
// Make newform 6270.2.a.l in Magma, downloaded from the LMFDB on 13 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6270_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_6270_2_a_l();".
// 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_6270_2_a_l();".
// 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_6270_a();"
function MakeCharacter_6270_a()
    N := 6270;
    order := 1;
    char_gens := [4181, 5017, 2851, 3631];
    v := [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_6270_a_Hecke(Kf)
    return MakeCharacter_6270_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], [1], [-4], [-1], [2], [6], [1], [0], [-6], [-4], [2], [6], [8], [0], [-6], [0], [14], [-4], [-12], [-10], [8], [0], [-6], [-10], [6], [-16], [12], [2], [6], [8], [12], [18], [-4], [6], [8], [14], [20], [24], [18], [0], [-10], [24], [14], [18], [-16], [-4], [8], [-12], [-10], [6], [-24], [14], [12], [30], [-12], [6], [-16], [2], [30], [8], [-6], [-28], [-24], [26], [-6], [8], [14], [0], [-10], [18], [-24], [8], [-22], [8], [24], [6], [14], [-30], [-10], [12], [-10], [24], [38], [8], [36], [-6], [14], [30], [8], [12], [-24], [-16], [-36], [-4], [-12], [-18], [42], [-28], [-10], [-4], [42], [12], [-42], [-4], [-22], [12], [6], [-12], [-10], [-40], [-22], [18], [20], [-40], [18], [-4], [0], [-18], [36], [-10], [-34], [-6], [-36], [-4], [6], [-10], [24], [-16], [50], [20], [48], [-4], [-34], [-30], [-22], [18], [44], [18], [-30], [-28], [30], [-16], [-12], [38], [-12], [26], [6], [-4], [0], [-46], [18], [-52], [24], [20], [12], [8], [-30], [-34], [-30], [-12], [-42], [-4], [0], [-18], [-24], [-4], [26], [-34], [54], [60], [-10], [-24], [14], [-28], [42], [20], [6], [8], [-10], [56], [-36], [-46], [54], [-24], [54], [2], [-4], [14], [0], [-34], [48], [32], [6], [-24], [30], [50], [-34], [-66], [36], [18], [56], [2], [-10], [12], [-42], [-16], [-12], [42], [32], [-10], [54], [8], [-60], [0], [2], [-4], [-66], [-72], [6], [50], [8], [-54], [-16], [36], [-34], [-42], [12], [8], [-12], [-10], [20], [32], [-18], [44], [-12], [50], [-6], [-12], [-60], [60], [-52], [-16], [-34], [-66], [24], [-16], [-72], [20], [-12], [-22], [-78], [0], [26], [42], [-36], [-10], [-4], [-54], [14], [8], [12], [38], [2], [42], [8], [30], [42], [-4], [-54], [-10], [20], [-70], [32], [14], [32], [-60], [-70], [38], [-12], [24], [80], [-12], [26], [32], [72], [-70], [-54], [-64], [6], [30], [-60], [54], [-36], [50], [-42], [-28], [30], [-36], [56], [62], [-78], [-16], [36], [-4], [38], [-12], [-34], [-24], [-70], [60], [54], [-66], [-76], [60], [38], [12], [-72], [26], [-6], [-28], [-70], [66], [32], [-42], [74], [-64], [20], [-24], [66], [62], [-30], [-40], [-24], [20], [60], [-58], [78], [-22], [-52], [26], [-6], [-54], [20], [-78], [-60], [-58], [-28], [-24], [6], [20], [38], [-18], [8], [-22], [-42], [-96], [-36], [-18], [24], [-46], [-78], [-48], [-60], [68], [-82], [42], [8], [14], [36], [56], [-36], [18], [-64], [-34], [-24], [48], [-58], [66], [-10], [-18], [18], [-52], [54], [-76], [-24], [-40], [86], [56], [24], [74], [42], [60], [-76], [-60], [-58], [-64], [-54], [-40], [102], [-70], [18], [-40], [54], [-30], [-40], [-82], [102], [68], [-36], [14], [-102], [36], [44], [-58], [-18], [24], [80], [18], [-48], [-90], [-70], [48], [-12], [-34], [66], [60], [90], [-100], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6270_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6270_2_a_l();".
// 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_6270_2_a_l(:prec:=1)
    chi := MakeCharacter_6270_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_6270_2_a_l();".
// 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_6270_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6270_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;