LMFDB - Newform orbit 1176.3.f.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(97,1176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1176.97"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1176.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.0436790888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 66 x^{14} + 136 x^{13} + 3441 x^{12} - 4512 x^{11} - 100322 x^{10} + \cdots + 13841287201 $ x^16 - 4x^15 - 66x^14 + 136x^13 + 3441x^12 - 4512x^11 - 100322x^10 + 24868x^9 + 2325060x^8 + 260012x^7 - 34273754x^6 - 26559312x^5 + 402103713x^4 + 286444088x^3 - 2633783418x^2 - 3060285788*x + 13841287201
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 7^{2} $
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_1 q^{3} + (\beta_{11} - \beta_{2}) q^{5} - 3 q^{9} + (\beta_{15} - \beta_{9} + \beta_{7}) q^{11} + ( - \beta_{6} - \beta_{5} + \cdots + 2 \beta_{2}) q^{13} + (\beta_{15} - 2 \beta_{10}) q^{15}+ \cdots + ( - 3 \beta_{15} + 3 \beta_{9} - 3 \beta_{7}) q^{99}+O(q^{100}) $ q + b1 * q^3 + (b11 - b2) * q^5 - 3 * q^9 + (b15 - b9 + b7) * q^11 + (-b6 - b5 - 2b4 + 2b2) * q^13 + (b15 - 2b10) q^15 + (b12 + b11 + b4 + b2) * q^17 + (-b12 - 2b11 - b5 + b4 - 4b1) * q^19 + (-b15 + b14 + b13 - 4b10 - b8 - 7) q^23 + (b13 - 3b10 + 2b9 - b8 + b7 - 16) * q^25 - 3b1 q^27 + (-b14 - b13 - 9b10 + b8 - 5b7 + 7) * q^29 + (-2b11 - b6 - 4b5 + b4 + 8b2 + 4b1) * q^31 + (-3b11 - b6 + 2b4 + b2) * q^33 + (-2b14 + 4b13 - b10 - 8) * q^37 + (b13 + 6b10 + 3b9 + b7) * q^39 + (-b12 - b11 - 8b4 + b3 + 7b2 - 7b1) q^41 + (2b14 + b13 - 4b10 - b8 - 3b7 - 7) q^43 + (-3b11 + 3b2) * q^45 + (6b11 + b6 + 3b5 + 6b4 + b3 - 4b2 - 15b1) q^47 + (b15 - 3b14 + b13 + 4b10) * q^51 + (-2b15 + 3b13 - 6b9 + b8 + b7 + 7) q^53 + (b12 + 2b11 + 2b6 - 5b5 + 7b4 - 2b3 + 4b2 - 34b1) q^55 + (-2b15 + 3b14 - 2b10 - 2b7 + 12) * q^57 + (-2b12 + 6b11 + b6 - 5b5 - 2b4 - b3 + 8b2 + 11b1) * q^59 + (3b12 + 2b11 - 2b5 - 2b3 - 8b2 - 14b1) * q^61 + (2b15 + 6b14 + 9b13 - 5b10 - b9 + 3b8 - 5b7 - 7) * q^65 + (2b13 + 8b10 - 2b9 + 2b8 + 10b7 - 14) q^67 + (b12 + 3b11 + 4b5 + b4 - 3b3 + 3b2 - 7b1) q^69 + (-b15 - 6b14 + 2b13 + 24b10 - b9 - 2b8 - 3b7 + 10) q^71 + (-b12 + 2b11 + 3b6 + 7b5 + 9b4 + 2b3 + 2b2 + 14b1) q^73 + (2b6 + 3b5 + 5b4 - 3b3 + 3b2 - 16b1) * q^75 + (2b14 + 3b13 - 28b10 + 2b9 + b8 - 5b7 + 19) q^79 + 9 * q^81 + (b12 - 6b11 + b6 + 3b5 + 6b4 + 6b3 + 10b2 - 42b1) * q^83 + (-2b14 - 6b13 + 11b10 + 2b9 + 2b8 - 18b7 - 26) * q^85 + (-b12 - 4b5 - 16b4 + 3b3 + 9b2 + 7b1) q^87 + (-b11 - b6 - b5 - 11b4 + 6b3 - 29b2 - 14b1) * q^89 + (-2b15 + 4b13 + 22b10 + 3b9 - 2b7 - 12) q^93 + (-4b15 - b14 + 5b13 + 8b10 + b9 - b8 + 17b7 + 57) * q^95 + (-2b12 - 2b11 + 4b6 - 10b5 + b4 + 2b3 + 14b2 - 14b1) q^97 + (-3b15 + 3b9 - 3b7) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 112 q^{23} - 256 q^{25} + 112 q^{29} - 128 q^{37} - 112 q^{43} + 112 q^{53} + 192 q^{57} - 112 q^{65} - 224 q^{67} + 160 q^{71} + 304 q^{79} + 144 q^{81} - 416 q^{85} - 192 q^{93} + 912 q^{95}+O(q^{100}) $ 16 * q - 48 * q^9 - 112 * q^23 - 256 * q^25 + 112 * q^29 - 128 * q^37 - 112 * q^43 + 112 * q^53 + 192 * q^57 - 112 * q^65 - 224 * q^67 + 160 * q^71 + 304 * q^79 + 144 * q^81 - 416 * q^85 - 192 * q^93 + 912 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} - 66 x^{14} + 136 x^{13} + 3441 x^{12} - 4512 x^{11} - 100322 x^{10} + \cdots + 13841287201 $ x^16 - 4*x^15 - 66*x^14 + 136*x^13 + 3441*x^12 - 4512*x^11 - 100322*x^10 + 24868*x^9 + 2325060*x^8 + 260012*x^7 - 34273754*x^6 - 26559312*x^5 + 402103713*x^4 + 286444088*x^3 - 2633783418*x^2 - 3060285788*x + 13841287201 :

$\beta_{1}$	$=$	$ ( - 19\!\cdots\!15 \nu^{15} + \cdots + 10\!\cdots\!26 ) / 17\!\cdots\!08 $ (-1912863316623227926255615v^15 + 157590092883913634324899v^14 + 134553559303277107074448835v^13 + 234251144123174764943611906v^12 - 6130277557280544667044415252v^11 - 14461729974232158231161784788v^10 + 159912185005949024540997253830v^9 + 545979676810483959695503249362v^8 - 2977979150709146858658059920446v^7 - 12186728557549192183318227164384v^6 + 33617527462650444084998233859972v^5 + 185179542565865634914909315411884v^4 - 252337232856569126836178509648847v^3 - 1880544733090865400075974684200981v^2 + 634559294256824390217426688607659*v + 10256187601200275396626601089582626) / 1773367572721598053400461619726508
$\beta_{2}$	$=$	$ ( 4157815028 \nu^{15} + 226803899359 \nu^{14} + 2133995005098 \nu^{13} + \cdots + 83\!\cdots\!95 ) / 22\!\cdots\!88 $ (4157815028v^15 + 226803899359v^14 + 2133995005098v^13 - 3577907083693v^12 - 194574957455356v^11 - 251785193114712v^10 + 7427269664001284v^9 + 25329048895604970v^8 - 174617950806252252v^7 - 734432835451712126v^6 + 2170987437180418964v^5 + 14053447947667247712v^4 - 16760250856674216432v^3 - 134039054755482730609v^2 + 6950960411836071666*v + 838547261713169306795) / 2201283776407962288
$\beta_{3}$	$=$	$ ( 44\!\cdots\!44 \nu^{15} + \cdots - 95\!\cdots\!31 ) / 70\!\cdots\!32 $ (44577133160376255065376144v^15 - 220555033154806103711604155v^14 - 5162518189659835853593171310v^13 - 1398480263636267715714079687v^12 + 320037559142085455768573452996v^11 + 506269249423326287867215280000v^10 - 10396523739110943366632487895640v^9 - 33723238549339283137701106927834v^8 + 227594131074857164767569406685596v^7 + 898481249079677698841857006826730v^6 - 2761263003538818849762244026919996v^5 - 16372803797949565065076363564681064v^4 + 22273979345309546558116233878974992v^3 + 153294449370457062676807243567106621v^2 - 19505215404059807908493705706414798*v - 950259744626931678628322183969956131) / 7093470290886392213601846478906032
$\beta_{4}$	$=$	$ ( 35\!\cdots\!56 \nu^{15} + \cdots - 37\!\cdots\!35 ) / 30\!\cdots\!44 $ (350876609979969256v^15 - 314839517719341913v^14 - 30338192533773169860v^13 - 44191121410098384887v^12 + 1549533327556378967100v^11 + 3577544079010336198648v^10 - 45762730994300176778752v^9 - 163736549542244206139566v^8 + 922903483838461429315560v^7 + 4034994607096103269464178v^6 - 10976894065514184482337940v^5 - 68463674804877207710195904v^4 + 80791935520653988589771600v^3 + 643880032991142085134009727v^2 - 82695890302296909056068804*v - 3791320355935665433758353035) / 30444450498871083689172144
$\beta_{5}$	$=$	$ ( - 30\!\cdots\!34 \nu^{15} + \cdots - 68\!\cdots\!68 ) / 15\!\cdots\!72 $ (-306973687093272834v^15 - 2341845882705077043v^14 - 2335381153484933007v^13 + 72431438271686622148v^12 + 1095636592069061456828v^11 + 316504958534616322428v^10 - 51864459345420910242736v^9 - 172673828522119257827122v^8 + 1332108879663617997416670v^7 + 5524798313850637177278924v^6 - 16939219803884735639985900v^5 - 112196222387208274773118828v^4 + 133331510868660742710492106v^3 + 1072322688407929217180244141v^2 - 10994258454029827334769551*v - 6890044785342379369426823168) / 15222225249435541844586072
$\beta_{6}$	$=$	$ ( - 23\!\cdots\!76 \nu^{15} + \cdots + 15\!\cdots\!25 ) / 70\!\cdots\!32 $ (-231514427916488493337407776v^15 - 1733455067248079744739512633v^14 + 10214679112641706166553809780v^13 + 157226781508357260615129245545v^12 - 331515392769619765360327208300v^11 - 6121320413343608886064433975344v^10 + 753910213734035502343699081640v^9 + 165749808765107773640040630561626v^8 + 33169905817259220276438310109232v^7 - 2654287494610295459961479433255350v^6 - 2514956268210773951458815232408716v^5 + 35074258805899930411680195251600824v^4 + 20791733953680467665654632242420768v^3 - 247670370240854720314562854642074985v^2 - 314157982384904309237699920311886836*v + 1564448041519563467199870447103059725) / 7093470290886392213601846478906032
$\beta_{7}$	$=$	$ ( - 63\!\cdots\!81 \nu^{15} + \cdots + 16\!\cdots\!34 ) / 12\!\cdots\!88 $ (-631665189371406619381v^15 - 662731806327679783830v^14 + 45896053921093831036569v^13 + 120019826580783387396674v^12 - 1976331469897354311221712v^11 - 6600269659182099803449260v^10 + 49214663148263150564790974v^9 + 214815970942431709648955796v^8 - 809637498244974183112312146v^7 - 4339176996709522847929161620v^6 + 7752454423369146809886700056v^5 + 56204781468861895398832278612v^4 - 38820748755921741840658481025v^3 - 435224110296497347123756050894v^2 - 2284047358170699659989600023*v + 1621199449804323773844571142234) / 12361429409444795863651383888
$\beta_{8}$	$=$	$ ( - 19\!\cdots\!83 \nu^{15} + \cdots + 59\!\cdots\!44 ) / 29\!\cdots\!88 $ (-1977904712196288644663483v^15 - 2769722957512364628380798v^14 + 149409984430279586627424289v^13 + 428261367104891772462536040v^12 - 6363784319313105137418186736v^11 - 23044219197298363809623543500v^10 + 157606104512295979331056952090v^9 + 733311784154638203407022055236v^8 - 2521393500276621203318490294690v^7 - 14510580175776780026009382445600v^6 + 23287275663692885610347825390648v^5 + 187025408125574894936827734601364v^4 - 105898168300369055037469101430151v^3 - 1438886497144196814507575697277558v^2 - 226661812196939822079938688056335*v + 5909802439232478076010365883454944) / 29679792012076954868626972715088
$\beta_{9}$	$=$	$ ( - 10\!\cdots\!74 \nu^{15} + \cdots + 22\!\cdots\!53 ) / 14\!\cdots\!44 $ (-1006502610972175217701274v^15 + 1963090966618337093009667v^14 + 60172801086069857151363930v^13 + 56603888885511048850037121v^12 - 2676266678568374371698125256v^11 - 4173190082019966805123910172v^10 + 59786981142647818231259517112v^9 + 200122682769621564888085523374v^8 - 1012047728005356222362359265264v^7 - 3938100309019050873821892408162v^6 + 8641483696324724173668850139168v^5 + 58215664532299935899528339600524v^4 - 54534994898684519685122968171118v^3 - 432713300511231086780968267672233v^2 - 12392694366195431088310976681250*v + 2271201017297437099281162437917753) / 14839896006038477434313486357544
$\beta_{10}$	$=$	$ ( - 1434785603940 \nu^{15} - 1900762364347 \nu^{14} + 108526077128390 \nu^{13} + \cdots + 43\!\cdots\!37 ) / 17\!\cdots\!32 $ (-1434785603940v^15 - 1900762364347v^14 + 108526077128390v^13 + 305761124010809v^12 - 4634287092419732v^11 - 16451997826420712v^10 + 114795138909966604v^9 + 526200538323412638v^8 - 1850353679066317668v^7 - 10386497708859409786v^6 + 17232564720652874620v^5 + 134816715680204446528v^4 - 85343622288924149712v^3 - 1035286717988536754363v^2 - 62730574414649842274*v + 4389566035171884470337) / 17684664455170110832
$\beta_{11}$	$=$	$ ( - 56\!\cdots\!90 \nu^{15} + \cdots + 36\!\cdots\!79 ) / 50\!\cdots\!88 $ (-56642036536487516741853090v^15 + 550151708697427688717919888v^14 + 5339589179841001817623440687v^13 - 25287452299837460910124817147v^12 - 317977388004172791551211997668v^11 + 697465441230838467700662181552v^10 + 10546985420425006804161240032730v^9 - 3297036198827858759397698547816v^8 - 232156867712300005582266163178298v^7 - 170021979053373415333732792410316v^6 + 3153454980931729730823605041137892v^5 + 5511167207643408268595853084813256v^4 - 26288679602125517652377123936378056v^3 - 62234831915445886667336704892166816v^2 + 99558002793695105408018783007808747*v + 368001721378570303740925286314071079) / 506676449349028015257274748493288
$\beta_{12}$	$=$	$ ( - 14\!\cdots\!32 \nu^{15} + \cdots + 68\!\cdots\!25 ) / 70\!\cdots\!32 $ (-1472854576289224688583582932v^15 + 14177957572127991765826757489v^14 + 133589084361417673257027320778v^13 - 690003008951526503333372363083v^12 - 7773215150996317680650193558428v^11 + 20430379757482454873976389670928v^10 + 257161462634343739260316597165288v^9 - 198589541098943116821847136568546v^8 - 5623094838222282744605179943097828v^7 - 1769720747266380799577640938931206v^6 + 77514399110872335948478631841912932v^5 + 96381684867830338904919200498510584v^4 - 646787794636939669329030846321953980v^3 - 1201105157312055536571474551805175751v^2 + 2688744471733625881174945807099407370*v + 6882255985247260617442869406556167025) / 7093470290886392213601846478906032
$\beta_{13}$	$=$	$ ( - 20\!\cdots\!15 \nu^{15} + \cdots + 65\!\cdots\!67 ) / 61\!\cdots\!44 $ (-2071905375627045789315v^15 - 2844636481759773644052v^14 + 157733948702019596146844v^13 + 449567602203165069603211v^12 - 6725986828004979180742224v^11 - 24119392666738067511134672v^10 + 166503982597390241454746606v^9 + 769275909613848053991157224v^8 - 2673576826096713395870302104v^7 - 15125690906902891763241947574v^6 + 24775978266365312174929462416v^5 + 196415977025304441064798346320v^4 - 122495048060051089345118969155v^3 - 1506218980785069936042248431092v^2 - 105268335342765609660034458692*v + 6578801327497297958540519165267) / 6180714704722397931825691944
$\beta_{14}$	$=$	$ ( 57\!\cdots\!16 \nu^{15} + \cdots - 14\!\cdots\!99 ) / 14\!\cdots\!44 $ (5730429750414417486547616v^15 - 8387837512395634923289889v^14 - 342621869434030589706960120v^13 - 450827648548175188543646559v^12 + 15241117814157487179441706708v^11 + 28481573290805989685558266608v^10 - 335264482327370694439359112004v^9 - 1265651614149107256641538486422v^8 + 5603691824106242473550003664872v^7 + 24744082644026753800545489559254v^6 - 40472203172878863229619248162604v^5 - 369470593907517033386782732335856v^4 + 196791075824769178089247059918788v^3 + 2679332788071186052372761204838303v^2 + 1693175745450032130411663484905176*v - 14844141869843621540101047634453999) / 14839896006038477434313486357544
$\beta_{15}$	$=$	$ ( - 24\!\cdots\!43 \nu^{15} + \cdots + 68\!\cdots\!30 ) / 29\!\cdots\!88 $ (-24675891599013579715795443v^15 + 14838686075194091099329066v^14 + 1595010131572037797129207367v^13 + 2978567877087263366555261114v^12 - 69893464516229714397314465488v^11 - 172584784330508783944914657140v^10 + 1602393277760914094630074473010v^9 + 6576326829449575293667802076204v^8 - 26369882913046879467815598155742v^7 - 128941011796545922861347081774396v^6 + 208403287137328237973585046446840v^5 + 1819869954801539204965089402205612v^4 - 999328036173611395753652828474999v^3 - 13481330685024586094700848877694774v^2 - 5891490276046582196429841685375385*v + 68188616501971600168964380980553930) / 29679792012076954868626972715088

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

$\nu$	$=$	$ ( \beta_{13} - 7\beta_{8} + 3\beta_{7} - \beta_{5} - 3\beta_{4} + 7\beta_{3} + 7\beta _1 + 7 ) / 28 $ (b13 - 7b8 + 3b7 - b5 - 3b4 + 7b3 + 7*b1 + 7) / 28
$\nu^{2}$	$=$	$ ( - 5 \beta_{13} + 28 \beta_{10} - 7 \beta_{8} - \beta_{7} + 19 \beta_{5} + 15 \beta_{4} - 7 \beta_{3} + \cdots + 259 ) / 28 $ (-5b13 + 28b10 - 7b8 - b7 + 19b5 + 15b4 - 7b3 + 28b2 - 259b1 + 259) / 28
$\nu^{3}$	$=$	$ ( - 9 \beta_{13} - 14 \beta_{12} + 28 \beta_{11} + 42 \beta_{10} - 6 \beta_{7} - 15 \beta_{5} + \cdots + 392 ) / 14 $ (-9b13 - 14b12 + 28b11 + 42b10 - 6b7 - 15b5 - 66b4 + 126b3 - 14*b2 + 392) / 14
$\nu^{4}$	$=$	$ ( 56 \beta_{15} + 84 \beta_{14} + 235 \beta_{13} - 28 \beta_{12} + 56 \beta_{11} - 1120 \beta_{10} + \cdots - 4809 ) / 28 $ (56b15 + 84b14 + 235b13 - 28b12 + 56b11 - 1120b10 - 259b8 + 299b7 + 665b5 + 441b4 + 259b3 + 1064b2 - 4809*b1 - 4809) / 28
$\nu^{5}$	$=$	$ ( - 1064 \beta_{15} - 1638 \beta_{14} - 277 \beta_{13} - 546 \beta_{12} + 1064 \beta_{11} + 3332 \beta_{10} + \cdots + 12677 ) / 28 $ (-1064b15 - 1638b14 - 277b13 - 546b12 + 1064b11 + 3332b10 - 168b9 + 2275b8 - 1335b7 - 56b6 + 1557b5 + 23b4 + 2275b3 + 2268b2 - 12677*b1 + 12677) / 28
$\nu^{6}$	$=$	$ ( 3285 \beta_{13} - 854 \beta_{12} + 1666 \beta_{11} - 15631 \beta_{10} + 3198 \beta_{7} - 84 \beta_{6} + \cdots - 44891 ) / 7 $ (3285b13 - 854b12 + 1666b11 - 15631b10 + 3198b7 - 84b6 - 381b5 - 2340b4 + 3738b3 - 833b2 - 44891) / 7
$\nu^{7}$	$=$	$ ( - 29596 \beta_{15} - 46452 \beta_{14} + 36989 \beta_{13} + 15484 \beta_{12} - 29596 \beta_{11} + \cdots - 359065 ) / 28 $ (-29596b15 - 46452b14 + 36989b13 + 15484b12 - 29596b11 - 104468b10 - 8820b9 + 41153b8 + 11399b7 + 2940b6 + 78847b5 + 103457b4 - 41153b3 + 134064b2 - 359065*b1 - 359065) / 28
$\nu^{8}$	$=$	$ ( - 134064 \beta_{15} - 209916 \beta_{14} + 211955 \beta_{13} - 69972 \beta_{12} + 134064 \beta_{11} + \cdots - 1667897 ) / 28 $ (-134064b15 - 209916b14 + 211955b13 - 69972b12 + 134064b11 - 703416b10 - 37632b9 + 200109b8 + 111971b7 - 12544b6 - 497793b5 - 675121b4 + 200109b3 - 837480b2 + 1667897*b1 - 1667897) / 28
$\nu^{9}$	$=$	$ ( 872793 \beta_{13} + 372974 \beta_{12} - 703416 \beta_{11} - 4194456 \beta_{10} + 939366 \beta_{7} + \cdots - 9713956 ) / 14 $ (872793b13 + 372974b12 - 703416b11 - 4194456b10 + 939366b7 + 98000b6 + 58071b5 + 642142b4 - 733894b3 + 351708b2 - 9713956) / 14
$\nu^{10}$	$=$	$ ( - 4546164 \beta_{15} - 7205268 \beta_{14} - 1482169 \beta_{13} + 2401756 \beta_{12} - 4546164 \beta_{11} + \cdots + 29953399 ) / 28 $ (-4546164b15 - 7205268b14 - 1482169b13 + 2401756b12 - 4546164b11 + 19126324b10 - 1755768b9 + 5223925b8 - 5541377b7 + 585256b6 - 9929563b5 - 8207899b4 - 5223925b3 - 14580160b2 + 29953399*b1 + 29953399) / 28
$\nu^{11}$	$=$	$ ( 14580160 \beta_{15} + 23393706 \beta_{14} + 20406571 \beta_{13} + 7797902 \beta_{12} - 14580160 \beta_{11} + \cdots - 257879615 ) / 28 $ (14580160b15 + 23393706b14 + 20406571b13 + 7797902b12 - 14580160b11 - 139775048b10 + 7590324b9 - 12364737b8 + 33765293b7 + 2530108b6 - 80902371b5 - 83482145b4 - 12364737b3 - 125194888b2 + 257879615*b1 - 257879615) / 28
$\nu^{12}$	$=$	$ ( - 32178942 \beta_{13} + 37199652 \beta_{12} - 69887524 \beta_{11} + 159281262 \beta_{10} + \cdots + 243316647 ) / 7 $ (-32178942b13 + 37199652b12 - 69887524b11 + 159281262b10 - 42210372b7 + 10914568b6 + 5308146b5 + 61754852b4 - 67561088b3 + 34943762b2 + 243316647) / 7
$\nu^{13}$	$=$	$ ( 248675000 \beta_{15} + 401179464 \beta_{14} - 915679463 \beta_{13} - 133726488 \beta_{12} + \cdots + 6762290087 ) / 28 $ (248675000b15 + 401179464b14 - 915679463b13 - 133726488b12 + 248675000b11 + 3773585256b10 + 154075992b9 - 175755559b8 - 977233653b7 - 51358664b6 - 2397150225b5 - 3154849099b4 + 175755559b3 - 4022260256b2 + 6762290087*b1 + 6762290087) / 28
$\nu^{14}$	$=$	$ ( 4022260256 \beta_{15} + 6449368632 \beta_{14} - 2779727501 \beta_{13} + 2149789544 \beta_{12} + \cdots + 5809106275 ) / 28 $ (4022260256b15 + 6449368632b14 - 2779727501b13 + 2149789544b12 - 4022260256b11 + 2516097892b10 + 2133082896b9 - 3469022823b8 - 756225905b7 + 711027632b6 + 2955823939b5 + 7251663063b4 - 3469022823b3 + 6538358148b2 - 5809106275*b1 + 5809106275) / 28
$\nu^{15}$	$=$	$ ( - 21982496013 \beta_{13} - 1361370150 \beta_{12} + 2516097892 \beta_{11} + 108721988158 \beta_{10} + \cdots + 174879424552 ) / 14 $ (-21982496013b13 - 1361370150b12 + 2516097892b11 + 108721988158b10 - 28487269278b7 - 678197632b6 - 79795395b5 - 1735700186b4 + 1170041726b3 - 1258048946b2 + 174879424552) / 14

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$1$	$-1$

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} $

97.1

−4.17475	−	2.98764i
3.90339	+	1.67627i
−3.32621	−	2.49774i
3.49578	+	1.44094i
4.67475	+	2.12162i
−3.40339	−	2.54230i
3.82621	+	1.63171i
−2.99578	−	2.30697i
−2.99578	+	2.30697i
3.82621	−	1.63171i
−3.40339	+	2.54230i
4.67475	−	2.12162i
3.49578	−	1.44094i
−3.32621	+	2.49774i
3.90339	−	1.67627i
−4.17475	+	2.98764i

−

1.73205i

−

9.21569i

−3.00000

97.2

−

1.73205i

−

6.58897i

−3.00000

97.3

−

1.73205i

−

6.29930i

−3.00000

97.4

−

1.73205i

−

2.22784i

−3.00000

97.5

−

1.73205i

5.23547i

−3.00000

97.6

−

1.73205i

5.34294i

−3.00000

97.7

−

1.73205i

5.38055i

−3.00000

97.8

−

1.73205i

8.37285i

−3.00000

97.9

1.73205i

−

8.37285i

−3.00000

97.10

1.73205i

−

5.38055i

−3.00000

97.11

1.73205i

−

5.34294i

−3.00000

97.12

1.73205i

−

5.23547i

−3.00000

97.13

1.73205i

2.22784i

−3.00000

97.14

1.73205i

6.29930i

−3.00000

97.15

1.73205i

6.58897i

−3.00000

97.16

1.73205i

9.21569i

−3.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1176.3.f.d	✓	16
3.b	odd	2	1		3528.3.f.j		16
4.b	odd	2	1		2352.3.f.m		16
7.b	odd	2	1	inner	1176.3.f.d	✓	16
7.c	even	3	1		1176.3.z.g		16
7.c	even	3	1		1176.3.z.h		16
7.d	odd	6	1		1176.3.z.g		16
7.d	odd	6	1		1176.3.z.h		16
21.c	even	2	1		3528.3.f.j		16
28.d	even	2	1		2352.3.f.m		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1176.3.f.d	✓	16	1.a	even	1	1	trivial
1176.3.f.d	✓	16	7.b	odd	2	1	inner
1176.3.z.g		16	7.c	even	3	1
1176.3.z.g		16	7.d	odd	6	1
1176.3.z.h		16	7.c	even	3	1
1176.3.z.h		16	7.d	odd	6	1
2352.3.f.m		16	4.b	odd	2	1
2352.3.f.m		16	28.d	even	2	1
3528.3.f.j		16	3.b	odd	2	1
3528.3.f.j		16	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{16} + 328 T_{5}^{14} + 44784 T_{5}^{12} + 3316528 T_{5}^{10} + 145126008 T_{5}^{8} + \cdots + 1153226845456 $ T5^16 + 328*T5^14 + 44784*T5^12 + 3316528*T5^10 + 145126008*T5^8 + 3810762208*T5^6 + 57617807552*T5^4 + 440311927104*T5^2 + 1153226845456 acting on $S_{3}^{\mathrm{new}}(1176, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ T^{16} + \cdots + 1153226845456 $ T^16 + 328T^14 + 44784T^12 + 3316528T^10 + 145126008T^8 + 3810762208T^6 + 57617807552T^4 + 440311927104*T^2 + 1153226845456
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} - 684 T^{6} + \cdots + 614656)^{2} $ (T^8 - 684T^6 - 1456T^5 + 115420T^4 + 564480T^3 - 1370432T^2 - 1229312T + 614656)^2
$13$	$ T^{16} + \cdots + 65\!\cdots\!84 $ T^16 + 2128T^14 + 1817192T^12 + 798484192T^10 + 192820471960T^8 + 25463126072768T^6 + 1741768032633248T^4 + 55521022545093248*T^2 + 650909090772154384
$17$	$ T^{16} + \cdots + 412354916962576 $ T^16 + 2312T^14 + 2026992T^12 + 853134512T^10 + 176516302200T^8 + 15902442625760T^6 + 414696648578240T^4 + 2665509800579904*T^2 + 412354916962576
$19$	$ T^{16} + \cdots + 58\!\cdots\!84 $ T^16 + 3376T^14 + 4542288T^12 + 3134872448T^10 + 1194033037376T^8 + 250906264678400T^6 + 26869610801872896T^4 + 1122628740354408448*T^2 + 580497900263440384
$23$	$ (T^{8} + 56 T^{7} + \cdots - 60075776)^{2} $ (T^8 + 56T^7 - 532T^6 - 60928T^5 - 261924T^4 + 15107904T^3 + 151085376T^2 + 194216960*T - 60075776)^2
$29$	$ (T^{8} - 56 T^{7} + \cdots + 208436284528)^{2} $ (T^8 - 56T^7 - 3400T^6 + 178336T^5 + 3744432T^4 - 161226336T^3 - 1430852736T^2 + 37664765440*T + 208436284528)^2
$31$	$ T^{16} + \cdots + 36\!\cdots\!24 $ T^16 + 6960T^14 + 18384976T^12 + 23059175808T^10 + 13774848665664T^8 + 3267399419437056T^6 + 141035794634752000T^4 + 1770660684043911168*T^2 + 3679353008133505024
$37$	$ (T^{8} + 64 T^{7} + \cdots - 75056064752)^{2} $ (T^8 + 64T^7 - 2664T^6 - 171456T^5 + 2248536T^4 + 124103680T^3 - 399876256T^2 - 18465825536*T - 75056064752)^2
$41$	$ T^{16} + \cdots + 70\!\cdots\!76 $ T^16 + 10520T^14 + 43130992T^12 + 89927946064T^10 + 103002812881656T^8 + 64365342166449824T^6 + 19980389824074637504T^4 + 2293640527002526075072*T^2 + 7045013430369971132176
$43$	$ (T^{8} + 56 T^{7} + \cdots - 166473949184)^{2} $ (T^8 + 56T^7 - 3144T^6 - 166688T^5 + 3148304T^4 + 142110208T^3 - 693617664T^2 - 34762227712*T - 166473949184)^2
$47$	$ T^{16} + \cdots + 47\!\cdots\!84 $ T^16 + 22112T^14 + 169316976T^12 + 529955760896T^10 + 647692282575936T^8 + 194072245946408960T^6 + 10526342701640695808T^4 + 173983901707951669248*T^2 + 473749289056190070784
$53$	$ (T^{8} + \cdots - 32106868908032)^{2} $ (T^8 - 56T^7 - 14680T^6 + 1010656T^5 + 46265104T^4 - 4688834560T^3 + 63325048832T^2 + 1752471560192*T - 32106868908032)^2
$59$	$ T^{16} + \cdots + 20\!\cdots\!36 $ T^16 + 34912T^14 + 504664176T^12 + 3918755621120T^10 + 17772719821560896T^8 + 48006466576568643584T^6 + 75216257619403843387392T^4 + 62376106084920758548037632*T^2 + 20954584989393198604522356736
$61$	$ T^{16} + \cdots + 21\!\cdots\!04 $ T^16 + 31824T^14 + 411909416T^12 + 2792048627936T^10 + 10636522502583576T^8 + 22694500972119676352T^6 + 25309510711823477914272T^4 + 12536004328843264574802560*T^2 + 2195585137593314484059033104
$67$	$ (T^{8} + \cdots + 4621952057344)^{2} $ (T^8 + 112T^7 - 12608T^6 - 1113600T^5 + 49065856T^4 + 1534588928T^3 - 31140655104T^2 - 477828513792*T + 4621952057344)^2
$71$	$ (T^{8} + \cdots + 18962560708864)^{2} $ (T^8 - 80T^7 - 19148T^6 + 794768T^5 + 131832156T^4 - 421799168T^3 - 262246802240T^2 - 4336623946240*T + 18962560708864)^2
$73$	$ T^{16} + \cdots + 60\!\cdots\!56 $ T^16 + 37520T^14 + 541804648T^12 + 3806191526240T^10 + 13674449418597016T^8 + 24579153039659132096T^6 + 21125831410908918370720T^4 + 7439669478760811986139264*T^2 + 601996143217456658116976656
$79$	$ (T^{8} + \cdots + 4233495998464)^{2} $ (T^8 - 152T^7 - 4424T^6 + 1068128T^5 + 8203536T^4 - 1817725184T^3 - 3879100928T^2 + 720748261376*T + 4233495998464)^2
$83$	$ T^{16} + \cdots + 11\!\cdots\!84 $ T^16 + 83264T^14 + 2725920704T^12 + 44225119870976T^10 + 365462005702411264T^8 + 1385701369401780797440T^6 + 1673808147723390778081280T^4 + 281854566814324884949172224*T^2 + 11136234550220679520907689984
$89$	$ T^{16} + \cdots + 15\!\cdots\!64 $ T^16 + 74824T^14 + 2072826288T^12 + 26074923366448T^10 + 144719678739472632T^8 + 282869965601171541472T^6 + 48061380865597288889792T^4 + 1877416558957916918342976*T^2 + 1528859638903204701819664
$97$	$ T^{16} + \cdots + 16\!\cdots\!96 $ T^16 + 63632T^14 + 1377447592T^12 + 13856471793760T^10 + 69309394967255064T^8 + 162847754943486101696T^6 + 146015940801054545446048T^4 + 29378030670084052784682112*T^2 + 1630911426721555402413089296
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1176.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + (\beta_{11} - \beta_{2}) q^{5} - 3 q^{9} + (\beta_{15} - \beta_{9} + \beta_{7}) q^{11} + ( - \beta_{6} - \beta_{5} + \cdots + 2 \beta_{2}) q^{13} + (\beta_{15} - 2 \beta_{10}) q^{15}+ \cdots + ( - 3 \beta_{15} + 3 \beta_{9} - 3 \beta_{7}) q^{99}+O(q^{100}) \) q + b1 * q^3 + (b11 - b2) * q^5 - 3 * q^9 + (b15 - b9 + b7) * q^11 + (-b6 - b5 - 2b4 + 2b2) * q^13 + (b15 - 2b10) q^15 + (b12 + b11 + b4 + b2) * q^17 + (-b12 - 2b11 - b5 + b4 - 4b1) * q^19 + (-b15 + b14 + b13 - 4b10 - b8 - 7) q^23 + (b13 - 3b10 + 2b9 - b8 + b7 - 16) * q^25 - 3b1 q^27 + (-b14 - b13 - 9b10 + b8 - 5b7 + 7) * q^29 + (-2b11 - b6 - 4b5 + b4 + 8b2 + 4b1) * q^31 + (-3b11 - b6 + 2b4 + b2) * q^33 + (-2b14 + 4b13 - b10 - 8) * q^37 + (b13 + 6b10 + 3b9 + b7) * q^39 + (-b12 - b11 - 8b4 + b3 + 7b2 - 7b1) q^41 + (2b14 + b13 - 4b10 - b8 - 3b7 - 7) q^43 + (-3b11 + 3b2) * q^45 + (6b11 + b6 + 3b5 + 6b4 + b3 - 4b2 - 15b1) q^47 + (b15 - 3b14 + b13 + 4b10) * q^51 + (-2b15 + 3b13 - 6b9 + b8 + b7 + 7) q^53 + (b12 + 2b11 + 2b6 - 5b5 + 7b4 - 2b3 + 4b2 - 34b1) q^55 + (-2b15 + 3b14 - 2b10 - 2b7 + 12) * q^57 + (-2b12 + 6b11 + b6 - 5b5 - 2b4 - b3 + 8b2 + 11b1) * q^59 + (3b12 + 2b11 - 2b5 - 2b3 - 8b2 - 14b1) * q^61 + (2b15 + 6b14 + 9b13 - 5b10 - b9 + 3b8 - 5b7 - 7) * q^65 + (2b13 + 8b10 - 2b9 + 2b8 + 10b7 - 14) q^67 + (b12 + 3b11 + 4b5 + b4 - 3b3 + 3b2 - 7b1) q^69 + (-b15 - 6b14 + 2b13 + 24b10 - b9 - 2b8 - 3b7 + 10) q^71 + (-b12 + 2b11 + 3b6 + 7b5 + 9b4 + 2b3 + 2b2 + 14b1) q^73 + (2b6 + 3b5 + 5b4 - 3b3 + 3b2 - 16b1) * q^75 + (2b14 + 3b13 - 28b10 + 2b9 + b8 - 5b7 + 19) q^79 + 9 * q^81 + (b12 - 6b11 + b6 + 3b5 + 6b4 + 6b3 + 10b2 - 42b1) * q^83 + (-2b14 - 6b13 + 11b10 + 2b9 + 2b8 - 18b7 - 26) * q^85 + (-b12 - 4b5 - 16b4 + 3b3 + 9b2 + 7b1) q^87 + (-b11 - b6 - b5 - 11b4 + 6b3 - 29b2 - 14b1) * q^89 + (-2b15 + 4b13 + 22b10 + 3b9 - 2b7 - 12) q^93 + (-4b15 - b14 + 5b13 + 8b10 + b9 - b8 + 17b7 + 57) * q^95 + (-2b12 - 2b11 + 4b6 - 10b5 + b4 + 2b3 + 14b2 - 14b1) q^97 + (-3b15 + 3b9 - 3b7) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 112 q^{23} - 256 q^{25} + 112 q^{29} - 128 q^{37} - 112 q^{43} + 112 q^{53} + 192 q^{57} - 112 q^{65} - 224 q^{67} + 160 q^{71} + 304 q^{79} + 144 q^{81} - 416 q^{85} - 192 q^{93} + 912 q^{95}+O(q^{100}) \) 16 * q - 48 * q^9 - 112 * q^23 - 256 * q^25 + 112 * q^29 - 128 * q^37 - 112 * q^43 + 112 * q^53 + 192 * q^57 - 112 * q^65 - 224 * q^67 + 160 * q^71 + 304 * q^79 + 144 * q^81 - 416 * q^85 - 192 * q^93 + 912 * q^95

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	1176.3.f.d

Level	$1176$
Weight	$3$
Character orbit	1176.f
Analytic conductor	$32.044$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(32.0436790888\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} - 66 x^{14} + 136 x^{13} + 3441 x^{12} - 4512 x^{11} - 100322 x^{10} + \cdots + 13841287201 \) x^16 - 4x^15 - 66x^14 + 136x^13 + 3441x^12 - 4512x^11 - 100322x^10 + 24868x^9 + 2325060x^8 + 260012x^7 - 34273754x^6 - 26559312x^5 + 402103713x^4 + 286444088x^3 - 2633783418x^2 - 3060285788*x + 13841287201
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{12}\cdot 7^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 19\!\cdots\!15 \nu^{15} + \cdots + 10\!\cdots\!26 ) / 17\!\cdots\!08 \) (-1912863316623227926255615v^15 + 157590092883913634324899v^14 + 134553559303277107074448835v^13 + 234251144123174764943611906v^12 - 6130277557280544667044415252v^11 - 14461729974232158231161784788v^10 + 159912185005949024540997253830v^9 + 545979676810483959695503249362v^8 - 2977979150709146858658059920446v^7 - 12186728557549192183318227164384v^6 + 33617527462650444084998233859972v^5 + 185179542565865634914909315411884v^4 - 252337232856569126836178509648847v^3 - 1880544733090865400075974684200981v^2 + 634559294256824390217426688607659*v + 10256187601200275396626601089582626) / 1773367572721598053400461619726508
\(\beta_{2}\)	\(=\)	\( ( 4157815028 \nu^{15} + 226803899359 \nu^{14} + 2133995005098 \nu^{13} + \cdots + 83\!\cdots\!95 ) / 22\!\cdots\!88 \) (4157815028v^15 + 226803899359v^14 + 2133995005098v^13 - 3577907083693v^12 - 194574957455356v^11 - 251785193114712v^10 + 7427269664001284v^9 + 25329048895604970v^8 - 174617950806252252v^7 - 734432835451712126v^6 + 2170987437180418964v^5 + 14053447947667247712v^4 - 16760250856674216432v^3 - 134039054755482730609v^2 + 6950960411836071666*v + 838547261713169306795) / 2201283776407962288
\(\beta_{3}\)	\(=\)	\( ( 44\!\cdots\!44 \nu^{15} + \cdots - 95\!\cdots\!31 ) / 70\!\cdots\!32 \) (44577133160376255065376144v^15 - 220555033154806103711604155v^14 - 5162518189659835853593171310v^13 - 1398480263636267715714079687v^12 + 320037559142085455768573452996v^11 + 506269249423326287867215280000v^10 - 10396523739110943366632487895640v^9 - 33723238549339283137701106927834v^8 + 227594131074857164767569406685596v^7 + 898481249079677698841857006826730v^6 - 2761263003538818849762244026919996v^5 - 16372803797949565065076363564681064v^4 + 22273979345309546558116233878974992v^3 + 153294449370457062676807243567106621v^2 - 19505215404059807908493705706414798*v - 950259744626931678628322183969956131) / 7093470290886392213601846478906032
\(\beta_{4}\)	\(=\)	\( ( 35\!\cdots\!56 \nu^{15} + \cdots - 37\!\cdots\!35 ) / 30\!\cdots\!44 \) (350876609979969256v^15 - 314839517719341913v^14 - 30338192533773169860v^13 - 44191121410098384887v^12 + 1549533327556378967100v^11 + 3577544079010336198648v^10 - 45762730994300176778752v^9 - 163736549542244206139566v^8 + 922903483838461429315560v^7 + 4034994607096103269464178v^6 - 10976894065514184482337940v^5 - 68463674804877207710195904v^4 + 80791935520653988589771600v^3 + 643880032991142085134009727v^2 - 82695890302296909056068804*v - 3791320355935665433758353035) / 30444450498871083689172144
\(\beta_{5}\)	\(=\)	\( ( - 30\!\cdots\!34 \nu^{15} + \cdots - 68\!\cdots\!68 ) / 15\!\cdots\!72 \) (-306973687093272834v^15 - 2341845882705077043v^14 - 2335381153484933007v^13 + 72431438271686622148v^12 + 1095636592069061456828v^11 + 316504958534616322428v^10 - 51864459345420910242736v^9 - 172673828522119257827122v^8 + 1332108879663617997416670v^7 + 5524798313850637177278924v^6 - 16939219803884735639985900v^5 - 112196222387208274773118828v^4 + 133331510868660742710492106v^3 + 1072322688407929217180244141v^2 - 10994258454029827334769551*v - 6890044785342379369426823168) / 15222225249435541844586072
\(\beta_{6}\)	\(=\)	\( ( - 23\!\cdots\!76 \nu^{15} + \cdots + 15\!\cdots\!25 ) / 70\!\cdots\!32 \) (-231514427916488493337407776v^15 - 1733455067248079744739512633v^14 + 10214679112641706166553809780v^13 + 157226781508357260615129245545v^12 - 331515392769619765360327208300v^11 - 6121320413343608886064433975344v^10 + 753910213734035502343699081640v^9 + 165749808765107773640040630561626v^8 + 33169905817259220276438310109232v^7 - 2654287494610295459961479433255350v^6 - 2514956268210773951458815232408716v^5 + 35074258805899930411680195251600824v^4 + 20791733953680467665654632242420768v^3 - 247670370240854720314562854642074985v^2 - 314157982384904309237699920311886836*v + 1564448041519563467199870447103059725) / 7093470290886392213601846478906032
\(\beta_{7}\)	\(=\)	\( ( - 63\!\cdots\!81 \nu^{15} + \cdots + 16\!\cdots\!34 ) / 12\!\cdots\!88 \) (-631665189371406619381v^15 - 662731806327679783830v^14 + 45896053921093831036569v^13 + 120019826580783387396674v^12 - 1976331469897354311221712v^11 - 6600269659182099803449260v^10 + 49214663148263150564790974v^9 + 214815970942431709648955796v^8 - 809637498244974183112312146v^7 - 4339176996709522847929161620v^6 + 7752454423369146809886700056v^5 + 56204781468861895398832278612v^4 - 38820748755921741840658481025v^3 - 435224110296497347123756050894v^2 - 2284047358170699659989600023*v + 1621199449804323773844571142234) / 12361429409444795863651383888
\(\beta_{8}\)	\(=\)	\( ( - 19\!\cdots\!83 \nu^{15} + \cdots + 59\!\cdots\!44 ) / 29\!\cdots\!88 \) (-1977904712196288644663483v^15 - 2769722957512364628380798v^14 + 149409984430279586627424289v^13 + 428261367104891772462536040v^12 - 6363784319313105137418186736v^11 - 23044219197298363809623543500v^10 + 157606104512295979331056952090v^9 + 733311784154638203407022055236v^8 - 2521393500276621203318490294690v^7 - 14510580175776780026009382445600v^6 + 23287275663692885610347825390648v^5 + 187025408125574894936827734601364v^4 - 105898168300369055037469101430151v^3 - 1438886497144196814507575697277558v^2 - 226661812196939822079938688056335*v + 5909802439232478076010365883454944) / 29679792012076954868626972715088
\(\beta_{9}\)	\(=\)	\( ( - 10\!\cdots\!74 \nu^{15} + \cdots + 22\!\cdots\!53 ) / 14\!\cdots\!44 \) (-1006502610972175217701274v^15 + 1963090966618337093009667v^14 + 60172801086069857151363930v^13 + 56603888885511048850037121v^12 - 2676266678568374371698125256v^11 - 4173190082019966805123910172v^10 + 59786981142647818231259517112v^9 + 200122682769621564888085523374v^8 - 1012047728005356222362359265264v^7 - 3938100309019050873821892408162v^6 + 8641483696324724173668850139168v^5 + 58215664532299935899528339600524v^4 - 54534994898684519685122968171118v^3 - 432713300511231086780968267672233v^2 - 12392694366195431088310976681250*v + 2271201017297437099281162437917753) / 14839896006038477434313486357544
\(\beta_{10}\)	\(=\)	\( ( - 1434785603940 \nu^{15} - 1900762364347 \nu^{14} + 108526077128390 \nu^{13} + \cdots + 43\!\cdots\!37 ) / 17\!\cdots\!32 \) (-1434785603940v^15 - 1900762364347v^14 + 108526077128390v^13 + 305761124010809v^12 - 4634287092419732v^11 - 16451997826420712v^10 + 114795138909966604v^9 + 526200538323412638v^8 - 1850353679066317668v^7 - 10386497708859409786v^6 + 17232564720652874620v^5 + 134816715680204446528v^4 - 85343622288924149712v^3 - 1035286717988536754363v^2 - 62730574414649842274*v + 4389566035171884470337) / 17684664455170110832
\(\beta_{11}\)	\(=\)	\( ( - 56\!\cdots\!90 \nu^{15} + \cdots + 36\!\cdots\!79 ) / 50\!\cdots\!88 \) (-56642036536487516741853090v^15 + 550151708697427688717919888v^14 + 5339589179841001817623440687v^13 - 25287452299837460910124817147v^12 - 317977388004172791551211997668v^11 + 697465441230838467700662181552v^10 + 10546985420425006804161240032730v^9 - 3297036198827858759397698547816v^8 - 232156867712300005582266163178298v^7 - 170021979053373415333732792410316v^6 + 3153454980931729730823605041137892v^5 + 5511167207643408268595853084813256v^4 - 26288679602125517652377123936378056v^3 - 62234831915445886667336704892166816v^2 + 99558002793695105408018783007808747*v + 368001721378570303740925286314071079) / 506676449349028015257274748493288
\(\beta_{12}\)	\(=\)	\( ( - 14\!\cdots\!32 \nu^{15} + \cdots + 68\!\cdots\!25 ) / 70\!\cdots\!32 \) (-1472854576289224688583582932v^15 + 14177957572127991765826757489v^14 + 133589084361417673257027320778v^13 - 690003008951526503333372363083v^12 - 7773215150996317680650193558428v^11 + 20430379757482454873976389670928v^10 + 257161462634343739260316597165288v^9 - 198589541098943116821847136568546v^8 - 5623094838222282744605179943097828v^7 - 1769720747266380799577640938931206v^6 + 77514399110872335948478631841912932v^5 + 96381684867830338904919200498510584v^4 - 646787794636939669329030846321953980v^3 - 1201105157312055536571474551805175751v^2 + 2688744471733625881174945807099407370*v + 6882255985247260617442869406556167025) / 7093470290886392213601846478906032
\(\beta_{13}\)	\(=\)	\( ( - 20\!\cdots\!15 \nu^{15} + \cdots + 65\!\cdots\!67 ) / 61\!\cdots\!44 \) (-2071905375627045789315v^15 - 2844636481759773644052v^14 + 157733948702019596146844v^13 + 449567602203165069603211v^12 - 6725986828004979180742224v^11 - 24119392666738067511134672v^10 + 166503982597390241454746606v^9 + 769275909613848053991157224v^8 - 2673576826096713395870302104v^7 - 15125690906902891763241947574v^6 + 24775978266365312174929462416v^5 + 196415977025304441064798346320v^4 - 122495048060051089345118969155v^3 - 1506218980785069936042248431092v^2 - 105268335342765609660034458692*v + 6578801327497297958540519165267) / 6180714704722397931825691944
\(\beta_{14}\)	\(=\)	\( ( 57\!\cdots\!16 \nu^{15} + \cdots - 14\!\cdots\!99 ) / 14\!\cdots\!44 \) (5730429750414417486547616v^15 - 8387837512395634923289889v^14 - 342621869434030589706960120v^13 - 450827648548175188543646559v^12 + 15241117814157487179441706708v^11 + 28481573290805989685558266608v^10 - 335264482327370694439359112004v^9 - 1265651614149107256641538486422v^8 + 5603691824106242473550003664872v^7 + 24744082644026753800545489559254v^6 - 40472203172878863229619248162604v^5 - 369470593907517033386782732335856v^4 + 196791075824769178089247059918788v^3 + 2679332788071186052372761204838303v^2 + 1693175745450032130411663484905176*v - 14844141869843621540101047634453999) / 14839896006038477434313486357544
\(\beta_{15}\)	\(=\)	\( ( - 24\!\cdots\!43 \nu^{15} + \cdots + 68\!\cdots\!30 ) / 29\!\cdots\!88 \) (-24675891599013579715795443v^15 + 14838686075194091099329066v^14 + 1595010131572037797129207367v^13 + 2978567877087263366555261114v^12 - 69893464516229714397314465488v^11 - 172584784330508783944914657140v^10 + 1602393277760914094630074473010v^9 + 6576326829449575293667802076204v^8 - 26369882913046879467815598155742v^7 - 128941011796545922861347081774396v^6 + 208403287137328237973585046446840v^5 + 1819869954801539204965089402205612v^4 - 999328036173611395753652828474999v^3 - 13481330685024586094700848877694774v^2 - 5891490276046582196429841685375385*v + 68188616501971600168964380980553930) / 29679792012076954868626972715088

\(\nu\)	\(=\)	\( ( \beta_{13} - 7\beta_{8} + 3\beta_{7} - \beta_{5} - 3\beta_{4} + 7\beta_{3} + 7\beta _1 + 7 ) / 28 \) (b13 - 7b8 + 3b7 - b5 - 3b4 + 7b3 + 7*b1 + 7) / 28
\(\nu^{2}\)	\(=\)	\( ( - 5 \beta_{13} + 28 \beta_{10} - 7 \beta_{8} - \beta_{7} + 19 \beta_{5} + 15 \beta_{4} - 7 \beta_{3} + \cdots + 259 ) / 28 \) (-5b13 + 28b10 - 7b8 - b7 + 19b5 + 15b4 - 7b3 + 28b2 - 259b1 + 259) / 28
\(\nu^{3}\)	\(=\)	\( ( - 9 \beta_{13} - 14 \beta_{12} + 28 \beta_{11} + 42 \beta_{10} - 6 \beta_{7} - 15 \beta_{5} + \cdots + 392 ) / 14 \) (-9b13 - 14b12 + 28b11 + 42b10 - 6b7 - 15b5 - 66b4 + 126b3 - 14*b2 + 392) / 14
\(\nu^{4}\)	\(=\)	\( ( 56 \beta_{15} + 84 \beta_{14} + 235 \beta_{13} - 28 \beta_{12} + 56 \beta_{11} - 1120 \beta_{10} + \cdots - 4809 ) / 28 \) (56b15 + 84b14 + 235b13 - 28b12 + 56b11 - 1120b10 - 259b8 + 299b7 + 665b5 + 441b4 + 259b3 + 1064b2 - 4809*b1 - 4809) / 28
\(\nu^{5}\)	\(=\)	\( ( - 1064 \beta_{15} - 1638 \beta_{14} - 277 \beta_{13} - 546 \beta_{12} + 1064 \beta_{11} + 3332 \beta_{10} + \cdots + 12677 ) / 28 \) (-1064b15 - 1638b14 - 277b13 - 546b12 + 1064b11 + 3332b10 - 168b9 + 2275b8 - 1335b7 - 56b6 + 1557b5 + 23b4 + 2275b3 + 2268b2 - 12677*b1 + 12677) / 28
\(\nu^{6}\)	\(=\)	\( ( 3285 \beta_{13} - 854 \beta_{12} + 1666 \beta_{11} - 15631 \beta_{10} + 3198 \beta_{7} - 84 \beta_{6} + \cdots - 44891 ) / 7 \) (3285b13 - 854b12 + 1666b11 - 15631b10 + 3198b7 - 84b6 - 381b5 - 2340b4 + 3738b3 - 833b2 - 44891) / 7
\(\nu^{7}\)	\(=\)	\( ( - 29596 \beta_{15} - 46452 \beta_{14} + 36989 \beta_{13} + 15484 \beta_{12} - 29596 \beta_{11} + \cdots - 359065 ) / 28 \) (-29596b15 - 46452b14 + 36989b13 + 15484b12 - 29596b11 - 104468b10 - 8820b9 + 41153b8 + 11399b7 + 2940b6 + 78847b5 + 103457b4 - 41153b3 + 134064b2 - 359065*b1 - 359065) / 28
\(\nu^{8}\)	\(=\)	\( ( - 134064 \beta_{15} - 209916 \beta_{14} + 211955 \beta_{13} - 69972 \beta_{12} + 134064 \beta_{11} + \cdots - 1667897 ) / 28 \) (-134064b15 - 209916b14 + 211955b13 - 69972b12 + 134064b11 - 703416b10 - 37632b9 + 200109b8 + 111971b7 - 12544b6 - 497793b5 - 675121b4 + 200109b3 - 837480b2 + 1667897*b1 - 1667897) / 28
\(\nu^{9}\)	\(=\)	\( ( 872793 \beta_{13} + 372974 \beta_{12} - 703416 \beta_{11} - 4194456 \beta_{10} + 939366 \beta_{7} + \cdots - 9713956 ) / 14 \) (872793b13 + 372974b12 - 703416b11 - 4194456b10 + 939366b7 + 98000b6 + 58071b5 + 642142b4 - 733894b3 + 351708b2 - 9713956) / 14
\(\nu^{10}\)	\(=\)	\( ( - 4546164 \beta_{15} - 7205268 \beta_{14} - 1482169 \beta_{13} + 2401756 \beta_{12} - 4546164 \beta_{11} + \cdots + 29953399 ) / 28 \) (-4546164b15 - 7205268b14 - 1482169b13 + 2401756b12 - 4546164b11 + 19126324b10 - 1755768b9 + 5223925b8 - 5541377b7 + 585256b6 - 9929563b5 - 8207899b4 - 5223925b3 - 14580160b2 + 29953399*b1 + 29953399) / 28
\(\nu^{11}\)	\(=\)	\( ( 14580160 \beta_{15} + 23393706 \beta_{14} + 20406571 \beta_{13} + 7797902 \beta_{12} - 14580160 \beta_{11} + \cdots - 257879615 ) / 28 \) (14580160b15 + 23393706b14 + 20406571b13 + 7797902b12 - 14580160b11 - 139775048b10 + 7590324b9 - 12364737b8 + 33765293b7 + 2530108b6 - 80902371b5 - 83482145b4 - 12364737b3 - 125194888b2 + 257879615*b1 - 257879615) / 28
\(\nu^{12}\)	\(=\)	\( ( - 32178942 \beta_{13} + 37199652 \beta_{12} - 69887524 \beta_{11} + 159281262 \beta_{10} + \cdots + 243316647 ) / 7 \) (-32178942b13 + 37199652b12 - 69887524b11 + 159281262b10 - 42210372b7 + 10914568b6 + 5308146b5 + 61754852b4 - 67561088b3 + 34943762b2 + 243316647) / 7
\(\nu^{13}\)	\(=\)	\( ( 248675000 \beta_{15} + 401179464 \beta_{14} - 915679463 \beta_{13} - 133726488 \beta_{12} + \cdots + 6762290087 ) / 28 \) (248675000b15 + 401179464b14 - 915679463b13 - 133726488b12 + 248675000b11 + 3773585256b10 + 154075992b9 - 175755559b8 - 977233653b7 - 51358664b6 - 2397150225b5 - 3154849099b4 + 175755559b3 - 4022260256b2 + 6762290087*b1 + 6762290087) / 28
\(\nu^{14}\)	\(=\)	\( ( 4022260256 \beta_{15} + 6449368632 \beta_{14} - 2779727501 \beta_{13} + 2149789544 \beta_{12} + \cdots + 5809106275 ) / 28 \) (4022260256b15 + 6449368632b14 - 2779727501b13 + 2149789544b12 - 4022260256b11 + 2516097892b10 + 2133082896b9 - 3469022823b8 - 756225905b7 + 711027632b6 + 2955823939b5 + 7251663063b4 - 3469022823b3 + 6538358148b2 - 5809106275*b1 + 5809106275) / 28
\(\nu^{15}\)	\(=\)	\( ( - 21982496013 \beta_{13} - 1361370150 \beta_{12} + 2516097892 \beta_{11} + 108721988158 \beta_{10} + \cdots + 174879424552 ) / 14 \) (-21982496013b13 - 1361370150b12 + 2516097892b11 + 108721988158b10 - 28487269278b7 - 678197632b6 - 79795395b5 - 1735700186b4 + 1170041726b3 - 1258048946b2 + 174879424552) / 14

\(n\)	\(295\)	\(589\)	\(785\)	\(1081\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( T^{16} + \cdots + 1153226845456 \) T^16 + 328T^14 + 44784T^12 + 3316528T^10 + 145126008T^8 + 3810762208T^6 + 57617807552T^4 + 440311927104*T^2 + 1153226845456
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} - 684 T^{6} + \cdots + 614656)^{2} \) (T^8 - 684T^6 - 1456T^5 + 115420T^4 + 564480T^3 - 1370432T^2 - 1229312T + 614656)^2
$13$	\( T^{16} + \cdots + 65\!\cdots\!84 \) T^16 + 2128T^14 + 1817192T^12 + 798484192T^10 + 192820471960T^8 + 25463126072768T^6 + 1741768032633248T^4 + 55521022545093248*T^2 + 650909090772154384
$17$	\( T^{16} + \cdots + 412354916962576 \) T^16 + 2312T^14 + 2026992T^12 + 853134512T^10 + 176516302200T^8 + 15902442625760T^6 + 414696648578240T^4 + 2665509800579904*T^2 + 412354916962576
$19$	\( T^{16} + \cdots + 58\!\cdots\!84 \) T^16 + 3376T^14 + 4542288T^12 + 3134872448T^10 + 1194033037376T^8 + 250906264678400T^6 + 26869610801872896T^4 + 1122628740354408448*T^2 + 580497900263440384
$23$	\( (T^{8} + 56 T^{7} + \cdots - 60075776)^{2} \) (T^8 + 56T^7 - 532T^6 - 60928T^5 - 261924T^4 + 15107904T^3 + 151085376T^2 + 194216960*T - 60075776)^2
$29$	\( (T^{8} - 56 T^{7} + \cdots + 208436284528)^{2} \) (T^8 - 56T^7 - 3400T^6 + 178336T^5 + 3744432T^4 - 161226336T^3 - 1430852736T^2 + 37664765440*T + 208436284528)^2
$31$	\( T^{16} + \cdots + 36\!\cdots\!24 \) T^16 + 6960T^14 + 18384976T^12 + 23059175808T^10 + 13774848665664T^8 + 3267399419437056T^6 + 141035794634752000T^4 + 1770660684043911168*T^2 + 3679353008133505024
$37$	\( (T^{8} + 64 T^{7} + \cdots - 75056064752)^{2} \) (T^8 + 64T^7 - 2664T^6 - 171456T^5 + 2248536T^4 + 124103680T^3 - 399876256T^2 - 18465825536*T - 75056064752)^2
$41$	\( T^{16} + \cdots + 70\!\cdots\!76 \) T^16 + 10520T^14 + 43130992T^12 + 89927946064T^10 + 103002812881656T^8 + 64365342166449824T^6 + 19980389824074637504T^4 + 2293640527002526075072*T^2 + 7045013430369971132176
$43$	\( (T^{8} + 56 T^{7} + \cdots - 166473949184)^{2} \) (T^8 + 56T^7 - 3144T^6 - 166688T^5 + 3148304T^4 + 142110208T^3 - 693617664T^2 - 34762227712*T - 166473949184)^2
$47$	\( T^{16} + \cdots + 47\!\cdots\!84 \) T^16 + 22112T^14 + 169316976T^12 + 529955760896T^10 + 647692282575936T^8 + 194072245946408960T^6 + 10526342701640695808T^4 + 173983901707951669248*T^2 + 473749289056190070784
$53$	\( (T^{8} + \cdots - 32106868908032)^{2} \) (T^8 - 56T^7 - 14680T^6 + 1010656T^5 + 46265104T^4 - 4688834560T^3 + 63325048832T^2 + 1752471560192*T - 32106868908032)^2
$59$	\( T^{16} + \cdots + 20\!\cdots\!36 \) T^16 + 34912T^14 + 504664176T^12 + 3918755621120T^10 + 17772719821560896T^8 + 48006466576568643584T^6 + 75216257619403843387392T^4 + 62376106084920758548037632*T^2 + 20954584989393198604522356736
$61$	\( T^{16} + \cdots + 21\!\cdots\!04 \) T^16 + 31824T^14 + 411909416T^12 + 2792048627936T^10 + 10636522502583576T^8 + 22694500972119676352T^6 + 25309510711823477914272T^4 + 12536004328843264574802560*T^2 + 2195585137593314484059033104
$67$	\( (T^{8} + \cdots + 4621952057344)^{2} \) (T^8 + 112T^7 - 12608T^6 - 1113600T^5 + 49065856T^4 + 1534588928T^3 - 31140655104T^2 - 477828513792*T + 4621952057344)^2
$71$	\( (T^{8} + \cdots + 18962560708864)^{2} \) (T^8 - 80T^7 - 19148T^6 + 794768T^5 + 131832156T^4 - 421799168T^3 - 262246802240T^2 - 4336623946240*T + 18962560708864)^2
$73$	\( T^{16} + \cdots + 60\!\cdots\!56 \) T^16 + 37520T^14 + 541804648T^12 + 3806191526240T^10 + 13674449418597016T^8 + 24579153039659132096T^6 + 21125831410908918370720T^4 + 7439669478760811986139264*T^2 + 601996143217456658116976656
$79$	\( (T^{8} + \cdots + 4233495998464)^{2} \) (T^8 - 152T^7 - 4424T^6 + 1068128T^5 + 8203536T^4 - 1817725184T^3 - 3879100928T^2 + 720748261376*T + 4233495998464)^2
$83$	\( T^{16} + \cdots + 11\!\cdots\!84 \) T^16 + 83264T^14 + 2725920704T^12 + 44225119870976T^10 + 365462005702411264T^8 + 1385701369401780797440T^6 + 1673808147723390778081280T^4 + 281854566814324884949172224*T^2 + 11136234550220679520907689984
$89$	\( T^{16} + \cdots + 15\!\cdots\!64 \) T^16 + 74824T^14 + 2072826288T^12 + 26074923366448T^10 + 144719678739472632T^8 + 282869965601171541472T^6 + 48061380865597288889792T^4 + 1877416558957916918342976*T^2 + 1528859638903204701819664
$97$	\( T^{16} + \cdots + 16\!\cdots\!96 \) T^16 + 63632T^14 + 1377447592T^12 + 13856471793760T^10 + 69309394967255064T^8 + 162847754943486101696T^6 + 146015940801054545446048T^4 + 29378030670084052784682112*T^2 + 1630911426721555402413089296
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