LMFDB - Newform orbit 1078.2.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1077,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1078.1077");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.c (of order $2$, degree $1$, 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:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 $ x^16 - 6x^15 + 18x^14 - 36x^13 + 34x^12 + 18x^11 - 72x^10 + 132x^9 - 93x^8 - 102x^7 + 144x^6 - 432x^5 + 502x^4 + 288x^3 + 72x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{9} q^{2} + \beta_{14} q^{3} - q^{4} + (\beta_{11} - \beta_{8}) q^{5} - \beta_{13} q^{6} - \beta_{9} q^{8} + ( - \beta_{2} - 2) q^{9} + (\beta_{10} + \beta_{6}) q^{10} + ( - \beta_{12} + \beta_{9} - \beta_{3} - 1) q^{11}+ \cdots + ( - \beta_{15} + 5 \beta_{12} + \cdots + 4) q^{99}+O(q^{100}) $ q + b9 * q^2 + b14 * q^3 - q^4 + (b11 - b8) * q^5 - b13 * q^6 - b9 * q^8 + (-b2 - 2) * q^9 + (b10 + b6) * q^10 + (-b12 + b9 - b3 - 1) * q^11 - b14 * q^12 + (-b10 - b4) * q^13 + (-3b1 - 2) q^15 + q^16 + (-b13 - 2b10 - b6 + 2b4) * q^17 + (-b15 - 2b9) q^18 + (-b13 + b10 + 2b6 - b4) q^19 + (-b11 + b8) * q^20 + (-b9 + b7 + b1) * q^22 + (-b3 + b2 - 2) * q^23 + b13 * q^24 + (-b2 - 2b1 - 1) q^25 + (b11 + b5) * q^26 + (-b14 + 4b11 - 2b8 - 3b5) q^27 + (b12 - 2b9 + b7) q^29 + (-3b12 + b9) q^30 + (-b14 + 2b11 - 3b8 - 2b5) q^31 + b9 * q^32 + (-b14 - b13 + b11 - 3b10 + b8 - 2b6 - b5) * q^33 + (-b14 + 2b11 - b8 - 2b5) * q^34 + (b2 + 2) * q^36 + (-b3 + 2) * q^37 + (-b14 - b11 + 2b8 + b5) q^38 + (b15 + 4b12 - b9 + b7) q^39 + (-b10 - b6) * q^40 + (-3b13 + b10 - 2b4) * q^41 + (b15 + 3b12 + b9 + 3b7) * q^43 + (b12 - b9 + b3 + 1) * q^44 + (b14 - 6b11 + 3b8) * q^45 + (b15 - 2b9 + b7) q^46 - 2b5 q^47 + b14 * q^48 + (-b15 - 2b12 + b9) q^50 + (-3b15 + b12 - 7b9 + b7) * q^51 + (b10 + b4) * q^52 + (-2b3 + b2 + b1 + 4) q^53 + (b13 + 4b10 + 2b6 - 3b4) q^54 + (-b14 + 2b13 - 3b11 + 2b8 + b5 - b4) q^55 + (-2b12 - 4b9 + b7) * q^57 + (b3 - b1 + 1) * q^58 + (3b14 + 2b11 + b5) * q^59 + (3b1 + 2) q^60 + (-b13 - b10 - 3b6 - 2b4) * q^61 + (b13 + 2b10 + 3b6 - 2b4) q^62 - q^64 + (2b15 + 4b12 + 2b9 + b7) q^65 + (-b14 + b13 + 3b11 + b10 - 2b8 - b6 - b4) * q^66 + (2b3 + b1 - 1) q^67 + (b13 + 2b10 + b6 - 2b4) * q^68 + (-3b11 + 3b8 + 2b5) q^69 + (-2b3 - 2b2 - b1) * q^71 + (b15 + 2b9) q^72 + (-5b10 - b6 + 2b4) * q^73 + (2b9 + b7) q^74 + (-b14 - 2b11 + 2b8 - 3b5) q^75 + (b13 - b10 - 2b6 + b4) q^76 + (b3 - b2 - 4b1 - 3) q^78 + (3b15 + 2b12 - 3b9 + b7) q^79 + (b11 - b8) * q^80 + (2b3 + b2 - 4b1 - 1) * q^81 + (-3b14 - b11 + 2b5) * q^82 + (-3b13 + 2b10 - b6 - 2b4) q^83 + (-6b12 + 4b9 - 2b7) q^85 + (3b3 - b2 - 3b1 - 4) * q^86 + (2b13 + 4b10 + b6 - b4) * q^87 + (b9 - b7 - b1) * q^88 + (3b14 - 4b11 - b8 + b5) * q^89 + (-b13 - 6b10 - 3b6) * q^90 + (b3 - b2 + 2) * q^92 + (-b3 + 3b2 - 3b1 + 4) * q^93 - 2b4 q^94 + (-3b12 - 5b9 + b7) * q^95 - b13 * q^96 + (-2b14 + 3b11 - 2b8 + 4b5) * q^97 + (-b15 + 5b12 - 5b9 + b7 - b3 + 2b2 + b1 + 4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 32 q^{9} - 16 q^{11} - 8 q^{15} + 16 q^{16} - 8 q^{22} - 32 q^{23} + 32 q^{36} + 32 q^{37} + 16 q^{44} + 56 q^{53} + 24 q^{58} + 8 q^{60} - 16 q^{64} - 24 q^{67} + 8 q^{71} - 16 q^{78}+ \cdots + 56 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 - 32 * q^9 - 16 * q^11 - 8 * q^15 + 16 * q^16 - 8 * q^22 - 32 * q^23 + 32 * q^36 + 32 * q^37 + 16 * q^44 + 56 * q^53 + 24 * q^58 + 8 * q^60 - 16 * q^64 - 24 * q^67 + 8 * q^71 - 16 * q^78 + 16 * q^81 - 40 * q^86 + 8 * q^88 + 32 * q^92 + 88 * q^93 + 56 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 $ x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 34*x^12 + 18*x^11 - 72*x^10 + 132*x^9 - 93*x^8 - 102*x^7 + 144*x^6 - 432*x^5 + 502*x^4 + 288*x^3 + 72*x^2 + 12*x + 1 :

$\beta_{1}$	$=$	$ ( - 73568941 \nu^{15} - 2828429898 \nu^{14} + 10706428236 \nu^{13} - 20127265496 \nu^{12} + \cdots - 10982429182739 ) / 3707507912227 $ (-73568941v^15 - 2828429898v^14 + 10706428236v^13 - 20127265496v^12 + 35741021050v^11 - 17927851065v^10 + 58772639836v^9 - 308269732922v^8 + 50382616322v^7 - 35812385940v^6 - 245517720058v^5 + 1525793905272v^4 + 747026251899v^3 + 178512937629v^2 + 28424647318*v - 10982429182739) / 3707507912227
$\beta_{2}$	$=$	$ ( - 973484628 \nu^{15} + 16798064640 \nu^{14} - 57412057192 \nu^{13} + 106920971791 \nu^{12} + \cdots + 17707406591301 ) / 3707507912227 $ (-973484628v^15 + 16798064640v^14 - 57412057192v^13 + 106920971791v^12 - 159815035576v^11 + 58809142080v^10 - 208743822856v^9 + 1110018585936v^8 - 324206168264v^7 + 209105071620v^6 + 1201975196104v^5 - 5438378020432v^4 - 2697240781348v^3 - 647205045504v^2 - 103443810800*v + 17707406591301) / 3707507912227
$\beta_{3}$	$=$	$ ( - 4604477275 \nu^{15} + 26065356324 \nu^{14} - 75543186530 \nu^{13} + 146897647868 \nu^{12} + \cdots + 1817177370827 ) / 3707507912227 $ (-4604477275v^15 + 26065356324v^14 - 75543186530v^13 + 146897647868v^12 - 132924225826v^11 - 13305240150v^10 - 16121748394v^9 + 141233089912v^8 - 557871539990v^7 + 316601789265v^6 + 1303176243718v^5 - 490404558308v^4 - 369763065215v^3 - 98321373186v^2 - 17113540344*v + 1817177370827) / 3707507912227
$\beta_{4}$	$=$	$ ( - 275511061668 \nu^{15} + 1657750582996 \nu^{14} - 4984224043414 \nu^{13} + \cdots - 1555184012514 ) / 3707507912227 $ (-275511061668v^15 + 1657750582996v^14 - 4984224043414v^13 + 9981450357924v^12 - 9469500994272v^11 - 4940688808597v^10 + 20095109174076v^9 - 36700380318006v^8 + 25948536491036v^7 + 28399264833487v^6 - 41476514830836v^5 + 119515269818808v^4 - 137980607533908v^3 - 79280602131237v^2 - 12412885174910*v - 1555184012514) / 3707507912227
$\beta_{5}$	$=$	$ ( - 464283294072 \nu^{15} + 2897231896860 \nu^{14} - 9050241140038 \nu^{13} + \cdots - 1395125725490 ) / 3707507912227 $ (-464283294072v^15 + 2897231896860v^14 - 9050241140038v^13 + 18865423115472v^12 - 20243277302808v^11 - 3659327351235v^10 + 34524757354698v^9 - 69586426065626v^8 + 59597927208868v^7 + 33784753952673v^6 - 76446349441200v^5 + 218895832838984v^4 - 286618252654908v^3 - 66966547472787v^2 - 10425321369434*v - 1395125725490) / 3707507912227
$\beta_{6}$	$=$	$ ( 476129408728 \nu^{15} - 2861932186210 \nu^{14} + 8584036352598 \nu^{13} + \cdots + 2687120456379 ) / 3707507912227 $ (476129408728v^15 - 2861932186210v^14 + 8584036352598v^13 - 17121557060007v^12 + 16021649705480v^11 + 9119986400225v^10 - 35176977398796v^9 + 63105295993728v^8 - 43746959330988v^7 - 50114785825646v^6 + 71179229667320v^5 - 205114133957706v^4 + 238865093218054v^3 + 137040061864945v^2 + 21456435065178*v + 2687120456379) / 3707507912227
$\beta_{7}$	$=$	$ ( 770100281810 \nu^{15} - 4733263012640 \nu^{14} + 14562049553051 \nu^{13} + \cdots + 4887214330899 ) / 3707507912227 $ (770100281810v^15 - 4733263012640v^14 + 14562049553051v^13 - 29920520825502v^12 + 30799143245438v^11 + 8829439453946v^10 - 56098808754161v^9 + 110050939842954v^8 - 89459573480504v^7 - 62707181505525v^6 + 117223638772823v^5 - 351034936647114v^4 + 441822991616242v^3 + 152138648268578v^2 + 39169305360240*v + 4887214330899) / 3707507912227
$\beta_{8}$	$=$	$ ( - 805007695745 \nu^{15} + 5029032860754 \nu^{14} - 15724330059984 \nu^{13} + \cdots - 2408034273855 ) / 3707507912227 $ (-805007695745v^15 + 5029032860754v^14 - 15724330059984v^13 + 32793740003767v^12 - 35212340782612v^11 - 6414480346779v^10 + 60443911042560v^9 - 121464878425672v^8 + 103864347546612v^7 + 58759557056628v^6 - 132943428475514v^5 + 378138116252806v^4 - 494704696217447v^3 - 115575276518547v^2 - 17991168996474*v - 2408034273855) / 3707507912227
$\beta_{9}$	$=$	$ ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} + \cdots - 10233974547 ) / 3810388399 $ (-1616321538v^15 + 9938916556v^14 - 30594542475v^13 + 62871582510v^12 - 64714538406v^11 - 18641341380v^10 + 118271636061v^9 - 231117756606v^8 + 186162436800v^7 + 134499720676v^6 - 250255719435v^5 + 736991238906v^4 - 924410484114v^3 - 318408624800v^2 - 82003950372*v - 10233974547) / 3810388399
$\beta_{10}$	$=$	$ ( 854458 \nu^{15} - 5149288 \nu^{14} + 15489103 \nu^{13} - 31002870 \nu^{12} + 29353426 \nu^{11} + \cdots + 4877403 ) / 1828267 $ (854458v^15 - 5149288v^14 + 15489103v^13 - 31002870v^12 + 29353426v^11 + 15677868v^10 - 63062803v^9 + 114195864v^8 - 80446252v^7 - 89178588v^6 + 128785525v^5 - 370469382v^4 + 434157674v^3 + 248804552v^2 + 38956914*v + 4877403) / 1828267
$\beta_{11}$	$=$	$ ( 1570148876 \nu^{15} - 9814610316 \nu^{14} + 30686832232 \nu^{13} - 63994473524 \nu^{12} + \cdots + 4714210963 ) / 1973128213 $ (1570148876v^15 - 9814610316v^14 + 30686832232v^13 - 63994473524v^12 + 68737309508v^11 + 12449599248v^10 - 117627630280v^9 + 236328234138v^8 - 202589468876v^7 - 114627530376v^6 + 259195647184v^5 - 741062624316v^4 + 968316440200v^3 + 226232221584v^2 + 35218281816*v + 4714210963) / 1973128213
$\beta_{12}$	$=$	$ ( 3087253540240 \nu^{15} - 18982048227910 \nu^{14} + 58429841322972 \nu^{13} + \cdots + 19545417290592 ) / 3707507912227 $ (3087253540240v^15 - 18982048227910v^14 + 58429841322972v^13 - 120068214463500v^12 + 123576350616332v^11 + 35598713867733v^10 - 225839628061730v^9 + 441382360047234v^8 - 355634599263140v^7 - 256040217912758v^6 + 477587659531454v^5 - 1407573904107408v^4 + 1765495989661296v^3 + 608117198238559v^2 + 156616076158474*v + 19545417290592) / 3707507912227
$\beta_{13}$	$=$	$ ( 3131859052620 \nu^{15} - 18867452624394 \nu^{14} + 56728752984694 \nu^{13} + \cdots + 17856504555702 ) / 3707507912227 $ (3131859052620v^15 - 18867452624394v^14 + 56728752984694v^13 - 113478926578056v^12 + 107222867624596v^11 + 57967676009072v^10 - 231355635133834v^9 + 418078071142320v^8 - 293684812663328v^7 - 327591625067831v^6 + 471436953125158v^5 - 1356409325056608v^4 + 1589452776259132v^3 + 910890443850692v^2 + 142623474998120*v + 17856504555702) / 3707507912227
$\beta_{14}$	$=$	$ ( - 5329561831437 \nu^{15} + 33315128327418 \nu^{14} - 104171995568578 \nu^{13} + \cdots - 15992012287118 ) / 3707507912227 $ (-5329561831437v^15 + 33315128327418v^14 - 104171995568578v^13 + 217250160715424v^12 - 233356529166572v^11 - 42306147026280v^10 + 399610538807086v^9 - 802749774453640v^8 + 687914400857440v^7 + 389163059670207v^6 - 880025451774682v^5 + 2513669318791624v^4 - 3284887445773991v^3 - 767456463763692v^2 - 119471231971904*v - 15992012287118) / 3707507912227
$\beta_{15}$	$=$	$ ( - 7516121976528 \nu^{15} + 46209561787616 \nu^{14} - 142236306189628 \nu^{13} + \cdots - 47582417802291 ) / 3707507912227 $ (-7516121976528v^15 + 46209561787616v^14 - 142236306189628v^13 + 292274046112479v^12 - 300793116749828v^11 - 86647872654676v^10 + 549671926014968v^9 - 1074450180412176v^8 + 866011308655208v^7 + 622075435017104v^6 - 1161717303443108v^5 + 3426608236011048v^4 - 4298074501231946v^3 - 1480448402799748v^2 - 381276303387704*v - 47582417802291) / 3707507912227

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

$\nu$	$=$	$ ( \beta_{12} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 4 $ (b12 + b9 - b7 + b5 + b4 + b3 + b1 + 2) / 4
$\nu^{2}$	$=$	$ ( -\beta_{13} + \beta_{12} + 2\beta_{10} + 2\beta_{9} + \beta_{4} ) / 2 $ (-b13 + b12 + 2b10 + 2b9 + b4) / 2
$\nu^{3}$	$=$	$ ( -\beta_{14} - \beta_{13} - 2\beta_{11} + 2\beta_{10} + \beta_{8} + \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 2 $ (-b14 - b13 - 2b11 + 2b10 + b8 + b6 - 3b5 + 3b4) / 2
$\nu^{4}$	$=$	$ ( -5\beta_{14} - 9\beta_{11} + 3\beta_{8} - 5\beta_{5} + \beta_{2} + 6\beta _1 + 13 ) / 2 $ (-5b14 - 9b11 + 3b8 - 5b5 + b2 + 6*b1 + 13) / 2
$\nu^{5}$	$=$	$ ( 8 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} + 35 \beta_{12} - 14 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + \cdots + 64 ) / 4 $ (8b15 - 8b14 + 8b13 + 35b12 - 14b11 - 14b10 + 29b9 + 8b8 - 3b7 - 8b6 - 11b5 - 11b4 + 3b3 + 8b2 + 35*b1 + 64) / 4
$\nu^{6}$	$=$	$ 8\beta_{15} + 34\beta_{12} + 29\beta_{9} + \beta_{7} $ 8b15 + 34b12 + 29*b9 + b7
$\nu^{7}$	$=$	$ ( 51 \beta_{15} - 49 \beta_{14} - 49 \beta_{13} + 196 \beta_{12} - 82 \beta_{11} + 82 \beta_{10} + \cdots - 338 ) / 4 $ (51b15 - 49b14 - 49b13 + 196b12 - 82b11 + 82b10 + 142b9 + 51b8 + 2b7 + 51b6 - 47b5 + 47b4 + 2b3 - 51b2 - 196*b1 - 338) / 4
$\nu^{8}$	$=$	$ ( -148\beta_{14} - 245\beta_{11} + 150\beta_{8} - 118\beta_{5} + 10\beta_{3} - 50\beta_{2} - 188\beta _1 - 323 ) / 2 $ (-148b14 - 245b11 + 150b8 - 118b5 + 10b3 - 50b2 - 188*b1 - 323) / 2
$\nu^{9}$	$=$	$ ( - 278 \beta_{14} + 278 \beta_{13} - 456 \beta_{11} - 456 \beta_{10} + 298 \beta_{8} + \cdots - 223 \beta_{4} ) / 2 $ (-278b14 + 278b13 - 456b11 - 456b10 + 298b8 - 298b6 - 223b5 - 223b4) / 2
$\nu^{10}$	$=$	$ ( 288 \beta_{15} + 809 \beta_{13} + 1027 \beta_{12} - 1320 \beta_{10} + 674 \beta_{9} + \cdots - 599 \beta_{4} ) / 2 $ (288b15 + 809b13 + 1027b12 - 1320b10 + 674b9 + 70b7 - 864b6 - 599b4) / 2
$\nu^{11}$	$=$	$ ( 1673 \beta_{15} + 1533 \beta_{14} + 1533 \beta_{13} + 5864 \beta_{12} + 2490 \beta_{11} - 2490 \beta_{10} + \cdots - 9580 ) / 4 $ (1673b15 + 1533b14 + 1533b13 + 5864b12 + 2490b11 - 2490b10 + 3716b9 - 1673b8 + 408b7 - 1673b6 + 1125b5 - 1125b4 + 408b3 - 1673b2 - 5864*b1 - 9580) / 4
$\nu^{12}$	$=$	$ 428\beta_{3} - 1603\beta_{2} - 5576\beta _1 - 9071 $ 428b3 - 1603b2 - 5576*b1 - 9071
$\nu^{13}$	$=$	$ ( - 9210 \beta_{15} - 8354 \beta_{14} + 8354 \beta_{13} - 31783 \beta_{12} - 13502 \beta_{11} + \cdots - 51380 ) / 4 $ (-9210b15 - 8354b14 + 8354b13 - 31783b12 - 13502b11 - 13502b10 - 19597b9 + 9210b8 - 2489b7 - 9210b6 - 5865b5 - 5865b4 + 2489b3 - 9210b2 - 31783*b1 - 51380) / 4
$\nu^{14}$	$=$	$ ( - 8782 \beta_{15} + 23857 \beta_{13} - 30180 \beta_{12} - 38504 \beta_{10} - 18481 \beta_{9} + \cdots - 16480 \beta_{4} ) / 2 $ (-8782b15 + 23857b13 - 30180b12 - 38504b10 - 18481b9 - 2459b7 - 26346b6 - 16480b4) / 2
$\nu^{15}$	$=$	$ ( 45285 \beta_{14} + 45285 \beta_{13} + 73006 \beta_{11} - 73006 \beta_{10} - 50203 \beta_{8} + \cdots - 31097 \beta_{4} ) / 2 $ (45285b14 + 45285b13 + 73006b11 - 73006b10 - 50203b8 - 50203b6 + 31097b5 - 31097b4) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$.

$n$	$199$	$981$
$\chi(n)$	$-1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1077.1

−0.186243	+	0.0499037i
0.601150	−	2.24352i
−0.347596	+	1.29724i
1.60599	−	0.430324i
0.430324	−	1.60599i
−1.29724	+	0.347596i
2.24352	−	0.601150i
−0.0499037	+	0.186243i
−0.0499037	−	0.186243i
2.24352	+	0.601150i
−1.29724	−	0.347596i
0.430324	+	1.60599i
1.60599	+	0.430324i
−0.347596	−	1.29724i
0.601150	+	2.24352i
−0.186243	−	0.0499037i

−

1.00000i

−

3.12703i

−1.00000

2.20288i

−3.12703

1.00000i

−6.77832

2.20288

1077.2

−

1.00000i

−

2.59518i

−1.00000

−

3.53900i

−2.59518

1.00000i

−3.73495

−3.53900

1077.3

−

1.00000i

−

1.55924i

−1.00000

1.01909i

−1.55924

1.00000i

0.568783

1.01909

1077.4

−

1.00000i

−

1.02738i

−1.00000

−

1.25868i

−1.02738

1.00000i

1.94448

−1.25868

1077.5

−

1.00000i

1.02738i

−1.00000

1.25868i

1.02738

1.00000i

1.94448

1.25868

1077.6

−

1.00000i

1.55924i

−1.00000

−

1.01909i

1.55924

1.00000i

0.568783

−1.01909

1077.7

−

1.00000i

2.59518i

−1.00000

3.53900i

2.59518

1.00000i

−3.73495

3.53900

1077.8

−

1.00000i

3.12703i

−1.00000

−

2.20288i

3.12703

1.00000i

−6.77832

−2.20288

1077.9

1.00000i

−

3.12703i

−1.00000

2.20288i

3.12703

−

1.00000i

−6.77832

−2.20288

1077.10

1.00000i

−

2.59518i

−1.00000

−

3.53900i

2.59518

−

1.00000i

−3.73495

3.53900

1077.11

1.00000i

−

1.55924i

−1.00000

1.01909i

1.55924

−

1.00000i

0.568783

−1.01909

1077.12

1.00000i

−

1.02738i

−1.00000

−

1.25868i

1.02738

−

1.00000i

1.94448

1.25868

1077.13

1.00000i

1.02738i

−1.00000

1.25868i

−1.02738

−

1.00000i

1.94448

−1.25868

1077.14

1.00000i

1.55924i

−1.00000

−

1.01909i

−1.55924

−

1.00000i

0.568783

1.01909

1077.15

1.00000i

2.59518i

−1.00000

3.53900i

−2.59518

−

1.00000i

−3.73495

−3.53900

1077.16

1.00000i

3.12703i

−1.00000

−

2.20288i

−3.12703

−

1.00000i

−6.77832

2.20288

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
11.b	odd	2	1	inner
77.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1078.2.c.b		16
7.b	odd	2	1	inner	1078.2.c.b		16
7.c	even	3	1		154.2.i.a	✓	16
7.c	even	3	1		1078.2.i.c		16
7.d	odd	6	1		154.2.i.a	✓	16
7.d	odd	6	1		1078.2.i.c		16
11.b	odd	2	1	inner	1078.2.c.b		16
21.g	even	6	1		1386.2.bk.c		16
21.h	odd	6	1		1386.2.bk.c		16
28.f	even	6	1		1232.2.bn.b		16
28.g	odd	6	1		1232.2.bn.b		16
77.b	even	2	1	inner	1078.2.c.b		16
77.h	odd	6	1		154.2.i.a	✓	16
77.h	odd	6	1		1078.2.i.c		16
77.i	even	6	1		154.2.i.a	✓	16
77.i	even	6	1		1078.2.i.c		16
231.k	odd	6	1		1386.2.bk.c		16
231.l	even	6	1		1386.2.bk.c		16
308.m	odd	6	1		1232.2.bn.b		16
308.n	even	6	1		1232.2.bn.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
154.2.i.a	✓	16	7.c	even	3	1
154.2.i.a	✓	16	7.d	odd	6	1
154.2.i.a	✓	16	77.h	odd	6	1
154.2.i.a	✓	16	77.i	even	6	1
1078.2.c.b		16	1.a	even	1	1	trivial
1078.2.c.b		16	7.b	odd	2	1	inner
1078.2.c.b		16	11.b	odd	2	1	inner
1078.2.c.b		16	77.b	even	2	1	inner
1078.2.i.c		16	7.c	even	3	1
1078.2.i.c		16	7.d	odd	6	1
1078.2.i.c		16	77.h	odd	6	1
1078.2.i.c		16	77.i	even	6	1
1232.2.bn.b		16	28.f	even	6	1
1232.2.bn.b		16	28.g	odd	6	1
1232.2.bn.b		16	308.m	odd	6	1
1232.2.bn.b		16	308.n	even	6	1
1386.2.bk.c		16	21.g	even	6	1
1386.2.bk.c		16	21.h	odd	6	1
1386.2.bk.c		16	231.k	odd	6	1
1386.2.bk.c		16	231.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 20T_{3}^{6} + 126T_{3}^{4} + 272T_{3}^{2} + 169 $ T3^8 + 20*T3^6 + 126*T3^4 + 272*T3^2 + 169 acting on $S_{2}^{\mathrm{new}}(1078, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$3$	$ (T^{8} + 20 T^{6} + \cdots + 169)^{2} $ (T^8 + 20T^6 + 126T^4 + 272*T^2 + 169)^2
$5$	$ (T^{8} + 20 T^{6} + \cdots + 100)^{2} $ (T^8 + 20T^6 + 108T^4 + 188*T^2 + 100)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} + 8 T^{7} + \cdots + 14641)^{2} $ (T^8 + 8T^7 + 20T^6 - 40T^5 - 362T^4 - 440T^3 + 2420T^2 + 10648*T + 14641)^2
$13$	$ (T^{8} - 44 T^{6} + \cdots + 25)^{2} $ (T^8 - 44T^6 + 510T^4 - 908*T^2 + 25)^2
$17$	$ (T^{8} - 92 T^{6} + \cdots + 78400)^{2} $ (T^8 - 92T^6 + 2352T^4 - 23168*T^2 + 78400)^2
$19$	$ (T^{8} - 80 T^{6} + 1272 T^{4} + \cdots + 4)^{2} $ (T^8 - 80T^6 + 1272T^4 - 164*T^2 + 4)^2
$23$	$ (T^{4} + 8 T^{3} - 6 T^{2} + \cdots - 14)^{4} $ (T^4 + 8T^3 - 6T^2 - 34*T - 14)^4
$29$	$ (T^{8} + 36 T^{6} + 366 T^{4} + \cdots + 9)^{2} $ (T^8 + 36T^6 + 366T^4 + 900*T^2 + 9)^2
$31$	$ (T^{8} + 164 T^{6} + \cdots + 3136)^{2} $ (T^8 + 164T^6 + 7200T^4 + 40640*T^2 + 3136)^2
$37$	$ (T^{4} - 8 T^{3} + 12 T^{2} + \cdots - 2)^{4} $ (T^4 - 8T^3 + 12T^2 - 2*T - 2)^4
$41$	$ (T^{8} - 200 T^{6} + \cdots + 1674436)^{2} $ (T^8 - 200T^6 + 12588T^4 - 287612*T^2 + 1674436)^2
$43$	$ (T^{8} + 244 T^{6} + \cdots + 1201216)^{2} $ (T^8 + 244T^6 + 19536T^4 + 527488*T^2 + 1201216)^2
$47$	$ (T^{8} + 80 T^{6} + \cdots + 4096)^{2} $ (T^8 + 80T^6 + 1920T^4 + 14336*T^2 + 4096)^2
$53$	$ (T^{4} - 14 T^{3} + \cdots - 1082)^{4} $ (T^4 - 14T^3 + 24T^2 + 322*T - 1082)^4
$59$	$ (T^{8} + 296 T^{6} + \cdots + 2356225)^{2} $ (T^8 + 296T^6 + 19974T^4 + 448508*T^2 + 2356225)^2
$61$	$ (T^{8} - 300 T^{6} + \cdots + 1306449)^{2} $ (T^8 - 300T^6 + 26838T^4 - 632556*T^2 + 1306449)^2
$67$	$ (T^{4} + 6 T^{3} + \cdots + 417)^{4} $ (T^4 + 6T^3 - 54T^2 - 48*T + 417)^4
$71$	$ (T^{4} - 2 T^{3} + \cdots + 3262)^{4} $ (T^4 - 2T^3 - 174T^2 - 338*T + 3262)^4
$73$	$ (T^{8} - 204 T^{6} + \cdots + 125316)^{2} $ (T^8 - 204T^6 + 11868T^4 - 144972*T^2 + 125316)^2
$79$	$ (T^{8} + 400 T^{6} + \cdots + 33953929)^{2} $ (T^8 + 400T^6 + 46182T^4 + 2113108*T^2 + 33953929)^2
$83$	$ (T^{8} - 260 T^{6} + \cdots + 107584)^{2} $ (T^8 - 260T^6 + 9888T^4 - 77504*T^2 + 107584)^2
$89$	$ (T^{8} + 456 T^{6} + \cdots + 49617936)^{2} $ (T^8 + 456T^6 + 62088T^4 + 3072528*T^2 + 49617936)^2
$97$	$ (T^{8} + 644 T^{6} + \cdots + 496532089)^{2} $ (T^8 + 644T^6 + 145902T^4 + 14100500*T^2 + 496532089)^2
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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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{9} q^{2} + \beta_{14} q^{3} - q^{4} + (\beta_{11} - \beta_{8}) q^{5} - \beta_{13} q^{6} - \beta_{9} q^{8} + ( - \beta_{2} - 2) q^{9} + (\beta_{10} + \beta_{6}) q^{10} + ( - \beta_{12} + \beta_{9} - \beta_{3} - 1) q^{11}+ \cdots + ( - \beta_{15} + 5 \beta_{12} + \cdots + 4) q^{99}+O(q^{100}) \) q + b9 * q^2 + b14 * q^3 - q^4 + (b11 - b8) * q^5 - b13 * q^6 - b9 * q^8 + (-b2 - 2) * q^9 + (b10 + b6) * q^10 + (-b12 + b9 - b3 - 1) * q^11 - b14 * q^12 + (-b10 - b4) * q^13 + (-3b1 - 2) q^15 + q^16 + (-b13 - 2b10 - b6 + 2b4) * q^17 + (-b15 - 2b9) q^18 + (-b13 + b10 + 2b6 - b4) q^19 + (-b11 + b8) * q^20 + (-b9 + b7 + b1) * q^22 + (-b3 + b2 - 2) * q^23 + b13 * q^24 + (-b2 - 2b1 - 1) q^25 + (b11 + b5) * q^26 + (-b14 + 4b11 - 2b8 - 3b5) q^27 + (b12 - 2b9 + b7) q^29 + (-3b12 + b9) q^30 + (-b14 + 2b11 - 3b8 - 2b5) q^31 + b9 * q^32 + (-b14 - b13 + b11 - 3b10 + b8 - 2b6 - b5) * q^33 + (-b14 + 2b11 - b8 - 2b5) * q^34 + (b2 + 2) * q^36 + (-b3 + 2) * q^37 + (-b14 - b11 + 2b8 + b5) q^38 + (b15 + 4b12 - b9 + b7) q^39 + (-b10 - b6) * q^40 + (-3b13 + b10 - 2b4) * q^41 + (b15 + 3b12 + b9 + 3b7) * q^43 + (b12 - b9 + b3 + 1) * q^44 + (b14 - 6b11 + 3b8) * q^45 + (b15 - 2b9 + b7) q^46 - 2b5 q^47 + b14 * q^48 + (-b15 - 2b12 + b9) q^50 + (-3b15 + b12 - 7b9 + b7) * q^51 + (b10 + b4) * q^52 + (-2b3 + b2 + b1 + 4) q^53 + (b13 + 4b10 + 2b6 - 3b4) q^54 + (-b14 + 2b13 - 3b11 + 2b8 + b5 - b4) q^55 + (-2b12 - 4b9 + b7) * q^57 + (b3 - b1 + 1) * q^58 + (3b14 + 2b11 + b5) * q^59 + (3b1 + 2) q^60 + (-b13 - b10 - 3b6 - 2b4) * q^61 + (b13 + 2b10 + 3b6 - 2b4) q^62 - q^64 + (2b15 + 4b12 + 2b9 + b7) q^65 + (-b14 + b13 + 3b11 + b10 - 2b8 - b6 - b4) * q^66 + (2b3 + b1 - 1) q^67 + (b13 + 2b10 + b6 - 2b4) * q^68 + (-3b11 + 3b8 + 2b5) q^69 + (-2b3 - 2b2 - b1) * q^71 + (b15 + 2b9) q^72 + (-5b10 - b6 + 2b4) * q^73 + (2b9 + b7) q^74 + (-b14 - 2b11 + 2b8 - 3b5) q^75 + (b13 - b10 - 2b6 + b4) q^76 + (b3 - b2 - 4b1 - 3) q^78 + (3b15 + 2b12 - 3b9 + b7) q^79 + (b11 - b8) * q^80 + (2b3 + b2 - 4b1 - 1) * q^81 + (-3b14 - b11 + 2b5) * q^82 + (-3b13 + 2b10 - b6 - 2b4) q^83 + (-6b12 + 4b9 - 2b7) q^85 + (3b3 - b2 - 3b1 - 4) * q^86 + (2b13 + 4b10 + b6 - b4) * q^87 + (b9 - b7 - b1) * q^88 + (3b14 - 4b11 - b8 + b5) * q^89 + (-b13 - 6b10 - 3b6) * q^90 + (b3 - b2 + 2) * q^92 + (-b3 + 3b2 - 3b1 + 4) * q^93 - 2b4 q^94 + (-3b12 - 5b9 + b7) * q^95 - b13 * q^96 + (-2b14 + 3b11 - 2b8 + 4b5) * q^97 + (-b15 + 5b12 - 5b9 + b7 - b3 + 2b2 + b1 + 4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 32 q^{9} - 16 q^{11} - 8 q^{15} + 16 q^{16} - 8 q^{22} - 32 q^{23} + 32 q^{36} + 32 q^{37} + 16 q^{44} + 56 q^{53} + 24 q^{58} + 8 q^{60} - 16 q^{64} - 24 q^{67} + 8 q^{71} - 16 q^{78}+ \cdots + 56 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 - 32 * q^9 - 16 * q^11 - 8 * q^15 + 16 * q^16 - 8 * q^22 - 32 * q^23 + 32 * q^36 + 32 * q^37 + 16 * q^44 + 56 * q^53 + 24 * q^58 + 8 * q^60 - 16 * q^64 - 24 * q^67 + 8 * q^71 - 16 * q^78 + 16 * q^81 - 40 * q^86 + 8 * q^88 + 32 * q^92 + 88 * q^93 + 56 * q^99

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	1078.2.c.b

Level	$1078$
Weight	$2$
Character orbit	1078.c
Analytic conductor	$8.608$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(8.60787333789\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \) x^16 - 6x^15 + 18x^14 - 36x^13 + 34x^12 + 18x^11 - 72x^10 + 132x^9 - 93x^8 - 102x^7 + 144x^6 - 432x^5 + 502x^4 + 288x^3 + 72x^2 + 12*x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 73568941 \nu^{15} - 2828429898 \nu^{14} + 10706428236 \nu^{13} - 20127265496 \nu^{12} + \cdots - 10982429182739 ) / 3707507912227 \) (-73568941v^15 - 2828429898v^14 + 10706428236v^13 - 20127265496v^12 + 35741021050v^11 - 17927851065v^10 + 58772639836v^9 - 308269732922v^8 + 50382616322v^7 - 35812385940v^6 - 245517720058v^5 + 1525793905272v^4 + 747026251899v^3 + 178512937629v^2 + 28424647318*v - 10982429182739) / 3707507912227
\(\beta_{2}\)	\(=\)	\( ( - 973484628 \nu^{15} + 16798064640 \nu^{14} - 57412057192 \nu^{13} + 106920971791 \nu^{12} + \cdots + 17707406591301 ) / 3707507912227 \) (-973484628v^15 + 16798064640v^14 - 57412057192v^13 + 106920971791v^12 - 159815035576v^11 + 58809142080v^10 - 208743822856v^9 + 1110018585936v^8 - 324206168264v^7 + 209105071620v^6 + 1201975196104v^5 - 5438378020432v^4 - 2697240781348v^3 - 647205045504v^2 - 103443810800*v + 17707406591301) / 3707507912227
\(\beta_{3}\)	\(=\)	\( ( - 4604477275 \nu^{15} + 26065356324 \nu^{14} - 75543186530 \nu^{13} + 146897647868 \nu^{12} + \cdots + 1817177370827 ) / 3707507912227 \) (-4604477275v^15 + 26065356324v^14 - 75543186530v^13 + 146897647868v^12 - 132924225826v^11 - 13305240150v^10 - 16121748394v^9 + 141233089912v^8 - 557871539990v^7 + 316601789265v^6 + 1303176243718v^5 - 490404558308v^4 - 369763065215v^3 - 98321373186v^2 - 17113540344*v + 1817177370827) / 3707507912227
\(\beta_{4}\)	\(=\)	\( ( - 275511061668 \nu^{15} + 1657750582996 \nu^{14} - 4984224043414 \nu^{13} + \cdots - 1555184012514 ) / 3707507912227 \) (-275511061668v^15 + 1657750582996v^14 - 4984224043414v^13 + 9981450357924v^12 - 9469500994272v^11 - 4940688808597v^10 + 20095109174076v^9 - 36700380318006v^8 + 25948536491036v^7 + 28399264833487v^6 - 41476514830836v^5 + 119515269818808v^4 - 137980607533908v^3 - 79280602131237v^2 - 12412885174910*v - 1555184012514) / 3707507912227
\(\beta_{5}\)	\(=\)	\( ( - 464283294072 \nu^{15} + 2897231896860 \nu^{14} - 9050241140038 \nu^{13} + \cdots - 1395125725490 ) / 3707507912227 \) (-464283294072v^15 + 2897231896860v^14 - 9050241140038v^13 + 18865423115472v^12 - 20243277302808v^11 - 3659327351235v^10 + 34524757354698v^9 - 69586426065626v^8 + 59597927208868v^7 + 33784753952673v^6 - 76446349441200v^5 + 218895832838984v^4 - 286618252654908v^3 - 66966547472787v^2 - 10425321369434*v - 1395125725490) / 3707507912227
\(\beta_{6}\)	\(=\)	\( ( 476129408728 \nu^{15} - 2861932186210 \nu^{14} + 8584036352598 \nu^{13} + \cdots + 2687120456379 ) / 3707507912227 \) (476129408728v^15 - 2861932186210v^14 + 8584036352598v^13 - 17121557060007v^12 + 16021649705480v^11 + 9119986400225v^10 - 35176977398796v^9 + 63105295993728v^8 - 43746959330988v^7 - 50114785825646v^6 + 71179229667320v^5 - 205114133957706v^4 + 238865093218054v^3 + 137040061864945v^2 + 21456435065178*v + 2687120456379) / 3707507912227
\(\beta_{7}\)	\(=\)	\( ( 770100281810 \nu^{15} - 4733263012640 \nu^{14} + 14562049553051 \nu^{13} + \cdots + 4887214330899 ) / 3707507912227 \) (770100281810v^15 - 4733263012640v^14 + 14562049553051v^13 - 29920520825502v^12 + 30799143245438v^11 + 8829439453946v^10 - 56098808754161v^9 + 110050939842954v^8 - 89459573480504v^7 - 62707181505525v^6 + 117223638772823v^5 - 351034936647114v^4 + 441822991616242v^3 + 152138648268578v^2 + 39169305360240*v + 4887214330899) / 3707507912227
\(\beta_{8}\)	\(=\)	\( ( - 805007695745 \nu^{15} + 5029032860754 \nu^{14} - 15724330059984 \nu^{13} + \cdots - 2408034273855 ) / 3707507912227 \) (-805007695745v^15 + 5029032860754v^14 - 15724330059984v^13 + 32793740003767v^12 - 35212340782612v^11 - 6414480346779v^10 + 60443911042560v^9 - 121464878425672v^8 + 103864347546612v^7 + 58759557056628v^6 - 132943428475514v^5 + 378138116252806v^4 - 494704696217447v^3 - 115575276518547v^2 - 17991168996474*v - 2408034273855) / 3707507912227
\(\beta_{9}\)	\(=\)	\( ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} + \cdots - 10233974547 ) / 3810388399 \) (-1616321538v^15 + 9938916556v^14 - 30594542475v^13 + 62871582510v^12 - 64714538406v^11 - 18641341380v^10 + 118271636061v^9 - 231117756606v^8 + 186162436800v^7 + 134499720676v^6 - 250255719435v^5 + 736991238906v^4 - 924410484114v^3 - 318408624800v^2 - 82003950372*v - 10233974547) / 3810388399
\(\beta_{10}\)	\(=\)	\( ( 854458 \nu^{15} - 5149288 \nu^{14} + 15489103 \nu^{13} - 31002870 \nu^{12} + 29353426 \nu^{11} + \cdots + 4877403 ) / 1828267 \) (854458v^15 - 5149288v^14 + 15489103v^13 - 31002870v^12 + 29353426v^11 + 15677868v^10 - 63062803v^9 + 114195864v^8 - 80446252v^7 - 89178588v^6 + 128785525v^5 - 370469382v^4 + 434157674v^3 + 248804552v^2 + 38956914*v + 4877403) / 1828267
\(\beta_{11}\)	\(=\)	\( ( 1570148876 \nu^{15} - 9814610316 \nu^{14} + 30686832232 \nu^{13} - 63994473524 \nu^{12} + \cdots + 4714210963 ) / 1973128213 \) (1570148876v^15 - 9814610316v^14 + 30686832232v^13 - 63994473524v^12 + 68737309508v^11 + 12449599248v^10 - 117627630280v^9 + 236328234138v^8 - 202589468876v^7 - 114627530376v^6 + 259195647184v^5 - 741062624316v^4 + 968316440200v^3 + 226232221584v^2 + 35218281816*v + 4714210963) / 1973128213
\(\beta_{12}\)	\(=\)	\( ( 3087253540240 \nu^{15} - 18982048227910 \nu^{14} + 58429841322972 \nu^{13} + \cdots + 19545417290592 ) / 3707507912227 \) (3087253540240v^15 - 18982048227910v^14 + 58429841322972v^13 - 120068214463500v^12 + 123576350616332v^11 + 35598713867733v^10 - 225839628061730v^9 + 441382360047234v^8 - 355634599263140v^7 - 256040217912758v^6 + 477587659531454v^5 - 1407573904107408v^4 + 1765495989661296v^3 + 608117198238559v^2 + 156616076158474*v + 19545417290592) / 3707507912227
\(\beta_{13}\)	\(=\)	\( ( 3131859052620 \nu^{15} - 18867452624394 \nu^{14} + 56728752984694 \nu^{13} + \cdots + 17856504555702 ) / 3707507912227 \) (3131859052620v^15 - 18867452624394v^14 + 56728752984694v^13 - 113478926578056v^12 + 107222867624596v^11 + 57967676009072v^10 - 231355635133834v^9 + 418078071142320v^8 - 293684812663328v^7 - 327591625067831v^6 + 471436953125158v^5 - 1356409325056608v^4 + 1589452776259132v^3 + 910890443850692v^2 + 142623474998120*v + 17856504555702) / 3707507912227
\(\beta_{14}\)	\(=\)	\( ( - 5329561831437 \nu^{15} + 33315128327418 \nu^{14} - 104171995568578 \nu^{13} + \cdots - 15992012287118 ) / 3707507912227 \) (-5329561831437v^15 + 33315128327418v^14 - 104171995568578v^13 + 217250160715424v^12 - 233356529166572v^11 - 42306147026280v^10 + 399610538807086v^9 - 802749774453640v^8 + 687914400857440v^7 + 389163059670207v^6 - 880025451774682v^5 + 2513669318791624v^4 - 3284887445773991v^3 - 767456463763692v^2 - 119471231971904*v - 15992012287118) / 3707507912227
\(\beta_{15}\)	\(=\)	\( ( - 7516121976528 \nu^{15} + 46209561787616 \nu^{14} - 142236306189628 \nu^{13} + \cdots - 47582417802291 ) / 3707507912227 \) (-7516121976528v^15 + 46209561787616v^14 - 142236306189628v^13 + 292274046112479v^12 - 300793116749828v^11 - 86647872654676v^10 + 549671926014968v^9 - 1074450180412176v^8 + 866011308655208v^7 + 622075435017104v^6 - 1161717303443108v^5 + 3426608236011048v^4 - 4298074501231946v^3 - 1480448402799748v^2 - 381276303387704*v - 47582417802291) / 3707507912227

\(\nu\)	\(=\)	\( ( \beta_{12} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 4 \) (b12 + b9 - b7 + b5 + b4 + b3 + b1 + 2) / 4
\(\nu^{2}\)	\(=\)	\( ( -\beta_{13} + \beta_{12} + 2\beta_{10} + 2\beta_{9} + \beta_{4} ) / 2 \) (-b13 + b12 + 2b10 + 2b9 + b4) / 2
\(\nu^{3}\)	\(=\)	\( ( -\beta_{14} - \beta_{13} - 2\beta_{11} + 2\beta_{10} + \beta_{8} + \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 2 \) (-b14 - b13 - 2b11 + 2b10 + b8 + b6 - 3b5 + 3b4) / 2
\(\nu^{4}\)	\(=\)	\( ( -5\beta_{14} - 9\beta_{11} + 3\beta_{8} - 5\beta_{5} + \beta_{2} + 6\beta _1 + 13 ) / 2 \) (-5b14 - 9b11 + 3b8 - 5b5 + b2 + 6*b1 + 13) / 2
\(\nu^{5}\)	\(=\)	\( ( 8 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} + 35 \beta_{12} - 14 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + \cdots + 64 ) / 4 \) (8b15 - 8b14 + 8b13 + 35b12 - 14b11 - 14b10 + 29b9 + 8b8 - 3b7 - 8b6 - 11b5 - 11b4 + 3b3 + 8b2 + 35*b1 + 64) / 4
\(\nu^{6}\)	\(=\)	\( 8\beta_{15} + 34\beta_{12} + 29\beta_{9} + \beta_{7} \) 8b15 + 34b12 + 29*b9 + b7
\(\nu^{7}\)	\(=\)	\( ( 51 \beta_{15} - 49 \beta_{14} - 49 \beta_{13} + 196 \beta_{12} - 82 \beta_{11} + 82 \beta_{10} + \cdots - 338 ) / 4 \) (51b15 - 49b14 - 49b13 + 196b12 - 82b11 + 82b10 + 142b9 + 51b8 + 2b7 + 51b6 - 47b5 + 47b4 + 2b3 - 51b2 - 196*b1 - 338) / 4
\(\nu^{8}\)	\(=\)	\( ( -148\beta_{14} - 245\beta_{11} + 150\beta_{8} - 118\beta_{5} + 10\beta_{3} - 50\beta_{2} - 188\beta _1 - 323 ) / 2 \) (-148b14 - 245b11 + 150b8 - 118b5 + 10b3 - 50b2 - 188*b1 - 323) / 2
\(\nu^{9}\)	\(=\)	\( ( - 278 \beta_{14} + 278 \beta_{13} - 456 \beta_{11} - 456 \beta_{10} + 298 \beta_{8} + \cdots - 223 \beta_{4} ) / 2 \) (-278b14 + 278b13 - 456b11 - 456b10 + 298b8 - 298b6 - 223b5 - 223b4) / 2
\(\nu^{10}\)	\(=\)	\( ( 288 \beta_{15} + 809 \beta_{13} + 1027 \beta_{12} - 1320 \beta_{10} + 674 \beta_{9} + \cdots - 599 \beta_{4} ) / 2 \) (288b15 + 809b13 + 1027b12 - 1320b10 + 674b9 + 70b7 - 864b6 - 599b4) / 2
\(\nu^{11}\)	\(=\)	\( ( 1673 \beta_{15} + 1533 \beta_{14} + 1533 \beta_{13} + 5864 \beta_{12} + 2490 \beta_{11} - 2490 \beta_{10} + \cdots - 9580 ) / 4 \) (1673b15 + 1533b14 + 1533b13 + 5864b12 + 2490b11 - 2490b10 + 3716b9 - 1673b8 + 408b7 - 1673b6 + 1125b5 - 1125b4 + 408b3 - 1673b2 - 5864*b1 - 9580) / 4
\(\nu^{12}\)	\(=\)	\( 428\beta_{3} - 1603\beta_{2} - 5576\beta _1 - 9071 \) 428b3 - 1603b2 - 5576*b1 - 9071
\(\nu^{13}\)	\(=\)	\( ( - 9210 \beta_{15} - 8354 \beta_{14} + 8354 \beta_{13} - 31783 \beta_{12} - 13502 \beta_{11} + \cdots - 51380 ) / 4 \) (-9210b15 - 8354b14 + 8354b13 - 31783b12 - 13502b11 - 13502b10 - 19597b9 + 9210b8 - 2489b7 - 9210b6 - 5865b5 - 5865b4 + 2489b3 - 9210b2 - 31783*b1 - 51380) / 4
\(\nu^{14}\)	\(=\)	\( ( - 8782 \beta_{15} + 23857 \beta_{13} - 30180 \beta_{12} - 38504 \beta_{10} - 18481 \beta_{9} + \cdots - 16480 \beta_{4} ) / 2 \) (-8782b15 + 23857b13 - 30180b12 - 38504b10 - 18481b9 - 2459b7 - 26346b6 - 16480b4) / 2
\(\nu^{15}\)	\(=\)	\( ( 45285 \beta_{14} + 45285 \beta_{13} + 73006 \beta_{11} - 73006 \beta_{10} - 50203 \beta_{8} + \cdots - 31097 \beta_{4} ) / 2 \) (45285b14 + 45285b13 + 73006b11 - 73006b10 - 50203b8 - 50203b6 + 31097b5 - 31097b4) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1077.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$3$	\( (T^{8} + 20 T^{6} + \cdots + 169)^{2} \) (T^8 + 20T^6 + 126T^4 + 272*T^2 + 169)^2
$5$	\( (T^{8} + 20 T^{6} + \cdots + 100)^{2} \) (T^8 + 20T^6 + 108T^4 + 188*T^2 + 100)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} + 8 T^{7} + \cdots + 14641)^{2} \) (T^8 + 8T^7 + 20T^6 - 40T^5 - 362T^4 - 440T^3 + 2420T^2 + 10648*T + 14641)^2
$13$	\( (T^{8} - 44 T^{6} + \cdots + 25)^{2} \) (T^8 - 44T^6 + 510T^4 - 908*T^2 + 25)^2
$17$	\( (T^{8} - 92 T^{6} + \cdots + 78400)^{2} \) (T^8 - 92T^6 + 2352T^4 - 23168*T^2 + 78400)^2
$19$	\( (T^{8} - 80 T^{6} + 1272 T^{4} + \cdots + 4)^{2} \) (T^8 - 80T^6 + 1272T^4 - 164*T^2 + 4)^2
$23$	\( (T^{4} + 8 T^{3} - 6 T^{2} + \cdots - 14)^{4} \) (T^4 + 8T^3 - 6T^2 - 34*T - 14)^4
$29$	\( (T^{8} + 36 T^{6} + 366 T^{4} + \cdots + 9)^{2} \) (T^8 + 36T^6 + 366T^4 + 900*T^2 + 9)^2
$31$	\( (T^{8} + 164 T^{6} + \cdots + 3136)^{2} \) (T^8 + 164T^6 + 7200T^4 + 40640*T^2 + 3136)^2
$37$	\( (T^{4} - 8 T^{3} + 12 T^{2} + \cdots - 2)^{4} \) (T^4 - 8T^3 + 12T^2 - 2*T - 2)^4
$41$	\( (T^{8} - 200 T^{6} + \cdots + 1674436)^{2} \) (T^8 - 200T^6 + 12588T^4 - 287612*T^2 + 1674436)^2
$43$	\( (T^{8} + 244 T^{6} + \cdots + 1201216)^{2} \) (T^8 + 244T^6 + 19536T^4 + 527488*T^2 + 1201216)^2
$47$	\( (T^{8} + 80 T^{6} + \cdots + 4096)^{2} \) (T^8 + 80T^6 + 1920T^4 + 14336*T^2 + 4096)^2
$53$	\( (T^{4} - 14 T^{3} + \cdots - 1082)^{4} \) (T^4 - 14T^3 + 24T^2 + 322*T - 1082)^4
$59$	\( (T^{8} + 296 T^{6} + \cdots + 2356225)^{2} \) (T^8 + 296T^6 + 19974T^4 + 448508*T^2 + 2356225)^2
$61$	\( (T^{8} - 300 T^{6} + \cdots + 1306449)^{2} \) (T^8 - 300T^6 + 26838T^4 - 632556*T^2 + 1306449)^2
$67$	\( (T^{4} + 6 T^{3} + \cdots + 417)^{4} \) (T^4 + 6T^3 - 54T^2 - 48*T + 417)^4
$71$	\( (T^{4} - 2 T^{3} + \cdots + 3262)^{4} \) (T^4 - 2T^3 - 174T^2 - 338*T + 3262)^4
$73$	\( (T^{8} - 204 T^{6} + \cdots + 125316)^{2} \) (T^8 - 204T^6 + 11868T^4 - 144972*T^2 + 125316)^2
$79$	\( (T^{8} + 400 T^{6} + \cdots + 33953929)^{2} \) (T^8 + 400T^6 + 46182T^4 + 2113108*T^2 + 33953929)^2
$83$	\( (T^{8} - 260 T^{6} + \cdots + 107584)^{2} \) (T^8 - 260T^6 + 9888T^4 - 77504*T^2 + 107584)^2
$89$	\( (T^{8} + 456 T^{6} + \cdots + 49617936)^{2} \) (T^8 + 456T^6 + 62088T^4 + 3072528*T^2 + 49617936)^2
$97$	\( (T^{8} + 644 T^{6} + \cdots + 496532089)^{2} \) (T^8 + 644T^6 + 145902T^4 + 14100500*T^2 + 496532089)^2
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\(n\)	\(199\)	\(981\)
\(\chi(n)\)	\(-1\)	\(-1\)