
// Make newform 975.2.a.m in Magma, downloaded from the LMFDB on 03 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_975_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_975_2_a_m();".
// 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_975_2_a_m();".
// 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, -3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, -1, 1]];
    Rf_basisdens := [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_975_a();"
function MakeCharacter_975_a()
    N := 975;
    order := 1;
    char_gens := [326, 352, 301];
    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_975_a_Hecke(Kf)
    return MakeCharacter_975_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, 0, -1], [-1, 0, 0], [0, 0, 0], [1, -2, -1], [-2, 1, 2], [1, 0, 0], [-5, 2, 0], [1, 3, -1], [-1, -1, 3], [-3, 0, 0], [0, 3, -2], [-5, 1, 1], [-1, -5, 3], [-1, 3, 5], [-1, 0, -1], [2, -1, 3], [-5, -2, -5], [3, -6, -4], [2, -5, -2], [-3, 7, -1], [-3, 3, 5], [-1, -1, 5], [-1, -4, -7], [-4, -4, -6], [-16, 2, 0], [0, -1, -3], [-5, 5, -1], [6, 0, -2], [-5, -5, -7], [-10, 0, -2], [-3, 5, 1], [7, -7, 1], [2, 2, -4], [8, -2, -2], [-8, -6, 0], [5, -2, -3], [0, 5, -5], [5, -5, 3], [11, 5, -1], [4, 3, 5], [7, -11, -5], [-6, 13, 1], [8, 2, 2], [-15, 1, -1], [1, 1, 1], [-6, 0, 4], [3, 3, 11], [1, -1, -1], [-8, -7, -6], [-6, 8, 8], [6, 4, 6], [-13, 10, 5], [3, 3, -3], [-15, 3, 1], [-8, -3, 5], [-7, -9, -3], [-7, 8, 6], [-1, 0, 11], [-8, -2, -4], [0, -6, -14], [-6, 0, -12], [2, 12, 4], [9, -5, -3], [10, -4, -4], [-16, 3, 1], [-7, -1, -3], [17, 7, -1], [6, -1, 15], [4, -10, 2], [1, 7, 3], [-11, -11, 1], [-23, -2, 3], [-16, 12, 4], [-10, -1, 3], [4, -11, 4], [-3, -13, -13], [6, 8, -6], [2, -8, 2], [7, -7, 1], [13, -3, -3], [-1, -15, -9], [-18, 2, 2], [-9, -7, -3], [-4, -10, -12], [19, 7, 11], [15, 7, -7], [-10, 10, 6], [5, -3, -7], [4, -6, -14], [-2, -1, 4], [-12, 12, 14], [-10, -9, 0], [-4, 3, 0], [0, 18, 2], [-3, -2, -19], [-3, -5, 3], [15, -5, 9], [10, -4, 0], [-10, -12, -6], [-15, -13, 9], [7, -15, 7], [1, -7, 15], [11, 7, -1], [0, 21, -5], [25, -15, -7], [-4, -18, 8], [8, -3, 6], [4, -10, 6], [17, -5, -13], [-30, 9, 7], [4, -6, -16], [-10, 24, 12], [-34, 0, 2], [-23, -9, 1], [11, 1, 19], [7, -2, -4], [11, 13, 9], [24, -16, -10], [5, 8, -6], [-7, 9, 15], [-7, 1, 5], [-6, 11, 11], [-12, 10, -2], [8, -9, -2], [25, -10, -5], [3, 12, 12], [11, -17, -5], [1, 7, 5], [3, 1, 23], [6, -10, 6], [36, -1, -2], [18, -11, -12], [27, -1, -9], [-1, 10, 6], [-9, 17, 15], [2, 14, -12], [20, 2, 4], [6, 17, 0], [-27, 10, -6], [-16, 6, 4], [-11, -14, -3], [-17, 17, 5], [-8, 4, 10], [-14, 13, -4], [-17, 8, 18], [-13, 9, 5], [24, -4, 8], [6, -20, -12], [9, 9, 11], [16, 11, 28], [26, -14, -8], [18, -13, -9], [26, -10, -8], [33, 13, -9], [0, -12, 8], [5, -11, -7], [-12, -2, -16], [0, -2, 10], [31, -20, -2], [-15, -1, -7], [14, -3, -8], [32, 1, -9], [6, 13, 26], [-20, 14, 0], [-18, 26, 2], [-20, -15, -8], [-25, 3, -17], [13, -12, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_975_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_975_2_a_m();".
// 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_975_2_a_m(:prec:=3)
    chi := MakeCharacter_975_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(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_975_2_a_m();".
// 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_975_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_975_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, -1, 3, 1]>,<7,R![23, -13, -1, 1]>],Snew);
    return Vf;
end function;