
// Make newform 8004.2.a.j in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8004_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_8004_2_a_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8004_2_a_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [3888, -8424, -67608, -17988, 151028, 65278, -122604, -47547, 46229, 14400, -8965, -2028, 932, 130, -49, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1436096932911927060552, -5734576008998611033216, -44010301598073058173068, -985493942779777495944, 87862981358437561842546, 12943701821822810175386, -49998222735113343315999, -6849714181952244815581, 12059212146105833185194, 1061787358987892121509, -1434850612960902643364, -39868293740417462800, 82428009307722432792, -2343590043301536281, -1780989647760093687, 128089526929863711], [10187796734063718568848, 4140098642180225184060, 2778605399574854232452, -17036408551707978644276, -39843524562808100614170, 63124294156284455139465, 31123533131595448591055, -38488851452268663931116, -9960991537155374353069, 8893034704374805484880, 1415048553601951450082, -963378769466844341132, -88974714684163101079, 49357290400200766149, 2000567955267874333, -958087900246403592], [62507843402239991683728, -119911680097050427614324, -436002491424203250589680, 373450758905607661475524, 631086737220708958112600, -374464283660322591966429, -348051881480837120172498, 160820231928040018855981, 91025874133171328629977, -34218304176590055106445, -11804652191031635223666, 3820388029937201339968, 721401882595358991197, -211861740633534736190, -16191524997021342828, 4486141389040236929], [-6638130556491638529144, 63961903110921104602836, 66867424884872498482256, -209073282062352857065304, -128266188808560877480848, 210029308309103838088587, 90582649286250348855860, -88888441699606672234641, -28530527789909789588287, 18290876230917575384385, 4159658356999503153722, -1963933514692243560836, -273011890888993264339, 105192616124931196248, 6394679216719266718, -2170353582856352493], [3287618491616138537532, -7863407220333139897548, -61721460041205402182268, 33444989655224619931886, 117837296847755457978037, -56119572043326164111031, -74930972648825634056871, 30526909733759416311410, 21665705786600716578792, -7391050976096937823363, -3015546321065278419096, 901332302367398689427, 193579150520849398669, -53334067423612189198, -4488671540266881279, 1181647930064972053], [-41744720482603951993704, 63732795840155393435880, 458996610472572057501996, -366980592422283677211496, -875606750046801999312014, 511728863085525731674458, 552656578883574074980929, -257042371188445739715205, -157147318825198786079022, 59301366750646584632885, 21480555688367312916876, -6950409487206069267520, -1357434095246951691440, 397851089569802580887, 31111659866916402345, -8599410738229651769], [75381322683826038993408, -188592356674960101516348, -734875152379202721919584, 501014971463436711617372, 1275883347371748259536286, -590218313888419955236599, -763778398499676259472184, 286804021630695488156393, 209053689477697038444789, -66526615038749754213403, -27920257611472853652366, 7937358719018169248978, 1741577168182876800871, -463149040304201162734, -39636043673251564446, 10166089516631326123], [24368330568281801524128, -58482275477158954969404, -290455416050807336719528, 206406504455500178785888, 530394989727383036599272, -237361474884041971348545, -315727483540994364481618, 111080659953374901792933, 84691399626259266906305, -25318495554936950579361, -11102524477032892698670, 3000960197217652805056, 683105279786040092525, -174748567588173763710, -15405971411407983416, 3833924860594975845], [18853135064703346138200, -186121404779846364836280, -270657506289607141947036, 516577122221256491326894, 513309222385173256168109, -494990400061097230473426, -335507138819282232469092, 210015545205224104585822, 99213383165647751861865, -44614067074995671927885, -13994381788245243532422, 5019613746164753586637, 904938036516128835734, -282138336758875451579, -21065591888740470630, 6055690973521494323], [7696294671096247442904, -36722337309588432352164, -118630530405786786024672, 131479917376481033077250, 230632231887950394137473, -151733582643722727860283, -143065840395346199349174, 69581428527495728527523, 39771223605111128968242, -15313513871554454343490, -5342141610410401144176, 1746467731155074504273, 333589527564189827935, -98318269336507859293, -7587807286478686338, 2103742732645796968], [60226142682101943655944, -431698814922534389321508, -1029539354898820273092228, 1205285884230039076834004, 1908106630392058166642008, -1234507192266499635469335, -1144928931914839786999539, 543132566223574961590886, 310045122176464699783689, -117349102552007926729696, -40821134148960918199182, 13220075965007561551802, 2511604996502047287625, -736811796288429578563, -56547423258025579917, 15643727938311237652], [83307899572874489830104, -292239039888885554789364, -1043599212580558264999584, 932234121437022270337808, 1916731753201095741369916, -1026586069984405062827043, -1190123853407848150127478, 478818628546944868872983, 334561429000218091255911, -109453059483503010262423, -45539064879024801708138, 13050836157168499238396, 2878724443289246873407, -765501985844103172942, -66100600868765845944, 16902750278691257515], [73093928798098161025416, -330829154134196740304148, -982373741782699986430740, 1105580567571758766871040, 1870018247056491778135546, -1274906221868961120611337, -1173217721169586183526145, 594566658421572741407942, 331680872425345273285917, -132199513307530952648122, -45130633610269192266030, 15152969014719198018848, 2842406557507315492831, -854944160891350069327, -64997796606289655715, 18310254666719483314], [-193169117447690193359208, 517658891675791265035020, 1969768145562148310520360, -1594204885835619991562096, -3591447951250514794518160, 1814874225627403113533529, 2164766731936135949379246, -848072989308638681641817, -588180818937536644168137, 191723502462768280879765, 77765457791517503993622, -22457115817186195403960, -4807355654391981949669, 1292569355600345279962, 108682923995574038268, -28101973564733554969]];
    Rf_basisdens := [1, 1, 605960599794441010008, 1817881799383323030024, 5453645398149969090072, 1817881799383323030024, 908940899691661515012, 5453645398149969090072, 5453645398149969090072, 1817881799383323030024, 2726822699074984545036, 908940899691661515012, 5453645398149969090072, 5453645398149969090072, 5453645398149969090072, 5453645398149969090072];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8004_a();"
function MakeCharacter_8004_a()
    N := 8004;
    order := 1;
    char_gens := [4003, 2669, 3133, 553];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_8004_a_Hecke(Kf)
    return MakeCharacter_8004_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 1, 0, -1, 1, 0, 0, -1, 1, 0, 1, -1, 0], [1, 0, 0, 0, 0, 1, -1, 1, 0, -1, 0, 0, 0, 0, 1, 0], [0, 1, 0, 1, -1, 0, 1, -1, -1, 0, 0, -1, 0, 0, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, -1, 0, 0, 0], [3, -1, 1, -1, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, -2, 0], [2, 0, 1, 0, 0, -1, 1, -2, 0, 0, 0, 0, 0, -1, 0, 0], [3, 1, 0, 1, -1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 0, -1, 0, 0, -1, 0, -1, -1, 0, 0, -1, 0, 0, -1], [2, 0, 0, 0, 0, -1, 1, 0, 1, 0, -1, 0, 0, 1, -1, 0], [1, -1, -1, 0, 0, 0, 0, 1, -1, 0, -1, -1, 0, 0, 0, 0], [0, 1, -1, 2, -2, 1, 1, 1, 0, 1, 0, -1, 1, -1, 2, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, 1, 0, 1, 0, 0], [1, -1, -1, 0, 0, 0, -2, 1, -1, -1, 0, 0, 0, 0, 0, 0], [2, 0, 0, 2, -2, 1, 2, -2, -1, 1, 2, -1, 1, -2, 2, 0], [1, -1, 0, 1, -1, 0, -1, -1, 1, 0, 1, 0, 0, -1, 1, 1], [-1, 0, 0, 0, 0, 1, 1, -1, -1, 1, 0, 0, 1, 0, 0, 0], [4, 1, 1, 0, 0, -2, 0, -2, 0, 0, 1, 0, 1, -2, -1, 0], [3, 0, 1, -1, 1, 0, -1, -1, -1, -1, 1, 1, -1, -1, 1, 1], [2, -1, 0, -1, -1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1], [1, 0, 1, -1, 1, 1, -2, 2, 1, -1, 1, 1, -1, 0, 2, 1], [4, -3, -1, -2, 2, -1, -1, 1, 1, -1, -1, 0, -1, 2, -2, 0], [2, -1, 1, 0, 0, -2, 3, -2, -1, 1, -1, 0, 2, 0, -2, 0], [0, 0, -1, 0, -1, -1, 1, 1, 0, 0, -2, 0, 1, 1, 0, -1], [0, 2, -1, 1, -1, 2, 1, 2, 0, 0, 0, 0, -1, 1, 2, -1], [4, -1, 1, 1, -1, -3, 4, -4, 0, 1, -1, 0, 1, -2, -1, 0], [-3, 2, 0, -1, 2, 0, -1, 1, 1, -1, -1, 1, -1, 2, -1, 0], [4, -1, 0, 0, -1, -2, 4, -1, 2, 2, 0, 0, 1, 0, -2, 1], [3, -1, 2, -2, 1, -1, -1, 0, 0, 1, 1, 1, 0, -1, -2, 1], [2, -2, 0, 0, 2, -2, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0], [1, 1, 1, 0, -2, -1, 2, -3, 0, 1, 0, -1, 0, -1, 0, 0], [4, -1, -2, 3, -3, 1, 2, 2, -1, 1, 1, -2, 0, -1, 3, -1], [3, 1, 1, -1, 0, 0, -1, 1, -1, 0, 1, 0, 0, 0, 1, 0], [2, 0, 1, -1, -1, 0, -2, -2, 1, -1, 1, 2, 0, 0, 1, 1], [3, -1, 1, -1, 1, 0, -1, -1, 1, -1, -1, 2, 0, 0, -2, 0], [-1, -2, -1, 2, -1, 1, 1, 1, 0, 0, -1, -1, 1, 0, 1, -1], [-2, 0, 1, 1, 1, 0, -1, -3, -1, 0, 0, 0, 0, -1, -1, -1], [0, -1, -2, 1, -1, 0, 0, 2, -1, 1, 0, -2, 1, 0, 1, -1], [2, -1, 0, 0, 1, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, -1], [-4, -4, 0, -1, 2, 0, -3, 1, 1, 0, 0, 0, 1, 1, -1, 0], [3, 0, 0, 2, -2, 0, 4, 0, 2, 2, -1, -1, 2, 1, -1, 0], [1, -2, 1, -2, 0, -1, 1, -2, -1, 1, -1, 1, 1, 2, -3, 0], [3, 1, 0, -2, -1, 0, -2, 3, 1, 0, -1, 1, 0, 3, -1, -1], [-4, 2, -1, 1, -1, 2, 3, 1, -1, 0, -1, 0, 0, 0, 2, -1], [2, -1, 0, 2, 2, 0, 0, 1, 1, -1, 1, -1, -1, -3, 1, 1], [1, -3, 0, -1, 3, 0, -3, 1, 0, 1, 1, 0, 0, -2, 0, 1], [4, -1, -1, 0, 1, 1, -1, -1, 1, 0, 1, 0, -1, -2, 3, 0], [-4, -1, -2, 1, 1, 3, -2, 5, 0, -1, 0, -1, -2, 0, 3, -1], [2, 2, 0, -1, 1, 1, 0, 2, -2, 0, -1, 0, -1, 1, 0, -1], [0, -1, 0, -1, 3, 2, -3, 4, 1, -1, -1, 1, -2, 3, -1, 1], [4, -3, 1, -1, 1, -1, -2, -1, 1, 1, 0, 2, 0, 1, -2, 0], [2, 0, 0, 0, -2, -2, 2, -1, 2, 2, 2, -1, 1, -2, 1, 0], [7, 2, -1, 3, -3, -1, 2, 0, 0, 1, 0, -2, 1, -3, 1, -1], [2, 1, 2, -1, 3, -1, 0, 1, 0, 0, 1, 1, -1, 0, 1, 1], [3, -4, 2, -2, 0, 1, -5, -1, 0, -1, 3, 1, -1, -1, 2, 0], [-2, -2, -1, -3, 3, 1, -3, 3, 0, -1, -1, 0, -2, 4, -2, 1], [-1, -1, 0, 0, 1, 1, -1, 0, -2, -1, 1, 1, 0, -1, 2, 1], [8, -1, 2, 0, -1, 0, -2, 1, 4, 0, 0, 4, 2, 1, 0, 1], [-3, -1, -1, 0, 0, 0, -2, 5, 1, 0, -2, 0, 1, 3, -1, 0], [-6, 1, -2, -1, 3, -1, -2, 4, 1, -1, -2, 0, -2, 4, -1, -1], [3, -3, 0, 1, -1, 2, 1, 0, -1, -1, -2, 0, 0, 0, -1, -1], [0, 1, 0, -1, 1, 0, -3, 2, 0, -1, -2, 0, -2, 2, 0, -1], [-1, 0, -1, 1, 1, 0, -1, 0, -2, -1, 0, 1, -1, 1, 1, -1], [-4, 1, 1, -2, 2, 1, -1, -1, 0, -3, -1, 2, -2, 0, 0, 0], [1, 1, -2, 3, -3, 3, 2, 2, 0, 0, 0, -1, 1, 0, 2, 1], [-5, 1, -1, -2, 0, 1, -3, 2, -1, -2, -2, 0, 0, 3, -3, -2], [-4, 1, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, 1, 4, -3, -1], [4, -2, 0, -2, 2, -3, -3, 1, 1, -1, -1, 1, 1, 1, -3, 0], [-1, -3, 1, 0, 0, 3, -1, -3, -2, -2, 4, -1, -2, -3, 2, 0], [6, 2, 1, -1, -3, -1, 1, -4, -1, 3, 3, -1, 0, -2, 1, 1], [-7, -1, -1, 2, -2, 2, -2, -1, 0, 1, 0, -2, 2, -1, 2, 0], [9, 1, 0, 1, -1, -2, 3, -3, -1, 0, 1, -1, 1, -3, -2, 1], [2, -2, 0, -1, 2, 0, -1, 3, 1, -1, 0, 0, -2, 2, 0, 0], [-4, 0, 0, -2, 0, -2, 0, -5, 0, 0, 1, 0, 0, -1, -1, 2], [6, -1, 2, 0, -2, 1, 1, -2, 1, -3, 1, 1, 2, -1, -1, 1], [1, 1, 0, 1, 1, -1, -2, 0, -2, 0, 1, -2, 0, -1, 1, -1], [2, -2, 1, -2, 2, -2, 2, -2, -1, 0, -2, 0, 1, 3, -6, 1], [4, -2, 1, -2, 3, 0, 0, 2, 2, -1, 0, 2, -2, 1, -1, 1], [1, -2, -1, 1, -5, -1, 3, -3, -3, 3, -1, -2, 2, 2, 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1, -4, -2, 3, -1, -1, 1, 0, -3, 2], [6, 6, 0, -4, -4, -2, 0, 2, 3, -1, -3, 1, -1, 4, -1, 0], [-9, -3, -2, -3, 5, -1, -3, 1, 1, -3, 0, 0, -1, 0, -2, 1], [7, -2, 4, 2, -4, -4, 6, -11, 2, 1, 2, 5, 3, -3, -1, 4], [-3, -5, -1, 1, 2, 2, -3, 2, 1, 1, -1, -1, -1, 2, -1, 2], [-8, 5, -1, 0, 0, 0, 6, 1, -4, -2, 0, -4, -5, -1, 0, -2], [0, -11, 2, -3, 3, -3, 5, -1, 2, 0, -3, 3, 2, 1, -3, 3], [5, 3, -1, 4, 4, 1, 1, 2, -1, 0, 1, -1, -1, -2, -1, 0], [8, -4, -2, 4, -8, 0, 6, -6, 0, 4, 2, 0, 2, -4, 2, 2], [6, 0, 0, 0, 0, 4, -2, 5, -2, -3, 2, -1, -1, 0, 2, -2], [2, 1, 1, 0, 0, -5, -3, 2, 4, 0, 1, 1, -1, -2, 0, 2], [-4, 2, -5, 4, -5, -2, 6, -4, -8, 3, -1, -5, 4, -3, 1, -3], [-5, -2, -4, 4, -4, 6, -8, 7, -4, 1, 2, -1, 0, -2, 5, -2], [4, 3, -1, 2, -4, -3, 7, -3, 2, 5, -3, 0, 2, -2, 0, 2], [-1, -3, 0, -1, 3, 4, -5, 5, 1, -7, 2, 1, -3, 2, 0, -3], [1, 0, -1, 0, 0, 0, 2, 2, -3, 2, 1, 0, -1, -2, 3, -2], [-4, -1, -4, -1, 1, 0, 1, 7, 0, 2, -3, -3, -3, 1, 1, -1], [6, 3, 5, -3, 1, -4, 3, -11, -3, 2, 3, 1, 3, -6, -3, 1], [-7, 1, -5, 2, -4, 2, -2, 5, 4, 0, -2, -2, 0, 3, 3, 0], [3, -1, 0, 3, 1, 0, -1, 3, -2, -1, -1, -3, -2, -1, 3, 1], [-12, 5, 1, 3, -4, 0, 3, 4, 2, 1, 1, 1, 2, -1, 4, 0], [2, 4, -3, 3, -3, 1, 2, 6, 2, -1, 0, -4, 0, 1, 1, -3], [4, -10, -1, -4, 5, -1, -5, 3, 0, 1, -4, 0, 1, 1, -7, 3], [-6, 3, -1, 0, 0, 2, -4, 2, -1, 0, 2, -1, -1, 1, 3, -4], [14, 5, 0, -7, 3, -1, 0, 1, 3, -4, -5, 4, -4, 7, -3, -1], [-10, 2, 1, -5, 5, 3, -10, 1, -6, -5, 3, 1, -5, 0, -1, -1], [2, 4, -2, 1, -4, 5, 6, 4, 0, 1, -1, 1, -3, 3, 2, -2], [-6, 2, -1, -1, -1, 4, -3, -1, -4, -1, 1, -1, -4, -2, 5, -1], [-14, -4, -1, 0, -1, 1, -7, -1, -3, -3, 2, 0, 1, -4, 3, -1], [6, -2, 2, 1, -4, 3, 4, 0, 0, 3, 1, -1, 1, -1, 2, 0], [2, 3, 3, -1, 1, 2, -1, 4, 3, 1, 4, 2, -1, -1, 2, 2], [-2, -8, -2, 0, 0, 3, -3, 6, 3, 2, -1, 0, 2, -1, 3, 2], [8, 4, -1, 8, -7, -1, 1, 2, 3, 2, -3, 0, 2, 0, 3, -3], [4, 0, -3, 1, -3, 1, 0, 0, -3, 2, 3, -3, -4, -2, 4, 1], [-16, -1, 3, -4, 2, -2, 0, -5, -1, 0, 2, 1, 1, 1, -2, 0], [-4, -6, 1, 1, -1, -6, 7, -3, 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11, -7, 2, 4, -1, 3, 3, -2, 1, -1], [-19, 2, 1, -2, 3, 2, -8, 6, -5, -2, 2, -2, -2, -1, -1, -3], [21, 5, -2, -3, 3, 0, -1, 12, -5, 2, -6, 2, -4, 6, -7, -3], [19, -10, 2, -2, 2, -6, -5, 1, 1, 1, 1, 2, 2, -5, -7, 4], [5, -8, -1, -1, 4, 7, -10, 12, 2, -5, -1, 1, -4, 6, 5, -1], [12, 9, 0, 1, -3, 5, 6, 6, -1, 2, 0, -5, -1, 4, 4, 1], [-8, 3, -1, 1, 1, 0, 3, 6, 0, 3, -3, 0, -1, 0, -3, 3], [-13, 6, 1, 3, -3, 1, 11, -7, -3, -1, 0, -1, 3, -3, 1, 1], [-8, 2, -4, 1, -3, 0, -3, 3, -9, 3, -4, -9, -2, 2, 1, -7], [-3, 2, 3, -8, 5, 0, -2, 15, -1, 1, -1, 4, -2, 2, -3, 3], [2, -10, -1, -1, 3, -1, 4, -2, -4, -1, 0, -2, 6, 1, -9, 1], [0, -6, -4, 2, -2, 1, -6, 3, -1, 0, 3, 3, 1, -1, 6, 2], [-3, -7, 5, -1, -5, 2, -1, -5, 5, -3, 5, 2, 4, 0, -6, 2], [-1, -4, 6, -4, 2, -4, -4, -6, 1, 1, 3, 7, 0, 2, -4, 0], [-11, -2, -7, 9, -9, 0, 7, -3, 1, 5, 3, -2, 5, -2, 4, 3], [-23, 1, -1, -2, -1, 1, 1, 3, 2, -3, -5, 4, 1, 2, -1, 2], [-6, 5, -6, 1, -3, 4, 1, 7, 2, 2, -2, -1, -3, 6, 4, -3], [-6, 2, 0, 2, -2, 2, 2, 7, -1, 1, 4, -1, 0, -4, 4, -2], [-14, 3, 0, -1, 1, -4, 3, -5, 7, 1, -2, 1, -2, 0, 0, -1], [8, 1, -3, -2, 2, -1, 4, -8, -4, -4, -3, 1, 1, -2, -5, -2], [12, -1, 2, -4, -2, -2, -6, -6, 0, 2, 6, 0, 0, -2, -3, 2], [-6, -5, -4, 5, -3, 3, -6, -4, -2, 2, 1, -4, 3, -5, 4, 3], [-4, -4, -2, -4, 2, 2, -2, 5, 0, -6, 2, 1, 1, -4, 3, 0], [11, -1, 2, -9, 11, -1, 0, 1, 3, -3, -4, 6, -4, 7, -5, 3], [-8, 12, -3, 4, -6, -1, 13, -4, -4, 5, -2, -2, -2, 1, -2, -4], [0, -9, 2, 1, -1, -5, -4, -9, 3, 3, 3, 1, 9, -3, -4, 1], [-5, 9, -1, 8, -6, -2, 4, -3, -1, 4, 7, 0, 1, -8, 6, 0], [10, -4, -4, 4, -4, -4, 6, 0, 5, -1, 0, 0, 3, 2, 0, 0], [-11, 3, -4, -4, 2, 7, -1, 13, -1, -1, 3, 3, -6, 5, 2, 4], [-3, 12, -4, 2, 2, 6, 2, 5, -6, -3, -4, -1, -6, -2, 7, -4], [2, -11, 3, -7, 5, -5, -6, -8, 0, -5, -1, 4, -5, 4, -5, 2], [3, 6, -1, 6, -6, 1, 5, -5, -1, 3, 0, -8, 1, -1, 0, -7], [5, 2, 1, 7, -9, 1, 6, -3, -5, 2, 0, -5, 5, -3, 2, 1], [15, 0, 4, -2, 2, -2, 5, 1, 3, -2, -4, 3, 5, 8, -8, -2], [-2, 10, -1, 1, 3, -3, -4, 1, -4, 2, 2, -3, 1, -1, -2, -1], [-12, 7, -1, -3, 4, -3, 8, 0, 2, -2, -7, 0, -1, 4, -4, 0], [-9, -7, -3, 0, -2, -2, 2, -2, 5, 6, 1, -2, -1, -2, 4, 4], [9, -7, 2, 1, 3, -1, 4, -14, -4, -2, -1, 2, 1, 0, -5, -1], [-9, 7, -4, 5, -3, 9, 2, 6, -5, -2, -3, -2, -5, -1, 10, -3], [-2, -3, 5, -4, 5, 4, 0, -8, 5, -5, 2, 3, 3, -1, 2, 4], [0, 8, 0, -1, -2, 0, 7, -6, -5, 2, 3, -2, -3, 2, -2, -4], [-5, -2, -3, -1, -1, 2, -3, 2, 2, -1, -3, 0, -2, 6, 0, 1], [15, -6, 3, -7, 3, 2, 3, 4, -1, 0, 1, 2, 1, 0, 1, 3], [4, -7, 4, -11, 13, -7, -4, 7, 0, -1, -6, 5, -4, 4, -5, 1], [5, 3, 1, -2, 2, -3, -2, -1, 1, -8, -3, -2, 2, 4, -5, -4], [24, 0, 0, 6, -6, -2, 8, -12, -4, -1, 4, -2, 8, -8, -3, 4], [-17, -4, 0, 2, -8, 0, 4, 0, 4, 6, -1, 3, 2, 3, -3, 2], [6, 7, 2, -4, 7, 1, 3, 3, -5, -3, -1, 2, -1, 5, 0, -1], [-14, -1, -4, 3, 3, 3, -6, -1, -3, -3, -5, -1, 3, 5, -3, -5], [-3, -7, 7, -10, 0, -3, -4, 0, -4, 0, -2, 0, 1, 3, -9, -2], [-3, -2, -5, 10, -8, 7, 3, 8, 0, 3, 1, -5, 2, 0, 8, -6], [3, 2, 0, -4, 8, 6, -6, 5, 0, -2, -3, 3, -2, 7, -4, 2], [12, 0, -4, 4, 0, -5, -1, -2, 0, 1, 2, -7, 2, -10, 6, 0], [-9, -1, -4, 3, -1, 0, 5, 3, 7, -2, -6, 3, 1, 2, 1, 3], [-15, -1, -1, 3, 2, 8, -5, 11, 2, -2, 3, 2, -6, 3, 5, 0], [18, 6, -3, -3, 3, 5, -14, 8, -1, -4, 4, -2, -7, -1, 9, -1], [6, 12, 0, 12, -12, 5, 13, -7, -2, 0, 6, -4, 1, -12, 12, -2], [27, 6, 6, -6, 8, -7, -3, -5, 4, -4, -1, 5, -5, 1, -5, -4], [7, 1, 3, -6, 14, -5, -5, -1, -3, -4, 2, 1, -9, 1, -6, -2], [3, 1, -3, 0, -2, -1, -5, -1, -4, -2, -4, -1, 2, 5, -4, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8004_2_a_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_8004_2_a_j(:prec:=16)
    chi := MakeCharacter_8004_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8004_2_a_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_8004_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3888, -8424, -67608, -17988, 151028, 65278, -122604, -47547, 46229, 14400, -8965, -2028, 932, 130, -49, -3, 1]>,<7,R![-7776, 49248, 4752, -349536, 19952, 664560, 2976, -365696, 34702, 78570, -12919, -7172, 1462, 283, -65, -4, 1]>],Snew);
    return Vf;
end function;