
// Make newform 390.2.l.b in Magma, downloaded from the LMFDB on 03 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_390_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_390_l_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_l_b();".
// 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_l_b();".
// 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];
        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_l();"
function MakeCharacter_390_l()
    N := 390;
    order := 4;
    char_gens := [131, 157, 301];
    v := [2, 3, 4];
    // 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_l_Hecke();"
function MakeCharacter_390_l_Hecke( : Kf := false)
    N := 390;
    order := 4;
    char_gens := [131, 157, 301];
    char_values := [[-1, 0, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, -1, 0, -1], [0, -2, 0, -1], [2, 0, -2, 2], [0, 0, -4, 0], [0, -1, 0, 0], [3, 0, 3, 0], [0, 2, 4, 2], [0, 0, 0, -4], [6, 0, 0, 0], [2, 1, 0, -1], [-4, 0, 4, -2], [0, 0, -2, 0], [-3, 6, -3, 0], [-2, 6, -2, 0], [-2, 0, 2, -2], [4, -6, 0, 6], [2, 2, 0, -2], [2, 0, -2, 4], [0, 3, 10, 3], [-4, -10, -4, 0], [0, 6, 4, 6], [4, 0, -4, -4], [-14, 2, 0, -2], [2, 0, -2, -10], [0, 2, -6, 2], [4, 12, 4, 0], [-5, -10, -5, 0], [0, 5, -12, 5], [-7, 0, 7, 0], [-6, 0, 6, -4], [0, 12, -4, 12], [-8, -10, -8, 0], [0, -4, 4, -4], [-4, -13, 0, 13], [-6, 5, 0, -5], [12, 0, -12, -6], [8, -8, 8, 0], [-2, -2, -2, 0], [10, 0, -10, -10], [12, 4, 0, -4], [6, 0, 0, 0], [0, 2, 0, 2], [10, 2, 10, 0], [-8, 6, -8, 0], [0, 10, 8, 10], [-4, 4, 0, -4], [-4, -8, -4, 0], [-2, 0, -2, 0], [0, -7, -8, -7], [3, 0, -3, -8], [10, 7, 0, -7], [2, -4, 0, 4], [0, -4, 20, -4], [-7, 0, -7, 0], [-16, 0, 16, 0], [-18, 2, 0, -2], [2, -3, 0, 3], [-6, 0, 6, 14], [0, -2, 2, -2], [-13, -6, -13, 0], [-4, 0, 4, -12], [10, 0, -10, 4], [0, 0, 28, 0], [7, -8, 7, 0], [-4, 8, -4, 0], [-4, -20, 0, 20], [-23, 0, 23, 0], [5, -22, 5, 0], [0, -9, -20, -9], [-10, 0, 10, 22], [-22, -1, 0, 1], [0, 0, 0, -20], [12, -10, 12, 0], [0, 10, 24, 10], [4, 0, -4, 8], [2, 0, 0, 0], [-12, 0, 12, -4], [0, -14, 18, -14], [0, -2, 6, -2], [14, -6, 0, 6], [20, 5, 0, -5], [0, 5, -18, 5], [-5, -16, -5, 0], [0, 4, -24, 4], [-9, 0, 9, -14], [2, -14, 0, 14], [-14, 0, 14, 10], [0, 21, -8, 21], [0, 8, 0, 0], [-7, -14, -7, 0], [6, 13, 0, -13], [10, 0, -10, 10], [0, -14, 6, -14], [0, 0, 8, 0], [2, 0, -2, -20], [-20, -15, 0, 15], [0, -12, -8, -12], [1, -10, 1, 0], [0, -3, 0, 3], [-19, 0, 19, 2], [-8, 10, -8, 0], [15, 0, -15, -6], [24, -4, 0, 4], [26, 6, 0, -6], [8, 0, -8, 34], [-12, 28, -12, 0], [-6, 0, 6, 2], [-8, -6, 0, 6], [-12, -4, 0, 4], [6, 0, -6, -12], [16, -8, 16, 0], [18, -10, 18, 0], [0, -16, -4, -16], [14, 7, 0, -7], [0, 8, -22, 8], [8, 20, 8, 0], [-18, -16, -18, 0], [0, 0, 0, -6], [34, -6, 0, 6], [16, 23, 0, -23], [-17, 20, -17, 0], [8, -26, 8, 0], [-6, 0, 6, 4], [-32, -4, 0, 4], [0, 18, 10, 18], [0, 9, -16, 9], [-4, -6, 0, 6], [-14, 0, 14, 16], [0, -22, 0, 0], [0, 10, -4, 10], [12, 0, -12, 32], [8, 20, 0, -20], [2, 0, -2, -2], [0, -6, -18, -6], [0, -16, 14, -16], [12, 0, -12, -6], [14, 0, -14, 28], [14, -10, 14, 0], [0, -16, 0, 16], [8, 10, 0, -10], [0, -21, 16, -21], [-22, -16, -22, 0], [30, 4, 30, 0], [0, -6, 46, -6], [30, -7, 0, 7], [-24, 24, -24, 0], [23, -4, 23, 0], [0, -14, -18, -14], [-18, 0, 18, 6], [24, 0, -24, -16], [0, 0, -14, 0], [1, 30, 1, 0], [4, 0, 4, 0], [15, 0, -15, 14], [0, -10, 16, -10], [0, 24, -24, 24], [26, 22, 0, -22], [15, 0, -15, -24], [0, 19, 12, 19], [-10, 32, -10, 0], [-13, 0, 13, -12], [-8, 0, 8, -32], [0, 10, -6, 10], [20, 2, 20, 0], [-2, 0, 2, 2], [-8, 2, 0, -2], [-8, 0, 8, -42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_390_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_390_2_l_b();".
// 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_l_b(:prec:=4)
    chi := MakeCharacter_390_l();
    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_l_b();".
// 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_l_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_390_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, -32, 32, -8, 1]>,<17,R![18, -6, 1]>],Snew);
    return Vf;
end function;