
// Make newform 6720.2.a.d in Magma, downloaded from the LMFDB on 30 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6720_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_6720_2_a_d();".
// 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_6720_2_a_d();".
// 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_6720_a();"
function MakeCharacter_6720_a()
    N := 6720;
    order := 1;
    char_gens := [1471, 3781, 4481, 5377, 1921];
    v := [1, 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_6720_a_Hecke( : Kf := Kf)
    return MakeCharacter_6720_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 := 2;
    raw_aps := [[0], [-1], [-1], [-1], [-2], [-4], [2], [2], [-4], [-6], [2], [-10], [-10], [12], [8], [0], [-8], [2], [-12], [10], [4], [0], [-12], [2], [-8], [6], [-8], [4], [14], [4], [0], [12], [-8], [2], [-18], [4], [-8], [12], [16], [-2], [22], [22], [22], [-14], [-12], [10], [20], [-12], [-12], [26], [4], [-14], [30], [16], [14], [0], [-18], [6], [22], [-26], [-4], [30], [32], [-8], [-8], [-32], [-20], [2], [0], [10], [-30], [-10], [20], [34], [16], [24], [6], [20], [-30], [-14], [36], [26], [-30], [24], [10], [-16], [10], [22], [10], [-8], [-20], [24], [0], [30], [40], [8], [22], [30], [-40], [10], [28], [-40], [-12], [-6], [16], [32], [36], [42], [-26], [38], [36], [-2], [-20], [10], [0], [30], [16], [32], [-44], [-38], [-38], [-10], [30], [-16], [22], [42], [26], [-12], [-12], [-12], [-24], [-48], [32], [-14], [-22], [22], [-10], [-40], [-42], [-26], [-54], [50], [-40], [-12], [-50], [12], [-28], [-34], [-34], [-12], [22], [18], [-20], [8], [-12], [30], [-16], [-14], [-12], [-30], [-24], [36], [-16], [-24], [12], [-32], [56], [28], [14], [-18], [-26], [46], [58], [2], [46], [-26], [20], [6], [-20], [6], [-16], [40], [50], [2], [-64], [-26], [-10], [-16], [14], [-44], [56], [44], [-32], [2], [0], [-6], [14], [-46], [30], [-48], [-6], [42], [16], [30], [-40], [38], [-66], [12], [-2], [30], [22], [-18], [-16], [36], [12], [-2], [-8], [-46], [-8], [-52], [-38], [18], [-66], [-16], [-28], [-10], [62], [54], [-16], [54], [2], [-22], [12], [-66], [-40], [-32], [34], [-44], [22], [4], [-60], [-14], [0], [-26], [-34], [20], [4], [12], [-68], [-12], [62], [18], [12], [-46], [30], [-42], [-26], [12], [-18], [32], [-16], [-56], [26], [-16], [34], [30], [-30], [78], [20], [60], [-58], [76], [-48], [-24], [-72], [-44], [4], [-62], [-38], [64], [-48], [-32], [-8], [62], [8], [54], [14], [-56], [-22], [-30], [-46], [52], [-28], [60], [42], [-70], [-2], [-6], [40], [64], [68], [68], [-44], [32], [80], [50], [12], [-30], [66], [82], [0], [-14], [18], [60], [60], [6], [60], [-34], [40], [90], [-2], [-42], [-78], [32], [32], [-2], [16], [-36], [-12], [24], [-66], [12], [-26], [-44], [44], [12], [-22], [-18], [14], [-52], [-70], [28], [70], [20], [-4], [-86], [42], [-68], [-68], [-34], [-62], [46], [10], [-16], [78], [50], [60], [-52], [-60], [-88], [90], [-10], [0], [26], [72], [-26], [-14], [48], [-38], [-50], [6], [64], [22], [-2], [-2], [28], [14], [-40], [-26], [-12], [62], [84], [16], [80], [-46], [64], [64], [-76], [4], [24], [-26], [-38], [26], [-8], [2], [10], [30], [74], [56], [-102], [62], [44], [32], [-42], [-54], [86], [30], [58], [84], [88], [12], [8], [-72], [52], [-102], [62], [66], [20], [90], [-40], [-82], [12], [96], [0], [56], [30], [20], [-30], [-50], [-36], [2], [58], [4], [0], [-8], [30], [10], [-10], [60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6720_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6720_2_a_d();".
// 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_6720_2_a_d(:prec:=1)
    chi := MakeCharacter_6720_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(3067) - 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_6720_2_a_d();".
// 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_6720_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6720_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2, 1]>,<13,R![4, 1]>,<17,R![-2, 1]>,<19,R![-2, 1]>,<23,R![4, 1]>,<29,R![6, 1]>,<31,R![-2, 1]>],Snew);
    return Vf;
end function;