LMFDB - Newform orbit 6024.2.a.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.a (trivial)

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:	yes
Analytic conductor:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - q^{3} + \beta_1 q^{5} + \beta_{10} q^{7} + q^{9}+O(q^{10}) $ q - q^3 + b1 * q^5 + b10 * q^7 + q^9	$ q - q^{3} + \beta_1 q^{5} + \beta_{10} q^{7} + q^{9} - \beta_{8} q^{11} + \beta_{4} q^{13} - \beta_1 q^{15} + (\beta_{8} + \beta_{7} + \beta_{4} + 2) q^{17} + (\beta_{9} + \beta_{6} - \beta_{5}) q^{19} - \beta_{10} q^{21} + (\beta_{12} + \beta_{7}) q^{23} + (\beta_{2} + \beta_1 + 1) q^{25} - q^{27} + (\beta_{9} + \beta_{3} + 1) q^{29} + ( - \beta_{11} - \beta_{9} + \cdots - \beta_{6}) q^{31}+ \cdots - \beta_{8} q^{99}+O(q^{100}) $ q - q^3 + b1 * q^5 + b10 * q^7 + q^9 - b8 * q^11 + b4 * q^13 - b1 * q^15 + (b8 + b7 + b4 + 2) * q^17 + (b9 + b6 - b5) * q^19 - b10 * q^21 + (b12 + b7) * q^23 + (b2 + b1 + 1) * q^25 - q^27 + (b9 + b3 + 1) * q^29 + (-b11 - b9 - b8 - b6) * q^31 + b8 * q^33 + (-b12 + 2b10 - b7 + b4 - 2) q^35 + (b11 + b7 - b2 + b1) * q^37 - b4 * q^39 + (-b9 - b8 - b7 - b4 + 2) * q^41 + (b11 - b3 - b2 + b1) * q^43 + b1 * q^45 + (b13 + b10 - b7 + b6 - b5 - b3 + 1) * q^47 + (b12 - b11 + b5 + b3 + 2) * q^49 + (-b8 - b7 - b4 - 2) * q^51 + (-b12 + b10 + b4 - b3) * q^53 + (-b13 + b11 + 2b5 + b2) q^55 + (-b9 - b6 + b5) * q^57 + (-b13 + b9 - b4) * q^59 + (-b12 - b11 + b1 + 1) * q^61 + b10 * q^63 + (-b13 + b10 - b7 + b4 - 2b3 + b2 + b1 + 1) q^65 + (b13 - b11 - b9 - b8 + b7 - b6 + b5 + b4 + b3 - 3) * q^67 + (-b12 - b7) * q^69 + (-b13 - b12 - b11 + b10 - b9 + b8 + b5 + b3 + b2 - b1 + 2) * q^71 + (b13 + b11 + b8 + b7 - b4 - b2 + b1 + 1) * q^73 + (-b2 - b1 - 1) * q^75 + (b12 - b9 - b8 - 2b7 + b6 - b5 - b3 + 1) q^77 + (b13 + b12 - b10 + b9 + b7 - b4 + 2b3 - 1) q^79 + q^81 + (b8 + b7 - b6 - b5 - b4 + b3 - b2 + b1) * q^83 + (-b8 - 2b7 - b6 - b5 - b4 - b3 + 4b1) * q^85 + (-b9 - b3 - 1) * q^87 + (2b13 + b12 + b11 + b9 - b8 - b5 - 2b4 + b3 - b2 + b1 + 1) * q^89 + (b10 - b9 - b8 + b5 + b3 + b2 + b1 - 4) * q^91 + (b11 + b9 + b8 + b6) * q^93 + (2b11 + 2b9 - 3b7 - 2b5 - 3b4 - b2 + b1 + 2) q^95 + (-2b13 - b12 - b9 + b8 + b7 - b6 + b4 - 2b3 - b2 + b1 + 1) * q^97 - b8 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{14} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6*x^13 - 29*x^12 + 210*x^11 + 280*x^10 - 2796*x^9 - 863*x^8 + 17652*x^7 - 1300*x^6 - 53458*x^5 + 10892*x^4 + 67298*x^3 - 13761*x^2 - 22140*x - 3364 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - \nu - 6 $ v^2 - v - 6
$\beta_{3}$	$=$	$ ( - 34385 \nu^{13} + 26892 \nu^{12} + 1553117 \nu^{11} - 869644 \nu^{10} - 26472460 \nu^{9} + \cdots - 56474950 ) / 10615128 $ (-34385v^13 + 26892v^12 + 1553117v^11 - 869644v^10 - 26472460v^9 + 9438328v^8 + 207665199v^7 - 38974970v^6 - 726563800v^5 + 59731310v^4 + 933220260v^3 - 52788358v^2 - 253959291*v - 56474950) / 10615128
$\beta_{4}$	$=$	$ ( 891500 \nu^{13} - 2582249 \nu^{12} - 32877251 \nu^{11} + 81418106 \nu^{10} + 470527080 \nu^{9} + \cdots + 543511258 ) / 47768076 $ (891500v^13 - 2582249v^12 - 32877251v^11 + 81418106v^10 + 470527080v^9 - 921011508v^8 - 3245410102v^7 + 4526703895v^6 + 10740423537v^5 - 9653543531v^4 - 14580166089v^3 + 7501423177v^2 + 5250877601*v + 543511258) / 47768076
$\beta_{5}$	$=$	$ ( - 2285707 \nu^{13} + 5763184 \nu^{12} + 85005175 \nu^{11} - 177040372 \nu^{10} + \cdots - 2060172338 ) / 95536152 $ (-2285707v^13 + 5763184v^12 + 85005175v^11 - 177040372v^10 - 1224526764v^9 + 1920505680v^8 + 8476711133v^7 - 8770496906v^6 - 28100462208v^5 + 16436289310v^4 + 38536651212v^3 - 10547722286v^2 - 15515835721*v - 2060172338) / 95536152
$\beta_{6}$	$=$	$ ( - 510731 \nu^{13} + 1185871 \nu^{12} + 19439080 \nu^{11} - 36738588 \nu^{10} - 286607520 \nu^{9} + \cdots - 677427348 ) / 15922692 $ (-510731v^13 + 1185871v^12 + 19439080v^11 - 36738588v^10 - 286607520v^9 + 402727860v^8 + 2026352809v^7 - 1865943851v^6 - 6827363977v^5 + 3566414523v^4 + 9440736513v^3 - 2259965299v^2 - 3691152442*v - 677427348) / 15922692
$\beta_{7}$	$=$	$ ( 1750919 \nu^{13} - 4072364 \nu^{12} - 65938241 \nu^{11} + 124543394 \nu^{10} + 961333458 \nu^{9} + \cdots + 1930586878 ) / 47768076 $ (1750919v^13 - 4072364v^12 - 65938241v^11 + 124543394v^10 + 961333458v^9 - 1341558738v^8 - 6714256819v^7 + 6055282684v^6 + 22294893624v^5 - 11158864646v^4 - 30232129854v^3 + 7029931060v^2 + 11780644049*v + 1930586878) / 47768076
$\beta_{8}$	$=$	$ ( - 799400 \nu^{13} + 2007980 \nu^{12} + 29958653 \nu^{11} - 62535530 \nu^{10} - 434576058 \nu^{9} + \cdots - 636577414 ) / 15922692 $ (-799400v^13 + 2007980v^12 + 29958653v^11 - 62535530v^10 - 434576058v^9 + 693826704v^8 + 3020215300v^7 - 3303220780v^6 - 9973980471v^5 + 6716846546v^4 + 13368961119v^3 - 4982795950v^2 - 4904604035*v - 636577414) / 15922692
$\beta_{9}$	$=$	$ ( 2690857 \nu^{13} - 6588652 \nu^{12} - 101245663 \nu^{11} + 205940440 \nu^{10} + 1473998214 \nu^{9} + \cdots + 2458464146 ) / 47768076 $ (2690857v^13 - 6588652v^12 - 101245663v^11 + 205940440v^10 + 1473998214v^9 - 2300026932v^8 - 10278303353v^7 + 11083245044v^6 + 34072151028v^5 - 22949713318v^4 - 45888598104v^3 + 17130259784v^2 + 16722285277*v + 2458464146) / 47768076
$\beta_{10}$	$=$	$ ( - 1085369 \nu^{13} + 2654776 \nu^{12} + 40709620 \nu^{11} - 82526094 \nu^{10} - 590411514 \nu^{9} + \cdots - 798333708 ) / 15922692 $ (-1085369v^13 + 2654776v^12 + 40709620v^11 - 82526094v^10 - 590411514v^9 + 913528686v^8 + 4096485739v^7 - 4337434292v^6 - 13473230383v^5 + 8801355990v^4 + 17847143889v^3 - 6578394766v^2 - 6272928610*v - 798333708) / 15922692
$\beta_{11}$	$=$	$ ( 1359385 \nu^{13} - 3285059 \nu^{12} - 51337298 \nu^{11} + 101829564 \nu^{10} + 751000860 \nu^{9} + \cdots + 1300892340 ) / 15922692 $ (1359385v^13 - 3285059v^12 - 51337298v^11 + 101829564v^10 + 751000860v^9 - 1119757014v^8 - 5266901315v^7 + 5240205949v^6 + 17571182555v^5 - 10334058561v^4 - 23897723349v^3 + 7318586177v^2 + 9075081374*v + 1300892340) / 15922692
$\beta_{12}$	$=$	$ ( 1400227 \nu^{13} - 3427662 \nu^{12} - 52961826 \nu^{11} + 107080910 \nu^{10} + 776737884 \nu^{9} + \cdots + 1560309976 ) / 15922692 $ (1400227v^13 - 3427662v^12 - 52961826v^11 + 107080910v^10 + 776737884v^9 - 1192572510v^8 - 5472245579v^7 + 5699815236v^6 + 18422349217v^5 - 11564497352v^4 - 25553159163v^3 + 8209065926v^2 + 10105957332*v + 1560309976) / 15922692
$\beta_{13}$	$=$	$ ( - 6572812 \nu^{13} + 16236166 \nu^{12} + 247008691 \nu^{11} - 503783854 \nu^{10} + \cdots - 6511648574 ) / 47768076 $ (-6572812v^13 + 16236166v^12 + 247008691v^11 - 503783854v^10 - 3595411272v^9 + 5552233938v^8 + 25099371248v^7 - 26089480472v^6 - 83439350505v^5 + 51662191384v^4 + 113289156915v^3 - 36259637984v^2 - 43344239917*v - 6511648574) / 47768076

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + \beta _1 + 6 $ b2 + b1 + 6
$\nu^{3}$	$=$	$ \beta_{13} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 11\beta _1 + 2 $ b13 - b8 + b7 - b6 + b4 + 11*b1 + 2
$\nu^{4}$	$=$	$ - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 3 \beta_{5} + \cdots + 60 $ -b13 - b12 + b10 + b9 + b8 + b7 - b6 + 3b5 + b4 + b3 + 14b2 + 16*b1 + 60
$\nu^{5}$	$=$	$ 15 \beta_{13} + 2 \beta_{10} - 19 \beta_{8} + 15 \beta_{7} - 15 \beta_{6} + \beta_{5} + 13 \beta_{4} + \cdots + 44 $ 15b13 + 2b10 - 19b8 + 15b7 - 15b6 + b5 + 13b4 + b3 + 4b2 + 131b1 + 44
$\nu^{6}$	$=$	$ - 12 \beta_{13} - 16 \beta_{12} + 3 \beta_{11} + 17 \beta_{10} + 14 \beta_{9} + 9 \beta_{8} + \cdots + 678 $ -12b13 - 16b12 + 3b11 + 17b10 + 14b9 + 9b8 + 20b7 - 19b6 + 54b5 + 18b4 + 20b3 + 175b2 + 233*b1 + 678
$\nu^{7}$	$=$	$ 187 \beta_{13} - \beta_{12} - 5 \beta_{11} + 36 \beta_{10} - 7 \beta_{9} - 268 \beta_{8} + 203 \beta_{7} + \cdots + 746 $ 187b13 - b12 - 5b11 + 36b10 - 7b9 - 268b8 + 203b7 - 198b6 + 45b5 + 161b4 + 25b3 + 109b2 + 1602b1 + 746
$\nu^{8}$	$=$	$ - 102 \beta_{13} - 199 \beta_{12} + 45 \beta_{11} + 218 \beta_{10} + 125 \beta_{9} - \beta_{8} + \cdots + 8042 $ -102b13 - 199b12 + 45b11 + 218b10 + 125b9 - b8 + 332b7 - 321b6 + 800b5 + 272b4 + 312b3 + 2154b2 + 3272b1 + 8042
$\nu^{9}$	$=$	$ 2245 \beta_{13} - 3 \beta_{12} - 138 \beta_{11} + 473 \beta_{10} - 249 \beta_{9} - 3489 \beta_{8} + \cdots + 11464 $ 2245b13 - 3b12 - 138b11 + 473b10 - 249b9 - 3489b8 + 2717b7 - 2551b6 + 1074b5 + 2031b4 + 484b3 + 2089b2 + 19856*b1 + 11464
$\nu^{10}$	$=$	$ - 561 \beta_{13} - 2237 \beta_{12} + 393 \beta_{11} + 2508 \beta_{10} + 557 \beta_{9} - 1714 \beta_{8} + \cdots + 97700 $ -561b13 - 2237b12 + 393b11 + 2508b10 + 557b9 - 1714b8 + 5151b7 - 5090b6 + 11302b5 + 3885b4 + 4530b3 + 26606b2 + 45038*b1 + 97700
$\nu^{11}$	$=$	$ 26820 \beta_{13} + 346 \beta_{12} - 2758 \beta_{11} + 5426 \beta_{10} - 5826 \beta_{9} - 44574 \beta_{8} + \cdots + 167648 $ 26820b13 + 346b12 - 2758b11 + 5426b10 - 5826b9 - 44574b8 + 36376b7 - 32878b6 + 20445b5 + 25946b4 + 8453b3 + 34772b2 + 248237*b1 + 167648
$\nu^{12}$	$=$	$ 1937 \beta_{13} - 23447 \beta_{12} + 1039 \beta_{11} + 26896 \beta_{10} - 7911 \beta_{9} + \cdots + 1204036 $ 1937b13 - 23447b12 + 1039b11 + 26896b10 - 7911b9 - 43524b8 + 77287b7 - 77082b6 + 157729b5 + 54109b4 + 64193b3 + 331250b2 + 611874*b1 + 1204036
$\nu^{13}$	$=$	$ 322155 \beta_{13} + 11913 \beta_{12} - 48703 \beta_{11} + 55998 \beta_{10} - 113821 \beta_{9} + \cdots + 2379408 $ 322155b13 + 11913b12 - 48703b11 + 55998b10 - 113821b9 - 570870b8 + 487827b7 - 426928b6 + 348366b5 + 333155b4 + 139074b3 + 537923b2 + 3123648*b1 + 2379408

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

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} $

1.1

−3.30404
−3.23770
−2.26645
−2.15802
−1.55864
−0.379598
−0.198997
1.10854
1.26787
2.29106
3.55671
3.57026
3.63608
3.67292

−1.00000

−3.30404

−0.450333

1.00000

1.2

−1.00000

−3.23770

2.16912

1.00000

1.3

−1.00000

−2.26645

2.39900

1.00000

1.4

−1.00000

−2.15802

−0.0890340

1.00000

1.5

−1.00000

−1.55864

−1.20577

1.00000

1.6

−1.00000

−0.379598

−4.36268

1.00000

1.7

−1.00000

−0.198997

4.17683

1.00000

1.8

−1.00000

1.10854

5.12441

1.00000

1.9

−1.00000

1.26787

−0.0906464

1.00000

1.10

−1.00000

2.29106

−4.25127

1.00000

1.11

−1.00000

3.55671

2.50656

1.00000

1.12

−1.00000

3.57026

−1.78948

1.00000

1.13

−1.00000

3.63608

2.74254

1.00000

1.14

−1.00000

3.67292

−3.87925

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$3$	$1$
$251$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	6024.2.a.p	✓	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6024.2.a.p	✓	14	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6024))$:

$ T_{5}^{14} - 6 T_{5}^{13} - 29 T_{5}^{12} + 210 T_{5}^{11} + 280 T_{5}^{10} - 2796 T_{5}^{9} + \cdots - 3364 $

$ T_{7}^{14} - 3 T_{7}^{13} - 58 T_{7}^{12} + 169 T_{7}^{11} + 1199 T_{7}^{10} - 3477 T_{7}^{9} + \cdots + 432 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{14} $ T^14
$3$	$ (T + 1)^{14} $ (T + 1)^14
$5$	$ T^{14} - 6 T^{13} + \cdots - 3364 $ T^14 - 6T^13 - 29T^12 + 210T^11 + 280T^10 - 2796T^9 - 863T^8 + 17652T^7 - 1300T^6 - 53458T^5 + 10892T^4 + 67298T^3 - 13761T^2 - 22140*T - 3364
$7$	$ T^{14} - 3 T^{13} + \cdots + 432 $ T^14 - 3T^13 - 58T^12 + 169T^11 + 1199T^10 - 3477T^9 - 10497T^8 + 30681T^7 + 37177T^6 - 102972T^5 - 67332T^4 + 111196T^3 + 74828T^2 + 10592*T + 432
$11$	$ T^{14} - 7 T^{13} + \cdots + 254896 $ T^14 - 7T^13 - 62T^12 + 527T^11 + 845T^10 - 13280T^9 + 8447T^8 + 135907T^7 - 242882T^6 - 476691T^5 + 1422787T^4 - 194284T^3 - 1664200T^2 + 811392*T + 254896
$13$	$ T^{14} - 5 T^{13} + \cdots - 822144 $ T^14 - 5T^13 - 71T^12 + 364T^11 + 1807T^10 - 9531T^9 - 21330T^8 + 114530T^7 + 129567T^6 - 656012T^5 - 460324T^4 + 1654576T^3 + 988132T^2 - 1253248*T - 822144
$17$	$ T^{14} - 28 T^{13} + \cdots + 149472 $ T^14 - 28T^13 + 258T^12 - 270T^11 - 9899T^10 + 52192T^9 + 39087T^8 - 924248T^7 + 1488477T^6 + 4838172T^5 - 13798616T^4 - 3161528T^3 + 22630332T^2 + 3820176*T + 149472
$19$	$ T^{14} + \cdots - 146196576 $ T^14 + T^13 - 172T^12 - 125T^11 + 11665T^10 + 5997T^9 - 395441T^8 - 152079T^7 + 7015279T^6 + 2532474T^5 - 62047870T^4 - 27999112T^3 + 229689536T^2 + 130973504T - 146196576
$23$	$ T^{14} + 3 T^{13} + \cdots + 21681904 $ T^14 + 3T^13 - 169T^12 - 206T^11 + 10950T^10 - 5271T^9 - 320918T^8 + 618602T^7 + 3585549T^6 - 11717788T^5 - 2078629T^4 + 35415696T^3 - 19046564T^2 - 28179664*T + 21681904
$29$	$ T^{14} - 15 T^{13} + \cdots - 33209992 $ T^14 - 15T^13 - 58T^12 + 1665T^11 - 1256T^10 - 67202T^9 + 145352T^8 + 1192330T^7 - 3367505T^6 - 8633732T^5 + 31346934T^4 + 7988416T^3 - 102968804T^2 + 107986848*T - 33209992
$31$	$ T^{14} - 9 T^{13} + \cdots - 2750544 $ T^14 - 9T^13 - 154T^12 + 1033T^11 + 10010T^10 - 38586T^9 - 306252T^8 + 564444T^7 + 4244485T^6 - 3431232T^5 - 25290576T^4 + 6294144T^3 + 49570108T^2 + 14592112*T - 2750544
$37$	$ T^{14} + \cdots + 3046642304 $ T^14 - 4T^13 - 235T^12 + 736T^11 + 21525T^10 - 54837T^9 - 970865T^8 + 2230002T^7 + 22519810T^6 - 54260744T^5 - 250488864T^4 + 726888128T^3 + 854891968T^2 - 4007382784*T + 3046642304
$41$	$ T^{14} + \cdots + 678627008 $ T^14 - 32T^13 + 284T^12 + 920T^11 - 25312T^10 + 60765T^9 + 698108T^8 - 3022445T^7 - 8585523T^6 + 50187252T^5 + 55156844T^4 - 371254604T^3 - 238864708T^2 + 1037274240*T + 678627008
$43$	$ T^{14} + 3 T^{13} + \cdots + 33067616 $ T^14 + 3T^13 - 182T^12 - 463T^11 + 12522T^10 + 24870T^9 - 408038T^8 - 556784T^7 + 6419723T^6 + 5641108T^5 - 42183850T^4 - 38950344T^3 + 90382400T^2 + 119919840*T + 33067616
$47$	$ T^{14} + \cdots + 4493408256 $ T^14 - 15T^13 - 213T^12 + 3478T^11 + 15621T^10 - 298076T^9 - 350923T^8 + 11610264T^7 - 8293536T^6 - 200242728T^5 + 396403756T^4 + 1071685984T^3 - 3000439488T^2 - 820085760*T + 4493408256
$53$	$ T^{14} - 8 T^{13} + \cdots - 9448832 $ T^14 - 8T^13 - 153T^12 + 1423T^11 + 5976T^10 - 79855T^9 - 12705T^8 + 1844874T^7 - 3081245T^6 - 16461576T^5 + 48841486T^4 + 24720232T^3 - 194203984T^2 + 162227008*T - 9448832
$59$	$ T^{14} + 11 T^{13} + \cdots + 28297944 $ T^14 + 11T^13 - 189T^12 - 1922T^11 + 13144T^10 + 123611T^9 - 391234T^8 - 3580548T^7 + 4086849T^6 + 43793776T^5 + 4295027T^4 - 141721556T^3 - 46482270T^2 + 123301872*T + 28297944
$61$	$ T^{14} + \cdots + 190748952 $ T^14 - 25T^13 - 72T^12 + 5272T^11 - 11552T^10 - 364852T^9 + 1015583T^8 + 9392908T^7 - 13716821T^6 - 82248301T^5 + 77309247T^4 + 283682446T^3 - 200910750T^2 - 328519584*T + 190748952
$67$	$ T^{14} + \cdots - 301611224 $ T^14 + 20T^13 - 257T^12 - 7364T^11 - 2637T^10 + 708761T^9 + 1752044T^8 - 30955037T^7 - 73146569T^6 + 707691435T^5 + 683255553T^4 - 7356824595T^3 + 7223831081T^2 + 1360536916*T - 301611224
$71$	$ T^{14} + \cdots + 616855785984 $ T^14 - 33T^13 + 34T^12 + 10684T^11 - 122867T^10 - 555581T^9 + 19015943T^8 - 100674762T^7 - 474271622T^6 + 7947653728T^5 - 36163837048T^4 + 48755018848T^3 + 148497082176T^2 - 604176406272*T + 616855785984
$73$	$ T^{14} + \cdots - 29286680333 $ T^14 - 8T^13 - 392T^12 + 2585T^11 + 61968T^10 - 289292T^9 - 5132293T^8 + 12516486T^7 + 228781573T^6 - 65683780T^5 - 4693676710T^4 - 6505772071T^3 + 21379373866T^2 + 29643057376*T - 29286680333
$79$	$ T^{14} + 17 T^{13} + \cdots - 131584 $ T^14 + 17T^13 - 239T^12 - 4726T^11 + 2837T^10 + 234669T^9 + 172246T^8 - 3415898T^7 - 953031T^6 + 14293648T^5 + 5753296T^4 - 18137376T^3 - 13173760T^2 - 2425856*T - 131584
$83$	$ T^{14} + \cdots + 54999094464 $ T^14 + 4T^13 - 581T^12 - 2236T^11 + 125489T^10 + 479360T^9 - 12428748T^8 - 49299516T^7 + 562700946T^6 + 2430744208T^5 - 9472646967T^4 - 47511995664T^3 + 726757424T^2 + 112931970624*T + 54999094464
$89$	$ T^{14} + \cdots - 62604878976 $ T^14 - 14T^13 - 481T^12 + 6446T^11 + 83244T^10 - 987922T^9 - 6964625T^8 + 59686092T^7 + 318958816T^6 - 1292199944T^5 - 6619953772T^4 + 7238267104T^3 + 44786357296T^2 + 175049088*T - 62604878976
$97$	$ T^{14} + \cdots + 23118443764928 $ T^14 - 9T^13 - 796T^12 + 5014T^11 + 258638T^10 - 875636T^9 - 43322311T^8 + 25870724T^7 + 3841186894T^6 + 6992735424T^5 - 163571015884T^4 - 646018506048T^3 + 2161705353400T^2 + 15739319019360*T + 23118443764928
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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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\( q - q^{3} + \beta_1 q^{5} + \beta_{10} q^{7} + q^{9}+O(q^{10}) \) q - q^3 + b1 * q^5 + b10 * q^7 + q^9	\( q - q^{3} + \beta_1 q^{5} + \beta_{10} q^{7} + q^{9} - \beta_{8} q^{11} + \beta_{4} q^{13} - \beta_1 q^{15} + (\beta_{8} + \beta_{7} + \beta_{4} + 2) q^{17} + (\beta_{9} + \beta_{6} - \beta_{5}) q^{19} - \beta_{10} q^{21} + (\beta_{12} + \beta_{7}) q^{23} + (\beta_{2} + \beta_1 + 1) q^{25} - q^{27} + (\beta_{9} + \beta_{3} + 1) q^{29} + ( - \beta_{11} - \beta_{9} + \cdots - \beta_{6}) q^{31}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) q - q^3 + b1 * q^5 + b10 * q^7 + q^9 - b8 * q^11 + b4 * q^13 - b1 * q^15 + (b8 + b7 + b4 + 2) * q^17 + (b9 + b6 - b5) * q^19 - b10 * q^21 + (b12 + b7) * q^23 + (b2 + b1 + 1) * q^25 - q^27 + (b9 + b3 + 1) * q^29 + (-b11 - b9 - b8 - b6) * q^31 + b8 * q^33 + (-b12 + 2b10 - b7 + b4 - 2) q^35 + (b11 + b7 - b2 + b1) * q^37 - b4 * q^39 + (-b9 - b8 - b7 - b4 + 2) * q^41 + (b11 - b3 - b2 + b1) * q^43 + b1 * q^45 + (b13 + b10 - b7 + b6 - b5 - b3 + 1) * q^47 + (b12 - b11 + b5 + b3 + 2) * q^49 + (-b8 - b7 - b4 - 2) * q^51 + (-b12 + b10 + b4 - b3) * q^53 + (-b13 + b11 + 2b5 + b2) q^55 + (-b9 - b6 + b5) * q^57 + (-b13 + b9 - b4) * q^59 + (-b12 - b11 + b1 + 1) * q^61 + b10 * q^63 + (-b13 + b10 - b7 + b4 - 2b3 + b2 + b1 + 1) q^65 + (b13 - b11 - b9 - b8 + b7 - b6 + b5 + b4 + b3 - 3) * q^67 + (-b12 - b7) * q^69 + (-b13 - b12 - b11 + b10 - b9 + b8 + b5 + b3 + b2 - b1 + 2) * q^71 + (b13 + b11 + b8 + b7 - b4 - b2 + b1 + 1) * q^73 + (-b2 - b1 - 1) * q^75 + (b12 - b9 - b8 - 2b7 + b6 - b5 - b3 + 1) q^77 + (b13 + b12 - b10 + b9 + b7 - b4 + 2b3 - 1) q^79 + q^81 + (b8 + b7 - b6 - b5 - b4 + b3 - b2 + b1) * q^83 + (-b8 - 2b7 - b6 - b5 - b4 - b3 + 4b1) * q^85 + (-b9 - b3 - 1) * q^87 + (2b13 + b12 + b11 + b9 - b8 - b5 - 2b4 + b3 - b2 + b1 + 1) * q^89 + (b10 - b9 - b8 + b5 + b3 + b2 + b1 - 4) * q^91 + (b11 + b9 + b8 + b6) * q^93 + (2b11 + 2b9 - 3b7 - 2b5 - 3b4 - b2 + b1 + 2) q^95 + (-2b13 - b12 - b9 + b8 + b7 - b6 + b4 - 2b3 - b2 + b1 + 1) * q^97 - b8 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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	6024.2.a.p

Level	$6024$
Weight	$2$
Character orbit	6024.a
Self dual	yes
Analytic conductor	$48.102$
Analytic rank	$0$
Dimension	$14$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(48.1018821776\)
Analytic rank:	\(0\)
Dimension:	\(14\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{14} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 \) x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - \nu - 6 \) v^2 - v - 6
\(\beta_{3}\)	\(=\)	\( ( - 34385 \nu^{13} + 26892 \nu^{12} + 1553117 \nu^{11} - 869644 \nu^{10} - 26472460 \nu^{9} + \cdots - 56474950 ) / 10615128 \) (-34385v^13 + 26892v^12 + 1553117v^11 - 869644v^10 - 26472460v^9 + 9438328v^8 + 207665199v^7 - 38974970v^6 - 726563800v^5 + 59731310v^4 + 933220260v^3 - 52788358v^2 - 253959291*v - 56474950) / 10615128
\(\beta_{4}\)	\(=\)	\( ( 891500 \nu^{13} - 2582249 \nu^{12} - 32877251 \nu^{11} + 81418106 \nu^{10} + 470527080 \nu^{9} + \cdots + 543511258 ) / 47768076 \) (891500v^13 - 2582249v^12 - 32877251v^11 + 81418106v^10 + 470527080v^9 - 921011508v^8 - 3245410102v^7 + 4526703895v^6 + 10740423537v^5 - 9653543531v^4 - 14580166089v^3 + 7501423177v^2 + 5250877601*v + 543511258) / 47768076
\(\beta_{5}\)	\(=\)	\( ( - 2285707 \nu^{13} + 5763184 \nu^{12} + 85005175 \nu^{11} - 177040372 \nu^{10} + \cdots - 2060172338 ) / 95536152 \) (-2285707v^13 + 5763184v^12 + 85005175v^11 - 177040372v^10 - 1224526764v^9 + 1920505680v^8 + 8476711133v^7 - 8770496906v^6 - 28100462208v^5 + 16436289310v^4 + 38536651212v^3 - 10547722286v^2 - 15515835721*v - 2060172338) / 95536152
\(\beta_{6}\)	\(=\)	\( ( - 510731 \nu^{13} + 1185871 \nu^{12} + 19439080 \nu^{11} - 36738588 \nu^{10} - 286607520 \nu^{9} + \cdots - 677427348 ) / 15922692 \) (-510731v^13 + 1185871v^12 + 19439080v^11 - 36738588v^10 - 286607520v^9 + 402727860v^8 + 2026352809v^7 - 1865943851v^6 - 6827363977v^5 + 3566414523v^4 + 9440736513v^3 - 2259965299v^2 - 3691152442*v - 677427348) / 15922692
\(\beta_{7}\)	\(=\)	\( ( 1750919 \nu^{13} - 4072364 \nu^{12} - 65938241 \nu^{11} + 124543394 \nu^{10} + 961333458 \nu^{9} + \cdots + 1930586878 ) / 47768076 \) (1750919v^13 - 4072364v^12 - 65938241v^11 + 124543394v^10 + 961333458v^9 - 1341558738v^8 - 6714256819v^7 + 6055282684v^6 + 22294893624v^5 - 11158864646v^4 - 30232129854v^3 + 7029931060v^2 + 11780644049*v + 1930586878) / 47768076
\(\beta_{8}\)	\(=\)	\( ( - 799400 \nu^{13} + 2007980 \nu^{12} + 29958653 \nu^{11} - 62535530 \nu^{10} - 434576058 \nu^{9} + \cdots - 636577414 ) / 15922692 \) (-799400v^13 + 2007980v^12 + 29958653v^11 - 62535530v^10 - 434576058v^9 + 693826704v^8 + 3020215300v^7 - 3303220780v^6 - 9973980471v^5 + 6716846546v^4 + 13368961119v^3 - 4982795950v^2 - 4904604035*v - 636577414) / 15922692
\(\beta_{9}\)	\(=\)	\( ( 2690857 \nu^{13} - 6588652 \nu^{12} - 101245663 \nu^{11} + 205940440 \nu^{10} + 1473998214 \nu^{9} + \cdots + 2458464146 ) / 47768076 \) (2690857v^13 - 6588652v^12 - 101245663v^11 + 205940440v^10 + 1473998214v^9 - 2300026932v^8 - 10278303353v^7 + 11083245044v^6 + 34072151028v^5 - 22949713318v^4 - 45888598104v^3 + 17130259784v^2 + 16722285277*v + 2458464146) / 47768076
\(\beta_{10}\)	\(=\)	\( ( - 1085369 \nu^{13} + 2654776 \nu^{12} + 40709620 \nu^{11} - 82526094 \nu^{10} - 590411514 \nu^{9} + \cdots - 798333708 ) / 15922692 \) (-1085369v^13 + 2654776v^12 + 40709620v^11 - 82526094v^10 - 590411514v^9 + 913528686v^8 + 4096485739v^7 - 4337434292v^6 - 13473230383v^5 + 8801355990v^4 + 17847143889v^3 - 6578394766v^2 - 6272928610*v - 798333708) / 15922692
\(\beta_{11}\)	\(=\)	\( ( 1359385 \nu^{13} - 3285059 \nu^{12} - 51337298 \nu^{11} + 101829564 \nu^{10} + 751000860 \nu^{9} + \cdots + 1300892340 ) / 15922692 \) (1359385v^13 - 3285059v^12 - 51337298v^11 + 101829564v^10 + 751000860v^9 - 1119757014v^8 - 5266901315v^7 + 5240205949v^6 + 17571182555v^5 - 10334058561v^4 - 23897723349v^3 + 7318586177v^2 + 9075081374*v + 1300892340) / 15922692
\(\beta_{12}\)	\(=\)	\( ( 1400227 \nu^{13} - 3427662 \nu^{12} - 52961826 \nu^{11} + 107080910 \nu^{10} + 776737884 \nu^{9} + \cdots + 1560309976 ) / 15922692 \) (1400227v^13 - 3427662v^12 - 52961826v^11 + 107080910v^10 + 776737884v^9 - 1192572510v^8 - 5472245579v^7 + 5699815236v^6 + 18422349217v^5 - 11564497352v^4 - 25553159163v^3 + 8209065926v^2 + 10105957332*v + 1560309976) / 15922692
\(\beta_{13}\)	\(=\)	\( ( - 6572812 \nu^{13} + 16236166 \nu^{12} + 247008691 \nu^{11} - 503783854 \nu^{10} + \cdots - 6511648574 ) / 47768076 \) (-6572812v^13 + 16236166v^12 + 247008691v^11 - 503783854v^10 - 3595411272v^9 + 5552233938v^8 + 25099371248v^7 - 26089480472v^6 - 83439350505v^5 + 51662191384v^4 + 113289156915v^3 - 36259637984v^2 - 43344239917*v - 6511648574) / 47768076

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + \beta _1 + 6 \) b2 + b1 + 6
\(\nu^{3}\)	\(=\)	\( \beta_{13} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 11\beta _1 + 2 \) b13 - b8 + b7 - b6 + b4 + 11*b1 + 2
\(\nu^{4}\)	\(=\)	\( - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 3 \beta_{5} + \cdots + 60 \) -b13 - b12 + b10 + b9 + b8 + b7 - b6 + 3b5 + b4 + b3 + 14b2 + 16*b1 + 60
\(\nu^{5}\)	\(=\)	\( 15 \beta_{13} + 2 \beta_{10} - 19 \beta_{8} + 15 \beta_{7} - 15 \beta_{6} + \beta_{5} + 13 \beta_{4} + \cdots + 44 \) 15b13 + 2b10 - 19b8 + 15b7 - 15b6 + b5 + 13b4 + b3 + 4b2 + 131b1 + 44
\(\nu^{6}\)	\(=\)	\( - 12 \beta_{13} - 16 \beta_{12} + 3 \beta_{11} + 17 \beta_{10} + 14 \beta_{9} + 9 \beta_{8} + \cdots + 678 \) -12b13 - 16b12 + 3b11 + 17b10 + 14b9 + 9b8 + 20b7 - 19b6 + 54b5 + 18b4 + 20b3 + 175b2 + 233*b1 + 678
\(\nu^{7}\)	\(=\)	\( 187 \beta_{13} - \beta_{12} - 5 \beta_{11} + 36 \beta_{10} - 7 \beta_{9} - 268 \beta_{8} + 203 \beta_{7} + \cdots + 746 \) 187b13 - b12 - 5b11 + 36b10 - 7b9 - 268b8 + 203b7 - 198b6 + 45b5 + 161b4 + 25b3 + 109b2 + 1602b1 + 746
\(\nu^{8}\)	\(=\)	\( - 102 \beta_{13} - 199 \beta_{12} + 45 \beta_{11} + 218 \beta_{10} + 125 \beta_{9} - \beta_{8} + \cdots + 8042 \) -102b13 - 199b12 + 45b11 + 218b10 + 125b9 - b8 + 332b7 - 321b6 + 800b5 + 272b4 + 312b3 + 2154b2 + 3272b1 + 8042
\(\nu^{9}\)	\(=\)	\( 2245 \beta_{13} - 3 \beta_{12} - 138 \beta_{11} + 473 \beta_{10} - 249 \beta_{9} - 3489 \beta_{8} + \cdots + 11464 \) 2245b13 - 3b12 - 138b11 + 473b10 - 249b9 - 3489b8 + 2717b7 - 2551b6 + 1074b5 + 2031b4 + 484b3 + 2089b2 + 19856*b1 + 11464
\(\nu^{10}\)	\(=\)	\( - 561 \beta_{13} - 2237 \beta_{12} + 393 \beta_{11} + 2508 \beta_{10} + 557 \beta_{9} - 1714 \beta_{8} + \cdots + 97700 \) -561b13 - 2237b12 + 393b11 + 2508b10 + 557b9 - 1714b8 + 5151b7 - 5090b6 + 11302b5 + 3885b4 + 4530b3 + 26606b2 + 45038*b1 + 97700
\(\nu^{11}\)	\(=\)	\( 26820 \beta_{13} + 346 \beta_{12} - 2758 \beta_{11} + 5426 \beta_{10} - 5826 \beta_{9} - 44574 \beta_{8} + \cdots + 167648 \) 26820b13 + 346b12 - 2758b11 + 5426b10 - 5826b9 - 44574b8 + 36376b7 - 32878b6 + 20445b5 + 25946b4 + 8453b3 + 34772b2 + 248237*b1 + 167648
\(\nu^{12}\)	\(=\)	\( 1937 \beta_{13} - 23447 \beta_{12} + 1039 \beta_{11} + 26896 \beta_{10} - 7911 \beta_{9} + \cdots + 1204036 \) 1937b13 - 23447b12 + 1039b11 + 26896b10 - 7911b9 - 43524b8 + 77287b7 - 77082b6 + 157729b5 + 54109b4 + 64193b3 + 331250b2 + 611874*b1 + 1204036
\(\nu^{13}\)	\(=\)	\( 322155 \beta_{13} + 11913 \beta_{12} - 48703 \beta_{11} + 55998 \beta_{10} - 113821 \beta_{9} + \cdots + 2379408 \) 322155b13 + 11913b12 - 48703b11 + 55998b10 - 113821b9 - 570870b8 + 487827b7 - 426928b6 + 348366b5 + 333155b4 + 139074b3 + 537923b2 + 3123648*b1 + 2379408

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.14
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{14} \) T^14
$3$	\( (T + 1)^{14} \) (T + 1)^14
$5$	\( T^{14} - 6 T^{13} + \cdots - 3364 \) T^14 - 6T^13 - 29T^12 + 210T^11 + 280T^10 - 2796T^9 - 863T^8 + 17652T^7 - 1300T^6 - 53458T^5 + 10892T^4 + 67298T^3 - 13761T^2 - 22140*T - 3364
$7$	\( T^{14} - 3 T^{13} + \cdots + 432 \) T^14 - 3T^13 - 58T^12 + 169T^11 + 1199T^10 - 3477T^9 - 10497T^8 + 30681T^7 + 37177T^6 - 102972T^5 - 67332T^4 + 111196T^3 + 74828T^2 + 10592*T + 432
$11$	\( T^{14} - 7 T^{13} + \cdots + 254896 \) T^14 - 7T^13 - 62T^12 + 527T^11 + 845T^10 - 13280T^9 + 8447T^8 + 135907T^7 - 242882T^6 - 476691T^5 + 1422787T^4 - 194284T^3 - 1664200T^2 + 811392*T + 254896
$13$	\( T^{14} - 5 T^{13} + \cdots - 822144 \) T^14 - 5T^13 - 71T^12 + 364T^11 + 1807T^10 - 9531T^9 - 21330T^8 + 114530T^7 + 129567T^6 - 656012T^5 - 460324T^4 + 1654576T^3 + 988132T^2 - 1253248*T - 822144
$17$	\( T^{14} - 28 T^{13} + \cdots + 149472 \) T^14 - 28T^13 + 258T^12 - 270T^11 - 9899T^10 + 52192T^9 + 39087T^8 - 924248T^7 + 1488477T^6 + 4838172T^5 - 13798616T^4 - 3161528T^3 + 22630332T^2 + 3820176*T + 149472
$19$	\( T^{14} + \cdots - 146196576 \) T^14 + T^13 - 172T^12 - 125T^11 + 11665T^10 + 5997T^9 - 395441T^8 - 152079T^7 + 7015279T^6 + 2532474T^5 - 62047870T^4 - 27999112T^3 + 229689536T^2 + 130973504T - 146196576
$23$	\( T^{14} + 3 T^{13} + \cdots + 21681904 \) T^14 + 3T^13 - 169T^12 - 206T^11 + 10950T^10 - 5271T^9 - 320918T^8 + 618602T^7 + 3585549T^6 - 11717788T^5 - 2078629T^4 + 35415696T^3 - 19046564T^2 - 28179664*T + 21681904
$29$	\( T^{14} - 15 T^{13} + \cdots - 33209992 \) T^14 - 15T^13 - 58T^12 + 1665T^11 - 1256T^10 - 67202T^9 + 145352T^8 + 1192330T^7 - 3367505T^6 - 8633732T^5 + 31346934T^4 + 7988416T^3 - 102968804T^2 + 107986848*T - 33209992
$31$	\( T^{14} - 9 T^{13} + \cdots - 2750544 \) T^14 - 9T^13 - 154T^12 + 1033T^11 + 10010T^10 - 38586T^9 - 306252T^8 + 564444T^7 + 4244485T^6 - 3431232T^5 - 25290576T^4 + 6294144T^3 + 49570108T^2 + 14592112*T - 2750544
$37$	\( T^{14} + \cdots + 3046642304 \) T^14 - 4T^13 - 235T^12 + 736T^11 + 21525T^10 - 54837T^9 - 970865T^8 + 2230002T^7 + 22519810T^6 - 54260744T^5 - 250488864T^4 + 726888128T^3 + 854891968T^2 - 4007382784*T + 3046642304
$41$	\( T^{14} + \cdots + 678627008 \) T^14 - 32T^13 + 284T^12 + 920T^11 - 25312T^10 + 60765T^9 + 698108T^8 - 3022445T^7 - 8585523T^6 + 50187252T^5 + 55156844T^4 - 371254604T^3 - 238864708T^2 + 1037274240*T + 678627008
$43$	\( T^{14} + 3 T^{13} + \cdots + 33067616 \) T^14 + 3T^13 - 182T^12 - 463T^11 + 12522T^10 + 24870T^9 - 408038T^8 - 556784T^7 + 6419723T^6 + 5641108T^5 - 42183850T^4 - 38950344T^3 + 90382400T^2 + 119919840*T + 33067616
$47$	\( T^{14} + \cdots + 4493408256 \) T^14 - 15T^13 - 213T^12 + 3478T^11 + 15621T^10 - 298076T^9 - 350923T^8 + 11610264T^7 - 8293536T^6 - 200242728T^5 + 396403756T^4 + 1071685984T^3 - 3000439488T^2 - 820085760*T + 4493408256
$53$	\( T^{14} - 8 T^{13} + \cdots - 9448832 \) T^14 - 8T^13 - 153T^12 + 1423T^11 + 5976T^10 - 79855T^9 - 12705T^8 + 1844874T^7 - 3081245T^6 - 16461576T^5 + 48841486T^4 + 24720232T^3 - 194203984T^2 + 162227008*T - 9448832
$59$	\( T^{14} + 11 T^{13} + \cdots + 28297944 \) T^14 + 11T^13 - 189T^12 - 1922T^11 + 13144T^10 + 123611T^9 - 391234T^8 - 3580548T^7 + 4086849T^6 + 43793776T^5 + 4295027T^4 - 141721556T^3 - 46482270T^2 + 123301872*T + 28297944
$61$	\( T^{14} + \cdots + 190748952 \) T^14 - 25T^13 - 72T^12 + 5272T^11 - 11552T^10 - 364852T^9 + 1015583T^8 + 9392908T^7 - 13716821T^6 - 82248301T^5 + 77309247T^4 + 283682446T^3 - 200910750T^2 - 328519584*T + 190748952
$67$	\( T^{14} + \cdots - 301611224 \) T^14 + 20T^13 - 257T^12 - 7364T^11 - 2637T^10 + 708761T^9 + 1752044T^8 - 30955037T^7 - 73146569T^6 + 707691435T^5 + 683255553T^4 - 7356824595T^3 + 7223831081T^2 + 1360536916*T - 301611224
$71$	\( T^{14} + \cdots + 616855785984 \) T^14 - 33T^13 + 34T^12 + 10684T^11 - 122867T^10 - 555581T^9 + 19015943T^8 - 100674762T^7 - 474271622T^6 + 7947653728T^5 - 36163837048T^4 + 48755018848T^3 + 148497082176T^2 - 604176406272*T + 616855785984
$73$	\( T^{14} + \cdots - 29286680333 \) T^14 - 8T^13 - 392T^12 + 2585T^11 + 61968T^10 - 289292T^9 - 5132293T^8 + 12516486T^7 + 228781573T^6 - 65683780T^5 - 4693676710T^4 - 6505772071T^3 + 21379373866T^2 + 29643057376*T - 29286680333
$79$	\( T^{14} + 17 T^{13} + \cdots - 131584 \) T^14 + 17T^13 - 239T^12 - 4726T^11 + 2837T^10 + 234669T^9 + 172246T^8 - 3415898T^7 - 953031T^6 + 14293648T^5 + 5753296T^4 - 18137376T^3 - 13173760T^2 - 2425856*T - 131584
$83$	\( T^{14} + \cdots + 54999094464 \) T^14 + 4T^13 - 581T^12 - 2236T^11 + 125489T^10 + 479360T^9 - 12428748T^8 - 49299516T^7 + 562700946T^6 + 2430744208T^5 - 9472646967T^4 - 47511995664T^3 + 726757424T^2 + 112931970624*T + 54999094464
$89$	\( T^{14} + \cdots - 62604878976 \) T^14 - 14T^13 - 481T^12 + 6446T^11 + 83244T^10 - 987922T^9 - 6964625T^8 + 59686092T^7 + 318958816T^6 - 1292199944T^5 - 6619953772T^4 + 7238267104T^3 + 44786357296T^2 + 175049088*T - 62604878976
$97$	\( T^{14} + \cdots + 23118443764928 \) T^14 - 9T^13 - 796T^12 + 5014T^11 + 258638T^10 - 875636T^9 - 43322311T^8 + 25870724T^7 + 3841186894T^6 + 6992735424T^5 - 163571015884T^4 - 646018506048T^3 + 2161705353400T^2 + 15739319019360*T + 23118443764928
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\(2\)	\(1\)
\(3\)	\(1\)
\(251\)	\(-1\)