
// Make newform 448.6.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_448_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_448_6_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_448_6_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_448_a();"
function MakeCharacter_448_a()
    N := 448;
    order := 1;
    char_gens := [127, 197, 129];
    v := [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_448_a_Hecke(Kf)
    return MakeCharacter_448_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 := 6;
    raw_aps := [[0], [-14], [56], [49], [232], [140], [-1722], [-98], [-1824], [-3418], [7644], [10398], [-17962], [10880], [-9324], [-2262], [-2730], [-25648], [-48404], [58560], [68082], [-31784], [-20538], [-50582], [-58506], [-38696], [-53060], [-146324], [-92898], [-83354], [-60384], [-61586], [-204462], [-35406], [20226], [-70904], [-293524], [13192], [-493612], [-240716], [294932], [336980], [-358264], [-989554], [990050], [840756], [1150732], [824264], [74382], [-1131956], [-198726], [-482904], [805910], [430738], [-1176910], [-1290976], [1277556], [-1650544], [1064090], [-22342], [-2495738], [1931776], [-459074], [-667128], [-111034], [68778], [-564448], [2077294], [-53248], [2272004], [4006450], [-73784], [-1404312], [1603234], [-4770120], [2230788], [-4840242], [-995820], [-3316050], [3072734], [2814378], [-3058022], [-1937496], [3947902], [7417704], [1402692], [-590574], [-2904842], [922684], [-7182352], [-612570], [-2603300], [-5463088], [1640900], [2997964], [6894048], [-2304764], [-12096042], [5484430], [6717994], [-5002352], [-9019606], [12405106], [6488038], [-10228504], [2653378], [-14304374], [-10026534], [7522920], [3386250], [6908608], [9688958], [-7847418], [-10197194], [8362576], [1102830], [17135426], [54964], [485166], [-2721360], [2145248], [2927962], [13499220], [-5429724], [20827982], [-23514066], [19574670], [26115180], [-15412628], [16986816], [2015112], [15138072], [-7214008], [10969726], [19244190], [8211854], [-18618656], [26250070], [10037300], [14088390], [18143342], [21366922], [-17801736], [16292124], [2084992], [22784972], [22697500], [25289950], [-10394734], [-43339944], [-37165930], [9047850], [32967860], [16109884], [-44728564], [6605184], [30892984], [-4872154], [-32500398], [2640400], [-40817920], [-6719830], [2789792], [33359382], [-7600326], [5797596], [-12682496], [-14440048], [18092062], [-22403444], [34508244], [44165100], [30609120], [38484314], [44520700], [257138], [59523340], [22872926], [-4836048], [-20101648], [-48438632], [8380722], [8781526], [-55221334], [24112416], [-49001372], [-21272014], [-46715858], [-21571406], [24855572], [-10607198], [-23972940], [59591060], [-8304660], [-47556620], [-44726766], [-74355154], [-23852154], [-28848582], [-39241272], [-84534542], [44247364], [-38851680], [3423994], [27462974], [-8690528], [64587348], [-100377000], [48181758], [-81147878], [34800798], [-78811320], [80858888], [119242438], [26281192], [-30786098], [12981520], [-17168690], [-82294552], [-38844066], [-39004298], [18673592], [-107391406], [40512328], [-73902850], [-1292090], [-83139014], [111017416], [-33647936], [129603576], [60555662], [104136298], [128450320], [-75229286], [967806], [-79222752], [-31922618], [13235218], [164366820], [104915104], [8644224], [-5004594], [-107111424], [44996402], [51457098], [-40091884], [-96799696], [74500958], [19835824], [6698928], [56812810], [-40764318], [148621136], [191741578], [-31644256], [-95297884], [30936982], [21917322], [-26097288], [-52424722], [-47141480], [-80432728], [180053944], [-107736608], [-167100682], [-128458246], [-195434598], [205197034], [40600156], [-91592698], [225295700], [-27045472], [63156226], [50706064], [-83362958], [-183398768], [-41668448], [197576826], [121702270], [-219184042], [230757352], [-221800776], [126392812], [-94076080], [-175669858], [-109289184], [-66225662], [7902018], [-19624052], [-264054126], [64562210], [202322134], [-222052554], [-308761362], [-213825166], [11334260], [-220325056], [45937444], [141016610], [167678210], [-104608770], [-111088338], [-204266560], [284189992], [221365008], [-155685318], [-88282704], [-4863432], [-58199488], [-49163734], [317400888], [-216655874], [215030442], [-21138984], [-174142880], [-220327030], [-85163694], [574272], [-259941542], [-131108766], [-349363462], [-228141518], [-336301056], [-340445240], [424471734], [149519230], [243529360], [-223287106], [78842368], [98475606], [-23920390], [179971854], [140698852], [-178786566], [154289500], [94839262], [224401630], [142325106], [-34959526], [-167101228], [-371490526], [-331416202], [-391310892], [-69891712], [-398783522], [-385410464], [41030612], [-133867016], [427945224], [16480912], [-22770398], [-390482406], [387706898], [-367437084], [-299869630], [191548322], [-452285988], [59806278], [-24271726], [-172440144], [51410994], [-504028910], [-47305728], [386245052], [405410558], [470445238], [295903636], [285156080], [379937574], [-30924732], [50440194], [90990672], [574960146], [196823704], [-87416662], [61511758], [251584856], [300748826], [-226639434], [-28085442], [559059844], [209207238], [294887192], [373394834], [150627330], [425925304], [388792984], [537651128], [136029380], [88376820], [548920990], [372194088], [-218913540], [-717856006], [-229114568], [710761570], [-376944820], [610986726], [-104995812], [659115582], [-714635936], [-580539214], [163865264], [-4963938], [17774988], [-40778500], [-431984526], [392830622], [37552634], [-591109806], [-362159014], [244222198], [-199454008], [-131868068], [-505194198], [-259521444], [-418838760], [-3455732], [-575293194], [-738623424], [-620411058], [-37535008], [75936208], [-688612582], [300982262], [763452984], [556018432], [182454666], [836521434], [-494044572]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_448_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_448_6_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_448_6_a_c(:prec:=1)
    chi := MakeCharacter_448_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_448_6_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_448_6_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_448_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![14, 1]>,<5,R![-56, 1]>],Snew);
    return Vf;
end function;