
// Make newform 825.4.c.c in Magma, downloaded from the LMFDB on 17 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_c();".
// 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_c();".
// 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, 4], [0, 3], [0, 0], [0, -21], [11, 0], [0, -68], [0, -21], [-125, 0], [0, 137], [150, 0], [292, 0], [0, 349], [497, 0], [0, -208], [0, 369], [0, 542], [-235, 0], [482, 0], [0, 734], [587, 0], [0, -518], [1045, 0], [0, -608], [770, 0], [0, -1541], [827, 0], [0, -248], [0, -366], [-270, 0], [0, 1002], [0, 469], [-408, 0], [0, -2466], [-1020, 0], [-5, 0], [452, 0], [0, -1766], [0, -2068], [0, -3386], [0, 117], [-2995, 0], [4067, 0], [3047, 0], [0, 1232], [0, 4979], [-600, 0], [-2468, 0], [0, 5392], [0, -2366], [4645, 0], [0, -513], [-2690, 0], [-3728, 0], [2352, 0], [0, -3846], [0, 522], [-4020, 0], [6687, 0], [0, -3746], [-5883, 0], [0, -3943], [0, 1487], [0, 4844], [4632, 0], [0, 8437], [0, -3636], [-758, 0], [0, 7374], [0, 6524], [-3710, 0], [0, 2832], [7040, 0], [0, -6206], [0, 1962], [7960, 0], [0, -7188], [-7920, 0], [0, 9654], [1952, 0], [-9690, 0], [2935, 0], [12837, 0], [-6108, 0], [0, -9278], [-2455, 0], [0, -3503], [7630, 0], [0, 7414], [4982, 0], [0, 13422], [0, 15804], [9060, 0], [0, 2854], [-2278, 0], [-13290, 0], [0, 10762], [1570, 0], [-22638, 0], [0, -10273], [-6628, 0], [0, -1131], [0, 22954], [0, 5532], [25225, 0], [-2088, 0], [0, -7831], [0, 8199], [0, 9542], [-24705, 0], [15452, 0], [0, -6176], [0, -13198], [0, -19216], [-27700, 0], [-5108, 0], [322, 0], [0, 7432], [0, 1409], [0, -21548], [13380, 0], [10907, 0], [0, 17522], [0, -8306], [0, 18427], [-8278, 0], [-21923, 0], [11425, 0], [-7380, 0], [0, 10924], [0, -13578], [-2875, 0], [0, -9568], [-35048, 0], [0, -8226], [-6818, 0], [-29390, 0], [0, -34358], [0, -14291], [0, -11576], [12825, 0], [-36843, 0], [43962, 0], [0, 33522], [0, 1704], [-1750, 0], [15260, 0], [0, -878], [0, 35019], [1280, 0], [0, -16888], [0, -29836], [29292, 0], [0, 6532], [0, -20476], [0, 51914], [-41893, 0], [-495, 0], [16310, 0], [0, 18744], [-25553, 0], [0, 6879], [0, 13677], [0, 26984], [41937, 0], [0, 13504], [0, -33353], [-16978, 0], [0, 2714]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![16, 0, 1]>,<7,R![441, 0, 1]>],Snew);
    return Vf;
end function;