
// Make newform 300.3.b.c in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_300_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_300_b_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_300_3_b_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_300_3_b_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, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 4, 0, 2], [0, 8, 0, 2], [3, 0, 2, 0]];
    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_300_b();"
function MakeCharacter_300_b()
    N := 300;
    order := 2;
    char_gens := [151, 101, 277];
    v := [2, 1, 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_300_b_Hecke();"
function MakeCharacter_300_b_Hecke(Kf)
    N := 300;
    order := 2;
    char_gens := [151, 101, 277];
    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, 0, 0], [0, -1, 0, 1], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, -3, 0], [0, -4, 0, 0], [0, 0, 0, 6], [34, 0, 0, 0], [0, 0, 0, -18], [0, 0, -9, 0], [14, 0, 0, 0], [0, 28, 0, 0], [0, 0, 6, 0], [0, -4, 0, 0], [0, 0, 0, -18], [0, 0, 0, 18], [0, 0, 3, 0], [-46, 0, 0, 0], [0, 16, 0, 0], [0, 0, -12, 0], [0, 53, 0, 0], [22, 0, 0, 0], [0, 0, 0, 54], [0, 0, 24, 0], [0, 61, 0, 0], [0, 0, 39, 0], [0, 23, 0, 0], [0, 0, 0, 6], [-86, 0, 0, 0], [0, 0, 0, -42], [0, -17, 0, 0], [0, 0, -33, 0], [0, 0, 0, -18], [82, 0, 0, 0], [0, 0, 33, 0], [-46, 0, 0, 0], [0, 46, 0, 0], [0, -34, 0, 0], [0, 0, 0, 30], [0, 0, 0, 54], [0, 0, -63, 0], [194, 0, 0, 0], [0, 0, 18, 0], [0, -109, 0, 0], [0, 0, 0, 42], [34, 0, 0, 0], [-46, 0, 0, 0], [0, -199, 0, 0], [0, 0, 0, -150], [-86, 0, 0, 0], [0, 0, 0, -138], [0, 0, 0, 0], [134, 0, 0, 0], [0, 0, 69, 0], [0, 0, 0, 78], [0, 0, 0, -114], [0, 0, 75, 0], [2, 0, 0, 0], [0, -224, 0, 0], [0, 0, -42, 0], [0, -124, 0, 0], [0, 0, 0, 66], [0, 136, 0, 0], [0, 0, -12, 0], [0, -139, 0, 0], [0, 0, 0, 54], [-598, 0, 0, 0], [0, -179, 0, 0], [0, 0, 0, 114], [-518, 0, 0, 0], [0, 0, 0, 126], [0, 0, 78, 0], [0, -89, 0, 0], [0, 266, 0, 0], [-86, 0, 0, 0], [0, 0, 0, 54], [0, 0, 93, 0], [0, -14, 0, 0], [0, 0, 60, 0], [214, 0, 0, 0], [0, 0, -39, 0], [-238, 0, 0, 0], [0, 0, 54, 0], [0, 191, 0, 0], [274, 0, 0, 0], [0, 0, 0, -342], [0, 0, -174, 0], [0, 223, 0, 0], [0, 0, 21, 0], [0, -427, 0, 0], [0, 0, 0, 6], [0, 0, 12, 0], [0, 1, 0, 0], [0, 0, 105, 0], [514, 0, 0, 0], [0, 0, 0, 294], [0, 0, -69, 0], [0, 0, -150, 0], [0, 416, 0, 0], [314, 0, 0, 0], [0, 268, 0, 0], [0, 0, 0, 402], [0, 0, 0, -366], [0, 0, 0, 0], [-526, 0, 0, 0], [0, 61, 0, 0], [0, 0, 0, 102], [0, 0, 0, -330], [0, 0, -18, 0], [-766, 0, 0, 0], [0, 103, 0, 0], [0, 278, 0, 0], [0, 0, 0, 174], [514, 0, 0, 0], [1094, 0, 0, 0], [0, 0, -36, 0], [0, -202, 0, 0], [0, 0, 0, -522], [0, 0, 0, 342], [0, 0, 177, 0], [-118, 0, 0, 0], [0, -97, 0, 0], [0, 0, 0, -186], [0, 0, 0, -42], [122, 0, 0, 0], [0, 0, 129, 0], [-1166, 0, 0, 0], [0, 0, -48, 0], [0, -659, 0, 0], [0, 68, 0, 0], [682, 0, 0, 0], [0, 0, 0, 390], [-658, 0, 0, 0], [0, 556, 0, 0], [0, 0, 18, 0], [-206, 0, 0, 0], [0, 0, 0, 486], [0, -584, 0, 0], [0, 0, 0, 534], [0, 0, 204, 0], [674, 0, 0, 0], [0, 0, 303, 0], [0, -49, 0, 0], [0, 0, 0, -366], [-398, 0, 0, 0], [0, 0, -222, 0], [0, -334, 0, 0], [0, 0, 0, -138], [-278, 0, 0, 0], [0, 0, 0, -162], [0, 286, 0, 0], [0, 0, 180, 0], [0, 566, 0, 0], [0, 0, 0, -546], [0, 358, 0, 0], [0, 0, 282, 0], [-866, 0, 0, 0], [0, 0, 66, 0], [0, -509, 0, 0], [0, 0, -267, 0], [0, 0, 0, 330], [0, 0, 0, -690], [0, -707, 0, 0], [0, 0, -75, 0], [0, 0, 0, -18], [0, 0, 0, -210], [914, 0, 0, 0], [0, -632, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_300_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_300_3_b_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_300_3_b_c(:prec:=4)
    chi := MakeCharacter_300_b();
    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_300_3_b_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_300_3_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_300_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![4, 0, 1]>,<11,R![180, 0, 1]>],Snew);
    return Vf;
end function;