
// Make newform 525.4.d.f in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_525_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_525_d_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_525_4_d_f();".
// 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_525_4_d_f();".
// 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];
        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_525_d();"
function MakeCharacter_525_d()
    N := 525;
    order := 2;
    char_gens := [176, 127, 451];
    v := [2, 1, 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_525_d_Hecke();"
function MakeCharacter_525_d_Hecke(Kf)
    N := 525;
    order := 2;
    char_gens := [176, 127, 451];
    char_values := [[1, 0], [-1, 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 := 4;
    raw_aps := [[0, 0], [0, -3], [0, 0], [0, -7], [42, 0], [0, 20], [0, -66], [-38, 0], [0, 12], [258, 0], [146, 0], [0, -434], [-282, 0], [0, 20], [0, 72], [0, 336], [360, 0], [-682, 0], [0, -812], [810, 0], [0, -124], [-1136, 0], [0, 156], [1038, 0], [0, -1208], [546, 0], [0, -520], [0, -1212], [1078, 0], [0, -1452], [0, 1312], [-1356, 0], [0, 984], [394, 0], [1014, 0], [-1996, 0], [0, 2392], [0, 2036], [0, 3936], [0, 378], [222, 0], [-2590, 0], [2214, 0], [0, 4178], [0, 3060], [-2666, 0], [-1348, 0], [0, 3188], [0, 3396], [-5294, 0], [0, 852], [-4866, 0], [-2050, 0], [-1152, 0], [0, 6450], [0, 1968], [3894, 0], [7094, 0], [0, 3310], [7158, 0], [0, -5164], [0, -8598], [0, 448], [-5832, 0], [0, 9848], [0, 5616], [452, 0], [0, 2302], [0, -1584], [-8174, 0], [0, 8610], [2154, 0], [0, -6644], [0, 7958], [-3440, 0], [0, 12936], [14862, 0], [0, -10460], [-9150, 0], [4894, 0], [1668, 0], [-12418, 0], [15186, 0], [0, -5704], [17206, 0], [0, 3456], [-16074, 0], [0, -7526], [-2274, 0], [0, -10024], [0, 2460], [-19320, 0], [0, 12544], [-15510, 0], [14344, 0], [0, -21384], [7134, 0], [-19122, 0], [0, -15640], [2846, 0], [0, 4444], [0, -18552], [0, -16452], [-7722, 0], [2576, 0], [0, 2464], [0, -1452], [0, 10698], [8730, 0], [1910, 0], [0, 5596], [0, 28586], [0, 19236], [-6734, 0], [7184, 0], [510, 0], [0, -20752], [0, -21072], [0, 2892], [-750, 0], [30062, 0], [0, 15446], [0, 25110], [0, 7968], [-14398, 0], [-9234, 0], [-8030, 0], [-27060, 0], [0, 3724], [0, -5668], [16072, 0], [0, -8256], [-6352, 0], [0, -11558], [7770, 0], [-22646, 0], [0, -35502], [0, 17080], [0, -5730], [-2550, 0], [-27538, 0], [-19242, 0], [0, -11752], [0, 28692], [-28442, 0], [-20172, 0], [0, 19820], [0, 10290], [31606, 0], [0, 23172], [0, 15550], [28530, 0], [0, -28780], [0, -22872], [0, 10708], [1326, 0], [13696, 0], [-42354, 0], [0, -6644], [1350, 0], [0, -49320], [0, 5940], [0, -47216], [12552, 0], [0, -46908], [0, -46128], [-12184, 0], [0, 5164]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_525_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_525_4_d_f();".
// 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_525_4_d_f(:prec:=2)
    chi := MakeCharacter_525_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_525_4_d_f();".
// 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_525_4_d_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![0, 1]>,<11,R![-42, 1]>],Snew);
    return Vf;
end function;