LMFDB - Newform orbit 2160.2.bw.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(1151,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 5, 0]))

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

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

chi := DirichletCharacter("2160.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.bw (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:	$17.2476868366$
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} - 12x^{14} + 99x^{12} - 432x^{10} + 1368x^{8} - 2214x^{6} + 2511x^{4} - 486x^{2} + 81 $ x^16 - 12x^14 + 99x^12 - 432x^10 + 1368x^8 - 2214x^6 + 2511x^4 - 486*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 720)
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_1 q^{5} + ( - \beta_{8} - \beta_{3}) q^{7}+O(q^{10}) $ q + b1 * q^5 + (-b8 - b3) * q^7	$ q + \beta_1 q^{5} + ( - \beta_{8} - \beta_{3}) q^{7} + ( - \beta_{12} + \beta_{4}) q^{11} + (\beta_{9} + \beta_{7} - \beta_{2} + \cdots + 1) q^{13}+ \cdots + ( - 6 \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{97}+O(q^{100}) $ q + b1 * q^5 + (-b8 - b3) * q^7 + (-b12 + b4) * q^11 + (b9 + b7 - b2 + b1 + 1) * q^13 + (b10 + b9 - 2b7 + 2b6 + b5 - b1 + 1) * q^17 + (-b15 - b14) * q^19 + (-b15 - b13 + b11 - b4) * q^23 + (b6 + 1) * q^25 + (-b9 + b6 - b5 + b2 - 3) * q^29 + (-b15 - b14 + b12 - 2b11 - 2b8 + b4 + 2b3) q^31 + (b11 - b3) * q^35 + (b10 - b9 + 2b6 + b5 - 2b2 - 3b1 + 1) q^37 + (b9 - b7 + b2 - 2b1 + 1) q^41 + (-2b15 + 2b14 - b13 + b12 + b11 - b4 - b3) * q^43 + (-b15 - b14 + b13 + b11 - b8 + b4 + 2b3) q^47 + (-b10 - b9 + 7b7 - 2b6 - b5 + b2 + 4b1 - 1) q^49 + (-2b10 - 2b9 - 6b6 + 2b2 - 4) * q^53 + (-b15 + b12 - b4) * q^55 + (-2b14 - 2b13 - 2b12 - 2b8 - 2b3) q^59 + (-b10 - 2b9 - 4b6 - b5 - b2 + 3b1 - 4) q^61 + (b10 - b9 + b7) * q^65 + (-2b15 + b14 + b13 - b12 - b8 + 2b4 + b3) * q^67 + (-b15 + 3b14 + 2b13 - 2b12 + 2b11 - 2b8 - 2b4 - 2b3) q^71 + (-2b10 + 2b9 - 6b7 - 2b6 + 2b2 - 8b1 - 2) * q^73 + (b9 - b7 + b2 + b1 + 1) * q^77 + (-2b15 + 2b14 + 2b13 - 2b11 - 4b8 - 2b4 - 2b3) q^79 + (2b15 + 2b14 + 2b13 - 2b12 + 2b11 - b8 + 3b3) * q^83 + (b10 + b9 - b7 + 3b6 + b5 - b2 - b1 + 1) q^85 + (4b7 - 4b6 - 2b5 - 2b2 + 2b1 - 1) q^89 + (-b15 - b14 - 2b13 - 2b8) * q^91 + (-b14 - b12) * q^95 + (-6b7 + 2b6 + 6b1 + 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 4 q^{13} + 8 q^{25} - 48 q^{29} + 8 q^{37} + 12 q^{41} + 4 q^{49} - 28 q^{61} + 12 q^{65} - 32 q^{73} + 12 q^{77} - 12 q^{85} + 16 q^{97}+O(q^{100}) $ 16 * q + 4 * q^13 + 8 * q^25 - 48 * q^29 + 8 * q^37 + 12 * q^41 + 4 * q^49 - 28 * q^61 + 12 * q^65 - 32 * q^73 + 12 * q^77 - 12 * q^85 + 16 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 12x^{14} + 99x^{12} - 432x^{10} + 1368x^{8} - 2214x^{6} + 2511x^{4} - 486x^{2} + 81 $ x^16 - 12*x^14 + 99*x^12 - 432*x^10 + 1368*x^8 - 2214*x^6 + 2511*x^4 - 486*x^2 + 81 :

$\beta_{1}$	$=$	$ ( - 7562 \nu^{14} + 112407 \nu^{12} - 1024458 \nu^{10} + 5417334 \nu^{8} - 19376352 \nu^{6} + \cdots + 17730657 ) / 14108742 $ (-7562v^14 + 112407v^12 - 1024458v^10 + 5417334v^8 - 19376352v^6 + 40723992v^4 - 49249674*v^2 + 17730657) / 14108742
$\beta_{2}$	$=$	$ ( - 10649 \nu^{14} + 116868 \nu^{12} - 972708 \nu^{10} + 4196898 \nu^{8} - 15824394 \nu^{6} + \cdots + 55511892 ) / 14108742 $ (-10649v^14 + 116868v^12 - 972708v^10 + 4196898v^8 - 15824394v^6 + 36173628v^4 - 80683425*v^2 + 55511892) / 14108742
$\beta_{3}$	$=$	$ ( 23066 \nu^{15} - 316611 \nu^{13} + 2757234 \nu^{11} - 13759200 \nu^{9} + 47569968 \nu^{7} + \cdots - 83080323 \nu ) / 14108742 $ (23066v^15 - 316611v^13 + 2757234v^11 - 13759200v^9 + 47569968v^7 - 98907750v^5 + 127418292v^3 - 83080323v) / 14108742
$\beta_{4}$	$=$	$ ( - 16265 \nu^{15} + 211059 \nu^{13} - 1814727 \nu^{11} + 8791056 \nu^{9} - 30710736 \nu^{7} + \cdots + 62606763 \nu ) / 7054371 $ (-16265v^15 + 211059v^13 - 1814727v^11 + 8791056v^9 - 30710736v^7 + 64798893v^5 - 94569309v^3 + 62606763v) / 7054371
$\beta_{5}$	$=$	$ ( - 35317 \nu^{14} + 374094 \nu^{12} - 3029718 \nu^{10} + 11564082 \nu^{8} - 35760456 \nu^{6} + \cdots - 22109274 ) / 14108742 $ (-35317v^14 + 374094v^12 - 3029718v^10 + 11564082v^8 - 35760456v^6 + 40197330v^4 - 51229125*v^2 - 22109274) / 14108742
$\beta_{6}$	$=$	$ ( - 12193 \nu^{14} + 143136 \nu^{12} - 1173312 \nu^{10} + 4997712 \nu^{8} - 15676416 \nu^{6} + \cdots + 816480 ) / 4702914 $ (-12193v^14 + 143136v^12 - 1173312v^10 + 4997712v^8 - 15676416v^6 + 24022656v^4 - 28521639*v^2 + 816480) / 4702914
$\beta_{7}$	$=$	$ ( 23\nu^{14} - 285\nu^{12} + 2364\nu^{10} - 10530\nu^{8} + 33120\nu^{6} - 53730\nu^{4} + 55053\nu^{2} - 6237 ) / 7614 $ (23v^14 - 285v^12 + 2364v^10 - 10530v^8 + 33120v^6 - 53730v^4 + 55053*v^2 - 6237) / 7614
$\beta_{8}$	$=$	$ ( - 89531 \nu^{15} + 1036515 \nu^{13} - 8393466 \nu^{11} + 34735554 \nu^{9} - 104887242 \nu^{7} + \cdots - 53503173 \nu ) / 14108742 $ (-89531v^15 + 1036515v^13 - 8393466v^11 + 34735554v^9 - 104887242v^7 + 142694892v^5 - 135912519v^3 - 53503173v) / 14108742
$\beta_{9}$	$=$	$ ( - 2035 \nu^{14} + 24309 \nu^{12} - 198183 \nu^{10} + 857412 \nu^{8} - 2647413 \nu^{6} + \cdots + 1085481 ) / 414963 $ (-2035v^14 + 24309v^12 - 198183v^10 + 857412v^8 - 2647413v^6 + 4245399v^4 - 4317489*v^2 + 1085481) / 414963
$\beta_{10}$	$=$	$ ( - 38732 \nu^{14} + 470775 \nu^{12} - 3928947 \nu^{10} + 17606043 \nu^{8} - 57697884 \nu^{6} + \cdots + 23351733 ) / 7054371 $ (-38732v^14 + 470775v^12 - 3928947v^10 + 17606043v^8 - 57697884v^6 + 101233557v^4 - 121425453*v^2 + 23351733) / 7054371
$\beta_{11}$	$=$	$ ( - 128438 \nu^{15} + 1559679 \nu^{13} - 12946890 \nu^{11} + 57361014 \nu^{9} + \cdots + 90171063 \nu ) / 14108742 $ (-128438v^15 + 1559679v^13 - 12946890v^11 + 57361014v^9 - 183712176v^7 + 307533078v^5 - 355337712v^3 + 90171063v) / 14108742
$\beta_{12}$	$=$	$ ( 137965 \nu^{15} - 1558635 \nu^{13} + 12570342 \nu^{11} - 50857398 \nu^{9} + 153437364 \nu^{7} + \cdots + 81723087 \nu ) / 14108742 $ (137965v^15 - 1558635v^13 + 12570342v^11 - 50857398v^9 + 153437364v^7 - 199058364v^5 + 202613211v^3 + 81723087v) / 14108742
$\beta_{13}$	$=$	$ ( - 101627 \nu^{15} + 1219290 \nu^{13} - 10046262 \nu^{11} + 43726257 \nu^{9} - 137854539 \nu^{7} + \cdots + 36662949 \nu ) / 7054371 $ (-101627v^15 + 1219290v^13 - 10046262v^11 + 43726257v^9 - 137854539v^7 + 220266864v^5 - 244684827v^3 + 36662949v) / 7054371
$\beta_{14}$	$=$	$ ( 259723 \nu^{15} - 3081831 \nu^{13} + 25299132 \nu^{11} - 108722934 \nu^{9} + 339601068 \nu^{7} + \cdots + 14612319 \nu ) / 14108742 $ (259723v^15 - 3081831v^13 + 25299132v^11 - 108722934v^9 + 339601068v^7 - 521002314v^5 + 549259029v^3 + 14612319v) / 14108742
$\beta_{15}$	$=$	$ ( 174589 \nu^{15} - 2061234 \nu^{13} + 16877913 \nu^{11} - 72135288 \nu^{9} + 224710938 \nu^{7} + \cdots - 27853632 \nu ) / 7054371 $ (174589v^15 - 2061234v^13 + 16877913v^11 - 72135288v^9 + 224710938v^7 - 344051145v^5 + 372836331v^3 - 27853632v) / 7054371

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

$\nu$	$=$	$ ( -\beta_{15} + \beta_{14} + 2\beta_{13} - \beta_{12} - 2\beta_{11} - 4\beta_{8} - 2\beta_{4} - 2\beta_{3} ) / 6 $ (-b15 + b14 + 2b13 - b12 - 2b11 - 4b8 - 2b4 - 2*b3) / 6
$\nu^{2}$	$=$	$ ( \beta_{10} - 2\beta_{7} - 5\beta_{6} + \beta_{5} - 2\beta_1 ) / 2 $ (b10 - 2b7 - 5b6 + b5 - 2*b1) / 2
$\nu^{3}$	$=$	$ -\beta_{15} - \beta_{14} - \beta_{13} - 3\beta_{11} - \beta_{8} - 3\beta_{3} $ -b15 - b14 - b13 - 3b11 - b8 - 3b3
$\nu^{4}$	$=$	$ ( -2\beta_{10} + 5\beta_{9} - 12\beta_{7} - 28\beta_{6} + 7\beta_{5} + 7\beta_{2} - 28 ) / 2 $ (-2b10 + 5b9 - 12b7 - 28b6 + 7b5 + 7b2 - 28) / 2
$\nu^{5}$	$=$	$ ( -9\beta_{15} - 6\beta_{14} - 30\beta_{13} + 6\beta_{12} - 6\beta_{11} + 24\beta_{8} + 9\beta_{4} - 24\beta_{3} ) / 2 $ (-9b15 - 6b14 - 30b13 + 6b12 - 6b11 + 24b8 + 9b4 - 24b3) / 2
$\nu^{6}$	$=$	$ -21\beta_{10} + 21\beta_{9} + 3\beta_{7} - 12\beta_{6} + 9\beta_{5} + 12\beta_{2} + 39\beta _1 - 63 $ -21b10 + 21b9 + 3b7 - 12b6 + 9b5 + 12b2 + 39*b1 - 63
$\nu^{7}$	$=$	$ ( - 21 \beta_{15} + 21 \beta_{14} - 144 \beta_{13} + 39 \beta_{12} + 144 \beta_{11} + 156 \beta_{8} + \cdots + 12 \beta_{3} ) / 2 $ (-21b15 + 21b14 - 144b13 + 39b12 + 144b11 + 156b8 + 18b4 + 12b3) / 2
$\nu^{8}$	$=$	$ ( -117\beta_{10} + 126\beta_{9} + 504\beta_{7} + 657\beta_{6} - 117\beta_{5} - 126\beta_{2} + 432\beta _1 + 126 ) / 2 $ (-117b10 + 126b9 + 504b7 + 657b6 - 117b5 - 126b2 + 432*b1 + 126) / 2
$\nu^{9}$	$=$	$ 24\beta_{15} + 84\beta_{14} - 12\beta_{13} + 60\beta_{12} + 417\beta_{11} - 12\beta_{8} - 60\beta_{4} + 417\beta_{3} $ 24b15 + 84b14 - 12b13 + 60b12 + 417b11 - 12b8 - 60b4 + 417b3
$\nu^{10}$	$=$	$ ( 810 \beta_{10} - 585 \beta_{9} + 2574 \beta_{7} + 4266 \beta_{6} - 1395 \beta_{5} - 1395 \beta_{2} + \cdots + 4266 ) / 2 $ (810b10 - 585b9 + 2574b7 + 4266b6 - 1395b5 - 1395b2 - 594*b1 + 4266) / 2
$\nu^{11}$	$=$	$ ( 729 \beta_{15} + 72 \beta_{14} + 4554 \beta_{13} - 72 \beta_{12} - 522 \beta_{11} + \cdots + 5076 \beta_{3} ) / 2 $ (729b15 + 72b14 + 4554b13 - 72b12 - 522b11 - 5076b8 - 729b4 + 5076b3) / 2
$\nu^{12}$	$=$	$ 4005 \beta_{10} - 4005 \beta_{9} - 2106 \beta_{7} + 1503 \beta_{6} - 2502 \beta_{5} - 1503 \beta_{2} + \cdots + 9441 $ 4005b10 - 4005b9 - 2106b7 + 1503b6 - 2502b5 - 1503b2 - 9720*b1 + 9441
$\nu^{13}$	$=$	$ ( 3591 \beta_{15} - 3591 \beta_{14} + 29862 \beta_{13} - 3213 \beta_{12} - 29862 \beta_{11} + \cdots + 4590 \beta_{3} ) / 2 $ (3591b15 - 3591b14 + 29862b13 - 3213b12 - 29862b11 - 25272b8 + 378b4 + 4590b3) / 2
$\nu^{14}$	$=$	$ ( 15849 \beta_{10} - 30240 \beta_{9} - 117234 \beta_{7} - 119313 \beta_{6} + 15849 \beta_{5} + \cdots - 30240 ) / 2 $ (15849b10 - 30240b9 - 117234b7 - 119313b6 + 15849b5 + 30240b2 - 89586*b1 - 30240) / 2
$\nu^{15}$	$=$	$ 2673 \beta_{15} - 7236 \beta_{14} + 16497 \beta_{13} - 9909 \beta_{12} - 70983 \beta_{11} + \cdots - 70983 \beta_{3} $ 2673b15 - 7236b14 + 16497b13 - 9909b12 - 70983b11 + 16497b8 + 9909b4 - 70983b3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$-\beta_{6}$

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} $

1151.1

−2.08612	+	1.20442i
−1.20298	+	0.694538i
1.20298	−	0.694538i
2.08612	−	1.20442i
−0.384853	+	0.222195i
−1.74724	+	1.00877i
1.74724	−	1.00877i
0.384853	−	0.222195i
−2.08612	−	1.20442i
−1.20298	−	0.694538i
1.20298	+	0.694538i
2.08612	+	1.20442i
−0.384853	−	0.222195i
−1.74724	−	1.00877i
1.74724	+	1.00877i
0.384853	+	0.222195i

−0.866025

−

0.500000i

−1.46341

0.844901i

1151.2

−0.866025

−

0.500000i

−0.123122

0.0710842i

1151.3

−0.866025

−

0.500000i

0.123122

−

0.0710842i

1151.4

−0.866025

−

0.500000i

1.46341

−

0.844901i

1151.5

0.866025

0.500000i

−3.76026

2.17099i

1151.6

0.866025

0.500000i

−2.49072

1.43802i

1151.7

0.866025

0.500000i

2.49072

−

1.43802i

1151.8

0.866025

0.500000i

3.76026

−

2.17099i

1871.1

−0.866025

0.500000i

−1.46341

−

0.844901i

1871.2

−0.866025

0.500000i

−0.123122

−

0.0710842i

1871.3

−0.866025

0.500000i

0.123122

0.0710842i

1871.4

−0.866025

0.500000i

1.46341

0.844901i

1871.5

0.866025

−

0.500000i

−3.76026

−

2.17099i

1871.6

0.866025

−

0.500000i

−2.49072

−

1.43802i

1871.7

0.866025

−

0.500000i

2.49072

1.43802i

1871.8

0.866025

−

0.500000i

3.76026

2.17099i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
36.h	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2160.2.bw.b		16
3.b	odd	2	1		720.2.bw.b	✓	16
4.b	odd	2	1	inner	2160.2.bw.b		16
9.c	even	3	1		720.2.bw.b	✓	16
9.c	even	3	1		6480.2.h.c		16
9.d	odd	6	1	inner	2160.2.bw.b		16
9.d	odd	6	1		6480.2.h.c		16
12.b	even	2	1		720.2.bw.b	✓	16
36.f	odd	6	1		720.2.bw.b	✓	16
36.f	odd	6	1		6480.2.h.c		16
36.h	even	6	1	inner	2160.2.bw.b		16
36.h	even	6	1		6480.2.h.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
720.2.bw.b	✓	16	3.b	odd	2	1
720.2.bw.b	✓	16	9.c	even	3	1
720.2.bw.b	✓	16	12.b	even	2	1
720.2.bw.b	✓	16	36.f	odd	6	1
2160.2.bw.b		16	1.a	even	1	1	trivial
2160.2.bw.b		16	4.b	odd	2	1	inner
2160.2.bw.b		16	9.d	odd	6	1	inner
2160.2.bw.b		16	36.h	even	6	1	inner
6480.2.h.c		16	9.c	even	3	1
6480.2.h.c		16	9.d	odd	6	1
6480.2.h.c		16	36.f	odd	6	1
6480.2.h.c		16	36.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{16} - 30T_{7}^{14} + 666T_{7}^{12} - 6120T_{7}^{10} + 41247T_{7}^{8} - 104760T_{7}^{6} + 200394T_{7}^{4} - 4050T_{7}^{2} + 81 $ T7^16 - 30*T7^14 + 666*T7^12 - 6120*T7^10 + 41247*T7^8 - 104760*T7^6 + 200394*T7^4 - 4050*T7^2 + 81 acting on $S_{2}^{\mathrm{new}}(2160, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{4} - T^{2} + 1)^{4} $ (T^4 - T^2 + 1)^4
$7$	$ T^{16} - 30 T^{14} + \cdots + 81 $ T^16 - 30T^14 + 666T^12 - 6120T^10 + 41247T^8 - 104760T^6 + 200394T^4 - 4050*T^2 + 81
$11$	$ T^{16} + 36 T^{14} + \cdots + 20736 $ T^16 + 36T^14 + 1116T^12 + 5904T^10 + 21888T^8 + 41472T^6 + 57024T^4 + 41472*T^2 + 20736
$13$	$ (T^{8} - 2 T^{7} + 34 T^{6} + \cdots + 16)^{2} $ (T^8 - 2T^7 + 34T^6 - 92T^5 + 1048T^4 - 2264T^3 + 5896T^2 + 304*T + 16)^2
$17$	$ (T^{8} + 72 T^{6} + \cdots + 1296)^{2} $ (T^8 + 72T^6 + 828T^4 + 2592*T^2 + 1296)^2
$19$	$ (T^{8} + 60 T^{6} + \cdots + 144)^{2} $ (T^8 + 60T^6 + 756T^4 + 1440*T^2 + 144)^2
$23$	$ T^{16} + \cdots + 11433811041 $ T^16 + 90T^14 + 5634T^12 + 167400T^10 + 3519927T^8 + 48000600T^6 + 479965986T^4 + 2915953830*T^2 + 11433811041
$29$	$ (T^{4} + 12 T^{3} + 45 T^{2} + \cdots + 9)^{4} $ (T^4 + 12T^3 + 45T^2 - 36*T + 9)^4
$31$	$ T^{16} - 156 T^{14} + \cdots + 20736 $ T^16 - 156T^14 + 18396T^12 - 920304T^10 + 34789248T^8 - 18772992T^6 + 9180864T^4 - 456192*T^2 + 20736
$37$	$ (T^{4} - 2 T^{3} + \cdots - 284)^{4} $ (T^4 - 2T^3 - 102T^2 + 508*T - 284)^4
$41$	$ (T^{8} - 6 T^{7} + \cdots + 9801)^{2} $ (T^8 - 6T^7 - 18T^6 + 180T^5 + 531T^4 - 7020T^3 + 21222T^2 - 23166*T + 9801)^2
$43$	$ T^{16} + \cdots + 303595776 $ T^16 - 168T^14 + 19404T^12 - 1185408T^10 + 52881408T^8 - 1301057856T^6 + 21802447296T^4 - 2581818624*T^2 + 303595776
$47$	$ T^{16} + \cdots + 83156680161 $ T^16 + 198T^14 + 28674T^12 + 1876248T^10 + 89932023T^8 + 984569256T^6 + 7851562146T^4 + 30090151674*T^2 + 83156680161
$53$	$ (T^{4} + 144 T^{2} + 2304)^{4} $ (T^4 + 144*T^2 + 2304)^4
$59$	$ T^{16} + \cdots + 77720518656 $ T^16 + 216T^14 + 39456T^12 + 1396224T^10 + 34391808T^8 + 451878912T^6 + 4311097344T^4 + 22159982592*T^2 + 77720518656
$61$	$ (T^{8} + 14 T^{7} + \cdots + 6713281)^{2} $ (T^8 + 14T^7 + 226T^6 + 1496T^5 + 16903T^4 + 101288T^3 + 840034T^2 + 2482178*T + 6713281)^2
$67$	$ T^{16} + \cdots + 6439662447201 $ T^16 - 306T^14 + 70146T^12 - 6281064T^10 + 410490423T^8 - 9098217432T^6 + 145996644834T^4 - 1150666487262*T^2 + 6439662447201
$71$	$ (T^{8} - 468 T^{6} + \cdots + 17539344)^{2} $ (T^8 - 468T^6 + 66420T^4 - 2654496*T^2 + 17539344)^2
$73$	$ (T^{4} + 8 T^{3} + \cdots - 704)^{4} $ (T^4 + 8T^3 - 192T^2 - 832*T - 704)^4
$79$	$ T^{16} + \cdots + 18\!\cdots\!36 $ T^16 - 480T^14 + 149184T^12 - 27648000T^10 + 3738304512T^8 - 328635187200T^6 + 20983934877696T^4 - 777462963240960*T^2 + 18815867877851136
$83$	$ T^{16} + \cdots + 707275355433201 $ T^16 + 378T^14 + 96714T^12 + 13213368T^10 + 1303923663T^8 + 77749267176T^6 + 3264176402586T^4 + 56365922444454*T^2 + 707275355433201
$89$	$ (T^{8} + 324 T^{6} + \cdots + 14220441)^{2} $ (T^8 + 324T^6 + 35982T^4 + 1499796*T^2 + 14220441)^2
$97$	$ (T^{4} - 4 T^{3} + \cdots + 10816)^{4} $ (T^4 - 4T^3 + 120T^2 + 416*T + 10816)^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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.bw (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{5} + ( - \beta_{8} - \beta_{3}) q^{7}+O(q^{10}) \) q + b1 * q^5 + (-b8 - b3) * q^7	\( q + \beta_1 q^{5} + ( - \beta_{8} - \beta_{3}) q^{7} + ( - \beta_{12} + \beta_{4}) q^{11} + (\beta_{9} + \beta_{7} - \beta_{2} + \cdots + 1) q^{13}+ \cdots + ( - 6 \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{97}+O(q^{100}) \) q + b1 * q^5 + (-b8 - b3) * q^7 + (-b12 + b4) * q^11 + (b9 + b7 - b2 + b1 + 1) * q^13 + (b10 + b9 - 2b7 + 2b6 + b5 - b1 + 1) * q^17 + (-b15 - b14) * q^19 + (-b15 - b13 + b11 - b4) * q^23 + (b6 + 1) * q^25 + (-b9 + b6 - b5 + b2 - 3) * q^29 + (-b15 - b14 + b12 - 2b11 - 2b8 + b4 + 2b3) q^31 + (b11 - b3) * q^35 + (b10 - b9 + 2b6 + b5 - 2b2 - 3b1 + 1) q^37 + (b9 - b7 + b2 - 2b1 + 1) q^41 + (-2b15 + 2b14 - b13 + b12 + b11 - b4 - b3) * q^43 + (-b15 - b14 + b13 + b11 - b8 + b4 + 2b3) q^47 + (-b10 - b9 + 7b7 - 2b6 - b5 + b2 + 4b1 - 1) q^49 + (-2b10 - 2b9 - 6b6 + 2b2 - 4) * q^53 + (-b15 + b12 - b4) * q^55 + (-2b14 - 2b13 - 2b12 - 2b8 - 2b3) q^59 + (-b10 - 2b9 - 4b6 - b5 - b2 + 3b1 - 4) q^61 + (b10 - b9 + b7) * q^65 + (-2b15 + b14 + b13 - b12 - b8 + 2b4 + b3) * q^67 + (-b15 + 3b14 + 2b13 - 2b12 + 2b11 - 2b8 - 2b4 - 2b3) q^71 + (-2b10 + 2b9 - 6b7 - 2b6 + 2b2 - 8b1 - 2) * q^73 + (b9 - b7 + b2 + b1 + 1) * q^77 + (-2b15 + 2b14 + 2b13 - 2b11 - 4b8 - 2b4 - 2b3) q^79 + (2b15 + 2b14 + 2b13 - 2b12 + 2b11 - b8 + 3b3) * q^83 + (b10 + b9 - b7 + 3b6 + b5 - b2 - b1 + 1) q^85 + (4b7 - 4b6 - 2b5 - 2b2 + 2b1 - 1) q^89 + (-b15 - b14 - 2b13 - 2b8) * q^91 + (-b14 - b12) * q^95 + (-6b7 + 2b6 + 6b1 + 2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 4 q^{13} + 8 q^{25} - 48 q^{29} + 8 q^{37} + 12 q^{41} + 4 q^{49} - 28 q^{61} + 12 q^{65} - 32 q^{73} + 12 q^{77} - 12 q^{85} + 16 q^{97}+O(q^{100}) \) 16 * q + 4 * q^13 + 8 * q^25 - 48 * q^29 + 8 * q^37 + 12 * q^41 + 4 * q^49 - 28 * q^61 + 12 * q^65 - 32 * q^73 + 12 * q^77 - 12 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	2160.2.bw.b

Level	$2160$
Weight	$2$
Character orbit	2160.bw
Analytic conductor	$17.248$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(17.2476868366\)
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:	\( x^{16} - 12x^{14} + 99x^{12} - 432x^{10} + 1368x^{8} - 2214x^{6} + 2511x^{4} - 486x^{2} + 81 \) x^16 - 12x^14 + 99x^12 - 432x^10 + 1368x^8 - 2214x^6 + 2511x^4 - 486*x^2 + 81
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{4} \)
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 7562 \nu^{14} + 112407 \nu^{12} - 1024458 \nu^{10} + 5417334 \nu^{8} - 19376352 \nu^{6} + \cdots + 17730657 ) / 14108742 \) (-7562v^14 + 112407v^12 - 1024458v^10 + 5417334v^8 - 19376352v^6 + 40723992v^4 - 49249674*v^2 + 17730657) / 14108742
\(\beta_{2}\)	\(=\)	\( ( - 10649 \nu^{14} + 116868 \nu^{12} - 972708 \nu^{10} + 4196898 \nu^{8} - 15824394 \nu^{6} + \cdots + 55511892 ) / 14108742 \) (-10649v^14 + 116868v^12 - 972708v^10 + 4196898v^8 - 15824394v^6 + 36173628v^4 - 80683425*v^2 + 55511892) / 14108742
\(\beta_{3}\)	\(=\)	\( ( 23066 \nu^{15} - 316611 \nu^{13} + 2757234 \nu^{11} - 13759200 \nu^{9} + 47569968 \nu^{7} + \cdots - 83080323 \nu ) / 14108742 \) (23066v^15 - 316611v^13 + 2757234v^11 - 13759200v^9 + 47569968v^7 - 98907750v^5 + 127418292v^3 - 83080323v) / 14108742
\(\beta_{4}\)	\(=\)	\( ( - 16265 \nu^{15} + 211059 \nu^{13} - 1814727 \nu^{11} + 8791056 \nu^{9} - 30710736 \nu^{7} + \cdots + 62606763 \nu ) / 7054371 \) (-16265v^15 + 211059v^13 - 1814727v^11 + 8791056v^9 - 30710736v^7 + 64798893v^5 - 94569309v^3 + 62606763v) / 7054371
\(\beta_{5}\)	\(=\)	\( ( - 35317 \nu^{14} + 374094 \nu^{12} - 3029718 \nu^{10} + 11564082 \nu^{8} - 35760456 \nu^{6} + \cdots - 22109274 ) / 14108742 \) (-35317v^14 + 374094v^12 - 3029718v^10 + 11564082v^8 - 35760456v^6 + 40197330v^4 - 51229125*v^2 - 22109274) / 14108742
\(\beta_{6}\)	\(=\)	\( ( - 12193 \nu^{14} + 143136 \nu^{12} - 1173312 \nu^{10} + 4997712 \nu^{8} - 15676416 \nu^{6} + \cdots + 816480 ) / 4702914 \) (-12193v^14 + 143136v^12 - 1173312v^10 + 4997712v^8 - 15676416v^6 + 24022656v^4 - 28521639*v^2 + 816480) / 4702914
\(\beta_{7}\)	\(=\)	\( ( 23\nu^{14} - 285\nu^{12} + 2364\nu^{10} - 10530\nu^{8} + 33120\nu^{6} - 53730\nu^{4} + 55053\nu^{2} - 6237 ) / 7614 \) (23v^14 - 285v^12 + 2364v^10 - 10530v^8 + 33120v^6 - 53730v^4 + 55053*v^2 - 6237) / 7614
\(\beta_{8}\)	\(=\)	\( ( - 89531 \nu^{15} + 1036515 \nu^{13} - 8393466 \nu^{11} + 34735554 \nu^{9} - 104887242 \nu^{7} + \cdots - 53503173 \nu ) / 14108742 \) (-89531v^15 + 1036515v^13 - 8393466v^11 + 34735554v^9 - 104887242v^7 + 142694892v^5 - 135912519v^3 - 53503173v) / 14108742
\(\beta_{9}\)	\(=\)	\( ( - 2035 \nu^{14} + 24309 \nu^{12} - 198183 \nu^{10} + 857412 \nu^{8} - 2647413 \nu^{6} + \cdots + 1085481 ) / 414963 \) (-2035v^14 + 24309v^12 - 198183v^10 + 857412v^8 - 2647413v^6 + 4245399v^4 - 4317489*v^2 + 1085481) / 414963
\(\beta_{10}\)	\(=\)	\( ( - 38732 \nu^{14} + 470775 \nu^{12} - 3928947 \nu^{10} + 17606043 \nu^{8} - 57697884 \nu^{6} + \cdots + 23351733 ) / 7054371 \) (-38732v^14 + 470775v^12 - 3928947v^10 + 17606043v^8 - 57697884v^6 + 101233557v^4 - 121425453*v^2 + 23351733) / 7054371
\(\beta_{11}\)	\(=\)	\( ( - 128438 \nu^{15} + 1559679 \nu^{13} - 12946890 \nu^{11} + 57361014 \nu^{9} + \cdots + 90171063 \nu ) / 14108742 \) (-128438v^15 + 1559679v^13 - 12946890v^11 + 57361014v^9 - 183712176v^7 + 307533078v^5 - 355337712v^3 + 90171063v) / 14108742
\(\beta_{12}\)	\(=\)	\( ( 137965 \nu^{15} - 1558635 \nu^{13} + 12570342 \nu^{11} - 50857398 \nu^{9} + 153437364 \nu^{7} + \cdots + 81723087 \nu ) / 14108742 \) (137965v^15 - 1558635v^13 + 12570342v^11 - 50857398v^9 + 153437364v^7 - 199058364v^5 + 202613211v^3 + 81723087v) / 14108742
\(\beta_{13}\)	\(=\)	\( ( - 101627 \nu^{15} + 1219290 \nu^{13} - 10046262 \nu^{11} + 43726257 \nu^{9} - 137854539 \nu^{7} + \cdots + 36662949 \nu ) / 7054371 \) (-101627v^15 + 1219290v^13 - 10046262v^11 + 43726257v^9 - 137854539v^7 + 220266864v^5 - 244684827v^3 + 36662949v) / 7054371
\(\beta_{14}\)	\(=\)	\( ( 259723 \nu^{15} - 3081831 \nu^{13} + 25299132 \nu^{11} - 108722934 \nu^{9} + 339601068 \nu^{7} + \cdots + 14612319 \nu ) / 14108742 \) (259723v^15 - 3081831v^13 + 25299132v^11 - 108722934v^9 + 339601068v^7 - 521002314v^5 + 549259029v^3 + 14612319v) / 14108742
\(\beta_{15}\)	\(=\)	\( ( 174589 \nu^{15} - 2061234 \nu^{13} + 16877913 \nu^{11} - 72135288 \nu^{9} + 224710938 \nu^{7} + \cdots - 27853632 \nu ) / 7054371 \) (174589v^15 - 2061234v^13 + 16877913v^11 - 72135288v^9 + 224710938v^7 - 344051145v^5 + 372836331v^3 - 27853632v) / 7054371

\(\nu\)	\(=\)	\( ( -\beta_{15} + \beta_{14} + 2\beta_{13} - \beta_{12} - 2\beta_{11} - 4\beta_{8} - 2\beta_{4} - 2\beta_{3} ) / 6 \) (-b15 + b14 + 2b13 - b12 - 2b11 - 4b8 - 2b4 - 2*b3) / 6
\(\nu^{2}\)	\(=\)	\( ( \beta_{10} - 2\beta_{7} - 5\beta_{6} + \beta_{5} - 2\beta_1 ) / 2 \) (b10 - 2b7 - 5b6 + b5 - 2*b1) / 2
\(\nu^{3}\)	\(=\)	\( -\beta_{15} - \beta_{14} - \beta_{13} - 3\beta_{11} - \beta_{8} - 3\beta_{3} \) -b15 - b14 - b13 - 3b11 - b8 - 3b3
\(\nu^{4}\)	\(=\)	\( ( -2\beta_{10} + 5\beta_{9} - 12\beta_{7} - 28\beta_{6} + 7\beta_{5} + 7\beta_{2} - 28 ) / 2 \) (-2b10 + 5b9 - 12b7 - 28b6 + 7b5 + 7b2 - 28) / 2
\(\nu^{5}\)	\(=\)	\( ( -9\beta_{15} - 6\beta_{14} - 30\beta_{13} + 6\beta_{12} - 6\beta_{11} + 24\beta_{8} + 9\beta_{4} - 24\beta_{3} ) / 2 \) (-9b15 - 6b14 - 30b13 + 6b12 - 6b11 + 24b8 + 9b4 - 24b3) / 2
\(\nu^{6}\)	\(=\)	\( -21\beta_{10} + 21\beta_{9} + 3\beta_{7} - 12\beta_{6} + 9\beta_{5} + 12\beta_{2} + 39\beta _1 - 63 \) -21b10 + 21b9 + 3b7 - 12b6 + 9b5 + 12b2 + 39*b1 - 63
\(\nu^{7}\)	\(=\)	\( ( - 21 \beta_{15} + 21 \beta_{14} - 144 \beta_{13} + 39 \beta_{12} + 144 \beta_{11} + 156 \beta_{8} + \cdots + 12 \beta_{3} ) / 2 \) (-21b15 + 21b14 - 144b13 + 39b12 + 144b11 + 156b8 + 18b4 + 12b3) / 2
\(\nu^{8}\)	\(=\)	\( ( -117\beta_{10} + 126\beta_{9} + 504\beta_{7} + 657\beta_{6} - 117\beta_{5} - 126\beta_{2} + 432\beta _1 + 126 ) / 2 \) (-117b10 + 126b9 + 504b7 + 657b6 - 117b5 - 126b2 + 432*b1 + 126) / 2
\(\nu^{9}\)	\(=\)	\( 24\beta_{15} + 84\beta_{14} - 12\beta_{13} + 60\beta_{12} + 417\beta_{11} - 12\beta_{8} - 60\beta_{4} + 417\beta_{3} \) 24b15 + 84b14 - 12b13 + 60b12 + 417b11 - 12b8 - 60b4 + 417b3
\(\nu^{10}\)	\(=\)	\( ( 810 \beta_{10} - 585 \beta_{9} + 2574 \beta_{7} + 4266 \beta_{6} - 1395 \beta_{5} - 1395 \beta_{2} + \cdots + 4266 ) / 2 \) (810b10 - 585b9 + 2574b7 + 4266b6 - 1395b5 - 1395b2 - 594*b1 + 4266) / 2
\(\nu^{11}\)	\(=\)	\( ( 729 \beta_{15} + 72 \beta_{14} + 4554 \beta_{13} - 72 \beta_{12} - 522 \beta_{11} + \cdots + 5076 \beta_{3} ) / 2 \) (729b15 + 72b14 + 4554b13 - 72b12 - 522b11 - 5076b8 - 729b4 + 5076b3) / 2
\(\nu^{12}\)	\(=\)	\( 4005 \beta_{10} - 4005 \beta_{9} - 2106 \beta_{7} + 1503 \beta_{6} - 2502 \beta_{5} - 1503 \beta_{2} + \cdots + 9441 \) 4005b10 - 4005b9 - 2106b7 + 1503b6 - 2502b5 - 1503b2 - 9720*b1 + 9441
\(\nu^{13}\)	\(=\)	\( ( 3591 \beta_{15} - 3591 \beta_{14} + 29862 \beta_{13} - 3213 \beta_{12} - 29862 \beta_{11} + \cdots + 4590 \beta_{3} ) / 2 \) (3591b15 - 3591b14 + 29862b13 - 3213b12 - 29862b11 - 25272b8 + 378b4 + 4590b3) / 2
\(\nu^{14}\)	\(=\)	\( ( 15849 \beta_{10} - 30240 \beta_{9} - 117234 \beta_{7} - 119313 \beta_{6} + 15849 \beta_{5} + \cdots - 30240 ) / 2 \) (15849b10 - 30240b9 - 117234b7 - 119313b6 + 15849b5 + 30240b2 - 89586*b1 - 30240) / 2
\(\nu^{15}\)	\(=\)	\( 2673 \beta_{15} - 7236 \beta_{14} + 16497 \beta_{13} - 9909 \beta_{12} - 70983 \beta_{11} + \cdots - 70983 \beta_{3} \) 2673b15 - 7236b14 + 16497b13 - 9909b12 - 70983b11 + 16497b8 + 9909b4 - 70983b3

\(n\)	\(271\)	\(1297\)	\(1621\)	\(2081\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(-\beta_{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1151.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{4} - T^{2} + 1)^{4} \) (T^4 - T^2 + 1)^4
$7$	\( T^{16} - 30 T^{14} + \cdots + 81 \) T^16 - 30T^14 + 666T^12 - 6120T^10 + 41247T^8 - 104760T^6 + 200394T^4 - 4050*T^2 + 81
$11$	\( T^{16} + 36 T^{14} + \cdots + 20736 \) T^16 + 36T^14 + 1116T^12 + 5904T^10 + 21888T^8 + 41472T^6 + 57024T^4 + 41472*T^2 + 20736
$13$	\( (T^{8} - 2 T^{7} + 34 T^{6} + \cdots + 16)^{2} \) (T^8 - 2T^7 + 34T^6 - 92T^5 + 1048T^4 - 2264T^3 + 5896T^2 + 304*T + 16)^2
$17$	\( (T^{8} + 72 T^{6} + \cdots + 1296)^{2} \) (T^8 + 72T^6 + 828T^4 + 2592*T^2 + 1296)^2
$19$	\( (T^{8} + 60 T^{6} + \cdots + 144)^{2} \) (T^8 + 60T^6 + 756T^4 + 1440*T^2 + 144)^2
$23$	\( T^{16} + \cdots + 11433811041 \) T^16 + 90T^14 + 5634T^12 + 167400T^10 + 3519927T^8 + 48000600T^6 + 479965986T^4 + 2915953830*T^2 + 11433811041
$29$	\( (T^{4} + 12 T^{3} + 45 T^{2} + \cdots + 9)^{4} \) (T^4 + 12T^3 + 45T^2 - 36*T + 9)^4
$31$	\( T^{16} - 156 T^{14} + \cdots + 20736 \) T^16 - 156T^14 + 18396T^12 - 920304T^10 + 34789248T^8 - 18772992T^6 + 9180864T^4 - 456192*T^2 + 20736
$37$	\( (T^{4} - 2 T^{3} + \cdots - 284)^{4} \) (T^4 - 2T^3 - 102T^2 + 508*T - 284)^4
$41$	\( (T^{8} - 6 T^{7} + \cdots + 9801)^{2} \) (T^8 - 6T^7 - 18T^6 + 180T^5 + 531T^4 - 7020T^3 + 21222T^2 - 23166*T + 9801)^2
$43$	\( T^{16} + \cdots + 303595776 \) T^16 - 168T^14 + 19404T^12 - 1185408T^10 + 52881408T^8 - 1301057856T^6 + 21802447296T^4 - 2581818624*T^2 + 303595776
$47$	\( T^{16} + \cdots + 83156680161 \) T^16 + 198T^14 + 28674T^12 + 1876248T^10 + 89932023T^8 + 984569256T^6 + 7851562146T^4 + 30090151674*T^2 + 83156680161
$53$	\( (T^{4} + 144 T^{2} + 2304)^{4} \) (T^4 + 144*T^2 + 2304)^4
$59$	\( T^{16} + \cdots + 77720518656 \) T^16 + 216T^14 + 39456T^12 + 1396224T^10 + 34391808T^8 + 451878912T^6 + 4311097344T^4 + 22159982592*T^2 + 77720518656
$61$	\( (T^{8} + 14 T^{7} + \cdots + 6713281)^{2} \) (T^8 + 14T^7 + 226T^6 + 1496T^5 + 16903T^4 + 101288T^3 + 840034T^2 + 2482178*T + 6713281)^2
$67$	\( T^{16} + \cdots + 6439662447201 \) T^16 - 306T^14 + 70146T^12 - 6281064T^10 + 410490423T^8 - 9098217432T^6 + 145996644834T^4 - 1150666487262*T^2 + 6439662447201
$71$	\( (T^{8} - 468 T^{6} + \cdots + 17539344)^{2} \) (T^8 - 468T^6 + 66420T^4 - 2654496*T^2 + 17539344)^2
$73$	\( (T^{4} + 8 T^{3} + \cdots - 704)^{4} \) (T^4 + 8T^3 - 192T^2 - 832*T - 704)^4
$79$	\( T^{16} + \cdots + 18\!\cdots\!36 \) T^16 - 480T^14 + 149184T^12 - 27648000T^10 + 3738304512T^8 - 328635187200T^6 + 20983934877696T^4 - 777462963240960*T^2 + 18815867877851136
$83$	\( T^{16} + \cdots + 707275355433201 \) T^16 + 378T^14 + 96714T^12 + 13213368T^10 + 1303923663T^8 + 77749267176T^6 + 3264176402586T^4 + 56365922444454*T^2 + 707275355433201
$89$	\( (T^{8} + 324 T^{6} + \cdots + 14220441)^{2} \) (T^8 + 324T^6 + 35982T^4 + 1499796*T^2 + 14220441)^2
$97$	\( (T^{4} - 4 T^{3} + \cdots + 10816)^{4} \) (T^4 - 4T^3 + 120T^2 + 416*T + 10816)^4
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