
// Make newform 945.2.bv.d in Magma, downloaded from the LMFDB on 23 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_945_bv();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_945_bv_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_945_2_bv_d();".
// 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_945_2_bv_d();".
// 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, -1, 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_945_bv();"
function MakeCharacter_945_bv()
    N := 945;
    order := 12;
    char_gens := [596, 757, 136];
    v := [8, 9, 2];
    // 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_945_bv_Hecke();"
function MakeCharacter_945_bv_Hecke(Kf)
    N := 945;
    order := 12;
    char_gens := [596, 757, 136];
    char_values := [[0, 0, -1, 0], [0, 0, 0, 1], [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, 1, 1, 0], [0, 0, 0, 0], [2, -1, -2, 0], [1, 0, 2, 0], [0, -2, -2, -2], [2, 0, -2, -2], [-2, 2, 2, 0], [0, -1, 5, -1], [2, -2, -3, -1], [4, 0, 4, 0], [3, 0, -6, 5], [0, 3, -3, -3], [-4, 6, 2, -6], [-3, 3, 0, -3], [-6, -9, 9, 6], [1, -1, 3, 4], [0, 4, 0, -2], [-6, 0, 12, 1], [5, -3, -3, 5], [6, 0, 0, 0], [-5, -5, 1, 4], [-2, 0, 4, -8], [1, 1, -1, 0], [0, -3, -4, -3], [-3, 3, 5, 2], [-18, -2, 9, 2], [4, 4, -7, 3], [2, -1, -1, -1], [0, -2, 0, 0], [6, 3, -3, 3], [2, -13, -13, 2], [-4, 12, -4, 0], [-4, 4, 0, 0], [0, -4, 0, -4], [5, -12, 5, 0], [0, -1, -3, -1], [-1, 0, 0, 1], [-4, -5, -1, 1], [3, -11, -14, 14], [10, 0, 0, 10], [-9, 3, -9, 0], [2, 0, -4, -8], [9, 10, 0, -5], [-13, -4, 4, 13], [12, 4, 4, 12], [6, -8, -6, 16], [0, -4, -2, -4], [4, 4, -5, 1], [10, 7, -3, 3], [4, -5, -4, 10], [10, 5, -5, 5], [0, 22, 0, -22], [-4, 13, 2, -13], [8, 0, -16, 8], [17, -17, -11, 6], [-3, -19, -16, 16], [-3, -4, 3, 8], [-4, -8, 2, 8], [-6, -6, 12, 12], [-12, -1, 12, 2], [-2, 13, 13, -2], [-16, 7, 9, 9], [-19, 4, -4, 19], [0, 0, 0, 16], [-3, 6, 6, -3], [-3, -4, -4, -3], [27, -10, 0, 5], [2, -6, 4, 4], [8, 5, 5, 8], [22, 7, -22, -14], [16, -14, -2, -2], [-2, -19, 1, 19], [-5, -3, 2, -2], [-3, 3, 3, 0], [-2, 0, 4, -16], [-9, -9, 2, 7], [6, -4, -3, 4], [0, 24, 24, -24], [13, 6, -13, -12], [-5, 12, 0, -6], [0, 0, 8, 0], [-4, -11, 4, 22], [-19, -3, 19, 6], [18, -6, -6, 18], [8, -16, 0, 8], [-3, 10, -10, 3], [-9, 0, 18, -14], [-21, 10, 10, -21], [-3, 4, -3, 0], [13, 21, 8, -8], [-15, -4, 11, -11], [0, 1, -7, 1], [-23, -23, 21, 2], [0, -4, -10, -4], [-4, 14, -4, 0], [-21, 8, 8, -21], [18, 0, -18, 0], [-17, -14, -17, 0], [1, -5, 4, 4], [0, 4, 3, 4], [-1, -1, -21, 22], [9, 9, -3, -6], [-22, 5, 5, -22], [-12, 0, 24, -2], [4, -28, 0, 14], [-20, 20, 8, -12], [-14, 14, 21, 7], [0, 2, -2, -2], [15, 0, -30, -5], [8, 10, 8, 0], [3, -3, -3, 0], [3, -3, -23, -20], [4, -7, 3, 3], [0, 15, -1, 15], [20, 20, 0, -10], [0, -17, -2, -17], [-15, 21, -6, -6], [-19, 19, -5, -24], [15, -15, -17, -2], [0, -26, 0, 0], [0, 0, 0, -11], [8, -8, -16, -8], [22, 38, -38, -22], [15, -15, 4, 19], [9, 0, -18, 5], [44, 4, 0, -2], [-16, 0, 32, -21], [0, -7, -19, -7], [14, -14, -1, 13], [-12, -12, 2, 10], [-6, -14, -6, 0], [-4, -5, -1, 1], [35, 7, -35, -14], [-11, 2, 2, -11], [-9, -4, -9, 0], [0, -15, 10, -15], [11, 11, -27, 16], [-8, -9, 9, 8], [-4, 29, 33, -33], [-26, 3, -26, 0], [8, 0, -16, 12], [16, 10, 0, -5], [-17, 8, -8, 17], [-5, -12, -12, -5], [27, 10, -27, -20], [-19, 11, 19, -22], [26, 26, -16, -10], [-22, -13, 9, -9], [17, 15, -17, -30], [-27, -8, 19, -19], [-7, -7, 3, 4], [-12, 0, 24, -3], [-11, -37, 37, 11], [-2, 2, -9, -11], [16, -7, -9, -9], [27, -7, -27, 14], [-12, -42, 6, 42], [19, 20, 0, -10], [-8, -4, 4, 8], [-15, 0, 30, 12], [5, -17, -17, 5], [-21, 0, 0, 21], [17, -22, 5, 5], [8, -4, -4, 4], [-16, 36, 36, -16], [-12, 19, -7, -7], [14, 2, -14, -4], [-30, -21, 9, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_945_bv_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_945_2_bv_d();".
// 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_945_2_bv_d(:prec:=4)
    chi := MakeCharacter_945_bv();
    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_945_2_bv_d();".
// 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_945_2_bv_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_945_bv();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 2, 2, -2, 1]>,<11,R![64, -32, 24, 4, 1]>,<13,R![16, -32, 20, -4, 1]>],Snew);
    return Vf;
end function;