
// Make newform 6675.2.a.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6675_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_6675_2_a_h();".
// 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_6675_2_a_h();".
// 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_6675_a();"
function MakeCharacter_6675_a()
    N := 6675;
    order := 1;
    char_gens := [4451, 802, 2851];
    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_6675_a_Hecke(Kf)
    return MakeCharacter_6675_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 := [[1], [-1], [0], [0], [4], [6], [6], [8], [-8], [6], [-4], [-2], [-6], [-4], [0], [2], [0], [14], [-12], [0], [6], [16], [-12], [1], [6], [-10], [-16], [4], [14], [-6], [16], [12], [18], [20], [6], [12], [2], [-20], [8], [2], [-4], [-2], [-20], [10], [-2], [-24], [-16], [16], [-12], [-26], [6], [20], [10], [4], [14], [32], [-18], [-8], [2], [10], [-12], [22], [-4], [-32], [18], [-6], [12], [-30], [-12], [-34], [-14], [-12], [32], [2], [8], [-16], [6], [-2], [-14], [10], [0], [-26], [12], [-6], [12], [12], [-30], [10], [14], [-8], [28], [-16], [8], [-8], [0], [24], [-34], [18], [12], [38], [20], [22], [12], [18], [0], [34], [-36], [-6], [44], [10], [8], [10], [18], [-4], [-16], [-46], [-20], [24], [6], [12], [6], [6], [-26], [-36], [44], [14], [22], [-20], [-32], [34], [16], [-8], [-8], [34], [42], [-14], [30], [-36], [-38], [10], [28], [6], [16], [44], [38], [-20], [-2], [-6], [56], [-24], [22], [18], [4], [24], [-44], [40], [-36], [-30], [46], [30], [28], [-14], [40], [-20], [22], [-16], [-4], [-14], [-22], [18], [-20], [30], [-52], [-38], [-52], [-54], [52], [-26], [56], [-34], [8], [-24], [-14], [34], [16], [-66], [-30], [-12], [-6], [12], [-58], [20], [-8], [-42], [36], [-66], [2], [46], [-22], [-56], [30], [52], [-14], [10], [-48], [-58], [52], [-52], [42], [60], [2], [-26], [-64], [-44], [-56], [-30], [-16], [-38], [-8], [-18], [-10], [32], [-30], [32], [-20], [-26], [-42], [-36], [-24], [-48], [-18], [32], [8], [-54], [-20], [-24], [50], [-6], [-48], [16], [-44], [28], [0], [14], [54], [60], [16], [-24], [-40], [-24], [42], [-14], [72], [10], [18], [-60], [14], [-60], [38], [-10], [-40], [-36], [-58], [26], [18], [36], [-50], [-46], [-52], [66], [62], [-44], [74], [-48], [50], [32], [-4], [-2], [10], [8], [24], [12], [-32], [-26], [12], [48], [38], [18], [-24], [-14], [78], [36], [-10], [-72], [-30], [62], [52], [54], [-20], [-68], [42], [82], [12], [4], [-20], [2], [60], [14], [24], [-66], [-72], [-26], [-30], [-4], [-64], [42], [-32], [-24], [-46], [-54], [-20], [-50], [-50], [-64], [-42], [-62], [-24], [-84], [24], [-42], [14], [-66], [-52], [-36], [32], [12], [30], [42], [-78], [-8], [26], [-42], [6], [4], [54], [-12], [-26], [60], [12], [-74], [52], [50], [-2], [-8], [6], [-74], [-72], [-12], [-14], [0], [-22], [34], [-48], [48], [28], [82], [30], [80], [-46], [-60], [36], [-72], [86], [60], [-74], [12], [-48], [66], [42], [14], [62], [-6], [-8], [-30], [36], [0], [48], [70], [-12], [-72], [-62], [6], [24], [100], [-4], [-54], [-24], [42], [32], [6], [46], [26], [-56], [-2], [70], [-80], [30], [18], [4], [-24], [98], [-78], [-36], [-16], [-10], [-34], [-12], [88], [54], [8], [-66], [82], [24], [-60], [-90], [-14], [-92], [58], [60], [52]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6675_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6675_2_a_h();".
// 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_6675_2_a_h(:prec:=1)
    chi := MakeCharacter_6675_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_6675_2_a_h();".
// 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_6675_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6675_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<7,R![0, 1]>,<11,R![-4, 1]>,<13,R![-6, 1]>],Snew);
    return Vf;
end function;