LMFDB - Newform orbit 4842.2.a.t

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 33 x^{11} + 200 x^{10} + 236 x^{9} - 2569 x^{8} + 1311 x^{7} + 11583 x^{6} + \cdots + 2160 $ x^13 - 5x^12 - 33x^11 + 200x^10 + 236x^9 - 2569x^8 + 1311x^7 + 11583x^6 - 15582x^5 - 10013x^4 + 24596x^3 - 6260x^2 - 5472x + 2160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
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 $1,\beta_1,\ldots,\beta_{12}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - q^{2} + q^{4} - \beta_1 q^{5} - \beta_{3} q^{7} - q^{8} + \beta_1 q^{10} - \beta_{10} q^{11} + (\beta_{2} + 1) q^{13} + \beta_{3} q^{14} + q^{16} + ( - \beta_{8} - 1) q^{17} + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots + 2) q^{19}+ \cdots + (\beta_{12} + 2 \beta_{10} - \beta_{9} + \cdots - 4) q^{98}+O(q^{100}) $ q - q^2 + q^4 - b1 * q^5 - b3 * q^7 - q^8 + b1 * q^10 - b10 * q^11 + (b2 + 1) * q^13 + b3 * q^14 + q^16 + (-b8 - 1) * q^17 + (b11 - b10 - b9 - b4 + 2) * q^19 - b1 * q^20 + b10 * q^22 + b5 * q^23 + (b12 - b9 + b5 + b4 + 2) * q^25 + (-b2 - 1) * q^26 - b3 * q^28 + (b12 + b11 - b10 - b7 - b5 - b3 - b1) * q^29 + (b9 - b8) * q^31 - q^32 + (b8 + 1) * q^34 + (-b9 + b8 + b7 - b4 + b3 + b2 - b1 + 1) * q^35 + (-b12 - b11 + b9 + 2) * q^37 + (-b11 + b10 + b9 + b4 - 2) * q^38 + b1 * q^40 + (-b12 + b10 - b9 + b7 + b5 + b1 - 3) * q^41 + (-b7 - b5 - b3 - b2 + 1) * q^43 - b10 * q^44 - b5 * q^46 + (-b11 + b9 - b7 + b6 - b5 + b4 - b3 + b2 - b1) * q^47 + (-b12 - 2b10 + b9 + b8 + b7 - 2b5 - b4 - b3 - b1 + 4) * q^49 + (-b12 + b9 - b5 - b4 - 2) * q^50 + (b2 + 1) * q^52 + (-b12 + b9 - b6 - 2) * q^53 + (-b11 + 2b10 + b9 - b8 + b7 - b6 + b5 + b4 + b3 - 2) q^55 + b3 * q^56 + (-b12 - b11 + b10 + b7 + b5 + b3 + b1) * q^58 + (b10 - b9 - b6 + b4) * q^59 + (-b11 + b10 - b7 + 2b6 + b5 - b3 + b1 - 1) q^61 + (-b9 + b8) * q^62 + q^64 + (-b10 + b8 + b7 - b6 - b5 - 2b4 - b3 - b2 - b1 - 1) q^65 + (b11 - b10 - b5 - b2 - b1 + 4) * q^67 + (-b8 - 1) * q^68 + (b9 - b8 - b7 + b4 - b3 - b2 + b1 - 1) * q^70 + (-b11 + 2b10 + 2b9 - b7 + b6 + b5 + 2b4 - b3 + b2 - b1 - 2) q^71 + (b12 + b11 + b8 + b4 - 2b1 + 4) q^73 + (b12 + b11 - b9 - 2) * q^74 + (b11 - b10 - b9 - b4 + 2) * q^76 + (b12 + b11 - b9 + b6 + b5 - b4 - b2 + b1 - 1) * q^77 + (-b10 - b9 + b8 - 2b6 - b2 - 2b1 + 5) * q^79 - b1 * q^80 + (b12 - b10 + b9 - b7 - b5 - b1 + 3) * q^82 + (b12 + b11 - b10 - b9 + b8 + b7 + b6 - b5 - b3 - b1 + 2) * q^83 + (b12 - b11 - b8 - b7 - b6 + b5 + b3 - b2 + b1 + 2) * q^85 + (b7 + b5 + b3 + b2 - 1) * q^86 + b10 * q^88 + (b12 + b11 + b8 - 2b7 + 2b6 - b5 - 2b3 - 2b1 + 2) * q^89 + (-2b11 + 3b10 + b9 + 3b6 + b4 - 2b3 + 2b2 - 2b1) * q^91 + b5 * q^92 + (b11 - b9 + b7 - b6 + b5 - b4 + b3 - b2 + b1) * q^94 + (-2b11 + 2b10 - b6 - b4 - 1) * q^95 + (-b12 + b11 - 2b10 - 2b9 + b8 + 2b7 - b6 + b5 - 2b4 + 2b3 - b2 + b1 + 3) q^97 + (b12 + 2b10 - b9 - b8 - b7 + 2b5 + b4 + b3 + b1 - 4) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{2} + 13 q^{4} - 5 q^{5} + 6 q^{7} - 13 q^{8} + 5 q^{10} - 5 q^{11} + 8 q^{13} - 6 q^{14} + 13 q^{16} - 7 q^{17} + 13 q^{19} - 5 q^{20} + 5 q^{22} + 4 q^{23} + 26 q^{25} - 8 q^{26} + 6 q^{28}+ \cdots - 39 q^{98}+O(q^{100}) $ 13 * q - 13 * q^2 + 13 * q^4 - 5 * q^5 + 6 * q^7 - 13 * q^8 + 5 * q^10 - 5 * q^11 + 8 * q^13 - 6 * q^14 + 13 * q^16 - 7 * q^17 + 13 * q^19 - 5 * q^20 + 5 * q^22 + 4 * q^23 + 26 * q^25 - 8 * q^26 + 6 * q^28 - 22 * q^29 + 8 * q^31 - 13 * q^32 + 7 * q^34 - 8 * q^35 + 36 * q^37 - 13 * q^38 + 5 * q^40 - 16 * q^41 + 14 * q^43 - 5 * q^44 - 4 * q^46 - 4 * q^47 + 39 * q^49 - 26 * q^50 + 8 * q^52 - 21 * q^53 - 6 * q^56 + 22 * q^58 + 4 * q^59 + 8 * q^61 - 8 * q^62 + 13 * q^64 - 24 * q^65 + 40 * q^67 - 7 * q^68 + 8 * q^70 - 7 * q^71 + 31 * q^73 - 36 * q^74 + 13 * q^76 - 10 * q^77 + 43 * q^79 - 5 * q^80 + 16 * q^82 + 10 * q^83 + 30 * q^85 - 14 * q^86 + 5 * q^88 + 2 * q^89 + 24 * q^91 + 4 * q^92 + 4 * q^94 - 2 * q^95 + 27 * q^97 - 39 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{13} - 5 x^{12} - 33 x^{11} + 200 x^{10} + 236 x^{9} - 2569 x^{8} + 1311 x^{7} + 11583 x^{6} + \cdots + 2160 $ x^13 - 5*x^12 - 33*x^11 + 200*x^10 + 236*x^9 - 2569*x^8 + 1311*x^7 + 11583*x^6 - 15582*x^5 - 10013*x^4 + 24596*x^3 - 6260*x^2 - 5472*x + 2160 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 48024874805 \nu^{12} - 112287409573 \nu^{11} - 1941343452888 \nu^{10} + \cdots - 283838423867382 ) / 124747362130182 $ (48024874805v^12 - 112287409573v^11 - 1941343452888v^10 + 4661678813218v^9 + 25746664037524v^8 - 64568707789679v^7 - 124740902028006v^6 + 366728280584655v^5 + 132349851534843v^4 - 924186446242159v^3 + 165915501592513v^2 + 1231589887634822v - 283838423867382) / 124747362130182
$\beta_{3}$	$=$	$ ( 852354528985 \nu^{12} - 5565998476235 \nu^{11} - 24931795402263 \nu^{10} + \cdots - 71\!\cdots\!68 ) / 498989448520728 $ (852354528985v^12 - 5565998476235v^11 - 24931795402263v^10 + 222999309674426v^9 + 67809369284036v^8 - 2889498022765681v^7 + 2936240649919401v^6 + 13437755317500921v^5 - 22501654361868084v^4 - 14476242953328965v^3 + 33905371325920238v^2 - 790543441394096v - 7135321253071368) / 498989448520728
$\beta_{4}$	$=$	$ ( - 1462385429459 \nu^{12} + 5438514414121 \nu^{11} + 55460849135577 \nu^{10} + \cdots + 726463997646072 ) / 498989448520728 $ (-1462385429459v^12 + 5438514414121v^11 + 55460849135577v^10 - 223613158488046v^9 - 638992003731088v^8 + 3023577814303631v^7 + 2002626917600661v^6 - 15457619371574655v^5 + 3319327641718824v^4 + 24015438167285407v^3 - 8450625764052706v^2 - 8414011116504308v + 726463997646072) / 498989448520728
$\beta_{5}$	$=$	$ ( - 868230374567 \nu^{12} + 4292066532829 \nu^{11} + 29608543880865 \nu^{10} + \cdots + 45\!\cdots\!64 ) / 249494724260364 $ (-868230374567v^12 + 4292066532829v^11 + 29608543880865v^10 - 172210781998990v^9 - 242022329130154v^8 + 2231221511993039v^7 - 673104802752525v^6 - 10347700457769675v^5 + 11446839915598068v^4 + 10942647896603347v^3 - 19205954143548196v^2 + 1141012497025972v + 4525884532262664) / 249494724260364
$\beta_{6}$	$=$	$ ( - 630139679479 \nu^{12} + 2701324988981 \nu^{11} + 22362606192965 \nu^{10} + \cdots + 36\!\cdots\!24 ) / 166329816173576 $ (-630139679479v^12 + 2701324988981v^11 + 22362606192965v^10 - 110188878729306v^9 - 212412736035108v^8 + 1469426488671711v^7 + 8099800770765v^6 - 7291875723961147v^5 + 5911876575648176v^4 + 10367983048606231v^3 - 11323162532461150v^2 - 3433794836880676v + 3638197188330624) / 166329816173576
$\beta_{7}$	$=$	$ ( - 2000957877725 \nu^{12} + 7635503796289 \nu^{11} + 71656431054273 \nu^{10} + \cdots + 84\!\cdots\!24 ) / 498989448520728 $ (-2000957877725v^12 + 7635503796289v^11 + 71656431054273v^10 - 312948904996228v^9 - 704096950929424v^8 + 4189399857925685v^7 + 444893315979717v^6 - 20821591731301911v^5 + 16637899023589626v^4 + 29137144742255173v^3 - 32407998154263220v^2 - 7321545315875216v + 8429509843089624) / 498989448520728
$\beta_{8}$	$=$	$ ( 647904046508 \nu^{12} - 2958462557782 \nu^{11} - 22330056820125 \nu^{10} + \cdots - 31\!\cdots\!74 ) / 124747362130182 $ (647904046508v^12 - 2958462557782v^11 - 22330056820125v^10 + 119976211894435v^9 + 192212383085599v^8 - 1583507423251541v^7 + 320616638739144v^6 + 7674059886042198v^5 - 7502314288114329v^4 - 9916668415011091v^3 + 12693921759640060v^2 + 1750282976092010v - 3142157756066574) / 124747362130182
$\beta_{9}$	$=$	$ ( 4401723597281 \nu^{12} - 18847913017729 \nu^{11} - 158350218453045 \nu^{10} + \cdots - 10\!\cdots\!12 ) / 498989448520728 $ (4401723597281v^12 - 18847913017729v^11 - 158350218453045v^10 + 765561425553280v^9 + 1574339848381408v^8 - 10134784365728573v^7 - 1402922813124657v^6 + 49457201715504891v^5 - 32721323671713066v^4 - 65658815154377377v^3 + 56842152070588012v^2 + 14201661636087044v - 10796184896235312) / 498989448520728
$\beta_{10}$	$=$	$ ( 6609202784399 \nu^{12} - 27174425518447 \nu^{11} - 241966416215319 \nu^{10} + \cdots - 18\!\cdots\!48 ) / 498989448520728 $ (6609202784399v^12 - 27174425518447v^11 - 241966416215319v^10 + 1105175609150380v^9 + 2528452212144112v^8 - 14665541151353999v^7 - 4151436180006291v^6 + 71978940006584853v^5 - 40440321513996066v^4 - 97437878729532859v^3 + 79210205380248472v^2 + 20306966993208644v - 18201659224281048) / 498989448520728
$\beta_{11}$	$=$	$ ( 2233298029201 \nu^{12} - 9837337220151 \nu^{11} - 80661040634447 \nu^{10} + \cdots - 70\!\cdots\!48 ) / 166329816173576 $ (2233298029201v^12 - 9837337220151v^11 - 80661040634447v^10 + 399969919876814v^9 + 808935967676248v^8 - 5312697579214149v^7 - 799851335767539v^6 + 26171112127655033v^5 - 16573051963403412v^4 - 36020155348323261v^3 + 30470208695623234v^2 + 8635958312368788v - 7065047705357648) / 166329816173576
$\beta_{12}$	$=$	$ ( 3800284887937 \nu^{12} - 16435280248754 \nu^{11} - 136514077675176 \nu^{10} + \cdots - 12\!\cdots\!04 ) / 249494724260364 $ (3800284887937v^12 - 16435280248754v^11 - 136514077675176v^10 + 666798074019653v^9 + 1348688255186402v^8 - 8810402602009141v^7 - 1029670062610134v^6 + 42805111001309448v^5 - 29467165572314013v^4 - 55779774557434739v^3 + 52101837785128919v^2 + 10166823879269704v - 12033672049025904) / 249494724260364

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{12} - \beta_{9} + \beta_{5} + \beta_{4} + 7 $ b12 - b9 + b5 + b4 + 7
$\nu^{3}$	$=$	$ \beta_{12} + \beta_{11} - \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - \beta_{7} + 3 \beta_{6} + \cdots + 12 \beta_1 $ b12 + b11 - b10 - 3b9 + 2b8 - b7 + 3b6 - b5 - 2b4 - b3 + b2 + 12*b1
$\nu^{4}$	$=$	$ 16 \beta_{12} + \beta_{11} + \beta_{10} - 14 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + \cdots + 88 $ 16b12 + b11 + b10 - 14b9 - 3b8 - 3b7 + 4b6 + 20b5 + 22b4 + 3b3 + 2*b2 - b1 + 88
$\nu^{5}$	$=$	$ 23 \beta_{12} + 27 \beta_{11} - 34 \beta_{10} - 64 \beta_{9} + 41 \beta_{8} - 27 \beta_{7} + 67 \beta_{6} + \cdots + 12 $ 23b12 + 27b11 - 34b10 - 64b9 + 41b8 - 27b7 + 67b6 - 34b5 - 52b4 - 29b3 + 35b2 + 165b1 + 12
$\nu^{6}$	$=$	$ 248 \beta_{12} + 35 \beta_{11} + 30 \beta_{10} - 196 \beta_{9} - 82 \beta_{8} - 84 \beta_{7} + \cdots + 1293 $ 248b12 + 35b11 + 30b10 - 196b9 - 82b8 - 84b7 + 100b6 + 364b5 + 449b4 + 68b3 + 48b2 - 43b1 + 1293
$\nu^{7}$	$=$	$ 459 \beta_{12} + 587 \beta_{11} - 828 \beta_{10} - 1208 \beta_{9} + 733 \beta_{8} - 601 \beta_{7} + \cdots + 306 $ 459b12 + 587b11 - 828b10 - 1208b9 + 733b8 - 601b7 + 1293b6 - 848b5 - 1134b4 - 669b3 + 779b2 + 2492b1 + 306
$\nu^{8}$	$=$	$ 3971 \beta_{12} + 827 \beta_{11} + 690 \beta_{10} - 2813 \beta_{9} - 1728 \beta_{8} - 1770 \beta_{7} + \cdots + 20654 $ 3971b12 + 827b11 + 690b10 - 2813b9 - 1728b8 - 1770b7 + 1994b6 + 6623b5 + 8844b4 + 1254b3 + 886b2 - 1257b1 + 20654
$\nu^{9}$	$=$	$ 8680 \beta_{12} + 11760 \beta_{11} - 17751 \beta_{10} - 22241 \beta_{9} + 12943 \beta_{8} - 12250 \beta_{7} + \cdots + 5438 $ 8680b12 + 11760b11 - 17751b10 - 22241b9 + 12943b8 - 12250b7 + 23968b6 - 18605b5 - 23480b4 - 13934b3 + 15318b2 + 40428b1 + 5438
$\nu^{10}$	$=$	$ 65619 \beta_{12} + 16566 \beta_{11} + 15253 \beta_{10} - 41161 \beta_{9} - 33879 \beta_{8} + \cdots + 347744 $ 65619b12 + 16566b11 + 15253b10 - 41161b9 - 33879b8 - 33681b7 + 36500b6 + 122103b5 + 171110b4 + 22841b3 + 14740b2 - 31132b1 + 347744
$\nu^{11}$	$=$	$ 159165 \beta_{12} + 225687 \beta_{11} - 357471 \beta_{10} - 406499 \beta_{9} + 231564 \beta_{8} + \cdots + 76578 $ 159165b12 + 225687b11 - 357471b10 - 406499b9 + 231564b8 - 237723b7 + 437921b6 - 384669b5 - 474590b4 - 276001b3 + 288293b2 + 690021b1 + 76578
$\nu^{12}$	$=$	$ 1110603 \beta_{12} + 302889 \beta_{11} + 336235 \beta_{10} - 607559 \beta_{9} - 651855 \beta_{8} + \cdots + 6052913 $ 1110603b12 + 302889b11 + 336235b10 - 607559b9 - 651855b8 - 610351b7 + 637084b6 + 2278899b5 + 3281157b4 + 429363b3 + 229116b2 - 708671b1 + 6052913

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

4.19382
3.33953
3.18568
2.58186
1.70551
1.17142
0.681990
0.502643
−0.515478
−1.28478
−2.75362
−3.42451
−4.38407

−1.00000

1.00000

−4.19382

0.923616

−1.00000

4.19382

1.2

−1.00000

1.00000

−3.33953

−2.30049

−1.00000

3.33953

1.3

−1.00000

1.00000

−3.18568

2.09569

−1.00000

3.18568

1.4

−1.00000

1.00000

−2.58186

4.73956

−1.00000

2.58186

1.5

−1.00000

1.00000

−1.70551

3.49407

−1.00000

1.70551

1.6

−1.00000

1.00000

−1.17142

−4.65932

−1.00000

1.17142

1.7

−1.00000

1.00000

−0.681990

−1.45298

−1.00000

0.681990

1.8

−1.00000

1.00000

−0.502643

3.57806

−1.00000

0.502643

1.9

−1.00000

1.00000

0.515478

−4.54762

−1.00000

−0.515478

1.10

−1.00000

1.00000

1.28478

−0.550024

−1.00000

−1.28478

1.11

−1.00000

1.00000

2.75362

4.94283

−1.00000

−2.75362

1.12

−1.00000

1.00000

3.42451

1.03411

−1.00000

−3.42451

1.13

−1.00000

1.00000

4.38407

−1.29752

−1.00000

−4.38407

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$269$	$ -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	4842.2.a.t	✓	13
3.b	odd	2	1		4842.2.a.u	yes	13

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4842.2.a.t	✓	13	1.a	even	1	1	trivial
4842.2.a.u	yes	13	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4842))$:

$ T_{5}^{13} + 5 T_{5}^{12} - 33 T_{5}^{11} - 200 T_{5}^{10} + 236 T_{5}^{9} + 2569 T_{5}^{8} + \cdots - 2160 $

$ T_{7}^{13} - 6 T_{7}^{12} - 47 T_{7}^{11} + 316 T_{7}^{10} + 632 T_{7}^{9} - 5514 T_{7}^{8} + \cdots - 29632 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T + 1)^{13} $ (T + 1)^13
$3$	$ T^{13} $ T^13
$5$	$ T^{13} + 5 T^{12} + \cdots - 2160 $ T^13 + 5T^12 - 33T^11 - 200T^10 + 236T^9 + 2569T^8 + 1311T^7 - 11583T^6 - 15582T^5 + 10013T^4 + 24596T^3 + 6260T^2 - 5472T - 2160
$7$	$ T^{13} - 6 T^{12} + \cdots - 29632 $ T^13 - 6T^12 - 47T^11 + 316T^10 + 632T^9 - 5514T^8 - 1536T^7 + 36488T^6 - 6718T^5 - 98326T^4 + 22415T^3 + 101148T^2 - 18976T - 29632
$11$	$ T^{13} + 5 T^{12} + \cdots + 927 $ T^13 + 5T^12 - 57T^11 - 260T^10 + 1190T^9 + 4387T^8 - 12486T^7 - 28782T^6 + 65294T^5 + 57550T^4 - 113681T^3 - 26235T^2 + 33972T + 927
$13$	$ T^{13} - 8 T^{12} + \cdots - 5982080 $ T^13 - 8T^12 - 79T^11 + 742T^10 + 1964T^9 - 25818T^8 - 8943T^7 + 416982T^6 - 286728T^5 - 3095656T^4 + 3547536T^3 + 8662688T^2 - 9351424T - 5982080
$17$	$ T^{13} + 7 T^{12} + \cdots - 48 $ T^13 + 7T^12 - 73T^11 - 502T^10 + 1396T^9 + 9891T^8 - 4883T^7 - 60901T^6 - 37180T^5 + 28277T^4 + 17052T^3 - 2540T^2 - 1152T - 48
$19$	$ T^{13} - 13 T^{12} + \cdots + 57304240 $ T^13 - 13T^12 - 63T^11 + 1452T^10 - 1516T^9 - 54003T^8 + 173823T^7 + 722301T^6 - 3660612T^5 - 1824481T^4 + 26088772T^3 - 17415816T^2 - 52716176T + 57304240
$23$	$ T^{13} - 4 T^{12} + \cdots + 4992 $ T^13 - 4T^12 - 114T^11 + 318T^10 + 4285T^9 - 6612T^8 - 57153T^7 + 29216T^6 + 272440T^5 + 93752T^4 - 174304T^3 - 51680T^2 + 30528T + 4992
$29$	$ T^{13} + 22 T^{12} + \cdots + 22591272 $ T^13 + 22T^12 + 11T^11 - 2628T^10 - 11730T^9 + 106588T^8 + 680228T^7 - 1552808T^6 - 14267556T^5 + 1439136T^4 + 105149289T^3 + 81856646T^2 - 107121480T + 22591272
$31$	$ T^{13} - 8 T^{12} + \cdots - 6792584 $ T^13 - 8T^12 - 106T^11 + 902T^10 + 3772T^9 - 37468T^8 - 46380T^7 + 708604T^6 - 70121T^5 - 6152730T^4 + 4731379T^3 + 20375636T^2 - 21309060T - 6792584
$37$	$ T^{13} - 36 T^{12} + \cdots + 61649536 $ T^13 - 36T^12 + 412T^11 - 524T^10 - 20401T^9 + 98094T^8 + 276709T^7 - 2228474T^6 - 763996T^5 + 18070736T^4 - 3223408T^3 - 57343808T^2 + 11016576T + 61649536
$41$	$ T^{13} + \cdots + 509491560 $ T^13 + 16T^12 - 195T^11 - 3590T^10 + 12324T^9 + 274272T^8 - 387794T^7 - 8859172T^6 + 10398520T^5 + 120247142T^4 - 156918119T^3 - 556053988T^2 + 659228124T + 509491560
$43$	$ T^{13} + \cdots + 455294232 $ T^13 - 14T^12 - 154T^11 + 2282T^10 + 10082T^9 - 141128T^8 - 392514T^7 + 4042456T^6 + 9887097T^5 - 50487662T^4 - 138132781T^3 + 164485628T^2 + 678403588T + 455294232
$47$	$ T^{13} + \cdots - 324618624 $ T^13 + 4T^12 - 326T^11 - 684T^10 + 38405T^9 + 14090T^8 - 1955917T^7 + 1595130T^6 + 42539296T^5 - 65448560T^4 - 325113952T^3 + 641667072T^2 + 114746304T - 324618624
$53$	$ T^{13} + 21 T^{12} + \cdots + 117504 $ T^13 + 21T^12 - 52T^11 - 3113T^10 - 5519T^9 + 140725T^8 + 220391T^7 - 2463468T^6 + 427942T^5 + 4979815T^4 - 393704T^3 - 1877040T^2 - 114048T + 117504
$59$	$ T^{13} - 4 T^{12} + \cdots + 9122304 $ T^13 - 4T^12 - 335T^11 + 2060T^10 + 29708T^9 - 273208T^8 - 98933T^7 + 7743004T^6 - 26844676T^5 + 11079272T^4 + 104576320T^3 - 198990592T^2 + 100596480T + 9122304
$61$	$ T^{13} + \cdots - 1266383872 $ T^13 - 8T^12 - 482T^11 + 3254T^10 + 86271T^9 - 406616T^8 - 7623691T^7 + 15811264T^6 + 345073388T^5 + 225998368T^4 - 6173700464T^3 - 19367101120T^2 - 16531115520T - 1266383872
$67$	$ T^{13} - 40 T^{12} + \cdots + 136808 $ T^13 - 40T^12 + 484T^11 - 200T^10 - 34498T^9 + 203536T^8 + 7242T^7 - 3135064T^6 + 9812631T^5 - 11988810T^4 + 5234091T^3 + 894442T^2 - 1129960T + 136808
$71$	$ T^{13} + \cdots + 12272931297 $ T^13 + 7T^12 - 534T^11 - 2801T^10 + 102274T^9 + 285470T^8 - 9250926T^7 - 1159690T^6 + 382067753T^5 - 876102901T^4 - 4506750733T^3 + 21118613370T^2 - 28641932442T + 12272931297
$73$	$ T^{13} + \cdots - 967485841 $ T^13 - 31T^12 + 58T^11 + 5855T^10 - 41154T^9 - 223004T^8 + 1909538T^7 + 4319090T^6 - 33465809T^5 - 54343509T^4 + 249784201T^3 + 384909492T^2 - 626887560T - 967485841
$79$	$ T^{13} + \cdots - 8777596642736 $ T^13 - 43T^12 + 50T^11 + 20051T^10 - 179345T^9 - 3472133T^8 + 43261977T^7 + 284862582T^6 - 4256184188T^5 - 12868870975T^4 + 189683660784T^3 + 427160509876T^2 - 3276965248496T - 8777596642736
$83$	$ T^{13} + \cdots - 47304380544 $ T^13 - 10T^12 - 639T^11 + 4926T^10 + 165534T^9 - 858664T^8 - 21778997T^7 + 57984310T^6 + 1462532052T^5 - 549543680T^4 - 41348583792T^3 - 58122572608T^2 + 128182474752T - 47304380544
$89$	$ T^{13} + \cdots - 947117568 $ T^13 - 2T^12 - 596T^11 + 2716T^10 + 117342T^9 - 708016T^8 - 8520812T^7 + 50410686T^6 + 287027369T^5 - 1247814004T^4 - 4192592959T^3 + 6405363184T^2 + 2444671296T - 947117568
$97$	$ T^{13} + \cdots - 305719279329 $ T^13 - 27T^12 - 318T^11 + 13519T^10 - 18778T^9 - 1909176T^8 + 7515118T^7 + 117218094T^6 - 452737705T^5 - 3879655829T^4 + 9204866337T^3 + 65037124704T^2 - 33002448580T - 305719279329
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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:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\( q - q^{2} + q^{4} - \beta_1 q^{5} - \beta_{3} q^{7} - q^{8} + \beta_1 q^{10} - \beta_{10} q^{11} + (\beta_{2} + 1) q^{13} + \beta_{3} q^{14} + q^{16} + ( - \beta_{8} - 1) q^{17} + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots + 2) q^{19}+ \cdots + (\beta_{12} + 2 \beta_{10} - \beta_{9} + \cdots - 4) q^{98}+O(q^{100}) \) q - q^2 + q^4 - b1 * q^5 - b3 * q^7 - q^8 + b1 * q^10 - b10 * q^11 + (b2 + 1) * q^13 + b3 * q^14 + q^16 + (-b8 - 1) * q^17 + (b11 - b10 - b9 - b4 + 2) * q^19 - b1 * q^20 + b10 * q^22 + b5 * q^23 + (b12 - b9 + b5 + b4 + 2) * q^25 + (-b2 - 1) * q^26 - b3 * q^28 + (b12 + b11 - b10 - b7 - b5 - b3 - b1) * q^29 + (b9 - b8) * q^31 - q^32 + (b8 + 1) * q^34 + (-b9 + b8 + b7 - b4 + b3 + b2 - b1 + 1) * q^35 + (-b12 - b11 + b9 + 2) * q^37 + (-b11 + b10 + b9 + b4 - 2) * q^38 + b1 * q^40 + (-b12 + b10 - b9 + b7 + b5 + b1 - 3) * q^41 + (-b7 - b5 - b3 - b2 + 1) * q^43 - b10 * q^44 - b5 * q^46 + (-b11 + b9 - b7 + b6 - b5 + b4 - b3 + b2 - b1) * q^47 + (-b12 - 2b10 + b9 + b8 + b7 - 2b5 - b4 - b3 - b1 + 4) * q^49 + (-b12 + b9 - b5 - b4 - 2) * q^50 + (b2 + 1) * q^52 + (-b12 + b9 - b6 - 2) * q^53 + (-b11 + 2b10 + b9 - b8 + b7 - b6 + b5 + b4 + b3 - 2) q^55 + b3 * q^56 + (-b12 - b11 + b10 + b7 + b5 + b3 + b1) * q^58 + (b10 - b9 - b6 + b4) * q^59 + (-b11 + b10 - b7 + 2b6 + b5 - b3 + b1 - 1) q^61 + (-b9 + b8) * q^62 + q^64 + (-b10 + b8 + b7 - b6 - b5 - 2b4 - b3 - b2 - b1 - 1) q^65 + (b11 - b10 - b5 - b2 - b1 + 4) * q^67 + (-b8 - 1) * q^68 + (b9 - b8 - b7 + b4 - b3 - b2 + b1 - 1) * q^70 + (-b11 + 2b10 + 2b9 - b7 + b6 + b5 + 2b4 - b3 + b2 - b1 - 2) q^71 + (b12 + b11 + b8 + b4 - 2b1 + 4) q^73 + (b12 + b11 - b9 - 2) * q^74 + (b11 - b10 - b9 - b4 + 2) * q^76 + (b12 + b11 - b9 + b6 + b5 - b4 - b2 + b1 - 1) * q^77 + (-b10 - b9 + b8 - 2b6 - b2 - 2b1 + 5) * q^79 - b1 * q^80 + (b12 - b10 + b9 - b7 - b5 - b1 + 3) * q^82 + (b12 + b11 - b10 - b9 + b8 + b7 + b6 - b5 - b3 - b1 + 2) * q^83 + (b12 - b11 - b8 - b7 - b6 + b5 + b3 - b2 + b1 + 2) * q^85 + (b7 + b5 + b3 + b2 - 1) * q^86 + b10 * q^88 + (b12 + b11 + b8 - 2b7 + 2b6 - b5 - 2b3 - 2b1 + 2) * q^89 + (-2b11 + 3b10 + b9 + 3b6 + b4 - 2b3 + 2b2 - 2b1) * q^91 + b5 * q^92 + (b11 - b9 + b7 - b6 + b5 - b4 + b3 - b2 + b1) * q^94 + (-2b11 + 2b10 - b6 - b4 - 1) * q^95 + (-b12 + b11 - 2b10 - 2b9 + b8 + 2b7 - b6 + b5 - 2b4 + 2b3 - b2 + b1 + 3) q^97 + (b12 + 2b10 - b9 - b8 - b7 + 2b5 + b4 + b3 + b1 - 4) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} + 13 q^{4} - 5 q^{5} + 6 q^{7} - 13 q^{8} + 5 q^{10} - 5 q^{11} + 8 q^{13} - 6 q^{14} + 13 q^{16} - 7 q^{17} + 13 q^{19} - 5 q^{20} + 5 q^{22} + 4 q^{23} + 26 q^{25} - 8 q^{26} + 6 q^{28}+ \cdots - 39 q^{98}+O(q^{100}) \) 13 * q - 13 * q^2 + 13 * q^4 - 5 * q^5 + 6 * q^7 - 13 * q^8 + 5 * q^10 - 5 * q^11 + 8 * q^13 - 6 * q^14 + 13 * q^16 - 7 * q^17 + 13 * q^19 - 5 * q^20 + 5 * q^22 + 4 * q^23 + 26 * q^25 - 8 * q^26 + 6 * q^28 - 22 * q^29 + 8 * q^31 - 13 * q^32 + 7 * q^34 - 8 * q^35 + 36 * q^37 - 13 * q^38 + 5 * q^40 - 16 * q^41 + 14 * q^43 - 5 * q^44 - 4 * q^46 - 4 * q^47 + 39 * q^49 - 26 * q^50 + 8 * q^52 - 21 * q^53 - 6 * q^56 + 22 * q^58 + 4 * q^59 + 8 * q^61 - 8 * q^62 + 13 * q^64 - 24 * q^65 + 40 * q^67 - 7 * q^68 + 8 * q^70 - 7 * q^71 + 31 * q^73 - 36 * q^74 + 13 * q^76 - 10 * q^77 + 43 * q^79 - 5 * q^80 + 16 * q^82 + 10 * q^83 + 30 * q^85 - 14 * q^86 + 5 * q^88 + 2 * q^89 + 24 * q^91 + 4 * q^92 + 4 * q^94 - 2 * q^95 + 27 * q^97 - 39 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	4842.2.a.t

Level	$4842$
Weight	$2$
Character orbit	4842.a
Self dual	yes
Analytic conductor	$38.664$
Analytic rank	$0$
Dimension	$13$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(38.6635646587\)
Analytic rank:	\(0\)
Dimension:	\(13\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{13} - 5 x^{12} - 33 x^{11} + 200 x^{10} + 236 x^{9} - 2569 x^{8} + 1311 x^{7} + 11583 x^{6} + \cdots + 2160 \) x^13 - 5x^12 - 33x^11 + 200x^10 + 236x^9 - 2569x^8 + 1311x^7 + 11583x^6 - 15582x^5 - 10013x^4 + 24596x^3 - 6260x^2 - 5472x + 2160
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 48024874805 \nu^{12} - 112287409573 \nu^{11} - 1941343452888 \nu^{10} + \cdots - 283838423867382 ) / 124747362130182 \) (48024874805v^12 - 112287409573v^11 - 1941343452888v^10 + 4661678813218v^9 + 25746664037524v^8 - 64568707789679v^7 - 124740902028006v^6 + 366728280584655v^5 + 132349851534843v^4 - 924186446242159v^3 + 165915501592513v^2 + 1231589887634822v - 283838423867382) / 124747362130182
\(\beta_{3}\)	\(=\)	\( ( 852354528985 \nu^{12} - 5565998476235 \nu^{11} - 24931795402263 \nu^{10} + \cdots - 71\!\cdots\!68 ) / 498989448520728 \) (852354528985v^12 - 5565998476235v^11 - 24931795402263v^10 + 222999309674426v^9 + 67809369284036v^8 - 2889498022765681v^7 + 2936240649919401v^6 + 13437755317500921v^5 - 22501654361868084v^4 - 14476242953328965v^3 + 33905371325920238v^2 - 790543441394096v - 7135321253071368) / 498989448520728
\(\beta_{4}\)	\(=\)	\( ( - 1462385429459 \nu^{12} + 5438514414121 \nu^{11} + 55460849135577 \nu^{10} + \cdots + 726463997646072 ) / 498989448520728 \) (-1462385429459v^12 + 5438514414121v^11 + 55460849135577v^10 - 223613158488046v^9 - 638992003731088v^8 + 3023577814303631v^7 + 2002626917600661v^6 - 15457619371574655v^5 + 3319327641718824v^4 + 24015438167285407v^3 - 8450625764052706v^2 - 8414011116504308v + 726463997646072) / 498989448520728
\(\beta_{5}\)	\(=\)	\( ( - 868230374567 \nu^{12} + 4292066532829 \nu^{11} + 29608543880865 \nu^{10} + \cdots + 45\!\cdots\!64 ) / 249494724260364 \) (-868230374567v^12 + 4292066532829v^11 + 29608543880865v^10 - 172210781998990v^9 - 242022329130154v^8 + 2231221511993039v^7 - 673104802752525v^6 - 10347700457769675v^5 + 11446839915598068v^4 + 10942647896603347v^3 - 19205954143548196v^2 + 1141012497025972v + 4525884532262664) / 249494724260364
\(\beta_{6}\)	\(=\)	\( ( - 630139679479 \nu^{12} + 2701324988981 \nu^{11} + 22362606192965 \nu^{10} + \cdots + 36\!\cdots\!24 ) / 166329816173576 \) (-630139679479v^12 + 2701324988981v^11 + 22362606192965v^10 - 110188878729306v^9 - 212412736035108v^8 + 1469426488671711v^7 + 8099800770765v^6 - 7291875723961147v^5 + 5911876575648176v^4 + 10367983048606231v^3 - 11323162532461150v^2 - 3433794836880676v + 3638197188330624) / 166329816173576
\(\beta_{7}\)	\(=\)	\( ( - 2000957877725 \nu^{12} + 7635503796289 \nu^{11} + 71656431054273 \nu^{10} + \cdots + 84\!\cdots\!24 ) / 498989448520728 \) (-2000957877725v^12 + 7635503796289v^11 + 71656431054273v^10 - 312948904996228v^9 - 704096950929424v^8 + 4189399857925685v^7 + 444893315979717v^6 - 20821591731301911v^5 + 16637899023589626v^4 + 29137144742255173v^3 - 32407998154263220v^2 - 7321545315875216v + 8429509843089624) / 498989448520728
\(\beta_{8}\)	\(=\)	\( ( 647904046508 \nu^{12} - 2958462557782 \nu^{11} - 22330056820125 \nu^{10} + \cdots - 31\!\cdots\!74 ) / 124747362130182 \) (647904046508v^12 - 2958462557782v^11 - 22330056820125v^10 + 119976211894435v^9 + 192212383085599v^8 - 1583507423251541v^7 + 320616638739144v^6 + 7674059886042198v^5 - 7502314288114329v^4 - 9916668415011091v^3 + 12693921759640060v^2 + 1750282976092010v - 3142157756066574) / 124747362130182
\(\beta_{9}\)	\(=\)	\( ( 4401723597281 \nu^{12} - 18847913017729 \nu^{11} - 158350218453045 \nu^{10} + \cdots - 10\!\cdots\!12 ) / 498989448520728 \) (4401723597281v^12 - 18847913017729v^11 - 158350218453045v^10 + 765561425553280v^9 + 1574339848381408v^8 - 10134784365728573v^7 - 1402922813124657v^6 + 49457201715504891v^5 - 32721323671713066v^4 - 65658815154377377v^3 + 56842152070588012v^2 + 14201661636087044v - 10796184896235312) / 498989448520728
\(\beta_{10}\)	\(=\)	\( ( 6609202784399 \nu^{12} - 27174425518447 \nu^{11} - 241966416215319 \nu^{10} + \cdots - 18\!\cdots\!48 ) / 498989448520728 \) (6609202784399v^12 - 27174425518447v^11 - 241966416215319v^10 + 1105175609150380v^9 + 2528452212144112v^8 - 14665541151353999v^7 - 4151436180006291v^6 + 71978940006584853v^5 - 40440321513996066v^4 - 97437878729532859v^3 + 79210205380248472v^2 + 20306966993208644v - 18201659224281048) / 498989448520728
\(\beta_{11}\)	\(=\)	\( ( 2233298029201 \nu^{12} - 9837337220151 \nu^{11} - 80661040634447 \nu^{10} + \cdots - 70\!\cdots\!48 ) / 166329816173576 \) (2233298029201v^12 - 9837337220151v^11 - 80661040634447v^10 + 399969919876814v^9 + 808935967676248v^8 - 5312697579214149v^7 - 799851335767539v^6 + 26171112127655033v^5 - 16573051963403412v^4 - 36020155348323261v^3 + 30470208695623234v^2 + 8635958312368788v - 7065047705357648) / 166329816173576
\(\beta_{12}\)	\(=\)	\( ( 3800284887937 \nu^{12} - 16435280248754 \nu^{11} - 136514077675176 \nu^{10} + \cdots - 12\!\cdots\!04 ) / 249494724260364 \) (3800284887937v^12 - 16435280248754v^11 - 136514077675176v^10 + 666798074019653v^9 + 1348688255186402v^8 - 8810402602009141v^7 - 1029670062610134v^6 + 42805111001309448v^5 - 29467165572314013v^4 - 55779774557434739v^3 + 52101837785128919v^2 + 10166823879269704v - 12033672049025904) / 249494724260364

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{12} - \beta_{9} + \beta_{5} + \beta_{4} + 7 \) b12 - b9 + b5 + b4 + 7
\(\nu^{3}\)	\(=\)	\( \beta_{12} + \beta_{11} - \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - \beta_{7} + 3 \beta_{6} + \cdots + 12 \beta_1 \) b12 + b11 - b10 - 3b9 + 2b8 - b7 + 3b6 - b5 - 2b4 - b3 + b2 + 12*b1
\(\nu^{4}\)	\(=\)	\( 16 \beta_{12} + \beta_{11} + \beta_{10} - 14 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + \cdots + 88 \) 16b12 + b11 + b10 - 14b9 - 3b8 - 3b7 + 4b6 + 20b5 + 22b4 + 3b3 + 2*b2 - b1 + 88
\(\nu^{5}\)	\(=\)	\( 23 \beta_{12} + 27 \beta_{11} - 34 \beta_{10} - 64 \beta_{9} + 41 \beta_{8} - 27 \beta_{7} + 67 \beta_{6} + \cdots + 12 \) 23b12 + 27b11 - 34b10 - 64b9 + 41b8 - 27b7 + 67b6 - 34b5 - 52b4 - 29b3 + 35b2 + 165b1 + 12
\(\nu^{6}\)	\(=\)	\( 248 \beta_{12} + 35 \beta_{11} + 30 \beta_{10} - 196 \beta_{9} - 82 \beta_{8} - 84 \beta_{7} + \cdots + 1293 \) 248b12 + 35b11 + 30b10 - 196b9 - 82b8 - 84b7 + 100b6 + 364b5 + 449b4 + 68b3 + 48b2 - 43b1 + 1293
\(\nu^{7}\)	\(=\)	\( 459 \beta_{12} + 587 \beta_{11} - 828 \beta_{10} - 1208 \beta_{9} + 733 \beta_{8} - 601 \beta_{7} + \cdots + 306 \) 459b12 + 587b11 - 828b10 - 1208b9 + 733b8 - 601b7 + 1293b6 - 848b5 - 1134b4 - 669b3 + 779b2 + 2492b1 + 306
\(\nu^{8}\)	\(=\)	\( 3971 \beta_{12} + 827 \beta_{11} + 690 \beta_{10} - 2813 \beta_{9} - 1728 \beta_{8} - 1770 \beta_{7} + \cdots + 20654 \) 3971b12 + 827b11 + 690b10 - 2813b9 - 1728b8 - 1770b7 + 1994b6 + 6623b5 + 8844b4 + 1254b3 + 886b2 - 1257b1 + 20654
\(\nu^{9}\)	\(=\)	\( 8680 \beta_{12} + 11760 \beta_{11} - 17751 \beta_{10} - 22241 \beta_{9} + 12943 \beta_{8} - 12250 \beta_{7} + \cdots + 5438 \) 8680b12 + 11760b11 - 17751b10 - 22241b9 + 12943b8 - 12250b7 + 23968b6 - 18605b5 - 23480b4 - 13934b3 + 15318b2 + 40428b1 + 5438
\(\nu^{10}\)	\(=\)	\( 65619 \beta_{12} + 16566 \beta_{11} + 15253 \beta_{10} - 41161 \beta_{9} - 33879 \beta_{8} + \cdots + 347744 \) 65619b12 + 16566b11 + 15253b10 - 41161b9 - 33879b8 - 33681b7 + 36500b6 + 122103b5 + 171110b4 + 22841b3 + 14740b2 - 31132b1 + 347744
\(\nu^{11}\)	\(=\)	\( 159165 \beta_{12} + 225687 \beta_{11} - 357471 \beta_{10} - 406499 \beta_{9} + 231564 \beta_{8} + \cdots + 76578 \) 159165b12 + 225687b11 - 357471b10 - 406499b9 + 231564b8 - 237723b7 + 437921b6 - 384669b5 - 474590b4 - 276001b3 + 288293b2 + 690021b1 + 76578
\(\nu^{12}\)	\(=\)	\( 1110603 \beta_{12} + 302889 \beta_{11} + 336235 \beta_{10} - 607559 \beta_{9} - 651855 \beta_{8} + \cdots + 6052913 \) 1110603b12 + 302889b11 + 336235b10 - 607559b9 - 651855b8 - 610351b7 + 637084b6 + 2278899b5 + 3281157b4 + 429363b3 + 229116b2 - 708671b1 + 6052913

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.13
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T + 1)^{13} \) (T + 1)^13
$3$	\( T^{13} \) T^13
$5$	\( T^{13} + 5 T^{12} + \cdots - 2160 \) T^13 + 5T^12 - 33T^11 - 200T^10 + 236T^9 + 2569T^8 + 1311T^7 - 11583T^6 - 15582T^5 + 10013T^4 + 24596T^3 + 6260T^2 - 5472T - 2160
$7$	\( T^{13} - 6 T^{12} + \cdots - 29632 \) T^13 - 6T^12 - 47T^11 + 316T^10 + 632T^9 - 5514T^8 - 1536T^7 + 36488T^6 - 6718T^5 - 98326T^4 + 22415T^3 + 101148T^2 - 18976T - 29632
$11$	\( T^{13} + 5 T^{12} + \cdots + 927 \) T^13 + 5T^12 - 57T^11 - 260T^10 + 1190T^9 + 4387T^8 - 12486T^7 - 28782T^6 + 65294T^5 + 57550T^4 - 113681T^3 - 26235T^2 + 33972T + 927
$13$	\( T^{13} - 8 T^{12} + \cdots - 5982080 \) T^13 - 8T^12 - 79T^11 + 742T^10 + 1964T^9 - 25818T^8 - 8943T^7 + 416982T^6 - 286728T^5 - 3095656T^4 + 3547536T^3 + 8662688T^2 - 9351424T - 5982080
$17$	\( T^{13} + 7 T^{12} + \cdots - 48 \) T^13 + 7T^12 - 73T^11 - 502T^10 + 1396T^9 + 9891T^8 - 4883T^7 - 60901T^6 - 37180T^5 + 28277T^4 + 17052T^3 - 2540T^2 - 1152T - 48
$19$	\( T^{13} - 13 T^{12} + \cdots + 57304240 \) T^13 - 13T^12 - 63T^11 + 1452T^10 - 1516T^9 - 54003T^8 + 173823T^7 + 722301T^6 - 3660612T^5 - 1824481T^4 + 26088772T^3 - 17415816T^2 - 52716176T + 57304240
$23$	\( T^{13} - 4 T^{12} + \cdots + 4992 \) T^13 - 4T^12 - 114T^11 + 318T^10 + 4285T^9 - 6612T^8 - 57153T^7 + 29216T^6 + 272440T^5 + 93752T^4 - 174304T^3 - 51680T^2 + 30528T + 4992
$29$	\( T^{13} + 22 T^{12} + \cdots + 22591272 \) T^13 + 22T^12 + 11T^11 - 2628T^10 - 11730T^9 + 106588T^8 + 680228T^7 - 1552808T^6 - 14267556T^5 + 1439136T^4 + 105149289T^3 + 81856646T^2 - 107121480T + 22591272
$31$	\( T^{13} - 8 T^{12} + \cdots - 6792584 \) T^13 - 8T^12 - 106T^11 + 902T^10 + 3772T^9 - 37468T^8 - 46380T^7 + 708604T^6 - 70121T^5 - 6152730T^4 + 4731379T^3 + 20375636T^2 - 21309060T - 6792584
$37$	\( T^{13} - 36 T^{12} + \cdots + 61649536 \) T^13 - 36T^12 + 412T^11 - 524T^10 - 20401T^9 + 98094T^8 + 276709T^7 - 2228474T^6 - 763996T^5 + 18070736T^4 - 3223408T^3 - 57343808T^2 + 11016576T + 61649536
$41$	\( T^{13} + \cdots + 509491560 \) T^13 + 16T^12 - 195T^11 - 3590T^10 + 12324T^9 + 274272T^8 - 387794T^7 - 8859172T^6 + 10398520T^5 + 120247142T^4 - 156918119T^3 - 556053988T^2 + 659228124T + 509491560
$43$	\( T^{13} + \cdots + 455294232 \) T^13 - 14T^12 - 154T^11 + 2282T^10 + 10082T^9 - 141128T^8 - 392514T^7 + 4042456T^6 + 9887097T^5 - 50487662T^4 - 138132781T^3 + 164485628T^2 + 678403588T + 455294232
$47$	\( T^{13} + \cdots - 324618624 \) T^13 + 4T^12 - 326T^11 - 684T^10 + 38405T^9 + 14090T^8 - 1955917T^7 + 1595130T^6 + 42539296T^5 - 65448560T^4 - 325113952T^3 + 641667072T^2 + 114746304T - 324618624
$53$	\( T^{13} + 21 T^{12} + \cdots + 117504 \) T^13 + 21T^12 - 52T^11 - 3113T^10 - 5519T^9 + 140725T^8 + 220391T^7 - 2463468T^6 + 427942T^5 + 4979815T^4 - 393704T^3 - 1877040T^2 - 114048T + 117504
$59$	\( T^{13} - 4 T^{12} + \cdots + 9122304 \) T^13 - 4T^12 - 335T^11 + 2060T^10 + 29708T^9 - 273208T^8 - 98933T^7 + 7743004T^6 - 26844676T^5 + 11079272T^4 + 104576320T^3 - 198990592T^2 + 100596480T + 9122304
$61$	\( T^{13} + \cdots - 1266383872 \) T^13 - 8T^12 - 482T^11 + 3254T^10 + 86271T^9 - 406616T^8 - 7623691T^7 + 15811264T^6 + 345073388T^5 + 225998368T^4 - 6173700464T^3 - 19367101120T^2 - 16531115520T - 1266383872
$67$	\( T^{13} - 40 T^{12} + \cdots + 136808 \) T^13 - 40T^12 + 484T^11 - 200T^10 - 34498T^9 + 203536T^8 + 7242T^7 - 3135064T^6 + 9812631T^5 - 11988810T^4 + 5234091T^3 + 894442T^2 - 1129960T + 136808
$71$	\( T^{13} + \cdots + 12272931297 \) T^13 + 7T^12 - 534T^11 - 2801T^10 + 102274T^9 + 285470T^8 - 9250926T^7 - 1159690T^6 + 382067753T^5 - 876102901T^4 - 4506750733T^3 + 21118613370T^2 - 28641932442T + 12272931297
$73$	\( T^{13} + \cdots - 967485841 \) T^13 - 31T^12 + 58T^11 + 5855T^10 - 41154T^9 - 223004T^8 + 1909538T^7 + 4319090T^6 - 33465809T^5 - 54343509T^4 + 249784201T^3 + 384909492T^2 - 626887560T - 967485841
$79$	\( T^{13} + \cdots - 8777596642736 \) T^13 - 43T^12 + 50T^11 + 20051T^10 - 179345T^9 - 3472133T^8 + 43261977T^7 + 284862582T^6 - 4256184188T^5 - 12868870975T^4 + 189683660784T^3 + 427160509876T^2 - 3276965248496T - 8777596642736
$83$	\( T^{13} + \cdots - 47304380544 \) T^13 - 10T^12 - 639T^11 + 4926T^10 + 165534T^9 - 858664T^8 - 21778997T^7 + 57984310T^6 + 1462532052T^5 - 549543680T^4 - 41348583792T^3 - 58122572608T^2 + 128182474752T - 47304380544
$89$	\( T^{13} + \cdots - 947117568 \) T^13 - 2T^12 - 596T^11 + 2716T^10 + 117342T^9 - 708016T^8 - 8520812T^7 + 50410686T^6 + 287027369T^5 - 1247814004T^4 - 4192592959T^3 + 6405363184T^2 + 2444671296T - 947117568
$97$	\( T^{13} + \cdots - 305719279329 \) T^13 - 27T^12 - 318T^11 + 13519T^10 - 18778T^9 - 1909176T^8 + 7515118T^7 + 117218094T^6 - 452737705T^5 - 3879655829T^4 + 9204866337T^3 + 65037124704T^2 - 33002448580T - 305719279329
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\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( +1 \)
\(269\)	\( -1 \)