LMFDB - Newform orbit 5520.2.be.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (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:	$44.0774219157$
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} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} + \cdots + 64 $ x^16 - 4x^15 + 8x^14 + 28x^13 + 373x^12 - 920x^11 + 1088x^10 - 168x^9 + 16460x^8 - 45408x^7 + 62624x^6 - 18048x^5 + 2160x^4 - 1664x^3 + 6272x^2 - 896*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
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_{10} q^{3} + \beta_{10} q^{5} + (\beta_{4} + 1) q^{7} - q^{9}+O(q^{10}) $ q + b10 * q^3 + b10 * q^5 + (b4 + 1) * q^7 - q^9	$ q + \beta_{10} q^{3} + \beta_{10} q^{5} + (\beta_{4} + 1) q^{7} - q^{9} - \beta_{6} q^{11} + ( - \beta_{3} + 1) q^{13} - q^{15} + (\beta_{14} + \beta_1) q^{17} + (\beta_{9} + \beta_{4} + \beta_{3}) q^{19} - \beta_{14} q^{21} + ( - \beta_{15} + \beta_{5} - 1) q^{23} - q^{25} - \beta_{10} q^{27} + ( - \beta_{6} + \beta_{2}) q^{29} + (\beta_{15} + \beta_{14} + \cdots + \beta_1) q^{31}+ \cdots + \beta_{6} q^{99}+O(q^{100}) $ q + b10 * q^3 + b10 * q^5 + (b4 + 1) * q^7 - q^9 - b6 * q^11 + (-b3 + 1) * q^13 - q^15 + (b14 + b1) * q^17 + (b9 + b4 + b3) * q^19 - b14 * q^21 + (-b15 + b5 - 1) * q^23 - q^25 - b10 * q^27 + (-b6 + b2) * q^29 + (b15 + b14 + b13 + b12 + b11 + b1) * q^31 + (b13 - b10) * q^33 - b14 * q^35 + (-b15 - b12 - b10) * q^37 + (b10 + b8) * q^39 + (b9 - b7 + b6 - b5) * q^41 + (b9 - b7 - b4 - b3) * q^43 - b10 * q^45 + (b14 + b13 - b12 + b1) * q^47 + (-b9 - b7 + 2b6 - 2b5 + b4 + b3 - b2 + 1) * q^49 + (-b5 + b4 + 1) * q^51 + (-b15 + b14 + b12 + b11 - b8) * q^53 + (b13 - b10) * q^55 + (-b14 - b11 - b10 - b8) * q^57 + (b15 + b14 - 2b10 + 2b1) * q^59 + (b14 + b12 - 2b10 + b8 + b1) q^61 + (-b4 - 1) * q^63 + (b10 + b8) * q^65 + (-b9 + b7 - 2b6 + 2b5 - 2b4 - b3 + 1) q^67 + (-b10 + b7 + b1) * q^69 + (-2b15 - 2b13 + b11 + b10 - b8) * q^71 + (2b7 - 2b6 + b3 + 3) * q^73 - b10 * q^75 + (-b9 - b7 - 2b6 - 3b4 + 1) * q^77 + (-b7 + 3b6 - b5 + b4 - b3 - b2 - 2) q^79 + q^81 + (-2b7 - b5 + b2) q^83 + (-b5 + b4 + 1) * q^85 + (b13 + b12 - b10 - b8) * q^87 + (-2b15 - 2b14 - 2b13 + 2b10 - 2b1) q^89 + (-b9 - b7 + b6 - b5 + 2b3 - b2 + 2) q^91 + (b9 - b7 + b6 - b5 + b4 + b3 - b2) * q^93 + (-b14 - b11 - b10 - b8) * q^95 + (-2b15 - b14 - b13 + b11 + b8) q^97 + b6 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) $ 16 * q + 8 * q^7 - 16 * q^9	$ 16 q + 8 q^{7} - 16 q^{9} - 8 q^{11} + 8 q^{13} - 16 q^{15} - 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} + 4 q^{51} - 8 q^{63} + 16 q^{67} + 40 q^{73} + 24 q^{77} - 32 q^{79} + 16 q^{81} + 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) $ 16 * q + 8 * q^7 - 16 * q^9 - 8 * q^11 + 8 * q^13 - 16 * q^15 - 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 + 4 * q^51 - 8 * q^63 + 16 * q^67 + 40 * q^73 + 24 * q^77 - 32 * q^79 + 16 * q^81 + 4 * q^85 + 48 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} + \cdots + 64 $ x^16 - 4*x^15 + 8*x^14 + 28*x^13 + 373*x^12 - 920*x^11 + 1088*x^10 - 168*x^9 + 16460*x^8 - 45408*x^7 + 62624*x^6 - 18048*x^5 + 2160*x^4 - 1664*x^3 + 6272*x^2 - 896*x + 64 :

$\beta_{1}$	$=$	$ ( - 22\!\cdots\!31 \nu^{15} + \cdots + 94\!\cdots\!12 ) / 34\!\cdots\!80 $ (-22393307326694017831v^15 + 79205853823130329892v^14 - 175071232472117903174v^13 - 526628042123880972776v^12 - 9091844016900447538055v^11 + 15856188225716940095240v^10 - 27782783728011205760238v^9 + 38959469703544326596092v^8 - 449503757829354878585096v^7 + 825981882874189932557056v^6 - 1621157387025078645325152v^5 + 2003438432384731993177024v^4 - 3755942242288309796882992v^3 + 1286677423391995969163200v^2 - 1332221403888018014991392*v + 94236309393562936299712) / 346931609914134839972480
$\beta_{2}$	$=$	$ ( 25\!\cdots\!12 \nu^{15} + \cdots - 31\!\cdots\!04 ) / 86\!\cdots\!20 $ (25569479030280566412v^15 - 135876466122824388149v^14 + 345190768387595378903v^13 + 430477237088200538532v^12 + 8635195911545456863880v^11 - 35865907280266927386845v^10 + 61329501638043960405891v^9 - 42773987978022886175024v^8 + 435067467376470129669992v^7 - 1712413047279119646456252v^6 + 3217485504535708805590224v^5 - 2744439062106117518346848v^4 + 1007413334526259350866064v^3 - 99682082229361135065120v^2 + 4173357676598058424304*v - 311438671188616313007904) / 86732902478533709993120
$\beta_{3}$	$=$	$ ( - 57\!\cdots\!99 \nu^{15} + \cdots - 72\!\cdots\!32 ) / 17\!\cdots\!40 $ (-57299681650542813799v^15 + 310762723733366637388v^14 - 797609807237825424606v^13 - 889272987826610412594v^12 - 19198824197009526898095v^11 + 82758888276838958876400v^10 - 141781898740870143152142v^9 + 115423269614068896266898v^8 - 965514734333909107594024v^7 + 3940945426884564607889904v^6 - 7511743716854025606700288v^5 + 6833395910542852553834256v^4 - 2355692976644302317089648v^3 + 226667030484296414298880v^2 - 8823571164277865133088*v - 72370372478847279136032) / 173465804957067419986240
$\beta_{4}$	$=$	$ ( - 43\!\cdots\!39 \nu^{15} + \cdots + 37\!\cdots\!88 ) / 10\!\cdots\!40 $ (-439455554879435383139v^15 + 2241122015124087814768v^14 - 5561764669179045278226v^13 - 7995018565696568059124v^12 - 151253320213586326551555v^11 + 581239161047249858179860v^10 - 964687428832831710042042v^9 + 700416946484716590460688v^8 - 7432691074169667193750664v^7 + 27883629363643201451158784v^6 - 51191725696677861835307008v^5 + 43448370476958442988560896v^4 - 15979835659401989048171248v^3 + 1654837676509585409533440v^2 - 76921847512402003069408*v + 3743822249151046185922688) / 1040794829742404519917440
$\beta_{5}$	$=$	$ ( - 37\!\cdots\!43 \nu^{15} + \cdots + 17\!\cdots\!88 ) / 69\!\cdots\!96 $ (-37104260317816956443v^15 + 151526112378019231664v^14 - 321061485849858930146v^13 - 974908431881953254128v^12 - 13835641266715917580139v^11 + 34947175687175212714580v^10 - 47875325984313322571514v^9 + 16736683687098622304212v^8 - 625533344564557583278984v^7 + 1734559757843010960652096v^6 - 2643931979850558069220992v^5 + 1293457952188598334348928v^4 - 803716592544779228231536v^3 + 117111471819228528626944v^2 - 8800959934029242894816*v + 17380390028458691226688) / 69386321982826967994496
$\beta_{6}$	$=$	$ ( 13\!\cdots\!66 \nu^{15} + \cdots + 84\!\cdots\!08 ) / 17\!\cdots\!40 $ (139701703083084843466v^15 - 352863803987486548657v^14 + 350397790926717630654v^13 + 5332779684253886100096v^12 + 58315309974111227789410v^11 - 50079493872898502763745v^10 - 16471505765686500866422v^9 + 148038962812064696378968v^8 + 2321667818194567952846876v^7 - 2953484078021429244894616v^6 + 313489945615288993111112v^5 + 7777687897607258688480656v^4 - 107340536214601554404528v^3 - 309337175284090395834640v^2 + 46133565624937104022592*v + 844306681429309503644608) / 173465804957067419986240
$\beta_{7}$	$=$	$ ( 34\!\cdots\!24 \nu^{15} + \cdots + 16\!\cdots\!24 ) / 34\!\cdots\!48 $ (34009524100498954924v^15 - 93876249017534482861v^14 + 116684157278614951436v^13 + 1236916786086591809948v^12 + 13968426198284995440156v^11 - 15179392427921230632309v^10 + 3150971691525804233672v^9 + 28027087138017880071620v^8 + 566002065946927646456116v^7 - 850100044846283359262720v^6 + 430569310029225112211832v^5 + 1425441230980083855222048v^4 + 89009254303847462446576v^3 - 80106538518255115634544v^2 + 10748186561286905391968*v + 163776380549154061343424) / 34693160991413483997248
$\beta_{8}$	$=$	$ ( - 22\!\cdots\!08 \nu^{15} + \cdots + 55\!\cdots\!96 ) / 17\!\cdots\!40 $ (-223356151832978209908v^15 + 880505055132253233071v^14 - 1717983330107770032102v^13 - 6445092595785024774798v^12 - 83457016995930477068460v^11 + 200933097563041183630095v^10 - 225167864663606741739234v^9 + 623219353097996059986v^8 - 3638831396467228504089708v^7 + 9878598436726013996313568v^6 - 13129113754973767703626776v^5 + 2128792998735165940548672v^4 + 1801469580584110661948304v^3 - 1858437542654510627821200v^2 - 738358944973925573408256*v + 55233526678960890115296) / 173465804957067419986240
$\beta_{9}$	$=$	$ ( 14\!\cdots\!01 \nu^{15} + \cdots - 19\!\cdots\!72 ) / 10\!\cdots\!40 $ (1467508247969570962901v^15 - 5233002171242895530902v^14 + 9692947659699732558114v^13 + 44025670310952507825176v^12 + 569272045850750951194965v^11 - 1098043412608702503400770v^10 + 1190786430340385736616338v^9 - 91948359191815807646372v^8 + 24499444218239838062427296v^7 - 56117189935335908355064496v^6 + 71186232994978237024199152v^5 - 11673278022187112926811424v^4 + 20939510598395836955100592v^3 - 4166312496714838271935200v^2 + 391637599271987046216352*v - 1932955519447723140567872) / 1040794829742404519917440
$\beta_{10}$	$=$	$ ( - 18\!\cdots\!28 \nu^{15} + \cdots + 81\!\cdots\!96 ) / 81\!\cdots\!60 $ (-1856365266029439028v^15 + 7283657711690606901v^14 - 14306448104402867992v^13 - 53067119951003531693v^12 - 696404637902714743190v^11 + 1653992395133468254305v^10 - 1899083034361956529764v^9 + 161278877668106868691v^8 - 30518839909115709877418v^7 + 81918401702948551124388v^6 - 110192195306696806738916v^5 + 24908556001061761521512v^4 - 970114662909385438936v^3 + 507745037561589969520v^2 - 11216975671400262226896*v + 817166959957868757296) / 810587873631156168160
$\beta_{11}$	$=$	$ ( - 31\!\cdots\!31 \nu^{15} + \cdots + 18\!\cdots\!32 ) / 10\!\cdots\!40 $ (-3161330175133885254831v^15 + 12185535405329615195492v^14 - 24185853183290763727854v^13 - 89828524897078368355296v^12 - 1195903546080911591781535v^11 + 2713239457625169804239320v^10 - 3308875375372790427186838v^9 + 486411393437322269222212v^8 - 52314610125801861039516216v^7 + 135911502779325343997747216v^6 - 189225266087830922589685312v^5 + 52211043531645188945835584v^4 - 23214628329792689250941552v^3 - 1211545795163318363457600v^2 - 25987382272721391873487392*v + 1876980550567798417556032) / 1040794829742404519917440
$\beta_{12}$	$=$	$ ( 22\!\cdots\!75 \nu^{15} + \cdots - 11\!\cdots\!28 ) / 34\!\cdots\!48 $ (229762909361918816175v^15 - 901020790322433139847v^14 + 1773054069533715792634v^13 + 6545072874457425453552v^12 + 86281763985010554218963v^11 - 204495788443437004457355v^10 + 235921791902117618025326v^9 - 27015465087264750298364v^8 + 3791267880997429381455816v^7 - 10142129746831373977569608v^6 + 13689244198922517790399968v^5 - 3408341800529148310923648v^4 + 838043210186307474026992v^3 - 749426359272182604170864v^2 + 1617327804848636083628576*v - 117283563291563124756928) / 34693160991413483997248
$\beta_{13}$	$=$	$ ( 16\!\cdots\!81 \nu^{15} + \cdots - 71\!\cdots\!92 ) / 17\!\cdots\!40 $ (1647786134581718440981v^15 - 6489058375714485668502v^14 + 12726226693636926236734v^13 + 47105813531921546323536v^12 + 617241953105613212778205v^11 - 1479479863038053949806330v^10 + 1679834085428337891531678v^9 - 137215035263440896446332v^8 + 27082925975940619456063336v^7 - 73166458764427204113160696v^6 + 97753879874209701251464832v^5 - 21834891657963530120968224v^4 + 263816343263634857645072v^3 - 3441505923093088420226080v^2 + 9764350823638301822300192*v - 711791183647280820656192) / 173465804957067419986240
$\beta_{14}$	$=$	$ ( 10\!\cdots\!79 \nu^{15} + \cdots - 49\!\cdots\!28 ) / 10\!\cdots\!40 $ (10521267573052040758179v^15 - 41256382875221750700458v^14 + 81105100589352480044706v^13 + 300354967909233596033724v^12 + 3949213686776762072093155v^11 - 9362653473555985745334190v^10 + 10782076755025973164713922v^9 - 1044812632511988236449528v^8 + 173235171883211922179021424v^7 - 464019093201352332432477584v^6 + 625505202864271357582465648v^5 - 147200343497065264843608896v^4 + 18778730745673106956448528v^3 - 14771338934975639530906080v^2 + 67813411026733915122914208*v - 4930276316315155162179328) / 1040794829742404519917440
$\beta_{15}$	$=$	$ ( - 20\!\cdots\!19 \nu^{15} + \cdots + 86\!\cdots\!48 ) / 17\!\cdots\!40 $ (-2014702384092608461819v^15 + 7931243684916089679668v^14 - 15554783592512782860496v^13 - 57610372955257229156194v^12 - 754745724599480418409495v^11 + 1807616767329908505829800v^10 - 2054011490558481066866392v^9 + 163825187962440470325338v^8 - 33105615204951311090397084v^7 + 89399980582648206502360744v^6 - 119495653065311981325644808v^5 + 26514840899483907674201376v^4 + 80965926607595525060032v^3 + 3458967843824083491365280v^2 - 11810244640222325675946208*v + 861240134394725537790048) / 173465804957067419986240

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

$\nu$	$=$	$ ( \beta_{15} + \beta_{13} - \beta_{10} - \beta_{7} + \beta_{6} ) / 2 $ (b15 + b13 - b10 - b7 + b6) / 2
$\nu^{2}$	$=$	$ \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - 6\beta_{10} + 2\beta_{8} $ b15 + b14 - b12 + b11 - 6b10 + 2b8
$\nu^{3}$	$=$	$ ( 15 \beta_{15} + 5 \beta_{14} + 15 \beta_{13} - 7 \beta_{12} + 2 \beta_{11} - 18 \beta_{10} - 2 \beta_{9} + \cdots - 8 ) / 2 $ (15b15 + 5b14 + 15b13 - 7b12 + 2b11 - 18b10 - 2b9 + 6b8 + 15b7 - 15b6 - b5 - 5b4 + b3 - 7b2 - b1 - 8) / 2
$\nu^{4}$	$=$	$ -23\beta_{9} + 37\beta_{7} - 16\beta_{6} + 4\beta_{5} - 35\beta_{4} - 15\beta_{3} - 29\beta_{2} - 125 $ -23b9 + 37b7 - 16b6 + 4b5 - 35b4 - 15b3 - 29*b2 - 125
$\nu^{5}$	$=$	$ ( - 333 \beta_{15} - 181 \beta_{14} - 297 \beta_{13} + 219 \beta_{12} - 82 \beta_{11} + 524 \beta_{10} + \cdots - 408 ) / 2 $ (-333b15 - 181b14 - 297b13 + 219b12 - 82b11 + 524b10 - 82b9 - 222b8 + 333b7 - 297b6 - 29b5 - 181b4 - 3b3 - 219b2 + 29*b1 - 408) / 2
$\nu^{6}$	$=$	$ - 1101 \beta_{15} - 923 \beta_{14} - 638 \beta_{13} + 793 \beta_{12} - 545 \beta_{11} + 2482 \beta_{10} + \cdots - 82 \beta_1 $ -1101b15 - 923b14 - 638b13 + 793b12 - 545b11 + 2482b10 - 1068b8 - 82b1
$\nu^{7}$	$=$	$ ( - 8539 \beta_{15} - 5455 \beta_{14} - 6931 \beta_{13} + 6049 \beta_{12} - 2698 \beta_{11} + \cdots + 13796 ) / 2 $ (-8539b15 - 5455b14 - 6931b13 + 6049b12 - 2698b11 + 15272b10 + 2698b9 - 6650b8 - 8539b7 + 6931b6 + 555b5 + 5455b4 + 601b3 + 6049b2 + 555*b1 + 13796) / 2
$\nu^{8}$	$=$	$ 13535 \beta_{9} - 30705 \beta_{7} + 19868 \beta_{6} - 1292 \beta_{5} + 23987 \beta_{4} + 5879 \beta_{3} + \cdots + 67757 $ 13535b9 - 30705b7 + 19868b6 - 1292b5 + 23987b4 + 5879b3 + 21689*b2 + 67757
$\nu^{9}$	$=$	$ ( 228693 \beta_{15} + 156049 \beta_{14} + 175233 \beta_{13} - 163999 \beta_{12} + 80530 \beta_{11} + \cdots + 411416 ) / 2 $ (228693b15 + 156049b14 + 175233b13 - 163999b12 + 80530b11 - 430600b10 + 80530b9 + 187902b8 - 228693b7 + 175233b6 + 9441b5 + 156049b4 + 23903b3 + 163999b2 - 9441*b1 + 411416) / 2
$\nu^{10}$	$=$	$ 840209 \beta_{15} + 632431 \beta_{14} + 572066 \beta_{13} - 592981 \beta_{12} + 348673 \beta_{11} + \cdots + 17622 \beta_1 $ 840209b15 + 632431b14 + 572066b13 - 592981b12 + 348673b11 - 1685906b10 + 731924b8 + 17622b1
$\nu^{11}$	$=$	$ ( 6197443 \beta_{15} + 4357551 \beta_{14} + 4599939 \beta_{13} - 4440465 \beta_{12} + 2294850 \beta_{11} + \cdots - 11675244 ) / 2 $ (6197443b15 + 4357551b14 + 4599939b13 - 4440465b12 + 2294850b11 - 11917632b10 - 2294850b9 + 5196474b8 + 6197443b7 - 4599939b6 - 157907b5 - 4357551b4 - 756009b3 - 4440465b2 - 157907*b1 - 11675244) / 2
$\nu^{12}$	$=$	$ - 9198311 \beta_{9} + 22872101 \beta_{7} - 15968640 \beta_{6} + 198152 \beta_{5} - 16890359 \beta_{4} + \cdots - 46197501 $ -9198311b9 + 22872101b7 - 15968640b6 + 198152b5 - 16890359b4 - 3493971b3 - 16184341*b2 - 46197501
$\nu^{13}$	$=$	$ ( - 168436077 \beta_{15} - 120154729 \beta_{14} - 122923041 \beta_{13} + 120385895 \beta_{12} + \cdots - 324105616 ) / 2 $ (-168436077b15 - 120154729b14 - 122923041b13 + 120385895b12 - 63909658b11 + 326873928b10 - 63909658b9 - 142418846b8 + 168436077b7 - 122923041b6 - 2751249b5 - 120154729b4 - 22032951b3 - 120385895b2 + 2751249*b1 - 324105616) / 2
$\nu^{14}$	$=$	$ - 621856845 \beta_{15} - 454761563 \beta_{14} - 439706318 \beta_{13} + 441047113 \beta_{12} + \cdots - 1156906 \beta_1 $ -621856845b15 - 454761563b14 - 439706318b13 + 441047113b12 - 246060185b11 + 1223537266b10 - 532130908b8 - 1156906b1
$\nu^{15}$	$=$	$ ( - 4580463835 \beta_{15} - 3291035767 \beta_{14} - 3313391155 \beta_{13} + 3267676777 \beta_{12} + \cdots + 8903892116 ) / 2 $ (-4580463835b15 - 3291035767b14 - 3313391155b13 + 3267676777b12 - 1759193050b11 + 8926247504b10 + 1759193050b9 - 3887314778b8 - 4580463835b7 + 3313391155b6 + 51934243b5 + 3291035767b4 + 619638001b3 + 3267676777b2 + 51934243*b1 + 8903892116) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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} $

1471.1

1.49885	+	1.49885i
−0.373815	−	0.373815i
1.30491	+	1.30491i
−2.02116	−	2.02116i
−2.64546	−	2.64546i
3.68710	+	3.68710i
0.476829	+	0.476829i
0.0727486	+	0.0727486i
1.49885	−	1.49885i
−0.373815	+	0.373815i
1.30491	−	1.30491i
−2.02116	+	2.02116i
−2.64546	+	2.64546i
3.68710	−	3.68710i
0.476829	−	0.476829i
0.0727486	−	0.0727486i

−

1.00000i

−

1.00000i

−4.84428

−1.00000

1471.2

−

1.00000i

−

1.00000i

−1.61153

−1.00000

1471.3

−

1.00000i

−

1.00000i

−0.482745

−1.00000

1471.4

−

1.00000i

−

1.00000i

−0.279423

−1.00000

1471.5

−

1.00000i

−

1.00000i

1.23448

−1.00000

1471.6

−

1.00000i

−

1.00000i

1.58474

−1.00000

1471.7

−

1.00000i

−

1.00000i

3.79952

−1.00000

1471.8

−

1.00000i

−

1.00000i

4.59925

−1.00000

1471.9

1.00000i

−4.84428

−1.00000

1471.10

1.00000i

−1.61153

−1.00000

1471.11

1.00000i

−0.482745

−1.00000

1471.12

1.00000i

−0.279423

−1.00000

1471.13

1.00000i

1.23448

−1.00000

1471.14

1.00000i

1.58474

−1.00000

1471.15

1.00000i

3.79952

−1.00000

1471.16

1.00000i

4.59925

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
92.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5520.2.be.b	yes	16
4.b	odd	2	1		5520.2.be.a	✓	16
23.b	odd	2	1		5520.2.be.a	✓	16
92.b	even	2	1	inner	5520.2.be.b	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5520.2.be.a	✓	16	4.b	odd	2	1
5520.2.be.a	✓	16	23.b	odd	2	1
5520.2.be.b	yes	16	1.a	even	1	1	trivial
5520.2.be.b	yes	16	92.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} - 4T_{7}^{7} - 25T_{7}^{6} + 108T_{7}^{5} + 39T_{7}^{4} - 316T_{7}^{3} + 37T_{7}^{2} + 164T_{7} + 36 $ T7^8 - 4*T7^7 - 25*T7^6 + 108*T7^5 + 39*T7^4 - 316*T7^3 + 37*T7^2 + 164*T7 + 36 acting on $S_{2}^{\mathrm{new}}(5520, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$5$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$7$	$ (T^{8} - 4 T^{7} - 25 T^{6} + \cdots + 36)^{2} $ (T^8 - 4T^7 - 25T^6 + 108T^5 + 39T^4 - 316T^3 + 37T^2 + 164*T + 36)^2
$11$	$ (T^{8} + 4 T^{7} + \cdots + 800)^{2} $ (T^8 + 4T^7 - 40T^6 - 216T^5 + 60T^4 + 1984T^3 + 3988T^2 + 3040*T + 800)^2
$13$	$ (T^{8} - 4 T^{7} + \cdots + 592)^{2} $ (T^8 - 4T^7 - 40T^6 + 92T^5 + 352T^4 - 624T^3 - 780T^2 + 816*T + 592)^2
$17$	$ T^{16} + 94 T^{14} + \cdots + 1354896 $ T^16 + 94T^14 + 3247T^12 + 51764T^10 + 401575T^8 + 1606646T^6 + 3360361T^4 + 3447736*T^2 + 1354896
$19$	$ (T^{8} - 102 T^{6} + \cdots - 17672)^{2} $ (T^8 - 102T^6 + 64T^5 + 2696T^4 - 3528T^3 - 13804T^2 + 31760T - 17672)^2
$23$	$ T^{16} + \cdots + 78310985281 $ T^16 + 12T^15 + 72T^14 + 540T^13 + 2844T^12 + 6828T^11 + 21880T^10 + 8284T^9 - 600954T^8 + 190532T^7 + 11574520T^6 + 83076276T^5 + 795867804T^4 + 3475625220T^3 + 10658584008T^2 + 40857905364*T + 78310985281
$29$	$ (T^{8} + 2 T^{7} + \cdots + 540)^{2} $ (T^8 + 2T^7 - 113T^6 - 64T^5 + 3563T^4 - 3190T^3 - 20403T^2 + 14020*T + 540)^2
$31$	$ T^{16} + \cdots + 10223636544 $ T^16 + 258T^14 + 26695T^12 + 1436588T^10 + 43467807T^8 + 740738018T^6 + 6607070457T^4 + 24444959440*T^2 + 10223636544
$37$	$ T^{16} + \cdots + 56667802500 $ T^16 + 250T^14 + 24035T^12 + 1197664T^10 + 34217739T^8 + 575371058T^6 + 5549557933T^4 + 27998279316*T^2 + 56667802500
$41$	$ (T^{8} - 2 T^{7} + \cdots - 127396)^{2} $ (T^8 - 2T^7 - 141T^6 + 608T^5 + 4375T^4 - 29090T^3 + 22381T^2 + 108340*T - 127396)^2
$43$	$ (T^{8} - 208 T^{6} + \cdots + 2022592)^{2} $ (T^8 - 208T^6 + 48T^5 + 14000T^4 - 6560T^3 - 329776T^2 + 210496T + 2022592)^2
$47$	$ T^{16} + \cdots + 1006281834496 $ T^16 + 384T^14 + 57928T^12 + 4435512T^10 + 187253328T^8 + 4486194464T^6 + 59719495440T^4 + 400649540608*T^2 + 1006281834496
$53$	$ T^{16} + \cdots + 1337788330384 $ T^16 + 502T^14 + 103103T^12 + 11131204T^10 + 672051079T^8 + 22004448414T^6 + 334723406649T^4 + 1343036772264*T^2 + 1337788330384
$59$	$ T^{16} + \cdots + 99130522500 $ T^16 + 342T^14 + 44987T^12 + 2988528T^10 + 108621347T^8 + 2153638294T^6 + 21456226381T^4 + 86338855300*T^2 + 99130522500
$61$	$ T^{16} + \cdots + 713958400 $ T^16 + 488T^14 + 93600T^12 + 8915784T^10 + 438981792T^8 + 10653617664T^6 + 116846287376T^4 + 464349666560*T^2 + 713958400
$67$	$ (T^{8} - 8 T^{7} + \cdots + 1963932)^{2} $ (T^8 - 8T^7 - 225T^6 + 1260T^5 + 16391T^4 - 51744T^3 - 414555T^2 + 180556*T + 1963932)^2
$71$	$ T^{16} + \cdots + 925232372100 $ T^16 + 830T^14 + 269771T^12 + 43210960T^10 + 3489208643T^8 + 126576746382T^6 + 1381219933533T^4 + 2463835005316*T^2 + 925232372100
$73$	$ (T^{8} - 20 T^{7} + \cdots + 36784)^{2} $ (T^8 - 20T^7 - 88T^6 + 2404T^5 + 2576T^4 - 28864T^3 - 11628T^2 + 73040*T + 36784)^2
$79$	$ (T^{8} + 16 T^{7} + \cdots - 958464)^{2} $ (T^8 + 16T^7 - 282T^6 - 4304T^5 + 11680T^4 + 136704T^3 - 34304T^2 - 1085440*T - 958464)^2
$83$	$ (T^{8} - 317 T^{6} + \cdots - 700572)^{2} $ (T^8 - 317T^6 - 412T^5 + 21231T^4 - 15792T^3 - 453931T^2 + 1298396T - 700572)^2
$89$	$ T^{16} + \cdots + 71156629504 $ T^16 + 488T^14 + 89520T^12 + 7763968T^10 + 335213312T^8 + 7033284608T^6 + 64438915072T^4 + 188835102720*T^2 + 71156629504
$97$	$ T^{16} + \cdots + 10557973504 $ T^16 + 772T^14 + 202452T^12 + 23048576T^10 + 1145918016T^8 + 21868471936T^6 + 94700347136T^4 + 106855160832*T^2 + 10557973504
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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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{10} q^{3} + \beta_{10} q^{5} + (\beta_{4} + 1) q^{7} - q^{9}+O(q^{10}) \) q + b10 * q^3 + b10 * q^5 + (b4 + 1) * q^7 - q^9	\( q + \beta_{10} q^{3} + \beta_{10} q^{5} + (\beta_{4} + 1) q^{7} - q^{9} - \beta_{6} q^{11} + ( - \beta_{3} + 1) q^{13} - q^{15} + (\beta_{14} + \beta_1) q^{17} + (\beta_{9} + \beta_{4} + \beta_{3}) q^{19} - \beta_{14} q^{21} + ( - \beta_{15} + \beta_{5} - 1) q^{23} - q^{25} - \beta_{10} q^{27} + ( - \beta_{6} + \beta_{2}) q^{29} + (\beta_{15} + \beta_{14} + \cdots + \beta_1) q^{31}+ \cdots + \beta_{6} q^{99}+O(q^{100}) \) q + b10 * q^3 + b10 * q^5 + (b4 + 1) * q^7 - q^9 - b6 * q^11 + (-b3 + 1) * q^13 - q^15 + (b14 + b1) * q^17 + (b9 + b4 + b3) * q^19 - b14 * q^21 + (-b15 + b5 - 1) * q^23 - q^25 - b10 * q^27 + (-b6 + b2) * q^29 + (b15 + b14 + b13 + b12 + b11 + b1) * q^31 + (b13 - b10) * q^33 - b14 * q^35 + (-b15 - b12 - b10) * q^37 + (b10 + b8) * q^39 + (b9 - b7 + b6 - b5) * q^41 + (b9 - b7 - b4 - b3) * q^43 - b10 * q^45 + (b14 + b13 - b12 + b1) * q^47 + (-b9 - b7 + 2b6 - 2b5 + b4 + b3 - b2 + 1) * q^49 + (-b5 + b4 + 1) * q^51 + (-b15 + b14 + b12 + b11 - b8) * q^53 + (b13 - b10) * q^55 + (-b14 - b11 - b10 - b8) * q^57 + (b15 + b14 - 2b10 + 2b1) * q^59 + (b14 + b12 - 2b10 + b8 + b1) q^61 + (-b4 - 1) * q^63 + (b10 + b8) * q^65 + (-b9 + b7 - 2b6 + 2b5 - 2b4 - b3 + 1) q^67 + (-b10 + b7 + b1) * q^69 + (-2b15 - 2b13 + b11 + b10 - b8) * q^71 + (2b7 - 2b6 + b3 + 3) * q^73 - b10 * q^75 + (-b9 - b7 - 2b6 - 3b4 + 1) * q^77 + (-b7 + 3b6 - b5 + b4 - b3 - b2 - 2) q^79 + q^81 + (-2b7 - b5 + b2) q^83 + (-b5 + b4 + 1) * q^85 + (b13 + b12 - b10 - b8) * q^87 + (-2b15 - 2b14 - 2b13 + 2b10 - 2b1) q^89 + (-b9 - b7 + b6 - b5 + 2b3 - b2 + 2) q^91 + (b9 - b7 + b6 - b5 + b4 + b3 - b2) * q^93 + (-b14 - b11 - b10 - b8) * q^95 + (-2b15 - b14 - b13 + b11 + b8) q^97 + b6 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) \) 16 * q + 8 * q^7 - 16 * q^9	\( 16 q + 8 q^{7} - 16 q^{9} - 8 q^{11} + 8 q^{13} - 16 q^{15} - 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} + 4 q^{51} - 8 q^{63} + 16 q^{67} + 40 q^{73} + 24 q^{77} - 32 q^{79} + 16 q^{81} + 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) \) 16 * q + 8 * q^7 - 16 * q^9 - 8 * q^11 + 8 * q^13 - 16 * q^15 - 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 + 4 * q^51 - 8 * q^63 + 16 * q^67 + 40 * q^73 + 24 * q^77 - 32 * q^79 + 16 * q^81 + 4 * q^85 + 48 * q^91 + 8 * 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	5520.2.be.b

Level	$5520$
Weight	$2$
Character orbit	5520.be
Analytic conductor	$44.077$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(44.0774219157\)
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:	\( x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} + \cdots + 64 \) x^16 - 4x^15 + 8x^14 + 28x^13 + 373x^12 - 920x^11 + 1088x^10 - 168x^9 + 16460x^8 - 45408x^7 + 62624x^6 - 18048x^5 + 2160x^4 - 1664x^3 + 6272x^2 - 896*x + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 22\!\cdots\!31 \nu^{15} + \cdots + 94\!\cdots\!12 ) / 34\!\cdots\!80 \) (-22393307326694017831v^15 + 79205853823130329892v^14 - 175071232472117903174v^13 - 526628042123880972776v^12 - 9091844016900447538055v^11 + 15856188225716940095240v^10 - 27782783728011205760238v^9 + 38959469703544326596092v^8 - 449503757829354878585096v^7 + 825981882874189932557056v^6 - 1621157387025078645325152v^5 + 2003438432384731993177024v^4 - 3755942242288309796882992v^3 + 1286677423391995969163200v^2 - 1332221403888018014991392*v + 94236309393562936299712) / 346931609914134839972480
\(\beta_{2}\)	\(=\)	\( ( 25\!\cdots\!12 \nu^{15} + \cdots - 31\!\cdots\!04 ) / 86\!\cdots\!20 \) (25569479030280566412v^15 - 135876466122824388149v^14 + 345190768387595378903v^13 + 430477237088200538532v^12 + 8635195911545456863880v^11 - 35865907280266927386845v^10 + 61329501638043960405891v^9 - 42773987978022886175024v^8 + 435067467376470129669992v^7 - 1712413047279119646456252v^6 + 3217485504535708805590224v^5 - 2744439062106117518346848v^4 + 1007413334526259350866064v^3 - 99682082229361135065120v^2 + 4173357676598058424304*v - 311438671188616313007904) / 86732902478533709993120
\(\beta_{3}\)	\(=\)	\( ( - 57\!\cdots\!99 \nu^{15} + \cdots - 72\!\cdots\!32 ) / 17\!\cdots\!40 \) (-57299681650542813799v^15 + 310762723733366637388v^14 - 797609807237825424606v^13 - 889272987826610412594v^12 - 19198824197009526898095v^11 + 82758888276838958876400v^10 - 141781898740870143152142v^9 + 115423269614068896266898v^8 - 965514734333909107594024v^7 + 3940945426884564607889904v^6 - 7511743716854025606700288v^5 + 6833395910542852553834256v^4 - 2355692976644302317089648v^3 + 226667030484296414298880v^2 - 8823571164277865133088*v - 72370372478847279136032) / 173465804957067419986240
\(\beta_{4}\)	\(=\)	\( ( - 43\!\cdots\!39 \nu^{15} + \cdots + 37\!\cdots\!88 ) / 10\!\cdots\!40 \) (-439455554879435383139v^15 + 2241122015124087814768v^14 - 5561764669179045278226v^13 - 7995018565696568059124v^12 - 151253320213586326551555v^11 + 581239161047249858179860v^10 - 964687428832831710042042v^9 + 700416946484716590460688v^8 - 7432691074169667193750664v^7 + 27883629363643201451158784v^6 - 51191725696677861835307008v^5 + 43448370476958442988560896v^4 - 15979835659401989048171248v^3 + 1654837676509585409533440v^2 - 76921847512402003069408*v + 3743822249151046185922688) / 1040794829742404519917440
\(\beta_{5}\)	\(=\)	\( ( - 37\!\cdots\!43 \nu^{15} + \cdots + 17\!\cdots\!88 ) / 69\!\cdots\!96 \) (-37104260317816956443v^15 + 151526112378019231664v^14 - 321061485849858930146v^13 - 974908431881953254128v^12 - 13835641266715917580139v^11 + 34947175687175212714580v^10 - 47875325984313322571514v^9 + 16736683687098622304212v^8 - 625533344564557583278984v^7 + 1734559757843010960652096v^6 - 2643931979850558069220992v^5 + 1293457952188598334348928v^4 - 803716592544779228231536v^3 + 117111471819228528626944v^2 - 8800959934029242894816*v + 17380390028458691226688) / 69386321982826967994496
\(\beta_{6}\)	\(=\)	\( ( 13\!\cdots\!66 \nu^{15} + \cdots + 84\!\cdots\!08 ) / 17\!\cdots\!40 \) (139701703083084843466v^15 - 352863803987486548657v^14 + 350397790926717630654v^13 + 5332779684253886100096v^12 + 58315309974111227789410v^11 - 50079493872898502763745v^10 - 16471505765686500866422v^9 + 148038962812064696378968v^8 + 2321667818194567952846876v^7 - 2953484078021429244894616v^6 + 313489945615288993111112v^5 + 7777687897607258688480656v^4 - 107340536214601554404528v^3 - 309337175284090395834640v^2 + 46133565624937104022592*v + 844306681429309503644608) / 173465804957067419986240
\(\beta_{7}\)	\(=\)	\( ( 34\!\cdots\!24 \nu^{15} + \cdots + 16\!\cdots\!24 ) / 34\!\cdots\!48 \) (34009524100498954924v^15 - 93876249017534482861v^14 + 116684157278614951436v^13 + 1236916786086591809948v^12 + 13968426198284995440156v^11 - 15179392427921230632309v^10 + 3150971691525804233672v^9 + 28027087138017880071620v^8 + 566002065946927646456116v^7 - 850100044846283359262720v^6 + 430569310029225112211832v^5 + 1425441230980083855222048v^4 + 89009254303847462446576v^3 - 80106538518255115634544v^2 + 10748186561286905391968*v + 163776380549154061343424) / 34693160991413483997248
\(\beta_{8}\)	\(=\)	\( ( - 22\!\cdots\!08 \nu^{15} + \cdots + 55\!\cdots\!96 ) / 17\!\cdots\!40 \) (-223356151832978209908v^15 + 880505055132253233071v^14 - 1717983330107770032102v^13 - 6445092595785024774798v^12 - 83457016995930477068460v^11 + 200933097563041183630095v^10 - 225167864663606741739234v^9 + 623219353097996059986v^8 - 3638831396467228504089708v^7 + 9878598436726013996313568v^6 - 13129113754973767703626776v^5 + 2128792998735165940548672v^4 + 1801469580584110661948304v^3 - 1858437542654510627821200v^2 - 738358944973925573408256*v + 55233526678960890115296) / 173465804957067419986240
\(\beta_{9}\)	\(=\)	\( ( 14\!\cdots\!01 \nu^{15} + \cdots - 19\!\cdots\!72 ) / 10\!\cdots\!40 \) (1467508247969570962901v^15 - 5233002171242895530902v^14 + 9692947659699732558114v^13 + 44025670310952507825176v^12 + 569272045850750951194965v^11 - 1098043412608702503400770v^10 + 1190786430340385736616338v^9 - 91948359191815807646372v^8 + 24499444218239838062427296v^7 - 56117189935335908355064496v^6 + 71186232994978237024199152v^5 - 11673278022187112926811424v^4 + 20939510598395836955100592v^3 - 4166312496714838271935200v^2 + 391637599271987046216352*v - 1932955519447723140567872) / 1040794829742404519917440
\(\beta_{10}\)	\(=\)	\( ( - 18\!\cdots\!28 \nu^{15} + \cdots + 81\!\cdots\!96 ) / 81\!\cdots\!60 \) (-1856365266029439028v^15 + 7283657711690606901v^14 - 14306448104402867992v^13 - 53067119951003531693v^12 - 696404637902714743190v^11 + 1653992395133468254305v^10 - 1899083034361956529764v^9 + 161278877668106868691v^8 - 30518839909115709877418v^7 + 81918401702948551124388v^6 - 110192195306696806738916v^5 + 24908556001061761521512v^4 - 970114662909385438936v^3 + 507745037561589969520v^2 - 11216975671400262226896*v + 817166959957868757296) / 810587873631156168160
\(\beta_{11}\)	\(=\)	\( ( - 31\!\cdots\!31 \nu^{15} + \cdots + 18\!\cdots\!32 ) / 10\!\cdots\!40 \) (-3161330175133885254831v^15 + 12185535405329615195492v^14 - 24185853183290763727854v^13 - 89828524897078368355296v^12 - 1195903546080911591781535v^11 + 2713239457625169804239320v^10 - 3308875375372790427186838v^9 + 486411393437322269222212v^8 - 52314610125801861039516216v^7 + 135911502779325343997747216v^6 - 189225266087830922589685312v^5 + 52211043531645188945835584v^4 - 23214628329792689250941552v^3 - 1211545795163318363457600v^2 - 25987382272721391873487392*v + 1876980550567798417556032) / 1040794829742404519917440
\(\beta_{12}\)	\(=\)	\( ( 22\!\cdots\!75 \nu^{15} + \cdots - 11\!\cdots\!28 ) / 34\!\cdots\!48 \) (229762909361918816175v^15 - 901020790322433139847v^14 + 1773054069533715792634v^13 + 6545072874457425453552v^12 + 86281763985010554218963v^11 - 204495788443437004457355v^10 + 235921791902117618025326v^9 - 27015465087264750298364v^8 + 3791267880997429381455816v^7 - 10142129746831373977569608v^6 + 13689244198922517790399968v^5 - 3408341800529148310923648v^4 + 838043210186307474026992v^3 - 749426359272182604170864v^2 + 1617327804848636083628576*v - 117283563291563124756928) / 34693160991413483997248
\(\beta_{13}\)	\(=\)	\( ( 16\!\cdots\!81 \nu^{15} + \cdots - 71\!\cdots\!92 ) / 17\!\cdots\!40 \) (1647786134581718440981v^15 - 6489058375714485668502v^14 + 12726226693636926236734v^13 + 47105813531921546323536v^12 + 617241953105613212778205v^11 - 1479479863038053949806330v^10 + 1679834085428337891531678v^9 - 137215035263440896446332v^8 + 27082925975940619456063336v^7 - 73166458764427204113160696v^6 + 97753879874209701251464832v^5 - 21834891657963530120968224v^4 + 263816343263634857645072v^3 - 3441505923093088420226080v^2 + 9764350823638301822300192*v - 711791183647280820656192) / 173465804957067419986240
\(\beta_{14}\)	\(=\)	\( ( 10\!\cdots\!79 \nu^{15} + \cdots - 49\!\cdots\!28 ) / 10\!\cdots\!40 \) (10521267573052040758179v^15 - 41256382875221750700458v^14 + 81105100589352480044706v^13 + 300354967909233596033724v^12 + 3949213686776762072093155v^11 - 9362653473555985745334190v^10 + 10782076755025973164713922v^9 - 1044812632511988236449528v^8 + 173235171883211922179021424v^7 - 464019093201352332432477584v^6 + 625505202864271357582465648v^5 - 147200343497065264843608896v^4 + 18778730745673106956448528v^3 - 14771338934975639530906080v^2 + 67813411026733915122914208*v - 4930276316315155162179328) / 1040794829742404519917440
\(\beta_{15}\)	\(=\)	\( ( - 20\!\cdots\!19 \nu^{15} + \cdots + 86\!\cdots\!48 ) / 17\!\cdots\!40 \) (-2014702384092608461819v^15 + 7931243684916089679668v^14 - 15554783592512782860496v^13 - 57610372955257229156194v^12 - 754745724599480418409495v^11 + 1807616767329908505829800v^10 - 2054011490558481066866392v^9 + 163825187962440470325338v^8 - 33105615204951311090397084v^7 + 89399980582648206502360744v^6 - 119495653065311981325644808v^5 + 26514840899483907674201376v^4 + 80965926607595525060032v^3 + 3458967843824083491365280v^2 - 11810244640222325675946208*v + 861240134394725537790048) / 173465804957067419986240

\(\nu\)	\(=\)	\( ( \beta_{15} + \beta_{13} - \beta_{10} - \beta_{7} + \beta_{6} ) / 2 \) (b15 + b13 - b10 - b7 + b6) / 2
\(\nu^{2}\)	\(=\)	\( \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - 6\beta_{10} + 2\beta_{8} \) b15 + b14 - b12 + b11 - 6b10 + 2b8
\(\nu^{3}\)	\(=\)	\( ( 15 \beta_{15} + 5 \beta_{14} + 15 \beta_{13} - 7 \beta_{12} + 2 \beta_{11} - 18 \beta_{10} - 2 \beta_{9} + \cdots - 8 ) / 2 \) (15b15 + 5b14 + 15b13 - 7b12 + 2b11 - 18b10 - 2b9 + 6b8 + 15b7 - 15b6 - b5 - 5b4 + b3 - 7b2 - b1 - 8) / 2
\(\nu^{4}\)	\(=\)	\( -23\beta_{9} + 37\beta_{7} - 16\beta_{6} + 4\beta_{5} - 35\beta_{4} - 15\beta_{3} - 29\beta_{2} - 125 \) -23b9 + 37b7 - 16b6 + 4b5 - 35b4 - 15b3 - 29*b2 - 125
\(\nu^{5}\)	\(=\)	\( ( - 333 \beta_{15} - 181 \beta_{14} - 297 \beta_{13} + 219 \beta_{12} - 82 \beta_{11} + 524 \beta_{10} + \cdots - 408 ) / 2 \) (-333b15 - 181b14 - 297b13 + 219b12 - 82b11 + 524b10 - 82b9 - 222b8 + 333b7 - 297b6 - 29b5 - 181b4 - 3b3 - 219b2 + 29*b1 - 408) / 2
\(\nu^{6}\)	\(=\)	\( - 1101 \beta_{15} - 923 \beta_{14} - 638 \beta_{13} + 793 \beta_{12} - 545 \beta_{11} + 2482 \beta_{10} + \cdots - 82 \beta_1 \) -1101b15 - 923b14 - 638b13 + 793b12 - 545b11 + 2482b10 - 1068b8 - 82b1
\(\nu^{7}\)	\(=\)	\( ( - 8539 \beta_{15} - 5455 \beta_{14} - 6931 \beta_{13} + 6049 \beta_{12} - 2698 \beta_{11} + \cdots + 13796 ) / 2 \) (-8539b15 - 5455b14 - 6931b13 + 6049b12 - 2698b11 + 15272b10 + 2698b9 - 6650b8 - 8539b7 + 6931b6 + 555b5 + 5455b4 + 601b3 + 6049b2 + 555*b1 + 13796) / 2
\(\nu^{8}\)	\(=\)	\( 13535 \beta_{9} - 30705 \beta_{7} + 19868 \beta_{6} - 1292 \beta_{5} + 23987 \beta_{4} + 5879 \beta_{3} + \cdots + 67757 \) 13535b9 - 30705b7 + 19868b6 - 1292b5 + 23987b4 + 5879b3 + 21689*b2 + 67757
\(\nu^{9}\)	\(=\)	\( ( 228693 \beta_{15} + 156049 \beta_{14} + 175233 \beta_{13} - 163999 \beta_{12} + 80530 \beta_{11} + \cdots + 411416 ) / 2 \) (228693b15 + 156049b14 + 175233b13 - 163999b12 + 80530b11 - 430600b10 + 80530b9 + 187902b8 - 228693b7 + 175233b6 + 9441b5 + 156049b4 + 23903b3 + 163999b2 - 9441*b1 + 411416) / 2
\(\nu^{10}\)	\(=\)	\( 840209 \beta_{15} + 632431 \beta_{14} + 572066 \beta_{13} - 592981 \beta_{12} + 348673 \beta_{11} + \cdots + 17622 \beta_1 \) 840209b15 + 632431b14 + 572066b13 - 592981b12 + 348673b11 - 1685906b10 + 731924b8 + 17622b1
\(\nu^{11}\)	\(=\)	\( ( 6197443 \beta_{15} + 4357551 \beta_{14} + 4599939 \beta_{13} - 4440465 \beta_{12} + 2294850 \beta_{11} + \cdots - 11675244 ) / 2 \) (6197443b15 + 4357551b14 + 4599939b13 - 4440465b12 + 2294850b11 - 11917632b10 - 2294850b9 + 5196474b8 + 6197443b7 - 4599939b6 - 157907b5 - 4357551b4 - 756009b3 - 4440465b2 - 157907*b1 - 11675244) / 2
\(\nu^{12}\)	\(=\)	\( - 9198311 \beta_{9} + 22872101 \beta_{7} - 15968640 \beta_{6} + 198152 \beta_{5} - 16890359 \beta_{4} + \cdots - 46197501 \) -9198311b9 + 22872101b7 - 15968640b6 + 198152b5 - 16890359b4 - 3493971b3 - 16184341*b2 - 46197501
\(\nu^{13}\)	\(=\)	\( ( - 168436077 \beta_{15} - 120154729 \beta_{14} - 122923041 \beta_{13} + 120385895 \beta_{12} + \cdots - 324105616 ) / 2 \) (-168436077b15 - 120154729b14 - 122923041b13 + 120385895b12 - 63909658b11 + 326873928b10 - 63909658b9 - 142418846b8 + 168436077b7 - 122923041b6 - 2751249b5 - 120154729b4 - 22032951b3 - 120385895b2 + 2751249*b1 - 324105616) / 2
\(\nu^{14}\)	\(=\)	\( - 621856845 \beta_{15} - 454761563 \beta_{14} - 439706318 \beta_{13} + 441047113 \beta_{12} + \cdots - 1156906 \beta_1 \) -621856845b15 - 454761563b14 - 439706318b13 + 441047113b12 - 246060185b11 + 1223537266b10 - 532130908b8 - 1156906b1
\(\nu^{15}\)	\(=\)	\( ( - 4580463835 \beta_{15} - 3291035767 \beta_{14} - 3313391155 \beta_{13} + 3267676777 \beta_{12} + \cdots + 8903892116 ) / 2 \) (-4580463835b15 - 3291035767b14 - 3313391155b13 + 3267676777b12 - 1759193050b11 + 8926247504b10 + 1759193050b9 - 3887314778b8 - 4580463835b7 + 3313391155b6 + 51934243b5 + 3291035767b4 + 619638001b3 + 3267676777b2 + 51934243*b1 + 8903892116) / 2

\(n\)	\(1201\)	\(1381\)	\(1841\)	\(4417\)	\(4831\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1471.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$5$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$7$	\( (T^{8} - 4 T^{7} - 25 T^{6} + \cdots + 36)^{2} \) (T^8 - 4T^7 - 25T^6 + 108T^5 + 39T^4 - 316T^3 + 37T^2 + 164*T + 36)^2
$11$	\( (T^{8} + 4 T^{7} + \cdots + 800)^{2} \) (T^8 + 4T^7 - 40T^6 - 216T^5 + 60T^4 + 1984T^3 + 3988T^2 + 3040*T + 800)^2
$13$	\( (T^{8} - 4 T^{7} + \cdots + 592)^{2} \) (T^8 - 4T^7 - 40T^6 + 92T^5 + 352T^4 - 624T^3 - 780T^2 + 816*T + 592)^2
$17$	\( T^{16} + 94 T^{14} + \cdots + 1354896 \) T^16 + 94T^14 + 3247T^12 + 51764T^10 + 401575T^8 + 1606646T^6 + 3360361T^4 + 3447736*T^2 + 1354896
$19$	\( (T^{8} - 102 T^{6} + \cdots - 17672)^{2} \) (T^8 - 102T^6 + 64T^5 + 2696T^4 - 3528T^3 - 13804T^2 + 31760T - 17672)^2
$23$	\( T^{16} + \cdots + 78310985281 \) T^16 + 12T^15 + 72T^14 + 540T^13 + 2844T^12 + 6828T^11 + 21880T^10 + 8284T^9 - 600954T^8 + 190532T^7 + 11574520T^6 + 83076276T^5 + 795867804T^4 + 3475625220T^3 + 10658584008T^2 + 40857905364*T + 78310985281
$29$	\( (T^{8} + 2 T^{7} + \cdots + 540)^{2} \) (T^8 + 2T^7 - 113T^6 - 64T^5 + 3563T^4 - 3190T^3 - 20403T^2 + 14020*T + 540)^2
$31$	\( T^{16} + \cdots + 10223636544 \) T^16 + 258T^14 + 26695T^12 + 1436588T^10 + 43467807T^8 + 740738018T^6 + 6607070457T^4 + 24444959440*T^2 + 10223636544
$37$	\( T^{16} + \cdots + 56667802500 \) T^16 + 250T^14 + 24035T^12 + 1197664T^10 + 34217739T^8 + 575371058T^6 + 5549557933T^4 + 27998279316*T^2 + 56667802500
$41$	\( (T^{8} - 2 T^{7} + \cdots - 127396)^{2} \) (T^8 - 2T^7 - 141T^6 + 608T^5 + 4375T^4 - 29090T^3 + 22381T^2 + 108340*T - 127396)^2
$43$	\( (T^{8} - 208 T^{6} + \cdots + 2022592)^{2} \) (T^8 - 208T^6 + 48T^5 + 14000T^4 - 6560T^3 - 329776T^2 + 210496T + 2022592)^2
$47$	\( T^{16} + \cdots + 1006281834496 \) T^16 + 384T^14 + 57928T^12 + 4435512T^10 + 187253328T^8 + 4486194464T^6 + 59719495440T^4 + 400649540608*T^2 + 1006281834496
$53$	\( T^{16} + \cdots + 1337788330384 \) T^16 + 502T^14 + 103103T^12 + 11131204T^10 + 672051079T^8 + 22004448414T^6 + 334723406649T^4 + 1343036772264*T^2 + 1337788330384
$59$	\( T^{16} + \cdots + 99130522500 \) T^16 + 342T^14 + 44987T^12 + 2988528T^10 + 108621347T^8 + 2153638294T^6 + 21456226381T^4 + 86338855300*T^2 + 99130522500
$61$	\( T^{16} + \cdots + 713958400 \) T^16 + 488T^14 + 93600T^12 + 8915784T^10 + 438981792T^8 + 10653617664T^6 + 116846287376T^4 + 464349666560*T^2 + 713958400
$67$	\( (T^{8} - 8 T^{7} + \cdots + 1963932)^{2} \) (T^8 - 8T^7 - 225T^6 + 1260T^5 + 16391T^4 - 51744T^3 - 414555T^2 + 180556*T + 1963932)^2
$71$	\( T^{16} + \cdots + 925232372100 \) T^16 + 830T^14 + 269771T^12 + 43210960T^10 + 3489208643T^8 + 126576746382T^6 + 1381219933533T^4 + 2463835005316*T^2 + 925232372100
$73$	\( (T^{8} - 20 T^{7} + \cdots + 36784)^{2} \) (T^8 - 20T^7 - 88T^6 + 2404T^5 + 2576T^4 - 28864T^3 - 11628T^2 + 73040*T + 36784)^2
$79$	\( (T^{8} + 16 T^{7} + \cdots - 958464)^{2} \) (T^8 + 16T^7 - 282T^6 - 4304T^5 + 11680T^4 + 136704T^3 - 34304T^2 - 1085440*T - 958464)^2
$83$	\( (T^{8} - 317 T^{6} + \cdots - 700572)^{2} \) (T^8 - 317T^6 - 412T^5 + 21231T^4 - 15792T^3 - 453931T^2 + 1298396T - 700572)^2
$89$	\( T^{16} + \cdots + 71156629504 \) T^16 + 488T^14 + 89520T^12 + 7763968T^10 + 335213312T^8 + 7033284608T^6 + 64438915072T^4 + 188835102720*T^2 + 71156629504
$97$	\( T^{16} + \cdots + 10557973504 \) T^16 + 772T^14 + 202452T^12 + 23048576T^10 + 1145918016T^8 + 21868471936T^6 + 94700347136T^4 + 106855160832*T^2 + 10557973504
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