
// Make newform 3675.2.a.br in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3675_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_3675_2_a_br();".
// 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_3675_2_a_br();".
// 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
        poly := [4, 5, -7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, 0, 1, 0], [0, -6, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_3675_a();"
function MakeCharacter_3675_a()
    N := 3675;
    order := 1;
    char_gens := [1226, 1177, 2551];
    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_3675_a_Hecke(Kf)
    return MakeCharacter_3675_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 := [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2, -1, 0, 1], [2, 1, 1, -1], [2, -1, 0, -1], [0, 2, -1, 0], [0, 0, 2, 0], [4, 0, 0, 0], [1, 3, -1, 1], [-3, 2, -2, 0], [2, -1, 2, -1], [-1, -1, 1, 1], [-2, -1, -2, 1], [2, -3, 0, 1], [2, -1, -2, 1], [-5, 0, 0, 0], [0, 0, 1, 0], [6, -4, -2, 0], [-1, 2, 0, -2], [1, 3, 0, 1], [-2, 3, -2, 1], [-2, -1, 0, -1], [3, 2, 2, 0], [4, -1, 0, -1], [5, -1, -2, -1], [4, 3, 0, 1], [6, -1, 1, -1], [-2, -2, 0, 2], [1, -1, -2, 3], [6, 0, 4, 0], [0, 2, -2, -2], [-9, 5, 0, -1], [4, 5, -4, 1], [1, -3, -2, 3], [1, -4, -2, -2], [-5, 1, -2, -1], [-4, 7, 0, 1], [6, -2, 0, 2], [0, 4, -4, 0], [5, -7, -1, 1], [-8, -3, -2, 3], [9, 1, -1, 1], [2, 0, 2, -4], [-3, 1, 2, -5], [5, 5, -4, 1], [9, 5, -2, 3], [-2, -6, -2, 2], [10, 1, 5, -3], [-16, 1, 0, 1], [-2, -4, 2, 0], [5, -2, -6, 0], [-12, 1, 2, -1], [-18, 5, 2, 1], [10, 8, -2, 0], [0, 1, 0, -3], [16, -4, 2, -2], [-6, 5, -1, 1], [10, 1, 0, -3], [-4, 4, -5, 4], [-10, -4, 2, 0], [1, -3, -1, -1], [-8, 4, -2, 0], [5, 11, -1, -1], [-2, 6, 4, 2], [-7, -7, 2, -3], [-1, 5, 1, 1], [-2, 9, -2, 3], [-2, -2, 2, -4], [-12, -5, 4, -1], [10, 1, -2, 3], [17, -1, 1, 3], [2, 3, 1, 3], [8, 6, -1, -2], [0, -6, 2, -6], [14, 3, 6, -1], [1, -5, -1, -5], [2, 1, 2, 1], [-14, 3, -1, 5], [12, -7, 6, -9], [-7, 2, 2, -4], [0, -2, -2, 2], [-3, 3, 3, -5], [3, -7, 0, -3], [18, 6, 8, -6], [-4, 3, 2, 3], [-5, 12, -2, 2], [6, -4, 4, 4], [-1, 3, -8, 5], [-10, 4, 4, -4], [-4, -7, -8, 3], [-23, 1, -1, -1], [-4, 0, 2, 4], [-8, 4, 3, 2], [0, -10, 4, 2], [2, -8, -2, -4], [2, 6, 8, -2], [17, 1, 7, -1], [14, 5, -5, 3], [4, -4, 2, -6], [6, -2, 2, 2], [-14, 3, 0, 1], [16, -9, 4, -5], [8, -6, 9, -6], [-7, 1, 1, 3], [6, 0, 4, 0], [16, -4, 2, 0], [28, 1, 4, -1], [14, -5, 1, -7], [19, -5, -4, 1], [-2, 2, -6, 0], [-6, -8, 6, -4], [7, -5, 1, 1], [-5, -3, 6, 3], [10, 14, -2, 2], [-7, 9, -8, -1], [6, 3, 8, -7], [-14, 1, 2, 1], [2, -7, 0, -5], [-25, 6, -2, 4], [-7, -2, -8, -2], [-8, -12, 2, 4], [6, -7, -4, 7], [1, -11, -6, 5], [18, 1, 8, 1], [1, -8, -2, 2], [8, 13, 6, -5], [4, 6, -1, 0], [-17, 6, -4, 6], [-5, 1, 2, -7], [36, -2, -2, -2], [-17, 17, 2, 1], [-9, 0, -2, -2], [20, -7, 6, -3], [11, -9, -7, 3], [-4, 2, -4, 6], [15, -3, -4, 1], [4, -4, 6, 0], [12, -4, 6, 0], [-31, -5, -8, 7], [20, -7, 8, -3], [9, -3, 0, 5], [-30, -6, 2, 2], [-3, -8, 2, 2], [2, 2, -4, 2], [-35, 1, -7, 3], [-32, 0, -2, 0], [36, 0, 4, -4], [10, -7, -2, 3], [5, -8, 4, 4], [26, 10, -6, 2], [16, -10, 7, 4], [-18, 6, -8, -6], [-25, -13, -2, 3], [8, -6, -2, -6], [-1, -3, -7, 7], [-6, 3, 2, -1], [5, -15, 7, 5], [-2, -6, 8, -6], [14, -12, 4, 8], [-16, 5, 0, 5], [-25, 15, 8, 1], [24, -5, 2, -7], [16, 3, 0, 3], [-14, 11, -2, -7], [-19, -5, -3, -1], [-19, -8, 2, -2], [-15, -9, -1, 7], [-14, -2, -6, -2], [-10, 4, -14, 4], [-5, -6, -6, 0], [-16, -7, 8, 3], [-10, 3, 7, -1], [0, 2, 7, 0], [2, -8, 2, 4], [-13, 7, 1, 1], [16, -7, 8, 5], [17, 3, 4, 7], [3, -5, 5, 7], [-17, -3, -11, 3], [-24, 8, -2, 0], [7, -9, -3, -1], [-26, -5, -4, 7], [0, -9, 4, -3], [-8, -1, 2, -9], [-17, -14, 6, 0], [-11, -1, 1, 3], [-15, 13, 7, -3], [4, 17, -10, 3], [6, -9, 7, -5], [-6, -9, 8, -7], [-21, 13, -4, -7], [18, -8, -8, 0], [-16, -4, -8, 0], [0, -16, 12, -4], [7, 6, 6, 0], [6, 0, -2, -6], [-12, 7, -18, -1], [-10, 14, 0, 10], [30, -3, 4, -3], [9, -11, 2, -7], [2, 6, -10, -4], [-14, -4, -6, -2], [0, 7, 0, -7], [-18, -5, -6, -1], [-7, 3, -12, 3], [24, 3, -4, 5], [-24, 2, -4, -2], [-1, -11, -10, -1], [18, -15, -5, -1], [0, 23, -6, 3], [21, -7, 4, 1], [-10, 7, 8, -3], [-12, 17, 2, -5], [-25, -2, -8, 2], [5, -3, 2, -9], [12, 0, 2, 12], [2, -12, 2, -12], [-32, 4, -2, 0], [37, 3, 7, -9], [16, -6, -7, 2], [-4, -18, -6, 2], [0, 0, -4, 4], [-14, -13, -2, -3], [27, -6, 0, -2], [22, 6, 0, -2], [10, 1, 6, -1], [17, -3, 3, -9], [-34, 4, -12, 4], [-34, 9, 1, -1], [36, -4, -4, 0], [11, 5, 1, -5], [12, 2, 4, 2], [37, 1, 14, -7], [-8, -15, 0, 7], [17, -18, 6, 8], [-2, -15, 12, -7], [-6, -22, 4, -2], [-22, -6, 6, 2], [-6, -11, -10, -1], [-23, 7, -5, -3], [-4, 18, 1, 0], [19, 8, 0, 8], [12, 0, -12, 0], [4, 3, 0, -7], [-3, -17, -1, 1], [40, -1, 8, -3], [-5, 5, -2, 3], [2, 9, 16, -5], [39, -6, -6, 4], [20, -4, 12, -8], [-20, -12, 8, 4], [10, -9, 7, 1], [24, 7, 2, -9], [-14, -12, 4, 4], [21, 13, -7, 7], [7, 21, -10, -3], [32, -15, 0, 5], [10, -16, 14, -2], [4, 4, -3, -8], [-22, 12, 2, -12], [17, 12, -2, 6], [-35, 11, 7, -1], [0, 12, -4, 12], [-9, -5, 12, 1], [16, 3, -8, 11], [54, -1, -4, 3], [-16, 4, -8, 0], [-24, 12, 6, 8], [-22, -9, 9, -3], [4, 8, -12, 12], [11, -4, 2, 2], [4, -22, 1, -4], [23, -4, -10, 6], [1, 23, 2, 1], [6, -4, 4, -8], [-37, 5, -3, 9], [6, 19, -11, 5], [-26, 2, -4, 2], [-6, -1, -12, 5], [-11, -3, 2, -3], [-32, 3, 8, 5], [26, -11, -1, 13], [7, 15, -8, 7], [-4, 17, 2, -9], [-30, 9, 5, 7], [-24, 9, -2, -3], [3, -3, 6, 1], [-20, 0, -4, -8], [28, -3, 16, -15], [-10, 16, -20, 8], [-14, -2, -16, 6], [-38, 3, 2, 1], [-2, 9, 5, 9], [36, -10, 2, 10], [17, -3, -1, 3], [26, -10, -8, -6], [-12, -3, 14, 3], [-45, -5, 8, 1], [26, 2, -14, 0], [30, 8, -2, 0], [-21, -9, 0, -1], [-16, -26, 4, -2], [-9, 1, -12, 7], [38, 1, 5, -9], [6, -6, 12, -6], [-2, 1, -3, -11], [-4, -4, 6, -12], [-23, -4, -4, -4], [-4, -4, 4, -12], [-12, -10, -12, -2], [22, -27, -2, 1], [29, 11, -10, 5], [-12, -4, -14, 8], [5, -9, -13, -5], [20, -8, 0, -8], [-36, 9, -8, 3], [-2, -25, -7, 5], [-14, -28, 8, -4], [11, 5, -4, -5], [1, -5, 7, -13], [42, 10, 0, -6], [4, -10, 7, 0], [-4, -7, 10, -3], [-11, -8, -2, 2], [-1, -7, -9, 1], [-48, -4, 5, 2], [18, -18, -4, 6], [0, 12, -4, -4], [42, 13, 3, -1], [44, -10, 6, -10], [-31, 19, 1, -1], [-4, -5, -8, 5], [31, -21, -3, -7], [-12, 6, -4, -2], [15, 2, 14, -4], [-4, -7, 0, 1], [-35, 13, -5, 13], [45, -1, 0, 1], [-19, -3, -5, 9], [6, 9, -2, -11], [8, 1, -6, 13], [-16, 16, -9, 10], [-24, -2, -2, -14], [-34, 12, -16, 8], [6, -3, -5, 7], [7, 1, -4, 1], [-28, -16, -4, 0], [24, 0, 8, 4], [5, 11, 14, -3], [22, -12, -2, 10], [12, -21, -6, 11], [23, 7, -8, -1], [34, -11, 13, -9], [28, 1, -2, 9], [-32, -15, 2, 3], [6, 24, 4, -12], [40, -6, -6, 10], [0, 6, 6, 2], [1, -4, -4, 0], [-2, -3, 4, 5], [50, -14, -6, -2], [-10, -7, 14, 7], [-35, -11, -16, 11], [-19, 4, 8, 8], [22, -9, 4, 7], [9, -19, 14, 1], [-13, 26, -2, -8], [-48, -7, -6, 7], [17, 3, -9, 5], [-2, 9, -10, -9], [16, 16, -2, 0], [1, 11, 6, -1], [-5, -5, 7, 1], [-10, -12, -2, 0], [28, -9, 22, -11], [-23, 6, 0, -6], [-28, 26, -4, -6], [30, -15, 7, 13], [30, -28, -4, 12], [-38, 7, 16, 3], [16, -6, -4, 14], [-4, 7, -14, 11], [28, -22, -12, 2], [0, -26, -4, 10], [12, 0, 9, 10], [-26, 2, 10, 4], [-13, -11, -2, 13], [-34, 7, 0, -11], [-34, 9, 5, -7], [34, -8, 8, -4], [-14, -22, 12, -6], [-16, -10, 5, 2], [-30, -2, 0, 6], [3, -28, -10, 6], [7, 9, 0, -7], [38, 2, 18, -18], [-24, -8, 3, -8], [20, -6, -4, -10], [-22, 8, 2, -22], [-24, 8, -6, -8], [-37, 9, 0, 9], [4, -12, 0, -16], [2, 3, 4, -5], [7, 3, -4, -3], [-1, 14, 10, -16], [0, -21, 8, -13], [-32, 4, -7, -6], [12, -15, -8, 3], [21, -8, 12, 8], [-48, -3, -16, 9], [-24, 7, -20, 13], [9, 5, 6, -17], [-1, 1, 17, -3], [-18, 16, -10, 0], [-52, 8, -10, -4], [20, -4, 23, -8], [4, -8, 14, -12], [2, -10, 12, -2], [2, 12, 18, -8], [-19, -7, 23, 1], [8, -5, 18, 1], [2, 13, 16, -1], [-30, 14, -6, 12], [-6, -27, 0, 5], [54, -14, -6, -6], [-40, 22, 4, -2], [11, 3, 10, -3], [-2, -9, 12, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3675_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3675_2_a_br();".
// 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_3675_2_a_br(:prec:=4)
    chi := MakeCharacter_3675_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_3675_2_a_br();".
// 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_3675_2_a_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3675_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, -5, -7, 1, 1]>,<11,R![-328, 180, -8, -8, 1]>,<13,R![-194, 179, -19, -7, 1]>],Snew);
    return Vf;
end function;