
// Make newform 975.2.bp.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_975_bp();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_975_bp_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_975_2_bp_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_975_2_bp_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, -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_975_bp();"
function MakeCharacter_975_bp()
    N := 975;
    order := 12;
    char_gens := [326, 352, 301];
    v := [6, 6, 5];
    // 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_975_bp_Hecke();"
function MakeCharacter_975_bp_Hecke(Kf)
    N := 975;
    order := 12;
    char_gens := [326, 352, 301];
    char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0], [0, 1, 0, 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, 0, 0], [-2, 0, 1, 0], [0, 0, 0, 0], [-1, 1, -2, -3], [0, 0, 0, 0], [-1, 0, -3, 0], [0, 0, 0, 0], [-5, 5, 3, -2], [0, 0, 0, 0], [0, 0, 0, 0], [-1, 5, -5, 1], [-4, -7, -3, 3], [0, 0, 0, 0], [0, -1, 0, 2], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -5, 0, 10], [-7, -2, 9, 9], [0, 0, 0, 0], [-8, 1, -1, 8], [0, 14, 0, -7], [0, 0, 0, 0], [0, 0, 0, 0], [3, 3, 8, -11], [0, 0, 0, 0], [0, 18, 0, -9], [0, 0, 0, 0], [-12, -5, 5, 12], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-7, 0, 7, 0], [0, 0, 0, 0], [-9, 14, 14, -9], [0, 0, 0, -11], [14, -14, -11, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [4, 0, -8, 0], [0, 0, 0, 0], [-9, 7, 16, -16], [0, 0, 0, 0], [0, -17, 0, 0], [0, -15, 0, -15], [-6, -11, 17, 17], [0, 0, 0, 0], [-17, 12, 12, -17], [0, 0, 0, 0], [0, 0, 0, 0], [-15, -15, -1, 16], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [9, 10, -19, -19], [0, -12, 0, 24], [0, 0, 0, 0], [0, 0, 25, 0], [0, 0, 0, 0], [-18, -1, 1, 18], [0, 0, 0, 0], [19, 0, -38, 0], [0, 0, 0, 0], [-21, 21, 10, -11], [-29, 0, 0, 0], [0, 0, 0, 0], [17, 3, -20, -20], [0, 0, 0, 0], [0, 0, 0, 0], [0, 31, 0, -31], [-21, 0, -21, 0], [-7, -22, -15, 15], [0, 0, 0, 0], [0, 0, 0, 0], [12, -12, 11, 23], [0, 0, 0, 0], [8, -8, -23, -15], [0, 0, 0, 0], [-1, -21, 21, 1], [0, 0, 0, 0], [-35, 0, 35, 0], [46, 0, -23, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-7, 24, -17, -17], [0, 0, 0, 0], [-21, 1, -1, 21], [0, 0, 0, 0], [0, 0, 0, 0], [-23, -23, 21, 2], [0, 0, 0, 0], [-25, -18, 18, 25], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 8, 0, -8], [4, -25, -25, 4], [0, 0, 0, 41], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 16], [-19, -8, -8, -19], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 24, 0, 24], [0, 20, 0, 0], [-19, 28, -9, -9], [0, 0, 0, 0], [22, -27, -27, 22], [-15, -15, -14, 29], [0, 0, 0, 0], [-29, -29, 18, 11], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [9, -29, 20, 20], [0, -29, 0, -29], [0, 0, 0, 0], [0, 0, 0, 0], [-19, -30, -11, 11], [0, 0, 0, 0], [28, 28, -25, -3], [0, 0, 0, 0], [49, 0, 0, 0], [19, -31, -31, 19], [30, -23, -7, -7], [0, 0, 0, 0], [-20, 0, 10, 0], [56, 0, -28, 0], [0, 0, 0, 0], [32, 17, -15, 15], [0, 0, 0, 0], [-27, 27, -2, -29], [0, 0, 0, 0], [0, 0, 0, 0], [25, 31, -31, -25], [0, 0, 0, 0], [0, -14, 0, 28], [0, 0, 0, 0], [0, -53, 0, 53], [0, 0, 0, 0], [31, 27, -27, -31], [0, 0, 0, 0], [0, 66, 0, -33], [0, 0, 0, 0], [3, 3, -31, 28], [0, 0, 0, 0], [55, 0, 0, 0], [0, 0, 0, 0], [0, 0, 40, 0], [0, 0, 0, 0], [0, 18, 0, -36], [0, 0, 0, 0], [-32, 0, 64, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [7, -34, -34, 7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -44, 0], [0, 59, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_975_bp_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_975_2_bp_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_975_2_bp_a(:prec:=4)
    chi := MakeCharacter_975_bp();
    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_bp_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_975_2_bp_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_975_bp();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![169, 130, 41, 8, 1]>],Snew);
    return Vf;
end function;