573420 \nu^{4} + \cdots - 141632 ) / 81600 \) (-109v^14 - 1573v^12 - 9540v^10 - 49569v^8 - 243633v^6 - 573420v^4 - 168896*v^2 - 141632) / 81600
\(\beta_{3}\)	\(=\)	\( ( 2131 \nu^{14} + 19327 \nu^{12} + 90360 \nu^{10} + 520671 \nu^{8} + 2062467 \nu^{6} + 1325280 \nu^{4} + \cdots + 594368 ) / 1468800 \) (2131v^14 + 19327v^12 + 90360v^10 + 520671v^8 + 2062467v^6 + 1325280v^4 - 2434336*v^2 + 594368) / 1468800
\(\beta_{4}\)	\(=\)	\( ( 2143 \nu^{14} + 19731 \nu^{12} + 97680 \nu^{10} + 575163 \nu^{8} + 2303751 \nu^{6} + 2598840 \nu^{4} + \cdots + 1293504 ) / 489600 \) (2143v^14 + 19731v^12 + 97680v^10 + 575163v^8 + 2303751v^6 + 2598840v^4 + 1577792*v^2 + 1293504) / 489600
\(\beta_{5}\)	\(=\)	\( ( 6253 \nu^{15} + 50321 \nu^{13} + 223080 \nu^{11} + 1388673 \nu^{9} + 4996941 \nu^{7} + \cdots + 4934464 \nu ) / 1958400 \) (6253v^15 + 50321v^13 + 223080v^11 + 1388673v^9 + 4996941v^7 + 973440v^5 + 832832v^3 + 4934464v) / 1958400
\(\beta_{6}\)	\(=\)	\( ( 19129 \nu^{15} + 152653 \nu^{13} + 682440 \nu^{11} + 4248189 \nu^{9} + 15077913 \nu^{7} + \cdots - 15936448 \nu ) / 5875200 \) (19129v^15 + 152653v^13 + 682440v^11 + 4248189v^9 + 15077913v^7 + 2977920v^5 + 2547776v^3 - 15936448v) / 5875200
\(\beta_{7}\)	\(=\)	\( ( 1127 \nu^{15} + 9779 \nu^{13} + 45720 \nu^{11} + 273507 \nu^{9} + 1043559 \nu^{7} + \cdots + 300736 \nu ) / 345600 \) (1127v^15 + 9779v^13 + 45720v^11 + 273507v^9 + 1043559v^7 + 670560v^5 - 19712v^3 + 300736v) / 345600
\(\beta_{8}\)	\(=\)	\( ( - 2739 \nu^{14} - 23023 \nu^{12} - 107640 \nu^{10} - 656799 \nu^{8} - 2456883 \nu^{6} + \cdots - 1034432 ) / 326400 \) (-2739v^14 - 23023v^12 - 107640v^10 - 656799v^8 - 2456883v^6 - 1578720v^4 - 1674816*v^2 - 1034432) / 326400
\(\beta_{9}\)	\(=\)	\( ( - 2361 \nu^{15} - 22877 \nu^{13} - 116760 \nu^{11} - 674301 \nu^{9} - 2791017 \nu^{7} + \cdots - 1576768 \nu ) / 652800 \) (-2361v^15 - 22877v^13 - 116760v^11 - 674301v^9 - 2791017v^7 - 3745680v^5 - 1915584v^3 - 1576768v) / 652800
\(\beta_{10}\)	\(=\)	\( ( 473 \nu^{14} + 4081 \nu^{12} + 19080 \nu^{10} + 114693 \nu^{8} + 435501 \nu^{6} + 279840 \nu^{4} + \cdots + 125504 ) / 40800 \) (473v^14 + 4081v^12 + 19080v^10 + 114693v^8 + 435501v^6 + 279840v^4 + 58312*v^2 + 125504) / 40800
\(\beta_{11}\)	\(=\)	\( ( 11621 \nu^{14} + 95837 \nu^{12} + 439260 \nu^{10} + 2705961 \nu^{8} + 9941577 \nu^{6} + \cdots + 3563008 ) / 734400 \) (11621v^14 + 95837v^12 + 439260v^10 + 2705961v^8 + 9941577v^6 + 4710180v^4 + 4845424*v^2 + 3563008) / 734400
\(\beta_{12}\)	\(=\)	\( ( - 281 \nu^{15} - 2173 \nu^{13} - 9400 \nu^{11} - 59773 \nu^{9} - 206089 \nu^{7} + 12368 \nu^{5} + \cdots - 80448 \nu ) / 39168 \) (-281v^15 - 2173v^13 - 9400v^11 - 59773v^9 - 206089v^7 + 12368v^5 - 56672v^3 - 80448v) / 39168
\(\beta_{13}\)	\(=\)	\( ( - 1433 \nu^{15} - 12397 \nu^{13} - 57960 \nu^{11} - 347133 \nu^{9} - 1322937 \nu^{7} + \cdots - 381248 \nu ) / 195840 \) (-1433v^15 - 12397v^13 - 57960v^11 - 347133v^9 - 1322937v^7 - 850080v^5 + 24608v^3 - 381248v) / 195840
\(\beta_{14}\)	\(=\)	\( ( - 7839 \nu^{15} - 68923 \nu^{13} - 332040 \nu^{11} - 1987899 \nu^{9} - 7704783 \nu^{7} + \cdots - 4298432 \nu ) / 652800 \) (-7839v^15 - 68923v^13 - 332040v^11 - 1987899v^9 - 7704783v^7 - 6903120v^5 - 5265216v^3 - 4298432v) / 652800
\(\beta_{15}\)	\(=\)	\( ( 6439 \nu^{15} + 52163 \nu^{13} + 235560 \nu^{11} + 1462179 \nu^{9} + 5282103 \nu^{7} + \cdots + 1757632 \nu ) / 391680 \) (6439v^15 + 52163v^13 + 235560v^11 + 1462179v^9 + 5282103v^7 + 1775280v^5 + 2249216v^3 + 1757632v) / 391680

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{13} + \beta_{12} + \beta_{9} - \beta_{7} - 2\beta_{6} - \beta_{5} ) / 4 \) (b15 - b13 + b12 + b9 - b7 - 2*b6 - b5) / 4
\(\nu^{2}\)	\(=\)	\( ( -3\beta_{11} + 2\beta_{10} - 5\beta_{8} - 3\beta_{4} - 3\beta_{3} + 3\beta _1 - 5 ) / 4 \) (-3b11 + 2b10 - 5b8 - 3b4 - 3b3 + 3b1 - 5) / 4
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{15} - 2\beta_{14} + 5\beta_{13} - \beta_{12} + 2\beta_{9} + 10\beta_{7} + 2\beta_{6} + 2\beta_{5} ) / 2 \) (-2b15 - 2b14 + 5b13 - b12 + 2b9 + 10b7 + 2b6 + 2*b5) / 2
\(\nu^{4}\)	\(=\)	\( ( -9\beta_{11} - \beta_{8} + 33\beta_{4} + 9\beta_{3} + 18\beta_{2} - 9\beta _1 + 1 ) / 4 \) (-9b11 - b8 + 33b4 + 9b3 + 18b2 - 9*b1 + 1) / 4
\(\nu^{5}\)	\(=\)	\( ( -33\beta_{15} - 10\beta_{14} - 33\beta_{13} - 63\beta_{12} - 11\beta_{9} - 63\beta_{7} - 33\beta_{5} ) / 4 \) (-33b15 - 10b14 - 33b13 - 63b12 - 11b9 - 63b7 - 33*b5) / 4
\(\nu^{6}\)	\(=\)	\( ( 45\beta_{11} - 40\beta_{10} + 20\beta_{8} - 45\beta_{4} + 45\beta_{3} - 40\beta_{2} + 36\beta _1 - 81 ) / 2 \) (45b11 - 40b10 + 20b8 - 45b4 + 45b3 - 40b2 + 36*b1 - 81) / 2
\(\nu^{7}\)	\(=\)	\( ( -17\beta_{15} + 17\beta_{13} + 143\beta_{12} - 17\beta_{9} - 143\beta_{7} + 182\beta_{6} + 389\beta_{5} ) / 4 \) (-17b15 + 17b13 + 143b12 - 17b9 - 143b7 + 182b6 + 389*b5) / 4
\(\nu^{8}\)	\(=\)	\( ( 255\beta_{11} + 126\beta_{10} + 697\beta_{8} + 255\beta_{4} - 537\beta_{3} - 255\beta _1 + 697 ) / 4 \) (255b11 + 126b10 + 697b8 + 255b4 - 537b3 - 255b1 + 697) / 4
\(\nu^{9}\)	\(=\)	\( ( 646 \beta_{15} + 578 \beta_{14} - 85 \beta_{13} + 289 \beta_{12} - 646 \beta_{9} - 190 \beta_{7} + \cdots - 646 \beta_{5} ) / 2 \) (646b15 + 578b14 - 85b13 + 289b12 - 646b9 - 190b7 - 578b6 - 646b5) / 2
\(\nu^{10}\)	\(=\)	\( ( -891\beta_{11} - 3203\beta_{8} - 3597\beta_{4} + 891\beta_{3} - 1210\beta_{2} - 891\beta _1 + 3203 ) / 4 \) (-891b11 - 3203b8 - 3597b4 + 891b3 - 1210b2 - 891b1 + 3203) / 4
\(\nu^{11}\)	\(=\)	\( ( 693\beta_{15} - 4094\beta_{14} + 693\beta_{13} + 5643\beta_{12} + 9847\beta_{9} + 5643\beta_{7} + 693\beta_{5} ) / 4 \) (693b15 - 4094b14 + 693b13 + 5643b12 + 9847b9 + 5643b7 + 693*b5) / 4
\(\nu^{12}\)	\(=\)	\( ( - 7245 \beta_{11} + 6480 \beta_{10} - 3240 \beta_{8} + 7245 \beta_{4} - 7245 \beta_{3} + 6480 \beta_{2} + \cdots - 7567 ) / 2 \) (-7245b11 + 6480b10 - 3240b8 + 7245b4 - 7245b3 + 6480b2 + 3384*b1 - 7567) / 2
\(\nu^{13}\)	\(=\)	\( ( - 21347 \beta_{15} + 21347 \beta_{13} - 47267 \beta_{12} - 21347 \beta_{9} + 47267 \beta_{7} + \cdots - 20305 \beta_{5} ) / 4 \) (-21347b15 + 21347b13 - 47267b12 - 21347b9 + 47267b7 + 466b6 - 20305*b5) / 4
\(\nu^{14}\)	\(=\)	\( ( 21957\beta_{11} - 68614\beta_{10} - 19517\beta_{8} + 21957\beta_{4} + 131469\beta_{3} - 21957\beta _1 - 19517 ) / 4 \) (21957b11 - 68614b10 - 19517b8 + 21957b4 + 131469b3 - 21957b1 - 19517) / 4
\(\nu^{15}\)	\(=\)	\( ( - 60698 \beta_{15} - 54290 \beta_{14} - 83875 \beta_{13} - 27145 \beta_{12} + 60698 \beta_{9} + \cdots + 60698 \beta_{5} ) / 2 \) (-60698b15 - 54290b14 - 83875b13 - 27145b12 + 60698b9 - 187550b7 + 54290b6 + 60698b5) / 2

\(n\)	\(133\)	\(1409\)	\(1729\)	\(2047\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 287.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + T^{6} - 8 T^{4} + \cdots + 81)^{2} \) (T^8 + T^6 - 8T^4 + 9T^2 + 81)^2
$5$	\( (T^{4} - 4 T^{3} - 7 T^{2} + \cdots + 4)^{4} \) (T^4 - 4T^3 - 7T^2 + 22*T + 4)^4
$7$	\( (T^{4} + 12 T^{2} + 16)^{4} \) (T^4 + 12*T^2 + 16)^4
$11$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$13$	\( (T^{4} + 20 T^{2} + 16)^{4} \) (T^4 + 20*T^2 + 16)^4
$17$	\( (T^{8} + 84 T^{6} + \cdots + 102400)^{2} \) (T^8 + 84T^6 + 2384T^4 + 26880*T^2 + 102400)^2
$19$	\( (T^{8} - 60 T^{6} + \cdots + 256)^{2} \) (T^8 - 60T^6 + 512T^4 - 960*T^2 + 256)^2
$23$	\( (T^{8} - 98 T^{6} + \cdots + 16)^{2} \) (T^8 - 98T^6 + 2409T^4 - 6692*T^2 + 16)^2
$29$	\( T^{16} \) T^16
$31$	\( (T^{8} + 162 T^{6} + \cdots + 160000)^{2} \) (T^8 + 162T^6 + 7361T^4 + 106800*T^2 + 160000)^2
$37$	\( (T^{8} + 174 T^{6} + \cdots + 6400)^{2} \) (T^8 + 174T^6 + 7409T^4 + 91920*T^2 + 6400)^2
$41$	\( (T^{8} + 180 T^{6} + \cdots + 65536)^{2} \) (T^8 + 180T^6 + 9872T^4 + 153600*T^2 + 65536)^2
$43$	\( (T^{8} - 240 T^{6} + \cdots + 2027776)^{2} \) (T^8 - 240T^6 + 18272T^4 - 464640*T^2 + 2027776)^2
$47$	\( (T^{8} - 252 T^{6} + \cdots + 2611456)^{2} \) (T^8 - 252T^6 + 18944T^4 - 466368*T^2 + 2611456)^2
$53$	\( (T^{4} + 2 T^{3} + \cdots + 400)^{4} \) (T^4 + 2T^3 - 64T^2 + 40*T + 400)^4
$59$	\( (T^{8} + 282 T^{6} + \cdots + 10758400)^{2} \) (T^8 + 282T^6 + 26441T^4 + 951840*T^2 + 10758400)^2
$61$	\( (T^{8} + 372 T^{6} + \cdots + 14258176)^{2} \) (T^8 + 372T^6 + 41744T^4 + 1417728*T^2 + 14258176)^2
$67$	\( (T^{8} - 126 T^{6} + \cdots + 32400)^{2} \) (T^8 - 126T^6 + 3609T^4 - 22680*T^2 + 32400)^2
$71$	\( (T^{8} - 378 T^{6} + \cdots + 8271376)^{2} \) (T^8 - 378T^6 + 46769T^4 - 1943172*T^2 + 8271376)^2
$73$	\( (T^{4} + 14 T^{3} + \cdots - 6320)^{4} \) (T^4 + 14T^3 - 96T^2 - 1960*T - 6320)^4
$79$	\( (T^{8} + 180 T^{6} + \cdots + 20736)^{2} \) (T^8 + 180T^6 + 4608T^4 + 25920*T^2 + 20736)^2
$83$	\( (T^{8} + 504 T^{6} + \cdots + 48664576)^{2} \) (T^8 + 504T^6 + 77456T^4 + 3650304*T^2 + 48664576)^2
$89$	\( (T^{8} + 350 T^{6} + \cdots + 65536)^{2} \) (T^8 + 350T^6 + 36177T^4 + 1137920*T^2 + 65536)^2
$97$	\( (T^{4} + 22 T^{3} + \cdots - 1844)^{4} \) (T^4 + 22T^3 + 89T^2 - 352*T - 1844)^4
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