
// Make newform 900.3.c.m in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_900_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_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_900_3_c_m();".
// 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_900_3_c_m();".
// 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 := [2, -7, 8, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [4, -6, 2, -1], [-5, 8, -2, 1], [-9, 22, -4, 3]];
    Rf_basisdens := [1, 1, 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_900_c();"
function MakeCharacter_900_c()
    N := 900;
    order := 2;
    char_gens := [451, 101, 577];
    v := [1, 2, 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_900_c_Hecke();"
function MakeCharacter_900_c_Hecke(Kf)
    N := 900;
    order := 2;
    char_gens := [451, 101, 577];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 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 := 3;
    raw_aps := [[0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 2, 0, -2], [0, 1, -1, -2], [0, 0, -6, 2], [5, 0, -6, 2], [0, 3, 7, 4], [0, 6, 2, -4], [-12, 0, 6, -2], [0, 2, -2, -4], [30, 0, 6, -2], [-17, 0, 0, 0], [0, 12, 10, -2], [0, -8, -14, -6], [-30, 0, -24, 8], [0, -24, -6, 18], [-38, 0, 30, -10], [0, 11, 1, -10], [0, 32, 18, -14], [-55, 0, -6, 2], [0, 2, -12, -14], [0, 17, 11, -6], [53, 0, 6, -2], [-90, 0, 36, -12], [58, 0, 30, -10], [0, 16, 26, 10], [0, 23, 5, -18], [102, 0, -6, 2], [-25, 0, -24, 8], [0, -10, -18, -8], [0, -12, -38, -26], [145, 0, -36, 12], [0, -49, -31, 18], [-52, 0, 66, -22], [0, -10, -40, -30], [90, 0, 36, -12], [0, -45, -43, 2], [0, 24, 54, 30], [80, 0, -24, 8], [0, -7, 17, 24], [22, 0, 120, -40], [0, -46, -54, -8], [35, 0, -66, 22], [-110, 0, 84, -28], [0, 20, -20, -40], [0, 69, 31, -38], [0, 4, 8, 4], [0, -20, -82, -62], [-108, 0, 84, -28], [110, 0, -24, 8], [0, 84, 46, -38], [97, 0, 90, -30], [0, -65, 15, 80], [-90, 0, 84, -28], [0, -50, 18, 68], [-272, 0, -84, 28], [0, -62, -38, 24], [100, 0, -54, 18], [98, 0, 60, -20], [0, -57, 39, 96], [-10, 0, 186, -62], [0, 17, -15, -32], [0, 86, 64, -22], [-250, 0, 84, -28], [380, 0, -96, 32], [0, -53, 53, 106], [205, 0, 96, -32], [0, -73, 21, 94], [-98, 0, -126, 42], [-530, 0, 36, -12], [0, 86, 84, -2], [0, 116, 36, -80], [370, 0, 84, -28], [0, 77, 113, 36], [0, -38, -104, -66], [158, 0, -24, 8], [-140, 0, -24, 8], [-117, 0, 90, -30], [457, 0, -36, 12], [0, 17, -27, -44], [452, 0, -180, 60], [0, 78, 172, 94], [315, 0, -126, 42], [0, 36, 34, -2], [0, 133, 65, -68], [63, 0, -234, 78], [115, 0, -324, 108], [388, 0, -240, 80], [0, -20, -168, -148], [0, 4, -6, -10], [0, -182, -58, 124], [0, 94, 78, -16], [0, -76, -74, 2], [0, -60, -90, -30], [0, -76, -56, 20], [58, 0, -24, 8], [23, 0, -360, 120], [0, 29, -117, -146], [392, 0, -240, 80], [0, 233, 169, -64], [360, 0, 114, -38], [0, 160, 18, -142], [43, 0, 216, -72], [0, 268, 182, -86], [325, 0, -264, 88], [0, -99, 37, 136], [-985, 0, 36, -12], [0, -210, -240, -30], [-513, 0, 30, -10], [0, -328, -160, 168], [350, 0, 24, -8], [-170, 0, -336, 112], [0, 68, 42, -26], [0, 162, 238, 76], [-242, 0, -180, 60], [0, -148, -50, 98], [0, 252, 136, -116], [670, 0, -54, 18], [0, 311, 109, -202], [132, 0, -270, 90], [70, 0, 24, -8], [200, 0, -36, 12], [0, 227, 81, -146], [0, -89, 39, 128], [48, 0, 30, -10], [-668, 0, -276, 92], [0, 82, 308, 226], [0, 100, -78, -178], [-160, 0, 114, -38], [0, -84, -46, 38], [0, -42, -70, -28], [0, -180, -70, 110], [440, 0, -384, 128], [-537, 0, 120, -40], [-443, 0, 114, -38], [180, 0, -294, 98], [0, -196, -182, 14], [990, 0, -216, 72], [318, 0, -564, 188], [0, 44, 6, -38], [818, 0, 180, -60], [0, -196, -152, 44], [0, -75, -187, -112], [-488, 0, 354, -118], [0, -16, 196, 212], [530, 0, -276, 92], [385, 0, 24, -8], [0, 119, 161, 42], [0, 60, 278, 218], [1500, 0, -24, 8], [-502, 0, -420, 140], [0, -7, -71, -64], [0, -4, -148, -144], [0, -108, -190, -82], [0, 76, 124, 48], [0, 198, 62, -136], [258, 0, -24, 8], [-295, 0, 606, -202], [968, 0, 150, -50], [0, -88, -134, -46], [-355, 0, -504, 168], [0, -104, 176, 280], [0, -423, -177, 246], [-175, 0, -426, 142], [0, -398, -184, 214], [0, -94, -56, 38], [260, 0, 276, -92]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_900_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_c_m();".
// 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_900_3_c_m(:prec:=4)
    chi := MakeCharacter_900_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_900_3_c_m();".
// 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_900_3_c_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![2000, 0, 136, 0, 1]>,<13,R![-160, 4, 1]>,<17,R![-155, -6, 1]>],Snew);
    return Vf;
end function;