
// Make newform 325.4.c.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_325_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_325_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_325_4_c_b();".
// 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_325_4_c_b();".
// 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;
    Rf_num := [[1, 0], [0, 3]];
    Rf_basisdens := [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_325_c();"
function MakeCharacter_325_c()
    N := 325;
    order := 2;
    char_gens := [27, 301];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_325_c_Hecke();"
function MakeCharacter_325_c_Hecke(Kf)
    N := 325;
    order := 2;
    char_gens := [27, 301];
    char_values := [[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], [1, 0], [0, 0], [0, -5], [0, -16], [-26, -13], [45, 0], [0, -2], [-162, 0], [-144, 0], [0, -88], [0, -101], [0, 64], [97, 0], [0, -37], [414, 0], [0, 174], [376, 0], [0, -12], [0, -119], [0, 366], [-830, 0], [0, -146], [0, -146], [0, -284], [396, 0], [-182, 0], [612, 0], [0, -361], [-90, 0], [-2086, 0], [-1467, 0], [0, 138], [-2419, 0], [0, 310], [0, -561], [-1874, 0], [0, 398], [0, 796], [1566, 0], [-657, 0], [-1222, 0], [1260, 0], [0, -114], [0, 27], [1996, 0], [2833, 0], [0, -1169], [0, -76], [0, 1831], [-3627, 0], [0, -2025], [0, 70], [7092, 0], [5805, 0], [-792, 0], [5472, 0], [0, 777], [1384, 0], [0, -1354], [3764, 0], [0, -1409], [0, -102], [-2106, 0], [-10051, 0], [0, -718], [0, -3590], [2171, 0], [7047, 0], [0, -2291], [0, -3106], [0, 1376], [2536, 0], [92, 0], [0, -3394], [0, -193], [2106, 0], [0, 658], [0, 3962], [0, -418], [5823, 0], [0, 2447], [0, 2495], [15203, 0], [-1762, 0], [7317, 0], [0, -1672], [0, 3290], [0, 4847], [0, 704], [-3276, 0], [0, -5151], [0, -1220], [747, 0], [0, 5268], [12078, 0], [0, -5370], [3915, 0], [-16184, 0], [0, -2641], [14389, 0], [0, 3461], [16425, 0], [12213, 0], [-6383, 0], [0, 2142], [0, -7110], [0, -4028], [2394, 0], [-21971, 0], [15406, 0], [0, -3210], [0, 4916], [0, -1224], [0, -6625], [1710, 0], [0, -5484], [-25902, 0], [-18108, 0], [-32904, 0], [0, 5106], [7729, 0], [-19242, 0], [0, -7506], [0, -3056], [1170, 0], [0, -554], [30960, 0], [8372, 0], [0, -913], [0, -2252], [0, 9881], [18128, 0], [6410, 0], [0, 9430], [0, 9320], [0, -1883], [0, -252], [-31194, 0], [17055, 0], [0, -11840], [0, -365], [2554, 0], [0, -7174], [-13124, 0], [0, -7808], [0, -10359], [19422, 0], [1744, 0], [0, 6393], [0, 9739], [-15633, 0], [-30589, 0], [25884, 0], [12305, 0], [29772, 0], [47644, 0], [0, -7308], [-32398, 0], [0, -699], [0, 6672], [-24993, 0], [0, -13653], [-48933, 0], [0, 15796], [0, -5601], [-57526, 0], [25000, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_325_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_325_4_c_b();".
// 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_325_4_c_b(:prec:=2)
    chi := MakeCharacter_325_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_325_4_c_b();".
// 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_325_4_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_325_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![9, 0, 1]>,<3,R![-1, 1]>],Snew);
    return Vf;
end function;