
// Make newform 900.3.c.p 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_p();".
// 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_p();".
// 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 := [256, 0, -32, 0, 4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0], [-16, 0, 4, 0, 6, 0, 1, 0], [-1, 0, 2, 0, 0, 0, 0, 0], [0, -2, 0, 1, 0, 0, 0, 0], [0, 32, 0, -28, 0, 14, 0, 1], [0, 32, 0, -4, 0, 2, 0, -1], [16, 0, -4, 0, 2, 0, -1, 0]];
    Rf_basisdens := [1, 1, 16, 1, 1, 64, 64, 16];
    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, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, -1, -1, 0], [-2, 0, 0, -1, 0, 0, 0, -2], [0, -1, 0, 0, -1, 1, -3, 0], [1, 0, 1, -1, 0, 0, 0, 2], [0, 2, 0, 0, 3, 3, -5, 0], [0, -1, 0, 0, -3, 3, -1, 0], [-2, 0, -1, 2, 0, 0, 0, -4], [-4, 0, 0, -2, 0, 0, 0, -4], [0, 3, 0, 0, -1, 1, 13, 0], [5, 0, -1, -5, 0, 0, 0, 10], [0, -4, 0, 0, -2, -2, 14, 0], [0, 5, 0, 0, 3, -3, 17, 0], [0, 0, 0, 0, -6, -6, -6, 0], [12, 0, 0, 7, 0, 0, 0, 14], [5, 0, 7, -5, 0, 0, 0, 10], [0, 10, 0, 0, 1, 1, -39, 0], [-40, 0, 0, -10, 0, 0, 0, -20], [6, 0, -8, -6, 0, 0, 0, 12], [0, 4, 0, 0, -10, -10, -26, 0], [0, 8, 0, 0, 4, -4, 28, 0], [-15, 0, 0, 0, 0, 0, 0, 0], [0, 13, 0, 0, -1, 1, 53, 0], [10, 0, 8, -10, 0, 0, 0, 20], [0, -8, 0, 0, -1, -1, 31, 0], [2, 0, 0, -5, 0, 0, 0, -10], [0, 8, 0, 0, -4, 4, 36, 0], [0, 0, -16, 0, 0, 0, 0, 0], [0, 20, 0, 0, 9, 9, -71, 0], [0, -6, 0, 0, 10, -10, -34, 0], [2, 0, 24, -2, 0, 0, 0, 4], [0, 10, 0, 0, 10, -10, 30, 0], [-4, 0, -15, 4, 0, 0, 0, -8], [12, 0, 0, -9, 0, 0, 0, -18], [-5, 0, -23, 5, 0, 0, 0, -10], [0, -6, 0, 0, 21, 21, 45, 0], [0, 5, 0, 0, -13, 13, 33, 0], [0, 20, 0, 0, 16, 16, -64, 0], [-60, 0, 0, 1, 0, 0, 0, 2], [0, 10, 0, 0, -23, -23, -63, 0], [-75, 0, 0, 10, 0, 0, 0, 20], [0, -10, 0, 0, -2, 2, -38, 0], [-8, 0, -31, 8, 0, 0, 0, -16], [13, 0, 1, -13, 0, 0, 0, 26], [-20, 0, 23, 20, 0, 0, 0, -40], [0, -12, 0, 0, -11, -11, 37, 0], [146, 0, 0, 19, 0, 0, 0, 38], [0, 11, 0, 0, 11, -11, 33, 0], [0, -10, 0, 0, -15, -15, 25, 0], [55, 0, 0, -6, 0, 0, 0, -12], [0, 20, 0, 0, 14, 14, -66, 0], [0, -22, 0, 0, -18, 18, -70, 0], [0, -2, 0, 0, -17, -17, -9, 0], [0, 11, 0, 0, -7, 7, 51, 0], [22, 0, 24, -22, 0, 0, 0, 44], [116, 0, 0, 33, 0, 0, 0, 66], [0, -13, 0, 0, -29, 29, -23, 0], [-15, 0, 9, 15, 0, 0, 0, -30], [0, 16, 0, 0, 20, -20, 44, 0], [-15, 0, 17, 15, 0, 0, 0, -30], [0, 0, 0, 0, 28, 28, 28, 0], [-121, 0, 0, 32, 0, 0, 0, 64], [0, 14, 0, 0, 14, -14, 42, 0], [-4, 0, 16, 4, 0, 0, 0, -8], [-197, 0, 0, -26, 0, 0, 0, -52], [0, -4, 0, 0, 14, 14, 30, 0], [118, 0, 0, -12, 0, 0, 0, -24], [0, 3, 0, 0, 7, -7, 5, 0], [0, 10, 0, 0, -23, -23, -63, 0], [-30, 0, -31, 30, 0, 0, 0, -60], [264, 0, 0, -13, 0, 0, 0, -26], [5, 0, 15, -5, 0, 0, 0, 10], [0, 8, 0, 0, -2, -2, -34, 0], [0, -17, 0, 0, -11, 11, -57, 0], [228, 0, 0, -1, 0, 0, 0, -2], [0, -27, 0, 0, 9, -9, -117, 0], [-167, 0, 0, 48, 0, 0, 0, 96], [0, -20, 0, 0, 49, 49, 129, 0], [-256, 0, 0, 6, 0, 0, 0, 12], [0, -10, 0, 0, 27, 27, 67, 0], [-423, 0, 0, 6, 0, 0, 0, 12], [40, 0, 31, -40, 0, 0, 0, 80], [0, -24, 0, 0, 34, 34, 130, 0], [0, 30, 0, 0, 30, -30, 90, 0], [-4, 0, 0, 18, 0, 0, 0, 36], [0, -53, 0, 0, -19, 19, -193, 0], [30, 0, -8, -30, 0, 0, 0, 60], [0, 64, 0, 0, 19, 19, -237, 0], [0, 0, 0, 0, 50, 50, 50, 0], [0, 0, 49, 0, 0, 0, 0, 0], [0, -60, 0, 0, -20, -20, 220, 0], [-21, 0, -87, 21, 0, 0, 0, -42], [0, -30, 0, 0, 11, 11, 131, 0], [0, -38, 0, 0, 26, -26, -178, 0], [0, -9, 0, 0, 43, -43, -79, 0], [45, 0, 89, -45, 0, 0, 0, 90], [-176, 0, 0, 13, 0, 0, 0, 26], [-10, 0, 40, 10, 0, 0, 0, -20], [0, 52, 0, 0, -12, 12, 220, 0], [0, -32, 0, 0, -35, -35, 93, 0], [0, -9, 0, 0, 23, -23, -59, 0], [35, 0, 97, -35, 0, 0, 0, 70], [-437, 0, 0, 4, 0, 0, 0, 8], [0, 28, 0, 0, 63, 63, -49, 0], [0, -51, 0, 0, 29, -29, -233, 0], [0, -60, 0, 0, -42, -42, 198, 0], [623, 0, 0, 40, 0, 0, 0, 80], [70, 0, 24, -70, 0, 0, 0, 140], [468, 0, 0, -26, 0, 0, 0, -52], [0, 49, 0, 0, 49, -49, 147, 0], [33, 0, -71, -33, 0, 0, 0, 66], [-50, 0, -41, 50, 0, 0, 0, -100], [0, -54, 0, 0, -22, 22, -194, 0], [50, 0, -8, -50, 0, 0, 0, 100], [0, 40, 0, 0, 12, 12, -148, 0], [0, 48, 0, 0, 52, -52, 140, 0], [0, -60, 0, 0, -71, -71, 169, 0], [-528, 0, 0, 42, 0, 0, 0, 84], [-848, 0, 0, -14, 0, 0, 0, -28], [0, -3, 0, 0, -37, 37, 25, 0], [0, -52, 0, 0, -16, -16, 192, 0], [-48, 0, -160, 48, 0, 0, 0, -96], [0, -29, 0, 0, -27, 27, -89, 0], [630, 0, 0, 31, 0, 0, 0, 62], [0, 30, 0, 0, -1, -1, -121, 0], [-60, 0, 31, 60, 0, 0, 0, -120], [810, 0, 0, -20, 0, 0, 0, -40], [68, 0, -16, -68, 0, 0, 0, 136], [0, 20, 0, 0, 72, 72, -8, 0], [44, 0, 32, -44, 0, 0, 0, 88], [152, 0, 0, -19, 0, 0, 0, -38], [0, -44, 0, 0, -12, 12, -164, 0], [-289, 0, 0, 32, 0, 0, 0, 64], [0, -61, 0, 0, -15, 15, -229, 0], [35, 0, -71, -35, 0, 0, 0, 70], [0, -54, 0, 0, 6, -6, -222, 0], [0, 55, 0, 0, -45, 45, 265, 0], [-51, 0, 25, 51, 0, 0, 0, -102], [0, 12, 0, 0, 16, -16, 32, 0], [-10, 0, -137, 10, 0, 0, 0, -20], [0, 44, 0, 0, 3, 3, -173, 0], [240, 0, 0, -26, 0, 0, 0, -52], [0, -30, 0, 0, -17, -17, 103, 0], [1046, 0, 0, 53, 0, 0, 0, 106], [0, -74, 0, 0, -110, 110, -186, 0], [-100, 0, 48, 100, 0, 0, 0, -200], [0, -40, 0, 0, 90, 90, 250, 0], [406, 0, 0, -67, 0, 0, 0, -134], [0, 34, 0, 0, -38, 38, 174, 0], [-25, 0, 31, 25, 0, 0, 0, -50], [0, -104, 0, 0, 26, 26, 442, 0], [-80, 0, -32, 80, 0, 0, 0, -160], [0, 40, 0, 0, -10, -10, -170, 0], [-82, 0, -55, 82, 0, 0, 0, -164], [0, -35, 0, 0, -15, 15, -125, 0], [59, 0, 0, 92, 0, 0, 0, 184], [0, 68, 0, 0, 64, -64, 208, 0], [0, -48, 0, 0, 47, 47, 239, 0], [0, 36, 0, 0, 20, -20, 124, 0], [70, 0, 152, -70, 0, 0, 0, 140], [0, 60, 0, 0, -12, -12, -252, 0], [0, -50, 0, 0, 2, -2, -202, 0], [0, 84, 0, 0, 64, 64, -272, 0], [46, 0, 95, -46, 0, 0, 0, 92], [150, 0, 0, 100, 0, 0, 0, 200]];
    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_p();".
// 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_p(:prec:=8)
    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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![1625, 0, 118, 0, 1]>,<13,R![-115, 2, 1]>,<17,R![40768, 0, -424, 0, 1]>],Snew);
    return Vf;
end function;