LMFDB - Newform orbit 1458.2.c.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,2,Mod(487,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("1458.487");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1458.c (of order $3$, 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:	$11.6421886147$
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} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 $ x^12 - 6x^11 + 33x^10 - 110x^9 + 318x^8 - 678x^7 + 1225x^6 - 1698x^5 + 1905x^4 - 1584x^3 + 936x^2 - 342*x + 57
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{7} $
Twist minimal:	no (minimal twist has level 54)
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_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{10} + \beta_{2}) q^{7} + q^{8}+O(q^{10}) $ q + b2 * q^2 + (-b2 - 1) * q^4 + (b4 - b1 - 1) * q^5 + (-b10 + b2) * q^7 + q^8	$ q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{10} + \beta_{2}) q^{7} + q^{8} + (\beta_1 + 1) q^{10} + ( - \beta_{7} + \beta_{2}) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 2) q^{13}+ \cdots + (\beta_{11} + 2 \beta_{9} + \beta_{6} + \cdots + 3) q^{98}+O(q^{100}) $ q + b2 * q^2 + (-b2 - 1) * q^4 + (b4 - b1 - 1) * q^5 + (-b10 + b2) * q^7 + q^8 + (b1 + 1) * q^10 + (-b7 + b2) * q^11 + (-b7 - b6 - b5 - b2 - 2) * q^13 + (-b11 + b10 - b2 - 1) * q^14 + b2 * q^16 + (-b11 - b9 + b3 + 1) * q^17 + (-b3 - b1 + 1) * q^19 - b4 * q^20 + (-b9 + b7 + b6 - b2) * q^22 + (2b8 - b7 - b6 - b5 + b2) q^23 + (b10 + 2b8 - b7 - 3b5 - 2b3 + 3b2) * q^25 + (b6 + 2) * q^26 + (b11 + 1) * q^28 + (b10 + b7 + b5) * q^29 + (-2b9 + b7 + b6 - b5 - b4 - 3b2 + b1 - 1) * q^31 + (-b2 - 1) * q^32 + (b10 + b8 - b5 - b3 + b2) * q^34 + (b11 - 2b9 + 2b3 + b1) * q^35 + (-b11 - 2b3 - b1 + 2) q^37 + (-b8 + b4 + b3 + 2b2) q^38 + (b4 - b1 - 1) * q^40 + (b11 - b10 + b8 - b4 - b2 + b1) * q^41 + (b10 + b8 + b7 + b4 - b3 + 2b2) q^43 + (b9 - b6) * q^44 + (b6 - 2b3) q^46 + (-b10 + 2b5 + b4 - b2) q^47 + (-b11 + b10 - 2b9 - b7 - b6 - 3b5 + b4 - b2 - b1 - 3) * q^49 + (b11 - b10 + 2b9 - 2b8 + b7 + b6 + 3b5 - 3b2 - 2) * q^50 + (b7 + b5 + b2) * q^52 + (b11 - b6 - b1 - 2) * q^53 + (-b11 - b6 - 2b3 + 1) q^55 + (-b10 + b2) * q^56 + (b11 - b10 - b7 - b6 - b5 - 1) * q^58 + (b8 + b7 + b6 + b5 - 2b4 - b2 + 2b1 + 2) * q^59 + (2b8 + b7 - b5 - b4 - 2b3 + b2) * q^61 + (2b9 - b6 - b1 + 1) q^62 + q^64 + (2b8 + 2b7 + 2b5 - b4 - 2b3 - 2b2) q^65 + (b11 - b10 - 3b9 + 2b7 + 2b6 - b5 - b4 - 5b2 + b1 - 2) * q^67 + (b11 - b10 + b9 - b8 + b5 - b2 - 1) * q^68 + (-b10 + 2b8 - 2b5 - b4 - 2b3 - b2) q^70 + (b11 - 2b9 + b6 - b1 - 2) q^71 + (-b9 - b6 + b3 + 2b1 + 4) q^73 + (b10 - 2b8 + b4 + 2b3 + 3b2) q^74 + (b8 - b4 - 2b2 + b1 - 1) q^76 + (-b11 + b10 - 4b9 + 2b7 + 2b6 - 2b5 - b4 - b2 + b1 + 2) * q^77 + (-2b10 + 2b8 + b7 + b5 - b4 - 2b3 - b2) q^79 + (b1 + 1) * q^80 + (-b11 - b3 - b1) * q^82 + (-2b10 - b7 + b5 - 4b2) * q^83 + (-b11 + b10 - 2b9 + 2b8 - b7 - b6 - 3b5 + 3b4 + 2b2 - 3b1 - 2) * q^85 + (b11 - b10 + b9 - b8 - b7 - b6 - b4 - 2b2 + b1 - 2) q^86 + (-b7 + b2) * q^88 + (-2b11 + b6 - 4b3 - b1 - 2) * q^89 + (3b11 - 2b9 + 1) * q^91 + (-2b8 + b7 + b5 + 2b3 - b2) * q^92 + (-b11 + b10 - 2b9 - 2b5 - b4 + b2 + b1 + 2) * q^94 + (-2b9 + 2b8 + b7 + b6 - b5 + b4 + 5b2 - b1 + 5) q^95 + (-b10 + 2b8 - 3b7 - 4b5 - b4 - 2b3 + 4b2) q^97 + (b11 + 2b9 + b6 + b1 + 3) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q - 6 * q^2 - 6 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	$ 12 q - 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} + 6 q^{10} - 3 q^{11} - 9 q^{13} - 6 q^{14} - 6 q^{16} + 12 q^{17} + 18 q^{19} - 3 q^{20} - 3 q^{22} + 3 q^{23} - 15 q^{25} + 18 q^{26} + 12 q^{28} - 3 q^{29} - 12 q^{31} - 6 q^{32} - 6 q^{34} - 6 q^{35} + 30 q^{37} - 9 q^{38} - 3 q^{40} - 3 q^{41} - 12 q^{43} + 6 q^{44} - 6 q^{46} + 9 q^{47} - 12 q^{49} - 15 q^{50} - 9 q^{52} - 12 q^{53} + 18 q^{55} - 6 q^{56} - 3 q^{58} + 3 q^{59} - 12 q^{61} + 24 q^{62} + 12 q^{64} + 3 q^{65} - 21 q^{67} - 6 q^{68} + 3 q^{70} - 24 q^{71} + 42 q^{73} - 15 q^{74} - 9 q^{76} + 3 q^{77} + 6 q^{80} + 6 q^{82} + 27 q^{83} - 12 q^{86} - 3 q^{88} - 24 q^{89} + 12 q^{91} + 3 q^{92} + 9 q^{94} + 30 q^{95} - 18 q^{97} + 24 q^{98}+O(q^{100}) $ 12 * q - 6 * q^2 - 6 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 + 6 * q^10 - 3 * q^11 - 9 * q^13 - 6 * q^14 - 6 * q^16 + 12 * q^17 + 18 * q^19 - 3 * q^20 - 3 * q^22 + 3 * q^23 - 15 * q^25 + 18 * q^26 + 12 * q^28 - 3 * q^29 - 12 * q^31 - 6 * q^32 - 6 * q^34 - 6 * q^35 + 30 * q^37 - 9 * q^38 - 3 * q^40 - 3 * q^41 - 12 * q^43 + 6 * q^44 - 6 * q^46 + 9 * q^47 - 12 * q^49 - 15 * q^50 - 9 * q^52 - 12 * q^53 + 18 * q^55 - 6 * q^56 - 3 * q^58 + 3 * q^59 - 12 * q^61 + 24 * q^62 + 12 * q^64 + 3 * q^65 - 21 * q^67 - 6 * q^68 + 3 * q^70 - 24 * q^71 + 42 * q^73 - 15 * q^74 - 9 * q^76 + 3 * q^77 + 6 * q^80 + 6 * q^82 + 27 * q^83 - 12 * q^86 - 3 * q^88 - 24 * q^89 + 12 * q^91 + 3 * q^92 + 9 * q^94 + 30 * q^95 - 18 * q^97 + 24 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 $ x^12 - 6*x^11 + 33*x^10 - 110*x^9 + 318*x^8 - 678*x^7 + 1225*x^6 - 1698*x^5 + 1905*x^4 - 1584*x^3 + 936*x^2 - 342*x + 57 :

$\beta_{1}$	$=$	$ ( -\nu^{6} + 3\nu^{5} - 14\nu^{4} + 23\nu^{3} - 41\nu^{2} + 30\nu - 11 ) / 2 $ (-v^6 + 3v^5 - 14v^4 + 23v^3 - 41v^2 + 30*v - 11) / 2
$\beta_{2}$	$=$	$ ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + \cdots + 842 ) / 218 $ (6v^11 - 33v^10 + 127v^9 - 324v^8 + 438v^7 - 252v^6 - 1278v^5 + 3234v^4 - 5701v^3 + 5358v^2 - 3477*v + 842) / 218
$\beta_{3}$	$=$	$ ( \nu^{8} - 4\nu^{7} + 18\nu^{6} - 40\nu^{5} + 82\nu^{4} - 102\nu^{3} + 102\nu^{2} - 57\nu + 15 ) / 2 $ (v^8 - 4v^7 + 18v^6 - 40v^5 + 82v^4 - 102v^3 + 102v^2 - 57*v + 15) / 2
$\beta_{4}$	$=$	$ ( - 2 \nu^{11} + 11 \nu^{10} - 115 \nu^{9} + 435 \nu^{8} - 1781 \nu^{7} + 4226 \nu^{6} - 9493 \nu^{5} + \cdots + 882 ) / 218 $ (-2v^11 + 11v^10 - 115v^9 + 435v^8 - 1781v^7 + 4226v^6 - 9493v^5 + 13637v^4 - 16775v^3 + 12275v^2 - 5163*v + 882) / 218
$\beta_{5}$	$=$	$ ( 81 \nu^{11} - 282 \nu^{10} + 1224 \nu^{9} - 1867 \nu^{8} + 1771 \nu^{7} + 5754 \nu^{6} - 22703 \nu^{5} + \cdots + 5154 ) / 218 $ (81v^11 - 282v^10 + 1224v^9 - 1867v^8 + 1771v^7 + 5754v^6 - 22703v^5 + 45512v^4 - 62085v^3 + 50751v^2 - 25194*v + 5154) / 218
$\beta_{6}$	$=$	$ ( - \nu^{10} + 5 \nu^{9} - 24 \nu^{8} + 66 \nu^{7} - 154 \nu^{6} + 252 \nu^{5} - 318 \nu^{4} + 283 \nu^{3} + \cdots - 6 ) / 2 $ (-v^10 + 5v^9 - 24v^8 + 66v^7 - 154v^6 + 252v^5 - 318v^4 + 283v^3 - 148v^2 + 39*v - 6) / 2
$\beta_{7}$	$=$	$ ( - 2 \nu^{11} - 98 \nu^{10} + 539 \nu^{9} - 2726 \nu^{8} + 8247 \nu^{7} - 20517 \nu^{6} + 38140 \nu^{5} + \cdots - 4786 ) / 218 $ (-2v^11 - 98v^10 + 539v^9 - 2726v^8 + 8247v^7 - 20517v^6 + 38140v^5 - 54815v^4 + 60615v^3 - 45277v^2 + 22632*v - 4786) / 218
$\beta_{8}$	$=$	$ ( - 78 \nu^{11} + 429 \nu^{10} - 2196 \nu^{9} + 6719 \nu^{8} - 17684 \nu^{7} + 34014 \nu^{6} + \cdots + 3333 ) / 218 $ (-78v^11 + 429v^10 - 2196v^9 + 6719v^8 - 17684v^7 + 34014v^6 - 54236v^5 + 65432v^4 - 61047v^3 + 40545v^2 - 16929*v + 3333) / 218
$\beta_{9}$	$=$	$ ( - 3 \nu^{10} + 15 \nu^{9} - 73 \nu^{8} + 202 \nu^{7} - 483 \nu^{6} + 805 \nu^{5} - 1072 \nu^{4} + \cdots - 30 ) / 2 $ (-3v^10 + 15v^9 - 73v^8 + 202v^7 - 483v^6 + 805v^5 - 1072v^4 + 1008v^3 - 627v^2 + 228v - 30) / 2
$\beta_{10}$	$=$	$ ( 277 \nu^{11} - 1251 \nu^{10} + 6281 \nu^{9} - 16266 \nu^{8} + 39841 \nu^{7} - 62864 \nu^{6} + \cdots - 731 ) / 218 $ (277v^11 - 1251v^10 + 6281v^9 - 16266v^8 + 39841v^7 - 62864v^6 + 86405v^5 - 77744v^4 + 52922v^3 - 21433v^2 + 5976*v - 731) / 218
$\beta_{11}$	$=$	$ ( 5 \nu^{10} - 25 \nu^{9} + 123 \nu^{8} - 342 \nu^{7} + 831 \nu^{6} - 1401 \nu^{5} + 1922 \nu^{4} + \cdots + 98 ) / 2 $ (5v^10 - 25v^9 + 123v^8 - 342v^7 + 831v^6 - 1401v^5 + 1922v^4 - 1858v^3 + 1255v^2 - 510v + 98) / 2

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

$\nu$	$=$	$ ( -\beta_{9} - 2\beta_{8} - 2\beta_{5} + 6\beta_{4} + \beta_{3} + 3\beta_{2} - 3\beta _1 + 3 ) / 9 $ (-b9 - 2b8 - 2b5 + 6b4 + b3 + 3b2 - 3*b1 + 3) / 9
$\nu^{2}$	$=$	$ ( 3\beta_{11} + 5\beta_{9} - 2\beta_{8} - 3\beta_{6} - 2\beta_{5} + 6\beta_{4} - 2\beta_{3} + 3\beta_{2} - 24 ) / 9 $ (3b11 + 5b9 - 2b8 - 3b6 - 2b5 + 6b4 - 2b3 + 3b2 - 24) / 9
$\nu^{3}$	$=$	$ ( 6 \beta_{11} - 3 \beta_{10} + 14 \beta_{9} + \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + 13 \beta_{5} + \cdots - 39 ) / 9 $ (6b11 - 3b10 + 14b9 + b8 + 3b7 - 3b6 + 13b5 - 21b4 - 5b3 - 30b2 + 15b1 - 39) / 9
$\nu^{4}$	$=$	$ ( - 9 \beta_{11} - 6 \beta_{10} - 10 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + 28 \beta_{5} + \cdots + 54 ) / 9 $ (-9b11 - 6b10 - 10b9 + 4b8 + 6b7 + 6b6 + 28b5 - 48b4 + 13b3 - 63b2 + 3*b1 + 54) / 9
$\nu^{5}$	$=$	$ ( - 48 \beta_{11} + 21 \beta_{10} - 88 \beta_{9} + 28 \beta_{8} - 12 \beta_{7} + 9 \beta_{6} - 53 \beta_{5} + \cdots + 234 ) / 9 $ (-48b11 + 21b10 - 88b9 + 28b8 - 12b7 + 9b6 - 53b5 + 60b4 + 31b3 + 126b2 - 90*b1 + 234) / 9
$\nu^{6}$	$=$	$ ( - 3 \beta_{11} + 78 \beta_{10} - 37 \beta_{9} + 73 \beta_{8} - 51 \beta_{7} - 3 \beta_{6} - 230 \beta_{5} + \cdots + 24 ) / 9 $ (-3b11 + 78b10 - 37b9 + 73b8 - 51b7 - 3b6 - 230b5 + 303b4 - 92b3 + 537b2 - 75*b1 + 24) / 9
$\nu^{7}$	$=$	$ ( 294 \beta_{11} - 63 \beta_{10} + 464 \beta_{9} - 140 \beta_{8} + 30 \beta_{7} + 36 \beta_{6} + \cdots - 1080 ) / 9 $ (294b11 - 63b10 + 464b9 - 140b8 + 30b7 + 36b6 + 97b5 - 42b4 - 287b3 - 225b2 + 504*b1 - 1080) / 9
$\nu^{8}$	$=$	$ ( 354 \beta_{11} - 630 \beta_{10} + 683 \beta_{9} - 890 \beta_{8} + 372 \beta_{7} + 66 \beta_{6} + \cdots - 1314 ) / 9 $ (354b11 - 630b10 + 683b9 - 890b8 + 372b7 + 66b6 + 1528b5 - 1698b4 + 451b3 - 3555b2 + 879*b1 - 1314) / 9
$\nu^{9}$	$=$	$ ( - 1425 \beta_{11} - 282 \beta_{10} - 2032 \beta_{9} - 176 \beta_{8} + 174 \beta_{7} - 507 \beta_{6} + \cdots + 4011 ) / 9 $ (-1425b11 - 282b10 - 2032b9 - 176b8 + 174b7 - 507b6 + 934b5 - 1326b4 + 2455b3 - 2262b2 - 2361*b1 + 4011) / 9
$\nu^{10}$	$=$	$ ( - 3735 \beta_{11} + 3891 \beta_{10} - 6043 \beta_{9} + 6283 \beta_{8} - 2307 \beta_{7} - 1344 \beta_{6} + \cdots + 10989 ) / 9 $ (-3735b11 + 3891b10 - 6043b9 + 6283b8 - 2307b7 - 1344b6 - 8543b5 + 8475b4 - 725b3 + 19431b2 - 7593*b1 + 10989) / 9
$\nu^{11}$	$=$	$ ( 4797 \beta_{11} + 6081 \beta_{10} + 6134 \beta_{9} + 8893 \beta_{8} - 3555 \beta_{7} + 2661 \beta_{6} + \cdots - 9981 ) / 9 $ (4797b11 + 6081b10 + 6134b9 + 8893b8 - 3555b7 + 2661b6 - 14387b5 + 15597b4 - 17300b3 + 33435b2 + 7464*b1 - 9981) / 9

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times$.

$n$	$731$
$\chi(n)$	$-1 - \beta_{2}$

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} $

487.1

0.500000	−	0.677980i
0.500000	−	1.74095i
0.500000	−	1.80139i
0.500000	−	0.168222i
0.500000	+	1.96356i
0.500000	+	2.42499i
0.500000	+	0.677980i
0.500000	+	1.74095i
0.500000	+	1.80139i
0.500000	+	0.168222i
0.500000	−	1.96356i
0.500000	−	2.42499i

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.77684

−

3.07758i

0.159750

−

0.276696i

1.00000

3.55368

487.2

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.58406

−

2.74367i

−1.84286

3.19193i

1.00000

3.16812

487.3

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.04401

−

1.80828i

2.00610

−

3.47467i

1.00000

2.08802

487.4

−0.500000

0.866025i

−0.500000

−

0.866025i

0.370360

0.641483i

−2.06641

3.57913i

1.00000

−0.740720

487.5

−0.500000

0.866025i

−0.500000

−

0.866025i

0.510796

0.884725i

−1.33340

2.30951i

1.00000

−1.02159

487.6

−0.500000

0.866025i

−0.500000

−

0.866025i

2.02375

3.50524i

0.0768150

−

0.133048i

1.00000

−4.04750

973.1

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.77684

3.07758i

0.159750

0.276696i

1.00000

3.55368

973.2

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.58406

2.74367i

−1.84286

−

3.19193i

1.00000

3.16812

973.3

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.04401

1.80828i

2.00610

3.47467i

1.00000

2.08802

973.4

−0.500000

−

0.866025i

−0.500000

0.866025i

0.370360

−

0.641483i

−2.06641

−

3.57913i

1.00000

−0.740720

973.5

−0.500000

−

0.866025i

−0.500000

0.866025i

0.510796

−

0.884725i

−1.33340

−

2.30951i

1.00000

−1.02159

973.6

−0.500000

−

0.866025i

−0.500000

0.866025i

2.02375

−

3.50524i

0.0768150

0.133048i

1.00000

−4.04750

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1458.2.c.f		12
3.b	odd	2	1		1458.2.c.g		12
9.c	even	3	1		1458.2.a.g		6
9.c	even	3	1	inner	1458.2.c.f		12
9.d	odd	6	1		1458.2.a.f		6
9.d	odd	6	1		1458.2.c.g		12
27.e	even	9	2		54.2.e.b	✓	12
27.e	even	9	2		486.2.e.f		12
27.e	even	9	2		486.2.e.h		12
27.f	odd	18	2		162.2.e.b		12
27.f	odd	18	2		486.2.e.e		12
27.f	odd	18	2		486.2.e.g		12
108.j	odd	18	2		432.2.u.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.2.e.b	✓	12	27.e	even	9	2
162.2.e.b		12	27.f	odd	18	2
432.2.u.b		12	108.j	odd	18	2
486.2.e.e		12	27.f	odd	18	2
486.2.e.f		12	27.e	even	9	2
486.2.e.g		12	27.f	odd	18	2
486.2.e.h		12	27.e	even	9	2
1458.2.a.f		6	9.d	odd	6	1
1458.2.a.g		6	9.c	even	3	1
1458.2.c.f		12	1.a	even	1	1	trivial
1458.2.c.f		12	9.c	even	3	1	inner
1458.2.c.g		12	3.b	odd	2	1
1458.2.c.g		12	9.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} + 3 T_{5}^{11} + 27 T_{5}^{10} + 60 T_{5}^{9} + 459 T_{5}^{8} + 918 T_{5}^{7} + 3429 T_{5}^{6} + \cdots + 5184 $ T5^12 + 3*T5^11 + 27*T5^10 + 60*T5^9 + 459*T5^8 + 918*T5^7 + 3429*T5^6 + 1620*T5^5 + 6156*T5^4 - 4320*T5^3 + 14256*T5^2 - 7776*T5 + 5184 acting on $S_{2}^{\mathrm{new}}(1458, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + 3 T^{11} + \cdots + 5184 $ T^12 + 3T^11 + 27T^10 + 60T^9 + 459T^8 + 918T^7 + 3429T^6 + 1620T^5 + 6156T^4 - 4320T^3 + 14256T^2 - 7776*T + 5184
$7$	$ T^{12} + 6 T^{11} + \cdots + 64 $ T^12 + 6T^11 + 45T^10 + 148T^9 + 801T^8 + 2349T^7 + 8727T^6 + 12762T^5 + 20196T^4 - 9824T^3 + 4272T^2 - 576*T + 64
$11$	$ T^{12} + 3 T^{11} + \cdots + 81 $ T^12 + 3T^11 + 36T^10 + 33T^9 + 729T^8 + 594T^7 + 7641T^6 - 5346T^5 + 34101T^4 + 14877T^3 + 5022T^2 + 729*T + 81
$13$	$ T^{12} + 9 T^{11} + \cdots + 23104 $ T^12 + 9T^11 + 78T^10 + 373T^9 + 2106T^8 + 8685T^7 + 35889T^6 + 95148T^5 + 202788T^4 + 226048T^3 + 184176T^2 + 78432*T + 23104
$17$	$ (T^{6} - 6 T^{5} + \cdots + 333)^{2} $ (T^6 - 6T^5 - 18T^4 + 132T^3 - 45T^2 - 378*T + 333)^2
$19$	$ (T^{6} - 9 T^{5} + \cdots + 307)^{2} $ (T^6 - 9T^5 + 3T^4 + 119T^3 - 189T^2 - 159*T + 307)^2
$23$	$ T^{12} - 3 T^{11} + \cdots + 5184 $ T^12 - 3T^11 + 72T^10 - 465T^9 + 5184T^8 - 22545T^7 + 90423T^6 - 144342T^5 + 226800T^4 + 173448T^3 + 274752T^2 + 38880*T + 5184
$29$	$ T^{12} + 3 T^{11} + \cdots + 5184 $ T^12 + 3T^11 + 63T^10 - 264T^9 + 2565T^8 - 3942T^7 + 13149T^6 + 9882T^5 + 35316T^4 + 7344T^3 + 14256T^2 + 5184
$31$	$ T^{12} + 12 T^{11} + \cdots + 4032064 $ T^12 + 12T^11 + 165T^10 + 562T^9 + 5409T^8 + 13671T^7 + 130605T^6 + 188820T^5 + 1314180T^4 + 1373440T^3 + 9828336T^2 + 6240864*T + 4032064
$37$	$ (T^{6} - 15 T^{5} + \cdots + 11944)^{2} $ (T^6 - 15T^5 + 821T^3 - 3066T^2 - 1008T + 11944)^2
$41$	$ T^{12} + 3 T^{11} + \cdots + 2653641 $ T^12 + 3T^11 + 54T^10 + 303T^9 + 2673T^8 + 11124T^7 + 47655T^6 + 121986T^5 + 363123T^4 + 725409T^3 + 1735668T^2 + 2155167*T + 2653641
$43$	$ T^{12} + 12 T^{11} + \cdots + 49674304 $ T^12 + 12T^11 + 183T^10 + 922T^9 + 10593T^8 + 49365T^7 + 412269T^6 + 959292T^5 + 4071636T^4 + 6362176T^3 + 27174000T^2 + 33069216*T + 49674304
$47$	$ T^{12} - 9 T^{11} + \cdots + 419904 $ T^12 - 9T^11 + 153T^10 - 954T^9 + 14661T^8 - 96228T^7 + 497421T^6 - 1484244T^5 + 3280500T^4 - 4105728T^3 + 3674160T^2 - 1469664*T + 419904
$53$	$ (T^{6} + 6 T^{5} - 63 T^{4} + \cdots - 72)^{2} $ (T^6 + 6T^5 - 63T^4 + 3T^3 + 414T^2 - 216*T - 72)^2
$59$	$ T^{12} - 3 T^{11} + \cdots + 82464561 $ T^12 - 3T^11 + 126T^10 + 345T^9 + 10071T^8 + 32616T^7 + 476145T^6 + 2605608T^5 + 14183991T^4 + 40586049T^3 + 92615238T^2 + 101752605*T + 82464561
$61$	$ T^{12} + 12 T^{11} + \cdots + 1000000 $ T^12 + 12T^11 + 183T^10 + 166T^9 + 5085T^8 + 8703T^7 + 86649T^6 + 99720T^5 + 762300T^4 + 1138000T^3 + 4170000T^2 + 2100000*T + 1000000
$67$	$ T^{12} + \cdots + 249393368449 $ T^12 + 21T^11 + 480T^10 + 5107T^9 + 73071T^8 + 611424T^7 + 7227609T^6 + 46245528T^5 + 415015335T^4 + 1988154427T^3 + 15501520200T^2 + 52003291269*T + 249393368449
$71$	$ (T^{6} + 12 T^{5} + \cdots - 22104)^{2} $ (T^6 + 12T^5 - 81T^4 - 975T^3 + 1386T^2 + 12636*T - 22104)^2
$73$	$ (T^{6} - 21 T^{5} + \cdots - 269)^{2} $ (T^6 - 21T^5 + 51T^4 + 587T^3 + 501T^2 - 363*T - 269)^2
$79$	$ T^{12} + \cdots + 591851584 $ T^12 + 213T^10 - 698T^9 + 39393T^8 - 97965T^7 + 1443345T^6 - 7979904T^5 + 49140612T^4 - 158181872T^3 + 412898256T^2 - 574821984T + 591851584
$83$	$ T^{12} + \cdots + 13756474944 $ T^12 - 27T^11 + 558T^10 - 7191T^9 + 84402T^8 - 796311T^7 + 7230465T^6 - 52467588T^5 + 326204172T^4 - 1450675008T^3 + 4977856944T^2 - 10070347680*T + 13756474944
$89$	$ (T^{6} + 12 T^{5} + \cdots - 356229)^{2} $ (T^6 + 12T^5 - 270T^4 - 2109T^3 + 20502T^2 + 44469*T - 356229)^2
$97$	$ T^{12} + \cdots + 373532435929 $ T^12 + 18T^11 + 528T^10 + 4216T^9 + 110835T^8 + 935460T^7 + 13634358T^6 + 83450898T^5 + 895766949T^4 + 5166157000T^3 + 36829153215T^2 + 119002717176*T + 373532435929
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1458.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{10} + \beta_{2}) q^{7} + q^{8}+O(q^{10}) \) q + b2 * q^2 + (-b2 - 1) * q^4 + (b4 - b1 - 1) * q^5 + (-b10 + b2) * q^7 + q^8	\( q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{10} + \beta_{2}) q^{7} + q^{8} + (\beta_1 + 1) q^{10} + ( - \beta_{7} + \beta_{2}) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 2) q^{13}+ \cdots + (\beta_{11} + 2 \beta_{9} + \beta_{6} + \cdots + 3) q^{98}+O(q^{100}) \) q + b2 * q^2 + (-b2 - 1) * q^4 + (b4 - b1 - 1) * q^5 + (-b10 + b2) * q^7 + q^8 + (b1 + 1) * q^10 + (-b7 + b2) * q^11 + (-b7 - b6 - b5 - b2 - 2) * q^13 + (-b11 + b10 - b2 - 1) * q^14 + b2 * q^16 + (-b11 - b9 + b3 + 1) * q^17 + (-b3 - b1 + 1) * q^19 - b4 * q^20 + (-b9 + b7 + b6 - b2) * q^22 + (2b8 - b7 - b6 - b5 + b2) q^23 + (b10 + 2b8 - b7 - 3b5 - 2b3 + 3b2) * q^25 + (b6 + 2) * q^26 + (b11 + 1) * q^28 + (b10 + b7 + b5) * q^29 + (-2b9 + b7 + b6 - b5 - b4 - 3b2 + b1 - 1) * q^31 + (-b2 - 1) * q^32 + (b10 + b8 - b5 - b3 + b2) * q^34 + (b11 - 2b9 + 2b3 + b1) * q^35 + (-b11 - 2b3 - b1 + 2) q^37 + (-b8 + b4 + b3 + 2b2) q^38 + (b4 - b1 - 1) * q^40 + (b11 - b10 + b8 - b4 - b2 + b1) * q^41 + (b10 + b8 + b7 + b4 - b3 + 2b2) q^43 + (b9 - b6) * q^44 + (b6 - 2b3) q^46 + (-b10 + 2b5 + b4 - b2) q^47 + (-b11 + b10 - 2b9 - b7 - b6 - 3b5 + b4 - b2 - b1 - 3) * q^49 + (b11 - b10 + 2b9 - 2b8 + b7 + b6 + 3b5 - 3b2 - 2) * q^50 + (b7 + b5 + b2) * q^52 + (b11 - b6 - b1 - 2) * q^53 + (-b11 - b6 - 2b3 + 1) q^55 + (-b10 + b2) * q^56 + (b11 - b10 - b7 - b6 - b5 - 1) * q^58 + (b8 + b7 + b6 + b5 - 2b4 - b2 + 2b1 + 2) * q^59 + (2b8 + b7 - b5 - b4 - 2b3 + b2) * q^61 + (2b9 - b6 - b1 + 1) q^62 + q^64 + (2b8 + 2b7 + 2b5 - b4 - 2b3 - 2b2) q^65 + (b11 - b10 - 3b9 + 2b7 + 2b6 - b5 - b4 - 5b2 + b1 - 2) * q^67 + (b11 - b10 + b9 - b8 + b5 - b2 - 1) * q^68 + (-b10 + 2b8 - 2b5 - b4 - 2b3 - b2) q^70 + (b11 - 2b9 + b6 - b1 - 2) q^71 + (-b9 - b6 + b3 + 2b1 + 4) q^73 + (b10 - 2b8 + b4 + 2b3 + 3b2) q^74 + (b8 - b4 - 2b2 + b1 - 1) q^76 + (-b11 + b10 - 4b9 + 2b7 + 2b6 - 2b5 - b4 - b2 + b1 + 2) * q^77 + (-2b10 + 2b8 + b7 + b5 - b4 - 2b3 - b2) q^79 + (b1 + 1) * q^80 + (-b11 - b3 - b1) * q^82 + (-2b10 - b7 + b5 - 4b2) * q^83 + (-b11 + b10 - 2b9 + 2b8 - b7 - b6 - 3b5 + 3b4 + 2b2 - 3b1 - 2) * q^85 + (b11 - b10 + b9 - b8 - b7 - b6 - b4 - 2b2 + b1 - 2) q^86 + (-b7 + b2) * q^88 + (-2b11 + b6 - 4b3 - b1 - 2) * q^89 + (3b11 - 2b9 + 1) * q^91 + (-2b8 + b7 + b5 + 2b3 - b2) * q^92 + (-b11 + b10 - 2b9 - 2b5 - b4 + b2 + b1 + 2) * q^94 + (-2b9 + 2b8 + b7 + b6 - b5 + b4 + 5b2 - b1 + 5) q^95 + (-b10 + 2b8 - 3b7 - 4b5 - b4 - 2b3 + 4b2) q^97 + (b11 + 2b9 + b6 + b1 + 3) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q - 6 * q^2 - 6 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q - 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} + 6 q^{10} - 3 q^{11} - 9 q^{13} - 6 q^{14} - 6 q^{16} + 12 q^{17} + 18 q^{19} - 3 q^{20} - 3 q^{22} + 3 q^{23} - 15 q^{25} + 18 q^{26} + 12 q^{28} - 3 q^{29} - 12 q^{31} - 6 q^{32} - 6 q^{34} - 6 q^{35} + 30 q^{37} - 9 q^{38} - 3 q^{40} - 3 q^{41} - 12 q^{43} + 6 q^{44} - 6 q^{46} + 9 q^{47} - 12 q^{49} - 15 q^{50} - 9 q^{52} - 12 q^{53} + 18 q^{55} - 6 q^{56} - 3 q^{58} + 3 q^{59} - 12 q^{61} + 24 q^{62} + 12 q^{64} + 3 q^{65} - 21 q^{67} - 6 q^{68} + 3 q^{70} - 24 q^{71} + 42 q^{73} - 15 q^{74} - 9 q^{76} + 3 q^{77} + 6 q^{80} + 6 q^{82} + 27 q^{83} - 12 q^{86} - 3 q^{88} - 24 q^{89} + 12 q^{91} + 3 q^{92} + 9 q^{94} + 30 q^{95} - 18 q^{97} + 24 q^{98}+O(q^{100}) \) 12 * q - 6 * q^2 - 6 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 + 6 * q^10 - 3 * q^11 - 9 * q^13 - 6 * q^14 - 6 * q^16 + 12 * q^17 + 18 * q^19 - 3 * q^20 - 3 * q^22 + 3 * q^23 - 15 * q^25 + 18 * q^26 + 12 * q^28 - 3 * q^29 - 12 * q^31 - 6 * q^32 - 6 * q^34 - 6 * q^35 + 30 * q^37 - 9 * q^38 - 3 * q^40 - 3 * q^41 - 12 * q^43 + 6 * q^44 - 6 * q^46 + 9 * q^47 - 12 * q^49 - 15 * q^50 - 9 * q^52 - 12 * q^53 + 18 * q^55 - 6 * q^56 - 3 * q^58 + 3 * q^59 - 12 * q^61 + 24 * q^62 + 12 * q^64 + 3 * q^65 - 21 * q^67 - 6 * q^68 + 3 * q^70 - 24 * q^71 + 42 * q^73 - 15 * q^74 - 9 * q^76 + 3 * q^77 + 6 * q^80 + 6 * q^82 + 27 * q^83 - 12 * q^86 - 3 * q^88 - 24 * q^89 + 12 * q^91 + 3 * q^92 + 9 * q^94 + 30 * q^95 - 18 * q^97 + 24 * q^98

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	1458.2.c.f

Level	$1458$
Weight	$2$
Character orbit	1458.c
Analytic conductor	$11.642$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(11.6421886147\)
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:	\( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 \) x^12 - 6x^11 + 33x^10 - 110x^9 + 318x^8 - 678x^7 + 1225x^6 - 1698x^5 + 1905x^4 - 1584x^3 + 936x^2 - 342*x + 57
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{7} \)
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{6} + 3\nu^{5} - 14\nu^{4} + 23\nu^{3} - 41\nu^{2} + 30\nu - 11 ) / 2 \) (-v^6 + 3v^5 - 14v^4 + 23v^3 - 41v^2 + 30*v - 11) / 2
\(\beta_{2}\)	\(=\)	\( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + \cdots + 842 ) / 218 \) (6v^11 - 33v^10 + 127v^9 - 324v^8 + 438v^7 - 252v^6 - 1278v^5 + 3234v^4 - 5701v^3 + 5358v^2 - 3477*v + 842) / 218
\(\beta_{3}\)	\(=\)	\( ( \nu^{8} - 4\nu^{7} + 18\nu^{6} - 40\nu^{5} + 82\nu^{4} - 102\nu^{3} + 102\nu^{2} - 57\nu + 15 ) / 2 \) (v^8 - 4v^7 + 18v^6 - 40v^5 + 82v^4 - 102v^3 + 102v^2 - 57*v + 15) / 2
\(\beta_{4}\)	\(=\)	\( ( - 2 \nu^{11} + 11 \nu^{10} - 115 \nu^{9} + 435 \nu^{8} - 1781 \nu^{7} + 4226 \nu^{6} - 9493 \nu^{5} + \cdots + 882 ) / 218 \) (-2v^11 + 11v^10 - 115v^9 + 435v^8 - 1781v^7 + 4226v^6 - 9493v^5 + 13637v^4 - 16775v^3 + 12275v^2 - 5163*v + 882) / 218
\(\beta_{5}\)	\(=\)	\( ( 81 \nu^{11} - 282 \nu^{10} + 1224 \nu^{9} - 1867 \nu^{8} + 1771 \nu^{7} + 5754 \nu^{6} - 22703 \nu^{5} + \cdots + 5154 ) / 218 \) (81v^11 - 282v^10 + 1224v^9 - 1867v^8 + 1771v^7 + 5754v^6 - 22703v^5 + 45512v^4 - 62085v^3 + 50751v^2 - 25194*v + 5154) / 218
\(\beta_{6}\)	\(=\)	\( ( - \nu^{10} + 5 \nu^{9} - 24 \nu^{8} + 66 \nu^{7} - 154 \nu^{6} + 252 \nu^{5} - 318 \nu^{4} + 283 \nu^{3} + \cdots - 6 ) / 2 \) (-v^10 + 5v^9 - 24v^8 + 66v^7 - 154v^6 + 252v^5 - 318v^4 + 283v^3 - 148v^2 + 39*v - 6) / 2
\(\beta_{7}\)	\(=\)	\( ( - 2 \nu^{11} - 98 \nu^{10} + 539 \nu^{9} - 2726 \nu^{8} + 8247 \nu^{7} - 20517 \nu^{6} + 38140 \nu^{5} + \cdots - 4786 ) / 218 \) (-2v^11 - 98v^10 + 539v^9 - 2726v^8 + 8247v^7 - 20517v^6 + 38140v^5 - 54815v^4 + 60615v^3 - 45277v^2 + 22632*v - 4786) / 218
\(\beta_{8}\)	\(=\)	\( ( - 78 \nu^{11} + 429 \nu^{10} - 2196 \nu^{9} + 6719 \nu^{8} - 17684 \nu^{7} + 34014 \nu^{6} + \cdots + 3333 ) / 218 \) (-78v^11 + 429v^10 - 2196v^9 + 6719v^8 - 17684v^7 + 34014v^6 - 54236v^5 + 65432v^4 - 61047v^3 + 40545v^2 - 16929*v + 3333) / 218
\(\beta_{9}\)	\(=\)	\( ( - 3 \nu^{10} + 15 \nu^{9} - 73 \nu^{8} + 202 \nu^{7} - 483 \nu^{6} + 805 \nu^{5} - 1072 \nu^{4} + \cdots - 30 ) / 2 \) (-3v^10 + 15v^9 - 73v^8 + 202v^7 - 483v^6 + 805v^5 - 1072v^4 + 1008v^3 - 627v^2 + 228v - 30) / 2
\(\beta_{10}\)	\(=\)	\( ( 277 \nu^{11} - 1251 \nu^{10} + 6281 \nu^{9} - 16266 \nu^{8} + 39841 \nu^{7} - 62864 \nu^{6} + \cdots - 731 ) / 218 \) (277v^11 - 1251v^10 + 6281v^9 - 16266v^8 + 39841v^7 - 62864v^6 + 86405v^5 - 77744v^4 + 52922v^3 - 21433v^2 + 5976*v - 731) / 218
\(\beta_{11}\)	\(=\)	\( ( 5 \nu^{10} - 25 \nu^{9} + 123 \nu^{8} - 342 \nu^{7} + 831 \nu^{6} - 1401 \nu^{5} + 1922 \nu^{4} + \cdots + 98 ) / 2 \) (5v^10 - 25v^9 + 123v^8 - 342v^7 + 831v^6 - 1401v^5 + 1922v^4 - 1858v^3 + 1255v^2 - 510v + 98) / 2

\(\nu\)	\(=\)	\( ( -\beta_{9} - 2\beta_{8} - 2\beta_{5} + 6\beta_{4} + \beta_{3} + 3\beta_{2} - 3\beta _1 + 3 ) / 9 \) (-b9 - 2b8 - 2b5 + 6b4 + b3 + 3b2 - 3*b1 + 3) / 9
\(\nu^{2}\)	\(=\)	\( ( 3\beta_{11} + 5\beta_{9} - 2\beta_{8} - 3\beta_{6} - 2\beta_{5} + 6\beta_{4} - 2\beta_{3} + 3\beta_{2} - 24 ) / 9 \) (3b11 + 5b9 - 2b8 - 3b6 - 2b5 + 6b4 - 2b3 + 3b2 - 24) / 9
\(\nu^{3}\)	\(=\)	\( ( 6 \beta_{11} - 3 \beta_{10} + 14 \beta_{9} + \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + 13 \beta_{5} + \cdots - 39 ) / 9 \) (6b11 - 3b10 + 14b9 + b8 + 3b7 - 3b6 + 13b5 - 21b4 - 5b3 - 30b2 + 15b1 - 39) / 9
\(\nu^{4}\)	\(=\)	\( ( - 9 \beta_{11} - 6 \beta_{10} - 10 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + 28 \beta_{5} + \cdots + 54 ) / 9 \) (-9b11 - 6b10 - 10b9 + 4b8 + 6b7 + 6b6 + 28b5 - 48b4 + 13b3 - 63b2 + 3*b1 + 54) / 9
\(\nu^{5}\)	\(=\)	\( ( - 48 \beta_{11} + 21 \beta_{10} - 88 \beta_{9} + 28 \beta_{8} - 12 \beta_{7} + 9 \beta_{6} - 53 \beta_{5} + \cdots + 234 ) / 9 \) (-48b11 + 21b10 - 88b9 + 28b8 - 12b7 + 9b6 - 53b5 + 60b4 + 31b3 + 126b2 - 90*b1 + 234) / 9
\(\nu^{6}\)	\(=\)	\( ( - 3 \beta_{11} + 78 \beta_{10} - 37 \beta_{9} + 73 \beta_{8} - 51 \beta_{7} - 3 \beta_{6} - 230 \beta_{5} + \cdots + 24 ) / 9 \) (-3b11 + 78b10 - 37b9 + 73b8 - 51b7 - 3b6 - 230b5 + 303b4 - 92b3 + 537b2 - 75*b1 + 24) / 9
\(\nu^{7}\)	\(=\)	\( ( 294 \beta_{11} - 63 \beta_{10} + 464 \beta_{9} - 140 \beta_{8} + 30 \beta_{7} + 36 \beta_{6} + \cdots - 1080 ) / 9 \) (294b11 - 63b10 + 464b9 - 140b8 + 30b7 + 36b6 + 97b5 - 42b4 - 287b3 - 225b2 + 504*b1 - 1080) / 9
\(\nu^{8}\)	\(=\)	\( ( 354 \beta_{11} - 630 \beta_{10} + 683 \beta_{9} - 890 \beta_{8} + 372 \beta_{7} + 66 \beta_{6} + \cdots - 1314 ) / 9 \) (354b11 - 630b10 + 683b9 - 890b8 + 372b7 + 66b6 + 1528b5 - 1698b4 + 451b3 - 3555b2 + 879*b1 - 1314) / 9
\(\nu^{9}\)	\(=\)	\( ( - 1425 \beta_{11} - 282 \beta_{10} - 2032 \beta_{9} - 176 \beta_{8} + 174 \beta_{7} - 507 \beta_{6} + \cdots + 4011 ) / 9 \) (-1425b11 - 282b10 - 2032b9 - 176b8 + 174b7 - 507b6 + 934b5 - 1326b4 + 2455b3 - 2262b2 - 2361*b1 + 4011) / 9
\(\nu^{10}\)	\(=\)	\( ( - 3735 \beta_{11} + 3891 \beta_{10} - 6043 \beta_{9} + 6283 \beta_{8} - 2307 \beta_{7} - 1344 \beta_{6} + \cdots + 10989 ) / 9 \) (-3735b11 + 3891b10 - 6043b9 + 6283b8 - 2307b7 - 1344b6 - 8543b5 + 8475b4 - 725b3 + 19431b2 - 7593*b1 + 10989) / 9
\(\nu^{11}\)	\(=\)	\( ( 4797 \beta_{11} + 6081 \beta_{10} + 6134 \beta_{9} + 8893 \beta_{8} - 3555 \beta_{7} + 2661 \beta_{6} + \cdots - 9981 ) / 9 \) (4797b11 + 6081b10 + 6134b9 + 8893b8 - 3555b7 + 2661b6 - 14387b5 + 15597b4 - 17300b3 + 33435b2 + 7464*b1 - 9981) / 9

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 487.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + 3 T^{11} + \cdots + 5184 \) T^12 + 3T^11 + 27T^10 + 60T^9 + 459T^8 + 918T^7 + 3429T^6 + 1620T^5 + 6156T^4 - 4320T^3 + 14256T^2 - 7776*T + 5184
$7$	\( T^{12} + 6 T^{11} + \cdots + 64 \) T^12 + 6T^11 + 45T^10 + 148T^9 + 801T^8 + 2349T^7 + 8727T^6 + 12762T^5 + 20196T^4 - 9824T^3 + 4272T^2 - 576*T + 64
$11$	\( T^{12} + 3 T^{11} + \cdots + 81 \) T^12 + 3T^11 + 36T^10 + 33T^9 + 729T^8 + 594T^7 + 7641T^6 - 5346T^5 + 34101T^4 + 14877T^3 + 5022T^2 + 729*T + 81
$13$	\( T^{12} + 9 T^{11} + \cdots + 23104 \) T^12 + 9T^11 + 78T^10 + 373T^9 + 2106T^8 + 8685T^7 + 35889T^6 + 95148T^5 + 202788T^4 + 226048T^3 + 184176T^2 + 78432*T + 23104
$17$	\( (T^{6} - 6 T^{5} + \cdots + 333)^{2} \) (T^6 - 6T^5 - 18T^4 + 132T^3 - 45T^2 - 378*T + 333)^2
$19$	\( (T^{6} - 9 T^{5} + \cdots + 307)^{2} \) (T^6 - 9T^5 + 3T^4 + 119T^3 - 189T^2 - 159*T + 307)^2
$23$	\( T^{12} - 3 T^{11} + \cdots + 5184 \) T^12 - 3T^11 + 72T^10 - 465T^9 + 5184T^8 - 22545T^7 + 90423T^6 - 144342T^5 + 226800T^4 + 173448T^3 + 274752T^2 + 38880*T + 5184
$29$	\( T^{12} + 3 T^{11} + \cdots + 5184 \) T^12 + 3T^11 + 63T^10 - 264T^9 + 2565T^8 - 3942T^7 + 13149T^6 + 9882T^5 + 35316T^4 + 7344T^3 + 14256T^2 + 5184
$31$	\( T^{12} + 12 T^{11} + \cdots + 4032064 \) T^12 + 12T^11 + 165T^10 + 562T^9 + 5409T^8 + 13671T^7 + 130605T^6 + 188820T^5 + 1314180T^4 + 1373440T^3 + 9828336T^2 + 6240864*T + 4032064
$37$	\( (T^{6} - 15 T^{5} + \cdots + 11944)^{2} \) (T^6 - 15T^5 + 821T^3 - 3066T^2 - 1008T + 11944)^2
$41$	\( T^{12} + 3 T^{11} + \cdots + 2653641 \) T^12 + 3T^11 + 54T^10 + 303T^9 + 2673T^8 + 11124T^7 + 47655T^6 + 121986T^5 + 363123T^4 + 725409T^3 + 1735668T^2 + 2155167*T + 2653641
$43$	\( T^{12} + 12 T^{11} + \cdots + 49674304 \) T^12 + 12T^11 + 183T^10 + 922T^9 + 10593T^8 + 49365T^7 + 412269T^6 + 959292T^5 + 4071636T^4 + 6362176T^3 + 27174000T^2 + 33069216*T + 49674304
$47$	\( T^{12} - 9 T^{11} + \cdots + 419904 \) T^12 - 9T^11 + 153T^10 - 954T^9 + 14661T^8 - 96228T^7 + 497421T^6 - 1484244T^5 + 3280500T^4 - 4105728T^3 + 3674160T^2 - 1469664*T + 419904
$53$	\( (T^{6} + 6 T^{5} - 63 T^{4} + \cdots - 72)^{2} \) (T^6 + 6T^5 - 63T^4 + 3T^3 + 414T^2 - 216*T - 72)^2
$59$	\( T^{12} - 3 T^{11} + \cdots + 82464561 \) T^12 - 3T^11 + 126T^10 + 345T^9 + 10071T^8 + 32616T^7 + 476145T^6 + 2605608T^5 + 14183991T^4 + 40586049T^3 + 92615238T^2 + 101752605*T + 82464561
$61$	\( T^{12} + 12 T^{11} + \cdots + 1000000 \) T^12 + 12T^11 + 183T^10 + 166T^9 + 5085T^8 + 8703T^7 + 86649T^6 + 99720T^5 + 762300T^4 + 1138000T^3 + 4170000T^2 + 2100000*T + 1000000
$67$	\( T^{12} + \cdots + 249393368449 \) T^12 + 21T^11 + 480T^10 + 5107T^9 + 73071T^8 + 611424T^7 + 7227609T^6 + 46245528T^5 + 415015335T^4 + 1988154427T^3 + 15501520200T^2 + 52003291269*T + 249393368449
$71$	\( (T^{6} + 12 T^{5} + \cdots - 22104)^{2} \) (T^6 + 12T^5 - 81T^4 - 975T^3 + 1386T^2 + 12636*T - 22104)^2
$73$	\( (T^{6} - 21 T^{5} + \cdots - 269)^{2} \) (T^6 - 21T^5 + 51T^4 + 587T^3 + 501T^2 - 363*T - 269)^2
$79$	\( T^{12} + \cdots + 591851584 \) T^12 + 213T^10 - 698T^9 + 39393T^8 - 97965T^7 + 1443345T^6 - 7979904T^5 + 49140612T^4 - 158181872T^3 + 412898256T^2 - 574821984T + 591851584
$83$	\( T^{12} + \cdots + 13756474944 \) T^12 - 27T^11 + 558T^10 - 7191T^9 + 84402T^8 - 796311T^7 + 7230465T^6 - 52467588T^5 + 326204172T^4 - 1450675008T^3 + 4977856944T^2 - 10070347680*T + 13756474944
$89$	\( (T^{6} + 12 T^{5} + \cdots - 356229)^{2} \) (T^6 + 12T^5 - 270T^4 - 2109T^3 + 20502T^2 + 44469*T - 356229)^2
$97$	\( T^{12} + \cdots + 373532435929 \) T^12 + 18T^11 + 528T^10 + 4216T^9 + 110835T^8 + 935460T^7 + 13634358T^6 + 83450898T^5 + 895766949T^4 + 5166157000T^3 + 36829153215T^2 + 119002717176*T + 373532435929
show more
show less

\(n\)	\(731\)
\(\chi(n)\)	\(-1 - \beta_{2}\)