LMFDB - Newform orbit 1680.2.dx.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(31,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("1680.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.dx (of order $6$, degree $2$, 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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 47 x^{10} - 180 x^{9} + 724 x^{8} - 1882 x^{7} + 4759 x^{6} - 8380 x^{5} + \cdots + 4468 $ x^12 - 6x^11 + 47x^10 - 180x^9 + 724x^8 - 1882x^7 + 4759x^6 - 8380x^5 + 13729x^4 - 15360x^3 + 15460x^2 - 8912*x + 4468
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{11} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{11} - 1) q^{9}+O(q^{10}) $ q + b11 * q^3 + b1 * q^5 - b4 * q^7 + (-b11 - 1) * q^9	$ q + \beta_{11} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{11} - 1) q^{9} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{10} + \beta_{9} + \cdots - \beta_1) q^{99}+O(q^{100}) $ q + b11 * q^3 + b1 * q^5 - b4 * q^7 + (-b11 - 1) * q^9 + (-b9 - b8 + b6 - b5 + b3 - b2) * q^11 + (2b11 + b8 + b7 + b2 - b1 + 1) q^13 + (b2 - b1) * q^15 + (-b9 - b7 - b5 + b3 + b2) * q^17 + (b9 + b7 + b6 + b5 + b4 - b3 + 2b2 - b1) q^19 + (b9 + b4) * q^21 + (-b11 + b9 + b8 + b7 + 2b6 - b5 + b4 + b3 + 1) q^23 - b11 * q^25 + q^27 + (-b6 + b5 - b4 - b3 - 2b2 - 2b1 - 1) * q^29 + (-b10 + 2b8 + b7 + 2b6 + b5 + b4 - b3 - b2 + 2b1) q^31 + (b10 - b7 - b6 + b5 - b4 + b1) * q^33 - b8 * q^35 + (-2b11 - b9 - b4 + b3 - 2) q^37 + (-b11 - b8 + b6 - b5 - b2 - 2) * q^39 + (-b10 + b9 - b8 - b7 + b4 - b3 - b2 + b1) * q^41 + (-2b11 + b10 - b9 + b8 + b7 + 2b6 + 2b5 - b4 + b3 - 3b2 + 3b1 - 1) q^43 - b2 * q^45 + (-b11 - b10 + b9 + b7 + b6 + b5 - b3 + 2b2 - b1 - 1) q^47 + (3b11 + b10 + b9 + b5 - 2b3 + b2 - 3b1 + 1) q^49 + (b10 - b8 - b6 - b4 - b1) * q^51 + (-b11 - b10 + b9 - b8 - 2b7 - b6 + b5 + b4 - b2 + 2b1) * q^53 + (-b10 - b9 + b6 - b5 - 1) * q^55 + (-b10 - b9 + b8 - b7 - b2 - b1) * q^57 + (b11 + b10 - b9 + b8 + b7 + b6 - b4 - 2b2 + 4b1) * q^59 + (-b11 + b9 + b7 + b6 - b5 + b4 + b3 + 5b1 + 1) q^61 - b9 * q^63 + (b11 + b9 + b4 - b3 + 2b2 - b1 + 1) q^65 + (-b10 - b9 - 2b8 + 2b6 - 2b5 - b4 - 2b2) * q^67 + (2b11 + b10 - b9 - 2b8 - 2b7 - b6 - b5 + 1) q^69 + (6b11 + b6 + b5 + b4 - b3 + 4b2 - 4b1 + 3) q^71 + (-3b11 - b9 - 3b8 + 3b6 - 3b5 + b3 - 3b2 - 6) q^73 + (b11 + 1) * q^75 + (b11 + 2b10 - 2b9 - 2b8 - 3b7 - 2b6 - 2b5 - 2b4 + 3b3 + 4b2 - 5b1 - 2) * q^77 + (-b11 + 2b10 + b8 - b7 + b5 - 2b4 - 4b1 + 1) q^79 + b11 * q^81 + (3b10 + 3b9 - b8 + b7 - b6 + b5 - b4 - b3 - b2 - b1 - 4) * q^83 + (b6 - b5 + b4 + b3 + 1) * q^85 + (-b11 - b10 + b9 + b8 + b7 + b6 + b4 - 2b2 + 4b1) * q^87 + (-b11 - b10 - b9 + 3b8 - b7 + 2b6 + b5 - b3 + 4b1 + 1) q^89 + (-3b11 + 2b9 + b8 + b7 + 3b6 + 2b4 + 4b2 + 2b1 - 1) * q^91 + (-b9 - b8 - 2b7 - b6 - 2b5 - b4 + b3 + 2b2 - b1) q^93 + (b11 - b10 + b8 - b6 + b5 - b4 - b3 + 2) * q^95 + (2b11 - b10 + b9 + b8 + b7 + 2b6 + 2b5 - 3b2 + 3b1 + 1) q^97 + (-b10 + b9 + b8 + b7 + b4 - b3 + b2 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 2 q^{7} - 6 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 + 2 * q^7 - 6 * q^9	$ 12 q - 6 q^{3} + 2 q^{7} - 6 q^{9} - 6 q^{17} + 2 q^{19} - 4 q^{21} + 6 q^{23} + 6 q^{25} + 12 q^{27} - 4 q^{29} - 4 q^{31} + 4 q^{35} - 10 q^{37} - 18 q^{39} - 4 q^{49} + 6 q^{51} + 4 q^{53} - 12 q^{55} - 4 q^{57} - 6 q^{59} + 12 q^{61} + 2 q^{63} + 4 q^{65} + 6 q^{67} - 54 q^{73} + 6 q^{75} - 36 q^{77} + 12 q^{79} - 6 q^{81} - 44 q^{83} + 4 q^{85} + 2 q^{87} + 6 q^{89} - 8 q^{91} - 4 q^{93} + 24 q^{95}+O(q^{100}) $ 12 * q - 6 * q^3 + 2 * q^7 - 6 * q^9 - 6 * q^17 + 2 * q^19 - 4 * q^21 + 6 * q^23 + 6 * q^25 + 12 * q^27 - 4 * q^29 - 4 * q^31 + 4 * q^35 - 10 * q^37 - 18 * q^39 - 4 * q^49 + 6 * q^51 + 4 * q^53 - 12 * q^55 - 4 * q^57 - 6 * q^59 + 12 * q^61 + 2 * q^63 + 4 * q^65 + 6 * q^67 - 54 * q^73 + 6 * q^75 - 36 * q^77 + 12 * q^79 - 6 * q^81 - 44 * q^83 + 4 * q^85 + 2 * q^87 + 6 * q^89 - 8 * q^91 - 4 * q^93 + 24 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 6 x^{11} + 47 x^{10} - 180 x^{9} + 724 x^{8} - 1882 x^{7} + 4759 x^{6} - 8380 x^{5} + \cdots + 4468 $ x^12 - 6*x^11 + 47*x^10 - 180*x^9 + 724*x^8 - 1882*x^7 + 4759*x^6 - 8380*x^5 + 13729*x^4 - 15360*x^3 + 15460*x^2 - 8912*x + 4468 :

$\beta_{1}$	$=$	$ ( 1864 \nu^{11} + 25751 \nu^{10} - 104079 \nu^{9} + 1221120 \nu^{8} - 3898574 \nu^{7} + \cdots + 92792416 ) / 4791672 $ (1864v^11 + 25751v^10 - 104079v^9 + 1221120v^8 - 3898574v^7 + 17581312v^6 - 38300322v^5 + 97380167v^4 - 132626665v^3 + 192767056v^2 - 128227422*v + 92792416) / 4791672
$\beta_{2}$	$=$	$ ( - 1864 \nu^{11} + 46255 \nu^{10} - 255951 \nu^{9} + 1750764 \nu^{8} - 5828782 \nu^{7} + \cdots + 98612624 ) / 4791672 $ (-1864v^11 + 46255v^10 - 255951v^9 + 1750764v^8 - 5828782v^7 + 22008896v^6 - 47936682v^5 + 111535423v^4 - 152104697v^3 + 209720300v^2 - 144753870*v + 98612624) / 4791672
$\beta_{3}$	$=$	$ ( 4354 \nu^{11} + 100427 \nu^{10} - 486483 \nu^{9} + 4358760 \nu^{8} - 14640170 \nu^{7} + \cdots + 221152144 ) / 4791672 $ (4354v^11 + 100427v^10 - 486483v^9 + 4358760v^8 - 14640170v^7 + 57463768v^6 - 128631360v^5 + 291902159v^4 - 410822593v^3 + 534284836v^2 - 378089934*v + 221152144) / 4791672
$\beta_{4}$	$=$	$ ( - 4354 \nu^{11} + 148321 \nu^{10} - 757257 \nu^{9} + 5218038 \nu^{8} - 16204582 \nu^{7} + \cdots + 176595908 ) / 4791672 $ (-4354v^11 + 148321v^10 - 757257v^9 + 5218038v^8 - 16204582v^7 + 59264504v^6 - 118820532v^5 + 264638761v^4 - 323337491v^3 + 430719938v^2 - 256309110*v + 176595908) / 4791672
$\beta_{5}$	$=$	$ ( 3557 \nu^{11} - 13563 \nu^{10} + 122758 \nu^{9} - 293039 \nu^{8} + 1257560 \nu^{7} + \cdots + 13135482 ) / 798612 $ (3557v^11 - 13563v^10 + 122758v^9 - 293039v^8 + 1257560v^7 - 1553118v^6 + 3531607v^5 + 1581521v^4 - 5172298v^3 + 19041413v^2 - 16869582*v + 13135482) / 798612
$\beta_{6}$	$=$	$ ( 3557 \nu^{11} - 25564 \nu^{10} + 182763 \nu^{9} - 788353 \nu^{8} + 2878786 \nu^{7} + \cdots - 14772298 ) / 798612 $ (3557v^11 - 25564v^10 + 182763v^9 - 788353v^8 + 2878786v^7 - 8151486v^6 + 17904441v^5 - 33237744v^4 + 42282929v^3 - 47241201v^2 + 27828688*v - 14772298) / 798612
$\beta_{7}$	$=$	$ ( 10456 \nu^{11} - 52053 \nu^{10} + 426675 \nu^{9} - 1368172 \nu^{8} + 5585586 \nu^{7} + \cdots + 15673912 ) / 1597224 $ (10456v^11 - 52053v^10 + 426675v^9 - 1368172v^8 + 5585586v^7 - 11612596v^6 + 28762722v^5 - 34908725v^4 + 52412649v^3 - 21345688v^2 + 17567254*v + 15673912) / 1597224
$\beta_{8}$	$=$	$ ( 10456 \nu^{11} - 62963 \nu^{10} + 481225 \nu^{9} - 1854758 \nu^{8} + 7204630 \nu^{7} + \cdots - 51152020 ) / 1597224 $ (10456v^11 - 62963v^10 + 481225v^9 - 1854758v^8 + 7204630v^7 - 18917932v^6 + 45241186v^5 - 80719815v^4 + 118350507v^3 - 132459674v^2 + 98205246*v - 51152020) / 1597224
$\beta_{9}$	$=$	$ ( - 33232 \nu^{11} + 153319 \nu^{10} - 1290501 \nu^{9} + 3865296 \nu^{8} - 16098454 \nu^{7} + \cdots - 57858232 ) / 4791672 $ (-33232v^11 + 153319v^10 - 1290501v^9 + 3865296v^8 - 16098454v^7 + 31160180v^6 - 78839142v^5 + 87177751v^4 - 136407143v^3 + 39881876v^2 - 37032810*v - 57858232) / 4791672
$\beta_{10}$	$=$	$ ( 33232 \nu^{11} - 212233 \nu^{10} + 1585071 \nu^{9} - 6333138 \nu^{8} + 24202402 \nu^{7} + \cdots - 165321092 ) / 4791672 $ (33232v^11 - 212233v^10 + 1585071v^9 - 6333138v^8 + 24202402v^7 - 64858988v^6 + 152808942v^5 - 273863125v^4 + 395360357v^3 - 438487454v^2 + 317227794*v - 165321092) / 4791672
$\beta_{11}$	$=$	$ ( - 218 \nu^{11} + 1199 \nu^{10} - 9101 \nu^{9} + 31962 \nu^{8} - 121122 \nu^{7} + 283164 \nu^{6} + \cdots + 215540 ) / 26184 $ (-218v^11 + 1199v^10 - 9101v^9 + 31962v^8 - 121122v^7 + 283164v^6 - 647132v^5 + 978503v^4 - 1335755v^3 + 1147202v^2 - 785966*v + 215540) / 26184

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

$\nu$	$=$	$ ( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{2} + \beta _1 + 1 ) / 2 $ (-b10 + b9 + b8 + b7 - b2 + b1 + 1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{6} - 2\beta_{5} + \beta_{2} + 3\beta _1 - 9 ) / 2 $ (-b10 + b9 + b8 + b7 + 2b6 - 2b5 + b2 + 3*b1 - 9) / 2
$\nu^{3}$	$=$	$ ( - 6 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 7 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + \cdots - 17 ) / 2 $ (-6b11 + 3b10 - 3b9 - 7b8 - 7b7 + 4b6 - 2b5 + 4b4 - 4b3 + 5b2 + b1 - 17) / 2
$\nu^{4}$	$=$	$ ( - 12 \beta_{11} + \beta_{10} - 13 \beta_{9} - 13 \beta_{8} - 17 \beta_{7} - 16 \beta_{6} + 20 \beta_{5} + \cdots + 49 ) / 2 $ (-12b11 + b10 - 13b9 - 13b8 - 17b7 - 16b6 + 20b5 + 8b4 - 8b3 - 17b2 - 27b1 + 49) / 2
$\nu^{5}$	$=$	$ ( 72 \beta_{11} - 19 \beta_{10} - 11 \beta_{9} + 55 \beta_{8} + 45 \beta_{7} - 62 \beta_{6} + 38 \beta_{5} + \cdots + 197 ) / 2 $ (72b11 - 19b10 - 11b9 + 55b8 + 45b7 - 62b6 + 38b5 - 44b4 + 44b3 - 95b2 - 25*b1 + 197) / 2
$\nu^{6}$	$=$	$ ( 246 \beta_{11} + 55 \beta_{10} + 115 \beta_{9} + 147 \beta_{8} + 229 \beta_{7} + 94 \beta_{6} + \cdots - 231 ) / 2 $ (246b11 + 55b10 + 115b9 + 147b8 + 229b7 + 94b6 - 176b5 - 150b4 + 154b3 + 61b2 + 297*b1 - 231) / 2
$\nu^{7}$	$=$	$ ( - 632 \beta_{11} + 279 \beta_{10} + 421 \beta_{9} - 471 \beta_{8} - 149 \beta_{7} + 740 \beta_{6} + \cdots - 2135 ) / 2 $ (-632b11 + 279b10 + 421b9 - 471b8 - 149b7 + 740b6 - 562b5 + 366b4 - 352b3 + 1301b2 + 379*b1 - 2135) / 2
$\nu^{8}$	$=$	$ ( - 3704 \beta_{11} - 857 \beta_{10} - 603 \beta_{9} - 1719 \beta_{8} - 2587 \beta_{7} - 262 \beta_{6} + \cdots - 65 ) / 2 $ (-3704b11 - 857b10 - 603b9 - 1719b8 - 2587b7 - 262b6 + 1366b5 + 2120b4 - 2208b3 + 1345b2 - 3469*b1 - 65) / 2
$\nu^{9}$	$=$	$ ( 3682 \beta_{11} - 4559 \beta_{10} - 6337 \beta_{9} + 4103 \beta_{8} - 1777 \beta_{7} - 8120 \beta_{6} + \cdots + 21501 ) / 2 $ (3682b11 - 4559b10 - 6337b9 + 4103b8 - 1777b7 - 8120b6 + 7450b5 - 2088b4 + 1608b3 - 13577b2 - 6577*b1 + 21501) / 2
$\nu^{10}$	$=$	$ ( 47972 \beta_{11} + 8031 \beta_{10} - 2271 \beta_{9} + 21121 \beta_{8} + 25589 \beta_{7} - 4696 \beta_{6} + \cdots + 24103 ) / 2 $ (47972b11 + 8031b10 - 2271b9 + 21121b8 + 25589b7 - 4696b6 - 7528b5 - 26112b4 + 27036b3 - 35111b2 + 37699*b1 + 24103) / 2
$\nu^{11}$	$=$	$ ( 6676 \beta_{11} + 66127 \beta_{10} + 73463 \beta_{9} - 31595 \beta_{8} + 50531 \beta_{7} + 83946 \beta_{6} + \cdots - 198093 ) / 2 $ (6676b11 + 66127b10 + 73463b9 - 31595b8 + 50531b7 + 83946b6 - 89810b5 - 1800b4 + 11436b3 + 111219b2 + 107593*b1 - 198093) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1 + \beta_{11}$	$1$	$1$	$1$	$-1$

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} $

31.1

0.500000	−	2.78149i
0.500000	+	3.30094i
0.500000	+	1.34658i
0.500000	−	1.65580i
0.500000	−	0.993257i
0.500000	+	2.51508i
0.500000	+	2.78149i
0.500000	−	3.30094i
0.500000	−	1.34658i
0.500000	+	1.65580i
0.500000	+	0.993257i
0.500000	−	2.51508i

−0.500000

0.866025i

−0.866025

0.500000i

−1.82843

−

1.91229i

−0.500000

−

0.866025i

31.2

−0.500000

0.866025i

−0.866025

0.500000i

−1.18186

2.36711i

−0.500000

−

0.866025i

31.3

−0.500000

0.866025i

−0.866025

0.500000i

2.64426

−

0.0887915i

−0.500000

−

0.866025i

31.4

−0.500000

0.866025i

0.866025

−

0.500000i

−1.44071

−

2.21909i

−0.500000

−

0.866025i

31.5

−0.500000

0.866025i

0.866025

−

0.500000i

0.341894

2.62357i

−0.500000

−

0.866025i

31.6

−0.500000

0.866025i

0.866025

−

0.500000i

2.46484

0.961549i

−0.500000

−

0.866025i

271.1

−0.500000

−

0.866025i

−0.866025

−

0.500000i

−1.82843

1.91229i

−0.500000

0.866025i

271.2

−0.500000

−

0.866025i

−0.866025

−

0.500000i

−1.18186

−

2.36711i

−0.500000

0.866025i

271.3

−0.500000

−

0.866025i

−0.866025

−

0.500000i

2.64426

0.0887915i

−0.500000

0.866025i

271.4

−0.500000

−

0.866025i

0.866025

0.500000i

−1.44071

2.21909i

−0.500000

0.866025i

271.5

−0.500000

−

0.866025i

0.866025

0.500000i

0.341894

−

2.62357i

−0.500000

0.866025i

271.6

−0.500000

−

0.866025i

0.866025

0.500000i

2.46484

−

0.961549i

−0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1680.2.dx.e	✓	12
4.b	odd	2	1		1680.2.dx.g	yes	12
7.d	odd	6	1		1680.2.dx.g	yes	12
28.f	even	6	1	inner	1680.2.dx.e	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1680.2.dx.e	✓	12	1.a	even	1	1	trivial
1680.2.dx.e	✓	12	28.f	even	6	1	inner
1680.2.dx.g	yes	12	4.b	odd	2	1
1680.2.dx.g	yes	12	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{12} - 39 T_{11}^{10} + 1191 T_{11}^{8} + 306 T_{11}^{7} - 12690 T_{11}^{6} - 5940 T_{11}^{5} + \cdots + 1296 $ T11^12 - 39*T11^10 + 1191*T11^8 + 306*T11^7 - 12690*T11^6 - 5940*T11^5 + 105120*T11^4 + 130680*T11^3 + 40392*T11^2 - 14256*T11 + 1296 acting on $S_{2}^{\mathrm{new}}(1680, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$5$	$ (T^{4} - T^{2} + 1)^{3} $ (T^4 - T^2 + 1)^3
$7$	$ T^{12} - 2 T^{11} + \cdots + 117649 $ T^12 - 2T^11 + 4T^10 - 52T^9 + 100T^8 - 242T^7 + 1250T^6 - 1694T^5 + 4900T^4 - 17836T^3 + 9604T^2 - 33614*T + 117649
$11$	$ T^{12} - 39 T^{10} + \cdots + 1296 $ T^12 - 39T^10 + 1191T^8 + 306T^7 - 12690T^6 - 5940T^5 + 105120T^4 + 130680T^3 + 40392T^2 - 14256*T + 1296
$13$	$ T^{12} + 82 T^{10} + \cdots + 62001 $ T^12 + 82T^10 + 2271T^8 + 24988T^6 + 105871T^4 + 161970*T^2 + 62001
$17$	$ T^{12} + 6 T^{11} + \cdots + 9216 $ T^12 + 6T^11 - 26T^10 - 228T^9 + 1032T^8 + 2520T^7 - 9224T^6 - 16944T^5 + 86416T^4 - 70080T^3 - 8832T^2 + 23040*T + 9216
$19$	$ T^{12} - 2 T^{11} + \cdots + 165649 $ T^12 - 2T^11 + 77T^10 - 54T^9 + 4570T^8 - 3658T^7 + 78805T^6 + 68390T^5 + 909370T^4 - 104646T^3 + 427949T^2 + 78958*T + 165649
$23$	$ T^{12} + \cdots + 401120784 $ T^12 - 6T^11 - 95T^10 + 642T^9 + 7491T^8 - 71502T^7 - 39650T^6 + 2068164T^5 - 1531664T^4 - 56708328T^3 + 280556808T^2 - 535228272*T + 401120784
$29$	$ (T^{6} + 2 T^{5} + \cdots + 144)^{2} $ (T^6 + 2T^5 - 66T^4 + 20T^3 + 1180T^2 - 2400*T + 144)^2
$31$	$ T^{12} + 4 T^{11} + \cdots + 14107536 $ T^12 + 4T^11 + 114T^10 - 248T^9 + 6991T^8 - 21468T^7 + 303234T^6 - 1265832T^5 + 7664961T^4 - 15962580T^3 + 39156012T^2 + 19966896*T + 14107536
$37$	$ T^{12} + 10 T^{11} + \cdots + 2217121 $ T^12 + 10T^11 + 89T^10 + 414T^9 + 2014T^8 + 6266T^7 + 25405T^6 + 61070T^5 + 195406T^4 + 274362T^3 + 783881T^2 + 711742*T + 2217121
$41$	$ T^{12} + 166 T^{10} + \cdots + 345744 $ T^12 + 166T^10 + 8253T^8 + 113008T^6 + 443116T^4 + 667968*T^2 + 345744
$43$	$ T^{12} + \cdots + 574848576 $ T^12 + 448T^10 + 71262T^8 + 4659616T^6 + 105059185T^4 + 656654688*T^2 + 574848576
$47$	$ T^{12} + 111 T^{10} + \cdots + 22581504 $ T^12 + 111T^10 + 600T^9 + 10569T^8 + 39492T^7 + 293976T^6 + 849024T^5 + 5454576T^4 + 13699584T^3 + 30015360T^2 + 29424384T + 22581504
$53$	$ T^{12} - 4 T^{11} + \cdots + 2985984 $ T^12 - 4T^11 + 187T^10 - 220T^9 + 24769T^8 - 24460T^7 + 1285096T^6 + 3731936T^5 + 37971328T^4 + 17839872T^3 + 17570304T^2 - 4478976*T + 2985984
$59$	$ T^{12} + 6 T^{11} + \cdots + 2985984 $ T^12 + 6T^11 + 162T^10 - 348T^9 + 11928T^8 - 47880T^7 + 758952T^6 - 3968496T^5 + 24181776T^4 - 60286464T^3 + 141647616T^2 + 19906560*T + 2985984
$61$	$ T^{12} - 12 T^{11} + \cdots + 50466816 $ T^12 - 12T^11 - 56T^10 + 1248T^9 + 4128T^8 - 104448T^7 + 188032T^6 + 2350848T^5 - 3837440T^4 - 38424576T^3 + 151480320T^2 - 144580608*T + 50466816
$67$	$ T^{12} - 6 T^{11} + \cdots + 50808384 $ T^12 - 6T^11 - 150T^10 + 972T^9 + 20559T^8 - 115560T^7 - 511326T^6 + 2962494T^5 + 17601849T^4 + 7859916T^3 - 29391336T^2 - 13087008*T + 50808384
$71$	$ T^{12} + \cdots + 3798503424 $ T^12 + 424T^10 + 69756T^8 + 5563744T^6 + 215533456T^4 + 3306152448*T^2 + 3798503424
$73$	$ T^{12} + \cdots + 9820017216 $ T^12 + 54T^11 + 1126T^10 + 8316T^9 - 43125T^8 - 821400T^7 + 2014006T^6 + 55197834T^5 - 73105535T^4 - 1736038500T^3 + 8786291064T^2 - 15013044000*T + 9820017216
$79$	$ T^{12} - 12 T^{11} + \cdots + 1296 $ T^12 - 12T^11 - 150T^10 + 2376T^9 + 37263T^8 - 49824T^7 - 163782T^6 + 234684T^5 + 857313T^4 - 67176T^3 - 31860T^2 + 2592*T + 1296
$83$	$ (T^{6} + 22 T^{5} + \cdots - 191664)^{2} $ (T^6 + 22T^5 - 114T^4 - 3404T^3 + 5740T^2 + 90048*T - 191664)^2
$89$	$ T^{12} + \cdots + 2285604864 $ T^12 - 6T^11 - 318T^10 + 1980T^9 + 89544T^8 - 1178136T^7 + 1195848T^6 + 45697392T^5 - 42287472T^4 - 1597142016T^3 + 7127495424T^2 - 6719127552*T + 2285604864
$97$	$ T^{12} + \cdots + 5780865024 $ T^12 + 520T^10 + 96432T^8 + 7758976T^6 + 270803200T^4 + 3303383040*T^2 + 5780865024
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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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.dx (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{11} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{11} - 1) q^{9}+O(q^{10}) \) q + b11 * q^3 + b1 * q^5 - b4 * q^7 + (-b11 - 1) * q^9	\( q + \beta_{11} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{11} - 1) q^{9} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{10} + \beta_{9} + \cdots - \beta_1) q^{99}+O(q^{100}) \) q + b11 * q^3 + b1 * q^5 - b4 * q^7 + (-b11 - 1) * q^9 + (-b9 - b8 + b6 - b5 + b3 - b2) * q^11 + (2b11 + b8 + b7 + b2 - b1 + 1) q^13 + (b2 - b1) * q^15 + (-b9 - b7 - b5 + b3 + b2) * q^17 + (b9 + b7 + b6 + b5 + b4 - b3 + 2b2 - b1) q^19 + (b9 + b4) * q^21 + (-b11 + b9 + b8 + b7 + 2b6 - b5 + b4 + b3 + 1) q^23 - b11 * q^25 + q^27 + (-b6 + b5 - b4 - b3 - 2b2 - 2b1 - 1) * q^29 + (-b10 + 2b8 + b7 + 2b6 + b5 + b4 - b3 - b2 + 2b1) q^31 + (b10 - b7 - b6 + b5 - b4 + b1) * q^33 - b8 * q^35 + (-2b11 - b9 - b4 + b3 - 2) q^37 + (-b11 - b8 + b6 - b5 - b2 - 2) * q^39 + (-b10 + b9 - b8 - b7 + b4 - b3 - b2 + b1) * q^41 + (-2b11 + b10 - b9 + b8 + b7 + 2b6 + 2b5 - b4 + b3 - 3b2 + 3b1 - 1) q^43 - b2 * q^45 + (-b11 - b10 + b9 + b7 + b6 + b5 - b3 + 2b2 - b1 - 1) q^47 + (3b11 + b10 + b9 + b5 - 2b3 + b2 - 3b1 + 1) q^49 + (b10 - b8 - b6 - b4 - b1) * q^51 + (-b11 - b10 + b9 - b8 - 2b7 - b6 + b5 + b4 - b2 + 2b1) * q^53 + (-b10 - b9 + b6 - b5 - 1) * q^55 + (-b10 - b9 + b8 - b7 - b2 - b1) * q^57 + (b11 + b10 - b9 + b8 + b7 + b6 - b4 - 2b2 + 4b1) * q^59 + (-b11 + b9 + b7 + b6 - b5 + b4 + b3 + 5b1 + 1) q^61 - b9 * q^63 + (b11 + b9 + b4 - b3 + 2b2 - b1 + 1) q^65 + (-b10 - b9 - 2b8 + 2b6 - 2b5 - b4 - 2b2) * q^67 + (2b11 + b10 - b9 - 2b8 - 2b7 - b6 - b5 + 1) q^69 + (6b11 + b6 + b5 + b4 - b3 + 4b2 - 4b1 + 3) q^71 + (-3b11 - b9 - 3b8 + 3b6 - 3b5 + b3 - 3b2 - 6) q^73 + (b11 + 1) * q^75 + (b11 + 2b10 - 2b9 - 2b8 - 3b7 - 2b6 - 2b5 - 2b4 + 3b3 + 4b2 - 5b1 - 2) * q^77 + (-b11 + 2b10 + b8 - b7 + b5 - 2b4 - 4b1 + 1) q^79 + b11 * q^81 + (3b10 + 3b9 - b8 + b7 - b6 + b5 - b4 - b3 - b2 - b1 - 4) * q^83 + (b6 - b5 + b4 + b3 + 1) * q^85 + (-b11 - b10 + b9 + b8 + b7 + b6 + b4 - 2b2 + 4b1) * q^87 + (-b11 - b10 - b9 + 3b8 - b7 + 2b6 + b5 - b3 + 4b1 + 1) q^89 + (-3b11 + 2b9 + b8 + b7 + 3b6 + 2b4 + 4b2 + 2b1 - 1) * q^91 + (-b9 - b8 - 2b7 - b6 - 2b5 - b4 + b3 + 2b2 - b1) q^93 + (b11 - b10 + b8 - b6 + b5 - b4 - b3 + 2) * q^95 + (2b11 - b10 + b9 + b8 + b7 + 2b6 + 2b5 - 3b2 + 3b1 + 1) q^97 + (-b10 + b9 + b8 + b7 + b4 - b3 + b2 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 + 2 * q^7 - 6 * q^9	\( 12 q - 6 q^{3} + 2 q^{7} - 6 q^{9} - 6 q^{17} + 2 q^{19} - 4 q^{21} + 6 q^{23} + 6 q^{25} + 12 q^{27} - 4 q^{29} - 4 q^{31} + 4 q^{35} - 10 q^{37} - 18 q^{39} - 4 q^{49} + 6 q^{51} + 4 q^{53} - 12 q^{55} - 4 q^{57} - 6 q^{59} + 12 q^{61} + 2 q^{63} + 4 q^{65} + 6 q^{67} - 54 q^{73} + 6 q^{75} - 36 q^{77} + 12 q^{79} - 6 q^{81} - 44 q^{83} + 4 q^{85} + 2 q^{87} + 6 q^{89} - 8 q^{91} - 4 q^{93} + 24 q^{95}+O(q^{100}) \) 12 * q - 6 * q^3 + 2 * q^7 - 6 * q^9 - 6 * q^17 + 2 * q^19 - 4 * q^21 + 6 * q^23 + 6 * q^25 + 12 * q^27 - 4 * q^29 - 4 * q^31 + 4 * q^35 - 10 * q^37 - 18 * q^39 - 4 * q^49 + 6 * q^51 + 4 * q^53 - 12 * q^55 - 4 * q^57 - 6 * q^59 + 12 * q^61 + 2 * q^63 + 4 * q^65 + 6 * q^67 - 54 * q^73 + 6 * q^75 - 36 * q^77 + 12 * q^79 - 6 * q^81 - 44 * q^83 + 4 * q^85 + 2 * q^87 + 6 * q^89 - 8 * q^91 - 4 * q^93 + 24 * q^95

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	1680.2.dx.e

Level	$1680$
Weight	$2$
Character orbit	1680.dx
Analytic conductor	$13.415$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(13.4148675396\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 6 x^{11} + 47 x^{10} - 180 x^{9} + 724 x^{8} - 1882 x^{7} + 4759 x^{6} - 8380 x^{5} + \cdots + 4468 \) x^12 - 6x^11 + 47x^10 - 180x^9 + 724x^8 - 1882x^7 + 4759x^6 - 8380x^5 + 13729x^4 - 15360x^3 + 15460x^2 - 8912*x + 4468
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 1864 \nu^{11} + 25751 \nu^{10} - 104079 \nu^{9} + 1221120 \nu^{8} - 3898574 \nu^{7} + \cdots + 92792416 ) / 4791672 \) (1864v^11 + 25751v^10 - 104079v^9 + 1221120v^8 - 3898574v^7 + 17581312v^6 - 38300322v^5 + 97380167v^4 - 132626665v^3 + 192767056v^2 - 128227422*v + 92792416) / 4791672
\(\beta_{2}\)	\(=\)	\( ( - 1864 \nu^{11} + 46255 \nu^{10} - 255951 \nu^{9} + 1750764 \nu^{8} - 5828782 \nu^{7} + \cdots + 98612624 ) / 4791672 \) (-1864v^11 + 46255v^10 - 255951v^9 + 1750764v^8 - 5828782v^7 + 22008896v^6 - 47936682v^5 + 111535423v^4 - 152104697v^3 + 209720300v^2 - 144753870*v + 98612624) / 4791672
\(\beta_{3}\)	\(=\)	\( ( 4354 \nu^{11} + 100427 \nu^{10} - 486483 \nu^{9} + 4358760 \nu^{8} - 14640170 \nu^{7} + \cdots + 221152144 ) / 4791672 \) (4354v^11 + 100427v^10 - 486483v^9 + 4358760v^8 - 14640170v^7 + 57463768v^6 - 128631360v^5 + 291902159v^4 - 410822593v^3 + 534284836v^2 - 378089934*v + 221152144) / 4791672
\(\beta_{4}\)	\(=\)	\( ( - 4354 \nu^{11} + 148321 \nu^{10} - 757257 \nu^{9} + 5218038 \nu^{8} - 16204582 \nu^{7} + \cdots + 176595908 ) / 4791672 \) (-4354v^11 + 148321v^10 - 757257v^9 + 5218038v^8 - 16204582v^7 + 59264504v^6 - 118820532v^5 + 264638761v^4 - 323337491v^3 + 430719938v^2 - 256309110*v + 176595908) / 4791672
\(\beta_{5}\)	\(=\)	\( ( 3557 \nu^{11} - 13563 \nu^{10} + 122758 \nu^{9} - 293039 \nu^{8} + 1257560 \nu^{7} + \cdots + 13135482 ) / 798612 \) (3557v^11 - 13563v^10 + 122758v^9 - 293039v^8 + 1257560v^7 - 1553118v^6 + 3531607v^5 + 1581521v^4 - 5172298v^3 + 19041413v^2 - 16869582*v + 13135482) / 798612
\(\beta_{6}\)	\(=\)	\( ( 3557 \nu^{11} - 25564 \nu^{10} + 182763 \nu^{9} - 788353 \nu^{8} + 2878786 \nu^{7} + \cdots - 14772298 ) / 798612 \) (3557v^11 - 25564v^10 + 182763v^9 - 788353v^8 + 2878786v^7 - 8151486v^6 + 17904441v^5 - 33237744v^4 + 42282929v^3 - 47241201v^2 + 27828688*v - 14772298) / 798612
\(\beta_{7}\)	\(=\)	\( ( 10456 \nu^{11} - 52053 \nu^{10} + 426675 \nu^{9} - 1368172 \nu^{8} + 5585586 \nu^{7} + \cdots + 15673912 ) / 1597224 \) (10456v^11 - 52053v^10 + 426675v^9 - 1368172v^8 + 5585586v^7 - 11612596v^6 + 28762722v^5 - 34908725v^4 + 52412649v^3 - 21345688v^2 + 17567254*v + 15673912) / 1597224
\(\beta_{8}\)	\(=\)	\( ( 10456 \nu^{11} - 62963 \nu^{10} + 481225 \nu^{9} - 1854758 \nu^{8} + 7204630 \nu^{7} + \cdots - 51152020 ) / 1597224 \) (10456v^11 - 62963v^10 + 481225v^9 - 1854758v^8 + 7204630v^7 - 18917932v^6 + 45241186v^5 - 80719815v^4 + 118350507v^3 - 132459674v^2 + 98205246*v - 51152020) / 1597224
\(\beta_{9}\)	\(=\)	\( ( - 33232 \nu^{11} + 153319 \nu^{10} - 1290501 \nu^{9} + 3865296 \nu^{8} - 16098454 \nu^{7} + \cdots - 57858232 ) / 4791672 \) (-33232v^11 + 153319v^10 - 1290501v^9 + 3865296v^8 - 16098454v^7 + 31160180v^6 - 78839142v^5 + 87177751v^4 - 136407143v^3 + 39881876v^2 - 37032810*v - 57858232) / 4791672
\(\beta_{10}\)	\(=\)	\( ( 33232 \nu^{11} - 212233 \nu^{10} + 1585071 \nu^{9} - 6333138 \nu^{8} + 24202402 \nu^{7} + \cdots - 165321092 ) / 4791672 \) (33232v^11 - 212233v^10 + 1585071v^9 - 6333138v^8 + 24202402v^7 - 64858988v^6 + 152808942v^5 - 273863125v^4 + 395360357v^3 - 438487454v^2 + 317227794*v - 165321092) / 4791672
\(\beta_{11}\)	\(=\)	\( ( - 218 \nu^{11} + 1199 \nu^{10} - 9101 \nu^{9} + 31962 \nu^{8} - 121122 \nu^{7} + 283164 \nu^{6} + \cdots + 215540 ) / 26184 \) (-218v^11 + 1199v^10 - 9101v^9 + 31962v^8 - 121122v^7 + 283164v^6 - 647132v^5 + 978503v^4 - 1335755v^3 + 1147202v^2 - 785966*v + 215540) / 26184

\(\nu\)	\(=\)	\( ( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{2} + \beta _1 + 1 ) / 2 \) (-b10 + b9 + b8 + b7 - b2 + b1 + 1) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{6} - 2\beta_{5} + \beta_{2} + 3\beta _1 - 9 ) / 2 \) (-b10 + b9 + b8 + b7 + 2b6 - 2b5 + b2 + 3*b1 - 9) / 2
\(\nu^{3}\)	\(=\)	\( ( - 6 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 7 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + \cdots - 17 ) / 2 \) (-6b11 + 3b10 - 3b9 - 7b8 - 7b7 + 4b6 - 2b5 + 4b4 - 4b3 + 5b2 + b1 - 17) / 2
\(\nu^{4}\)	\(=\)	\( ( - 12 \beta_{11} + \beta_{10} - 13 \beta_{9} - 13 \beta_{8} - 17 \beta_{7} - 16 \beta_{6} + 20 \beta_{5} + \cdots + 49 ) / 2 \) (-12b11 + b10 - 13b9 - 13b8 - 17b7 - 16b6 + 20b5 + 8b4 - 8b3 - 17b2 - 27b1 + 49) / 2
\(\nu^{5}\)	\(=\)	\( ( 72 \beta_{11} - 19 \beta_{10} - 11 \beta_{9} + 55 \beta_{8} + 45 \beta_{7} - 62 \beta_{6} + 38 \beta_{5} + \cdots + 197 ) / 2 \) (72b11 - 19b10 - 11b9 + 55b8 + 45b7 - 62b6 + 38b5 - 44b4 + 44b3 - 95b2 - 25*b1 + 197) / 2
\(\nu^{6}\)	\(=\)	\( ( 246 \beta_{11} + 55 \beta_{10} + 115 \beta_{9} + 147 \beta_{8} + 229 \beta_{7} + 94 \beta_{6} + \cdots - 231 ) / 2 \) (246b11 + 55b10 + 115b9 + 147b8 + 229b7 + 94b6 - 176b5 - 150b4 + 154b3 + 61b2 + 297*b1 - 231) / 2
\(\nu^{7}\)	\(=\)	\( ( - 632 \beta_{11} + 279 \beta_{10} + 421 \beta_{9} - 471 \beta_{8} - 149 \beta_{7} + 740 \beta_{6} + \cdots - 2135 ) / 2 \) (-632b11 + 279b10 + 421b9 - 471b8 - 149b7 + 740b6 - 562b5 + 366b4 - 352b3 + 1301b2 + 379*b1 - 2135) / 2
\(\nu^{8}\)	\(=\)	\( ( - 3704 \beta_{11} - 857 \beta_{10} - 603 \beta_{9} - 1719 \beta_{8} - 2587 \beta_{7} - 262 \beta_{6} + \cdots - 65 ) / 2 \) (-3704b11 - 857b10 - 603b9 - 1719b8 - 2587b7 - 262b6 + 1366b5 + 2120b4 - 2208b3 + 1345b2 - 3469*b1 - 65) / 2
\(\nu^{9}\)	\(=\)	\( ( 3682 \beta_{11} - 4559 \beta_{10} - 6337 \beta_{9} + 4103 \beta_{8} - 1777 \beta_{7} - 8120 \beta_{6} + \cdots + 21501 ) / 2 \) (3682b11 - 4559b10 - 6337b9 + 4103b8 - 1777b7 - 8120b6 + 7450b5 - 2088b4 + 1608b3 - 13577b2 - 6577*b1 + 21501) / 2
\(\nu^{10}\)	\(=\)	\( ( 47972 \beta_{11} + 8031 \beta_{10} - 2271 \beta_{9} + 21121 \beta_{8} + 25589 \beta_{7} - 4696 \beta_{6} + \cdots + 24103 ) / 2 \) (47972b11 + 8031b10 - 2271b9 + 21121b8 + 25589b7 - 4696b6 - 7528b5 - 26112b4 + 27036b3 - 35111b2 + 37699*b1 + 24103) / 2
\(\nu^{11}\)	\(=\)	\( ( 6676 \beta_{11} + 66127 \beta_{10} + 73463 \beta_{9} - 31595 \beta_{8} + 50531 \beta_{7} + 83946 \beta_{6} + \cdots - 198093 ) / 2 \) (6676b11 + 66127b10 + 73463b9 - 31595b8 + 50531b7 + 83946b6 - 89810b5 - 1800b4 + 11436b3 + 111219b2 + 107593*b1 - 198093) / 2

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(1 + \beta_{11}\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$5$	\( (T^{4} - T^{2} + 1)^{3} \) (T^4 - T^2 + 1)^3
$7$	\( T^{12} - 2 T^{11} + \cdots + 117649 \) T^12 - 2T^11 + 4T^10 - 52T^9 + 100T^8 - 242T^7 + 1250T^6 - 1694T^5 + 4900T^4 - 17836T^3 + 9604T^2 - 33614*T + 117649
$11$	\( T^{12} - 39 T^{10} + \cdots + 1296 \) T^12 - 39T^10 + 1191T^8 + 306T^7 - 12690T^6 - 5940T^5 + 105120T^4 + 130680T^3 + 40392T^2 - 14256*T + 1296
$13$	\( T^{12} + 82 T^{10} + \cdots + 62001 \) T^12 + 82T^10 + 2271T^8 + 24988T^6 + 105871T^4 + 161970*T^2 + 62001
$17$	\( T^{12} + 6 T^{11} + \cdots + 9216 \) T^12 + 6T^11 - 26T^10 - 228T^9 + 1032T^8 + 2520T^7 - 9224T^6 - 16944T^5 + 86416T^4 - 70080T^3 - 8832T^2 + 23040*T + 9216
$19$	\( T^{12} - 2 T^{11} + \cdots + 165649 \) T^12 - 2T^11 + 77T^10 - 54T^9 + 4570T^8 - 3658T^7 + 78805T^6 + 68390T^5 + 909370T^4 - 104646T^3 + 427949T^2 + 78958*T + 165649
$23$	\( T^{12} + \cdots + 401120784 \) T^12 - 6T^11 - 95T^10 + 642T^9 + 7491T^8 - 71502T^7 - 39650T^6 + 2068164T^5 - 1531664T^4 - 56708328T^3 + 280556808T^2 - 535228272*T + 401120784
$29$	\( (T^{6} + 2 T^{5} + \cdots + 144)^{2} \) (T^6 + 2T^5 - 66T^4 + 20T^3 + 1180T^2 - 2400*T + 144)^2
$31$	\( T^{12} + 4 T^{11} + \cdots + 14107536 \) T^12 + 4T^11 + 114T^10 - 248T^9 + 6991T^8 - 21468T^7 + 303234T^6 - 1265832T^5 + 7664961T^4 - 15962580T^3 + 39156012T^2 + 19966896*T + 14107536
$37$	\( T^{12} + 10 T^{11} + \cdots + 2217121 \) T^12 + 10T^11 + 89T^10 + 414T^9 + 2014T^8 + 6266T^7 + 25405T^6 + 61070T^5 + 195406T^4 + 274362T^3 + 783881T^2 + 711742*T + 2217121
$41$	\( T^{12} + 166 T^{10} + \cdots + 345744 \) T^12 + 166T^10 + 8253T^8 + 113008T^6 + 443116T^4 + 667968*T^2 + 345744
$43$	\( T^{12} + \cdots + 574848576 \) T^12 + 448T^10 + 71262T^8 + 4659616T^6 + 105059185T^4 + 656654688*T^2 + 574848576
$47$	\( T^{12} + 111 T^{10} + \cdots + 22581504 \) T^12 + 111T^10 + 600T^9 + 10569T^8 + 39492T^7 + 293976T^6 + 849024T^5 + 5454576T^4 + 13699584T^3 + 30015360T^2 + 29424384T + 22581504
$53$	\( T^{12} - 4 T^{11} + \cdots + 2985984 \) T^12 - 4T^11 + 187T^10 - 220T^9 + 24769T^8 - 24460T^7 + 1285096T^6 + 3731936T^5 + 37971328T^4 + 17839872T^3 + 17570304T^2 - 4478976*T + 2985984
$59$	\( T^{12} + 6 T^{11} + \cdots + 2985984 \) T^12 + 6T^11 + 162T^10 - 348T^9 + 11928T^8 - 47880T^7 + 758952T^6 - 3968496T^5 + 24181776T^4 - 60286464T^3 + 141647616T^2 + 19906560*T + 2985984
$61$	\( T^{12} - 12 T^{11} + \cdots + 50466816 \) T^12 - 12T^11 - 56T^10 + 1248T^9 + 4128T^8 - 104448T^7 + 188032T^6 + 2350848T^5 - 3837440T^4 - 38424576T^3 + 151480320T^2 - 144580608*T + 50466816
$67$	\( T^{12} - 6 T^{11} + \cdots + 50808384 \) T^12 - 6T^11 - 150T^10 + 972T^9 + 20559T^8 - 115560T^7 - 511326T^6 + 2962494T^5 + 17601849T^4 + 7859916T^3 - 29391336T^2 - 13087008*T + 50808384
$71$	\( T^{12} + \cdots + 3798503424 \) T^12 + 424T^10 + 69756T^8 + 5563744T^6 + 215533456T^4 + 3306152448*T^2 + 3798503424
$73$	\( T^{12} + \cdots + 9820017216 \) T^12 + 54T^11 + 1126T^10 + 8316T^9 - 43125T^8 - 821400T^7 + 2014006T^6 + 55197834T^5 - 73105535T^4 - 1736038500T^3 + 8786291064T^2 - 15013044000*T + 9820017216
$79$	\( T^{12} - 12 T^{11} + \cdots + 1296 \) T^12 - 12T^11 - 150T^10 + 2376T^9 + 37263T^8 - 49824T^7 - 163782T^6 + 234684T^5 + 857313T^4 - 67176T^3 - 31860T^2 + 2592*T + 1296
$83$	\( (T^{6} + 22 T^{5} + \cdots - 191664)^{2} \) (T^6 + 22T^5 - 114T^4 - 3404T^3 + 5740T^2 + 90048*T - 191664)^2
$89$	\( T^{12} + \cdots + 2285604864 \) T^12 - 6T^11 - 318T^10 + 1980T^9 + 89544T^8 - 1178136T^7 + 1195848T^6 + 45697392T^5 - 42287472T^4 - 1597142016T^3 + 7127495424T^2 - 6719127552*T + 2285604864
$97$	\( T^{12} + \cdots + 5780865024 \) T^12 + 520T^10 + 96432T^8 + 7758976T^6 + 270803200T^4 + 3303383040*T^2 + 5780865024
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