LMFDB - Newform orbit 1456.2.j.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1119,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1456.1119");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.j (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:	$11.6262185343$
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} - 20 x^{13} - 9 x^{12} + 42 x^{11} + 200 x^{10} - 26 x^{9} - 305 x^{8} - 922 x^{7} + 3202 x^{6} + \cdots + 1444 $ x^16 - 20x^13 - 9x^12 + 42x^11 + 200x^10 - 26x^9 - 305x^8 - 922x^7 + 3202x^6 + 1994x^5 + 5245x^4 - 12388x^3 + 11552x^2 - 5776*x + 1444
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
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_{14} q^{5} + (\beta_{13} - \beta_{2}) q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) $ q + b10 * q^3 + b14 * q^5 + (b13 - b2) * q^7 + (-b4 + 1) * q^9	$ q + \beta_{10} q^{3} + \beta_{14} q^{5} + (\beta_{13} - \beta_{2}) q^{7} + ( - \beta_{4} + 1) q^{9} + ( - \beta_{12} + \beta_{11} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{12} + 3 \beta_{11} + \cdots - 3 \beta_{2}) q^{99}+O(q^{100}) $ q + b10 * q^3 + b14 * q^5 + (b13 - b2) * q^7 + (-b4 + 1) * q^9 + (-b12 + b11 - b7 - b2) * q^11 + b6 * q^13 + (-b12 - b11 - b9) * q^15 + (b14 + b8 - b5) * q^17 + b15 * q^19 + (-b8 - 2b6 - b3 - b1) q^21 + (-b12 + b9 - b7 - b2) * q^23 + (-b4 + 2b3 + b1 + 1) q^25 + (-2b15 + 2b9 + 2b7 - 2b2) * q^27 - b1 * q^29 + (2b15 + b13 - b10) q^31 + (-b8 - 2b5) q^33 + (-2b12 - b11 - b10 - b9 + b7 - b2) q^35 + (-b3 + b1) * q^37 + b9 * q^39 + 2b6 q^41 + (-2b12 + b11 - 3b9 - b7 - b2) * q^43 + (-b14 + b8) * q^45 + (b15 - 2b13 - 2b10) * q^47 + (-b6 - b5 - 2b3 + b1) q^49 + (-3b12 - b11) q^51 + (b4 + 2b3 - b1 - 2) q^53 + b10 * q^55 + (-b3 + b1 + 2) * q^57 + (-b15 - 3b13 - b10 + b9 + b7 - b2) q^59 + (-2b14 + 2b5) * q^61 + (-b15 + 2b11 + b10 - 2b9 - b2) * q^63 - b3 * q^65 + (-4b12 - 2b11 - b9 - b7 - b2) * q^67 + (b14 + b8 + 6b6 - b5) q^69 + (-b12 + b11 - 2b9 - b7 - b2) q^71 + (-b14 + b8 - 4b6 + 2b5) * q^73 + (-2b15 - 4b13 + 2b10) q^75 + (-b14 - b8 - 5b6 + 2b5 + b4 + b1 - 1) * q^77 + (b11 + 5b9 - b7 - b2) q^79 + (-b4 + 2b3 - 2b1 + 1) * q^81 + (-2b15 + b13 - b10 + 4b9 + 4b7 - 4b2) * q^83 + (5b3 + b1 - 2) q^85 + (-2b15 + 2b13 + b10 + 2b9 + 2b7 - 2b2) q^87 + (-b14 - 3b8 + 2b5) * q^89 + (b12 + b9 + b7) * q^91 + (2b4 - 3b3 + b1 - 2) * q^93 + (-2b12 - b11 - 3b9 + b7 + b2) * q^95 + (-b14 + b8 + 8b6) q^97 + (-b12 + 3b11 - 4b9 - 3b7 - 3b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^9	$ 16 q + 16 q^{9} + 8 q^{21} + 4 q^{25} + 4 q^{29} + 4 q^{49} - 36 q^{53} + 32 q^{57} + 4 q^{65} - 20 q^{77} + 16 q^{81} - 56 q^{85} - 24 q^{93}+O(q^{100}) $ 16 * q + 16 * q^9 + 8 * q^21 + 4 * q^25 + 4 * q^29 + 4 * q^49 - 36 * q^53 + 32 * q^57 + 4 * q^65 - 20 * q^77 + 16 * q^81 - 56 * q^85 - 24 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 20 x^{13} - 9 x^{12} + 42 x^{11} + 200 x^{10} - 26 x^{9} - 305 x^{8} - 922 x^{7} + 3202 x^{6} + \cdots + 1444 $ x^16 - 20*x^13 - 9*x^12 + 42*x^11 + 200*x^10 - 26*x^9 - 305*x^8 - 922*x^7 + 3202*x^6 + 1994*x^5 + 5245*x^4 - 12388*x^3 + 11552*x^2 - 5776*x + 1444 :

$\beta_{1}$	$=$	$ ( - 330884890797308 \nu^{15} + 333264059208032 \nu^{14} + \cdots - 21\!\cdots\!96 ) / 43\!\cdots\!71 $ (-330884890797308v^15 + 333264059208032v^14 + 4369062423822408v^13 - 11770929614855239v^12 - 6262650430740754v^11 - 65649964185615916v^10 + 132902753064381314v^9 + 695542358388864183v^8 - 130710188995476258v^7 - 1966164603768739206v^6 - 2601946107404908726v^5 - 6259479307192338841v^4 + 14170621886636037060v^3 - 12910318298846225776v^2 + 6358559149605259456*v - 21254896069628680996) / 43553384718210384971
$\beta_{2}$	$=$	$ ( 27\!\cdots\!46 \nu^{15} + \cdots - 17\!\cdots\!90 ) / 34\!\cdots\!68 $ (2797306923089646v^15 - 50410599187008667v^14 - 27793898447731926v^13 - 145619215172822774v^12 + 928502366573805726v^11 + 1342809762262945843v^10 + 61330361941546208v^9 - 10993460216006236778v^8 - 10402156488270326052v^7 + 519740252781248655v^6 + 76271671094791272732v^5 - 95582079695322441032v^4 - 243261725287245164048v^3 - 521610947930885782667v^2 + 396215463926237851610*v - 175762235356854476590) / 348427077745683079768
$\beta_{3}$	$=$	$ ( 596302979571489 \nu^{15} + \cdots - 16\!\cdots\!10 ) / 43\!\cdots\!71 $ (596302979571489v^15 - 2408321452308105v^14 + 2257250482435665v^13 - 16875594708254810v^12 + 28828371225275988v^11 - 22002462731207535v^10 + 84804304535980238v^9 - 98859236779734907v^8 + 693858853425432069v^7 - 1193497085748766184v^6 - 59968894845592941v^5 - 9738095301372332232v^4 + 15956609625892394620v^3 - 12877686011418907496v^2 + 5871856218274514092*v - 160305379219223765110) / 43553384718210384971
$\beta_{4}$	$=$	$ ( - 808613637925352 \nu^{15} + \cdots + 69\!\cdots\!96 ) / 43\!\cdots\!71 $ (-808613637925352v^15 + 3570314174314916v^14 - 3385683297089204v^13 + 7511305277552021v^12 - 40827664028205820v^11 + 25914635934461428v^10 - 3111873183218312v^9 + 398958546636167394v^8 - 1173778899325497448v^7 + 758130686009451288v^6 - 697732175620796108v^5 + 9148023168514365317v^4 - 13795323179744330204v^3 + 10651964713348849704v^2 - 4693337485216940392*v + 69400295876981561496) / 43553384718210384971
$\beta_{5}$	$=$	$ ( - 13\!\cdots\!64 \nu^{15} - 569667636501451 \nu^{14} + \cdots + 36\!\cdots\!24 ) / 43\!\cdots\!71 $ (-13709277512370864v^15 - 569667636501451v^14 + 1126717902825571v^13 + 262407477353572830v^12 + 139980138566112511v^11 - 593213465941061626v^10 - 2595428844221441869v^9 + 160090197569250419v^8 + 4109206411753100895v^7 + 12348934270628799505v^6 - 40064957323312630616v^5 - 30999953679673547508v^4 - 75138891480570207396v^3 + 103321710358935973340v^2 - 106398555508774939664*v + 36254199048274600224) / 43553384718210384971
$\beta_{6}$	$=$	$ ( - 8028712719625 \nu^{15} - 3301751612019 \nu^{14} - 2073103843696 \nu^{13} + \cdots + 25\!\cdots\!92 ) / 13\!\cdots\!86 $ (-8028712719625v^15 - 3301751612019v^14 - 2073103843696v^13 + 158347575802740v^12 + 135099563802431v^11 - 259661129100837v^10 - 1678693144632070v^9 - 477409024734272v^8 + 1996909868000245v^7 + 7797780385544053v^6 - 22151517513117282v^5 - 23353931677898732v^4 - 52732344152074391v^3 + 73125420706972383v^2 - 73849786995418794*v + 25010323220274292) / 13304837243992786
$\beta_{7}$	$=$	$ ( 25\!\cdots\!47 \nu^{15} + \cdots + 75\!\cdots\!50 ) / 34\!\cdots\!68 $ (252754393612004347v^15 + 293320952930598205v^14 + 173607009011865608v^13 - 4980351412684863606v^12 - 8117236549774141891v^11 + 4587940941167171973v^10 + 59918915044829922410v^9 + 58270998490037617308v^8 - 49118173197755541141v^7 - 313524540307121236103v^6 + 480774855546617800856v^5 + 1267128556233124366066v^4 + 2448887570502452094569v^3 - 1094070335325341741411v^2 + 304384334255727469296*v + 75095193198479905750) / 348427077745683079768
$\beta_{8}$	$=$	$ ( 40\!\cdots\!08 \nu^{15} + \cdots - 12\!\cdots\!80 ) / 43\!\cdots\!71 $ (40071333187594308v^15 + 15950319079849650v^14 + 16009024334685726v^13 - 793691780755996531v^12 - 688132724949362514v^11 + 1254927395221609956v^10 + 8385829278100735692v^9 + 2686747740696576836v^8 - 9439467645815624310v^7 - 41674308245988965520v^6 + 111456211860625610848v^5 + 117219709381075666099v^4 + 279127772211575969674v^3 - 383620560489443306814v^2 + 383459563917392519260*v - 129511084049765424380) / 43553384718210384971
$\beta_{9}$	$=$	$ ( - 19\!\cdots\!61 \nu^{15} + \cdots + 76\!\cdots\!44 ) / 17\!\cdots\!84 $ (-199088474879778461v^15 - 190579121959814030v^14 - 75134223233945246v^13 + 3957964588407960996v^12 + 5540628896398517645v^11 - 5008012096630241988v^10 - 46757520462056517266v^9 - 35240198700076968714v^8 + 48549209617534329141v^7 + 238088364586111896928v^6 - 437859400118788231048v^5 - 908021936022700627154v^4 - 1694707577252222613217v^3 + 1236834976249366787710v^2 - 534928488216602259658*v + 76022586424915671544) / 174213538872841539884
$\beta_{10}$	$=$	$ ( - 21\!\cdots\!13 \nu^{15} + \cdots + 57\!\cdots\!72 ) / 17\!\cdots\!84 $ (-213986302562958813v^15 - 37123675168988920v^14 + 63532817512910602v^13 + 4351674254755246016v^12 + 2644511527914101153v^11 - 9863695956159331522v^10 - 46244396586083858026v^9 + 392791354227930186v^8 + 79924532942154291993v^7 + 219117728821437940258v^6 - 669598798734913127756v^5 - 601248006104522035962v^4 - 1035429016434839930577v^3 + 2719643801318082186980v^2 - 1654080934413063906574*v + 571454671832301642772) / 174213538872841539884
$\beta_{11}$	$=$	$ ( - 44\!\cdots\!85 \nu^{15} + \cdots + 17\!\cdots\!40 ) / 34\!\cdots\!68 $ (-446889063679734685v^15 - 418248084143414474v^14 - 230914060661692034v^13 + 8944521621143156844v^12 + 12402575006713654657v^11 - 10545452601562898112v^10 - 105089310610941035922v^9 - 82430169755733381346v^8 + 104036636838139083581v^7 + 546974354697196539264v^6 - 970817101373808544860v^5 - 2069256895081387876306v^4 - 3893663355841280989781v^3 + 2855376697234000298774v^2 - 1238890116866753790210*v + 178392372899461363640) / 348427077745683079768
$\beta_{12}$	$=$	$ ( - 61\!\cdots\!13 \nu^{15} + \cdots + 23\!\cdots\!72 ) / 34\!\cdots\!68 $ (-613352697007439013v^15 - 555391484455355498v^14 - 320453112102248162v^13 + 12175799216011155732v^12 + 16790683032083180505v^11 - 14573630351672087856v^10 - 141002498997437573698v^9 - 109287803390380983378v^8 + 134131480980359545093v^7 + 727180395346749124928v^6 - 1349920125868726637228v^5 - 2698765645659401746858v^4 - 5162584624964283468141v^3 + 3759957512575523354598v^2 - 1623425060868750093746*v + 230409127973654947672) / 348427077745683079768
$\beta_{13}$	$=$	$ ( - 64\!\cdots\!29 \nu^{15} + \cdots + 17\!\cdots\!88 ) / 34\!\cdots\!68 $ (-641589360298928529v^15 - 75093827873454372v^14 + 126957629525809490v^13 + 12975038224090822376v^12 + 7314118053981992405v^11 - 29017666905152987150v^10 - 135217279459017139946v^9 + 4647193940629066626v^8 + 233022176661159289701v^7 + 640878831313461226338v^6 - 2028858441997178392116v^5 - 1713885488498083531250v^4 - 3271276180788197697293v^3 + 8319465645877692077044v^2 - 5039271531010257753270*v + 1734698516581671489988) / 348427077745683079768
$\beta_{14}$	$=$	$ ( 95\!\cdots\!67 \nu^{15} + \cdots - 29\!\cdots\!16 ) / 43\!\cdots\!71 $ (95367647723687467v^15 + 36921465088978525v^14 + 27608263971888119v^13 - 1885605201045803695v^12 - 1584897593464019130v^11 + 3134916076954496588v^10 + 19927539538000256037v^9 + 5570252311116602657v^8 - 23789562744002839630v^7 - 94623762211796238796v^6 + 266013035432982623932v^5 + 276096638893497014183v^4 + 633768327324659790845v^3 - 876552519116085897609v^2 + 883653770944961675150*v - 299137488731410593216) / 43553384718210384971
$\beta_{15}$	$=$	$ ( - 92\!\cdots\!35 \nu^{15} + \cdots + 24\!\cdots\!92 ) / 34\!\cdots\!68 $ (-921923842355198835v^15 - 130319512490056708v^14 + 265470941411846742v^13 + 18620220524641757216v^12 + 11159566023359173887v^11 - 42526500318332268866v^10 - 196557521778341254734v^9 + 3260793313099278806v^8 + 338060114326559361135v^7 + 931990909126053578662v^6 - 2861091226481884222076v^5 - 2534948390888421222398v^4 - 4488395890526226921047v^3 + 11679585297646623849948v^2 - 7095546843363709558082*v + 2448738309938197456492) / 348427077745683079768

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

$\nu$	$=$	$ ( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{4} ) / 4 $ (-b15 + b13 + b12 - b11 + b10 + b8 + b4) / 4
$\nu^{2}$	$=$	$ ( \beta_{14} + \beta_{10} - 2\beta_{9} - \beta_{8} - 2\beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{2} ) / 2 $ (b14 + b10 - 2b9 - b8 - 2b7 + 2b6 - b5 + 2b2) / 2
$\nu^{3}$	$=$	$ ( - \beta_{15} + 6 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} + \beta_{11} + 7 \beta_{10} - 6 \beta_{9} + \cdots + 16 ) / 4 $ (-b15 + 6b14 - 3b13 + 3b12 + b11 + 7b10 - 6b9 - 3b8 + 16b6 + 2b5 + 3b4 + 6b3 - 2*b1 + 16) / 4
$\nu^{4}$	$=$	$ ( \beta_{12} - 9\beta_{11} + 6\beta_{9} - 4\beta_{7} + 5\beta_{4} + 4\beta_{3} - 4\beta_{2} - 6\beta _1 + 4 ) / 2 $ (b12 - 9b11 + 6b9 - 4b7 + 5b4 + 4b3 - 4b2 - 6*b1 + 4) / 2
$\nu^{5}$	$=$	$ ( 5 \beta_{15} + 18 \beta_{14} - 19 \beta_{13} - 19 \beta_{12} + 5 \beta_{11} + 37 \beta_{10} - 66 \beta_{9} + \cdots - 56 ) / 4 $ (5b15 + 18b14 - 19b13 - 19b12 + 5b11 + 37b10 - 66b9 - 5b8 - 108b7 + 56b6 + 4b5 - 5b4 - 18b3 + 4b1 - 56) / 4
$\nu^{6}$	$=$	$ ( - 12 \beta_{15} + 51 \beta_{14} - 8 \beta_{13} + 37 \beta_{10} + 2 \beta_{9} - 67 \beta_{8} + \cdots - 2 \beta_{2} ) / 2 $ (-12b15 + 51b14 - 8b13 + 37b10 + 2b9 - 67b8 + 2b7 + 66b6 + 33b5 - 2b2) / 2
$\nu^{7}$	$=$	$ ( 81 \beta_{15} + 130 \beta_{14} + 65 \beta_{13} - 65 \beta_{12} - 81 \beta_{11} - 343 \beta_{10} + \cdots + 368 ) / 4 $ (81b15 + 130b14 + 65b13 - 65b12 - 81b11 - 343b10 + 262b9 - 39b8 + 368b6 + 100b5 + 39b4 + 130b3 - 404b2 - 100b1 + 368) / 4
$\nu^{8}$	$=$	$ - 51 \beta_{12} - 67 \beta_{11} + 19 \beta_{9} - 208 \beta_{7} - 13 \beta_{4} - 83 \beta_{3} + \cdots - 250 $ -51b12 - 67b11 + 19b9 - 208b7 - 13b4 - 83b3 - 208b2 + 72b1 - 250
$\nu^{9}$	$=$	$ ( \beta_{15} + 1190 \beta_{14} + 75 \beta_{13} + 75 \beta_{12} + \beta_{11} - 39 \beta_{10} - 470 \beta_{9} + \cdots - 1928 ) / 4 $ (b15 + 1190b14 + 75b13 + 75b12 + b11 - 39b10 - 470b9 - 1371b8 - 432b7 + 1928b6 + 582b5 - 1371b4 - 1190b3 + 582b1 - 1928) / 4
$\nu^{10}$	$=$	$ ( 380 \beta_{15} + 1533 \beta_{14} + 562 \beta_{13} - 2571 \beta_{10} + 2568 \beta_{9} + \cdots - 2568 \beta_{2} ) / 2 $ (380b15 + 1533b14 + 562b13 - 2571b10 + 2568b9 - 765b8 + 2568b7 + 4110b6 + 629b5 - 2568b2) / 2
$\nu^{11}$	$=$	$ ( 4981 \beta_{15} - 3036 \beta_{14} + 755 \beta_{13} - 755 \beta_{12} - 4981 \beta_{11} - 14369 \beta_{10} + \cdots - 6972 ) / 4 $ (4981b15 - 3036b14 + 755b13 - 755b12 - 4981b11 - 14369b10 + 9388b9 + 2045b8 - 6972b6 - 1988b5 - 2045b4 - 3036b3 - 14544b2 + 1988b1 - 6972) / 4
$\nu^{12}$	$=$	$ ( 67 \beta_{12} + 861 \beta_{11} - 3746 \beta_{9} - 4124 \beta_{7} - 11299 \beta_{4} - 14416 \beta_{3} + \cdots - 32124 ) / 2 $ (67b12 + 861b11 - 3746b9 - 4124b7 - 11299b4 - 14416b3 - 4124b2 + 6094b1 - 32124) / 2
$\nu^{13}$	$=$	$ ( 11889 \beta_{15} + 32308 \beta_{14} + 6845 \beta_{13} + 6845 \beta_{12} + 11889 \beta_{11} + \cdots - 67204 ) / 4 $ (11889b15 + 32308b14 + 6845b13 + 6845b12 + 11889b11 - 53665b10 + 59316b9 - 28963b8 + 101092b7 + 67204b6 + 11610b5 - 28963b4 - 32308b3 + 11610b1 - 67204) / 4
$\nu^{14}$	$=$	$ ( 46950 \beta_{15} - 28377 \beta_{14} + 10472 \beta_{13} - 147929 \beta_{10} + 89504 \beta_{9} + \cdots - 89504 \beta_{2} ) / 2 $ (46950b15 - 28377b14 + 10472b13 - 147929b10 + 89504b9 + 32703b8 + 89504b7 - 47790b6 - 9829b5 - 89504b2) / 2
$\nu^{15}$	$=$	$ ( 16149 \beta_{15} - 323516 \beta_{14} + 13845 \beta_{13} - 13845 \beta_{12} - 16149 \beta_{11} + \cdots - 774388 ) / 4 $ (16149b15 - 323516b14 + 13845b13 - 13845b12 - 16149b11 - 84289b10 + 68140b9 + 217371b8 - 774388b6 - 144154b5 - 217371b4 - 323516b3 - 162876b2 + 144154b1 - 774388) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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} $

1119.1

−2.01498	−	0.824242i
−2.01498	+	0.824242i
1.47303	+	1.22455i
1.47303	−	1.22455i
0.247483	−	0.537458i
0.247483	+	0.537458i
−0.615409	+	2.42121i
−0.615409	−	2.42121i
2.42121	−	0.615409i
2.42121	+	0.615409i
0.537458	−	0.247483i
0.537458	+	0.247483i
−1.22455	−	1.47303i
−1.22455	+	1.47303i
−0.824242	−	2.01498i
−0.824242	+	2.01498i

−3.11102

−

0.739531i

−0.953027

−

2.46814i

6.67845

1119.2

−3.11102

0.739531i

−0.953027

2.46814i

6.67845

1119.3

−1.87164

−

0.330638i

2.52093

0.803059i

0.503039

1119.4

−1.87164

0.330638i

2.52093

−

0.803059i

0.503039

1119.5

−1.55888

−

3.64989i

−2.63133

0.275914i

−0.569882

1119.6

−1.55888

3.64989i

−2.63133

−

0.275914i

−0.569882

1119.7

−0.623212

−

2.24100i

0.559261

2.58597i

−2.61161

1119.8

−0.623212

2.24100i

0.559261

−

2.58597i

−2.61161

1119.9

0.623212

−

2.24100i

−0.559261

−

2.58597i

−2.61161

1119.10

0.623212

2.24100i

−0.559261

2.58597i

−2.61161

1119.11

1.55888

−

3.64989i

2.63133

−

0.275914i

−0.569882

1119.12

1.55888

3.64989i

2.63133

0.275914i

−0.569882

1119.13

1.87164

−

0.330638i

−2.52093

−

0.803059i

0.503039

1119.14

1.87164

0.330638i

−2.52093

0.803059i

0.503039

1119.15

3.11102

−

0.739531i

0.953027

2.46814i

6.67845

1119.16

3.11102

0.739531i

0.953027

−

2.46814i

6.67845

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1456.2.j.a	✓	16
4.b	odd	2	1	inner	1456.2.j.a	✓	16
7.b	odd	2	1	inner	1456.2.j.a	✓	16
28.d	even	2	1	inner	1456.2.j.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1456.2.j.a	✓	16	1.a	even	1	1	trivial
1456.2.j.a	✓	16	4.b	odd	2	1	inner
1456.2.j.a	✓	16	7.b	odd	2	1	inner
1456.2.j.a	✓	16	28.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 16T_{3}^{6} + 72T_{3}^{4} - 108T_{3}^{2} + 32 $ T3^8 - 16*T3^6 + 72*T3^4 - 108*T3^2 + 32 acting on $S_{2}^{\mathrm{new}}(1456, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 16 T^{6} + \cdots + 32)^{2} $ (T^8 - 16T^6 + 72T^4 - 108*T^2 + 32)^2
$5$	$ (T^{8} + 19 T^{6} + 79 T^{4} + \cdots + 4)^{2} $ (T^8 + 19T^6 + 79T^4 + 45*T^2 + 4)^2
$7$	$ T^{16} - 2 T^{14} + \cdots + 5764801 $ T^16 - 2T^14 - 96T^12 + 2T^10 + 6462T^8 + 98T^6 - 230496T^4 - 235298*T^2 + 5764801
$11$	$ (T^{8} + 50 T^{6} + 580 T^{4} + \cdots + 8)^{2} $ (T^8 + 50T^6 + 580T^4 + 148*T^2 + 8)^2
$13$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$17$	$ (T^{8} + 68 T^{6} + \cdots + 44944)^{2} $ (T^8 + 68T^6 + 1552T^4 + 14244*T^2 + 44944)^2
$19$	$ (T^{8} - 39 T^{6} + \cdots + 1250)^{2} $ (T^8 - 39T^6 + 509T^4 - 2325*T^2 + 1250)^2
$23$	$ (T^{8} + 57 T^{6} + \cdots + 200)^{2} $ (T^8 + 57T^6 + 963T^4 + 4555*T^2 + 200)^2
$29$	$ (T^{4} - T^{3} - 21 T^{2} + \cdots - 16)^{4} $ (T^4 - T^3 - 21T^2 + 45T - 16)^4
$31$	$ (T^{8} - 131 T^{6} + \cdots + 458882)^{2} $ (T^8 - 131T^6 + 5825T^4 - 99085*T^2 + 458882)^2
$37$	$ (T^{4} - 38 T^{2} + \cdots + 316)^{4} $ (T^4 - 38T^2 - 10T + 316)^4
$41$	$ (T^{2} + 4)^{8} $ (T^2 + 4)^8
$43$	$ (T^{8} + 217 T^{6} + \cdots + 1248200)^{2} $ (T^8 + 217T^6 + 11923T^4 + 218795*T^2 + 1248200)^2
$47$	$ (T^{8} - 159 T^{6} + \cdots + 1140050)^{2} $ (T^8 - 159T^6 + 8317T^4 - 167805*T^2 + 1140050)^2
$53$	$ (T^{4} + 9 T^{3} + \cdots - 1046)^{4} $ (T^4 + 9T^3 - 47T^2 - 577*T - 1046)^4
$59$	$ (T^{8} - 286 T^{6} + \cdots + 15814688)^{2} $ (T^8 - 286T^6 + 28016T^4 - 1127792*T^2 + 15814688)^2
$61$	$ (T^{8} + 192 T^{6} + \cdots + 102400)^{2} $ (T^8 + 192T^6 + 11584T^4 + 216320*T^2 + 102400)^2
$67$	$ (T^{8} + 346 T^{6} + \cdots + 22044800)^{2} $ (T^8 + 346T^6 + 37992T^4 + 1613280*T^2 + 22044800)^2
$71$	$ (T^{8} + 122 T^{6} + \cdots + 6728)^{2} $ (T^8 + 122T^6 + 1492T^4 + 5716*T^2 + 6728)^2
$73$	$ (T^{8} + 311 T^{6} + \cdots + 538756)^{2} $ (T^8 + 311T^6 + 24555T^4 + 547021*T^2 + 538756)^2
$79$	$ (T^{8} + 369 T^{6} + \cdots + 35011712)^{2} $ (T^8 + 369T^6 + 47515T^4 + 2410515*T^2 + 35011712)^2
$83$	$ (T^{8} - 491 T^{6} + \cdots + 25647122)^{2} $ (T^8 - 491T^6 + 68705T^4 - 2746165*T^2 + 25647122)^2
$89$	$ (T^{8} + 415 T^{6} + \cdots + 96786244)^{2} $ (T^8 + 415T^6 + 62571T^4 + 4073189*T^2 + 96786244)^2
$97$	$ (T^{8} + 335 T^{6} + \cdots + 4813636)^{2} $ (T^8 + 335T^6 + 35043T^4 + 1198445*T^2 + 4813636)^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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{10} q^{3} + \beta_{14} q^{5} + (\beta_{13} - \beta_{2}) q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) \) q + b10 * q^3 + b14 * q^5 + (b13 - b2) * q^7 + (-b4 + 1) * q^9	\( q + \beta_{10} q^{3} + \beta_{14} q^{5} + (\beta_{13} - \beta_{2}) q^{7} + ( - \beta_{4} + 1) q^{9} + ( - \beta_{12} + \beta_{11} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{12} + 3 \beta_{11} + \cdots - 3 \beta_{2}) q^{99}+O(q^{100}) \) q + b10 * q^3 + b14 * q^5 + (b13 - b2) * q^7 + (-b4 + 1) * q^9 + (-b12 + b11 - b7 - b2) * q^11 + b6 * q^13 + (-b12 - b11 - b9) * q^15 + (b14 + b8 - b5) * q^17 + b15 * q^19 + (-b8 - 2b6 - b3 - b1) q^21 + (-b12 + b9 - b7 - b2) * q^23 + (-b4 + 2b3 + b1 + 1) q^25 + (-2b15 + 2b9 + 2b7 - 2b2) * q^27 - b1 * q^29 + (2b15 + b13 - b10) q^31 + (-b8 - 2b5) q^33 + (-2b12 - b11 - b10 - b9 + b7 - b2) q^35 + (-b3 + b1) * q^37 + b9 * q^39 + 2b6 q^41 + (-2b12 + b11 - 3b9 - b7 - b2) * q^43 + (-b14 + b8) * q^45 + (b15 - 2b13 - 2b10) * q^47 + (-b6 - b5 - 2b3 + b1) q^49 + (-3b12 - b11) q^51 + (b4 + 2b3 - b1 - 2) q^53 + b10 * q^55 + (-b3 + b1 + 2) * q^57 + (-b15 - 3b13 - b10 + b9 + b7 - b2) q^59 + (-2b14 + 2b5) * q^61 + (-b15 + 2b11 + b10 - 2b9 - b2) * q^63 - b3 * q^65 + (-4b12 - 2b11 - b9 - b7 - b2) * q^67 + (b14 + b8 + 6b6 - b5) q^69 + (-b12 + b11 - 2b9 - b7 - b2) q^71 + (-b14 + b8 - 4b6 + 2b5) * q^73 + (-2b15 - 4b13 + 2b10) q^75 + (-b14 - b8 - 5b6 + 2b5 + b4 + b1 - 1) * q^77 + (b11 + 5b9 - b7 - b2) q^79 + (-b4 + 2b3 - 2b1 + 1) * q^81 + (-2b15 + b13 - b10 + 4b9 + 4b7 - 4b2) * q^83 + (5b3 + b1 - 2) q^85 + (-2b15 + 2b13 + b10 + 2b9 + 2b7 - 2b2) q^87 + (-b14 - 3b8 + 2b5) * q^89 + (b12 + b9 + b7) * q^91 + (2b4 - 3b3 + b1 - 2) * q^93 + (-2b12 - b11 - 3b9 + b7 + b2) * q^95 + (-b14 + b8 + 8b6) q^97 + (-b12 + 3b11 - 4b9 - 3b7 - 3b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^9	\( 16 q + 16 q^{9} + 8 q^{21} + 4 q^{25} + 4 q^{29} + 4 q^{49} - 36 q^{53} + 32 q^{57} + 4 q^{65} - 20 q^{77} + 16 q^{81} - 56 q^{85} - 24 q^{93}+O(q^{100}) \) 16 * q + 16 * q^9 + 8 * q^21 + 4 * q^25 + 4 * q^29 + 4 * q^49 - 36 * q^53 + 32 * q^57 + 4 * q^65 - 20 * q^77 + 16 * q^81 - 56 * q^85 - 24 * q^93

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	1456.2.j.a

Level	$1456$
Weight	$2$
Character orbit	1456.j
Analytic conductor	$11.626$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(11.6262185343\)
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:	\( x^{16} - 20 x^{13} - 9 x^{12} + 42 x^{11} + 200 x^{10} - 26 x^{9} - 305 x^{8} - 922 x^{7} + 3202 x^{6} + \cdots + 1444 \) x^16 - 20x^13 - 9x^12 + 42x^11 + 200x^10 - 26x^9 - 305x^8 - 922x^7 + 3202x^6 + 1994x^5 + 5245x^4 - 12388x^3 + 11552x^2 - 5776*x + 1444
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 330884890797308 \nu^{15} + 333264059208032 \nu^{14} + \cdots - 21\!\cdots\!96 ) / 43\!\cdots\!71 \) (-330884890797308v^15 + 333264059208032v^14 + 4369062423822408v^13 - 11770929614855239v^12 - 6262650430740754v^11 - 65649964185615916v^10 + 132902753064381314v^9 + 695542358388864183v^8 - 130710188995476258v^7 - 1966164603768739206v^6 - 2601946107404908726v^5 - 6259479307192338841v^4 + 14170621886636037060v^3 - 12910318298846225776v^2 + 6358559149605259456*v - 21254896069628680996) / 43553384718210384971
\(\beta_{2}\)	\(=\)	\( ( 27\!\cdots\!46 \nu^{15} + \cdots - 17\!\cdots\!90 ) / 34\!\cdots\!68 \) (2797306923089646v^15 - 50410599187008667v^14 - 27793898447731926v^13 - 145619215172822774v^12 + 928502366573805726v^11 + 1342809762262945843v^10 + 61330361941546208v^9 - 10993460216006236778v^8 - 10402156488270326052v^7 + 519740252781248655v^6 + 76271671094791272732v^5 - 95582079695322441032v^4 - 243261725287245164048v^3 - 521610947930885782667v^2 + 396215463926237851610*v - 175762235356854476590) / 348427077745683079768
\(\beta_{3}\)	\(=\)	\( ( 596302979571489 \nu^{15} + \cdots - 16\!\cdots\!10 ) / 43\!\cdots\!71 \) (596302979571489v^15 - 2408321452308105v^14 + 2257250482435665v^13 - 16875594708254810v^12 + 28828371225275988v^11 - 22002462731207535v^10 + 84804304535980238v^9 - 98859236779734907v^8 + 693858853425432069v^7 - 1193497085748766184v^6 - 59968894845592941v^5 - 9738095301372332232v^4 + 15956609625892394620v^3 - 12877686011418907496v^2 + 5871856218274514092*v - 160305379219223765110) / 43553384718210384971
\(\beta_{4}\)	\(=\)	\( ( - 808613637925352 \nu^{15} + \cdots + 69\!\cdots\!96 ) / 43\!\cdots\!71 \) (-808613637925352v^15 + 3570314174314916v^14 - 3385683297089204v^13 + 7511305277552021v^12 - 40827664028205820v^11 + 25914635934461428v^10 - 3111873183218312v^9 + 398958546636167394v^8 - 1173778899325497448v^7 + 758130686009451288v^6 - 697732175620796108v^5 + 9148023168514365317v^4 - 13795323179744330204v^3 + 10651964713348849704v^2 - 4693337485216940392*v + 69400295876981561496) / 43553384718210384971
\(\beta_{5}\)	\(=\)	\( ( - 13\!\cdots\!64 \nu^{15} - 569667636501451 \nu^{14} + \cdots + 36\!\cdots\!24 ) / 43\!\cdots\!71 \) (-13709277512370864v^15 - 569667636501451v^14 + 1126717902825571v^13 + 262407477353572830v^12 + 139980138566112511v^11 - 593213465941061626v^10 - 2595428844221441869v^9 + 160090197569250419v^8 + 4109206411753100895v^7 + 12348934270628799505v^6 - 40064957323312630616v^5 - 30999953679673547508v^4 - 75138891480570207396v^3 + 103321710358935973340v^2 - 106398555508774939664*v + 36254199048274600224) / 43553384718210384971
\(\beta_{6}\)	\(=\)	\( ( - 8028712719625 \nu^{15} - 3301751612019 \nu^{14} - 2073103843696 \nu^{13} + \cdots + 25\!\cdots\!92 ) / 13\!\cdots\!86 \) (-8028712719625v^15 - 3301751612019v^14 - 2073103843696v^13 + 158347575802740v^12 + 135099563802431v^11 - 259661129100837v^10 - 1678693144632070v^9 - 477409024734272v^8 + 1996909868000245v^7 + 7797780385544053v^6 - 22151517513117282v^5 - 23353931677898732v^4 - 52732344152074391v^3 + 73125420706972383v^2 - 73849786995418794*v + 25010323220274292) / 13304837243992786
\(\beta_{7}\)	\(=\)	\( ( 25\!\cdots\!47 \nu^{15} + \cdots + 75\!\cdots\!50 ) / 34\!\cdots\!68 \) (252754393612004347v^15 + 293320952930598205v^14 + 173607009011865608v^13 - 4980351412684863606v^12 - 8117236549774141891v^11 + 4587940941167171973v^10 + 59918915044829922410v^9 + 58270998490037617308v^8 - 49118173197755541141v^7 - 313524540307121236103v^6 + 480774855546617800856v^5 + 1267128556233124366066v^4 + 2448887570502452094569v^3 - 1094070335325341741411v^2 + 304384334255727469296*v + 75095193198479905750) / 348427077745683079768
\(\beta_{8}\)	\(=\)	\( ( 40\!\cdots\!08 \nu^{15} + \cdots - 12\!\cdots\!80 ) / 43\!\cdots\!71 \) (40071333187594308v^15 + 15950319079849650v^14 + 16009024334685726v^13 - 793691780755996531v^12 - 688132724949362514v^11 + 1254927395221609956v^10 + 8385829278100735692v^9 + 2686747740696576836v^8 - 9439467645815624310v^7 - 41674308245988965520v^6 + 111456211860625610848v^5 + 117219709381075666099v^4 + 279127772211575969674v^3 - 383620560489443306814v^2 + 383459563917392519260*v - 129511084049765424380) / 43553384718210384971
\(\beta_{9}\)	\(=\)	\( ( - 19\!\cdots\!61 \nu^{15} + \cdots + 76\!\cdots\!44 ) / 17\!\cdots\!84 \) (-199088474879778461v^15 - 190579121959814030v^14 - 75134223233945246v^13 + 3957964588407960996v^12 + 5540628896398517645v^11 - 5008012096630241988v^10 - 46757520462056517266v^9 - 35240198700076968714v^8 + 48549209617534329141v^7 + 238088364586111896928v^6 - 437859400118788231048v^5 - 908021936022700627154v^4 - 1694707577252222613217v^3 + 1236834976249366787710v^2 - 534928488216602259658*v + 76022586424915671544) / 174213538872841539884
\(\beta_{10}\)	\(=\)	\( ( - 21\!\cdots\!13 \nu^{15} + \cdots + 57\!\cdots\!72 ) / 17\!\cdots\!84 \) (-213986302562958813v^15 - 37123675168988920v^14 + 63532817512910602v^13 + 4351674254755246016v^12 + 2644511527914101153v^11 - 9863695956159331522v^10 - 46244396586083858026v^9 + 392791354227930186v^8 + 79924532942154291993v^7 + 219117728821437940258v^6 - 669598798734913127756v^5 - 601248006104522035962v^4 - 1035429016434839930577v^3 + 2719643801318082186980v^2 - 1654080934413063906574*v + 571454671832301642772) / 174213538872841539884
\(\beta_{11}\)	\(=\)	\( ( - 44\!\cdots\!85 \nu^{15} + \cdots + 17\!\cdots\!40 ) / 34\!\cdots\!68 \) (-446889063679734685v^15 - 418248084143414474v^14 - 230914060661692034v^13 + 8944521621143156844v^12 + 12402575006713654657v^11 - 10545452601562898112v^10 - 105089310610941035922v^9 - 82430169755733381346v^8 + 104036636838139083581v^7 + 546974354697196539264v^6 - 970817101373808544860v^5 - 2069256895081387876306v^4 - 3893663355841280989781v^3 + 2855376697234000298774v^2 - 1238890116866753790210*v + 178392372899461363640) / 348427077745683079768
\(\beta_{12}\)	\(=\)	\( ( - 61\!\cdots\!13 \nu^{15} + \cdots + 23\!\cdots\!72 ) / 34\!\cdots\!68 \) (-613352697007439013v^15 - 555391484455355498v^14 - 320453112102248162v^13 + 12175799216011155732v^12 + 16790683032083180505v^11 - 14573630351672087856v^10 - 141002498997437573698v^9 - 109287803390380983378v^8 + 134131480980359545093v^7 + 727180395346749124928v^6 - 1349920125868726637228v^5 - 2698765645659401746858v^4 - 5162584624964283468141v^3 + 3759957512575523354598v^2 - 1623425060868750093746*v + 230409127973654947672) / 348427077745683079768
\(\beta_{13}\)	\(=\)	\( ( - 64\!\cdots\!29 \nu^{15} + \cdots + 17\!\cdots\!88 ) / 34\!\cdots\!68 \) (-641589360298928529v^15 - 75093827873454372v^14 + 126957629525809490v^13 + 12975038224090822376v^12 + 7314118053981992405v^11 - 29017666905152987150v^10 - 135217279459017139946v^9 + 4647193940629066626v^8 + 233022176661159289701v^7 + 640878831313461226338v^6 - 2028858441997178392116v^5 - 1713885488498083531250v^4 - 3271276180788197697293v^3 + 8319465645877692077044v^2 - 5039271531010257753270*v + 1734698516581671489988) / 348427077745683079768
\(\beta_{14}\)	\(=\)	\( ( 95\!\cdots\!67 \nu^{15} + \cdots - 29\!\cdots\!16 ) / 43\!\cdots\!71 \) (95367647723687467v^15 + 36921465088978525v^14 + 27608263971888119v^13 - 1885605201045803695v^12 - 1584897593464019130v^11 + 3134916076954496588v^10 + 19927539538000256037v^9 + 5570252311116602657v^8 - 23789562744002839630v^7 - 94623762211796238796v^6 + 266013035432982623932v^5 + 276096638893497014183v^4 + 633768327324659790845v^3 - 876552519116085897609v^2 + 883653770944961675150*v - 299137488731410593216) / 43553384718210384971
\(\beta_{15}\)	\(=\)	\( ( - 92\!\cdots\!35 \nu^{15} + \cdots + 24\!\cdots\!92 ) / 34\!\cdots\!68 \) (-921923842355198835v^15 - 130319512490056708v^14 + 265470941411846742v^13 + 18620220524641757216v^12 + 11159566023359173887v^11 - 42526500318332268866v^10 - 196557521778341254734v^9 + 3260793313099278806v^8 + 338060114326559361135v^7 + 931990909126053578662v^6 - 2861091226481884222076v^5 - 2534948390888421222398v^4 - 4488395890526226921047v^3 + 11679585297646623849948v^2 - 7095546843363709558082*v + 2448738309938197456492) / 348427077745683079768

\(\nu\)	\(=\)	\( ( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{4} ) / 4 \) (-b15 + b13 + b12 - b11 + b10 + b8 + b4) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{14} + \beta_{10} - 2\beta_{9} - \beta_{8} - 2\beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{2} ) / 2 \) (b14 + b10 - 2b9 - b8 - 2b7 + 2b6 - b5 + 2b2) / 2
\(\nu^{3}\)	\(=\)	\( ( - \beta_{15} + 6 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} + \beta_{11} + 7 \beta_{10} - 6 \beta_{9} + \cdots + 16 ) / 4 \) (-b15 + 6b14 - 3b13 + 3b12 + b11 + 7b10 - 6b9 - 3b8 + 16b6 + 2b5 + 3b4 + 6b3 - 2*b1 + 16) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{12} - 9\beta_{11} + 6\beta_{9} - 4\beta_{7} + 5\beta_{4} + 4\beta_{3} - 4\beta_{2} - 6\beta _1 + 4 ) / 2 \) (b12 - 9b11 + 6b9 - 4b7 + 5b4 + 4b3 - 4b2 - 6*b1 + 4) / 2
\(\nu^{5}\)	\(=\)	\( ( 5 \beta_{15} + 18 \beta_{14} - 19 \beta_{13} - 19 \beta_{12} + 5 \beta_{11} + 37 \beta_{10} - 66 \beta_{9} + \cdots - 56 ) / 4 \) (5b15 + 18b14 - 19b13 - 19b12 + 5b11 + 37b10 - 66b9 - 5b8 - 108b7 + 56b6 + 4b5 - 5b4 - 18b3 + 4b1 - 56) / 4
\(\nu^{6}\)	\(=\)	\( ( - 12 \beta_{15} + 51 \beta_{14} - 8 \beta_{13} + 37 \beta_{10} + 2 \beta_{9} - 67 \beta_{8} + \cdots - 2 \beta_{2} ) / 2 \) (-12b15 + 51b14 - 8b13 + 37b10 + 2b9 - 67b8 + 2b7 + 66b6 + 33b5 - 2b2) / 2
\(\nu^{7}\)	\(=\)	\( ( 81 \beta_{15} + 130 \beta_{14} + 65 \beta_{13} - 65 \beta_{12} - 81 \beta_{11} - 343 \beta_{10} + \cdots + 368 ) / 4 \) (81b15 + 130b14 + 65b13 - 65b12 - 81b11 - 343b10 + 262b9 - 39b8 + 368b6 + 100b5 + 39b4 + 130b3 - 404b2 - 100b1 + 368) / 4
\(\nu^{8}\)	\(=\)	\( - 51 \beta_{12} - 67 \beta_{11} + 19 \beta_{9} - 208 \beta_{7} - 13 \beta_{4} - 83 \beta_{3} + \cdots - 250 \) -51b12 - 67b11 + 19b9 - 208b7 - 13b4 - 83b3 - 208b2 + 72b1 - 250
\(\nu^{9}\)	\(=\)	\( ( \beta_{15} + 1190 \beta_{14} + 75 \beta_{13} + 75 \beta_{12} + \beta_{11} - 39 \beta_{10} - 470 \beta_{9} + \cdots - 1928 ) / 4 \) (b15 + 1190b14 + 75b13 + 75b12 + b11 - 39b10 - 470b9 - 1371b8 - 432b7 + 1928b6 + 582b5 - 1371b4 - 1190b3 + 582b1 - 1928) / 4
\(\nu^{10}\)	\(=\)	\( ( 380 \beta_{15} + 1533 \beta_{14} + 562 \beta_{13} - 2571 \beta_{10} + 2568 \beta_{9} + \cdots - 2568 \beta_{2} ) / 2 \) (380b15 + 1533b14 + 562b13 - 2571b10 + 2568b9 - 765b8 + 2568b7 + 4110b6 + 629b5 - 2568b2) / 2
\(\nu^{11}\)	\(=\)	\( ( 4981 \beta_{15} - 3036 \beta_{14} + 755 \beta_{13} - 755 \beta_{12} - 4981 \beta_{11} - 14369 \beta_{10} + \cdots - 6972 ) / 4 \) (4981b15 - 3036b14 + 755b13 - 755b12 - 4981b11 - 14369b10 + 9388b9 + 2045b8 - 6972b6 - 1988b5 - 2045b4 - 3036b3 - 14544b2 + 1988b1 - 6972) / 4
\(\nu^{12}\)	\(=\)	\( ( 67 \beta_{12} + 861 \beta_{11} - 3746 \beta_{9} - 4124 \beta_{7} - 11299 \beta_{4} - 14416 \beta_{3} + \cdots - 32124 ) / 2 \) (67b12 + 861b11 - 3746b9 - 4124b7 - 11299b4 - 14416b3 - 4124b2 + 6094b1 - 32124) / 2
\(\nu^{13}\)	\(=\)	\( ( 11889 \beta_{15} + 32308 \beta_{14} + 6845 \beta_{13} + 6845 \beta_{12} + 11889 \beta_{11} + \cdots - 67204 ) / 4 \) (11889b15 + 32308b14 + 6845b13 + 6845b12 + 11889b11 - 53665b10 + 59316b9 - 28963b8 + 101092b7 + 67204b6 + 11610b5 - 28963b4 - 32308b3 + 11610b1 - 67204) / 4
\(\nu^{14}\)	\(=\)	\( ( 46950 \beta_{15} - 28377 \beta_{14} + 10472 \beta_{13} - 147929 \beta_{10} + 89504 \beta_{9} + \cdots - 89504 \beta_{2} ) / 2 \) (46950b15 - 28377b14 + 10472b13 - 147929b10 + 89504b9 + 32703b8 + 89504b7 - 47790b6 - 9829b5 - 89504b2) / 2
\(\nu^{15}\)	\(=\)	\( ( 16149 \beta_{15} - 323516 \beta_{14} + 13845 \beta_{13} - 13845 \beta_{12} - 16149 \beta_{11} + \cdots - 774388 ) / 4 \) (16149b15 - 323516b14 + 13845b13 - 13845b12 - 16149b11 - 84289b10 + 68140b9 + 217371b8 - 774388b6 - 144154b5 - 217371b4 - 323516b3 - 162876b2 + 144154b1 - 774388) / 4

\(n\)	\(561\)	\(911\)	\(1093\)	\(1249\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1119.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 16 T^{6} + \cdots + 32)^{2} \) (T^8 - 16T^6 + 72T^4 - 108*T^2 + 32)^2
$5$	\( (T^{8} + 19 T^{6} + 79 T^{4} + \cdots + 4)^{2} \) (T^8 + 19T^6 + 79T^4 + 45*T^2 + 4)^2
$7$	\( T^{16} - 2 T^{14} + \cdots + 5764801 \) T^16 - 2T^14 - 96T^12 + 2T^10 + 6462T^8 + 98T^6 - 230496T^4 - 235298*T^2 + 5764801
$11$	\( (T^{8} + 50 T^{6} + 580 T^{4} + \cdots + 8)^{2} \) (T^8 + 50T^6 + 580T^4 + 148*T^2 + 8)^2
$13$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$17$	\( (T^{8} + 68 T^{6} + \cdots + 44944)^{2} \) (T^8 + 68T^6 + 1552T^4 + 14244*T^2 + 44944)^2
$19$	\( (T^{8} - 39 T^{6} + \cdots + 1250)^{2} \) (T^8 - 39T^6 + 509T^4 - 2325*T^2 + 1250)^2
$23$	\( (T^{8} + 57 T^{6} + \cdots + 200)^{2} \) (T^8 + 57T^6 + 963T^4 + 4555*T^2 + 200)^2
$29$	\( (T^{4} - T^{3} - 21 T^{2} + \cdots - 16)^{4} \) (T^4 - T^3 - 21T^2 + 45T - 16)^4
$31$	\( (T^{8} - 131 T^{6} + \cdots + 458882)^{2} \) (T^8 - 131T^6 + 5825T^4 - 99085*T^2 + 458882)^2
$37$	\( (T^{4} - 38 T^{2} + \cdots + 316)^{4} \) (T^4 - 38T^2 - 10T + 316)^4
$41$	\( (T^{2} + 4)^{8} \) (T^2 + 4)^8
$43$	\( (T^{8} + 217 T^{6} + \cdots + 1248200)^{2} \) (T^8 + 217T^6 + 11923T^4 + 218795*T^2 + 1248200)^2
$47$	\( (T^{8} - 159 T^{6} + \cdots + 1140050)^{2} \) (T^8 - 159T^6 + 8317T^4 - 167805*T^2 + 1140050)^2
$53$	\( (T^{4} + 9 T^{3} + \cdots - 1046)^{4} \) (T^4 + 9T^3 - 47T^2 - 577*T - 1046)^4
$59$	\( (T^{8} - 286 T^{6} + \cdots + 15814688)^{2} \) (T^8 - 286T^6 + 28016T^4 - 1127792*T^2 + 15814688)^2
$61$	\( (T^{8} + 192 T^{6} + \cdots + 102400)^{2} \) (T^8 + 192T^6 + 11584T^4 + 216320*T^2 + 102400)^2
$67$	\( (T^{8} + 346 T^{6} + \cdots + 22044800)^{2} \) (T^8 + 346T^6 + 37992T^4 + 1613280*T^2 + 22044800)^2
$71$	\( (T^{8} + 122 T^{6} + \cdots + 6728)^{2} \) (T^8 + 122T^6 + 1492T^4 + 5716*T^2 + 6728)^2
$73$	\( (T^{8} + 311 T^{6} + \cdots + 538756)^{2} \) (T^8 + 311T^6 + 24555T^4 + 547021*T^2 + 538756)^2
$79$	\( (T^{8} + 369 T^{6} + \cdots + 35011712)^{2} \) (T^8 + 369T^6 + 47515T^4 + 2410515*T^2 + 35011712)^2
$83$	\( (T^{8} - 491 T^{6} + \cdots + 25647122)^{2} \) (T^8 - 491T^6 + 68705T^4 - 2746165*T^2 + 25647122)^2
$89$	\( (T^{8} + 415 T^{6} + \cdots + 96786244)^{2} \) (T^8 + 415T^6 + 62571T^4 + 4073189*T^2 + 96786244)^2
$97$	\( (T^{8} + 335 T^{6} + \cdots + 4813636)^{2} \) (T^8 + 335T^6 + 35043T^4 + 1198445*T^2 + 4813636)^2
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