
// Make newform 50.6.a.a in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_50_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_50_6_a_a();".
// 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_50_6_a_a();".
// 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_50_a();"
function MakeCharacter_50_a()
    N := 50;
    order := 1;
    char_gens := [27];
    v := [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_50_a_Hecke( : Kf := Kf)
    return MakeCharacter_50_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]];
    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 := [[-4], [-11], [0], [-142], [777], [884], [-27], [1145], [1854], [-4920], [1802], [13178], [-15123], [7844], [-6732], [3414], [33960], [47402], [-13177], [-7548], [-59821], [75830], [46299], [-30585], [104018], [-23898], [-22636], [60633], [-7090], [-128841], [141338], [80052], [32253], [394865], [-491400], [200402], [-22942], [-336241], [59748], [-60696], [-7995], [-454798], [-428298], [-835531], [678318], [-31900], [-423673], [398204], [-1257612], [203780], [823974], [-555960], [523577], [113127], [872958], [-1646466], [1788720], [1121402], [-598312], [-1530498], [-1797001], [754494], [-1966267], [-599298], [-721366], [102348], [1310477], [-804397], [-2883207], [1273550], [2830614], [-981090], [-4191052], [3234554], [1390355], [1141974], [3462990], [5940068], [-2274123], [-4295515], [1797045], [-257548], [2229102], [-4205851], [352640], [1283619], [-2103975], [825233], [4501452], [-1442116], [393348], [-9176970], [6605978], [38052], [6856700], [8200164], [4065810], [5284077], [2533829], [498752], [3002693], [-12737292], [-5970816], [-9269055], [-3895348], [7294163], [-8728197], [-13096281], [-13016850], [-9939973], [15643808], [9337934], [-5066802], [13767020], [2070602], [-17911398], [-17141356], [-8487732], [-2454786], [-5915565], [4330952], [-9139846], [4572288], [15322149], [7025477], [7911252], [-15448540], [23054370], [16290488], [12827924], [-12453460], [-26324706], [-17499448], [3463808], [-12617523], [5708045], [12082704], [-13763572], [-8777382], [-10204590], [-1173748], [-19806198], [-30672196], [25551963], [-9194020], [-15690960], [1601114], [-16461267], [19673585], [36806784], [-26959612], [34733502], [21618749], [4481628], [-33663892], [-1031748], [4101470], [7716030], [43845803], [-10021548], [27952968], [-23181111], [-1584352], [34455177], [-29359887], [8939574], [-17889898], [33051668]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_50_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_50_6_a_a();".
// 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_50_6_a_a(:prec:=1)
    chi := MakeCharacter_50_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(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_50_6_a_a();".
// 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_50_6_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_50_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![11, 1]>],Snew);
    return Vf;
end function;