
// Make newform 825.4.c.e in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_825_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_825_c_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_825_4_c_e();".
// 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_825_4_c_e();".
// 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_825_c();"
function MakeCharacter_825_c()
    N := 825;
    order := 2;
    char_gens := [551, 727, 376];
    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_825_c_Hecke();"
function MakeCharacter_825_c_Hecke(Kf)
    N := 825;
    order := 2;
    char_gens := [551, 727, 376];
    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, 1], [0, -3], [0, 0], [0, 36], [11, 0], [0, -2], [0, 66], [-140, 0], [0, 68], [-150, 0], [-128, 0], [0, -314], [-118, 0], [0, -172], [0, -324], [0, -82], [740, 0], [122, 0], [0, -124], [-988, 0], [0, -2], [-1100, 0], [0, 868], [470, 0], [0, 1186], [1502, 0], [0, -32], [0, 1116], [2190, 0], [0, 1398], [0, -44], [-1308, 0], [0, 1626], [-180, 0], [-1430, 0], [-1948, 0], [0, 646], [0, -3052], [0, 1216], [0, 3858], [380, 0], [-538, 0], [1412, 0], [0, 638], [0, 3686], [240, 0], [5092, 0], [0, 3808], [0, -44], [2650, 0], [0, 5718], [6520, 0], [-2438, 0], [-5268, 0], [0, -5574], [0, -2472], [-4410, 0], [-1308, 0], [0, 226], [522, 0], [0, 1508], [0, 4658], [0, 6476], [5652, 0], [0, 5638], [0, 1506], [-7268, 0], [0, -7254], [0, 2276], [-4610, 0], [0, -4602], [4160, 0], [0, -5024], [0, -10842], [1540, 0], [0, 9468], [10590, 0], [0, -7434], [11402, 0], [510, 0], [-2420, 0], [-8178, 0], [-6768, 0], [0, 478], [3260, 0], [0, -6812], [-15290, 0], [0, -12854], [6782, 0], [0, 6408], [0, 10476], [-1920, 0], [0, 8416], [17732, 0], [5580, 0], [0, -12512], [22390, 0], [882, 0], [0, -18172], [-17998, 0], [0, -4164], [0, 13366], [0, -24612], [70, 0], [-12348, 0], [0, 23426], [0, 21036], [0, 2798], [11100, 0], [15242, 0], [0, 13876], [0, -8722], [0, -8014], [-26020, 0], [-23528, 0], [-18998, 0], [0, 4348], [0, 31916], [0, -25682], [-14940, 0], [4502, 0], [0, -5362], [0, 14566], [0, -11852], [-1828, 0], [23182, 0], [33850, 0], [7140, 0], [0, 8896], [0, 13038], [-11620, 0], [0, 1328], [-37808, 0], [0, 34326], [4282, 0], [-24410, 0], [0, -8162], [0, 26596], [0, -24614], [-39930, 0], [2412, 0], [-6018, 0], [0, -34632], [0, -40044], [44090, 0], [23780, 0], [0, 4078], [0, 30666], [-4780, 0], [0, 9428], [0, 15266], [-5118, 0], [0, 44188], [0, -16864], [0, -37924], [-36628, 0], [21300, 0], [-31450, 0], [0, -6174], [20422, 0], [0, 12156], [0, -14442], [0, 33356], [3852, 0], [0, -37454], [0, 16228], [-23728, 0], [0, 41306]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_825_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_825_4_c_e();".
// 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_825_4_c_e(:prec:=2)
    chi := MakeCharacter_825_c();
    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_825_4_c_e();".
// 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_825_4_c_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, 0, 1]>,<7,R![1296, 0, 1]>],Snew);
    return Vf;
end function;