LMFDB - Newform orbit 1860.2.q.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1860,2,Mod(1141,1860)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1860, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1860.1141");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1860.q (of order $3$, 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:	$14.8521747760$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
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} - 3 x^{11} + 29 x^{10} + 405 x^{8} - 117 x^{7} + 2515 x^{6} + 1476 x^{5} + 7995 x^{4} - 2175 x^{3} + \cdots + 4 $ x^12 - 3x^11 + 29x^10 + 405x^8 - 117x^7 + 2515x^6 + 1476x^5 + 7995x^4 - 2175x^3 + 899x^2 + 54x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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_{7} q^{3} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{9} - \beta_{4}) q^{7} + (\beta_{7} - 1) q^{9} - \beta_{6} q^{11} + \beta_{11} q^{13} - q^{15} + (\beta_{11} + \beta_{9} - \beta_{7} + \cdots + \beta_1) q^{17}+ \cdots + (\beta_{6} - \beta_{5}) q^{99}+O(q^{100}) $ q - b7 * q^3 + (-b7 + 1) * q^5 + (-b9 - b4) * q^7 + (b7 - 1) * q^9 - b6 * q^11 + b11 * q^13 - q^15 + (b11 + b9 - b7 - b6 + b5 + b4 + b1) * q^17 + (-b11 - b9 + b7 - b4 + b3 - b2 - b1) * q^19 + b9 * q^21 + (-b8 - b5 - b3 - b2 - b1) * q^23 - b7 * q^25 + q^27 + (b5 + b4 - 2) * q^29 + (b11 + b8 - b5 + 1) * q^31 + b5 * q^33 - b4 * q^35 + (-b6 + b5 + b3 - b2) * q^37 + b1 * q^39 + (b11 - b10 - b9 + 2b7 - b3 - 2) q^41 + (b10 + 2b9 - b8 - 2b7 - b6 + b5 + 2b4 - b2) q^43 + b7 * q^45 + (b5 - b1 - 1) * q^47 + (-b9 + b8 + 3b7 + 2b6 - b2 - 3) * q^49 + (-b11 - b9 + b7 + b6 - 1) * q^51 + (-b11 + b10 + b9 + b8 - 4b7 + b6 + b3 - b2 + 4) q^53 + (-b6 + b5) * q^55 + (b11 - b10 + b9 - b8 - b7 - b3 + b2 + 1) * q^57 + (2b11 - b10 + b8 + 2b7 + b6 - b5 - b3 + 2b2 + 2b1) * q^59 + (-b10 - b8 + b5 + 1) * q^61 + b4 * q^63 + (b11 + b1) * q^65 + (2b11 - 2b10 - b8 - 2b7 - 2b6 - 2b3 + b2 + 2) q^67 + (b11 - b10 + b8 - b6 + b5 + b2 + b1) * q^69 + (-2b11 + 2b10 + b9 - 2b7 + b6 + 2b3 + 2) * q^71 + (b11 + b8 - 2b7 - b2 + 2) q^73 + (b7 - 1) * q^75 + (2b5 - b4 + 2b1 - 2) * q^77 + (2b11 - b10 + b9 + b8 - 3b7 - b6 + b5 + b4 + b2 + 2b1) q^79 - b7 * q^81 + (b11 - 2b9 + 5b7 - b6 - 5) * q^83 + (b5 + b4 + b1 - 1) * q^85 + (-b9 + 2b7 + b6 - b5 - b4) q^87 + (-b10 + 3b5 + b3 + b2 + 2b1) * q^89 + (b10 + b8 + 3b5 + 2b4 + b1 - 4) * q^91 + (-b7 - b6 + b5 + b3 + b1) * q^93 + (-b10 - b8 - b4 - b1 + 1) * q^95 + (b10 - 2b5 - 2b4 - b3 - b2 + b1 - 6) * q^97 + (b6 - b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 6 q^{5} - 2 q^{7} - 6 q^{9} - 2 q^{13} - 12 q^{15} - 2 q^{17} + 5 q^{19} - 2 q^{21} + 2 q^{23} - 6 q^{25} + 12 q^{27} - 20 q^{29} + 7 q^{31} - 4 q^{35} + 3 q^{37} + 4 q^{39} - 9 q^{41}+ \cdots - 76 q^{97}+O(q^{100}) $ 12 * q - 6 * q^3 + 6 * q^5 - 2 * q^7 - 6 * q^9 - 2 * q^13 - 12 * q^15 - 2 * q^17 + 5 * q^19 - 2 * q^21 + 2 * q^23 - 6 * q^25 + 12 * q^27 - 20 * q^29 + 7 * q^31 - 4 * q^35 + 3 * q^37 + 4 * q^39 - 9 * q^41 - 5 * q^43 + 6 * q^45 - 16 * q^47 - 16 * q^49 - 2 * q^51 + 21 * q^53 + 5 * q^57 + 10 * q^59 + 18 * q^61 + 4 * q^63 + 2 * q^65 + 14 * q^67 - q^69 + 8 * q^71 + 10 * q^73 - 6 * q^75 - 20 * q^77 - 15 * q^79 - 6 * q^81 - 28 * q^83 - 4 * q^85 + 10 * q^87 + 8 * q^89 - 42 * q^91 - 2 * q^93 + 10 * q^95 - 76 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 3 x^{11} + 29 x^{10} + 405 x^{8} - 117 x^{7} + 2515 x^{6} + 1476 x^{5} + 7995 x^{4} - 2175 x^{3} + \cdots + 4 $ x^12 - 3*x^11 + 29*x^10 + 405*x^8 - 117*x^7 + 2515*x^6 + 1476*x^5 + 7995*x^4 - 2175*x^3 + 899*x^2 + 54*x + 4 :

$\beta_{1}$	$=$	$ ( - 1715905482153 \nu^{11} + 4307423742224 \nu^{10} - 47611747561143 \nu^{9} + \cdots + 42\!\cdots\!92 ) / 13\!\cdots\!12 $ (-1715905482153v^11 + 4307423742224v^10 - 47611747561143v^9 - 22026732866505v^8 - 709213546702347v^7 - 102743655549935v^6 - 4360517418576414v^5 - 4117141587930315v^4 - 15690355272729300v^3 - 592617816912946v^2 - 35924442731196*v + 4271846952750392) / 1365970028461912
$\beta_{2}$	$=$	$ ( 3882514598555 \nu^{11} - 3654097505731 \nu^{10} + 83423074388066 \nu^{9} + \cdots + 34\!\cdots\!52 ) / 27\!\cdots\!24 $ (3882514598555v^11 - 3654097505731v^10 + 83423074388066v^9 + 244430501364486v^8 + 1429627582566179v^7 + 2699925694537313v^6 + 6684966121603390v^5 + 25288432225061936v^4 + 29330130510866487v^3 + 41103545291472447v^2 - 65901573804163378*v + 3433312164825252) / 2731940056923824
$\beta_{3}$	$=$	$ ( 21589580391119 \nu^{11} - 42806520136345 \nu^{10} + 558994369293702 \nu^{9} + \cdots + 46\!\cdots\!64 ) / 27\!\cdots\!24 $ (21589580391119v^11 - 42806520136345v^10 + 558994369293702v^9 + 636198184577068v^8 + 8721058123973357v^7 + 6251728966003151v^6 + 51200575755143796v^5 + 85706421659100116v^4 + 202385230336415205v^3 + 118618963619174607v^2 - 45417058990762410*v + 4666342363880964) / 2731940056923824
$\beta_{4}$	$=$	$ ( 25075145750338 \nu^{11} - 66873958754819 \nu^{10} + 701903809517062 \nu^{9} + \cdots + 79\!\cdots\!60 ) / 27\!\cdots\!24 $ (25075145750338v^11 - 66873958754819v^10 + 701903809517062v^9 + 240739291405050v^8 + 10158536438098922v^7 + 381127269019235v^6 + 62056919293496074v^5 + 57385505985515610v^4 + 213000797425722006v^3 + 8306088831935581v^2 + 503605401255266*v + 7906950258268260) / 2731940056923824
$\beta_{5}$	$=$	$ ( 12926849045593 \nu^{11} - 34706543270271 \nu^{10} + 362382059783535 \nu^{9} + \cdots + 20\!\cdots\!76 ) / 13\!\cdots\!12 $ (12926849045593v^11 - 34706543270271v^10 + 362382059783535v^9 + 119326970175441v^8 + 5224870434206255v^7 + 149627157387372v^6 + 31893781538332708v^5 + 29419880460172191v^4 + 108152265166927838v^3 + 4261135446720783v^2 + 258361942228046*v + 2038601673147076) / 1365970028461912
$\beta_{6}$	$=$	$ ( - 19387396113109 \nu^{11} + 60071660832906 \nu^{10} - 569315897346363 \nu^{9} + \cdots + 248215096289260 ) / 13\!\cdots\!12 $ (-19387396113109v^11 + 60071660832906v^10 - 569315897346363v^9 + 59279220527206v^8 - 7890565716131224v^7 + 3034790214632463v^6 - 49515347833257093v^5 - 23784347613834399v^4 - 155406757821982111v^3 + 54547029143567445v^2 - 29541366298068178*v + 248215096289260) / 1365970028461912
$\beta_{7}$	$=$	$ ( 5238566167851 \nu^{11} - 16062118176933 \nu^{10} + 152835571159827 \nu^{9} + \cdots + 275888522976714 ) / 341492507115478 $ (5238566167851v^11 - 16062118176933v^10 + 152835571159827v^9 - 9682882600990v^8 + 2118154416422211v^7 - 753607026816579v^6 + 13168025268563061v^5 + 6871939112567340v^4 + 41084778923472141v^3 - 14416318613667057v^2 + 4594113701208861*v + 275888522976714) / 341492507115478
$\beta_{8}$	$=$	$ ( 63056330068462 \nu^{11} - 189866016689827 \nu^{10} + \cdots + 26\!\cdots\!84 ) / 27\!\cdots\!24 $ (63056330068462v^11 - 189866016689827v^10 + 1836627282515480v^9 - 35370890723074v^8 + 25702434406061388v^7 - 7599284004898975v^6 + 161111967635570212v^5 + 91445839241821448v^4 + 517813700557797434v^3 - 130418067849461937v^2 + 113095392052649390*v + 2652850925687084) / 2731940056923824
$\beta_{9}$	$=$	$ ( 98159661622173 \nu^{11} - 301815112817580 \nu^{10} + \cdots - 12\!\cdots\!88 ) / 27\!\cdots\!24 $ (98159661622173v^11 - 301815112817580v^10 + 2868428648186826v^9 - 210767411495224v^8 + 39745421039833735v^7 - 14428483954028640v^6 + 247719128415491034v^5 + 126695924654145076v^4 + 774040902698758701v^3 - 271624806414541104v^2 + 110551499175246080*v - 1236062870602888) / 2731940056923824
$\beta_{10}$	$=$	$ ( - 104845333135128 \nu^{11} + 301468108669177 \nu^{10} + \cdots - 35\!\cdots\!84 ) / 27\!\cdots\!24 $ (-104845333135128v^11 + 301468108669177v^10 - 3006804001728090v^9 - 362667601270232v^8 - 42624135392662450v^7 + 6979472085342847v^6 - 264468020066549484v^5 - 186973249233009614v^4 - 872467357012614226v^3 + 116589361906317723v^2 - 113933842241118522*v - 3596040361480084) / 2731940056923824
$\beta_{11}$	$=$	$ ( 67181011336704 \nu^{11} - 205968556515918 \nu^{10} + \cdots - 841726563234580 ) / 13\!\cdots\!12 $ (67181011336704v^11 - 205968556515918v^10 + 1960283587208253v^9 - 125908385552566v^8 + 27174920983523613v^7 - 9682649987872563v^6 + 168928638792809103v^5 + 87771000218055947v^4 + 527132019349132626v^3 - 184967573186327505v^2 + 57157396458040110*v - 841726563234580) / 1365970028461912

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

$\nu$	$=$	$ ( \beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} + 2\beta_{6} - \beta_{3} + 1 ) / 3 $ (b11 - b10 - b9 - b7 + 2*b6 - b3 + 1) / 3
$\nu^{2}$	$=$	$ ( 5 \beta_{11} - 3 \beta_{10} + \beta_{9} + 3 \beta_{8} - 23 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 5 \beta_1 ) / 3 $ (5b11 - 3b10 + b9 + 3b8 - 23b7 + 4b6 - 4b5 + b4 - 4b3 + 7b2 + 5*b1) / 3
$\nu^{3}$	$=$	$ ( 12\beta_{10} + 35\beta_{8} - 34\beta_{5} - 23\beta_{4} + 23\beta_{3} + 23\beta_{2} + 23\beta _1 - 41 ) / 3 $ (12b10 + 35b8 - 34b5 - 23b4 + 23b3 + 23b2 + 23*b1 - 41) / 3
$\nu^{4}$	$=$	$ - 35 \beta_{11} + 65 \beta_{10} + 13 \beta_{9} + 32 \beta_{8} + 125 \beta_{7} - 44 \beta_{6} + \cdots - 125 $ -35b11 + 65b10 + 13b9 + 32b8 + 125b7 - 44b6 + 65b3 - 32b2 - 125
$\nu^{5}$	$=$	$ ( - 551 \beta_{11} + 587 \beta_{10} + 515 \beta_{9} - 587 \beta_{8} + 1241 \beta_{7} - 778 \beta_{6} + \cdots - 551 \beta_1 ) / 3 $ (-551b11 + 587b10 + 515b9 - 587b8 + 1241b7 - 778b6 + 778b5 + 515b4 + 408b3 - 995b2 - 551*b1) / 3
$\nu^{6}$	$=$	$ ( - 2444 \beta_{10} - 5267 \beta_{8} + 3668 \beta_{5} + 1747 \beta_{4} - 2823 \beta_{3} - 2823 \beta_{2} + \cdots + 8071 ) / 3 $ (-2444b10 - 5267b8 + 3668b5 + 1747b4 - 2823b3 - 2823b2 - 2665*b1 + 8071) / 3
$\nu^{7}$	$=$	$ ( 14047 \beta_{11} - 27067 \beta_{10} - 12391 \beta_{9} - 11700 \beta_{8} - 34921 \beta_{7} + 19826 \beta_{6} + \cdots + 34921 ) / 3 $ (14047b11 - 27067b10 - 12391b9 - 11700b8 - 34921b7 + 19826b6 - 27067b3 + 11700b2 + 34921) / 3
$\nu^{8}$	$=$	$ 23539 \beta_{11} - 25721 \beta_{10} - 17989 \beta_{9} + 25721 \beta_{8} - 66077 \beta_{7} + \cdots + 23539 \beta_1 $ 23539b11 - 25721b10 - 17989b9 + 25721b8 - 66077b7 + 33068b6 - 33068b5 - 17989b4 - 21304b3 + 47025b2 + 23539*b1
$\nu^{9}$	$=$	$ ( 320352 \beta_{10} + 727315 \beta_{8} - 521594 \beta_{5} - 314635 \beta_{4} + 406963 \beta_{3} + \cdots - 955705 ) / 3 $ (320352b10 + 727315b8 - 521594b5 - 314635b4 + 406963b3 + 406963b2 + 368887*b1 - 955705) / 3
$\nu^{10}$	$=$	$ ( - 1886153 \beta_{11} + 3770995 \beta_{10} + 1517003 \beta_{9} + 1691764 \beta_{8} + 5132327 \beta_{7} + \cdots - 5132327 ) / 3 $ (-1886153b11 + 3770995b10 + 1517003b9 + 1691764b8 + 5132327b7 - 2663572b6 + 3770995b3 - 1691764b2 - 5132327) / 3
$\nu^{11}$	$=$	$ ( - 9796751 \beta_{11} + 10833263 \beta_{10} + 8219663 \beta_{9} - 10833263 \beta_{8} + \cdots - 9796751 \beta_1 ) / 3 $ (-9796751b11 + 10833263b10 + 8219663b9 - 10833263b8 + 25816265b7 - 13861858b6 + 13861858b5 + 8219663b4 + 8637276b3 - 19470539b2 - 9796751*b1) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times$.

$n$	$931$	$1117$	$1241$	$1801$
$\chi(n)$	$1$	$1$	$1$	$-1 + \beta_{7}$

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} $

1141.1

−0.979695	+	1.69688i
−0.0310901	+	0.0538497i
2.58466	−	4.47676i
0.175815	−	0.304520i
1.38292	−	2.39529i
−1.63261	+	2.82776i
−0.979695	−	1.69688i
−0.0310901	−	0.0538497i
2.58466	+	4.47676i
0.175815	+	0.304520i
1.38292	+	2.39529i
−1.63261	−	2.82776i

−0.500000

−

0.866025i

0.500000

−

0.866025i

−2.40843

−

4.17152i

−0.500000

0.866025i

1141.2

−0.500000

−

0.866025i

0.500000

−

0.866025i

−1.45061

−

2.51254i

−0.500000

0.866025i

1141.3

−0.500000

−

0.866025i

0.500000

−

0.866025i

−0.527660

−

0.913934i

−0.500000

0.866025i

1141.4

−0.500000

−

0.866025i

0.500000

−

0.866025i

0.374535

0.648713i

−0.500000

0.866025i

1141.5

−0.500000

−

0.866025i

0.500000

−

0.866025i

0.600569

1.04022i

−0.500000

0.866025i

1141.6

−0.500000

−

0.866025i

0.500000

−

0.866025i

2.41160

4.17701i

−0.500000

0.866025i

1741.1

−0.500000

0.866025i

0.500000

0.866025i

−2.40843

4.17152i

−0.500000

−

0.866025i

1741.2

−0.500000

0.866025i

0.500000

0.866025i

−1.45061

2.51254i

−0.500000

−

0.866025i

1741.3

−0.500000

0.866025i

0.500000

0.866025i

−0.527660

0.913934i

−0.500000

−

0.866025i

1741.4

−0.500000

0.866025i

0.500000

0.866025i

0.374535

−

0.648713i

−0.500000

−

0.866025i

1741.5

−0.500000

0.866025i

0.500000

0.866025i

0.600569

−

1.04022i

−0.500000

−

0.866025i

1741.6

−0.500000

0.866025i

0.500000

0.866025i

2.41160

−

4.17701i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1860.2.q.i	✓	12
31.c	even	3	1	inner	1860.2.q.i	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1860.2.q.i	✓	12	1.a	even	1	1	trivial
1860.2.q.i	✓	12	31.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1860, [\chi])$:

$ T_{7}^{12} + 2 T_{7}^{11} + 31 T_{7}^{10} + 44 T_{7}^{9} + 737 T_{7}^{8} + 1019 T_{7}^{7} + 4591 T_{7}^{6} + \cdots + 4096 $

$ T_{11}^{12} + 28 T_{11}^{10} + 4 T_{11}^{9} + 605 T_{11}^{8} + 50 T_{11}^{7} + 4440 T_{11}^{6} + \cdots + 82944 $

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^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$7$	$ T^{12} + 2 T^{11} + \cdots + 4096 $ T^12 + 2T^11 + 31T^10 + 44T^9 + 737T^8 + 1019T^7 + 4591T^6 - 1514T^5 + 9116T^4 - 1232T^3 + 8896T^2 - 3584*T + 4096
$11$	$ T^{12} + 28 T^{10} + \cdots + 82944 $ T^12 + 28T^10 + 4T^9 + 605T^8 + 50T^7 + 4440T^6 - 694T^5 + 23965T^4 - 2226T^3 + 51588T^2 + 1728T + 82944
$13$	$ T^{12} + 2 T^{11} + \cdots + 82944 $ T^12 + 2T^11 + 49T^10 + 188T^9 + 2020T^8 + 6191T^7 + 30496T^6 + 57359T^5 + 241501T^4 + 382308T^3 + 1059120T^2 + 307584*T + 82944
$17$	$ T^{12} + 2 T^{11} + \cdots + 82944 $ T^12 + 2T^11 + 49T^10 + 188T^9 + 2020T^8 + 6191T^7 + 30496T^6 + 57359T^5 + 241501T^4 + 382308T^3 + 1059120T^2 + 307584*T + 82944
$19$	$ T^{12} - 5 T^{11} + \cdots + 15085456 $ T^12 - 5T^11 + 106T^10 - 347T^9 + 7259T^8 - 24137T^7 + 217381T^6 - 466150T^5 + 3780104T^4 - 9422950T^3 + 25670113T^2 - 21365884*T + 15085456
$23$	$ (T^{6} - T^{5} - 92 T^{4} + \cdots - 2592)^{2} $ (T^6 - T^5 - 92T^4 + 135T^3 + 1712T^2 - 1692T - 2592)^2
$29$	$ (T^{6} + 10 T^{5} + \cdots - 324)^{2} $ (T^6 + 10T^5 - 5T^4 - 141T^3 + 117T^2 + 324*T - 324)^2
$31$	$ T^{12} + \cdots + 887503681 $ T^12 - 7T^11 + 17T^10 - 165T^9 + 2458T^8 - 8606T^7 + 8248T^6 - 266786T^5 + 2362138T^4 - 4915515T^3 + 15699857T^2 - 200404057*T + 887503681
$37$	$ T^{12} - 3 T^{11} + \cdots + 6718464 $ T^12 - 3T^11 + 94T^10 - 125T^9 + 6494T^8 - 9460T^7 + 138153T^6 + 65246T^5 + 1684321T^4 - 466956T^3 + 4617648T^2 + 2892672*T + 6718464
$41$	$ T^{12} + 9 T^{11} + \cdots + 331776 $ T^12 + 9T^11 + 139T^10 + 542T^9 + 7976T^8 + 31192T^7 + 260544T^6 + 307648T^5 + 1861696T^4 + 3840T^3 + 12379392T^2 - 2018304*T + 331776
$43$	$ T^{12} + 5 T^{11} + \cdots + 87086224 $ T^12 + 5T^11 + 174T^10 + 567T^9 + 22738T^8 + 75110T^7 + 796399T^6 - 416860T^5 + 9279757T^4 + 911844T^3 + 48599292T^2 - 44756272*T + 87086224
$47$	$ (T^{6} + 8 T^{5} + \cdots + 4248)^{2} $ (T^6 + 8T^5 - 73T^4 - 835T^3 - 1895T^2 + 822*T + 4248)^2
$53$	$ T^{12} - 21 T^{11} + \cdots + 6718464 $ T^12 - 21T^11 + 443T^10 - 5136T^9 + 72625T^8 - 735582T^7 + 7431381T^6 - 47164140T^5 + 239470668T^4 - 673778736T^3 + 1356301584T^2 + 93778560*T + 6718464
$59$	$ T^{12} + \cdots + 153363456 $ T^12 - 10T^11 + 255T^10 - 3052T^9 + 55154T^8 - 514871T^7 + 4102564T^6 - 17514819T^5 + 58256317T^4 - 90798684T^3 + 117884592T^2 + 51566976*T + 153363456
$61$	$ (T^{6} - 9 T^{5} + \cdots - 446)^{2} $ (T^6 - 9T^5 - 55T^4 + 480T^3 + 266T^2 - 2829*T - 446)^2
$67$	$ T^{12} + \cdots + 27090526464 $ T^12 - 14T^11 + 371T^10 - 2448T^9 + 59639T^8 - 375891T^7 + 5920337T^6 - 22619360T^5 + 289086052T^4 - 1306653120T^3 + 8144355600T^2 - 15625047744*T + 27090526464
$71$	$ T^{12} + \cdots + 95340147984 $ T^12 - 8T^11 + 359T^10 - 1606T^9 + 80326T^8 - 340617T^7 + 9037710T^6 - 21604995T^5 + 649298349T^4 - 1407158568T^3 + 20571712236T^2 + 36015166080*T + 95340147984
$73$	$ T^{12} + \cdots + 7664652304 $ T^12 - 10T^11 + 240T^10 - 396T^9 + 21565T^8 + 1052T^7 + 1478128T^6 + 3387110T^5 + 49101649T^4 + 62337288T^3 + 797156004T^2 + 1184349344*T + 7664652304
$79$	$ T^{12} + \cdots + 936115216 $ T^12 + 15T^11 + 262T^10 + 3117T^9 + 41243T^8 + 405303T^7 + 3343081T^6 + 19770258T^5 + 91051940T^4 + 290344254T^3 + 688586137T^2 + 998928804*T + 936115216
$83$	$ T^{12} + \cdots + 37053170064 $ T^12 + 28T^11 + 685T^10 + 8974T^9 + 132028T^8 + 1535353T^7 + 16655494T^6 + 132100939T^5 + 838030717T^4 + 3761741424T^3 + 12783735756T^2 + 26947340064*T + 37053170064
$89$	$ (T^{6} - 4 T^{5} + \cdots - 130896)^{2} $ (T^6 - 4T^5 - 313T^4 + 1131T^3 + 22509T^2 - 86994*T - 130896)^2
$97$	$ (T^{6} + 38 T^{5} + \cdots + 6370992)^{2} $ (T^6 + 38T^5 + 26T^4 - 13696T^3 - 137315T^2 + 372018*T + 6370992)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1860.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{3} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{9} - \beta_{4}) q^{7} + (\beta_{7} - 1) q^{9} - \beta_{6} q^{11} + \beta_{11} q^{13} - q^{15} + (\beta_{11} + \beta_{9} - \beta_{7} + \cdots + \beta_1) q^{17}+ \cdots + (\beta_{6} - \beta_{5}) q^{99}+O(q^{100}) \) q - b7 * q^3 + (-b7 + 1) * q^5 + (-b9 - b4) * q^7 + (b7 - 1) * q^9 - b6 * q^11 + b11 * q^13 - q^15 + (b11 + b9 - b7 - b6 + b5 + b4 + b1) * q^17 + (-b11 - b9 + b7 - b4 + b3 - b2 - b1) * q^19 + b9 * q^21 + (-b8 - b5 - b3 - b2 - b1) * q^23 - b7 * q^25 + q^27 + (b5 + b4 - 2) * q^29 + (b11 + b8 - b5 + 1) * q^31 + b5 * q^33 - b4 * q^35 + (-b6 + b5 + b3 - b2) * q^37 + b1 * q^39 + (b11 - b10 - b9 + 2b7 - b3 - 2) q^41 + (b10 + 2b9 - b8 - 2b7 - b6 + b5 + 2b4 - b2) q^43 + b7 * q^45 + (b5 - b1 - 1) * q^47 + (-b9 + b8 + 3b7 + 2b6 - b2 - 3) * q^49 + (-b11 - b9 + b7 + b6 - 1) * q^51 + (-b11 + b10 + b9 + b8 - 4b7 + b6 + b3 - b2 + 4) q^53 + (-b6 + b5) * q^55 + (b11 - b10 + b9 - b8 - b7 - b3 + b2 + 1) * q^57 + (2b11 - b10 + b8 + 2b7 + b6 - b5 - b3 + 2b2 + 2b1) * q^59 + (-b10 - b8 + b5 + 1) * q^61 + b4 * q^63 + (b11 + b1) * q^65 + (2b11 - 2b10 - b8 - 2b7 - 2b6 - 2b3 + b2 + 2) q^67 + (b11 - b10 + b8 - b6 + b5 + b2 + b1) * q^69 + (-2b11 + 2b10 + b9 - 2b7 + b6 + 2b3 + 2) * q^71 + (b11 + b8 - 2b7 - b2 + 2) q^73 + (b7 - 1) * q^75 + (2b5 - b4 + 2b1 - 2) * q^77 + (2b11 - b10 + b9 + b8 - 3b7 - b6 + b5 + b4 + b2 + 2b1) q^79 - b7 * q^81 + (b11 - 2b9 + 5b7 - b6 - 5) * q^83 + (b5 + b4 + b1 - 1) * q^85 + (-b9 + 2b7 + b6 - b5 - b4) q^87 + (-b10 + 3b5 + b3 + b2 + 2b1) * q^89 + (b10 + b8 + 3b5 + 2b4 + b1 - 4) * q^91 + (-b7 - b6 + b5 + b3 + b1) * q^93 + (-b10 - b8 - b4 - b1 + 1) * q^95 + (b10 - 2b5 - 2b4 - b3 - b2 + b1 - 6) * q^97 + (b6 - b5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 6 q^{5} - 2 q^{7} - 6 q^{9} - 2 q^{13} - 12 q^{15} - 2 q^{17} + 5 q^{19} - 2 q^{21} + 2 q^{23} - 6 q^{25} + 12 q^{27} - 20 q^{29} + 7 q^{31} - 4 q^{35} + 3 q^{37} + 4 q^{39} - 9 q^{41}+ \cdots - 76 q^{97}+O(q^{100}) \) 12 * q - 6 * q^3 + 6 * q^5 - 2 * q^7 - 6 * q^9 - 2 * q^13 - 12 * q^15 - 2 * q^17 + 5 * q^19 - 2 * q^21 + 2 * q^23 - 6 * q^25 + 12 * q^27 - 20 * q^29 + 7 * q^31 - 4 * q^35 + 3 * q^37 + 4 * q^39 - 9 * q^41 - 5 * q^43 + 6 * q^45 - 16 * q^47 - 16 * q^49 - 2 * q^51 + 21 * q^53 + 5 * q^57 + 10 * q^59 + 18 * q^61 + 4 * q^63 + 2 * q^65 + 14 * q^67 - q^69 + 8 * q^71 + 10 * q^73 - 6 * q^75 - 20 * q^77 - 15 * q^79 - 6 * q^81 - 28 * q^83 - 4 * q^85 + 10 * q^87 + 8 * q^89 - 42 * q^91 - 2 * q^93 + 10 * q^95 - 76 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1860.2.q.i

Level	$1860$
Weight	$2$
Character orbit	1860.q
Analytic conductor	$14.852$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(14.8521747760\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 3 x^{11} + 29 x^{10} + 405 x^{8} - 117 x^{7} + 2515 x^{6} + 1476 x^{5} + 7995 x^{4} - 2175 x^{3} + \cdots + 4 \) x^12 - 3x^11 + 29x^10 + 405x^8 - 117x^7 + 2515x^6 + 1476x^5 + 7995x^4 - 2175x^3 + 899x^2 + 54x + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index:	\( 3^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - 1715905482153 \nu^{11} + 4307423742224 \nu^{10} - 47611747561143 \nu^{9} + \cdots + 42\!\cdots\!92 ) / 13\!\cdots\!12 \) (-1715905482153v^11 + 4307423742224v^10 - 47611747561143v^9 - 22026732866505v^8 - 709213546702347v^7 - 102743655549935v^6 - 4360517418576414v^5 - 4117141587930315v^4 - 15690355272729300v^3 - 592617816912946v^2 - 35924442731196*v + 4271846952750392) / 1365970028461912
\(\beta_{2}\)	\(=\)	\( ( 3882514598555 \nu^{11} - 3654097505731 \nu^{10} + 83423074388066 \nu^{9} + \cdots + 34\!\cdots\!52 ) / 27\!\cdots\!24 \) (3882514598555v^11 - 3654097505731v^10 + 83423074388066v^9 + 244430501364486v^8 + 1429627582566179v^7 + 2699925694537313v^6 + 6684966121603390v^5 + 25288432225061936v^4 + 29330130510866487v^3 + 41103545291472447v^2 - 65901573804163378*v + 3433312164825252) / 2731940056923824
\(\beta_{3}\)	\(=\)	\( ( 21589580391119 \nu^{11} - 42806520136345 \nu^{10} + 558994369293702 \nu^{9} + \cdots + 46\!\cdots\!64 ) / 27\!\cdots\!24 \) (21589580391119v^11 - 42806520136345v^10 + 558994369293702v^9 + 636198184577068v^8 + 8721058123973357v^7 + 6251728966003151v^6 + 51200575755143796v^5 + 85706421659100116v^4 + 202385230336415205v^3 + 118618963619174607v^2 - 45417058990762410*v + 4666342363880964) / 2731940056923824
\(\beta_{4}\)	\(=\)	\( ( 25075145750338 \nu^{11} - 66873958754819 \nu^{10} + 701903809517062 \nu^{9} + \cdots + 79\!\cdots\!60 ) / 27\!\cdots\!24 \) (25075145750338v^11 - 66873958754819v^10 + 701903809517062v^9 + 240739291405050v^8 + 10158536438098922v^7 + 381127269019235v^6 + 62056919293496074v^5 + 57385505985515610v^4 + 213000797425722006v^3 + 8306088831935581v^2 + 503605401255266*v + 7906950258268260) / 2731940056923824
\(\beta_{5}\)	\(=\)	\( ( 12926849045593 \nu^{11} - 34706543270271 \nu^{10} + 362382059783535 \nu^{9} + \cdots + 20\!\cdots\!76 ) / 13\!\cdots\!12 \) (12926849045593v^11 - 34706543270271v^10 + 362382059783535v^9 + 119326970175441v^8 + 5224870434206255v^7 + 149627157387372v^6 + 31893781538332708v^5 + 29419880460172191v^4 + 108152265166927838v^3 + 4261135446720783v^2 + 258361942228046*v + 2038601673147076) / 1365970028461912
\(\beta_{6}\)	\(=\)	\( ( - 19387396113109 \nu^{11} + 60071660832906 \nu^{10} - 569315897346363 \nu^{9} + \cdots + 248215096289260 ) / 13\!\cdots\!12 \) (-19387396113109v^11 + 60071660832906v^10 - 569315897346363v^9 + 59279220527206v^8 - 7890565716131224v^7 + 3034790214632463v^6 - 49515347833257093v^5 - 23784347613834399v^4 - 155406757821982111v^3 + 54547029143567445v^2 - 29541366298068178*v + 248215096289260) / 1365970028461912
\(\beta_{7}\)	\(=\)	\( ( 5238566167851 \nu^{11} - 16062118176933 \nu^{10} + 152835571159827 \nu^{9} + \cdots + 275888522976714 ) / 341492507115478 \) (5238566167851v^11 - 16062118176933v^10 + 152835571159827v^9 - 9682882600990v^8 + 2118154416422211v^7 - 753607026816579v^6 + 13168025268563061v^5 + 6871939112567340v^4 + 41084778923472141v^3 - 14416318613667057v^2 + 4594113701208861*v + 275888522976714) / 341492507115478
\(\beta_{8}\)	\(=\)	\( ( 63056330068462 \nu^{11} - 189866016689827 \nu^{10} + \cdots + 26\!\cdots\!84 ) / 27\!\cdots\!24 \) (63056330068462v^11 - 189866016689827v^10 + 1836627282515480v^9 - 35370890723074v^8 + 25702434406061388v^7 - 7599284004898975v^6 + 161111967635570212v^5 + 91445839241821448v^4 + 517813700557797434v^3 - 130418067849461937v^2 + 113095392052649390*v + 2652850925687084) / 2731940056923824
\(\beta_{9}\)	\(=\)	\( ( 98159661622173 \nu^{11} - 301815112817580 \nu^{10} + \cdots - 12\!\cdots\!88 ) / 27\!\cdots\!24 \) (98159661622173v^11 - 301815112817580v^10 + 2868428648186826v^9 - 210767411495224v^8 + 39745421039833735v^7 - 14428483954028640v^6 + 247719128415491034v^5 + 126695924654145076v^4 + 774040902698758701v^3 - 271624806414541104v^2 + 110551499175246080*v - 1236062870602888) / 2731940056923824
\(\beta_{10}\)	\(=\)	\( ( - 104845333135128 \nu^{11} + 301468108669177 \nu^{10} + \cdots - 35\!\cdots\!84 ) / 27\!\cdots\!24 \) (-104845333135128v^11 + 301468108669177v^10 - 3006804001728090v^9 - 362667601270232v^8 - 42624135392662450v^7 + 6979472085342847v^6 - 264468020066549484v^5 - 186973249233009614v^4 - 872467357012614226v^3 + 116589361906317723v^2 - 113933842241118522*v - 3596040361480084) / 2731940056923824
\(\beta_{11}\)	\(=\)	\( ( 67181011336704 \nu^{11} - 205968556515918 \nu^{10} + \cdots - 841726563234580 ) / 13\!\cdots\!12 \) (67181011336704v^11 - 205968556515918v^10 + 1960283587208253v^9 - 125908385552566v^8 + 27174920983523613v^7 - 9682649987872563v^6 + 168928638792809103v^5 + 87771000218055947v^4 + 527132019349132626v^3 - 184967573186327505v^2 + 57157396458040110*v - 841726563234580) / 1365970028461912

\(\nu\)	\(=\)	\( ( \beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} + 2\beta_{6} - \beta_{3} + 1 ) / 3 \) (b11 - b10 - b9 - b7 + 2*b6 - b3 + 1) / 3
\(\nu^{2}\)	\(=\)	\( ( 5 \beta_{11} - 3 \beta_{10} + \beta_{9} + 3 \beta_{8} - 23 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 5 \beta_1 ) / 3 \) (5b11 - 3b10 + b9 + 3b8 - 23b7 + 4b6 - 4b5 + b4 - 4b3 + 7b2 + 5*b1) / 3
\(\nu^{3}\)	\(=\)	\( ( 12\beta_{10} + 35\beta_{8} - 34\beta_{5} - 23\beta_{4} + 23\beta_{3} + 23\beta_{2} + 23\beta _1 - 41 ) / 3 \) (12b10 + 35b8 - 34b5 - 23b4 + 23b3 + 23b2 + 23*b1 - 41) / 3
\(\nu^{4}\)	\(=\)	\( - 35 \beta_{11} + 65 \beta_{10} + 13 \beta_{9} + 32 \beta_{8} + 125 \beta_{7} - 44 \beta_{6} + \cdots - 125 \) -35b11 + 65b10 + 13b9 + 32b8 + 125b7 - 44b6 + 65b3 - 32b2 - 125
\(\nu^{5}\)	\(=\)	\( ( - 551 \beta_{11} + 587 \beta_{10} + 515 \beta_{9} - 587 \beta_{8} + 1241 \beta_{7} - 778 \beta_{6} + \cdots - 551 \beta_1 ) / 3 \) (-551b11 + 587b10 + 515b9 - 587b8 + 1241b7 - 778b6 + 778b5 + 515b4 + 408b3 - 995b2 - 551*b1) / 3
\(\nu^{6}\)	\(=\)	\( ( - 2444 \beta_{10} - 5267 \beta_{8} + 3668 \beta_{5} + 1747 \beta_{4} - 2823 \beta_{3} - 2823 \beta_{2} + \cdots + 8071 ) / 3 \) (-2444b10 - 5267b8 + 3668b5 + 1747b4 - 2823b3 - 2823b2 - 2665*b1 + 8071) / 3
\(\nu^{7}\)	\(=\)	\( ( 14047 \beta_{11} - 27067 \beta_{10} - 12391 \beta_{9} - 11700 \beta_{8} - 34921 \beta_{7} + 19826 \beta_{6} + \cdots + 34921 ) / 3 \) (14047b11 - 27067b10 - 12391b9 - 11700b8 - 34921b7 + 19826b6 - 27067b3 + 11700b2 + 34921) / 3
\(\nu^{8}\)	\(=\)	\( 23539 \beta_{11} - 25721 \beta_{10} - 17989 \beta_{9} + 25721 \beta_{8} - 66077 \beta_{7} + \cdots + 23539 \beta_1 \) 23539b11 - 25721b10 - 17989b9 + 25721b8 - 66077b7 + 33068b6 - 33068b5 - 17989b4 - 21304b3 + 47025b2 + 23539*b1
\(\nu^{9}\)	\(=\)	\( ( 320352 \beta_{10} + 727315 \beta_{8} - 521594 \beta_{5} - 314635 \beta_{4} + 406963 \beta_{3} + \cdots - 955705 ) / 3 \) (320352b10 + 727315b8 - 521594b5 - 314635b4 + 406963b3 + 406963b2 + 368887*b1 - 955705) / 3
\(\nu^{10}\)	\(=\)	\( ( - 1886153 \beta_{11} + 3770995 \beta_{10} + 1517003 \beta_{9} + 1691764 \beta_{8} + 5132327 \beta_{7} + \cdots - 5132327 ) / 3 \) (-1886153b11 + 3770995b10 + 1517003b9 + 1691764b8 + 5132327b7 - 2663572b6 + 3770995b3 - 1691764b2 - 5132327) / 3
\(\nu^{11}\)	\(=\)	\( ( - 9796751 \beta_{11} + 10833263 \beta_{10} + 8219663 \beta_{9} - 10833263 \beta_{8} + \cdots - 9796751 \beta_1 ) / 3 \) (-9796751b11 + 10833263b10 + 8219663b9 - 10833263b8 + 25816265b7 - 13861858b6 + 13861858b5 + 8219663b4 + 8637276b3 - 19470539b2 - 9796751*b1) / 3

\(n\)	\(931\)	\(1117\)	\(1241\)	\(1801\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1 + \beta_{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1141.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$5$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$7$	\( T^{12} + 2 T^{11} + \cdots + 4096 \) T^12 + 2T^11 + 31T^10 + 44T^9 + 737T^8 + 1019T^7 + 4591T^6 - 1514T^5 + 9116T^4 - 1232T^3 + 8896T^2 - 3584*T + 4096
$11$	\( T^{12} + 28 T^{10} + \cdots + 82944 \) T^12 + 28T^10 + 4T^9 + 605T^8 + 50T^7 + 4440T^6 - 694T^5 + 23965T^4 - 2226T^3 + 51588T^2 + 1728T + 82944
$13$	\( T^{12} + 2 T^{11} + \cdots + 82944 \) T^12 + 2T^11 + 49T^10 + 188T^9 + 2020T^8 + 6191T^7 + 30496T^6 + 57359T^5 + 241501T^4 + 382308T^3 + 1059120T^2 + 307584*T + 82944
$17$	\( T^{12} + 2 T^{11} + \cdots + 82944 \) T^12 + 2T^11 + 49T^10 + 188T^9 + 2020T^8 + 6191T^7 + 30496T^6 + 57359T^5 + 241501T^4 + 382308T^3 + 1059120T^2 + 307584*T + 82944
$19$	\( T^{12} - 5 T^{11} + \cdots + 15085456 \) T^12 - 5T^11 + 106T^10 - 347T^9 + 7259T^8 - 24137T^7 + 217381T^6 - 466150T^5 + 3780104T^4 - 9422950T^3 + 25670113T^2 - 21365884*T + 15085456
$23$	\( (T^{6} - T^{5} - 92 T^{4} + \cdots - 2592)^{2} \) (T^6 - T^5 - 92T^4 + 135T^3 + 1712T^2 - 1692T - 2592)^2
$29$	\( (T^{6} + 10 T^{5} + \cdots - 324)^{2} \) (T^6 + 10T^5 - 5T^4 - 141T^3 + 117T^2 + 324*T - 324)^2
$31$	\( T^{12} + \cdots + 887503681 \) T^12 - 7T^11 + 17T^10 - 165T^9 + 2458T^8 - 8606T^7 + 8248T^6 - 266786T^5 + 2362138T^4 - 4915515T^3 + 15699857T^2 - 200404057*T + 887503681
$37$	\( T^{12} - 3 T^{11} + \cdots + 6718464 \) T^12 - 3T^11 + 94T^10 - 125T^9 + 6494T^8 - 9460T^7 + 138153T^6 + 65246T^5 + 1684321T^4 - 466956T^3 + 4617648T^2 + 2892672*T + 6718464
$41$	\( T^{12} + 9 T^{11} + \cdots + 331776 \) T^12 + 9T^11 + 139T^10 + 542T^9 + 7976T^8 + 31192T^7 + 260544T^6 + 307648T^5 + 1861696T^4 + 3840T^3 + 12379392T^2 - 2018304*T + 331776
$43$	\( T^{12} + 5 T^{11} + \cdots + 87086224 \) T^12 + 5T^11 + 174T^10 + 567T^9 + 22738T^8 + 75110T^7 + 796399T^6 - 416860T^5 + 9279757T^4 + 911844T^3 + 48599292T^2 - 44756272*T + 87086224
$47$	\( (T^{6} + 8 T^{5} + \cdots + 4248)^{2} \) (T^6 + 8T^5 - 73T^4 - 835T^3 - 1895T^2 + 822*T + 4248)^2
$53$	\( T^{12} - 21 T^{11} + \cdots + 6718464 \) T^12 - 21T^11 + 443T^10 - 5136T^9 + 72625T^8 - 735582T^7 + 7431381T^6 - 47164140T^5 + 239470668T^4 - 673778736T^3 + 1356301584T^2 + 93778560*T + 6718464
$59$	\( T^{12} + \cdots + 153363456 \) T^12 - 10T^11 + 255T^10 - 3052T^9 + 55154T^8 - 514871T^7 + 4102564T^6 - 17514819T^5 + 58256317T^4 - 90798684T^3 + 117884592T^2 + 51566976*T + 153363456
$61$	\( (T^{6} - 9 T^{5} + \cdots - 446)^{2} \) (T^6 - 9T^5 - 55T^4 + 480T^3 + 266T^2 - 2829*T - 446)^2
$67$	\( T^{12} + \cdots + 27090526464 \) T^12 - 14T^11 + 371T^10 - 2448T^9 + 59639T^8 - 375891T^7 + 5920337T^6 - 22619360T^5 + 289086052T^4 - 1306653120T^3 + 8144355600T^2 - 15625047744*T + 27090526464
$71$	\( T^{12} + \cdots + 95340147984 \) T^12 - 8T^11 + 359T^10 - 1606T^9 + 80326T^8 - 340617T^7 + 9037710T^6 - 21604995T^5 + 649298349T^4 - 1407158568T^3 + 20571712236T^2 + 36015166080*T + 95340147984
$73$	\( T^{12} + \cdots + 7664652304 \) T^12 - 10T^11 + 240T^10 - 396T^9 + 21565T^8 + 1052T^7 + 1478128T^6 + 3387110T^5 + 49101649T^4 + 62337288T^3 + 797156004T^2 + 1184349344*T + 7664652304
$79$	\( T^{12} + \cdots + 936115216 \) T^12 + 15T^11 + 262T^10 + 3117T^9 + 41243T^8 + 405303T^7 + 3343081T^6 + 19770258T^5 + 91051940T^4 + 290344254T^3 + 688586137T^2 + 998928804*T + 936115216
$83$	\( T^{12} + \cdots + 37053170064 \) T^12 + 28T^11 + 685T^10 + 8974T^9 + 132028T^8 + 1535353T^7 + 16655494T^6 + 132100939T^5 + 838030717T^4 + 3761741424T^3 + 12783735756T^2 + 26947340064*T + 37053170064
$89$	\( (T^{6} - 4 T^{5} + \cdots - 130896)^{2} \) (T^6 - 4T^5 - 313T^4 + 1131T^3 + 22509T^2 - 86994*T - 130896)^2
$97$	\( (T^{6} + 38 T^{5} + \cdots + 6370992)^{2} \) (T^6 + 38T^5 + 26T^4 - 13696T^3 - 137315T^2 + 372018*T + 6370992)^2
show more
show less