
// Make newform 671.2.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_671_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_671_2_a_c();".
// 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_671_2_a_c();".
// 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 := [-43, 643, 241, -6404, 1674, 21077, -11589, -28019, 20141, 16875, -15053, -4673, 5755, 387, -1177, 78, 122, -18, -5, 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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7406641729547, 50083912844543, -132758950019250, -300469578965604, 431277028203763, 474088918297633, -514007340828783, -283316198675975, 280962923414937, 61181608853165, -74752830871939, 4170318228153, 7816695786848, -3896315485489, 420015084215, 585160018840, -153963852608, -28397403169, 8867201804], [4643140412416, 133177244880778, 121134977398858, -794326099908997, -761921796525769, 1686914951359080, 1352625850873237, -1744313483302286, -1042517331908497, 976030566478054, 401624582318481, -311772296964757, -79490940840104, 56715822193005, 7280102119263, -5438380518759, -157290256641, 211585433401, -9925308116], [1737797285981, -47424793625471, 64006649549778, 372817093364514, -360901712332219, -777549565519077, 625158766804376, 692134937669964, -488697894791415, -323580558431117, 204922845192456, 86061802478689, -49478643130019, -13113396066991, 6910095072255, 1065729788540, -519467802678, -35753207242, 16270525372], [-2002160866471, 114678199755666, -12717902390185, -424688104742192, 256243772589666, 262742744285573, -483092267921848, 273746173403669, 311955896020022, -372792814215164, -72597790484259, 168673178705194, -4098672552276, -37022753494585, 4711323004499, 3973232345087, -747153495518, -165903350608, 37762226457], [-165641941058, -19813780702, 237228681118952, 179493755356720, -1222257229443266, -311958245977134, 2128704632011736, 119332311754315, -1708211052099464, 41070927048001, 725805637025464, -42177686787597, -172638105605592, 12033133930515, 22839785265251, -1494342049450, -1546829355593, 68688107954, 41025764486], [-246530814813, 23059854361824, -27572301495516, -272000230119563, 222408147075718, 689021693044756, -477531549805097, -743045495285101, 464054899866586, 427053290996220, -242182034413731, -139981328381100, 72759363883607, 26056387605249, -12599484691005, -2544906190927, 1163734380318, 100366607650, -44058942396], [8443491451865, -127878466080227, -236302917332312, 778848775823786, 850923535863497, -1317806710310478, -1236732975882466, 935542592921969, 949735366352185, -308367411917029, -423076214974525, 41171189048558, 113473718291916, 888172948827, -18089060782894, -711542003465, 1573939504037, 46120188886, -57136613064], [-25082161671395, 144837697384176, 587551380088491, -1114097846357755, -2559452836437059, 3081313386395895, 3918184841562988, -3851875830463548, -2686226489462764, 2447482008098315, 907306072440698, -851967355105863, -143000610338671, 164287162414465, 5490595341552, -16384821425649, 970372146851, 654428683551, -81254538506], [-5331850884506, 76490070141423, 94000023879346, -557432190432943, -249311959857695, 1394627423083597, 134975892121205, -1669129966062270, 148090368081498, 1054469865039305, -196989043393835, -369628093368947, 89628292185599, 71928744979165, -19906864553003, -7229272850781, 2163987038842, 290362672773, -91424960248], [-8426703234447, 3421690058101, -54746227366930, 170395081372216, 635720108720707, -1082982581796351, -1141430209449476, 1883801315016804, 700069942254179, -1408591428559113, -130776269174957, 538405030181766, -28689902332059, -110076102275048, 15508655859118, 11406225381357, -2253974970877, -467343546422, 110220246484], [-992900881709, 72688955170899, -170724575819309, -237488214060000, 866457444779580, -354335113278947, -1243226926847429, 1250741201795975, 652563422729939, -1103682120823352, -80531592282880, 452239469595452, -44248321607856, -95838102406759, 17708092485247, 10136775407328, -2391672731682, -420512816139, 112916661333], [-6942619491638, -131669286419725, -195065901784280, 957988530198262, 1136718336996770, -2591704634942325, -1683719333735166, 3296195916484018, 934401836140697, -2143032129859066, -157363017924653, 759359884142781, -39968403213030, -148126283460976, 19764923687112, 14870979403509, -2794615480726, -595936463557, 134524452259], [2625408551097, -29439287899993, -296741969124174, 490128717085397, 1541694664772399, -2308864017778112, -2136542032244486, 3844359687237313, 965965174706451, -2873186468193146, 23829240400472, 1105726169117787, -147385979907443, -227625432361619, 46258916319239, 23718172730414, -5844382212784, -975838766429, 268063867548], [-8519539396534, 99943970595050, 279954024772659, -1122823610741571, -1430231827654368, 4173706077317173, 1859899256299250, -6212724005068246, -537601793521600, 4379858371730078, -350590014826373, -1627427645433792, 276164257045428, 327363575480166, -73247230474246, -33566135421448, 8692512067202, 1365268186173, -386309035457], [-7512666382563, 75002246004675, -76416261707646, -877910452999009, 818360171228219, 3162740167704228, -2216000497099625, -4586218801226622, 2656196462753142, 3205416930871435, -1630922731137518, -1188615976491545, 553369806506455, 239031289075534, -104725686151822, -24506168703039, 10303884149056, 996406897672, -407596389827], [7601132183693, -115787993029379, -97910802478304, 1308086020776475, 270534203277648, -4402846506872819, 239835296100884, 6222910401208218, -1147835903097151, -4290881606498135, 1065197051025780, 1578531396957927, -447303281907149, -315991261600359, 96395574289456, 32317494993362, -10331601737881, -1312562946260, 433257329190]];
    Rf_basisdens := [1, 1, 1, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763, 2584793613763];
    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_671_a();"
function MakeCharacter_671_a()
    N := 671;
    order := 1;
    char_gens := [123, 551];
    v := [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_671_a_Hecke(Kf)
    return MakeCharacter_671_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, 1, 0, 0, 0, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 0, 0, -1, 1, -1, 0, 0, 0, -1, -1, -1, 0, -1, -1, 0, 0], [0, 0, 1, 0, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0], [2, -2, -1, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1], [-1, 1, 0, 0, 1, 0, 0, 0, 0, -1, -1, 0, 0, -1, 1, -1, 0, -1, 0], [1, 0, 0, -1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0], [0, 0, 0, -1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, -1, -1, -1], [0, 0, -1, 2, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, -1, 1, 0], [3, -2, 1, -1, 1, 0, -1, 1, 1, -1, -1, 2, 1, 0, 1, 0, 2, 0, 0], [2, -1, 0, 0, 1, -1, 0, 0, 0, -2, -1, 0, 0, 0, 1, -1, 1, 1, 1], [-3, 2, -1, 1, -1, -1, -1, 0, 0, -1, -1, 1, 0, -1, 0, 0, -1, -2, -1], [1, -1, 0, -1, 1, 1, -1, -1, 0, 1, 0, -1, -1, 0, 0, -1, -1, 0, 1], [0, -4, -1, 0, 1, -1, -1, 1, 0, 1, -1, 1, 1, 0, -1, 0, 1, 1, 2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 1, 1, 0, -1, 1, -1, -1, -1, 0, 0, -1, -2, 2, -2, 0, -1, 1], [0, -3, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 1, -1, 0, -2, 1, 0, 2], [2, -3, 2, 2, 1, 0, -1, 1, -1, -1, -2, 2, 3, -2, 2, -1, 2, -1, 1], [3, -1, 0, 1, -1, 1, -1, -1, 0, 1, 0, 2, 1, 0, -2, 1, -1, -2, -1], [-1, 0, 0, 3, -2, 0, -1, 1, -2, -1, -1, 1, 1, -2, -1, 1, 1, 1, -2], [0, -1, 2, 0, 1, -1, 0, 2, 0, 0, -1, 0, 2, 0, -1, 2, 2, 1, -1], [3, 1, 1, 2, 0, 3, 0, -1, -1, 2, 0, 1, 1, 0, 0, 2, 1, 0, 0], [1, 1, 2, -3, -1, -1, 3, 1, 2, -1, 0, -1, -1, 0, 0, 1, 2, 0, -1], [0, -1, 2, 3, -1, -1, 1, 1, 0, -1, -1, 0, 1, -1, -2, 1, 1, 1, -1], [2, 0, -1, -1, 1, 1, -1, -1, 1, 2, 1, 2, -1, 3, -1, 1, -2, 0, -1], [1, 2, 2, 2, -2, 1, 2, 0, 0, -1, 2, -2, -2, -1, 0, -1, 0, 0, -1], [0, -1, 0, 4, -1, 0, 0, 1, 0, -1, 0, 0, 3, -1, -2, 0, 2, 1, 1], [2, 0, 2, 3, 2, 0, -2, 0, -1, -3, -2, -1, 2, -3, 2, -1, 1, -1, 1], [0, -3, 2, 4, -1, 0, -1, 1, -1, -1, -2, 2, 3, 0, -2, 1, 2, 1, -1], [1, 1, 1, 1, 0, 0, 1, 0, 0, -1, 0, -3, 0, 1, -1, 0, 1, 3, 2], [3, 1, -1, 2, 2, 1, -2, 0, -1, 2, 0, 0, 2, 1, -1, 1, 1, 1, 1], [2, -4, -2, -3, 2, 0, -2, 0, 1, 3, 0, -1, 0, 1, 0, -1, -1, -1, 1], [5, 0, 1, -2, -1, 0, 0, 1, 0, -1, 0, 2, -1, 2, 1, 2, 0, 1, 0], [1, 3, -2, -1, 1, -1, 3, -3, 2, 1, 2, -3, -3, 0, 0, -1, -3, 0, 1], [-2, 1, 1, -1, 2, 1, -2, 0, -1, -2, -1, 5, 2, 0, 3, -1, 1, -2, 1], [0, 1, -3, -1, 2, 1, -2, 0, 0, 1, 1, -2, -2, 1, 1, -2, -2, -1, 2], [1, -4, 1, 2, 1, 0, -1, -1, -2, 0, -1, 0, 3, -3, -1, -1, 1, -1, 1], [-3, -1, 0, 1, 1, -1, -1, 1, -2, -1, -2, -1, 3, -2, 2, -1, 0, -2, -1], [-2, 1, 3, -2, -2, -1, 0, 2, -1, -4, -1, -1, -2, 0, 1, -1, -1, 0, -3], [-2, 0, 3, -1, 1, 2, 2, -2, 0, 1, 0, 2, 0, -1, 2, 0, 2, -2, 1], [5, -1, -1, 2, -2, -2, 1, 2, -1, -2, -2, 0, 2, 0, -1, 3, 2, 0, -1], [-1, 1, 3, -1, 0, 1, -1, 1, -1, 1, 0, 0, 1, 0, 0, 2, 0, -1, -3], [0, -1, 1, 0, 0, 0, -1, 1, 1, 0, 0, 4, 5, 1, 0, 1, 2, 1, 0], [5, 1, 0, 0, 1, 2, 1, -2, -1, 1, 1, 2, 2, 0, -1, 2, 2, -1, 0], [1, -2, -2, -5, 0, -1, 2, 0, 3, 4, 2, -3, -2, 2, -2, 0, -1, 1, 0], [3, 0, 0, 0, 2, 3, -2, -2, -1, 2, 2, 1, 0, 2, 0, 2, -1, 1, 0], [5, 3, 1, 2, 0, 2, 0, 0, -1, -1, 1, -2, 0, -1, 1, 0, 1, 0, 2], [8, -4, 2, 2, 2, 4, -3, 0, 0, 1, -1, 5, 4, 1, 0, 1, 3, 0, 1], [-1, 0, 2, 0, 0, -2, 1, 1, 1, 0, -3, 4, 1, -1, 1, 0, 0, -4, -1], [-1, 1, 1, -3, -2, -2, 0, 0, 0, 0, 1, 0, -2, 0, 1, 1, -4, -3, -3], [0, -2, -3, 1, -1, 0, 0, -2, 2, 3, 2, 0, -2, -1, -4, 0, 0, 0, 1], [0, -4, 0, 1, 0, 1, -1, 1, -2, 2, -1, 2, 3, 1, -1, 2, 1, -1, 1], [0, 0, 0, 2, 2, 1, -1, -1, -1, -1, 1, -4, 1, 2, 1, -3, 0, 2, 4], [6, -2, -3, -4, 3, 4, 1, -4, 4, 4, 2, 1, -2, 3, -2, 0, 3, 1, 4], [-2, 0, 1, -2, 1, 2, 0, 0, 1, 0, 0, 1, 0, 2, -2, 1, -1, -1, 0], [-3, 6, 1, 2, -3, 2, 2, -1, -1, 0, -1, 1, -1, -4, 1, 0, -3, -5, -3], [1, 1, 2, 2, -3, -1, 3, -1, -1, -2, 0, -4, -3, -5, 2, -2, -1, -1, -2], [-4, 2, 2, 1, 0, -1, -1, 5, -2, -2, -1, 1, 3, -1, 1, 0, 1, 1, -1], [-1, -1, 0, -2, 1, 0, 0, 2, 0, -3, -3, 4, 2, -1, 1, 1, 2, -1, 0], [4, -4, -2, -3, 2, 0, -3, 0, 3, 0, -1, 4, 0, 2, 0, 0, 2, -1, 2], [-3, 2, -5, -2, -1, 1, -2, 0, 1, 3, 2, -1, -4, 3, -2, 0, -3, 3, -1], [6, -3, 2, 0, 3, 0, -1, -1, 1, -1, 0, 2, 3, 0, 2, 1, 2, -1, 1], [-5, -1, 3, 1, 2, 1, -2, 0, 0, -1, -2, 1, 2, -2, -1, 0, 0, 0, 0], [4, -9, 1, 2, 2, 0, -3, 5, -2, -5, -6, 1, 5, 0, 0, 0, 5, 2, 1], [6, 0, -1, 0, 1, 2, 1, 1, 2, 4, 2, 0, -1, 2, -3, 1, 3, 2, 1], [-3, 3, -1, -2, -2, 2, 0, -2, 1, 3, 5, -3, -4, 5, -3, 0, -3, 4, -2], [4, 1, 0, 0, -3, -5, 3, 2, -1, -4, -1, -2, 0, -3, -1, -1, 4, 1, -1], [1, -4, 3, -1, 1, 1, 2, 0, 2, 0, -2, 2, 0, -2, -2, 1, 1, -4, 0], [2, 0, -1, 3, -1, -1, 0, 1, -2, -4, -5, 0, 3, -2, 1, -2, 2, 0, -1], [-3, 2, -5, -1, -1, 0, 1, -1, -1, 1, 1, 2, -1, 0, 1, 1, -5, -2, 0], [5, -1, -5, 0, 2, 2, 0, -2, -1, 1, -1, 5, 0, 1, 1, 1, 0, 0, 0], [1, 2, -1, 4, -1, 4, -2, -1, -2, 3, 0, 2, 5, 0, -3, 4, 2, 1, 0], [-2, 1, 4, 7, -3, 0, -3, 3, -2, -4, 0, 3, 5, -1, 0, 0, 3, 2, -4], [4, -3, 0, 0, 1, 4, -7, 1, 1, 1, 2, 6, 3, 4, 0, 1, -2, -1, 1], [-7, 0, 3, -2, -3, -5, 6, 0, -1, -1, 2, -7, -4, -5, 2, -4, -1, -1, -3], [5, 0, -7, -5, 3, -1, 4, -6, 2, 4, 4, -2, -6, 4, 0, 1, -4, 2, 2], [8, 5, -1, -2, 2, 2, 0, 0, 1, 1, 2, -2, -4, 2, 3, -4, -1, -1, 2], [-12, 1, -1, -2, 0, -2, -3, 1, 0, -3, -2, 3, -1, 0, 2, -2, -3, -2, -1], [8, -1, -1, 5, 2, 0, -3, 1, -1, 2, -2, 4, 3, 1, 0, 1, -2, 1, 0], [4, -1, -1, -3, 2, 0, -1, -1, 1, 0, -2, -4, -5, -1, 0, -3, -2, 1, 4], [6, -3, 0, -1, 1, -2, 1, 1, -2, -2, 2, -3, 1, 3, 0, 2, 3, 4, 0], [7, 1, 0, -3, 1, 8, -4, 0, 1, 6, 5, 3, -2, 10, -1, 4, -3, 0, 1], [1, 2, -3, -6, 3, 4, 1, -3, 4, 6, 5, -1, -3, 5, -5, 2, 0, 1, 1], [-8, 3, -4, 0, -3, -5, 2, 2, 2, -2, -1, 3, -2, -2, 1, -2, -2, -1, -1], [4, 1, -1, 3, -2, -4, -1, 3, -1, -4, -2, -2, -1, -3, 2, -1, -2, 1, -2], [-3, 6, 1, -2, -1, 2, 7, -3, 2, 2, 5, -5, -3, 1, -1, -1, -1, 5, 1], [4, 0, 6, 4, 0, 2, -1, 0, 0, -1, 1, -1, 4, -1, 0, 1, 3, -2, 3], [-5, -4, 2, -1, -2, -2, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -1, -2], [2, -5, 1, -6, 2, 3, -2, 2, 3, 2, -3, 5, 0, 6, 1, 5, 3, 0, 1], [-4, -1, 3, -3, 0, 0, -1, 3, -2, -3, -4, -3, -1, 2, 2, 2, -1, 0, -3], [0, -6, -2, 5, 2, 0, -2, 0, -3, -1, -2, -1, 4, -1, 4, -5, -3, -1, 3], [1, 4, 1, -1, 1, 1, 2, 0, 2, 2, 0, 2, 0, -2, -2, 3, 2, 0, -2], [7, -4, 3, -1, 1, 0, -5, 1, 1, 1, 3, -4, 1, 2, 1, -2, 0, 0, 2], [5, -4, 0, -4, 0, 6, -1, 3, 1, 2, 1, 2, -1, 5, 1, 0, 0, 0, 1], [3, -4, 1, -2, 3, 2, -1, -1, 2, 4, 1, -1, -1, 3, -3, 1, -1, 1, 1], [-6, -1, -2, 1, -3, -3, 0, 0, -1, -1, 3, -6, -2, 3, -1, -1, -5, 4, -2], [2, -1, -1, -1, -2, -2, -1, 3, 0, -3, 0, -5, -1, 2, 2, -4, -3, 2, -1], [3, 0, 5, 0, 1, -2, 1, 1, -1, -5, -1, 2, 3, -4, 1, 2, 6, 0, 0], [-4, 0, -3, -1, -1, 2, 1, -4, 0, 6, 3, -5, -6, 2, -4, -1, -3, 2, 0], [-6, 8, -2, -4, -6, -3, 7, -1, 3, -3, 5, -6, -7, 0, -1, 3, -2, 2, -2], [12, -5, -1, 4, 2, 2, -3, -1, -4, -1, -2, 1, 1, 2, 4, 0, -3, 0, -1], [-2, 2, -6, -3, -2, 1, 1, 1, 0, 4, 1, -1, -3, 1, -3, 6, -3, -3, -3], [4, 0, 3, 2, 3, -1, -3, -1, 0, -1, -1, 7, 5, 0, -1, 4, 3, 1, -2], [-14, 1, 3, 1, -6, -2, 5, 3, -3, -2, 4, -8, -3, -1, -2, 1, -2, 5, -2], [2, -2, -2, -1, 0, -1, -1, -1, 0, 0, -1, 3, -3, 1, -1, 2, -3, -1, -1], [-13, -1, -2, 1, -3, -2, 0, -2, -1, -2, -1, 1, 0, -2, 1, 0, -1, -4, -1], [5, -7, -3, -10, 4, 3, -1, -1, 4, 6, 0, 3, -3, 9, -2, 3, -1, 0, 2], [-10, -1, 2, 0, -1, -1, -2, 4, -2, -2, -3, 2, 2, -2, 5, -4, -2, -3, 3], [-4, 1, 3, 5, -4, -5, 0, 6, -2, -7, -5, -2, 2, -3, 1, 0, 0, 1, -6], [-10, 2, 0, -2, -4, -2, 6, 0, 0, -2, 2, 2, 0, -2, 2, 0, -2, -4, -4], [-1, -4, 0, -4, -2, 0, -2, 2, 2, 2, 1, 3, 2, 5, 0, 2, 0, -1, -2], [15, -4, -6, -6, 2, 4, -5, -3, 3, 6, 1, 4, -3, 7, -3, 4, 2, -2, 1], [-3, -5, -3, -9, 6, -3, -3, 1, 3, -1, -4, 2, -1, 4, 4, 0, -2, -1, 1], [-15, 5, 0, 3, 1, -3, -1, -1, 0, -3, -6, -1, -1, -8, 2, -5, -3, -2, 1], [-6, -5, 4, 4, 3, 0, -3, 1, -1, -3, -4, 2, 5, 0, 2, -3, 2, -1, 1], [-11, -1, 0, -1, -3, 0, 0, 0, -1, 0, 3, 1, 0, 2, -5, 4, -1, 2, -3], [-2, 0, 1, 5, -1, -4, 0, 2, -4, -3, -4, -2, 0, -9, 2, -4, 0, -2, 1], [12, -3, 0, -3, 3, 1, -2, 2, 3, -5, -3, -1, 2, 3, 1, -1, 5, 0, 2], [10, -9, 3, 5, 6, 2, -1, -1, -4, -3, -4, 3, 7, -6, 2, 0, 5, -2, 7], [-3, -2, 6, -8, -2, -4, 1, 5, 4, -3, 1, 1, 1, 6, -3, 5, 3, 3, -2], [-1, -2, 0, 6, -2, -3, 3, -3, -3, -3, -2, -1, -1, -5, 3, -1, -2, -3, -2], [0, 9, -3, -4, -2, 2, 5, -3, 4, 1, 4, -3, -5, 2, 0, 1, 2, -2, -1], [7, -3, -4, -3, -1, 2, 0, 0, 3, 2, 7, -4, 0, 6, -1, 0, -1, 4, -1], [15, 0, 7, -5, 1, -1, 4, 0, 4, -2, 2, -2, -4, 2, 2, -3, 4, 2, 0], [-4, -1, -4, 4, 1, -2, 2, -3, 0, -1, 0, -4, -5, -1, 0, -4, -6, 5, 1], [10, 1, -5, -8, 0, 3, 2, -4, 3, 4, 5, -3, -8, 6, -1, 3, -2, 0, 3], [-2, 4, -1, 1, -1, 3, -1, -3, -1, 2, 1, 8, -1, 3, -1, 1, -2, 2, -3], [0, 5, -5, -2, -4, -2, 5, -1, 2, 3, 4, -6, -7, -2, -6, 0, -1, 4, -3], [-2, 4, -1, -3, 3, -1, 3, -3, 1, 0, 1, -3, -1, -3, 3, -5, 0, -2, 5], [-5, -5, -2, 5, -1, -3, 2, -5, -1, 1, -2, -1, -1, -9, 0, -4, 3, -4, 3], [-9, 4, -4, 4, -4, -2, -1, 1, -3, 0, -1, -1, 1, 1, -3, 2, -2, 2, -3], [-9, 2, -1, 11, -7, 0, -2, 4, -6, -1, -3, -1, 2, -4, -6, 4, -4, 3, -5], [8, 2, 0, -9, -4, 0, 0, 4, 3, 3, 4, 2, -2, 9, -2, 5, 1, 3, -3], [-3, 0, -3, -4, -1, -3, 6, -2, -1, 1, 6, -5, -6, 3, 0, 1, -4, 5, 1], [-1, 9, 1, 2, 2, 4, 1, -3, -4, 3, 3, -6, 1, 1, -2, 2, -1, 3, 3], [2, 2, 1, 6, -3, -3, -2, 2, -3, -11, -1, -2, 2, -4, 2, -2, 3, 2, -3], [3, 7, 1, -3, -2, -1, 1, 3, 1, -3, -4, 4, -3, 0, 0, 6, 0, -3, -5], [2, -4, 2, -1, 4, 0, -6, 6, -2, -2, -2, 4, 10, 4, 2, 6, 0, 0, -2], [7, -6, -4, 0, 4, 2, -8, 2, 2, 4, 1, 5, 2, 5, 0, 0, -4, 1, 4], [1, 4, 0, 10, 0, -2, 1, -1, -5, 0, 1, -6, 9, -3, -1, 2, 2, 0, -1], [5, -5, -4, 2, 1, 4, -4, -4, -2, 7, 3, 2, 4, 5, -5, 1, -2, -1, 2], [-6, 5, 6, 7, -1, 3, -1, 2, -3, -4, -2, 2, 6, -2, 3, 2, 4, -4, -3], [-2, 1, -4, 3, -1, 1, -2, 4, -1, -1, -3, -1, 2, 1, -1, 5, 4, 2, -4], [-8, -2, -1, 9, -1, 2, -2, -4, -2, 1, 0, -4, 4, -1, -2, -4, -2, 0, 1], [-10, 7, 5, 2, -4, -4, 7, -3, 0, 1, 2, -1, -5, -6, 2, -6, -5, 0, 3], [-2, -3, -4, 5, 3, -3, -1, -2, -2, 1, -7, 7, 6, -4, 1, -2, 3, -4, 0], [4, -3, 1, -1, 4, 3, -4, 4, -2, -3, -5, -2, 4, 1, 3, 0, 3, -1, 2], [4, 6, 1, -5, -1, -5, 5, 1, -1, -2, -1, -2, 3, -5, 3, -1, 5, -2, -1], [-3, 3, 2, -2, -1, -2, 4, 4, 4, -3, -5, -2, -2, -7, 1, -1, 5, -3, -2], [-4, -5, 1, -5, 2, -3, 0, 4, -2, -1, 1, -2, 0, 1, 3, 0, 4, 3, 2], [6, -4, 5, -4, -3, -5, 2, 3, 2, -2, 2, -2, 3, 3, 1, 5, 6, 1, -3], [3, -5, 0, 3, 3, 6, -4, 0, -3, 4, 1, 3, 6, 0, -1, 2, 3, 2, 3], [1, -3, -6, -2, 1, 3, -1, -3, 1, 8, 2, -3, -3, 7, -2, 0, -4, -1, 2], [8, -2, 7, -9, 5, 2, 0, 0, 2, -3, 0, 2, 2, 3, 4, 6, 4, -2, 3], [5, -6, -2, 1, -4, 1, -4, 2, -3, -2, 4, 4, 0, 2, -2, -2, -3, 3, 0], [3, -6, -1, 6, 1, -4, -1, 3, -2, -4, -5, 1, 5, -3, 1, -3, 1, -3, -1], [-1, 5, -3, 7, -4, 2, 1, 0, -6, -1, -4, -5, 0, -3, -1, 0, -3, -1, -2], [-6, 5, -9, -4, -6, -5, 4, -4, 3, 6, 5, -3, -10, 4, -5, 3, -9, 0, -3], [6, -2, -3, 5, 3, 5, -7, 3, -3, 0, -3, 6, 9, 1, -3, 4, 5, 2, 1], [6, -3, 1, 0, 8, 2, 1, -5, 2, -1, -2, -3, 1, 0, 0, 0, -1, 2, 3], [-21, 7, -1, -1, -2, -4, 5, -2, 1, -4, -1, 2, -2, -5, 1, -2, -4, -3, -3], [25, -7, 3, -3, 4, 3, -5, 1, 1, 1, 2, 0, -1, 6, 2, -1, 1, 3, 3], [7, -2, 4, 4, -4, 4, -5, 1, -1, 2, 1, 12, 3, 1, -1, 6, 2, -6, -5], [0, -3, 1, 7, -6, -3, 0, 6, -2, -5, 1, -5, 4, -1, -1, -2, 4, 5, 0], [-7, -4, 4, 1, 0, -2, 5, -1, -1, 0, -1, 4, -1, -3, 5, -4, 2, -2, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_671_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_671_2_a_c();".
// 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_671_2_a_c(:prec:=19)
    chi := MakeCharacter_671_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(997) - 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_671_2_a_c();".
// 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_671_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_671_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-43, 643, 241, -6404, 1674, 21077, -11589, -28019, 20141, 16875, -15053, -4673, 5755, 387, -1177, 78, 122, -18, -5, 1]>],Snew);
    return Vf;
end function;