LMFDB - Newform orbit 1320.2.bw.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1320,2,Mod(361,1320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1320, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 6]))

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

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

chi := DirichletCharacter("1320.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1320.bw (of order $5$, degree $4$, 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:	$10.5402530668$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{5})$
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} + 26 x^{14} - 64 x^{13} + 486 x^{12} - 10 x^{11} + 6075 x^{10} + 9130 x^{9} + \cdots + 50410000 $ x^16 - 4x^15 + 26x^14 - 64x^13 + 486x^12 - 10x^11 + 6075x^10 + 9130x^9 + 72165x^8 + 32200x^7 + 420125x^6 + 573200x^5 + 3577025x^4 + 5167000x^3 + 17399000x^2 + 15265000*x + 50410000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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_{2} q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} - \beta_{6} q^{9}+O(q^{10}) $ q + b2 * q^3 - b3 * q^5 + b1 * q^7 - b6 * q^9	$ q + \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} - \beta_{6} q^{9} - \beta_{12} q^{11} + ( - \beta_{9} + \beta_{8} + \beta_{4}) q^{13} + (\beta_{6} + \beta_{3} - \beta_{2} - 1) q^{15} + (\beta_{14} - \beta_{4}) q^{17} + (\beta_{15} - 2 \beta_{13} - \beta_{12} + \cdots - 1) q^{19}+ \cdots - \beta_{15} q^{99}+O(q^{100}) $ q + b2 * q^3 - b3 * q^5 + b1 * q^7 - b6 * q^9 - b12 * q^11 + (-b9 + b8 + b4) * q^13 + (b6 + b3 - b2 - 1) * q^15 + (b14 - b4) * q^17 + (b15 - 2b13 - b12 - b10 - b9 + 2b8 - b7 + b6 - b5 + b4 + b2 - b1 - 1) * q^19 + b5 * q^21 + (-b14 + b10 - b9 + b5 + 1) * q^23 + b2 * q^25 - b3 * q^27 + (2b15 + 2b14 - 2b13 - b12 - 2b11 - 3b10 - b8 - 4b5 - 2b4 + b3 + 2b2 - 3b1 - 1) q^29 + (-b13 - 2b12 - b11 - b10 - b9 - b6 - b5 + b3 - b1) q^31 + (b13 + b12 + b10 + b5 - b2 + b1) * q^33 + (-b11 - b8 - b5 - b1) * q^35 + (2b15 - 2b13 - b12 - b10 - 2b9 + b8 + b6 + 2b3 + b2 - 2b1 - 2) q^37 + (b14 + b9 - b8 - b5 - b4) * q^39 + (-b8 - 2b4) q^41 + (b15 - b14 - b13 - b9 - 2b7 + b4 + b2 - b1) q^43 + q^45 + (b14 - b11 - b10 - 2b6 - b5 - b4 + 3b2 - b1 + 2) * q^47 + (b15 + b14 - b13 + b12 - b11 - b7 - b5 - 3b3 + b2) q^49 + (-b11 - b10 - b8 - b5 - b4 - b1) * q^51 + (b15 + b14 - b11 - b9 + b7 - b6 + b5 + b1) * q^53 + (b15 - b13 - b10 - b7 - b5 + b2 - b1) * q^55 + (b15 + b14 - b11 - b8 + b7 + b6 - b5 - b4 + b3 - b2 - b1) * q^57 + (-b14 + b13 + b12 + 2b11 + b10 - b7 + b6 + 3b5 + b4 + b3 - 2b2 + 3b1 - 1) * q^59 + (-b11 - b9 - b8 - b5 - b3) * q^61 + b8 * q^63 + (b14 - b10 + b9 - b5) * q^65 + (b15 - b14 - b13 - b9 + b8 - 2b7 - 2b6 + b5 + b4 - 2b3 + b2 + 3) q^67 + (b14 - b10 - b5 + b2) * q^69 + (-b15 - b14 + b13 - b12 - 2b11 - 2b9 + b8 + b7 + 2b5 - 2b3 - b2) * q^71 + (2b15 + b12 + 2b11 + b10 + b8 - 2b7 + 2b5 + 2b4 + 2b1) * q^73 - b6 * q^75 + (2b15 + 4b14 - 2b13 - b12 - b11 - 3b10 - b9 + 2b8 + 2b6 - 3b5 - b4 + 3b3 - b2 - b1 - 2) * q^77 + (-b13 - 2b12 - b11 - b10 - b9 + 2b6 - 2b5 - 3b3 + 4b2 - 2b1) * q^79 + (b6 + b3 - b2 - 1) * q^81 + (b15 - b14 - b13 + b12 + b8 - b7 + 2b4 - b2 - 2) q^83 + (-b14 + b11 + b10 + b8 + b5 + b4 + b1) * q^85 + (b13 - b12 - b11 - b10 + b9 - 3b8 + 2b7 - b6 - b5 - 2b4 - b3 - b2 - 2b1 + 1) * q^87 + (b15 + b14 + b13 - 2b12 - 2b11 - b10 - 5b8 + 2b7 + 2b6 - 2b5 - 3b4 + 2b3 - b2 - 4b1 + 2) q^89 + (b15 - 2b13 - 3b12 - b11 - 3b10 - b9 + b8 - 3b7 - 3b6 - 3b5 + 6b2 - 2b1 + 3) * q^91 + (b15 + b14 + b13 + b12 + b9 - b8 - b4 - 2b3 + 1) q^93 + (b13 + b12 + b9 - b8 - b7 - b6 + b1) * q^95 + (2b10 - b9 + b8 - 4b6 + b5 + b4 - 3b3 + 3b2 + b1) * q^97 - b15 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 - 4 * q^9	$ 16 q - 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} - q^{11} + q^{13} - 4 q^{15} - 3 q^{17} - 2 q^{19} - 6 q^{21} + 10 q^{23} - 4 q^{25} - 4 q^{27} - 5 q^{29} + 3 q^{31} - q^{33} - q^{35} - 25 q^{37} + q^{39} - 2 q^{41} - 2 q^{43} + 16 q^{45} + 14 q^{47} - 8 q^{49} + 2 q^{51} - 11 q^{53} + 4 q^{55} + 8 q^{57} - 7 q^{59} - 3 q^{61} + 4 q^{63} + 6 q^{65} + 32 q^{67} - 15 q^{71} - q^{73} - 4 q^{75} + 9 q^{77} - 15 q^{79} - 4 q^{81} - 17 q^{83} + 2 q^{85} + 26 q^{89} + 36 q^{91} + 3 q^{93} - 2 q^{95} - 45 q^{97} - q^{99}+O(q^{100}) $ 16 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 - 4 * q^9 - q^11 + q^13 - 4 * q^15 - 3 * q^17 - 2 * q^19 - 6 * q^21 + 10 * q^23 - 4 * q^25 - 4 * q^27 - 5 * q^29 + 3 * q^31 - q^33 - q^35 - 25 * q^37 + q^39 - 2 * q^41 - 2 * q^43 + 16 * q^45 + 14 * q^47 - 8 * q^49 + 2 * q^51 - 11 * q^53 + 4 * q^55 + 8 * q^57 - 7 * q^59 - 3 * q^61 + 4 * q^63 + 6 * q^65 + 32 * q^67 - 15 * q^71 - q^73 - 4 * q^75 + 9 * q^77 - 15 * q^79 - 4 * q^81 - 17 * q^83 + 2 * q^85 + 26 * q^89 + 36 * q^91 + 3 * q^93 - 2 * q^95 - 45 * q^97 - q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 26 x^{14} - 64 x^{13} + 486 x^{12} - 10 x^{11} + 6075 x^{10} + 9130 x^{9} + \cdots + 50410000 $ x^16 - 4*x^15 + 26*x^14 - 64*x^13 + 486*x^12 - 10*x^11 + 6075*x^10 + 9130*x^9 + 72165*x^8 + 32200*x^7 + 420125*x^6 + 573200*x^5 + 3577025*x^4 + 5167000*x^3 + 17399000*x^2 + 15265000*x + 50410000 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 23\!\cdots\!19 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 13\!\cdots\!00 $ (231196211924204894994582343537360719v^15 - 3891126478463668967478206872578522832v^14 - 7521846409685859315032900404741690922v^13 + 79384446092462658825684852037394986308v^12 - 527757386202093772858039737673682938602v^11 + 1239768196337368419879035250635397198874v^10 - 11102527516769149292757379408320093359735v^9 + 3313787574108830970963520570765890700990v^8 - 95952511181632046898071464568659570073145v^7 - 144777490443149765021972653110986161043340v^6 - 635497788451807885597461837477930447344725v^5 + 3232872985894635484453158552595168530857300v^4 - 5479831169500790822352654721639781046053225v^3 - 11284140975963347913444480079900366143760900v^2 - 28145165587005112940180242787836009413984500*v - 20590554250371201481386945742383792015536000) / 133263454953106729197724941422397876724261000
$\beta_{3}$	$=$	$ ( - 12\!\cdots\!95 \nu^{15} + \cdots - 81\!\cdots\!00 ) / 13\!\cdots\!00 $ (-1229425550691830124210033200324386795v^15 + 3906675262362929782416251410294068078v^14 - 29307440213945523440125942251885143122v^13 + 66875170418595551360356142510984704408v^12 - 511859698307252451199695895232273109462v^11 - 779762859276171784026135592266183269742v^10 - 5772089599603440458733969869679065577025v^9 - 22396141475651665519219555477810130982960v^8 - 63282903678318195771515552639692914314535v^7 - 136536987057051056921192891623998602805030v^6 - 278603603737799502084911668470867824629775v^5 - 808231702519433883942318373102548415159750v^4 - 2979971723789916752451337136810184679718775v^3 - 31043130116645591429473336926766352697942550v^2 - 10392767066883059326693464088948065831745500*v - 8100494367566636123549650006921242451225000) / 133263454953106729197724941422397876724261000
$\beta_{4}$	$=$	$ ( 15\!\cdots\!87 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 13\!\cdots\!00 $ (1558100810894047906268755448081079587v^15 - 117680946581846240879657792868364807812v^14 + 904705069555488462911152443566878038378v^13 - 5430527518055431176266868955688140027892v^12 + 22398136379495084237410987811867077917218v^11 - 99679081886928605321136425841882203638734v^10 + 251152083924555143671341017724280492221605v^9 - 906400150740378938665115510312260419791410v^8 + 1208344128469311670969681061301929265026335v^7 - 5302711325352164198377661485659368132165860v^6 + 13878440291893123475679872029076123365257975v^5 - 42902804993390640972822089097659018024014300v^4 + 62056267153922168065587404987999236228336875v^3 - 191456563044618409242136500271261564666959100v^2 + 76216814722401612010616332511607456161888500*v - 416878967241432325621658510110436813361754000) / 133263454953106729197724941422397876724261000
$\beta_{5}$	$=$	$ ( - 10\!\cdots\!59 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 46\!\cdots\!50 $ (-10444864897066371082746047529679859v^15 - 47651225069419671073563525833496724v^14 + 331623252308492155300486345154176311v^13 - 2253939243652314619103544917721268479v^12 + 4373521684706374890242891105882995796v^11 - 44038889099326387429047419525737886365v^10 + 4235796335355071407264027374189392030v^9 - 396608400757543285723082744331102839295v^8 - 535992987560243530425345790749609775335v^7 - 2579679918966424179954439241396352533150v^6 + 10916729990210145206486844905913920305375v^5 - 22207133096686513510107924750110946380050v^4 - 43939196489351107767188334679429256967725v^3 - 113266719993923781366869651348741368182625v^2 - 84928747976740102829152271889019730250250*v - 41037327616546368861538365977881527622500) / 469237517440516652104665286698584072972750
$\beta_{6}$	$=$	$ ( - 40\!\cdots\!31 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 13\!\cdots\!00 $ (-4010483871747679655640224068428554231v^15 + 29633282457878898421502202015066116096v^14 - 161720863870861558248323946852154551414v^13 + 614928691627373283817991549323719687496v^12 - 2826919886492170236587373820992871178434v^11 + 6513906240235564751854492357311006867142v^10 - 24417140630033434131498187422794235192505v^9 + 39294186422779204965687824606282756131130v^8 - 165323298627066253215579061501044824889755v^7 + 809104004659461757037565310268998560865680v^6 - 1343604930051194566465232234265908000919675v^5 + 3607532218178142373856679184418488740602300v^4 - 6157183177861961419355814190275097756500375v^3 + 28123503965180883331787353656304446805290800v^2 - 5040914179300837990137328301018478403198000*v + 164158969737465527417506503700009740804188000) / 133263454953106729197724941422397876724261000
$\beta_{7}$	$=$	$ ( 11\!\cdots\!27 \nu^{15} + \cdots - 82\!\cdots\!00 ) / 13\!\cdots\!00 $ (11045263679844584814698075376597838327v^15 - 192395613937667076877641408534781028020v^14 + 1256685006668740234901372171793323015450v^13 - 6839234181828357299328170861551326750200v^12 + 28299684956018815081630748066943563698770v^11 - 115397347606715319034546940242321487106822v^10 + 289733640246513990157828693176997877538225v^9 - 1046059476568422952093299337371854126564610v^8 + 1252471469812849875141989455456625350629935v^7 - 7817868058974388853257596931345778266877180v^6 + 12240101779022302102158468377629894109521275v^5 - 51614763466663484376062761008682727864445700v^4 + 46258069719805712616300985805198466606058575v^3 - 273733834169866399301580204563466907785886800v^2 + 8692323043219148646918036314340190571472500*v - 828499969385539095170840370248944187627650000) / 133263454953106729197724941422397876724261000
$\beta_{8}$	$=$	$ ( - 47\!\cdots\!83 \nu^{15} + \cdots - 71\!\cdots\!00 ) / 46\!\cdots\!50 $ (-47856855531296407742751076554055983v^15 + 202282687343034814090415919271169512v^14 - 1261470858575780936116257777972859918v^13 + 3090932129657739168824735646959837402v^12 - 22795075357458056180627077875446201848v^11 + 188208130867183885506430306657634645v^10 - 267288394971251831766489684334631888240v^9 - 436948133723961440611611649285569687290v^8 - 3303667553977947330806973680536612701070v^7 - 1201759178052110383312031346769853328800v^6 - 20796977371351804057991745142400478964125v^5 - 28832457371395360566410768690477293808600v^4 - 171991810318666000396057716260122487735450v^3 - 227948925018440282881503275582992734725250v^2 - 793588049435541751270614521494970496973250*v - 711860887235213138876139772146068376002500) / 469237517440516652104665286698584072972750
$\beta_{9}$	$=$	$ ( 14\!\cdots\!57 \nu^{15} + \cdots + 47\!\cdots\!00 ) / 13\!\cdots\!00 $ (14097637668305820406939720159340964557v^15 - 125180643021093970634109011932302469806v^14 + 687315505843852038003258667728754260754v^13 - 2188346475325085210911061803812335642876v^12 + 6904294899905528153100886990967773021394v^11 - 2103947173182343700196535639438132937478v^10 - 32978227095544359668248061942055529343545v^9 + 355109076267009148545509250733121236969000v^8 - 977100294253233405431911542676883614228635v^7 + 1846074811570591762929009614658308796369030v^6 - 593635750080887875312405789149548841284175v^5 + 14099829949380305176426568698478903925932950v^4 - 53638742157370419341220853163068718020940675v^3 + 132065837105859469287585255022599554716708050v^2 - 313463717058390853095767603719861308658003500*v + 476228961269637066495412444944153660420795000) / 133263454953106729197724941422397876724261000
$\beta_{10}$	$=$	$ ( 14\!\cdots\!15 \nu^{15} + \cdots + 87\!\cdots\!00 ) / 13\!\cdots\!00 $ (14464460997086249030347443199969035515v^15 + 25172805655044025784624880926154409792v^14 - 285244547015073367497162932724762694258v^13 + 3127836744417344812359018162310173837072v^12 - 10337915914347509335699848543360260061158v^11 + 83207117603188930471491342064426643172202v^10 - 149509370311928864084510586153983451940715v^9 + 1025855015303803209243240446985924037452270v^8 - 683071867052516121862077447743598271660065v^7 + 6554141026257373978890115177037535616294280v^6 - 10830317060380622757931666259398678916299825v^5 + 49409381360663227082063853333197739540458500v^4 - 10077209212627978955447100253184449487333725v^3 + 297173335836826820466029900004685018212242800v^2 - 247343598005160172933665039421572642288538000*v + 878839624656591288500187732097877880541668000) / 133263454953106729197724941422397876724261000
$\beta_{11}$	$=$	$ ( 10\!\cdots\!03 \nu^{15} + \cdots + 19\!\cdots\!00 ) / 93\!\cdots\!00 $ (109483532825708721633924660906883803v^15 - 290547261842708738221487033660212382v^14 + 1776539826438228430722486652188240098v^13 - 1070879297863123823622802302664610572v^12 + 31265240339988643670428683931595235078v^11 + 99649746590787615156391709581058983740v^10 + 447432759258869179273507987450121051965v^9 + 1846258077324683573388413387667968002090v^8 + 6996578235718817448368654276561505882345v^7 + 9238292319013165104341592794750828467700v^6 + 19031445629446156740015945947081498273375v^5 + 112063083800061824896706801469769121191850v^4 + 257983927022929250286491430184101756109325v^3 + 759882051219123149654752216986679050988000v^2 + 893676071660038507669896723202013167476500*v + 1942242500199118709569918062363057119475000) / 938475034881033304209330573397168145945500
$\beta_{12}$	$=$	$ ( - 16\!\cdots\!87 \nu^{15} + \cdots - 79\!\cdots\!00 ) / 13\!\cdots\!00 $ (-16528306583525942187270901302661856887v^15 - 2791909298400579646466157580784411356v^14 + 141145447928578575264752490084765243894v^13 - 2640293449616006441502061966007490839776v^12 + 8732279097569335716707329380499396050494v^11 - 81610714446296735243819396431367164409254v^10 + 145388543722661131044540646479830978761935v^9 - 1066685900375842188749459205251444154274730v^8 + 653853689555948679573503530186302482698985v^7 - 6742804067705945634770055557006862189391160v^6 + 7090382063380502686541534851991226159615525v^5 - 51723631133517658940050000136914898964035000v^4 + 5805694041817805571049935978616573021531225v^3 - 353037569860887095517409351328529262048517600v^2 + 155022376647098294293757502767227904382693000*v - 794490304693127996805102660745858980784082000) / 133263454953106729197724941422397876724261000
$\beta_{13}$	$=$	$ ( - 15\!\cdots\!68 \nu^{15} + \cdots - 35\!\cdots\!00 ) / 66\!\cdots\!00 $ (-15814087450922974030074602288533825868v^15 + 113599454380835567532060066022191323605v^14 - 601929141619333922102216183863866097580v^13 + 1855976465684522934095597706674825650430v^12 - 7749190479131322044156240587962434566750v^11 + 5874653386455799174880793211721291910678v^10 - 20367428956217252417658931675775536628620v^9 - 188312595517949395074796884240303485288275v^8 + 132534643574977450467027791603290986120060v^7 - 245243254288696856566588301616793955613955v^6 - 1600427115455035686993714049011825873002600v^5 - 9866168367540713003528087462921469308427525v^4 + 8449826337577258041206545713700261173982950v^3 - 56804740273171178670570071677581188197701675v^2 + 68538305350853533063166244223600059108893000*v - 354233686281450785534373283371064755299696500) / 66631727476553364598862470711198938362130500
$\beta_{14}$	$=$	$ ( 17\!\cdots\!33 \nu^{15} + \cdots + 23\!\cdots\!50 ) / 46\!\cdots\!50 $ (179471557548653003873331169450716633v^15 - 1073546426570984683766925481402442447v^14 + 6827171558397679366607500684422405513v^13 - 24676669263440274973384236546652090642v^12 + 132492091439823451666166987382224612008v^11 - 229111449753741415742834715935908035660v^10 + 1373060985959896037586264367333773588410v^9 - 190912605162007032297750136408807346720v^8 + 10701200455718194008448973184525138999645v^7 - 4745217532485532673763990491116926262350v^6 + 69073187250159094666860586220561598452650v^5 + 31313166266123354443902587227152785685700v^4 + 499499597013166569244340885125738939801075v^3 + 395520244740131064578245913783462654639375v^2 + 1070898180926519019926435581981453127771750*v + 2316617225046757971768175832540467536697750) / 469237517440516652104665286698584072972750
$\beta_{15}$	$=$	$ ( - 54\!\cdots\!93 \nu^{15} + \cdots - 11\!\cdots\!00 ) / 13\!\cdots\!00 $ (-54969658670651591210783770018244288893v^15 + 330649609269277427402049238667735661246v^14 - 1966338343803376290502638175874731163354v^13 + 6318369922573405022804241586516246241276v^12 - 32467824378858457865689551928324245264374v^11 + 35636148065139637692216866380189729318354v^10 - 268219031906457111954610098273459459284355v^9 - 379522928528132268208232735550964606991540v^8 - 1871038286613697997686500235057659549702465v^7 - 859292688433901478752255508926243274214690v^6 - 14015641346638606483286027628490099434059325v^5 - 30225466198745449130551897674067104897150850v^4 - 84670757432715020305089090731004968333456325v^3 - 236781696804063587306312950040902823485328150v^2 - 286388793616797682620722991911860666393490000*v - 1196666884608711405168504298095700373241176000) / 133263454953106729197724941422397876724261000

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{7} - \beta_{5} - 10\beta_{3} + \beta_{2} $ b15 + b14 - b13 + b12 - b11 - b7 - b5 - 10*b3 + b2
$\nu^{3}$	$=$	$ 3 \beta_{13} + 6 \beta_{12} - 13 \beta_{11} + 2 \beta_{10} + 6 \beta_{9} - 19 \beta_{8} + \cdots - 12 \beta_1 $ 3b13 + 6b12 - 13b11 + 2b10 + 6b9 - 19b8 + 4b6 - 12b5 - 3b4 - 7b3 + 4b2 - 12b1
$\nu^{4}$	$=$	$ - 23 \beta_{15} - 25 \beta_{14} + 46 \beta_{13} + 29 \beta_{12} - 4 \beta_{11} + 54 \beta_{10} + \cdots - 8 $ -23b15 - 25b14 + 46b13 + 29b12 - 4b11 + 54b10 + 23b9 - 56b8 + 29b7 + 8b6 + 54b5 - 5b4 + 85b2 + 19b1 - 8
$\nu^{5}$	$=$	$ - 108 \beta_{15} - 92 \beta_{14} + 88 \beta_{13} + 20 \beta_{12} + 20 \beta_{11} + 191 \beta_{10} + \cdots - 346 $ -108b15 - 92b14 + 88b13 + 20b12 + 20b11 + 191b10 - 83b9 - 43b8 + 176b7 + 217b6 + 439b5 - 68b4 + 217b3 - 88b2 + 45*b1 - 346
$\nu^{6}$	$=$	$ - 516 \beta_{15} - 562 \beta_{14} - 209 \beta_{13} - 467 \beta_{12} + 444 \beta_{11} + 28 \beta_{10} + \cdots - 2564 $ -516b15 - 562b14 - 209b13 - 467b12 + 444b11 + 28b10 - 771b9 + 237b8 + 725b7 + 2081b6 + 1006b5 - 230b4 + 2564b3 - 1872b2 - 702*b1 - 2564
$\nu^{7}$	$=$	$ - 3062 \beta_{15} - 4180 \beta_{14} - 1370 \beta_{13} - 3062 \beta_{12} + 6414 \beta_{11} - 4301 \beta_{9} + \cdots - 5672 $ -3062b15 - 4180b14 - 1370b13 - 3062b12 + 6414b11 - 4301b9 + 5713b8 + 846b7 + 4343b5 + 3334b4 + 9191b3 - 4302b2 - 5672
$\nu^{8}$	$=$	$ - 9706 \beta_{15} - 9706 \beta_{14} - 8120 \beta_{13} - 16240 \beta_{12} + 26036 \beta_{11} + \cdots + 11708 \beta_1 $ -9706b15 - 9706b14 - 8120b13 - 16240b12 + 26036b11 - 6313b10 - 19528b9 + 45564b8 - 9706b7 - 37571b6 + 11708b5 + 21114b4 + 18118b3 - 9998b2 + 11708*b1
$\nu^{9}$	$=$	$ 26257 \beta_{15} + 45995 \beta_{14} - 52514 \beta_{13} - 81006 \beta_{12} + 19231 \beta_{11} + \cdots + 144132 $ 26257b15 + 45995b14 - 52514b13 - 81006b12 + 19231b11 - 127001b10 - 26257b9 + 154569b8 - 81006b7 - 144132b6 - 127001b5 + 36660b4 - 38955b2 - 7026b1 + 144132
$\nu^{10}$	$=$	$ 440947 \beta_{15} + 483063 \beta_{14} - 228887 \beta_{13} - 212060 \beta_{12} - 212060 \beta_{11} + \cdots + 1306764 $ 440947b15 + 483063b14 - 228887b13 - 212060b12 - 212060b11 - 693469b10 + 252522b9 + 199347b8 - 457774b7 - 562428b6 - 1139346b5 + 16827b4 - 562428b3 + 228887b2 - 29540*b1 + 1306764
$\nu^{11}$	$=$	$ 2718104 \beta_{15} + 2840778 \beta_{14} - 626999 \beta_{13} + 732053 \beta_{12} - 2098846 \beta_{11} + \cdots + 5986216 $ 2718104b15 + 2840778b14 - 626999b13 + 732053b12 - 2098846b11 - 1745202b10 + 2213779b9 - 1118203b8 - 2091105b7 - 2332939b6 - 4939624b5 - 386150b4 - 5986216b3 + 2959938b2 + 760293*b1 + 5986216
$\nu^{12}$	$=$	$ 10995323 \beta_{15} + 11950605 \beta_{14} + 1285185 \beta_{13} + 10995323 \beta_{12} - 15958511 \beta_{11} + \cdots + 15935328 $ 10995323b15 + 11950605b14 + 1285185b13 + 10995323b12 - 15958511b11 + 13416839b9 - 18278487b8 - 4855069b7 - 16993302b5 - 7095536b4 - 31789259b3 + 14650143b2 + 15935328
$\nu^{13}$	$=$	$ 19477584 \beta_{15} + 19477584 \beta_{14} + 33980025 \beta_{13} + 67960050 \beta_{12} + \cdots - 13860002 \beta_1 $ 19477584b15 + 19477584b14 + 33980025b13 + 67960050b12 - 70614189b11 + 45753437b10 + 60565182b9 - 131179371b8 + 19477584b7 + 59053179b6 - 13860002b5 - 46062741b4 - 92694357b3 + 58714332b2 - 13860002*b1
$\nu^{14}$	$=$	$ - 115203478 \beta_{15} - 138006690 \beta_{14} + 230406956 \beta_{13} + 275856559 \beta_{12} + \cdots - 430303568 $ -115203478b15 - 138006690b14 + 230406956b13 + 275856559b12 - 134636799b11 + 413863249b10 + 115203478b9 - 527510461b8 + 275856559b7 + 430303568b6 + 413863249b5 - 167738645b4 + 126678650b2 - 19433321b1 - 430303568
$\nu^{15}$	$=$	$ - 1360821463 \beta_{15} - 1491428947 \beta_{14} + 854245383 \beta_{13} + 506576080 \beta_{12} + \cdots - 3844564986 $ -1360821463b15 - 1491428947b14 + 854245383b13 + 506576080b12 + 506576080b11 + 2015380556b10 - 654559093b9 - 987791228b8 + 1708490766b7 + 2353512797b6 + 3467407389b5 - 347669303b4 + 2353512797b3 - 854245383b2 - 133545845*b1 - 3844564986

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times$.

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1 - \beta_{2} + \beta_{3} + \beta_{6}$

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} $

361.1

−2.16250	−	1.57115i
−2.06162	−	1.49786i
2.26881	+	1.64839i
4.07335	+	2.95946i
−2.16250	+	1.57115i
−2.06162	+	1.49786i
2.26881	−	1.64839i
4.07335	−	2.95946i
−1.22009	−	3.75505i
−0.725441	−	2.23268i
0.640948	+	1.97263i
1.18655	+	3.65182i
−1.22009	+	3.75505i
−0.725441	+	2.23268i
0.640948	−	1.97263i
1.18655	−	3.65182i

−0.809017

0.587785i

0.309017

0.951057i

−2.16250

−

1.57115i

0.309017

−

0.951057i

361.2

−0.809017

0.587785i

0.309017

0.951057i

−2.06162

−

1.49786i

0.309017

−

0.951057i

361.3

−0.809017

0.587785i

0.309017

0.951057i

2.26881

1.64839i

0.309017

−

0.951057i

361.4

−0.809017

0.587785i

0.309017

0.951057i

4.07335

2.95946i

0.309017

−

0.951057i

841.1

−0.809017

−

0.587785i

0.309017

−

0.951057i

−2.16250

1.57115i

0.309017

0.951057i

841.2

−0.809017

−

0.587785i

0.309017

−

0.951057i

−2.06162

1.49786i

0.309017

0.951057i

841.3

−0.809017

−

0.587785i

0.309017

−

0.951057i

2.26881

−

1.64839i

0.309017

0.951057i

841.4

−0.809017

−

0.587785i

0.309017

−

0.951057i

4.07335

−

2.95946i

0.309017

0.951057i

961.1

0.309017

−

0.951057i

−0.809017

0.587785i

−1.22009

−

3.75505i

−0.809017

−

0.587785i

961.2

0.309017

−

0.951057i

−0.809017

0.587785i

−0.725441

−

2.23268i

−0.809017

−

0.587785i

961.3

0.309017

−

0.951057i

−0.809017

0.587785i

0.640948

1.97263i

−0.809017

−

0.587785i

961.4

0.309017

−

0.951057i

−0.809017

0.587785i

1.18655

3.65182i

−0.809017

−

0.587785i

1081.1

0.309017

0.951057i

−0.809017

−

0.587785i

−1.22009

3.75505i

−0.809017

0.587785i

1081.2

0.309017

0.951057i

−0.809017

−

0.587785i

−0.725441

2.23268i

−0.809017

0.587785i

1081.3

0.309017

0.951057i

−0.809017

−

0.587785i

0.640948

−

1.97263i

−0.809017

0.587785i

1081.4

0.309017

0.951057i

−0.809017

−

0.587785i

1.18655

−

3.65182i

−0.809017

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1320.2.bw.i	✓	16
11.c	even	5	1	inner	1320.2.bw.i	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1320.2.bw.i	✓	16	1.a	even	1	1	trivial
1320.2.bw.i	✓	16	11.c	even	5	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{16} - 4 T_{7}^{15} + 26 T_{7}^{14} - 64 T_{7}^{13} + 486 T_{7}^{12} - 10 T_{7}^{11} + \cdots + 50410000 $ T7^16 - 4*T7^15 + 26*T7^14 - 64*T7^13 + 486*T7^12 - 10*T7^11 + 6075*T7^10 + 9130*T7^9 + 72165*T7^8 + 32200*T7^7 + 420125*T7^6 + 573200*T7^5 + 3577025*T7^4 + 5167000*T7^3 + 17399000*T7^2 + 15265000*T7 + 50410000 acting on $S_{2}^{\mathrm{new}}(1320, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} $ (T^4 + T^3 + T^2 + T + 1)^4
$5$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} $ (T^4 + T^3 + T^2 + T + 1)^4
$7$	$ T^{16} - 4 T^{15} + \cdots + 50410000 $ T^16 - 4T^15 + 26T^14 - 64T^13 + 486T^12 - 10T^11 + 6075T^10 + 9130T^9 + 72165T^8 + 32200T^7 + 420125T^6 + 573200T^5 + 3577025T^4 + 5167000T^3 + 17399000T^2 + 15265000*T + 50410000
$11$	$ T^{16} + \cdots + 214358881 $ T^16 + T^15 + 11T^14 + 60T^13 + 539T^11 + 360T^10 - 2920T^9 + 11771T^8 - 32120T^7 + 43560T^6 + 717409T^5 + 9663060T^3 + 19487171T^2 + 19487171T + 214358881
$13$	$ T^{16} - T^{15} + \cdots + 609961 $ T^16 - T^15 + 69T^14 + 39T^13 + 2020T^12 + 2729T^11 + 32722T^10 + 73757T^9 + 369283T^8 + 1193752T^7 + 5199207T^6 + 19113768T^5 + 60849899T^4 + 107096953T^3 + 96868028T^2 + 12221088T + 609961
$17$	$ T^{16} + 3 T^{15} + \cdots + 256 $ T^16 + 3T^15 - 3T^14 - 17T^13 + 1488T^12 + 4303T^11 + 38890T^10 + 105635T^9 + 430906T^8 + 935445T^7 + 1284565T^6 + 709868T^5 + 440313T^4 - 164452T^3 + 34432T^2 - 3712*T + 256
$19$	$ T^{16} + \cdots + 1178136976 $ T^16 + 2T^15 + 59T^14 + 246T^13 + 2899T^12 + 16780T^11 + 153477T^10 + 1128712T^9 + 7860538T^8 + 39534982T^7 + 157784792T^6 + 439657570T^5 + 885295629T^4 + 935161366T^3 + 322234144T^2 - 1436528048*T + 1178136976
$23$	$ (T^{8} - 5 T^{7} + \cdots + 2945)^{2} $ (T^8 - 5T^7 - 48T^6 + 210T^5 + 751T^4 - 2280T^3 - 4090T^2 + 1735*T + 2945)^2
$29$	$ T^{16} + \cdots + 1861576444816 $ T^16 + 5T^15 + 209T^14 + 814T^13 + 19354T^12 + 90167T^11 + 1108459T^10 + 7170301T^9 + 60403726T^8 + 465307686T^7 + 3169969601T^6 + 18851348831T^5 + 105514245641T^4 + 456829276896T^3 + 1336087815056T^2 + 2292188008792*T + 1861576444816
$31$	$ T^{16} + \cdots + 116208400 $ T^16 - 3T^15 + 153T^14 - 14T^13 + 8608T^12 + 10329T^11 + 214897T^10 + 450696T^9 + 1575878T^8 - 7669906T^7 + 54996358T^6 + 62408588T^5 + 132067121T^4 + 245249270T^3 + 766930360T^2 - 491999200*T + 116208400
$37$	$ T^{16} + \cdots + 1770642241 $ T^16 + 25T^15 + 399T^14 + 4249T^13 + 39414T^12 + 274517T^11 + 1951894T^10 + 8340016T^9 + 37668201T^8 + 412708166T^7 + 3928719951T^6 + 16331410296T^5 + 27696422711T^4 - 11038405594T^3 + 21881831476T^2 - 4118566283*T + 1770642241
$41$	$ T^{16} + \cdots + 50357155216 $ T^16 + 2T^15 + 82T^14 + 636T^13 + 6776T^12 - 7858T^11 + 451415T^10 + 1931458T^9 + 28238439T^8 + 89950724T^7 + 451518195T^6 + 557924174T^5 + 1402559081T^4 - 471987402T^3 + 6055695528T^2 + 23440344224*T + 50357155216
$43$	$ (T^{8} + T^{7} + \cdots + 14764)^{2} $ (T^8 + T^7 - 115T^6 - 307T^5 + 2490T^4 + 4018T^3 - 21027T^2 + 6530T + 14764)^2
$47$	$ T^{16} - 14 T^{15} + \cdots + 76125625 $ T^16 - 14T^15 + 252T^14 - 2057T^13 + 19105T^12 - 106038T^11 + 634947T^10 - 2440276T^9 + 8304031T^8 - 11963475T^7 + 13831830T^6 + 49828775T^5 + 322746400T^4 - 560909125T^3 + 266192625T^2 + 269820625*T + 76125625
$53$	$ T^{16} + \cdots + 67207451536 $ T^16 + 11T^15 + 144T^14 + 1782T^13 + 29411T^12 + 157318T^11 + 2965499T^10 + 35847987T^9 + 444355201T^8 + 3468494683T^7 + 18698898781T^6 + 65821355854T^5 + 170072105269T^4 + 261610767042T^3 + 271425953556T^2 + 175754469800*T + 67207451536
$59$	$ T^{16} + \cdots + 2346009925561 $ T^16 + 7T^15 + 337T^14 + 2401T^13 + 48516T^12 + 346237T^11 + 2438120T^10 + 16428268T^9 + 108822529T^8 + 500589844T^7 + 3038996745T^6 + 21538592944T^5 + 129690693661T^4 + 541321315398T^3 + 1561841269058T^2 + 2710197927029*T + 2346009925561
$61$	$ T^{16} + \cdots + 7744000000 $ T^16 + 3T^15 + 78T^14 + 441T^13 + 4865T^12 + 12631T^11 + 222808T^10 + 775973T^9 + 10762821T^8 + 36536860T^7 + 261514760T^6 + 764337800T^5 + 4534328400T^4 - 7998576000T^3 + 7538240000T^2 - 2569600000*T + 7744000000
$67$	$ (T^{8} - 16 T^{7} + \cdots + 716)^{2} $ (T^8 - 16T^7 - 12T^6 + 727T^5 - 221T^4 - 8183T^3 - 4707T^2 + 1994*T + 716)^2
$71$	$ T^{16} + \cdots + 153561664000000 $ T^16 + 15T^15 + 297T^14 + 4851T^13 + 82814T^12 + 550479T^11 + 9416729T^10 + 125576877T^9 + 2062794887T^8 + 21768601538T^7 + 175541026424T^6 + 1020988731328T^5 + 4487585062416T^4 + 14831670596800T^3 + 42925041056000T^2 + 100181884800000*T + 153561664000000
$73$	$ T^{16} + \cdots + 5139542907136 $ T^16 + T^15 + 134T^14 - 2319T^13 + 28625T^12 - 130829T^11 + 9102112T^10 - 126409607T^9 + 1321441413T^8 - 8267065952T^7 + 38516934812T^6 - 107401494128T^5 + 242602410384T^4 - 19672130368T^3 + 314475066048T^2 + 1189823534592T + 5139542907136
$79$	$ T^{16} + \cdots + 16374785296 $ T^16 + 15T^15 + 333T^14 + 1615T^13 + 10173T^12 - 44370T^11 + 38031T^10 + 1464330T^9 + 27008215T^8 - 84599405T^7 + 72620786T^6 + 501393370T^5 + 3205894173T^4 + 5232091410T^3 + 10379475308T^2 + 3888825960*T + 16374785296
$83$	$ T^{16} + 17 T^{15} + \cdots + 7929856 $ T^16 + 17T^15 + 161T^14 + 653T^13 + 11670T^12 + 16037T^11 + 572943T^10 + 859149T^9 + 25207353T^8 + 136607856T^7 + 764517008T^6 - 2327679744T^5 + 3228780544T^4 - 573689856T^3 + 5843689472T^2 + 132284416*T + 7929856
$89$	$ (T^{8} - 13 T^{7} + \cdots - 178410500)^{2} $ (T^8 - 13T^7 - 514T^6 + 8065T^5 + 61290T^4 - 1403235T^3 + 2005045T^2 + 50621850*T - 178410500)^2
$97$	$ T^{16} + \cdots + 4505094400 $ T^16 + 45T^15 + 912T^14 + 9819T^13 + 55789T^12 + 148801T^11 + 1101674T^10 + 19334923T^9 + 177596017T^8 + 920326312T^7 + 3335972504T^6 + 4639743072T^5 + 13722178416T^4 + 11180974080T^3 + 22449023040T^2 + 7160361600*T + 4505094400
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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:	\( N \)	\(=\)	\( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1320.bw (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} - \beta_{6} q^{9}+O(q^{10}) \) q + b2 * q^3 - b3 * q^5 + b1 * q^7 - b6 * q^9	\( q + \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} - \beta_{6} q^{9} - \beta_{12} q^{11} + ( - \beta_{9} + \beta_{8} + \beta_{4}) q^{13} + (\beta_{6} + \beta_{3} - \beta_{2} - 1) q^{15} + (\beta_{14} - \beta_{4}) q^{17} + (\beta_{15} - 2 \beta_{13} - \beta_{12} + \cdots - 1) q^{19}+ \cdots - \beta_{15} q^{99}+O(q^{100}) \) q + b2 * q^3 - b3 * q^5 + b1 * q^7 - b6 * q^9 - b12 * q^11 + (-b9 + b8 + b4) * q^13 + (b6 + b3 - b2 - 1) * q^15 + (b14 - b4) * q^17 + (b15 - 2b13 - b12 - b10 - b9 + 2b8 - b7 + b6 - b5 + b4 + b2 - b1 - 1) * q^19 + b5 * q^21 + (-b14 + b10 - b9 + b5 + 1) * q^23 + b2 * q^25 - b3 * q^27 + (2b15 + 2b14 - 2b13 - b12 - 2b11 - 3b10 - b8 - 4b5 - 2b4 + b3 + 2b2 - 3b1 - 1) q^29 + (-b13 - 2b12 - b11 - b10 - b9 - b6 - b5 + b3 - b1) q^31 + (b13 + b12 + b10 + b5 - b2 + b1) * q^33 + (-b11 - b8 - b5 - b1) * q^35 + (2b15 - 2b13 - b12 - b10 - 2b9 + b8 + b6 + 2b3 + b2 - 2b1 - 2) q^37 + (b14 + b9 - b8 - b5 - b4) * q^39 + (-b8 - 2b4) q^41 + (b15 - b14 - b13 - b9 - 2b7 + b4 + b2 - b1) q^43 + q^45 + (b14 - b11 - b10 - 2b6 - b5 - b4 + 3b2 - b1 + 2) * q^47 + (b15 + b14 - b13 + b12 - b11 - b7 - b5 - 3b3 + b2) q^49 + (-b11 - b10 - b8 - b5 - b4 - b1) * q^51 + (b15 + b14 - b11 - b9 + b7 - b6 + b5 + b1) * q^53 + (b15 - b13 - b10 - b7 - b5 + b2 - b1) * q^55 + (b15 + b14 - b11 - b8 + b7 + b6 - b5 - b4 + b3 - b2 - b1) * q^57 + (-b14 + b13 + b12 + 2b11 + b10 - b7 + b6 + 3b5 + b4 + b3 - 2b2 + 3b1 - 1) * q^59 + (-b11 - b9 - b8 - b5 - b3) * q^61 + b8 * q^63 + (b14 - b10 + b9 - b5) * q^65 + (b15 - b14 - b13 - b9 + b8 - 2b7 - 2b6 + b5 + b4 - 2b3 + b2 + 3) q^67 + (b14 - b10 - b5 + b2) * q^69 + (-b15 - b14 + b13 - b12 - 2b11 - 2b9 + b8 + b7 + 2b5 - 2b3 - b2) * q^71 + (2b15 + b12 + 2b11 + b10 + b8 - 2b7 + 2b5 + 2b4 + 2b1) * q^73 - b6 * q^75 + (2b15 + 4b14 - 2b13 - b12 - b11 - 3b10 - b9 + 2b8 + 2b6 - 3b5 - b4 + 3b3 - b2 - b1 - 2) * q^77 + (-b13 - 2b12 - b11 - b10 - b9 + 2b6 - 2b5 - 3b3 + 4b2 - 2b1) * q^79 + (b6 + b3 - b2 - 1) * q^81 + (b15 - b14 - b13 + b12 + b8 - b7 + 2b4 - b2 - 2) q^83 + (-b14 + b11 + b10 + b8 + b5 + b4 + b1) * q^85 + (b13 - b12 - b11 - b10 + b9 - 3b8 + 2b7 - b6 - b5 - 2b4 - b3 - b2 - 2b1 + 1) * q^87 + (b15 + b14 + b13 - 2b12 - 2b11 - b10 - 5b8 + 2b7 + 2b6 - 2b5 - 3b4 + 2b3 - b2 - 4b1 + 2) q^89 + (b15 - 2b13 - 3b12 - b11 - 3b10 - b9 + b8 - 3b7 - 3b6 - 3b5 + 6b2 - 2b1 + 3) * q^91 + (b15 + b14 + b13 + b12 + b9 - b8 - b4 - 2b3 + 1) q^93 + (b13 + b12 + b9 - b8 - b7 - b6 + b1) * q^95 + (2b10 - b9 + b8 - 4b6 + b5 + b4 - 3b3 + 3b2 + b1) * q^97 - b15 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 - 4 * q^9	\( 16 q - 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} - q^{11} + q^{13} - 4 q^{15} - 3 q^{17} - 2 q^{19} - 6 q^{21} + 10 q^{23} - 4 q^{25} - 4 q^{27} - 5 q^{29} + 3 q^{31} - q^{33} - q^{35} - 25 q^{37} + q^{39} - 2 q^{41} - 2 q^{43} + 16 q^{45} + 14 q^{47} - 8 q^{49} + 2 q^{51} - 11 q^{53} + 4 q^{55} + 8 q^{57} - 7 q^{59} - 3 q^{61} + 4 q^{63} + 6 q^{65} + 32 q^{67} - 15 q^{71} - q^{73} - 4 q^{75} + 9 q^{77} - 15 q^{79} - 4 q^{81} - 17 q^{83} + 2 q^{85} + 26 q^{89} + 36 q^{91} + 3 q^{93} - 2 q^{95} - 45 q^{97} - q^{99}+O(q^{100}) \) 16 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 - 4 * q^9 - q^11 + q^13 - 4 * q^15 - 3 * q^17 - 2 * q^19 - 6 * q^21 + 10 * q^23 - 4 * q^25 - 4 * q^27 - 5 * q^29 + 3 * q^31 - q^33 - q^35 - 25 * q^37 + q^39 - 2 * q^41 - 2 * q^43 + 16 * q^45 + 14 * q^47 - 8 * q^49 + 2 * q^51 - 11 * q^53 + 4 * q^55 + 8 * q^57 - 7 * q^59 - 3 * q^61 + 4 * q^63 + 6 * q^65 + 32 * q^67 - 15 * q^71 - q^73 - 4 * q^75 + 9 * q^77 - 15 * q^79 - 4 * q^81 - 17 * q^83 + 2 * q^85 + 26 * q^89 + 36 * q^91 + 3 * q^93 - 2 * q^95 - 45 * q^97 - 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	1320.2.bw.i

Level	$1320$
Weight	$2$
Character orbit	1320.bw
Analytic conductor	$10.540$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(10.5402530668\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 26 x^{14} - 64 x^{13} + 486 x^{12} - 10 x^{11} + 6075 x^{10} + 9130 x^{9} + \cdots + 50410000 \) x^16 - 4x^15 + 26x^14 - 64x^13 + 486x^12 - 10x^11 + 6075x^10 + 9130x^9 + 72165x^8 + 32200x^7 + 420125x^6 + 573200x^5 + 3577025x^4 + 5167000x^3 + 17399000x^2 + 15265000*x + 50410000
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 23\!\cdots\!19 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 13\!\cdots\!00 \) (231196211924204894994582343537360719v^15 - 3891126478463668967478206872578522832v^14 - 7521846409685859315032900404741690922v^13 + 79384446092462658825684852037394986308v^12 - 527757386202093772858039737673682938602v^11 + 1239768196337368419879035250635397198874v^10 - 11102527516769149292757379408320093359735v^9 + 3313787574108830970963520570765890700990v^8 - 95952511181632046898071464568659570073145v^7 - 144777490443149765021972653110986161043340v^6 - 635497788451807885597461837477930447344725v^5 + 3232872985894635484453158552595168530857300v^4 - 5479831169500790822352654721639781046053225v^3 - 11284140975963347913444480079900366143760900v^2 - 28145165587005112940180242787836009413984500*v - 20590554250371201481386945742383792015536000) / 133263454953106729197724941422397876724261000
\(\beta_{3}\)	\(=\)	\( ( - 12\!\cdots\!95 \nu^{15} + \cdots - 81\!\cdots\!00 ) / 13\!\cdots\!00 \) (-1229425550691830124210033200324386795v^15 + 3906675262362929782416251410294068078v^14 - 29307440213945523440125942251885143122v^13 + 66875170418595551360356142510984704408v^12 - 511859698307252451199695895232273109462v^11 - 779762859276171784026135592266183269742v^10 - 5772089599603440458733969869679065577025v^9 - 22396141475651665519219555477810130982960v^8 - 63282903678318195771515552639692914314535v^7 - 136536987057051056921192891623998602805030v^6 - 278603603737799502084911668470867824629775v^5 - 808231702519433883942318373102548415159750v^4 - 2979971723789916752451337136810184679718775v^3 - 31043130116645591429473336926766352697942550v^2 - 10392767066883059326693464088948065831745500*v - 8100494367566636123549650006921242451225000) / 133263454953106729197724941422397876724261000
\(\beta_{4}\)	\(=\)	\( ( 15\!\cdots\!87 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 13\!\cdots\!00 \) (1558100810894047906268755448081079587v^15 - 117680946581846240879657792868364807812v^14 + 904705069555488462911152443566878038378v^13 - 5430527518055431176266868955688140027892v^12 + 22398136379495084237410987811867077917218v^11 - 99679081886928605321136425841882203638734v^10 + 251152083924555143671341017724280492221605v^9 - 906400150740378938665115510312260419791410v^8 + 1208344128469311670969681061301929265026335v^7 - 5302711325352164198377661485659368132165860v^6 + 13878440291893123475679872029076123365257975v^5 - 42902804993390640972822089097659018024014300v^4 + 62056267153922168065587404987999236228336875v^3 - 191456563044618409242136500271261564666959100v^2 + 76216814722401612010616332511607456161888500*v - 416878967241432325621658510110436813361754000) / 133263454953106729197724941422397876724261000
\(\beta_{5}\)	\(=\)	\( ( - 10\!\cdots\!59 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 46\!\cdots\!50 \) (-10444864897066371082746047529679859v^15 - 47651225069419671073563525833496724v^14 + 331623252308492155300486345154176311v^13 - 2253939243652314619103544917721268479v^12 + 4373521684706374890242891105882995796v^11 - 44038889099326387429047419525737886365v^10 + 4235796335355071407264027374189392030v^9 - 396608400757543285723082744331102839295v^8 - 535992987560243530425345790749609775335v^7 - 2579679918966424179954439241396352533150v^6 + 10916729990210145206486844905913920305375v^5 - 22207133096686513510107924750110946380050v^4 - 43939196489351107767188334679429256967725v^3 - 113266719993923781366869651348741368182625v^2 - 84928747976740102829152271889019730250250*v - 41037327616546368861538365977881527622500) / 469237517440516652104665286698584072972750
\(\beta_{6}\)	\(=\)	\( ( - 40\!\cdots\!31 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 13\!\cdots\!00 \) (-4010483871747679655640224068428554231v^15 + 29633282457878898421502202015066116096v^14 - 161720863870861558248323946852154551414v^13 + 614928691627373283817991549323719687496v^12 - 2826919886492170236587373820992871178434v^11 + 6513906240235564751854492357311006867142v^10 - 24417140630033434131498187422794235192505v^9 + 39294186422779204965687824606282756131130v^8 - 165323298627066253215579061501044824889755v^7 + 809104004659461757037565310268998560865680v^6 - 1343604930051194566465232234265908000919675v^5 + 3607532218178142373856679184418488740602300v^4 - 6157183177861961419355814190275097756500375v^3 + 28123503965180883331787353656304446805290800v^2 - 5040914179300837990137328301018478403198000*v + 164158969737465527417506503700009740804188000) / 133263454953106729197724941422397876724261000
\(\beta_{7}\)	\(=\)	\( ( 11\!\cdots\!27 \nu^{15} + \cdots - 82\!\cdots\!00 ) / 13\!\cdots\!00 \) (11045263679844584814698075376597838327v^15 - 192395613937667076877641408534781028020v^14 + 1256685006668740234901372171793323015450v^13 - 6839234181828357299328170861551326750200v^12 + 28299684956018815081630748066943563698770v^11 - 115397347606715319034546940242321487106822v^10 + 289733640246513990157828693176997877538225v^9 - 1046059476568422952093299337371854126564610v^8 + 1252471469812849875141989455456625350629935v^7 - 7817868058974388853257596931345778266877180v^6 + 12240101779022302102158468377629894109521275v^5 - 51614763466663484376062761008682727864445700v^4 + 46258069719805712616300985805198466606058575v^3 - 273733834169866399301580204563466907785886800v^2 + 8692323043219148646918036314340190571472500*v - 828499969385539095170840370248944187627650000) / 133263454953106729197724941422397876724261000
\(\beta_{8}\)	\(=\)	\( ( - 47\!\cdots\!83 \nu^{15} + \cdots - 71\!\cdots\!00 ) / 46\!\cdots\!50 \) (-47856855531296407742751076554055983v^15 + 202282687343034814090415919271169512v^14 - 1261470858575780936116257777972859918v^13 + 3090932129657739168824735646959837402v^12 - 22795075357458056180627077875446201848v^11 + 188208130867183885506430306657634645v^10 - 267288394971251831766489684334631888240v^9 - 436948133723961440611611649285569687290v^8 - 3303667553977947330806973680536612701070v^7 - 1201759178052110383312031346769853328800v^6 - 20796977371351804057991745142400478964125v^5 - 28832457371395360566410768690477293808600v^4 - 171991810318666000396057716260122487735450v^3 - 227948925018440282881503275582992734725250v^2 - 793588049435541751270614521494970496973250*v - 711860887235213138876139772146068376002500) / 469237517440516652104665286698584072972750
\(\beta_{9}\)	\(=\)	\( ( 14\!\cdots\!57 \nu^{15} + \cdots + 47\!\cdots\!00 ) / 13\!\cdots\!00 \) (14097637668305820406939720159340964557v^15 - 125180643021093970634109011932302469806v^14 + 687315505843852038003258667728754260754v^13 - 2188346475325085210911061803812335642876v^12 + 6904294899905528153100886990967773021394v^11 - 2103947173182343700196535639438132937478v^10 - 32978227095544359668248061942055529343545v^9 + 355109076267009148545509250733121236969000v^8 - 977100294253233405431911542676883614228635v^7 + 1846074811570591762929009614658308796369030v^6 - 593635750080887875312405789149548841284175v^5 + 14099829949380305176426568698478903925932950v^4 - 53638742157370419341220853163068718020940675v^3 + 132065837105859469287585255022599554716708050v^2 - 313463717058390853095767603719861308658003500*v + 476228961269637066495412444944153660420795000) / 133263454953106729197724941422397876724261000
\(\beta_{10}\)	\(=\)	\( ( 14\!\cdots\!15 \nu^{15} + \cdots + 87\!\cdots\!00 ) / 13\!\cdots\!00 \) (14464460997086249030347443199969035515v^15 + 25172805655044025784624880926154409792v^14 - 285244547015073367497162932724762694258v^13 + 3127836744417344812359018162310173837072v^12 - 10337915914347509335699848543360260061158v^11 + 83207117603188930471491342064426643172202v^10 - 149509370311928864084510586153983451940715v^9 + 1025855015303803209243240446985924037452270v^8 - 683071867052516121862077447743598271660065v^7 + 6554141026257373978890115177037535616294280v^6 - 10830317060380622757931666259398678916299825v^5 + 49409381360663227082063853333197739540458500v^4 - 10077209212627978955447100253184449487333725v^3 + 297173335836826820466029900004685018212242800v^2 - 247343598005160172933665039421572642288538000*v + 878839624656591288500187732097877880541668000) / 133263454953106729197724941422397876724261000
\(\beta_{11}\)	\(=\)	\( ( 10\!\cdots\!03 \nu^{15} + \cdots + 19\!\cdots\!00 ) / 93\!\cdots\!00 \) (109483532825708721633924660906883803v^15 - 290547261842708738221487033660212382v^14 + 1776539826438228430722486652188240098v^13 - 1070879297863123823622802302664610572v^12 + 31265240339988643670428683931595235078v^11 + 99649746590787615156391709581058983740v^10 + 447432759258869179273507987450121051965v^9 + 1846258077324683573388413387667968002090v^8 + 6996578235718817448368654276561505882345v^7 + 9238292319013165104341592794750828467700v^6 + 19031445629446156740015945947081498273375v^5 + 112063083800061824896706801469769121191850v^4 + 257983927022929250286491430184101756109325v^3 + 759882051219123149654752216986679050988000v^2 + 893676071660038507669896723202013167476500*v + 1942242500199118709569918062363057119475000) / 938475034881033304209330573397168145945500
\(\beta_{12}\)	\(=\)	\( ( - 16\!\cdots\!87 \nu^{15} + \cdots - 79\!\cdots\!00 ) / 13\!\cdots\!00 \) (-16528306583525942187270901302661856887v^15 - 2791909298400579646466157580784411356v^14 + 141145447928578575264752490084765243894v^13 - 2640293449616006441502061966007490839776v^12 + 8732279097569335716707329380499396050494v^11 - 81610714446296735243819396431367164409254v^10 + 145388543722661131044540646479830978761935v^9 - 1066685900375842188749459205251444154274730v^8 + 653853689555948679573503530186302482698985v^7 - 6742804067705945634770055557006862189391160v^6 + 7090382063380502686541534851991226159615525v^5 - 51723631133517658940050000136914898964035000v^4 + 5805694041817805571049935978616573021531225v^3 - 353037569860887095517409351328529262048517600v^2 + 155022376647098294293757502767227904382693000*v - 794490304693127996805102660745858980784082000) / 133263454953106729197724941422397876724261000
\(\beta_{13}\)	\(=\)	\( ( - 15\!\cdots\!68 \nu^{15} + \cdots - 35\!\cdots\!00 ) / 66\!\cdots\!00 \) (-15814087450922974030074602288533825868v^15 + 113599454380835567532060066022191323605v^14 - 601929141619333922102216183863866097580v^13 + 1855976465684522934095597706674825650430v^12 - 7749190479131322044156240587962434566750v^11 + 5874653386455799174880793211721291910678v^10 - 20367428956217252417658931675775536628620v^9 - 188312595517949395074796884240303485288275v^8 + 132534643574977450467027791603290986120060v^7 - 245243254288696856566588301616793955613955v^6 - 1600427115455035686993714049011825873002600v^5 - 9866168367540713003528087462921469308427525v^4 + 8449826337577258041206545713700261173982950v^3 - 56804740273171178670570071677581188197701675v^2 + 68538305350853533063166244223600059108893000*v - 354233686281450785534373283371064755299696500) / 66631727476553364598862470711198938362130500
\(\beta_{14}\)	\(=\)	\( ( 17\!\cdots\!33 \nu^{15} + \cdots + 23\!\cdots\!50 ) / 46\!\cdots\!50 \) (179471557548653003873331169450716633v^15 - 1073546426570984683766925481402442447v^14 + 6827171558397679366607500684422405513v^13 - 24676669263440274973384236546652090642v^12 + 132492091439823451666166987382224612008v^11 - 229111449753741415742834715935908035660v^10 + 1373060985959896037586264367333773588410v^9 - 190912605162007032297750136408807346720v^8 + 10701200455718194008448973184525138999645v^7 - 4745217532485532673763990491116926262350v^6 + 69073187250159094666860586220561598452650v^5 + 31313166266123354443902587227152785685700v^4 + 499499597013166569244340885125738939801075v^3 + 395520244740131064578245913783462654639375v^2 + 1070898180926519019926435581981453127771750*v + 2316617225046757971768175832540467536697750) / 469237517440516652104665286698584072972750
\(\beta_{15}\)	\(=\)	\( ( - 54\!\cdots\!93 \nu^{15} + \cdots - 11\!\cdots\!00 ) / 13\!\cdots\!00 \) (-54969658670651591210783770018244288893v^15 + 330649609269277427402049238667735661246v^14 - 1966338343803376290502638175874731163354v^13 + 6318369922573405022804241586516246241276v^12 - 32467824378858457865689551928324245264374v^11 + 35636148065139637692216866380189729318354v^10 - 268219031906457111954610098273459459284355v^9 - 379522928528132268208232735550964606991540v^8 - 1871038286613697997686500235057659549702465v^7 - 859292688433901478752255508926243274214690v^6 - 14015641346638606483286027628490099434059325v^5 - 30225466198745449130551897674067104897150850v^4 - 84670757432715020305089090731004968333456325v^3 - 236781696804063587306312950040902823485328150v^2 - 286388793616797682620722991911860666393490000*v - 1196666884608711405168504298095700373241176000) / 133263454953106729197724941422397876724261000

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{7} - \beta_{5} - 10\beta_{3} + \beta_{2} \) b15 + b14 - b13 + b12 - b11 - b7 - b5 - 10*b3 + b2
\(\nu^{3}\)	\(=\)	\( 3 \beta_{13} + 6 \beta_{12} - 13 \beta_{11} + 2 \beta_{10} + 6 \beta_{9} - 19 \beta_{8} + \cdots - 12 \beta_1 \) 3b13 + 6b12 - 13b11 + 2b10 + 6b9 - 19b8 + 4b6 - 12b5 - 3b4 - 7b3 + 4b2 - 12b1
\(\nu^{4}\)	\(=\)	\( - 23 \beta_{15} - 25 \beta_{14} + 46 \beta_{13} + 29 \beta_{12} - 4 \beta_{11} + 54 \beta_{10} + \cdots - 8 \) -23b15 - 25b14 + 46b13 + 29b12 - 4b11 + 54b10 + 23b9 - 56b8 + 29b7 + 8b6 + 54b5 - 5b4 + 85b2 + 19b1 - 8
\(\nu^{5}\)	\(=\)	\( - 108 \beta_{15} - 92 \beta_{14} + 88 \beta_{13} + 20 \beta_{12} + 20 \beta_{11} + 191 \beta_{10} + \cdots - 346 \) -108b15 - 92b14 + 88b13 + 20b12 + 20b11 + 191b10 - 83b9 - 43b8 + 176b7 + 217b6 + 439b5 - 68b4 + 217b3 - 88b2 + 45*b1 - 346
\(\nu^{6}\)	\(=\)	\( - 516 \beta_{15} - 562 \beta_{14} - 209 \beta_{13} - 467 \beta_{12} + 444 \beta_{11} + 28 \beta_{10} + \cdots - 2564 \) -516b15 - 562b14 - 209b13 - 467b12 + 444b11 + 28b10 - 771b9 + 237b8 + 725b7 + 2081b6 + 1006b5 - 230b4 + 2564b3 - 1872b2 - 702*b1 - 2564
\(\nu^{7}\)	\(=\)	\( - 3062 \beta_{15} - 4180 \beta_{14} - 1370 \beta_{13} - 3062 \beta_{12} + 6414 \beta_{11} - 4301 \beta_{9} + \cdots - 5672 \) -3062b15 - 4180b14 - 1370b13 - 3062b12 + 6414b11 - 4301b9 + 5713b8 + 846b7 + 4343b5 + 3334b4 + 9191b3 - 4302b2 - 5672
\(\nu^{8}\)	\(=\)	\( - 9706 \beta_{15} - 9706 \beta_{14} - 8120 \beta_{13} - 16240 \beta_{12} + 26036 \beta_{11} + \cdots + 11708 \beta_1 \) -9706b15 - 9706b14 - 8120b13 - 16240b12 + 26036b11 - 6313b10 - 19528b9 + 45564b8 - 9706b7 - 37571b6 + 11708b5 + 21114b4 + 18118b3 - 9998b2 + 11708*b1
\(\nu^{9}\)	\(=\)	\( 26257 \beta_{15} + 45995 \beta_{14} - 52514 \beta_{13} - 81006 \beta_{12} + 19231 \beta_{11} + \cdots + 144132 \) 26257b15 + 45995b14 - 52514b13 - 81006b12 + 19231b11 - 127001b10 - 26257b9 + 154569b8 - 81006b7 - 144132b6 - 127001b5 + 36660b4 - 38955b2 - 7026b1 + 144132
\(\nu^{10}\)	\(=\)	\( 440947 \beta_{15} + 483063 \beta_{14} - 228887 \beta_{13} - 212060 \beta_{12} - 212060 \beta_{11} + \cdots + 1306764 \) 440947b15 + 483063b14 - 228887b13 - 212060b12 - 212060b11 - 693469b10 + 252522b9 + 199347b8 - 457774b7 - 562428b6 - 1139346b5 + 16827b4 - 562428b3 + 228887b2 - 29540*b1 + 1306764
\(\nu^{11}\)	\(=\)	\( 2718104 \beta_{15} + 2840778 \beta_{14} - 626999 \beta_{13} + 732053 \beta_{12} - 2098846 \beta_{11} + \cdots + 5986216 \) 2718104b15 + 2840778b14 - 626999b13 + 732053b12 - 2098846b11 - 1745202b10 + 2213779b9 - 1118203b8 - 2091105b7 - 2332939b6 - 4939624b5 - 386150b4 - 5986216b3 + 2959938b2 + 760293*b1 + 5986216
\(\nu^{12}\)	\(=\)	\( 10995323 \beta_{15} + 11950605 \beta_{14} + 1285185 \beta_{13} + 10995323 \beta_{12} - 15958511 \beta_{11} + \cdots + 15935328 \) 10995323b15 + 11950605b14 + 1285185b13 + 10995323b12 - 15958511b11 + 13416839b9 - 18278487b8 - 4855069b7 - 16993302b5 - 7095536b4 - 31789259b3 + 14650143b2 + 15935328
\(\nu^{13}\)	\(=\)	\( 19477584 \beta_{15} + 19477584 \beta_{14} + 33980025 \beta_{13} + 67960050 \beta_{12} + \cdots - 13860002 \beta_1 \) 19477584b15 + 19477584b14 + 33980025b13 + 67960050b12 - 70614189b11 + 45753437b10 + 60565182b9 - 131179371b8 + 19477584b7 + 59053179b6 - 13860002b5 - 46062741b4 - 92694357b3 + 58714332b2 - 13860002*b1
\(\nu^{14}\)	\(=\)	\( - 115203478 \beta_{15} - 138006690 \beta_{14} + 230406956 \beta_{13} + 275856559 \beta_{12} + \cdots - 430303568 \) -115203478b15 - 138006690b14 + 230406956b13 + 275856559b12 - 134636799b11 + 413863249b10 + 115203478b9 - 527510461b8 + 275856559b7 + 430303568b6 + 413863249b5 - 167738645b4 + 126678650b2 - 19433321b1 - 430303568
\(\nu^{15}\)	\(=\)	\( - 1360821463 \beta_{15} - 1491428947 \beta_{14} + 854245383 \beta_{13} + 506576080 \beta_{12} + \cdots - 3844564986 \) -1360821463b15 - 1491428947b14 + 854245383b13 + 506576080b12 + 506576080b11 + 2015380556b10 - 654559093b9 - 987791228b8 + 1708490766b7 + 2353512797b6 + 3467407389b5 - 347669303b4 + 2353512797b3 - 854245383b2 - 133545845*b1 - 3844564986

\(n\)	\(661\)	\(881\)	\(991\)	\(1057\)	\(1201\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1 - \beta_{2} + \beta_{3} + \beta_{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 361.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) (T^4 + T^3 + T^2 + T + 1)^4
$5$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) (T^4 + T^3 + T^2 + T + 1)^4
$7$	\( T^{16} - 4 T^{15} + \cdots + 50410000 \) T^16 - 4T^15 + 26T^14 - 64T^13 + 486T^12 - 10T^11 + 6075T^10 + 9130T^9 + 72165T^8 + 32200T^7 + 420125T^6 + 573200T^5 + 3577025T^4 + 5167000T^3 + 17399000T^2 + 15265000*T + 50410000
$11$	\( T^{16} + \cdots + 214358881 \) T^16 + T^15 + 11T^14 + 60T^13 + 539T^11 + 360T^10 - 2920T^9 + 11771T^8 - 32120T^7 + 43560T^6 + 717409T^5 + 9663060T^3 + 19487171T^2 + 19487171T + 214358881
$13$	\( T^{16} - T^{15} + \cdots + 609961 \) T^16 - T^15 + 69T^14 + 39T^13 + 2020T^12 + 2729T^11 + 32722T^10 + 73757T^9 + 369283T^8 + 1193752T^7 + 5199207T^6 + 19113768T^5 + 60849899T^4 + 107096953T^3 + 96868028T^2 + 12221088T + 609961
$17$	\( T^{16} + 3 T^{15} + \cdots + 256 \) T^16 + 3T^15 - 3T^14 - 17T^13 + 1488T^12 + 4303T^11 + 38890T^10 + 105635T^9 + 430906T^8 + 935445T^7 + 1284565T^6 + 709868T^5 + 440313T^4 - 164452T^3 + 34432T^2 - 3712*T + 256
$19$	\( T^{16} + \cdots + 1178136976 \) T^16 + 2T^15 + 59T^14 + 246T^13 + 2899T^12 + 16780T^11 + 153477T^10 + 1128712T^9 + 7860538T^8 + 39534982T^7 + 157784792T^6 + 439657570T^5 + 885295629T^4 + 935161366T^3 + 322234144T^2 - 1436528048*T + 1178136976
$23$	\( (T^{8} - 5 T^{7} + \cdots + 2945)^{2} \) (T^8 - 5T^7 - 48T^6 + 210T^5 + 751T^4 - 2280T^3 - 4090T^2 + 1735*T + 2945)^2
$29$	\( T^{16} + \cdots + 1861576444816 \) T^16 + 5T^15 + 209T^14 + 814T^13 + 19354T^12 + 90167T^11 + 1108459T^10 + 7170301T^9 + 60403726T^8 + 465307686T^7 + 3169969601T^6 + 18851348831T^5 + 105514245641T^4 + 456829276896T^3 + 1336087815056T^2 + 2292188008792*T + 1861576444816
$31$	\( T^{16} + \cdots + 116208400 \) T^16 - 3T^15 + 153T^14 - 14T^13 + 8608T^12 + 10329T^11 + 214897T^10 + 450696T^9 + 1575878T^8 - 7669906T^7 + 54996358T^6 + 62408588T^5 + 132067121T^4 + 245249270T^3 + 766930360T^2 - 491999200*T + 116208400
$37$	\( T^{16} + \cdots + 1770642241 \) T^16 + 25T^15 + 399T^14 + 4249T^13 + 39414T^12 + 274517T^11 + 1951894T^10 + 8340016T^9 + 37668201T^8 + 412708166T^7 + 3928719951T^6 + 16331410296T^5 + 27696422711T^4 - 11038405594T^3 + 21881831476T^2 - 4118566283*T + 1770642241
$41$	\( T^{16} + \cdots + 50357155216 \) T^16 + 2T^15 + 82T^14 + 636T^13 + 6776T^12 - 7858T^11 + 451415T^10 + 1931458T^9 + 28238439T^8 + 89950724T^7 + 451518195T^6 + 557924174T^5 + 1402559081T^4 - 471987402T^3 + 6055695528T^2 + 23440344224*T + 50357155216
$43$	\( (T^{8} + T^{7} + \cdots + 14764)^{2} \) (T^8 + T^7 - 115T^6 - 307T^5 + 2490T^4 + 4018T^3 - 21027T^2 + 6530T + 14764)^2
$47$	\( T^{16} - 14 T^{15} + \cdots + 76125625 \) T^16 - 14T^15 + 252T^14 - 2057T^13 + 19105T^12 - 106038T^11 + 634947T^10 - 2440276T^9 + 8304031T^8 - 11963475T^7 + 13831830T^6 + 49828775T^5 + 322746400T^4 - 560909125T^3 + 266192625T^2 + 269820625*T + 76125625
$53$	\( T^{16} + \cdots + 67207451536 \) T^16 + 11T^15 + 144T^14 + 1782T^13 + 29411T^12 + 157318T^11 + 2965499T^10 + 35847987T^9 + 444355201T^8 + 3468494683T^7 + 18698898781T^6 + 65821355854T^5 + 170072105269T^4 + 261610767042T^3 + 271425953556T^2 + 175754469800*T + 67207451536
$59$	\( T^{16} + \cdots + 2346009925561 \) T^16 + 7T^15 + 337T^14 + 2401T^13 + 48516T^12 + 346237T^11 + 2438120T^10 + 16428268T^9 + 108822529T^8 + 500589844T^7 + 3038996745T^6 + 21538592944T^5 + 129690693661T^4 + 541321315398T^3 + 1561841269058T^2 + 2710197927029*T + 2346009925561
$61$	\( T^{16} + \cdots + 7744000000 \) T^16 + 3T^15 + 78T^14 + 441T^13 + 4865T^12 + 12631T^11 + 222808T^10 + 775973T^9 + 10762821T^8 + 36536860T^7 + 261514760T^6 + 764337800T^5 + 4534328400T^4 - 7998576000T^3 + 7538240000T^2 - 2569600000*T + 7744000000
$67$	\( (T^{8} - 16 T^{7} + \cdots + 716)^{2} \) (T^8 - 16T^7 - 12T^6 + 727T^5 - 221T^4 - 8183T^3 - 4707T^2 + 1994*T + 716)^2
$71$	\( T^{16} + \cdots + 153561664000000 \) T^16 + 15T^15 + 297T^14 + 4851T^13 + 82814T^12 + 550479T^11 + 9416729T^10 + 125576877T^9 + 2062794887T^8 + 21768601538T^7 + 175541026424T^6 + 1020988731328T^5 + 4487585062416T^4 + 14831670596800T^3 + 42925041056000T^2 + 100181884800000*T + 153561664000000
$73$	\( T^{16} + \cdots + 5139542907136 \) T^16 + T^15 + 134T^14 - 2319T^13 + 28625T^12 - 130829T^11 + 9102112T^10 - 126409607T^9 + 1321441413T^8 - 8267065952T^7 + 38516934812T^6 - 107401494128T^5 + 242602410384T^4 - 19672130368T^3 + 314475066048T^2 + 1189823534592T + 5139542907136
$79$	\( T^{16} + \cdots + 16374785296 \) T^16 + 15T^15 + 333T^14 + 1615T^13 + 10173T^12 - 44370T^11 + 38031T^10 + 1464330T^9 + 27008215T^8 - 84599405T^7 + 72620786T^6 + 501393370T^5 + 3205894173T^4 + 5232091410T^3 + 10379475308T^2 + 3888825960*T + 16374785296
$83$	\( T^{16} + 17 T^{15} + \cdots + 7929856 \) T^16 + 17T^15 + 161T^14 + 653T^13 + 11670T^12 + 16037T^11 + 572943T^10 + 859149T^9 + 25207353T^8 + 136607856T^7 + 764517008T^6 - 2327679744T^5 + 3228780544T^4 - 573689856T^3 + 5843689472T^2 + 132284416*T + 7929856
$89$	\( (T^{8} - 13 T^{7} + \cdots - 178410500)^{2} \) (T^8 - 13T^7 - 514T^6 + 8065T^5 + 61290T^4 - 1403235T^3 + 2005045T^2 + 50621850*T - 178410500)^2
$97$	\( T^{16} + \cdots + 4505094400 \) T^16 + 45T^15 + 912T^14 + 9819T^13 + 55789T^12 + 148801T^11 + 1101674T^10 + 19334923T^9 + 177596017T^8 + 920326312T^7 + 3335972504T^6 + 4639743072T^5 + 13722178416T^4 + 11180974080T^3 + 22449023040T^2 + 7160361600*T + 4505094400
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