
// Make newform 390.2.bd.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_390_bd();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_390_bd_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_390_2_bd_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_390_2_bd_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
        poly := [1, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_390_bd();"
function MakeCharacter_390_bd()
    N := 390;
    order := 12;
    char_gens := [131, 157, 301];
    v := [12, 3, 11];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_390_bd_Hecke();"
function MakeCharacter_390_bd_Hecke(Kf)
    N := 390;
    order := 12;
    char_gens := [131, 157, 301];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0, -1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 2, 0, -1, 0, 0], [0, -2, -1, 2, -1, 2, -1, -2], [-2, 0, 2, 1, 0, 2, -2, 1], [0, 3, 0, 2, 0, -3, 0, 0], [2, -4, 1, 0, -1, 0, 1, 0], [-4, -1, -2, 2, 2, 0, -2, -1], [3, -4, 3, 0, -2, 2, -1, 0], [-2, -3, -2, 1, -2, 4, 0, -4], [1, -2, -3, 0, -3, -2, 1, 0], [2, 1, -2, -3, -2, 2, 4, 2], [3, 0, 3, 4, 1, 0, -4, -4], [-2, -2, 0, 0, 2, 4, 2, 1], [4, 2, -2, 2, 0, 0, 1, -2], [2, 0, 8, 0, -8, 0, -2, 0], [2, 2, -6, 0, 4, -4, 4, -3], [0, 1, -4, -4, -2, -4, -4, 1], [0, 0, -4, -3, 0, -3, 0, 3], [4, 2, 2, 0, -6, -4, -6, 4], [-5, 2, 0, -2, 10, 0, 1, 2], [4, -4, 0, 4, -8, 2, -3, -2], [-4, -7, 8, -7, 0, 3, -4, 4], [-3, 4, -3, -4, 5, -2, -2, 4], [2, -2, 1, -4, -1, 4, -1, 2], [2, 2, 6, 1, 2, -1, 0, 1], [6, 2, -6, -4, -6, 2, 6, 0], [2, 2, 2, -5, 4, -1, -6, 5], [-8, 5, -2, -3, 2, -5, 8, 6], [-1, 10, -2, 4, -1, 0, 1, -2], [0, 0, 0, 0, 4, 6, 4, 0], [8, -4, -4, -4, 0, 5, 2, -1], [0, -6, -5, 8, -4, 8, -5, -6], [-12, -1, -6, -4, 6, 4, 6, 1], [-4, -1, -2, -12, 2, 0, -2, 6], [1, 2, -8, -2, 8, -2, -1, 4], [-2, 4, 4, 11, 4, 4, -2, 0], [8, -4, -2, -3, -4, 3, 2, 4], [-1, -2, -2, 2, 1, 0, 4, 0], [8, 0, -12, 0, 4, 0, 4, -2], [-4, -5, 2, -2, 4, 7, -4, 7], [2, -6, 0, 6, -4, 6, -2, 0], [0, -8, 1, 12, 5, 12, 1, -8], [-4, -6, 16, -10, -4, -4, 0, 4], [2, -2, 2, -1, 2, 1, 0, -1], [0, 4, -2, 4, -10, 4, -2, 4], [0, -8, -4, -1, 0, 9, 8, 9], [14, 2, -2, 0, -14, -2, 4, -2], [8, -3, 4, 2, -4, -2, -4, 3], [2, -4, 0, 8, 0, -4, 2, 0], [0, -8, -5, 12, -5, -8, 0, 0], [-4, -4, -8, -2, 8, 4, 4, 4], [1, -6, -2, -4, -3, 0, 3, 2], [-4, 5, -12, 3, 2, -3, 12, -5], [-2, 0, 2, 4, 11, -4, 9, 4], [-9, 0, 9, 0, 14, 2, 5, 0], [8, 7, 0, 4, -4, -4, 0, -7], [-2, 0, 2, -2, 3, -20, 1, -2], [-8, 12, -2, -6, 6, 0, -6, 3], [2, -8, -6, -6, 6, 8, -2, 12], [14, -2, 14, 0, -4, 1, -10, 0], [-12, -1, -10, -6, 6, 6, 10, 1], [-10, -1, -12, -1, 0, 9, 6, -8], [-3, 0, 0, 0, 6, 8, 11, 8], [1, 0, -8, 2, 8, 0, -1, -4], [-4, 4, 0, -4, 8, 0, -2, 4], [-6, 1, 8, 0, -2, -2, -2, -17], [-6, 6, 2, 6, -2, -6, 6, -12], [2, 8, -4, 0, 2, -16, 2, -10], [10, -8, 10, 12, 4, 4, -14, -12], [2, 4, 4, -8, -2, 4, -8, 4], [1, 6, 4, -16, 4, 6, 1, 0], [2, 4, 2, 8, 16, -2, -18, -8], [-8, -12, 8, -10, 16, 0, -16, 5], [-2, 0, 2, 3, 4, -17, 2, 3], [0, -8, 0, -2, -11, -2, 0, -8], [2, -2, 12, -2, 0, -13, -6, 15], [0, 4, 12, 1, -4, 1, 12, 4], [10, 0, -10, -2, 4, -18, 14, -2], [2, 22, 1, 12, -1, 0, 1, -6], [10, 0, 2, 7, 10, 7, 0, -7], [-4, -5, -4, -13, -4, -5, -4, 0], [4, 4, 4, 8, -16, -2, 12, -8], [-10, -6, -4, 0, 14, 12, 14, -12], [14, 8, -2, 2, -14, -10, 4, -10], [6, 9, 4, 1, -4, -9, -6, -2], [8, 2, 6, 0, -14, -4, -14, -14], [2, -18, -4, -18, 2, 0, 0, 0], [-4, 7, 16, 0, -12, -14, -12, 0], [4, 14, 0, -14, -8, -6, -4, 8], [6, 8, 0, -6, 0, -8, -6, 12], [9, 20, 9, -2, -11, -10, 2, 2], [14, -4, 1, 8, -7, -8, -1, 4], [2, -4, 14, 5, 2, 9, 0, -9], [-4, -6, 2, -4, -2, 6, 4, 8], [2, 18, -1, 20, -3, 0, 3, -10], [12, 0, -12, 9, 4, 12, 16, 9], [-9, -10, -2, -10, 0, 14, 1, -4], [-2, 0, 2, 6, -12, -17, -14, 6], [-6, -11, -10, 3, 10, 11, 6, -6], [10, 2, -8, 8, -8, 2, 10, 0], [-4, -4, -6, -16, 4, 20, 12, 20], [20, 3, -4, 0, -16, -6, -16, 7], [10, 6, -2, 0, -10, -6, 4, -6], [-2, -19, 0, 19, 4, 15, 10, -4], [-8, 4, 0, -4, 16, 0, 16, 4], [0, -6, -12, 2, 0, 8, 0, -8], [11, -14, 0, 14, -22, 14, -4, 0], [-8, 10, 0, -10, 16, 0, -12, 10], [-5, 0, 10, 2, 5, -2, -20, -2], [10, -8, 4, 0, -14, 16, -14, -4], [6, -3, 2, 7, -6, -4, -4, -4], [-12, -6, 13, -4, 6, 4, -13, 6], [-8, -13, 8, 3, 8, -13, -8, 0], [-4, 12, -1, 16, 3, 0, -3, -8], [-6, -14, 9, -6, 3, 6, -9, 14], [0, 3, 14, 2, -10, 2, 14, 3], [-7, 0, 7, 14, 3, 14, -4, 14], [2, 0, -2, -9, -10, 3, -8, -9], [-4, 16, 8, 7, 2, -7, -8, -16], [-2, 0, 2, -1, -6, -11, -8, -1], [1, -8, 1, 2, -21, 4, 20, -2], [20, -1, 8, 5, 8, -1, 20, 0], [-12, -12, 2, 0, 6, 0, -2, 12], [-8, 12, -8, 12, 2, -6, 6, -12], [-16, -2, 0, 2, 32, -1, 4, -3], [4, 2, -14, 0, 10, -4, 10, 24], [0, 10, -3, -18, -25, -18, -3, 10], [11, 22, -8, 0, 8, -22, -11, 0], [18, -17, -12, -17, 0, 7, 6, 10], [-20, -4, -20, -8, 10, 2, 10, 8], [-8, 6, -11, -18, 8, 12, 22, 12], [5, 2, -38, 2, 5, 0, 0, 0], [14, 12, 14, -2, -18, -6, 4, 2], [0, -8, 8, -32, 8, 0, -8, 16], [-17, 0, 17, 4, -6, 8, -23, 4], [0, -2, -6, 6, 18, 6, -6, -2], [0, -28, -4, 7, 2, 7, -4, -28], [18, 9, 16, -14, -2, 0, 2, 7], [-5, -10, -9, -16, -5, -6, 0, 6], [10, 9, 14, -9, 14, 9, 10, 0], [-14, 20, -14, -19, -6, -10, 20, 19], [-10, 6, 4, 0, 6, -12, 6, -18], [30, 3, -8, 3, 0, -1, 4, -2], [6, 8, -10, -8, -6, 0, 20, 0], [-8, 0, 17, 0, -9, 0, -9, 4], [-2, 8, 0, -8, 4, 17, 2, 25], [7, 2, 27, 6, -27, -2, -7, -12], [10, 6, 0, -6, -20, 0, -22, 6], [9, 2, 4, 2, 0, -16, -2, 14], [8, -19, 20, 3, -4, -3, -20, 19], [-2, -4, -6, -16, -2, -12, 0, 12], [4, 16, -2, 21, -2, 16, 4, 0], [3, -8, 3, -10, 12, 4, -15, 10], [-12, 0, 12, 13, 2, 2, -10, 13], [12, 20, -12, 20, 0, 2, 6, -22], [20, -10, -6, -4, -10, 4, 6, 10], [-2, 2, 15, -24, 17, 0, -17, 12], [-4, -20, 0, -2, 0, -20, -4, 0], [6, -6, -20, -8, -20, -6, 6, 0], [20, 16, 18, -20, -20, 4, -36, 4], [-25, -6, 5, 0, 20, 12, 20, -18], [0, 20, 0, -20, 0, -22, -10, -2], [0, 3, 0, 8, -4, 8, 0, 3], [-3, 8, 12, 0, -3, -8, 0, 8], [1, 10, 0, -10, -2, 24, 10, 34], [12, 10, -9, 20, -12, -30, 18, -30], [4, 16, -16, 0, 12, -32, 12, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_390_bd_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_390_2_bd_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_390_2_bd_a(:prec:=8)
    chi := MakeCharacter_390_bd();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_390_2_bd_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_390_2_bd_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_390_bd();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![8464, 8096, 6272, 2144, 700, 112, 32, 4, 1]>],Snew);
    return Vf;
end function;